**Three-Dimensional Magnetically-Oriented Microcrystal Array: A Large Sample for Neutron Diffraction Analysis**

T. Kimura1, F. Kimura1, K. Matsumoto1 and N. Metoki2 *1Kyoto University 2Japan Atomic Energy Agency Japan* 

### **1. Introduction**

178 Neutron Diffraction

Rietvild, H. M. (1969). Profile Refinement method for nuclear and magnetic

Schubert, K. (1971). Crystal structures of two-componental phases. Moscow: Metallurgiya,

Young, R. A., Wilas, D. B. (1982). Profile Scope Functions in Rietveld Refinements. *Journal of Applied Crystallography,*Vol. 2, (February 1982), pp. 65-71, ISSN 0021-8898.

translation from the German language, Russia.

ISSN 0021-8898.

structures.J*ournal of Applied Crystallography,*Vol. 2, part 2, (June 1969), pp. 65-71,

Structure determination of biomolecules is of great importance because the structure is closely related to biological functions. Some proteins are activated when binding with ligands; a drug molecule functions by binding to a specific protein site (Sousa et al., 2000; Graves et al., 1994). The structure determination is important not only for biomolecules but also for many inorganic, organic, and polymeric crystals that are key materials in materials and pharmaceutical sciences.

Currently, three major methods to solve the structure of molecules are known: solution nuclear magnetic resonance (NMR) (Herrmann et al., 2002), X-ray and neutron single crystal analysis (Blundell et al., 1976; Kendrew et al., 1958; Blake et al., 1965), and X-ray powder diffraction (Margiolaki et al, 2008; Hariss et al., 2004) methods. The solution NMR method has an advantage over diffraction methods because it does not require crystals. However, it can be applied only to proteins with lower molecular weights. X-ray and neutron singlecrystal analysis is the most powerful method, but it is sometimes difficult to grow a crystal (Ataka et al., 1986) sufficiently large for conventional or synchrotron single-crystal X-ray measurement. The size requirement is much more severe for neutron diffraction measurements (Niimura et al., 1999). The X-ray powder method can be performed if microcrystalline powders are available, but an appropriate refinement of many parameters is needed to obtain a reliable result. Preferential orientation of a powder sample is utilized to produce single-crystal-like diffraction data (Wessels et al., 1999). However, the data quality strongly depends on the quality of the orientation.

It is well known that feeble magnetic materials such as most biological, organic, polymeric and inorganic materials respond to applied magnetic fields, although the response is weak. A number of studies on the magnetic alignment of such materials have been reported (Maret et al., 1985; Asai et al., 2006). The study of the feeble interaction of diamagnetic materials with an applied magnetic field and its application are now recognized as an immerging area of science and technology, and named "Magneto-Science".

We have recently proposed a method that enables to convert a diamagnetic or paramagnetic microcrystalline powder to a "pseudo single crystal (PSC)" (T. Kimura et al. 2005; T. Kimura,

Three-Dimensional Magnetically-Oriented

principal diamagnetic susceptibility axes (

transformation matrix defined by the Eulerian angles (

direction and intensity on the *xy* plane as follows:

the intensity *B*) then the easy magnetization axis,

av

higher terms, we obtain (T. Kimura et al., 2005)

*y*

where the constant terms are not shown.

2 and ( + 

distribution to obtain the expression

function of the Eulerian angles. Expanding eq (3) around

 

 **B** = *B* <sup>t</sup>

coordinate system as follows

where 

Here, 

certain rate

expressed as

2000) where

Fluctuations of

 = 60/

expressed by the following equation:

a and 

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 181

We believe that the magnetic method we introduce here is of considerable use for the crystal structure determination of materials that do not grow large sizes necessary for the diffraction measurement. Especially, our method has advantages when applied to single crystal neutron diffraction analyses of proteins where a single crystal sample much larger

A crystal has a diamagnetic susceptibility tensor , which is expressed in terms of the

magnetic field **B**, it has a magnetic energy, expressed with respect to the laboratory

the transpose. We consider an applied elliptical magnetic field that periodically changes in

In the RRR, a crystal could be assumed to be placed in an time-averaged magnetic potential

*EV t* ( /2 ) ( /2 ) ) d (

Inserting eq (2) into eq (3) and performing the integration, we obtain *E*av as an analytical

av <sup>0</sup> 2 3 <sup>0</sup> 1 2 ( / 4 ) ( ) ( / 4 ) ( )( ) *E V Bb*

 <sup>2</sup> 2 2 0 B 2 3 2 /[ ] *<sup>y</sup>*

*kT Bb V*

 

2 /

 

0

(*bx* cos*t*, *by* sin

are the longer and shorter axes of the ellipse (*by* < *bx*), respectively. If

This condition is called "Rapid Rotation Regime (RRR)", defined as |

0 is the magnetic permeability of the vacuum, *V* is the volume of the crystallite, **A** is the

is the angular velocity of the field rotation, *B* is the intensity of the field, and *bx* and *by*

(the intrinsic rate of magnetic response of a particle exposed to the static field of

are magnetic anisotropy and the viscosity of the surrounding medium.

t

t

2 2 <sup>2</sup> 22 2 <sup>2</sup>

*V Bb b x y*

 

0

 

 

<sup>a</sup>*B*2 in the case that the particle shape is spherical (T. Kimura et al.,

, , 

1<0). When the crystal is exposed to a

**A A**)**B** (1)

, and the superscript t indicates

*t*, 0) (2)

is far larger than a


=0 and truncating the

(5)

(4)

is

1, cannot follow the rotation of the field.

**B A AB** (3)

 

==

)2 around the minimum are calculated using the Boltzmann

3<2<

0)t**B**(t

*E* =–(*V*/2

than that needed for the X-ray diffraction measurement is required.

**2. Theoretical background of preparation of a 3D-MOMA** 

2009a)(Fig. 1). A PSC is a composite in which microcrystals are oriented three-dimensionally in a resin matrix. A PSC is also referred to as a "three-dimensional magnetically-oriented microcrystal array" (3D-MOMA) (F. Kimura et al., 2011). This method enables us to perform a single-crystal analysis of a material through alignment of its powder sample. The composite is fabricated using magnetic alignment of a microcrystal suspension under an elliptically rotating magnetic field, followed by consolidation of the suspending matrix.

The degree of alignment strongly depends on the diamagnetic anisotropy of the crystal, applied field strength, and size of microcrystals to be aligned. If these conditions are appropriately satisfied, the obtained 3D-MOMA can produce well-separated diffraction spots that allow single-crystal analysis. This method can be applied to biaxial crystals including the triclinic, monoclinic, and orthorhombic systems which have three different magnetic susceptibility values. We only obtain fiber diffraction patterns for uniaxial crystals even if we apply the 3D-MOMA method. This method does not work for the cubic system.

However, the 3D-MOMA method has some drawbacks including the limitation of the suspending medium, the difficulty of recovering a precious sample, and the broadening of diffraction spots by consolidation. A solution to these problems may be measurement of the X-ray diffraction patterns directly from a three-dimensional magnetically-oriented microcrystal suspension (3D-MOMS) without consolidation of the suspending medium. Xray diffraction from magnetically oriented solutions of macromolecular assemblies was reported (Glucksman et al., 1986; Samulski et al., 1986). Also, small-angle X-ray scattering of colloidal platelets (van der Beek et al., 2006) and molecular aggregates (Gielen et al., 2009) under magnetic fields were reported. *In-situ* X-ray (Kohama et al., 2007) and neutron (Terada et al., 2008) diffraction measurements under applied magnetic fields were reported. We showed a preliminary result (Matsumoto et al., 2011) of the X-ray diffraction from a MOMS achieved with and without sample rotation in a static magnetic field.

Fig. 1. Schematic of conversion of microcrystals to "three-dimensional magnetically-oriented microcrystal array" (3D-MOMA) or "pseudo single crystal" (PSC).

2009a)(Fig. 1). A PSC is a composite in which microcrystals are oriented three-dimensionally in a resin matrix. A PSC is also referred to as a "three-dimensional magnetically-oriented microcrystal array" (3D-MOMA) (F. Kimura et al., 2011). This method enables us to perform a single-crystal analysis of a material through alignment of its powder sample. The composite is fabricated using magnetic alignment of a microcrystal suspension under an elliptically rotating

The degree of alignment strongly depends on the diamagnetic anisotropy of the crystal, applied field strength, and size of microcrystals to be aligned. If these conditions are appropriately satisfied, the obtained 3D-MOMA can produce well-separated diffraction spots that allow single-crystal analysis. This method can be applied to biaxial crystals including the triclinic, monoclinic, and orthorhombic systems which have three different magnetic susceptibility values. We only obtain fiber diffraction patterns for uniaxial crystals even if we apply the 3D-MOMA method. This method does not work for the cubic system. However, the 3D-MOMA method has some drawbacks including the limitation of the suspending medium, the difficulty of recovering a precious sample, and the broadening of diffraction spots by consolidation. A solution to these problems may be measurement of the X-ray diffraction patterns directly from a three-dimensional magnetically-oriented microcrystal suspension (3D-MOMS) without consolidation of the suspending medium. Xray diffraction from magnetically oriented solutions of macromolecular assemblies was reported (Glucksman et al., 1986; Samulski et al., 1986). Also, small-angle X-ray scattering of colloidal platelets (van der Beek et al., 2006) and molecular aggregates (Gielen et al., 2009) under magnetic fields were reported. *In-situ* X-ray (Kohama et al., 2007) and neutron (Terada et al., 2008) diffraction measurements under applied magnetic fields were reported. We showed a preliminary result (Matsumoto et al., 2011) of the X-ray diffraction from a

magnetic field, followed by consolidation of the suspending matrix.

MOMS achieved with and without sample rotation in a static magnetic field.

Fig. 1. Schematic of conversion of microcrystals to "three-dimensional magnetically-oriented

microcrystal array" (3D-MOMA) or "pseudo single crystal" (PSC).

We believe that the magnetic method we introduce here is of considerable use for the crystal structure determination of materials that do not grow large sizes necessary for the diffraction measurement. Especially, our method has advantages when applied to single crystal neutron diffraction analyses of proteins where a single crystal sample much larger than that needed for the X-ray diffraction measurement is required.

### **2. Theoretical background of preparation of a 3D-MOMA**

A crystal has a diamagnetic susceptibility tensor , which is expressed in terms of the principal diamagnetic susceptibility axes (3<2<1<0). When the crystal is exposed to a magnetic field **B**, it has a magnetic energy, expressed with respect to the laboratory coordinate system as follows

$$E = -(V/2\mu\_0)^t \mathbf{B}(^t\mathbf{A} \not\propto \mathbf{A})\mathbf{B} \tag{1}$$

where 0 is the magnetic permeability of the vacuum, *V* is the volume of the crystallite, **A** is the transformation matrix defined by the Eulerian angles (, , , and the superscript t indicates the transpose. We consider an applied elliptical magnetic field that periodically changes in direction and intensity on the *xy* plane as follows:

$$\mathbf{B} = B \begin{pmatrix} b\_x \cos \alpha \, t, \, b\_y \sin \alpha \, t, \, 0 \end{pmatrix} \tag{2}$$

Here, is the angular velocity of the field rotation, *B* is the intensity of the field, and *bx* and *by* are the longer and shorter axes of the ellipse (*by* < *bx*), respectively. If is far larger than a certain rate (the intrinsic rate of magnetic response of a particle exposed to the static field of the intensity *B*) then the easy magnetization axis, 1, cannot follow the rotation of the field. This condition is called "Rapid Rotation Regime (RRR)", defined as ||>>1/2, where is expressed as = 60/<sup>a</sup>*B*2 in the case that the particle shape is spherical (T. Kimura et al., 2000) where a and are magnetic anisotropy and the viscosity of the surrounding medium.

In the RRR, a crystal could be assumed to be placed in an time-averaged magnetic potential expressed by the following equation:

$$E\_{\rm av} = - (\alpha \slash \ 2\pi) \int\_0^{2\pi} (V \slash \ 2\mu\_0)^t \mathbf{B} (^t\mathbf{A} \lll \mathbf{A}) \mathbf{B} \, dt \tag{3}$$

Inserting eq (2) into eq (3) and performing the integration, we obtain *E*av as an analytical function of the Eulerian angles. Expanding eq (3) around ===0 and truncating the higher terms, we obtain (T. Kimura et al., 2005)

$$E\_{\rm av} \approx (V \,/ \, 4\mu\_0) \mathcal{B}^2 b\_y \,^2 (\chi\_2 - \chi\_3) \theta^2 + (V \,/ \, 4\mu\_0) \mathcal{B}^2 \left(b\_x^{\; 2} - b\_y^{\; 2}\right) (\chi\_1 - \chi\_2) (\phi + \phi)^2 \tag{4}$$

where the constant terms are not shown.

Fluctuations of 2 and ( + )2 around the minimum are calculated using the Boltzmann distribution to obtain the expression

$$<\left(\Delta\theta\right)^{2} \approx 2\,\mu\_{0}k\_{\text{B}}T \;/\left[\text{B}^{2}b\_{y}^{\text{-}}V\left(\varkappa\_{2}-\varkappa\_{\text{A}}\right)\right] \tag{5}$$

Three-Dimensional Magnetically-Oriented

and 

susceptibility axes (

1 and 

International Union of Crystallography)

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 183

For the monoclinic system (point group 2, space group *P*21), its twofold axis (*b* axis) coincides with one of the magnetic susceptibility axes (Nye, 1985). The crystallographic and magnetic axes are shown schematically in Fig. 2b, where the *b* axis andχaxis are placed perpendicular to the plane of the diagram (F. Kimura et al., 2010b). Since the three magnetic susceptibility axes are mutually perpendicular, the other two magnetic susceptibility axes are placed on the plane of the diagram. The *a* and *c* axes are on the same plane. However, there is no rule to relate theχaxes with the *a* and *c* axes. If the crystal is rotated by an angle ofπaround each of the magnetic susceptibility axes, four different crystal orientations are obtained, as shown in Figs. 2b (a)–(d). These four orientations have the same magnetic energy when they are placed in a given magnetic field. Because of the twofold symmetry (point group 2) along the *b* axis, the crystal orientations of Figs. 2b(a) and 2b(c) are identical; those of Figs. 2b(b) and 2b(d) are also identical. Thus, there are two different orientations with the same magnetic energy. In Table 1, possible crystal orientations with an equal magnetic energy are summarized for

Fig. 2b. Schematic diagram of a crystal belonging to the monoclinic system (point group 2) showing crystallographic axes and magnetic susceptibility axes. (a) The *b* axis (twofold axis)

around each of susceptibility axes to produce three additional possible orientations, (b), (c), and (d). All four orientations have the same magnetic energy under a given magnetic field. Because of the symmetry (point group 2), the crystal orientations of (a) and (c) are identical, and those of (b) and (d) are also identical. (Reprinted from (F. Kimura et al., 2010b): *J.* Appl. Cryst. Vol.143(2010) pp. 151-153, Part 1, DOI: 10.1107/S0021889809048006. Copyright 2010

plane of the diagram. The other crystallographic axes (*a* and *c* axes) and magnetic

2 axis (its direction coincides with that of the *b* axis) are placed perpendicular to the

<sup>3</sup> axes) are placed in the plane. The crystal (a) is rotated by

crystals belonging to the orthorhombic, monoclinic, and triclinic systems.

$$<\left(\Delta(\phi+\phi)\right)^{2} \gg 2\,\mu\_{0}k\_{\text{B}}T\left/\left[B^{2}(b\_{x}^{\cdot2}-b\_{y}^{\cdot2})V\left(\mathbb{X}\_{1}-\mathbb{X}\_{2}\right)\right] \tag{6}$$

where *k*B is the Boltzmann constant and *T* is the temperature. To obtain a good 3D-MOMA, we should minimize and equalize the values of ( <sup>2</sup> and (( + ))2 by appropriately choosing the values of *bx* and *by*. The crystallite volume *V* is also important. The smaller the *V*, the larger the fluctuations. Typically, the size necessary is about the order of micrometers, depending on the field strength used and the magnetic anisotropy of the crystal. To obtain a high quality diffraction pattern, the rotation speed is also important. It should satisfy ||>>1/2 (RRR) (T. Kimura et al., 2005).

In general, the susceptibility axes of a biaxial crystal do not necessarily coincide with the crystallographic *a*, *b*, *c* axes, except the orthorhombic system (Fig. 2a) (F. Kimura et al., 2010a). The orthorhombic system includes three point groups, that is, 222, *mm*2, and *mmm*. Both for 222 and *mmm*, three crystallographic axes are 2-fold axes that coincide with the susceptibility axes. As a result, the rotation about the susceptibility axes through an angle of π does not create new crystal orientations. On the other hand, *mm*2 has just one 2-fold axis. Therefore, the rotation about the susceptibility axes can produce a new orientation.

Fig. 2a. Crystal orientations obtained by a rotation about the susceptibility axis by angle of are magnetically equivalent. The orthorhombic system includes three point groups, *mm*2, 222, and *mmm*, as displayed with stereo diagrams. In the case of (a) *mm*2, an orientation different from the original one is produced by rotation about 1 or <sup>2</sup> . In a 3D-MOMA, these orientations coexist. For (b) 222 and (c) *mmm*, the rotations do not alter the original crystal orientation, resulting in a single orientation in a 3D-MOMA. (Reprinted from (F. Kimura et al., 2010a): Cryst. Growth Des*.* Vol.10(2010) pp. 48-51, No. 1, DOI: 10.1021/cg90132h. Copyright 2010 American Chemical Society)

 <sup>2</sup> 22 2 0 B 1 2 ( ) 2 /[ ( ) ] *x y*

where *k*B is the Boltzmann constant and *T* is the temperature. To obtain a good 3D-MOMA,

appropriately choosing the values of *bx* and *by*. The crystallite volume *V* is also important. The smaller the *V*, the larger the fluctuations. Typically, the size necessary is about the order of micrometers, depending on the field strength used and the magnetic anisotropy of the

In general, the susceptibility axes of a biaxial crystal do not necessarily coincide with the crystallographic *a*, *b*, *c* axes, except the orthorhombic system (Fig. 2a) (F. Kimura et al., 2010a). The orthorhombic system includes three point groups, that is, 222, *mm*2, and *mmm*. Both for 222 and *mmm*, three crystallographic axes are 2-fold axes that coincide with the susceptibility axes. As a result, the rotation about the susceptibility axes through an angle of π does not create new crystal orientations. On the other hand, *mm*2 has just one 2-fold axis.

*kT B b b V*

 

<sup>2</sup> and ((

(6)

is also important. It

))2 by

 + 

 

crystal. To obtain a high quality diffraction pattern, the rotation speed


Therefore, the rotation about the susceptibility axes can produce a new orientation.

Fig. 2a. Crystal orientations obtained by a rotation about the susceptibility axis by angle of

222, and *mmm*, as displayed with stereo diagrams. In the case of (a) *mm*2, an orientation

these orientations coexist. For (b) 222 and (c) *mmm*, the rotations do not alter the original crystal orientation, resulting in a single orientation in a 3D-MOMA. (Reprinted from (F.

Kimura et al., 2010a): Cryst. Growth Des*.* Vol.10(2010) pp. 48-51, No. 1, DOI:

different from the original one is produced by rotation about

10.1021/cg90132h. Copyright 2010 American Chemical Society)

are magnetically equivalent. The orthorhombic system includes three point groups, *mm*2,

1 or 

<sup>2</sup> . In a 3D-MOMA,

should satisfy |

we should minimize and equalize the values of (

For the monoclinic system (point group 2, space group *P*21), its twofold axis (*b* axis) coincides with one of the magnetic susceptibility axes (Nye, 1985). The crystallographic and magnetic axes are shown schematically in Fig. 2b, where the *b* axis andχaxis are placed perpendicular to the plane of the diagram (F. Kimura et al., 2010b). Since the three magnetic susceptibility axes are mutually perpendicular, the other two magnetic susceptibility axes are placed on the plane of the diagram. The *a* and *c* axes are on the same plane. However, there is no rule to relate theχaxes with the *a* and *c* axes. If the crystal is rotated by an angle ofπaround each of the magnetic susceptibility axes, four different crystal orientations are obtained, as shown in Figs. 2b (a)–(d). These four orientations have the same magnetic energy when they are placed in a given magnetic field. Because of the twofold symmetry (point group 2) along the *b* axis, the crystal orientations of Figs. 2b(a) and 2b(c) are identical; those of Figs. 2b(b) and 2b(d) are also identical. Thus, there are two different orientations with the same magnetic energy. In Table 1, possible crystal orientations with an equal magnetic energy are summarized for crystals belonging to the orthorhombic, monoclinic, and triclinic systems.

Fig. 2b. Schematic diagram of a crystal belonging to the monoclinic system (point group 2) showing crystallographic axes and magnetic susceptibility axes. (a) The *b* axis (twofold axis) and 2 axis (its direction coincides with that of the *b* axis) are placed perpendicular to the plane of the diagram. The other crystallographic axes (*a* and *c* axes) and magnetic susceptibility axes ( 1 and <sup>3</sup> axes) are placed in the plane. The crystal (a) is rotated by around each of susceptibility axes to produce three additional possible orientations, (b), (c), and (d). All four orientations have the same magnetic energy under a given magnetic field. Because of the symmetry (point group 2), the crystal orientations of (a) and (c) are identical, and those of (b) and (d) are also identical. (Reprinted from (F. Kimura et al., 2010b): *J.* Appl. Cryst. Vol.143(2010) pp. 151-153, Part 1, DOI: 10.1107/S0021889809048006. Copyright 2010 International Union of Crystallography)

Three-Dimensional Magnetically-Oriented

cos *B t <sup>x</sup>* 

and *B t <sup>y</sup>* sin

Electromagnets

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 185


0.35 Magnetic field

*By*


Fig. 3. Generation of amplitude modulated magnetic field. Two pairs of magnets generate

resulting in an elliptical magnetic field.

 (a) (b) Fig. 4. (a) Generation of modulated magnetic field by rotating a sample in a static field. Rotation speed is switched. (b) Diagram illustrating the frequency-modulated sample rotation. The *x-*, *y-*, and *z-*axes are laboratory coordinates; the *x-* and *y-*axes are on the horizontal plane and the *z* axis is on the vertical plane. The *x*'-, *y*'-, and *z*'-axes are imbedded in the sample that is rotated around the *z*'-axis (the *z-*axis coincides with the *z*'-axis). The

Micro-crystallites were mixed with a UV (Ultra Violet) curable monomer. The concentration of the crystallites in the monomer is 10-30 v/v%. The suspension is poured into any size of

rotation is performed non-uniformly so that the condition

coincides with the direction of the magnetic field.

**4. Sample preparation** 

**4.1 Preparation of 3D - MOMA** 

0

0

0 900

0 900

s < 

q is satisfied. The *x-*axis

time

time

0.35

*Bx*


Table 1. Number of possible orientations and the resultant point symmetry of Pseudo Single Crystal (PSC) for biaxial crystal systems with different point groups

### **3. Magnetic field used to prepare 3D-MOMA**

### **3.1 Generation of modulated magnetic field**

A 3D-MOMA of a biaxial crystal was prepared under modulated magnetic fields. In the previous section, the calculation was performed using the elliptical field. In the actual experiment, there are various versions for the elliptical field. We used two different dynamic magnetic fields (T. Kimura et al., 2009b). The first one is an amplitude modulated dynamic field (T. Kimura et al., 2005). This is generated by a four-pole electromagnet, in which one pair of poles generates a *Bx* field strength sinusoidally oscillating at , and the other generates *By* field strength oscillating at the same frequency as the previous one but with a phase shift of / 2 . This combination created a magnetic field expressed as ( cos , sin ,0) *B tB t x y* , shown in Fig. 3. The second one is a frequency modulated dynamic magnetic field (T. Kimura et al., 2006). In this setup, a sample is non-uniformly rotated in a uniform static magnetic field. A sample-rotating unit was placed in a uniform horizontal magnetic field, shown in Fig. 4. The rotation axis was vertical. The rotation was not uniform. In the figure, the *x*-, *y*-, and *z*-axes are laboratory coordinates and the *x*'-, *y*'-, and *z*'-axes are imbedded in the rotating plate. The *x*-axis is parallel to the magnetic field. The *z*'- and *z*-axes are parallel to vertical direction. The angle between the *x*'-axis before rotating and *x*-axis is / 2 . The *x*'-axis was rotating at the angular velocity s between 360 / 2 and /2 degree, and between 180 / 2 and 180 / 2 degree. On the other hand, it was rotating at the angular verocity <sup>q</sup> ( <sup>s</sup> ) between / 2 and 180 / 2 degree, and between 180 / 2 and 360 / 2 degree*.* 

The three parameters, s, q, and , must be selected appropriately in order to obtain sharp diffraction spots. The half-width of a spot is a result of the fluctuations of the 1 and 3 axes about the *x*' and *z*' axes, respectively. In terms of diffraction analysis, it is advantageous that the magnitudes of the two fluctuations are equal.

Number of orientations

1 4 *mmm* 

*m* 4 *mmm*  2/*m* 2 *mmm* 

*mm*2 2 *mmm mmm* 1 *mmm* 

. This combination created a magnetic field expressed as

 and 180 / 2 

, must be selected appropriately in order to obtain sharp

s between 360 / 2

degree. On the other hand, it was rotating

degree, and between

1 and 3 axes

, shown in Fig. 3. The second one is a frequency modulated dynamic

Point group of PSC

, and the

 and /2 

single crystal

Triclinic 1 4 222

Monoclinic 2 2 222

Orthorhombic 222 1 222

Crystal (PSC) for biaxial crystal systems with different point groups

**3. Magnetic field used to prepare 3D-MOMA** 

/ 2 . The *x*'-axis was rotating at the angular velocity

degree*.* 

<sup>q</sup> ( 

s, q, and 

the magnitudes of the two fluctuations are equal.

**3.1 Generation of modulated magnetic field** 

with a phase shift of / 2

degree, and between 180 / 2

 and 360 / 2 

at the angular verocity

The three parameters,

( cos , sin ,0) *B tB t x y* 

180 / 2 

Table 1. Number of possible orientations and the resultant point symmetry of Pseudo Single

A 3D-MOMA of a biaxial crystal was prepared under modulated magnetic fields. In the previous section, the calculation was performed using the elliptical field. In the actual experiment, there are various versions for the elliptical field. We used two different dynamic magnetic fields (T. Kimura et al., 2009b). The first one is an amplitude modulated dynamic field (T. Kimura et al., 2005). This is generated by a four-pole electromagnet, in

other generates *By* field strength oscillating at the same frequency as the previous one but

magnetic field (T. Kimura et al., 2006). In this setup, a sample is non-uniformly rotated in a uniform static magnetic field. A sample-rotating unit was placed in a uniform horizontal magnetic field, shown in Fig. 4. The rotation axis was vertical. The rotation was not uniform. In the figure, the *x*-, *y*-, and *z*-axes are laboratory coordinates and the *x*'-, *y*'-, and *z*'-axes are imbedded in the rotating plate. The *x*-axis is parallel to the magnetic field. The *z*'- and *z*-axes are parallel to vertical direction. The angle between the *x*'-axis before rotating and *x*-axis is

which one pair of poles generates a *Bx* field strength sinusoidally oscillating at

 and 180 / 2 

diffraction spots. The half-width of a spot is a result of the fluctuations of the

<sup>s</sup> ) between / 2

about the *x*' and *z*' axes, respectively. In terms of diffraction analysis, it is advantageous that

Crystal system Point group of

Fig. 3. Generation of amplitude modulated magnetic field. Two pairs of magnets generate cos *B t <sup>x</sup>* and *B t <sup>y</sup>* sinresulting in an elliptical magnetic field.

Fig. 4. (a) Generation of modulated magnetic field by rotating a sample in a static field. Rotation speed is switched. (b) Diagram illustrating the frequency-modulated sample rotation. The *x-*, *y-*, and *z-*axes are laboratory coordinates; the *x-* and *y-*axes are on the horizontal plane and the *z* axis is on the vertical plane. The *x*'-, *y*'-, and *z*'-axes are imbedded in the sample that is rotated around the *z*'-axis (the *z-*axis coincides with the *z*'-axis). The rotation is performed non-uniformly so that the condition s < q is satisfied. The *x-*axis coincides with the direction of the magnetic field.

### **4. Sample preparation**

### **4.1 Preparation of 3D - MOMA**

Micro-crystallites were mixed with a UV (Ultra Violet) curable monomer. The concentration of the crystallites in the monomer is 10-30 v/v%. The suspension is poured into any size of

Three-Dimensional Magnetically-Oriented

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 187

Fig. 6. X-ray diffraction pattern obtained for the sample prepared without the application of a magnetic field(top). X-ray diffraction patterns obtained for the L-alanine pseudo-single crystal sample. Patterns in (a), (b), and (c) were obtained with the alignment of the *a*, *b*, and *c*  axes, respectively, perpendicular to the X-ray beam; this is achieved by using an automatic crystal axis alignment system. An oscillation of 10was applied around the aligned axis. Some of the assignments were performed using a software program. Contrasts are different

between left and right halves. (Reprinted from (T. Kimura et al., 2006): *Langmuir*  Vol.22(2006) pp. 3464-3466. No. 8, DOI: 10.1021/la053479m. Copyright 2006 American

Chemical Society)

plastic container, exposed to a dynamic magnetic field, followed by UV light irradiation to polymerize the resin precursor to fix the alignment. See experimental details in Table 2.

### **4.2 Preparation of MOMS**

Typical experimental setting (KU model 10-1) is shown in Fig. 5. Sample suspension is poured into a capillary, and then it is placed on a rotating unit equipped with a pair of magnets.

Fig. 5. Experimental setting of *in-situ* X-ray diffraction measurement of a microcrystal suspension. A glass capillary containing the suspension is rotated at the rotation speed about the *z*-axis. The magnetic field **B** is applied parallel to the *x*-axis. The X-ray beam is impinged from the *y* direction, and the diffractions are detected by an imaging plate (IP). The azimuthal *β* angle is indicated. (Reprinted from (Matsumoto et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 945-948, No. 4, DOI: 10.1021/cg200090u. Copyright 2011 American Chemical Society)

### **5. X-ray results**

We fabricated 3D-MOMAs using frequency modulated magnetic field except LiCoPO4. We fabricated 3D-MOMAs of LiCoPO4 using both amplitude and frequency modulated magnetic fields. Alignment of these 3D-MOMAs are almost the same. Here, we only show the results obtained from the MOMA using frequency modulated magnetic field.

### **5.1 Alanine (T. Kimura et al., 2006)**

The top figure in Fig. 6 shows the XRD powder pattern obtained from the experiment in which magnetic field was not applied. A sparse ring pattern is observed in this case. Figures 6(a), (b) and (c) show the XRD patterns of the obtained pseudo single crystal. Patterns in (a), (b), and (c) verify the alignment of the *a*, *b*, and *c* axes, respectively. The *y*, *z*, and *x*axes imbedded on the sample correspond to the *a*, *b*, and *c* axes, respectively. This indicates that the *b* and *c* axes correspond to the hard and easy magnetization axes, respectively.

plastic container, exposed to a dynamic magnetic field, followed by UV light irradiation to polymerize the resin precursor to fix the alignment. See experimental details in Table 2.

poured into a capillary, and then it is placed on a rotating unit equipped with a pair of

10-1) is shown in Fig. 5. Sample suspension is

Fig. 5. Experimental setting of *in-situ* X-ray diffraction measurement of a microcrystal suspension. A glass capillary containing the suspension is rotated at the rotation speed

about the *z*-axis. The magnetic field **B** is applied parallel to the *x*-axis. The X-ray beam is impinged from the *y* direction, and the diffractions are detected by an imaging plate (IP). The azimuthal *β* angle is indicated. (Reprinted from (Matsumoto et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 945-948, No. 4, DOI: 10.1021/cg200090u. Copyright 2011 American

We fabricated 3D-MOMAs using frequency modulated magnetic field except LiCoPO4. We fabricated 3D-MOMAs of LiCoPO4 using both amplitude and frequency modulated magnetic fields. Alignment of these 3D-MOMAs are almost the same. Here, we only show

The top figure in Fig. 6 shows the XRD powder pattern obtained from the experiment in which magnetic field was not applied. A sparse ring pattern is observed in this case. Figures 6(a), (b) and (c) show the XRD patterns of the obtained pseudo single crystal. Patterns in (a), (b), and (c) verify the alignment of the *a*, *b*, and *c* axes, respectively. The *y*, *z*, and *x*axes imbedded on the sample correspond to the *a*, *b*, and *c* axes, respectively. This indicates that

the results obtained from the MOMA using frequency modulated magnetic field.

the *b* and *c* axes correspond to the hard and easy magnetization axes, respectively.

**4.2 Preparation of MOMS** 

magnets.

Chemical Society)

**5. X-ray results** 

**5.1 Alanine (T. Kimura et al., 2006)** 

Typical experimental setting (KU model

Fig. 6. X-ray diffraction pattern obtained for the sample prepared without the application of a magnetic field(top). X-ray diffraction patterns obtained for the L-alanine pseudo-single crystal sample. Patterns in (a), (b), and (c) were obtained with the alignment of the *a*, *b*, and *c*  axes, respectively, perpendicular to the X-ray beam; this is achieved by using an automatic crystal axis alignment system. An oscillation of 10was applied around the aligned axis. Some of the assignments were performed using a software program. Contrasts are different between left and right halves. (Reprinted from (T. Kimura et al., 2006): *Langmuir*  Vol.22(2006) pp. 3464-3466. No. 8, DOI: 10.1021/la053479m. Copyright 2006 American Chemical Society)

Three-Dimensional Magnetically-Oriented

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 189

Fig. 7. (a) A typical X-ray diffraction image (oscillation angle 5°) of a pseudo single crystal of LiCoPO4 is shown with diffraction spots enclosed by prediction rectangles. (b) An enlarged view of some diffraction spots is shown with Miller indices. (Reprinted from (T. Kimura et

> /degree

> > 90 90

> > 10 10 20

5 8

8

s /rpm

> 5 40

> 10 30 10

q /rpm

> 25 132

> 60 100 60

10.1107/S0021889809013430. Copyright 2009 International Union of Crystallography)

LiCoPO4 0.3 90 10 60 Sucrose 12 90 10 60

applied magnetic field / T

al., 2009b): *J.* Appl. Cryst. Vol.142(2009) pp. 535-537, Part 3, DOI:

L-alanine for X-ray L-alanine for neutron

Table 2. Preparation condition of MOMAs

Lysozyme

The diffraction spots are broad compared to those from a real single crystal, but they are sufficiently distinguishable to be assigned by the software program. Since the lattice parameter is the largest for the *b* axis, the spots in Fig. 6(b) are relatively less resolved. The half widths of the spots obtained by using an azimuthal plot are ca. 3for most of the spots in all of the patterns in Figs. 6 (a), (b), and (c). Incidentally, the half widths for an actual original single crystal are ca. 0.5. The fluctuations in and + are complex functions in the case of the frequency-modulated magnetic field used in the present study. However, they can be roughly estimated by using eqs (5) and (6). Using the values of *B*=5 T, *bx*=1, *by*=0.5, *T*=300 K, and a rough estimation of 1—2 and 2—3 =10-7, we can obtain the estimation for the fluctuation of and + ca. 1for *V* =1000 m3. This value should be compared with the half widths of the diffraction spots. The experimental values are larger than those obtained by the theoretical estimation. Using the values of the viscosity ( =1.2 Pa s) and the shape factor (*F* = 0.064, corresponding to the assumed aspect ratio of 11 for crystallites), we can obtain the estimation, =1 min. The time required for the completion of the alignment might be estimated as five times of , i.e., ca. 5 min. This value is far shorter than 2 h actually applied in the experiment. These discrepancies have several possible explanations, including the existence of smaller crystallites, imperfect dispersion, shrinkage of the resin during the UV cure, the flow caused by sample rotation, insufficient rotation speed, imperfect switching from one rotation to the other, etc.

### **5.2 LiCoPO4 (T. Kimura et al., 2009b)**

A typical diffraction of the 3D-MOMA image is shown in Fig. 7. The estimated mosaicity is 3.9°, which is larger than that observed for normal single crystals on the same instrument (around 0.8°). The cell constants and an orientation matrix for data collection corresponded to a primitive orthorhombic cell with dimensions *a* = 10.202 (6), *b* = 5.918 (3), *c* = 4.709 (2) Å , *V* = 284.3(3) Å3 and *Z* = 4. The space group was determined to be *Pnma* (No. 62). These results are the same as those reported by Kubel (1994), who used a 0.090 0.108 0.158 mm single crystal. The structure was solved by direct methods and expanded using Fourier techniques. The present result was compared with that from the literature, showing that the atomic coordinates determined in this study are in excellent agreement with those determined using a traditional single crystal. The *R*1 and *wR*2 values were 6.59 and 16.8%, respectively.

#### **5.3 Sucrose (F. Kimura et al., 2010b)**

The sucrose crystal belongs to the monoclinic system (point group 2, space group *P*21). Assuming an initial random orientation of crystallites in a suspension, the probability of finding two different orientations with the same magnetic energy is equal. These two orientations produce a diffraction pattern similar to that produced by a twin crystal. The diffraction image was analyzed using software designed for twin structures. The cell constants correspond to a primitive monoclinic cell with dimensions *a* = 7.7735(12), *b* = 8.7169(13), *c* = 10.8765(17) Å , = 102.936(4)°, *V* = 718.29(19) Å3 and *Z* = 2. The space group was determined to be *P*21 (No. 4). The structure was solved by direct methods and expanded using Fourier techniques. The *R*1 and *wR*2 values were 7.88 and 17.25%, respectively.

The diffraction spots are broad compared to those from a real single crystal, but they are sufficiently distinguishable to be assigned by the software program. Since the lattice parameter is the largest for the *b* axis, the spots in Fig. 6(b) are relatively less resolved. The half widths of the spots obtained by using an azimuthal plot are ca. 3for most of the spots in all of the patterns in Figs. 6 (a), (b), and (c). Incidentally, the half widths for an actual

the case of the frequency-modulated magnetic field used in the present study. However, they can be roughly estimated by using eqs (5) and (6). Using the values of *B*=5 T, *bx*=1,

compared with the half widths of the diffraction spots. The experimental values are larger

s) and the shape factor (*F* = 0.064, corresponding to the assumed aspect ratio of 11 for

than 2 h actually applied in the experiment. These discrepancies have several possible explanations, including the existence of smaller crystallites, imperfect dispersion, shrinkage of the resin during the UV cure, the flow caused by sample rotation, insufficient rotation

A typical diffraction of the 3D-MOMA image is shown in Fig. 7. The estimated mosaicity is 3.9°, which is larger than that observed for normal single crystals on the same instrument (around 0.8°). The cell constants and an orientation matrix for data collection corresponded to a primitive orthorhombic cell with dimensions *a* = 10.202 (6), *b* = 5.918 (3), *c* = 4.709 (2) Å , *V* = 284.3(3) Å3 and *Z* = 4. The space group was determined to be *Pnma* (No. 62). These results are the same as those reported by Kubel (1994), who used a 0.090 0.108 0.158 mm single crystal. The structure was solved by direct methods and expanded using Fourier techniques. The present result was compared with that from the literature, showing that the atomic coordinates determined in this study are in excellent agreement with those determined using a traditional single crystal. The *R*1 and *wR*2 values were 6.59 and 16.8%,

The sucrose crystal belongs to the monoclinic system (point group 2, space group *P*21). Assuming an initial random orientation of crystallites in a suspension, the probability of finding two different orientations with the same magnetic energy is equal. These two orientations produce a diffraction pattern similar to that produced by a twin crystal. The diffraction image was analyzed using software designed for twin structures. The cell constants correspond to a primitive monoclinic cell with dimensions *a* = 7.7735(12), *b* =

was determined to be *P*21 (No. 4). The structure was solved by direct methods and expanded using Fourier techniques. The *R*1 and *wR*2 values were 7.88 and 17.25%,

= 102.936(4)°, *V* = 718.29(19) Å3 and *Z* = 2. The space group

than those obtained by the theoretical estimation. Using the values of the viscosity (

1—2 and 2—

 and +

are complex functions in

3 =10-7, we can obtain the

=1.2 Pa

ca. 1for *V* =1000 m3. This value should be

=1 min. The time required for the completion of

, i.e., ca. 5 min. This value is far shorter

original single crystal are ca. 0.5. The fluctuations in

 and +

*by*=0.5, *T*=300 K, and a rough estimation of

crystallites), we can obtain the estimation,

**5.2 LiCoPO4 (T. Kimura et al., 2009b)** 

**5.3 Sucrose (F. Kimura et al., 2010b)** 

8.7169(13), *c* = 10.8765(17) Å ,

respectively.

respectively.

the alignment might be estimated as five times of

speed, imperfect switching from one rotation to the other, etc.

estimation for the fluctuation of

Fig. 7. (a) A typical X-ray diffraction image (oscillation angle 5°) of a pseudo single crystal of LiCoPO4 is shown with diffraction spots enclosed by prediction rectangles. (b) An enlarged view of some diffraction spots is shown with Miller indices. (Reprinted from (T. Kimura et al., 2009b): *J.* Appl. Cryst. Vol.142(2009) pp. 535-537, Part 3, DOI:

10.1107/S0021889809013430. Copyright 2009 International Union of Crystallography)


Table 2. Preparation condition of MOMAs

Three-Dimensional Magnetically-Oriented

International Union of Crystallography)

(here denoted as

we conclude that

and  1= *c*, 

that the cell dimensions increase in the order

2= *a*, and

shows a detailed diffraction pattern with resolution rings.

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 191

Fig. 8. Comparison of the structure determined in this study (blue) and the structure reported previously (red). (Reprinted from (F. Kimura et al., 2010b): *J.* Appl. Cryst. Vol.143(2010) pp. 151-153, Part 1, DOI: 10.1107/S0021889809048006. Copyright 2010

Fig. 9. Twin structure of a 3D-MOMA of sucrose. The *b* axis and one of the susceptibility axes

(Reprinted from (F. Kimura et al., 2010b): *J.* Appl. Cryst. Vol.143(2010) pp. 151-153, Part 1, DOI:

system, the magnetic axes correspond to the crystallographic axes. From the figure, we find

1, 2, and 

field has been reported for orthorhombic lysozyme crystals.( Sato et al., 2000). Figure 12

10.1107/S0021889809048006. Copyright 2010 International Union of Crystallography)

**<sup>2</sup>** ) are placed in the plane. The angle between the c axis and the

<sup>3</sup> ) are placed perpendicular to the plane of the diagram. The other axes (

3= *b*. The alignment of the *c* axis parallel to the magnetic

1 axis is ca 16º.

3. Combining the indexing results,

1

In Fig. 8, the structure solved from the MOMA is compared with that reported previously (Hanson et al., 1973). They show a good agreement. The twin structure is shown in Fig. 9. The magnetic susceptibility axes, as discussed previously (Fig. 9), are also shown. In principle, these magnetic susceptibility axes are attributed to each of the easy (largest) 1, hard (smallest) 3, and intermediate (medium) 2 magnetic susceptibility axes. The angle between the magnetic axis, 1, and crystallographic *c* axis is determined to be ca 16°. This value is different from a previously reported value of -1° 5' (Finke, 1909). The reason for the difference is unclear at present.

### **5.4 Lyzozyme (F. Kimura et al., 2011)**

The fluctuations of the 1 and 3 axes were estimated from the half-widths determined from the rocking curve and the azimuthal β scan, respectively. The rotation axis of the scan coincided with the 3 axis. The data was collected using scans at 1.0° steps. The three parameters, s, q, and *α*, must be selected appropriately in order to obtain sharp diffraction spots. The half-width of a spot is a result of the fluctuations of the 1 and 3 axes about the *x*' and *z*'- axes, respectively. In terms of diffraction analysis, it is advantageous that the magnitudes of the two fluctuations are equal. Sample rotation speed s > 1/(2τ)should be met (Rapid Rotation Regime (RRR)). Next, an appropriate choice of *α* andq is necessary in order to minimize and equalize the fluctuations of the 1 and 3 axes.

In alanine, LiCoPO4, and sucrose studies describing the single-crystal analysis of MOMAs, =90° was appropriate. However, a theoretical study (T. Kimura, 2009a) shows that the equalization cannot be achieved for some sets of (1, 2, 3) if *α* is fixed to 90°. Since the three values of the magnetic susceptibility of lysozyme are unknown, we need to find out an appropriate value of by trial and error. Table 3 summarizes the half-widths of a diffraction spot ((400) diffraction) obtained from MOMAs fabricated with some sets of parameters (s, q, *α*).


Table 3. Fluctuation of 1 and 3 for three sets of parameters(s, q, *α*), determined using the (400) diffraction spot.

We tested several combinations of the parameters, and =10°, s =10 rpm, and q =60 rpm was found to be the most suitable regarding minimization and equalization of the halfwidths. This condition was therefore chosen for the preparation of a MOMA for X-ray analysis. In Fig. 10, photograph and microphotograph of fabricated lysozyme MOMA are shown. In Fig. 11, diffraction patterns taken from three directions are shown. Well-separated diffraction spots were obtained. The directions of the magnetic axes as deduced from the preparation procedure are indicated in the figure. From the results of the indexing described in the next paragraph, the crystal belongs to the orthorhombic system. For this crystal

In Fig. 8, the structure solved from the MOMA is compared with that reported previously (Hanson et al., 1973). They show a good agreement. The twin structure is shown in Fig. 9. The magnetic susceptibility axes, as discussed previously (Fig. 9), are also shown. In principle, these magnetic susceptibility axes are attributed to each of the easy (largest)

value is different from a previously reported value of -1° 5' (Finke, 1909). The reason for the

and *z*'- axes, respectively. In terms of diffraction analysis, it is advantageous that the

In alanine, LiCoPO4, and sucrose studies describing the single-crystal analysis of MOMAs,

three values of the magnetic susceptibility of lysozyme are unknown, we need to find out

diffraction spot ((400) diffraction) obtained from MOMAs fabricated with some sets of

=90° was appropriate. However, a theoretical study (T. Kimura, 2009a) shows that the

the rocking curve and the azimuthal β scan, respectively. The rotation axis of the

3 axis. The data was collected using

3, and intermediate (medium)

spots. The half-width of a spot is a result of the fluctuations of the

magnitudes of the two fluctuations are equal. Sample rotation speed

met (Rapid Rotation Regime (RRR)). Next, an appropriate choice of *α* and

q /rpm Fluctuation of

10 10 60 4.5 4.3 10 30 100 5.7 NA 20 10 60 6 5.1

3 for three sets of parameters(

was found to be the most suitable regarding minimization and equalization of the halfwidths. This condition was therefore chosen for the preparation of a MOMA for X-ray analysis. In Fig. 10, photograph and microphotograph of fabricated lysozyme MOMA are shown. In Fig. 11, diffraction patterns taken from three directions are shown. Well-separated diffraction spots were obtained. The directions of the magnetic axes as deduced from the preparation procedure are indicated in the figure. From the results of the indexing described in the next paragraph, the crystal belongs to the orthorhombic system. For this crystal

hard (smallest)

difference is unclear at present.

**5.4 Lyzozyme (F. Kimura et al., 2011)** 

1 and 

order to minimize and equalize the fluctuations of the

equalization cannot be achieved for some sets of (

1 and 

We tested several combinations of the parameters, and

between the magnetic axis,

The fluctuations of the

s, 

an appropriate value of

s, q, *α*).

> s /rpm

Table 3. Fluctuation of

(400) diffraction spot.

parameters (

/degree

coincided with the

parameters,

1,

scan

3 axes about the *x*'

q is necessary in

s > 1/(2τ)should be

scans at 1.0° steps. The three

3) if *α* is fixed to 90°. Since the

q, *α*), determined using the

q =60 rpm

s =10 rpm, and

3 /degree

2 magnetic susceptibility axes. The angle

1, and crystallographic *c* axis is determined to be ca 16°. This

3 axes were estimated from the half-widths determined from

1 and 

by trial and error. Table 3 summarizes the half-widths of a

s, 

 =10°, 

1 /degree Fluctuation of

q, and *α*, must be selected appropriately in order to obtain sharp diffraction

1 and 3 axes.

1, 2, 

Fig. 8. Comparison of the structure determined in this study (blue) and the structure reported previously (red). (Reprinted from (F. Kimura et al., 2010b): *J.* Appl. Cryst. Vol.143(2010) pp. 151-153, Part 1, DOI: 10.1107/S0021889809048006. Copyright 2010 International Union of Crystallography)

Fig. 9. Twin structure of a 3D-MOMA of sucrose. The *b* axis and one of the susceptibility axes (here denoted as <sup>3</sup> ) are placed perpendicular to the plane of the diagram. The other axes ( 1 and **<sup>2</sup>** ) are placed in the plane. The angle between the c axis and the 1 axis is ca 16º. (Reprinted from (F. Kimura et al., 2010b): *J.* Appl. Cryst. Vol.143(2010) pp. 151-153, Part 1, DOI: 10.1107/S0021889809048006. Copyright 2010 International Union of Crystallography)

system, the magnetic axes correspond to the crystallographic axes. From the figure, we find that the cell dimensions increase in the order 1, 2, and 3. Combining the indexing results, we conclude that 1= *c*, 2= *a*, and 3= *b*. The alignment of the *c* axis parallel to the magnetic field has been reported for orthorhombic lysozyme crystals.( Sato et al., 2000). Figure 12 shows a detailed diffraction pattern with resolution rings.

Three-Dimensional Magnetically-Oriented

*b*

*a*

*a*

Society)

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 193

Fig. 12. X-ray diffraction image of the lysozyme MOMAwith resolution rings indicated. (Reprinted from (F. Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 12-15, No. 1,

*b b*

**0**

*b*

*a*

*a*

*a*

*a*

*a*

*b c*

**0 0**

Fig. 13. Comparison of the crystal structures of lysozyme: (a and b) the structure determined through a MOMA prepared in the present study; (c and d) the structure reproduced from the database (PDB code 2ZQ4). (Reprinted from (F. Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 12-15, No. 1, DOI: 10.1021/cg100790r. Copyright 2010 American Chemical

DOI: 10.1021/cg100790r. Copyright 2010 American Chemical Society)

(a) (b)

(c) (d)

Fig. 10. (a) Photograph (a division: 1 mm) of a magnetically oriented microcrystal array (MOMA) of lysozyme. (b) Microphotograph of MOMA that is composed of various sizes of microcrystals. (Reprinted from (F. Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 12-15, No. 1, DOI: 10.1021/cg100790r. Copyright 2010 American Chemical Society)

The X-ray results are summarized in Table 4. The indexing was determined as follows: space group, *P*212121; lattice constants, *a*=51.26 Å, *b*=59.79 Å , *c*=29.95 Å , and *V*=91803 Å3. This data was compared with that reported on the lysozyme single crystal (PDB code 2ZQ4) in Table 5. The cell dimensions obtained in this study are shorter than those reported in the literature. The shrinkage was attributed to the dehydration of the crystal. X-ray diffraction of dehydrated lysozyme crystals in triclinic (Kachalova et al., 1991) and monoclinic (Nagendra et al., 1998) forms has been reported. The graphical display is shown in Fig. 13 for easy comparison of the MOMA with the reported structure (PDB code 2ZQ4). A comparison of the CR positions between the present result and 2ZQ4 gave rmsd=0.755 Å , indicating that the shrinkage of the lattice was mainly due to the loss of water molecules and that the protein chain conformation remained essentially unchanged.

Fig. 10. (a) Photograph (a division: 1 mm) of a magnetically oriented microcrystal array (MOMA) of lysozyme. (b) Microphotograph of MOMA that is composed of various sizes of microcrystals. (Reprinted from (F. Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011) pp.

12-15, No. 1, DOI: 10.1021/cg100790r. Copyright 2010 American Chemical Society)

Fig. 11. X-ray diffraction of a lysozyme MOMA taken from three different orthogonal directions. (Reprinted from (F. Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 12-

The X-ray results are summarized in Table 4. The indexing was determined as follows: space group, *P*212121; lattice constants, *a*=51.26 Å, *b*=59.79 Å , *c*=29.95 Å , and *V*=91803 Å3. This data was compared with that reported on the lysozyme single crystal (PDB code 2ZQ4) in Table 5. The cell dimensions obtained in this study are shorter than those reported in the literature. The shrinkage was attributed to the dehydration of the crystal. X-ray diffraction of dehydrated lysozyme crystals in triclinic (Kachalova et al., 1991) and monoclinic (Nagendra et al., 1998) forms has been reported. The graphical display is shown in Fig. 13 for easy comparison of the MOMA with the reported structure (PDB code 2ZQ4). A comparison of the CR positions between the present result and 2ZQ4 gave rmsd=0.755 Å , indicating that the shrinkage of the lattice was mainly due to the loss of water molecules and

15, No. 1, DOI: 10.1021/cg100790r. Copyright 2010 American Chemical Society)

that the protein chain conformation remained essentially unchanged.

(a) (b)

Fig. 12. X-ray diffraction image of the lysozyme MOMAwith resolution rings indicated. (Reprinted from (F. Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 12-15, No. 1, DOI: 10.1021/cg100790r. Copyright 2010 American Chemical Society)

Fig. 13. Comparison of the crystal structures of lysozyme: (a and b) the structure determined through a MOMA prepared in the present study; (c and d) the structure reproduced from the database (PDB code 2ZQ4). (Reprinted from (F. Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 12-15, No. 1, DOI: 10.1021/cg100790r. Copyright 2010 American Chemical Society)

Three-Dimensional Magnetically-Oriented

**6.1 Alanine (F. Kimura et al., 2010a)** 

the microcrystals are aligned three-dimensionally.

(a)

(b)

**6. Neutron diffraction data** 

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 195

In measuring the pole figure, a 3D-MOMA sample was set on a goniometer approximately in the direction shown in Fig. 14(a). The angles χ and φ correspond to the operation angles of the goniometer. The angle φ runs from 0 to 180° at a given value of the angle χ that runs from 0 to -180°. At this setting, the diffractions corresponding to the (040), (002), (120), etc. are expected to appear at locations shown in the figure. In Fig. 14(b), the measured pole figure is displayed. Here, all the diffraction spots are displayed in the same figure. A half sphere was scanned so that spots for {120} were all observed. These spots appear around χ=0, -180° and φ=45, 135° because *a*/*b* = 1/2. On the other hand, a spot for (040) was only scanned on one side. Instead, a close examination of (040) was made by scanning in the vicinity of χ=0° and φ=90°. A contour plot of the intensity of (040) is shown in Fig. 15 as a function of χ and φ. The average fwhm over two angles is ca. 4°. A 2θ profile for the (040) diffraction is shown in Fig. 16 as a typical example. The peak is clearly distinguished from the background. This high signal-to-noise ratio was unexpected because no care was taken to reduce the background incoherent scattering when choosing the resin precursor. This might be attributed to the fact that the coherent diffraction is extremely increased because

Fig. 14. (a) Sample setting of the L-alanine PSC and the expected diffraction spots. Angle

runs from 0 to -180° and *φ* runs from 0 to 180°. (b) Pole figure obtained for the PSC to be compared with (a). (Reprinted from (F. Kimura et al., 2010a): *Cryst. Growth Des.*Vol.10(2010) pp. 48-51. No. 1, DOI: 10.1021/cg90132h. Copyright 2010 American Chemical Society)


Table 4. Summary of data collection and refinement for lysozyme MOMA. The values in highest resolution bin were indicated in the parentheses.

### **6. Neutron diffraction data**

194 Neutron Diffraction

A. Crystal data

Cell dimensions

B. Refinement statistics

 Wavelength (Å) 1.5418 Space group P212121

 a (Å) 51.26 b (Å) 59.79 c (Å) 29.95 V (Å3) 91, 803 Observed reflections 10, 604

Resolution (Å) 26.77 - 3.00 (3.16 - 3.00)

Resolution limits (Å) 25.86 - 3.00 (3.78-3.00)

Table 4. Summary of data collection and refinement for lysozyme MOMA. The values in

No. of reflections used 2, 989 (1, 423)

 Final R-factor 0.215 (0.253) Free R value 0.270 (0.292)

 completeness (%) 84.4 (81) No. of protein atoms 1, 001

No. of solvent molecules 0

Average B-factor (Å2) 51.0

 Bond distances (Å) 0.004 Bond angles (deg.) 0.743 Dihedrals (deg.) 22.596

highest resolution bin were indicated in the parentheses.

r.m.s. deviation from ideal geometry

Independent reflections 1, 828 (250) completeness (%) 89.6 (88.6) Rsym (%) 19.4 (48.7) Redundancy 5.8 (5.5) Mean I/ 9.3 (2.7)

### **6.1 Alanine (F. Kimura et al., 2010a)**

In measuring the pole figure, a 3D-MOMA sample was set on a goniometer approximately in the direction shown in Fig. 14(a). The angles χ and φ correspond to the operation angles of the goniometer. The angle φ runs from 0 to 180° at a given value of the angle χ that runs from 0 to -180°. At this setting, the diffractions corresponding to the (040), (002), (120), etc. are expected to appear at locations shown in the figure. In Fig. 14(b), the measured pole figure is displayed. Here, all the diffraction spots are displayed in the same figure. A half sphere was scanned so that spots for {120} were all observed. These spots appear around χ=0, -180° and φ=45, 135° because *a*/*b* = 1/2. On the other hand, a spot for (040) was only scanned on one side. Instead, a close examination of (040) was made by scanning in the vicinity of χ=0° and φ=90°. A contour plot of the intensity of (040) is shown in Fig. 15 as a function of χ and φ. The average fwhm over two angles is ca. 4°. A 2θ profile for the (040) diffraction is shown in Fig. 16 as a typical example. The peak is clearly distinguished from the background. This high signal-to-noise ratio was unexpected because no care was taken to reduce the background incoherent scattering when choosing the resin precursor. This might be attributed to the fact that the coherent diffraction is extremely increased because the microcrystals are aligned three-dimensionally.

Fig. 14. (a) Sample setting of the L-alanine PSC and the expected diffraction spots. Angle runs from 0 to -180° and *φ* runs from 0 to 180°. (b) Pole figure obtained for the PSC to be compared with (a). (Reprinted from (F. Kimura et al., 2010a): *Cryst. Growth Des.*Vol.10(2010) pp. 48-51. No. 1, DOI: 10.1021/cg90132h. Copyright 2010 American Chemical Society)

Three-Dimensional Magnetically-Oriented

structure analysis.

**2011)** 

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 197

Fig. 16. Diffraction profile for the (040) plane. Baseline is not subtracted. (Reprinted from (F.

After subtraction of the baseline, the intensity of the peak is calculated by integration and corrected for Lorentz factor; no absorption correction was performed. The structure refinement was performed with Shelxl-97 using the crystallographic data of neutron diffraction of L-alanine reported in the literature (Wilson et al., 2005), where the atom coordinates, the temperature factors, and the anisotropic temperature factors were fixed. The results are summarized in Table 6. The values of *R*1 and *wR*2 are 0.184 and 0.368, respectively. In the present preliminary study, the integration was performed only along the 2θ direction, and the integration with respect to χ and φ directions was not performed. In addition, absorption correction was not made. Hence, the intensity data could include some error. However, the experimental and calculated results appear to agree satisfactorily, indicating that the MOMA can provide diffraction data that will lead to a successful

**7. Possibility of MOMS for application of structure analysis (Matsumoto et al.,** 

rpm, the {002} diffractions did not appear, and the {120} diffractions appeared on the equator. This suggests that the reciprocal vectors **G**{002} are aligned in the direction of the applied magnetic field (||*x* ) and the reciprocal vectors **G**{120} are distributed randomly on the *yz* plane (see Fig. 5). Because the crystal has three mutually orthogonal 2-fold axes (*a*, *b*, and *c* axes), the diffractions belonging to {120} are not distinguishable in a MOMA, neither are the diffractions {002}. With an increase in rotation speed, the {120} diffractions on the

the same time, the {002} diffractions appeared on the meridian at this rotation speed. This indicates that the *b*\* axis was directed to the *z*-axis, and there was rotational symmetry about

equator disappeared and moved to locations around 45° and 135° when

= 0 rpm (static) and 20 rpm. At

3 axis (||*b*\*) in the direction of the rotation axis (||*z*) is

= 0

= 20 rpm. At

Kimura et al., 2010a): Cryst. Growth Des. Vol.10(2010) pp. 48-51, No. 1, DOI:

10.1021/cg90132h. Copyright 2010 American Chemical Society)

Figure 17 shows the diffraction patterns obtained for

the *b*\* axis. The alignment of the


Table 5. Comparison of crystal data of lysozyme.

Fig. 15. Contour plot of the diffraction spot for (040) shown in Fig. 4(b) as a function of and . Half maximum of the intensity is ca. 600 from which the average fwhm is estimated to be ca. 4°. (Reprinted from (F. Kimura et al., 2010a): Cryst. Growth Des. Vol.10(2010) pp. 48-51, No. 1, DOI: 10.1021/cg90132h. Copyright 2010 American Chemical Society)

Sample MOMA Single crystal

PDB Code present work 2ZQ4

Space group *P*212121 *P*212121

*a* (Å) 51.26 56.48

*b* (Å) 59.79 73.76

*c* (Å) 29.95 30.54

*V* (Å3) 91, 803 127, 229

Fig. 15. Contour plot of the diffraction spot for (040) shown in Fig. 4(b) as a function of

No. 1, DOI: 10.1021/cg90132h. Copyright 2010 American Chemical Society)

. Half maximum of the intensity is ca. 600 from which the average fwhm is estimated to be ca. 4°. (Reprinted from (F. Kimura et al., 2010a): Cryst. Growth Des. Vol.10(2010) pp. 48-51,

and

Cell dimensions

Table 5. Comparison of crystal data of lysozyme.

Fig. 16. Diffraction profile for the (040) plane. Baseline is not subtracted. (Reprinted from (F. Kimura et al., 2010a): Cryst. Growth Des. Vol.10(2010) pp. 48-51, No. 1, DOI: 10.1021/cg90132h. Copyright 2010 American Chemical Society)

After subtraction of the baseline, the intensity of the peak is calculated by integration and corrected for Lorentz factor; no absorption correction was performed. The structure refinement was performed with Shelxl-97 using the crystallographic data of neutron diffraction of L-alanine reported in the literature (Wilson et al., 2005), where the atom coordinates, the temperature factors, and the anisotropic temperature factors were fixed. The results are summarized in Table 6. The values of *R*1 and *wR*2 are 0.184 and 0.368, respectively. In the present preliminary study, the integration was performed only along the 2θ direction, and the integration with respect to χ and φ directions was not performed. In addition, absorption correction was not made. Hence, the intensity data could include some error. However, the experimental and calculated results appear to agree satisfactorily, indicating that the MOMA can provide diffraction data that will lead to a successful structure analysis.

### **7. Possibility of MOMS for application of structure analysis (Matsumoto et al., 2011)**

Figure 17 shows the diffraction patterns obtained for = 0 rpm (static) and 20 rpm. At = 0 rpm, the {002} diffractions did not appear, and the {120} diffractions appeared on the equator. This suggests that the reciprocal vectors **G**{002} are aligned in the direction of the applied magnetic field (||*x* ) and the reciprocal vectors **G**{120} are distributed randomly on the *yz* plane (see Fig. 5). Because the crystal has three mutually orthogonal 2-fold axes (*a*, *b*, and *c* axes), the diffractions belonging to {120} are not distinguishable in a MOMA, neither are the diffractions {002}. With an increase in rotation speed, the {120} diffractions on the equator disappeared and moved to locations around 45° and 135° when = 20 rpm. At the same time, the {002} diffractions appeared on the meridian at this rotation speed. This indicates that the *b*\* axis was directed to the *z*-axis, and there was rotational symmetry about the *b*\* axis. The alignment of the 3 axis (||*b*\*) in the direction of the rotation axis (||*z*) is

Three-Dimensional Magnetically-Oriented

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 199

Table 6. Calculated and measured *F*2 values for diffractions from (*hkl*) planes for L-alanine

MOMA.

12.86 10.42 8.26 1.02 0.44 3.55 77.06 3.96 5.95 0.26 1.15 0.00 95.88 19.92 0.00 8.57 1.99 10.09 8.99 56.46 0.42 13.51 4.90 23.59 23.75 1.88 0.06 2.53 2.48 14.04 8.01

<sup>2</sup> *F*<sup>o</sup>

2

9.50 8.69 7.89 1.06 1.43 5.42 82.74 3.58 3.62 0.37 0.89 -0.43 167.23 22.84 -0.10 10.76 0.97 22.55 12.59 33.00 1.51 11.27 2.57 18.36 21.53 2.84 -0.81 4.15 -0.81 16.37 7.80

*hkl F*<sup>c</sup>

Fig. 17. X-ray diffraction images of a magnetically oriented microcrystal suspension (MOMS) obtained at = 0 and 20 rpm. Solid and broken circles indicate the {120} and {002} diffraction spots, respectively. (Reprint from (Matsumoto et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 945-948, No. 4, DOI: 10.1021/cg200090u. Copyright 2011 American Chemical Society)

based on the behavior of a magnetically uniaxial particle (T. Kimura, 2009; Yamaguchi et al., 2010). We successfully obtained two fiber X-ray diffraction patterns from a microcrystal suspension (crystal size of ca. 2–20 μm) oriented in a magnetic field of 1 T generated by permanent magnets. Diffraction spots were sharp (ca. 2–3° in half width) and well separated. The results suggest a potential use of MOMS for crystal structure analysis of solid materials that do not grow into large crystals but are obtained in the form of microcrystal suspensions.

To summarize, we have introduced a new method, 3D-MOMA (three-dimensional magnetically oriented microcrystal array) method. We use a 3D-MOMA that is a composite in which microcrystals are three-dimensionally aligned. With a 3D-MOMA, we can obtain diffraction equivalent to that from a real single crystal, successfully determining the molecular structure. With this method we can convert a suspension of biaxial microcrystals with sizes of 10 m or larger to a 3D-MOMA of centimeter sizes by using magnetic fields of 2 T or more. Successful examples of single crystal X-ray analyses of several crystals, including inorganic, organic, and protein crystals, by using 3D-MOMA are presented. A preliminary result of neutron diffraction of 3D-MOMA is also reported. Though the application to the neutron diffraction is preliminary, the MOMA method has high potential for the neutron diffraction method because a large size 3D-MOMA can easily be prepared. The problems to be solved for the future development of this method in the area of neutron single crystal diffraction analyses are discussed.

Fig. 17. X-ray diffraction images of a magnetically oriented microcrystal suspension

diffraction spots, respectively. (Reprint from (Matsumoto et al., 2011): Cryst. Growth Des. Vol.11(2011) pp. 945-948, No. 4, DOI: 10.1021/cg200090u. Copyright 2011 American

based on the behavior of a magnetically uniaxial particle (T. Kimura, 2009; Yamaguchi et al., 2010). We successfully obtained two fiber X-ray diffraction patterns from a microcrystal suspension (crystal size of ca. 2–20 μm) oriented in a magnetic field of 1 T generated by permanent magnets. Diffraction spots were sharp (ca. 2–3° in half width) and well separated. The results suggest a potential use of MOMS for crystal structure analysis of solid materials that do not grow into large crystals but are obtained in the form of microcrystal

To summarize, we have introduced a new method, 3D-MOMA (three-dimensional magnetically oriented microcrystal array) method. We use a 3D-MOMA that is a composite in which microcrystals are three-dimensionally aligned. With a 3D-MOMA, we can obtain diffraction equivalent to that from a real single crystal, successfully determining the molecular structure. With this method we can convert a suspension of biaxial microcrystals with sizes of 10 m or larger to a 3D-MOMA of centimeter sizes by using magnetic fields of 2 T or more. Successful examples of single crystal X-ray analyses of several crystals, including inorganic, organic, and protein crystals, by using 3D-MOMA are presented. A preliminary result of neutron diffraction of 3D-MOMA is also reported. Though the application to the neutron diffraction is preliminary, the MOMA method has high potential for the neutron diffraction method because a large size 3D-MOMA can easily be prepared. The problems to be solved for the future development of this method in the area of neutron

= 0 and 20 rpm. Solid and broken circles indicate the {120} and {002}

(MOMS) obtained at

Chemical Society)

suspensions.

single crystal diffraction analyses are discussed.


Table 6. Calculated and measured *F*2 values for diffractions from (*hkl*) planes for L-alanine MOMA.

Three-Dimensional Magnetically-Oriented

Microcrystal Array: A Large Sample for Neutron Diffraction Analysis 201

Kimura, F.; Kimura, T.; Matsumoto, K.; Metoki, N. (2010a). Single-Crystal Neutron

Kimura, F.; Kimura, T.; Oshima, W., Maeyama, W.; Aburaya, K. (2010b). X-ray diffraction

Kimura, F.; Mizutani, K.; Mikami, B.; Kimura, T. (2011). Single-Crystal X-ray Diffraction

Kimura, T.; Yamato, M.; Koshimizu, W.; Koike, M.; Kawai, T. (2000). Magnetic Orientation of Polymer Fibers in Suspension, Langmuir, Vol. 16, No. 2, PP. 858-861. Kimura, T.; Yoshino, M. (2005). Three-Dimensional Crystal Alignment Using a Time-

Kimura, T.; Kimura, F.; Yoshino, M. (2006). Magnetic Alteration of Crystallite Alignment

Kimura, T. (2009a). Orientation of Feeble Magnetic Particles in Dynamic Magnetic Fields,

Kimura, T.; Chang, C.; Kimura, F.; Maeyama, M. (2009b). The pseudo-single-crystal method:

Kohama, T.; Takeuchi, H.; Usui, M.; Akiyama, J.; Sung, M.-G.; Iwai, K.; Shigeo, A. (2007). In-

Diffraction, *Mater. Trans.*, vol. 48, No. 11, pp. 2867-2871. ISSN 1345-9678. Maret, G. & Dransfeld, K. (1985). Biomolecules and Polymers in High Steady Magnetic

Margiolaki, I.; Wright, J. P. (2008). Powder crystallography on macromolecules, Acta

Matsumoto, K.; Kimura, F.; Tsukui, S., Kimura, T. (2011). X-ray Diffraction of a Magnetically

Nagendra, H. G.; Sukumar, N.; Vijayan, M. (1998). Role of water in plasticity, stability, and

Niimura, N. (1999). Neutrons expand the field of structural biology, Curr. Opin. Struct. Biol.,

Nye, J. F. (1985). Paramagnetic and diamagnetic susceptibility, in Physical Properties of Crystals, ch. 3, pp. 53-67. Oxford University Press, ISBN0-19-851165-5, New York Samulski, E. T. (1986). Magnetically Oriented Solutions, *Science*, Vol. 234, No. 4782, (12

Sato, T.; Yamada, Y.; Saijo, S.; Hori, T.; Hirose, R.; Tanaka, N.; Sazaki, G.; Nakajima, K.;

Igarashi, N.; Tanaka, M.; Matsuura, Y. (2000). Enhancement in the perfection of

Crystallogr. Section A, Vol. 64, Part 1, (January 2008), pp. 169-180.

hydration, *Proteins*, Vol. 32, Issue 2, (1 August 1998), pp. 229-240.

*Jpn. J. Appl. Phys.*, Vol. 48, No. 2, (Feb. 2009), pp. 020217(1-3).

*Crystal growth & Design*, Vol. 10, No. 1, pp. 48-51.

*J*. *Appl*. *Crystallogr*., Vol. *43*, pp. 151-153.

*Des*., Vol. *11, No. 1*, pp. 12-15.

4805-4808.

2006) pp. 3464-3466.

535–537. ISSN 0021-8898

Berlin: Springer Verlag

(March 08, 2011), pp. 945-948.

December 198), p. 1424.

Vol. 9, Issue 5, (1 October 1999), pp. 602-608.

Diffraction Study of Pseudo Single Crystal Prepared from Microcrystalline Powder,

study of a pseudo single crystal prepared from a crystal belonging to point group 2,

Study of a Magnetically Oriented Microcrystal Array of Lysozyme, *Cryst*. *Growth*.

Dependent Elliptic Magnetic Field, *Langmuir,* Vol. 21, No. 11, (April 30, 2005 ), pp.

Converting Powder to a Pseudo Single Crystal , *Langmuir*, Vol. 22, No. 8, (March 9,

a third approach to crystal structure determination, *J*. *Appl*. *Crystallogr*., Vol. 42, pp.

Situ Observation of Crystal Alignment under a Magnetic Field Using X-ray

Field, in Topics in Applied Physics, Vol. 57, F. Herlach, (Ed), ch. 4. pp. 143-204,

Oriented Microcrystal Suspension of L-Alanine, *Crystal growth & Design*, Vol. 11,

action of proteins: The crystal structures of lysozyme at very low levels of

### **8. References**


Asai, S.; Fujiwara, M.; Kimura, T.; Liang, Z.; Uyeda, C.; Wang, B.; Yamamoto, I.; Zhang, C.

Blake, C. C.; Koenig, D. F.; Mair, G. A.; North, A. C.; Phillips, D. C.; Sarma, V. R. (1965).

Finke, W. (1909), Magnetische Messungen an Platinmetallen und monoklinen Kristallen,

Gielen, J. C.; Wolffs, M.; Portale, G.; Bras, W.; Henze, O.; Kilbinger, A. F. M.; Feast, W. J.;

Glucksman, M. J.; Hay, R. D.; Makowski, L. (1986). X-ray diffraction from magnetically

Graves, b. J.; Crownther, R. L.; Chandran, C.; Rumberger, J. M.; Li, S. H.; Huang, K.-

Hanson, J. C., Sieker, L. C. & Jensen, L. H. (1973). Sucrose: X-ray refinement and comparison

Hariss, K. D. M.; Cheung, E. Y. (2004), How to determine structures when single crystals

Herrmann, T.; Güntert, P.; Wüthrich, K. (2002). Protein NMR Structure Determination with

Kachalova, G. S.; Morozov, V. N.; Morozova, T. Ya.; Myachin, E. T.; Vagin, A. A.;

Kendrew, J. C.; Bodo, G.; Dintzis, H. M.; Parrish, R. G.; Wyckoff, H.; Phillips, D. C. (1958). X-

(Eds), ch. 5. pp. 191-247, Tokyo: Kodansha Springer. ISSN 0933-033X Ataka, M.; Tanaka, S. (1986). The growth of large single crystals of lysozyme. Biopolymers,

Blundell, T. L.; Johnson, L. N. Protein Crystallography; Academic Press: 1976.

Vol. 336, Issue 1, (6. November 1909), PP. 149–168.

Vol. 25, No. 2, (February 1986), pp. 337–350.

doi:10.1038/206757a0.

March 1986), pp. 1273-1276.

1272-1276.

537.

797–808.

526-538

pp. 209-227.

1058), pp. 662-666.

Issue 1, (17 June 1991), pp. 91-94.

(2006). Magnetic Orientation, In Magneto-Science, M. Yamaguchi, Y. Tanimoto,

Structure of hen egg-white lysozyme. A three-dimensional Fourier synthesis at 2 Angstrom resolution, Nature, Vol. 206, (22 May 1965), pp. 757 – 761,

insbesondere der Eisen-, Kobult- und Nickelsalze; von Wilhelm Fink., *Ann. Phys.*,

Maan, J. C.; Schenning, A. P. H. J.; Christianen, P. C. M. (2009). Molecular Organization of Cylindrical Sexithiophene Aggregates Measured by X-ray Scattering and Magnetic Alignment, *Langmuir* Vol. 25, No. 3, (January 9, 2009), PP.

oriented solutions of macromolecular assemblies, *Science*, Vol. 231, no. 4743, (14

S.;Presky, D. H.; Familletti, P. C.; Wolitzky, B. A.; Burns, D. K. Nature 1994, 367,

with neutron refinement, *Acta Cryst. Section B*, Vol. 29, Issue 4, (April 1973), pp.

cannot be grown: opportunities for structure determination of molecular materials using powder diffraction data, Chem. Soc. Rev., Vol. 33, No. 8, (22 Sep 2004) pp.

Automated NOE Assignment Using the New Software CANDID and the Torsion Angle Dynamics Algorithm DYANA, J. Mol. Biol. Vol. 319, No. 1, (24 May 2002),

Strokopytov, B. V.; Nekrasov, Yu. V. (1991). Comparison of structures of dry and wet hen egg-white lysozyme molecule at 1.8 Å resolution, *FEBS Lett.*, Vol. 284,

ray crystallography — the first image of myoglobin, Nature, Vol. 181, (8 March,

**8. References** 


**10** 

*Algeria* 

Mahieddine Lahoubi

**Temperature Evolution of the Double Umbrella** 

*Badji Mokhtar-Annaba University, Faculty of Sciences, Department of Physics, Annaba* 

The rare earth iron garnets RE3Fe5O12 (REIG hereafter) have been discovered at Grenoble (France) (Bertaut & Forrat, 1956; Bertaut et al., 1956) then independently at Murray Hill (USA) (Geller & Gilleo, 1957a, 1957b). These most studied ferrimagnetic materials have a general formula {RE3+3}[Fe3+2](Fe3+3)O12 where RE3+ can be any trivalent rare earth ion or the Yttrium Y3+. The crystal structure is described by the cubic space group Ia <sup>3</sup> d-( <sup>10</sup> O*<sup>h</sup>* ) No. 230. Three type of brackets are used to indicate the different coordinations of the cations with respect to the oxygen O2– ions situated in the general positions x, y, z of the sites 96h(1) (Fig. 1). The RE3+ ions are located in the dodecahedral sites {24c}(222) whereas the two Fe3+

ions are distributed in the octahedral [16a]( 3 ) and tetrahedral (24d)( 4 ) sites.

Fig. 1. Crystallographic sites of YIG in the space group Ia 3 d (Geller & Gilleo, 1957b)

**1. Introduction** 

**Magnetic Structure in Terbium Iron Garnet** 

orthorhombic lysozyme crystals grown in a high magnetic field(10 T), *Acta Crystallogr. Section D*, vol. 56, Issue 8, (August 2000) pp. 1079-1083. ISSN 0907-4449


## **Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet**

Mahieddine Lahoubi

*Badji Mokhtar-Annaba University, Faculty of Sciences, Department of Physics, Annaba Algeria* 

### **1. Introduction**

202 Neutron Diffraction

Terada, N.; Suzuki, H. S.; Suzuki, T. S.; Kitazawa, H.; Sakka, Y.; Kaneko, K.; Metoki, N.

van der Beek, D.; Petukhov, A. V.; Davidson, P.; J. Ferr\_e, J.; Jamet, J. P.; Wensink, H. H.;

Wessels, T., Baerlocher, C. & McCusker, L. B. (1999). Single-Crystal-Like Diffraction Data

Wilson, C. C.; Myles, D.; Ghosh, M.; Johnson, L. N.; Wang, W. (2005). Neutron diffraction

Yamaguchi, M.; Ozawa, S.; Yamamoto, I. (2010). Dynamic Behavior of Magnetic Alignment

103, Issue 4, (10 November 2000), pp. 633-643

Vol. 73, Issue 4, (5 April 2006), pp. 041402(1-10).

Mar 2008), pp. 112507(1-3).

August, 2010), pp. 080213(1-3)

479.

1318-1322.

orthorhombic lysozyme crystals grown in a high magnetic field(10 T), *Acta Crystallogr. Section D*, vol. 56, Issue 8, (August 2000) pp. 1079-1083. ISSN 0907-4449 Sousa, M. C.; Trame, C. B.; Tsuruta, H.; Wilbanks, S. M.; Reddy, V. S.; McKay, D. B. (2000).

Crystal and Solution Structures of an HslUV Protease–Chaperone Complex, Vol.

(2008). *In situ* neutron diffraction study of aligning of crystal orientation in diamagnetic ceramics under magnetic fields, *Appl. Phys. Lett.*, Vol. 92, Issue 11, (17

Vroege, G. J.; Bras, W.; Lekkerkerker, H. N. W. (2006). Magnetic-field-induced orientational order in the isotropic phase of hard colloidal platelets, *Phys. Rev. E*,

from Polycrystalline Materials, Science, *Vol. 284 no. 5413, (16 April 1999),* pp. 477–

investigations of L- and D-alanine at different temperatures: the search for structural evidence for parity violation, *New J. Chem*., Vol. 29, (07 Sep 2005), pp.

in Rotating Field for Magnetically Weak Particles, Jpn. J. Appl. Phys. Vol. 49, (5

The rare earth iron garnets RE3Fe5O12 (REIG hereafter) have been discovered at Grenoble (France) (Bertaut & Forrat, 1956; Bertaut et al., 1956) then independently at Murray Hill (USA) (Geller & Gilleo, 1957a, 1957b). These most studied ferrimagnetic materials have a general formula {RE3+3}[Fe3+2](Fe3+3)O12 where RE3+ can be any trivalent rare earth ion or the Yttrium Y3+. The crystal structure is described by the cubic space group Ia <sup>3</sup> d-( <sup>10</sup> O*<sup>h</sup>* ) No. 230. Three type of brackets are used to indicate the different coordinations of the cations with respect to the oxygen O2– ions situated in the general positions x, y, z of the sites 96h(1) (Fig. 1). The RE3+ ions are located in the dodecahedral sites {24c}(222) whereas the two Fe3+ ions are distributed in the octahedral [16a]( 3 ) and tetrahedral (24d)( 4 ) sites.

Fig. 1. Crystallographic sites of YIG in the space group Ia 3 d (Geller & Gilleo, 1957b)

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 205

the next-generation spintronic devices. The fundamental properties of these REIG and their applications have been enormously reviewed previously and only a few authors are reported here (Dahlbäck, 2006; Geller, 1977; Guillot, 1995; Kazei et al., 1991). At low temperature when both crystal field acting on the RE3+ ions and the RE3+{c}**–**Fe3+(d) magnetic exchange anisotropies become important (Nekvasil & Veltrusky, 1990), the Néel model must be replaced by a new spin configuration in which the RE3+ magnetic moments cease to be antiparallel to the Fe3+ magnetization *M*Fe. So, neutron diffraction experiments have been performed previously to study, the non collinear magnetic structures of the RE3+ moments (RE = Dy, Er, Ho, Tb, Tm and Yb) which appear at liquid-helium temperatures (Bertaut et al., 1970; Guillot et al., 1982; Herpin et al., 1960; Hock et al., 1990, 1991; Lahoubi et al., 1984; Lahoubi, 1986, 2012; Pickart et al., 1970; Tchéou et al., 1970a, 1970b). Neutron diffractions experiments have been also made to follow the temperature dependence of these ''*umbrella*'' magnetic structures in HoIG (Guillot et al., 1983, 1984), ErIG (Hock et al., 1991), DyIG (Lahoubi et al., 2009, 2010).

In the present chapter, we will present the temperature evolution of the Tb3+ magnetic ordering in TbIG using powder neutron diffraction experiments combined with magnetic field magnetization measurements on single crystal (Lahoubi et al., 1984, 1985; Lahoubi, 1986; Lahoubi et al., 1997; Lahoubi, 2012). The experimental techniques are described in the section 2. The principle of the non-polarized neutrons diffraction with the preliminarily experiment at 614 K are introduced in the main section 3. The neutron diffraction results obtained at high and low temperatures are discussed using the predictions of the symmetry analysis and compared with the data of magnetization measurements in the sections 4 and 5 respectively. The "*Representation Analysis*" of Bertaut (Bertaut, 1968, 1971, 1972) is applied to the paramagnetic space group Ia 3 d for determining all possible ''*umbrella*'' magnetic structures in this "*cubic description*". The method of the so-called '*'symmetry lowering device"* (Bertaut, 1981) is required in the treatment for the determination of the best subgroups of Ia 3 d when the temperature is decreasing below *T*N until liquid-helium temperatures. The "*basis vectors of irreductible representations*" of the distorted space group R 3 c are chosen in the "*rhombohedral description*". The thermal variations of the parameters of the "*double umbrella*" magnetic structure constitute the section 6 which will be followed by a conclusion in the section 7.

The neutron diffraction experiments were measured on polycrystalline sample of TbIG owing to some severe extinctions which appear when a high quality single crystal is used (Bonnet et al., 1979). The first set of patterns have been recorded previously at the Centre d'Etudes Nucleaires de Grenoble (CENG) CEA Grenoble, France (Lahoubi et al., 1984; Lahoubi, 1986; Lahoubi et al., 1997) using the famous "Position Sensitive Detector" (PSD) detector (Convert et al., 1983; Roudaut et al., 1983). For the study, ten temperatures are chosen in the cryostat: 4.2, 20, 54, 68, 80, 109 ± 2, 127 ± 5, 160, 208 ± 2 and around *T*comp at 244 ± 10 K. In the furnace, the temperatures are: above *T*N (*T* = 614 K) and below *T*N (*T* = 283, 400 and 453 K). The time of counting for some temperatures has been in the order of ten hours. The patterns were recorded with a wavelength equal to 2.49 Å and filters to avoid /2 contaminations are used. The second set of patterns has been collected recently on the high flux diffractometer D1B at the Institut Laue−Langevin Grenoble, France. The value of the wavelength is equal to 2.52 Å and four temperatures 5, 13, 20 and 160 K have been chosen below room temperature. The resolution of the multidetector is 0.2°. For the magnetic study, the magnetization of a flux-

**2. Experimental techniques** 

It is a rather loose structure with a volume of 236.9 Å3 per formula unit, which has the great technical advantage that it is possible to accommodate a very large variety of cations in the garnet structure. Thus, it is feasible to achieve an enormous range of control of the magnetic properties in the garnet structure system. The largest of REIG that can be formed is SmIG with a lattice parameter of 12.529 Å and the smallest is LuIG with a lattice parameter of 12.283 Å. The REIG have became these famous magnetic compounds by illustrating the Néel theory of ferrimagnetism (Néel, 1948). The strongest superexchange interactions between the two iron sublattices Fe3+[a]–Fe3+(d) are antiferromagnetic and the magnetic moment vectors *m*a and *m*d are antiparallel. They make YIG, an ideal ferrimagnet with the Néel temperature (*T*N) equal to 560 K (Pauthenet, 1958a, 1958b). However, substitution of a magnetic rare earth ion for the diamagnetic Y3+ in YIG introduces a third sublattice in the crystallographic site {c} in which *m*RE are the magnetic moment vectors. In this three sublattices model only weaker and negative antiferromagnetic interactions RE3+{c}**–**Fe3+(d) exist. If *M*a, *M*d and *M*RE are the magnetizations of each sublattice, the total bulk ferrite magnetization of REIG is given by the following equation

$$M\text{(REIG)} = \begin{array}{c} \mid M\_{\text{RE}} - \text{ (}M\_{\text{d}} - M\_{\text{a}}\text{)} \text{)} \tag{1}$$

For the heavier RE3+ ions (Eu3+, ..., Yb3+)*, M*RE is antiparallel to the net resultant of the iron magnetizations *M*Fe = *M*d – *M*a. We can consider that if we have *M*Fe ≈ *M*(YIG) then the interactions between the rare earth ions are negligible and the equation (1) becomes in a first approximation

$$M\text{(REIG)} = \begin{array}{c} \mid M\_{\text{RE}} - M\_{\text{Fe}} \vert \approx \; \mid \; M\_{\text{RE}} - M\_{\text{YIC}} \vert \tag{2}$$

The magnetizations *M*a and *M*d are still given by the N.M.R values found in YIG (Gonano et al., 1967). Below *T*N which is nearly the same for all REIG compounds (554 ± 6) K (Pauthenet, 1958a, 1958b) the magnetization of the rare earth ions *M*RE can dominate the magnetization *M*Fe. If the temperature is decreasing, a rapid increasing of *M*RE is observed because of the large magnetic moment *m*RE. In heavy rare earth iron garnets, there exists a compensation temperature (*T*comp) or inversion temperature (*T*I) (Herpin, 1968) at which the bulk ferrite magnetization vanishes. For TbIG, *T*comp is equal to (243.5 ± 0.5 K) and (249.0 ± 0.5 K) for the single crystal and powder samples respectively (Lahoubi et al., 1985; Lahoubi, 1986). In the vicinity of *T*comp, the magnetic behavior is equivalent to that observed in the antiferromagnet compounds with the existence of the so-called field induced phase transitions which have been studied previously theoretically and experimentally (Zvezdin, 1995). In the Néel model, the RE3+ magnetic behavior is described by the pure free ion Brillouin function assuming that the superexchange interactions are represented by the isotropic Weiss molecular field coefficients.

The optical and magneto-optical (MO) properties of REIG and their substituted compounds have also received a substantial interest due to their strong Faraday and Kerr effects. The REIG had their first industrial use in bubble memories more than twenty years ago. Today, these MO materials are the key elements of several technical applications. There are used in Faraday rotators, optical isolators, holographic storage and magnetic field sensors. These applications can be enhanced by using photonic crystals with REIG and such research has yielded promising results. Recently, an additional interest to the REIG has been caused by the prospects for developing materials based on these ferrimagnets for hardware components in the next-generation spintronic devices. The fundamental properties of these REIG and their applications have been enormously reviewed previously and only a few authors are reported here (Dahlbäck, 2006; Geller, 1977; Guillot, 1995; Kazei et al., 1991). At low temperature when both crystal field acting on the RE3+ ions and the RE3+{c}**–**Fe3+(d) magnetic exchange anisotropies become important (Nekvasil & Veltrusky, 1990), the Néel model must be replaced by a new spin configuration in which the RE3+ magnetic moments cease to be antiparallel to the Fe3+ magnetization *M*Fe. So, neutron diffraction experiments have been performed previously to study, the non collinear magnetic structures of the RE3+ moments (RE = Dy, Er, Ho, Tb, Tm and Yb) which appear at liquid-helium temperatures (Bertaut et al., 1970; Guillot et al., 1982; Herpin et al., 1960; Hock et al., 1990, 1991; Lahoubi et al., 1984; Lahoubi, 1986, 2012; Pickart et al., 1970; Tchéou et al., 1970a, 1970b). Neutron diffractions experiments have been also made to follow the temperature dependence of these ''*umbrella*'' magnetic structures in HoIG (Guillot et al., 1983, 1984), ErIG (Hock et al., 1991), DyIG (Lahoubi et al., 2009, 2010).

In the present chapter, we will present the temperature evolution of the Tb3+ magnetic ordering in TbIG using powder neutron diffraction experiments combined with magnetic field magnetization measurements on single crystal (Lahoubi et al., 1984, 1985; Lahoubi, 1986; Lahoubi et al., 1997; Lahoubi, 2012). The experimental techniques are described in the section 2. The principle of the non-polarized neutrons diffraction with the preliminarily experiment at 614 K are introduced in the main section 3. The neutron diffraction results obtained at high and low temperatures are discussed using the predictions of the symmetry analysis and compared with the data of magnetization measurements in the sections 4 and 5 respectively. The "*Representation Analysis*" of Bertaut (Bertaut, 1968, 1971, 1972) is applied to the paramagnetic space group Ia 3 d for determining all possible ''*umbrella*'' magnetic structures in this "*cubic description*". The method of the so-called '*'symmetry lowering device"* (Bertaut, 1981) is required in the treatment for the determination of the best subgroups of Ia 3 d when the temperature is decreasing below *T*N until liquid-helium temperatures. The "*basis vectors of irreductible representations*" of the distorted space group R 3 c are chosen in the "*rhombohedral description*". The thermal variations of the parameters of the "*double umbrella*" magnetic structure constitute the section 6 which will be followed by a conclusion in the section 7.

### **2. Experimental techniques**

204 Neutron Diffraction

It is a rather loose structure with a volume of 236.9 Å3 per formula unit, which has the great technical advantage that it is possible to accommodate a very large variety of cations in the garnet structure. Thus, it is feasible to achieve an enormous range of control of the magnetic properties in the garnet structure system. The largest of REIG that can be formed is SmIG with a lattice parameter of 12.529 Å and the smallest is LuIG with a lattice parameter of 12.283 Å. The REIG have became these famous magnetic compounds by illustrating the Néel theory of ferrimagnetism (Néel, 1948). The strongest superexchange interactions between the two iron sublattices Fe3+[a]–Fe3+(d) are antiferromagnetic and the magnetic moment vectors *m*a and *m*d are antiparallel. They make YIG, an ideal ferrimagnet with the Néel temperature (*T*N) equal to 560 K (Pauthenet, 1958a, 1958b). However, substitution of a magnetic rare earth ion for the diamagnetic Y3+ in YIG introduces a third sublattice in the crystallographic site {c} in which *m*RE are the magnetic moment vectors. In this three sublattices model only weaker and negative antiferromagnetic interactions RE3+{c}**–**Fe3+(d) exist. If *M*a, *M*d and *M*RE are the magnetizations of each sublattice, the total bulk ferrite

For the heavier RE3+ ions (Eu3+, ..., Yb3+)*, M*RE is antiparallel to the net resultant of the iron magnetizations *M*Fe = *M*d – *M*a. We can consider that if we have *M*Fe ≈ *M*(YIG) then the interactions between the rare earth ions are negligible and the equation (1) becomes in a first

The magnetizations *M*a and *M*d are still given by the N.M.R values found in YIG (Gonano et al., 1967). Below *T*N which is nearly the same for all REIG compounds (554 ± 6) K (Pauthenet, 1958a, 1958b) the magnetization of the rare earth ions *M*RE can dominate the magnetization *M*Fe. If the temperature is decreasing, a rapid increasing of *M*RE is observed because of the large magnetic moment *m*RE. In heavy rare earth iron garnets, there exists a compensation temperature (*T*comp) or inversion temperature (*T*I) (Herpin, 1968) at which the bulk ferrite magnetization vanishes. For TbIG, *T*comp is equal to (243.5 ± 0.5 K) and (249.0 ± 0.5 K) for the single crystal and powder samples respectively (Lahoubi et al., 1985; Lahoubi, 1986). In the vicinity of *T*comp, the magnetic behavior is equivalent to that observed in the antiferromagnet compounds with the existence of the so-called field induced phase transitions which have been studied previously theoretically and experimentally (Zvezdin, 1995). In the Néel model, the RE3+ magnetic behavior is described by the pure free ion Brillouin function assuming that the superexchange interactions are represented by the

The optical and magneto-optical (MO) properties of REIG and their substituted compounds have also received a substantial interest due to their strong Faraday and Kerr effects. The REIG had their first industrial use in bubble memories more than twenty years ago. Today, these MO materials are the key elements of several technical applications. There are used in Faraday rotators, optical isolators, holographic storage and magnetic field sensors. These applications can be enhanced by using photonic crystals with REIG and such research has yielded promising results. Recently, an additional interest to the REIG has been caused by the prospects for developing materials based on these ferrimagnets for hardware components in

*M M MM* REIG – – │ │ RE d a (1)

*M MM MM* REIG – – │ ││ │ RE Fe RE YIG (2)

magnetization of REIG is given by the following equation

isotropic Weiss molecular field coefficients.

approximation

The neutron diffraction experiments were measured on polycrystalline sample of TbIG owing to some severe extinctions which appear when a high quality single crystal is used (Bonnet et al., 1979). The first set of patterns have been recorded previously at the Centre d'Etudes Nucleaires de Grenoble (CENG) CEA Grenoble, France (Lahoubi et al., 1984; Lahoubi, 1986; Lahoubi et al., 1997) using the famous "Position Sensitive Detector" (PSD) detector (Convert et al., 1983; Roudaut et al., 1983). For the study, ten temperatures are chosen in the cryostat: 4.2, 20, 54, 68, 80, 109 ± 2, 127 ± 5, 160, 208 ± 2 and around *T*comp at 244 ± 10 K. In the furnace, the temperatures are: above *T*N (*T* = 614 K) and below *T*N (*T* = 283, 400 and 453 K). The time of counting for some temperatures has been in the order of ten hours. The patterns were recorded with a wavelength equal to 2.49 Å and filters to avoid /2 contaminations are used. The second set of patterns has been collected recently on the high flux diffractometer D1B at the Institut Laue−Langevin Grenoble, France. The value of the wavelength is equal to 2.52 Å and four temperatures 5, 13, 20 and 160 K have been chosen below room temperature. The resolution of the multidetector is 0.2°. For the magnetic study, the magnetization of a flux-

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 207

In the ordered magnetic state (*T* < *T*N) a magnetic intensity (*I*M) is added to the intensity *I*N. We can measure finally the total diffracted intensity *I*(*T* < *T*N) with the following equation

This method is applied if we have a good counting statistics and also if *I*M is not too lower then *I*N. We will recall briefly the following useful expressions of the nuclear and magnetic

In the paramagnetic state, the nuclear intensity *I*N(*H*) can be calculated through the nuclear

<sup>2</sup> *F*<sup>N</sup> b exp(2 i )exp(–B | | /2) 

with *P*(*H*), the multiplicity of the reflector plane, and *H*, the corresponding scattering vector which is concentrated in the Bragg peaks (hkl) of the reciprocal lattice, bj and Bj being respectively the neutron scattering length and the Debye-Waller factor of the jth atom in the jth position vector *r*j among N, the total number of atoms in the crystallographic unit cell.

When the materials are magnetically ordered, the magnetic diffracted intensity *I*M(*H*) has the same form that found before for *I*N(*H*) but we consider only the perpendicular component to

<sup>M</sup> j j f ( )exp(2 )exp(–B | | /2)

with fj*(H)*, the magnetic form factor of the jth spin *S*j , and η = |γ|e2/2m0c = +0.02696 cm, the magnetic diffusion length (γ = – 1.91348). The magnetic structure factor has a vector form.

The observed diffracted intensity *I*obs is integrated on number of counts. The value (in barns) is corrected by the Lorentz factor *L*(*H*) and normalized by the scale factor *K* where the

*i*

j j

The diffracted magnetic intensity *I*M is then obtained by the difference

diffracted intensities for the case of a polycrystalline sample.

structure factor *F*N(*H*) with the following expressions

The nuclear structure factor appears as a scalar factor.

N

*H* of the magnetic structure factor *F*M(*H*). We can write these equations

 j j j 1

**3.1.3 Observed** *I***obs and calculated** *I***cal intensities** 

*N*

definitions are expressed respectively by the following equations

j j 1

**3.1.1 Nuclear intensity** *I***N** 

**3.1.2 Magnetic intensity** *I***<sup>M</sup>**

*I*(*T* < *T*N) = *I*N + *I*M (4)

 *I*M(*T* < *T*N) = *I* – *I*N (5)

<sup>2</sup> *I PF* N N *H HH* │ │ (6)

*H Hr H* (7)

2 2 <sup>2</sup> *I*MM M *H F H HF H H* | | –| | /| | │ │ (8)

*F H S H Hr H* (9)

2

grown single crystal of TbIG was measured in dc magnetic fields produced either by a superconducting coil up to 80 kOe or a Bitter coil up to 150 and 200 kOe. The first series of experiments was performed at the Louis Néel Laboratory of Grenoble, France (currently Néel Institut) and the second series at the Service National des Champs Intenses (SNCI) of Grenoble (currently LNCMI). In the 4.2–300 K temperature range, the external magnetic field was applied parallel to the crystallographic directions <111>, <110> and <100> successively. The spherical sample with 5.5 mm of diameter and 0.4313 g of weight is oriented along these three main crystallographic directions by the X-ray Laüe technique with an error less than 1°. The isothermal magnetizations *M*T(*H*) as a function of internal magnetic field *H* (the external magnetic field *H*ex minus the demagnetizing magnetic field of the sphere *H*d) are analyzed by the least-squares refinement technique. The measured spontaneous magnetizations *M*Smes(*T*) are reported in (μB/mol) where one mole is equal to 2(TbIG) formula units.

### **3. Principle of neutron diffraction**

We present here only the case of the diffraction of the non-polarized (or unpolarized) neutrons. This method was used firstly in the determination of the magnetic structure in MnO (Shull & Smart, 1949). A multitude of others followed after: approximately thousands of magnetic structures have now been solved. The use of polarized neutrons ten years after has been the next progress in magnetic neutron scattering (Nathans et al., 1959). In this second method, the incident neutron beam is polarized either up or down, and the neutron intensities scattered by the sample are compared for the two possible states of the incident polarization. Compared to the non-polarized neutrons experiments, an interference term between the nuclear and the magnetic amplitudes adds or subtracts to the intensities depending on the direction of the polarization. For small magnetic amplitudes, the enhancement of sensitivity is remarkable and we know in the case of ferromagnetic and ferrimagnetics to determine the form factor of the magnetic atoms and to reconstruct the spin (or magnetization) density within the cell. Such investigations are now very common: several hundred spin density investigations have already been performed. In YIG (Bonnet et al., 1979), the polarized neutrons were used in addition to the study of the covalency effects. A book edited recently (Chatterji, 2006) is mostly devoted to the application of polarized neutron scattering from magnetic materials.

#### **3.1 Non-polarized neutrons diffraction: Determination of the diffracted intensities**

The scattering of neutron by a magnetic atom is composed essentially of two terms: a nuclear neutron scattering and a magnetic neutron scattering. The first term is due to a "*neutron-nucleus*" nuclear interaction giving the nuclear diffraction which yields information on the spatial arrangement of the nuclei of the atoms in crystal. In the second term, the neutron has a magnetic moment which can interact with the unpaired electrons of the magnetic atoms through the "*dipole-dipole*" interaction conducting to the magnetic diffraction (Bacon, 1975). The nuclear and the magnetic neutron scattering are incoherent in our non-polarized neutrons diffraction. For a magnetic material in the paramagnetic state obtained at *T* > *T*N, there is no magnetic contribution and only the nuclear diffraction exists with the diffracted nuclear intensity (*I*N). The total diffracted intensity *I*(*T* > *T*N) is equal to

$$I\{T > T\_{\rm N}\}\ = I\_{\rm N} \tag{3}$$

In the ordered magnetic state (*T* < *T*N) a magnetic intensity (*I*M) is added to the intensity *I*N. We can measure finally the total diffracted intensity *I*(*T* < *T*N) with the following equation

$$I(T \le T\_{\rm N}) = I\_{\rm N} + I\_{\rm M} \tag{4}$$

The diffracted magnetic intensity *I*M is then obtained by the difference

$$I\_{\rm M}(T \le T\_{\rm N}) = I - I\_{\rm N} \tag{5}$$

This method is applied if we have a good counting statistics and also if *I*M is not too lower then *I*N. We will recall briefly the following useful expressions of the nuclear and magnetic diffracted intensities for the case of a polycrystalline sample.

#### **3.1.1 Nuclear intensity** *I***N**

206 Neutron Diffraction

grown single crystal of TbIG was measured in dc magnetic fields produced either by a superconducting coil up to 80 kOe or a Bitter coil up to 150 and 200 kOe. The first series of experiments was performed at the Louis Néel Laboratory of Grenoble, France (currently Néel Institut) and the second series at the Service National des Champs Intenses (SNCI) of Grenoble (currently LNCMI). In the 4.2–300 K temperature range, the external magnetic field was applied parallel to the crystallographic directions <111>, <110> and <100> successively. The spherical sample with 5.5 mm of diameter and 0.4313 g of weight is oriented along these three main crystallographic directions by the X-ray Laüe technique with an error less than 1°. The isothermal magnetizations *M*T(*H*) as a function of internal magnetic field *H* (the external magnetic field *H*ex minus the demagnetizing magnetic field of the sphere *H*d) are analyzed by the least-squares refinement technique. The measured spontaneous magnetizations *M*Smes(*T*)

We present here only the case of the diffraction of the non-polarized (or unpolarized) neutrons. This method was used firstly in the determination of the magnetic structure in MnO (Shull & Smart, 1949). A multitude of others followed after: approximately thousands of magnetic structures have now been solved. The use of polarized neutrons ten years after has been the next progress in magnetic neutron scattering (Nathans et al., 1959). In this second method, the incident neutron beam is polarized either up or down, and the neutron intensities scattered by the sample are compared for the two possible states of the incident polarization. Compared to the non-polarized neutrons experiments, an interference term between the nuclear and the magnetic amplitudes adds or subtracts to the intensities depending on the direction of the polarization. For small magnetic amplitudes, the enhancement of sensitivity is remarkable and we know in the case of ferromagnetic and ferrimagnetics to determine the form factor of the magnetic atoms and to reconstruct the spin (or magnetization) density within the cell. Such investigations are now very common: several hundred spin density investigations have already been performed. In YIG (Bonnet et al., 1979), the polarized neutrons were used in addition to the study of the covalency effects. A book edited recently (Chatterji, 2006) is mostly devoted to the application of polarized

**3.1 Non-polarized neutrons diffraction: Determination of the diffracted intensities** 

The scattering of neutron by a magnetic atom is composed essentially of two terms: a nuclear neutron scattering and a magnetic neutron scattering. The first term is due to a "*neutron-nucleus*" nuclear interaction giving the nuclear diffraction which yields information on the spatial arrangement of the nuclei of the atoms in crystal. In the second term, the neutron has a magnetic moment which can interact with the unpaired electrons of the magnetic atoms through the "*dipole-dipole*" interaction conducting to the magnetic diffraction (Bacon, 1975). The nuclear and the magnetic neutron scattering are incoherent in our non-polarized neutrons diffraction. For a magnetic material in the paramagnetic state obtained at *T* > *T*N, there is no magnetic contribution and only the nuclear diffraction exists with the diffracted nuclear intensity (*I*N). The total diffracted intensity *I*(*T* > *T*N) is equal to

*IT T I* N N (3)

are reported in (μB/mol) where one mole is equal to 2(TbIG) formula units.

**3. Principle of neutron diffraction** 

neutron scattering from magnetic materials.

In the paramagnetic state, the nuclear intensity *I*N(*H*) can be calculated through the nuclear structure factor *F*N(*H*) with the following expressions

$$I\_{\mathcal{N}}(\boldsymbol{H}) = P(\boldsymbol{H})|\,\,\mathbb{F}\_{\mathcal{N}}(\boldsymbol{H})|^{2} \tag{6}$$

$$F\_{\mathbf{N}}(\mathbf{H}) \triangleq \sum\_{\mathbf{j}=1}^{N} \mathbf{b}\_{\mathbf{j}} \exp(2\pi i \mathbf{H} \cdot \mathbf{r}\_{\mathbf{j}}) \exp(-\mathbf{B}\_{\mathbf{j}} \left|\mathbf{H}\right|^{2}/2) \tag{7}$$

with *P*(*H*), the multiplicity of the reflector plane, and *H*, the corresponding scattering vector which is concentrated in the Bragg peaks (hkl) of the reciprocal lattice, bj and Bj being respectively the neutron scattering length and the Debye-Waller factor of the jth atom in the jth position vector *r*j among N, the total number of atoms in the crystallographic unit cell. The nuclear structure factor appears as a scalar factor.

#### **3.1.2 Magnetic intensity** *I***<sup>M</sup>**

When the materials are magnetically ordered, the magnetic diffracted intensity *I*M(*H*) has the same form that found before for *I*N(*H*) but we consider only the perpendicular component to *H* of the magnetic structure factor *F*M(*H*). We can write these equations

$$I\_{\rm M}(H) \;=\; \vert \left| F\_{\rm M}(H) \right|^2 \; \vert \left| H \cdot F\_{\rm M}(H) \right|^2 \;/ \vert H \; \vert^2 \; \vert \tag{8}$$

$$F\_{\mathbf{M}}(H) = \underset{\mathbf{j} = 1}{\text{\textquotedblleft}} \underset{\mathbf{j}}{\text{\textquotedblleft}} \langle H \rangle \exp(2\pi i H \cdot \mathbf{r}\_{\mathbf{j}}) \exp(-\mathbf{B}\_{\mathbf{j}} \left| H \right|^{2}/2) \tag{9}$$

with fj*(H)*, the magnetic form factor of the jth spin *S*j , and η = |γ|e2/2m0c = +0.02696 cm, the magnetic diffusion length (γ = – 1.91348). The magnetic structure factor has a vector form.

#### **3.1.3 Observed** *I***obs and calculated** *I***cal intensities**

The observed diffracted intensity *I*obs is integrated on number of counts. The value (in barns) is corrected by the Lorentz factor *L*(*H*) and normalized by the scale factor *K* where the definitions are expressed respectively by the following equations

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 209

We have observed during the refinement that the calculated parameters *m*Tb, *m*a and *m*d of the three magnetic sublattices are depending highly of the choice of the magnetic form factors, fTb*(*hkl) and particularly those related to the two iron sublattices fa*(*hkl) and fd*(*hkl). We choose at first, the theoretical magnetic form factors determined in the Hartree-Fock description based on the free ion model for the Tb3+ ion, fTb*(*hkl) (Blume, et al., 1962) and those calculated (Watson & Fremann, 1961) fa*(*hkl) and fd*(*hkl) with the equality fa*(*hkl) = fd*(*hkl) for the iron sublattices. Secondly, two previous experimental magnetic form factors fa*(*hkl) and fd*(*hkl) are also tentatively used. The first values (Bonnet, 1976; Bonnet et al., 1979) obtained by polarized neutrons experiments on YIG single crystal indicate that fa*(*hkl) and fd*(*hkl) are different that the free ion value and fa*(*hkl) > fd*(*hkl). The second values of fa*(*hkl) and fd*(*hkl) with the relation fa*(*hkl) < fd*(*hkl) have been found previously by powder neutron diffraction experiments (Guillot et al., 1983). In the study of the "*umbrella structure*" at low temperature on HoIG sample prepared by grinding of single crystals (Guillot et al., 1984), an evaluation of *m*a and *m*<sup>d</sup> using the Bonnet' fa*(*hkl), fd*(*hkl) determinations was made. At 4.2 K, these moments were found equal respectively to 4,01 and 4,26 μB. These values are smaller than the theoretical ground state 6S5/2 saturated magnetic moment (5 μB). The observed reduction of the moments

Fig. 2. Neutron diffraction patterns at 453 and 400 K

Fig. 3. Neutron diffraction patterns at 283 and 5 K (D1B)

**4.1 Results and discussion** 

$$L\left(H\right) \,=\, 1/\sin\theta \,\sin\left(2\theta\right); K = I\_{\text{cal}} \,/\, I\_{\text{obs}}\tag{10}$$

The value of *K* is obtained by the refinement of the nuclear structure. For each *I*obs, an absolute error Δ*I*obs is found. She is associated to the sum of a statistical error with the appreciated one on the continu fund noise. The corresponding observed relative error is noted Δ*I*obs/*I*obs. Using the least square method, we can define the reliability factor *R*

$$R = \sum |I\_{\text{obs}} - I\_{\text{cal}}| / \sum |I\_{\text{obs}}| \tag{11}$$

For each calculated diffracted intensity *I*cal, a calculated relative error Δ*I*cal/*I*cal is attributed. The non-polarized neutron diffraction is then based in the comparison between *I*cal and *I*obs.

#### **3.2 Preliminarily neutron diffraction at 614 K**

In the pattern (not show) recorded at *T* = 614 K higher than *T*N (568 ± 2 K) (Pauthenet, 1958a, 1958b), only the nuclear contribution exists and the reflections (hkl) are indexed with the general extinction rule of the cubic space group Ia 3 d, h + k + l = 2n. Attention was paid to the thermal agitation of the jth atom in the different sites by introducing the corresponding isotropic Debye-Waller factors Bj. For the determination of the calculated nuclear intensities (*I*N)cal, the previous scattering lengths (Bacon, 1972) are used: b(O) = 0.580; b(Fe) = 0.95; b(Tb) = 0.76 (in units of 10–12 cm/atom). We can determine the number of refinement cycles and to choose the parameters with a sufficient number of iterations. The previous best values of the parameters x = –0.0279, y = 0.0555, z = 0.1505 found previously at *T* = 693 K (Tchéou et al., 1970c) for the general positions (96h) of the oxygen atoms are used as constant parameters and the refinement is made only on the scale factor *K* and after the corresponding isotropic Debye-Waller factors Bj. A good agreement with a reliability factor of order 10 % is found for the lattice parameter *a* = 12.470 ± 0.004 Å and *K* = 0.42 ± 0.02 with Bh(O) = 0.88; Bd(Fe) = Ba(Fe) = 0.82 and Bc(Tb) = 0.81. We observe that the observed intensities of the reflections (400) and (420) are lower than the corresponding calculated intensities and the refinement of the temperature parameters Bj has a little influence on the values of the observed intensities (for example when Bc(Tb) change from 0.80 to 0.30).

#### **4. Neutron diffraction study at high temperature**

The neutron diffraction patterns below *T*N are reported in Fig. 2 for 453 and 400 K. Both patterns at (283 K) and at 5 K (D1B) are presented for a useful comparison in Fig. 3. A magnetic intensity *I*M(hkl) is superimposed to the nuclear intensity *I*N(hkl): we have then, *I*(hkl) = *I*N(hkl) + *I*M(hkl). In these patterns, we have indexed all the reflections (hkl) in the chemical cell with the same extinction rule characterized by a wave vector *k* = 0. In these temperatures, the magnetic structure factors *F*M(a)(hkl), *F*M(d)(hkl) and *F*M(c)(hkl) associated to each magnetic sublattice of TbIG are used to describe the collinear ferrimagnetic state of the Néel model along the easy axis [111] found by magnetization measurements (Lahoubi et al., 1985 and refs. herein). Two types of reflections (hkl) appear in the patterns: (h = odd, k = odd, l = even) and h = even, k = even, l = even). For the related reflections (hkl) with (h = even, k = odd, l = odd) and (h = odd, k = even, l = odd) and cyclic permutations (c.p.) of h, k, l must be done in the expressions of the magnetic structure factors. A complete description can be found in the previous paper on the neutron diffraction of HoIG (Guillot et al., 1984).

Fig. 2. Neutron diffraction patterns at 453 and 400 K

Fig. 3. Neutron diffraction patterns at 283 and 5 K (D1B)

#### **4.1 Results and discussion**

208 Neutron Diffraction

The value of *K* is obtained by the refinement of the nuclear structure. For each *I*obs, an absolute error Δ*I*obs is found. She is associated to the sum of a statistical error with the appreciated one on the continu fund noise. The corresponding observed relative error is

*<sup>R</sup>* = *II I* obs cal obs / (11)

For each calculated diffracted intensity *I*cal, a calculated relative error Δ*I*cal/*I*cal is attributed. The non-polarized neutron diffraction is then based in the comparison between *I*cal and *I*obs.

In the pattern (not show) recorded at *T* = 614 K higher than *T*N (568 ± 2 K) (Pauthenet, 1958a, 1958b), only the nuclear contribution exists and the reflections (hkl) are indexed with the general extinction rule of the cubic space group Ia 3 d, h + k + l = 2n. Attention was paid to the thermal agitation of the jth atom in the different sites by introducing the corresponding isotropic Debye-Waller factors Bj. For the determination of the calculated nuclear intensities (*I*N)cal, the previous scattering lengths (Bacon, 1972) are used: b(O) = 0.580; b(Fe) = 0.95; b(Tb) = 0.76 (in units of 10–12 cm/atom). We can determine the number of refinement cycles and to choose the parameters with a sufficient number of iterations. The previous best values of the parameters x = –0.0279, y = 0.0555, z = 0.1505 found previously at *T* = 693 K (Tchéou et al., 1970c) for the general positions (96h) of the oxygen atoms are used as constant parameters and the refinement is made only on the scale factor *K* and after the corresponding isotropic Debye-Waller factors Bj. A good agreement with a reliability factor of order 10 % is found for the lattice parameter *a* = 12.470 ± 0.004 Å and *K* = 0.42 ± 0.02 with Bh(O) = 0.88; Bd(Fe) = Ba(Fe) = 0.82 and Bc(Tb) = 0.81. We observe that the observed intensities of the reflections (400) and (420) are lower than the corresponding calculated intensities and the refinement of the temperature parameters Bj has a little influence on the values of the observed intensities

The neutron diffraction patterns below *T*N are reported in Fig. 2 for 453 and 400 K. Both patterns at (283 K) and at 5 K (D1B) are presented for a useful comparison in Fig. 3. A magnetic intensity *I*M(hkl) is superimposed to the nuclear intensity *I*N(hkl): we have then, *I*(hkl) = *I*N(hkl) + *I*M(hkl). In these patterns, we have indexed all the reflections (hkl) in the chemical cell with the same extinction rule characterized by a wave vector *k* = 0. In these temperatures, the magnetic structure factors *F*M(a)(hkl), *F*M(d)(hkl) and *F*M(c)(hkl) associated to each magnetic sublattice of TbIG are used to describe the collinear ferrimagnetic state of the Néel model along the easy axis [111] found by magnetization measurements (Lahoubi et al., 1985 and refs. herein). Two types of reflections (hkl) appear in the patterns: (h = odd, k = odd, l = even) and h = even, k = even, l = even). For the related reflections (hkl) with (h = even, k = odd, l = odd) and (h = odd, k = even, l = odd) and cyclic permutations (c.p.) of h, k, l must be done in the expressions of the magnetic structure factors. A complete description can be found in the previous paper on the neutron diffraction of HoIG (Guillot et al., 1984).

noted Δ*I*obs/*I*obs. Using the least square method, we can define the reliability factor *R*

**3.2 Preliminarily neutron diffraction at 614 K** 

(for example when Bc(Tb) change from 0.80 to 0.30).

**4. Neutron diffraction study at high temperature** 

cal obs *L K H* 1 /sin sin 2 ; / *I I* (10)

We have observed during the refinement that the calculated parameters *m*Tb, *m*a and *m*d of the three magnetic sublattices are depending highly of the choice of the magnetic form factors, fTb*(*hkl) and particularly those related to the two iron sublattices fa*(*hkl) and fd*(*hkl). We choose at first, the theoretical magnetic form factors determined in the Hartree-Fock description based on the free ion model for the Tb3+ ion, fTb*(*hkl) (Blume, et al., 1962) and those calculated (Watson & Fremann, 1961) fa*(*hkl) and fd*(*hkl) with the equality fa*(*hkl) = fd*(*hkl) for the iron sublattices. Secondly, two previous experimental magnetic form factors fa*(*hkl) and fd*(*hkl) are also tentatively used. The first values (Bonnet, 1976; Bonnet et al., 1979) obtained by polarized neutrons experiments on YIG single crystal indicate that fa*(*hkl) and fd*(*hkl) are different that the free ion value and fa*(*hkl) > fd*(*hkl). The second values of fa*(*hkl) and fd*(*hkl) with the relation fa*(*hkl) < fd*(*hkl) have been found previously by powder neutron diffraction experiments (Guillot et al., 1983). In the study of the "*umbrella structure*" at low temperature on HoIG sample prepared by grinding of single crystals (Guillot et al., 1984), an evaluation of *m*a and *m*<sup>d</sup> using the Bonnet' fa*(*hkl), fd*(*hkl) determinations was made. At 4.2 K, these moments were found equal respectively to 4,01 and 4,26 μB. These values are smaller than the theoretical ground state 6S5/2 saturated magnetic moment (5 μB). The observed reduction of the moments

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 211

In contrast to previous findings at 1.5 K (Bertaut et al., 1970; Tchéou al., 1970a) and 4.2 K (Lahoubi et al., 1984), the small superstructure lines (200)\* and (600, 442)\* have been observed recently and confirmed only at 5 K (D1B) (Lahoubi, 2012) with a sensibility equal to 0.5 and 1% respectively, the line (110)\* being chosen as a reference. The same order of magnitude (1/276) by comparison with the previous result (Hock et al., 1990) was found for the line (200)\*. Above

5 K, they are not observed in the neutron diffraction pattern at 13 K (Fig. 4).

Fig. 5. Neutron diffraction patterns at 20 and 54 K

Fig. 6. Neutron diffraction patterns at 68 and 80 K

Fig. 7. Neutron diffraction patterns at 109 and 127 ± 5 K

is explained by covalent bonding for YIG (Bonnet, 1976; Fuess et al., 1976) or topological frustration for FeF3 (Ferey et al., 1986). When the proposed values (Guillot., et al 1983) are chosen in the refinement of the neutron diagrams at 453, 400 and 283 K, we observe that the calculated intensities of the reflections (211) and (220) which have a high magnetic contribution are lower than the observed intensities. Consequently, we shall consider in this work for the Fe3+ ions only the theoretical values of fa*(*hkl) and fd*(*hkl) (Watson & Fremann, 1961). In this condition, *m*a and *m*d are not considered as fitting parameters in the first cycle of the refinement and the N.M.R values are used (Gonano et al., 1967). Working in this hypothesis leads to the best values of *m*Tb at each temperature. Finally, in the second cycle of the refinement based on twenty reflections, the parameters *m*a and *m*d are fitted by the selfconsistent calculation of *m*Tb. At 453 and 400 K, we obtain respectively for *m*Tb the refined values 0.50 ± 0.10 μB and 0.60 ± 0.10 μB with a reliability factor *R* varying in the range of 11.8– 9.3 and 11.6–8.6 % if the refinement is makes only on the twelve first reflections (Lahoubi, 1986). The results lead to a good agreement between the bulk calculated magnetizations *M*<sup>S</sup> cal(TbIG) and the observed spontaneous magnetization *M*<sup>S</sup> obs(TbIG) (Pauthenet, 1958a, 1958b). The result at 283 K (1.15 μB with *R* = 8.5 %) (Lahoubi, 1986) is similar to that found previously (Bonnet, 1976; Fuess et al., 1976) but with a poor agreement for *m*a and *m*d which have been found lower by comparison with those determined by N.M.R (Gonano et al., 1967).

### **5. Neutron diffraction study at low temperature and symmetry analysis**

The neutron patterns recorded at 5 (D1B), 13 (D1B), 20, 54, 68, 80, 109, 127, 160, 208 and 244 K are presented from Fig. 3 up to Fig. 9.

### **5.1 Results and discussion**

At *T* = 5 K (Fig. 3), two types of reflection appear. In addition to the earlier reflections (hkl) observed previously, pure superstructure lines (hkl)**\*** forbidden by the nuclear space group Ia 3 d are present and we have *I* = *I*M(hkl)**\***: (110)**\***, (310)**\***, (411, 330)**\***, (433, 530)**\*** and (510)**\***.

Fig. 4. Neutron diffraction pattern at 13 K (D1B)

In contrast to previous findings at 1.5 K (Bertaut et al., 1970; Tchéou al., 1970a) and 4.2 K (Lahoubi et al., 1984), the small superstructure lines (200)\* and (600, 442)\* have been observed recently and confirmed only at 5 K (D1B) (Lahoubi, 2012) with a sensibility equal to 0.5 and 1% respectively, the line (110)\* being chosen as a reference. The same order of magnitude (1/276) by comparison with the previous result (Hock et al., 1990) was found for the line (200)\*. Above 5 K, they are not observed in the neutron diffraction pattern at 13 K (Fig. 4).

Fig. 5. Neutron diffraction patterns at 20 and 54 K

210 Neutron Diffraction

is explained by covalent bonding for YIG (Bonnet, 1976; Fuess et al., 1976) or topological frustration for FeF3 (Ferey et al., 1986). When the proposed values (Guillot., et al 1983) are chosen in the refinement of the neutron diagrams at 453, 400 and 283 K, we observe that the calculated intensities of the reflections (211) and (220) which have a high magnetic contribution are lower than the observed intensities. Consequently, we shall consider in this work for the Fe3+ ions only the theoretical values of fa*(*hkl) and fd*(*hkl) (Watson & Fremann, 1961). In this condition, *m*a and *m*d are not considered as fitting parameters in the first cycle of the refinement and the N.M.R values are used (Gonano et al., 1967). Working in this hypothesis leads to the best values of *m*Tb at each temperature. Finally, in the second cycle of the refinement based on twenty reflections, the parameters *m*a and *m*d are fitted by the selfconsistent calculation of *m*Tb. At 453 and 400 K, we obtain respectively for *m*Tb the refined values 0.50 ± 0.10 μB and 0.60 ± 0.10 μB with a reliability factor *R* varying in the range of 11.8– 9.3 and 11.6–8.6 % if the refinement is makes only on the twelve first reflections (Lahoubi, 1986). The results lead to a good agreement between the bulk calculated magnetizations *M*Scal(TbIG) and the observed spontaneous magnetization *M*Sobs(TbIG) (Pauthenet, 1958a, 1958b). The result at 283 K (1.15 μB with *R* = 8.5 %) (Lahoubi, 1986) is similar to that found previously (Bonnet, 1976; Fuess et al., 1976) but with a poor agreement for *m*a and *m*d which have been found lower by comparison with those determined by N.M.R (Gonano et al., 1967).

**5. Neutron diffraction study at low temperature and symmetry analysis** 

K are presented from Fig. 3 up to Fig. 9.

Fig. 4. Neutron diffraction pattern at 13 K (D1B)

**5.1 Results and discussion** 

The neutron patterns recorded at 5 (D1B), 13 (D1B), 20, 54, 68, 80, 109, 127, 160, 208 and 244

At *T* = 5 K (Fig. 3), two types of reflection appear. In addition to the earlier reflections (hkl) observed previously, pure superstructure lines (hkl)**\*** forbidden by the nuclear space group Ia 3 d are present and we have *I* = *I*M(hkl)**\***: (110)**\***, (310)**\***, (411, 330)**\***, (433, 530)**\*** and (510)**\***.

Fig. 6. Neutron diffraction patterns at 68 and 80 K

Fig. 7. Neutron diffraction patterns at 109 and 127 ± 5 K

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 213

Notations Dj, D'j ( j = 1-3)

Unprimed and primed (**'**) notations respectively for p = 1-4 and p**'** = 5-8 with c.p.

Numbered positions of Fe3+

**(1)** 0,0,0 **(9)** 1/2,1/2,1/2

**(2)** 0,1/2,1/2 **(10)** 1/2,0, 0

**(3)** 1/2,0,1/2 **(11)** 0,1/2,0

**(4)** 1/2,1/2,0 **(12)** 0,0,1/2

(**'**) **(5)** 1/4,1/4,1/4 **(13)** 3/4,3/4,3/4

(**'**) **(6)** 1/4,3/4,3/4 **(14)** 3/4,1/4,1/4

(**'**) **(7)** 3/4,1/4,3/4 **(15)** 1/4,3/4,1/4

(**'**) **(8)** 3/4,3/4,1/4 **(16)** 1/4,1/4,3/4

ions in site [16a]( 3 )

Numbered positions of Fe3+ ions in site (24d)( 4 )

> **(1)** 3/8,0,1/4 **(7)** 5/8,0,3/4 **(13)** 7/8,1/2,3/4 **(19)** 1/8,1/2,1/4

> **(4)** 1/8,0,3/4 **(10)** 7/8,0,1/4 **(16)** 5/8,1/2,1/4 **(22)** 3/8,1/2,3/4

> **(2)** 1/4,3/8,0 **(8)** 3/4,5/8,0 **(14)** 3/4,7/8,1/2 **(20)** 1/4,1/8,1/2

> **(5)** 3/4,1/8,0 **(11)** 1/4,7/8,0 **(17)** 1/4,5/8,1/2 **(23)** 3/4,3/8,1/2

> **(3)** 0,3/4,3/8 **(9)** 0,3/4,5/8 **(15)** 1/2,3/4,7/8 **(21)** 1/2,3/4,3/8

> **(6)** 0,3/4,1/8 **(12)** 0,1/4,7/8 **(18)** 1/2,1/4,5/8 **(24)** 1/2,3/4,3/8

Table 1. Notations, numbers and positions in the unit cell of the three magnetic sublattices.

Notations Cj , C'j ( j = 1-3) and {i = 1-6} (Wolf et al., 1962; Wolf, 1964 and Pearson et al., 1965)

Local axes in D2(222) symmetry

Numbered positions of RE3+

**(1)** 1/8,0,1/4 **(7)** 7/8,0,3/4 **(13)** 5/8,1/2,3/4 **(19)** 3/8,1/2,1/4

**(4)** 3/8,0,3/4 **(10)** 5/8,0,1/4 **(16)** 7/8,1/2,1/4 **(22)** 1/8,1/2,3/4

**(2)** 1/4,1/8,0 **(8)** 3/4,7/8,0 **(14)** 3/4,5/8,1/2 **(20)** 1/4,3/8,1/2

**(5)** 3/4,3/8,0 **(11)** 1/4,5/8,0 **(17)** 1/4,7/8,1/2 **(23)** 3/4,1/8,1/2

**(3)** 0,1/4,1/8 **(9)** 0,3/4,7/8 **(15)** 1/2,3/4,5/8 **(21)** 1/2,1/4,3/8

**(6)** 0,3/4,3/8 **(12)** 0,1/4,5/8 **(18)** 1/2,1/4,7/8 **(24)** 1/2,3/4,1/8 D1

D<sup>1</sup>

D2

D<sup>2</sup>

D3

D<sup>3</sup>

*U*, *V* and *W* are the local axes of the RE3+ in D2(222) symmetry

ions in site {24c}(222)

 100 011 011

 100 011 011

 010 101 101

 010 101 101

 001 110 110

 001 110 110

*U**g*<sup>z</sup> *V**g*x (*g*y) *W**g*y (*g*x)

C1 1 3

C<sup>1</sup> 2 4

C2 3 5

C<sup>2</sup> 4 6

C3 5 1

C<sup>3</sup> 6 2

Fig. 8. Neutron diffraction patterns at 160 and 208 K

All the superstructure lines appear without any ambiguity from 5 K up to 127 ± 5 K. They are indexed with the same extinction rule (h**\*** + k**\*** + l**\*** = 2n) and imply the signature of a change of the collinear magnetic structure of the RE3+{24c} ions of the Néel model to a non collinear magnetic structure with the wave vector *k* = 0. At 160 and 208 K, the superstructure line (110)**\*** is resolved with the best sensibility (0.3 and 1 respectively) which is equal to the ratio of the peak to background normalized to the intensity of the line (211) (see the details for 2θ up to 15° in the left of Fig. 9).

Fig. 9. Details at 160 and 208 K (left) and neutron diffraction pattern at 244 ± 10 K (right)

The chemical cell is equal to the magnetic cell, thus the primitive translation noted (11/2,1/2,1/2) of the crystallographic lattice (I) (Hahn, 1983) is a primitive translation of the magnetic lattice (I). Based on the numbered positions gathered on Table 1 we can write for the RE3+{24c} and Fe3+(24d) sublattices that *S*j = *S*j + *S*j+12 (j = 1-3); for Fe3+[16a] sublattice, we have *S*p = *S*p+ *S*p+8 (p = 1-4) and *S*<sup>p</sup>**'** = *S*<sup>p</sup>**'** + *S*<sup>p</sup>**'**+8 (p' = 5-8). It means that two spin vectors *S*j, *S*j+12, *S*p, *S*p+8 and *S*<sup>p</sup>**'**, *S*<sup>p</sup>**'**+8 are coupled ferromagnetically. To discuss the corresponding magnetic structure factors *F*M(a)(hkl)**\***, *F*M(d)(hkl)**\*** and *F*M(c)(hkl)**\*** associated to each magnetic sublattice in TbIG, the four earlier linear combinations of four spin vectors introduced by Bertaut (Bertaut, 1963) labeled *F*, *G*, *C* and *A* are used. These four magnetic modes (one ferromagnetic and three antiferromagnetics) form the "*basis of irreductible representations*".

All the superstructure lines appear without any ambiguity from 5 K up to 127 ± 5 K. They are indexed with the same extinction rule (h**\*** + k**\*** + l**\*** = 2n) and imply the signature of a change of the collinear magnetic structure of the RE3+{24c} ions of the Néel model to a non collinear magnetic structure with the wave vector *k* = 0. At 160 and 208 K, the superstructure line (110)**\*** is resolved with the best sensibility (0.3 and 1 respectively) which is equal to the ratio of the peak to background normalized to the intensity of the line

Fig. 9. Details at 160 and 208 K (left) and neutron diffraction pattern at 244 ± 10 K (right)

The chemical cell is equal to the magnetic cell, thus the primitive translation noted (11/2,1/2,1/2) of the crystallographic lattice (I) (Hahn, 1983) is a primitive translation of the magnetic lattice (I). Based on the numbered positions gathered on Table 1 we can write for the RE3+{24c} and Fe3+(24d) sublattices that *S*j = *S*j + *S*j+12 (j = 1-3); for Fe3+[16a] sublattice, we have *S*p = *S*p+ *S*p+8 (p = 1-4) and *S*<sup>p</sup>**'** = *S*<sup>p</sup>**'** + *S*<sup>p</sup>**'**+8 (p' = 5-8). It means that two spin vectors *S*j, *S*j+12, *S*p, *S*p+8 and *S*<sup>p</sup>**'**, *S*<sup>p</sup>**'**+8 are coupled ferromagnetically. To discuss the corresponding magnetic structure factors *F*M(a)(hkl)**\***, *F*M(d)(hkl)**\*** and *F*M(c)(hkl)**\*** associated to each magnetic sublattice in TbIG, the four earlier linear combinations of four spin vectors introduced by Bertaut (Bertaut, 1963) labeled *F*, *G*, *C* and *A* are used. These four magnetic modes (one ferromagnetic and three antiferromagnetics) form the "*basis of irreductible representations*".

Fig. 8. Neutron diffraction patterns at 160 and 208 K

(211) (see the details for 2θ up to 15° in the left of Fig. 9).


Table 1. Notations, numbers and positions in the unit cell of the three magnetic sublattices. *U*, *V* and *W* are the local axes of the RE3+ in D2(222) symmetry

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 215

<sup>M</sup> 23 1 *F* 200 – i *FF A* (18)

<sup>M</sup> 1 2 12 3 *F* 310 2 / 2(– ) i[ 2 / 2( – ) ] *GG CC A* (19)

M 2 3 2 <sup>3</sup> <sup>1</sup> *F* 411 2 / 2( – ) i[ 2 / 2( ) – ] *GG CC A* (20)

<sup>M</sup> 1 2 12 3 *F* 330 2 / 2(– ) i[ 2 / 2( ) – ] *GG CC A* (21)

<sup>M</sup> 1 2 12 3 *F* 510 2 / 2(– – ) i[ 2 / 2(– ) ] *GG CC A* (22)

<sup>M</sup> 2 3 23 1 *F* 433 2 / 2(– ) i[ 2 / 2( ) ] *GG CC A* (23)

<sup>M</sup> 1 2 12 3 *F* 530 2 / 2(– – ) i[ 2 / 2(– – ) – ] *GG CC A* (24)

<sup>M</sup> 2 3 23 1 *F* 431 2 / 2(– ) i[ 2 / 2( – ) – ] *GG CC A* (27)

<sup>M</sup> 1 3 13 2 *F* 541 2 / 2( ) i[ 2 / 2(– ) – ] *G G CC A* (28)

<sup>M</sup> 23 1 *F* 600 – – i *FF A* (25)

<sup>M</sup> 21 3 *F* 442 – i *FF A* (26)

(c) (j = 1-3). It must be noted that the superstructure line (510)**\***

(c) and *G*<sup>j</sup>

(c) are

 <sup>c</sup> \*

 <sup>c</sup> \*

 <sup>c</sup> \*

two reflections which are represented by their magnetic structure factors

From these expressions, one can observe that both magnetic modes *F*<sup>j</sup>

necessary in the description of the non collinear structures of the Tb3+ ions. The absence of the small superstructure lines (200)**\*** and (600, 442)**\*** above 13 K gives rise to the equality

appears at the same Bragg peak of the pure nuclear reflection (431) (θ = 30.7°). A magnetic contribution of the RE3+ ions exists for (431). We observe also a magnetic contribution for another reflection (541). It is very difficult to isolate only the magnetic contribution of these

Bertaut (Bertaut, 1963, 1968, 1971, 1972) has created a group theory method called "*Representation Analysis*" which has been widely used in the last four decades by Bertaut himself and by other researchers. The essential role is plaid by the "*Basis Vectors of Irreductible Representations*" of the paramagnetic space group Ia 3 d of TbIG and its highest subgroups.

The representation analysis of the cubic space group Ia d was applied in the past (Bertaut et al., 1970; Tchéou et al., 1970a) in order to determine the spin configurations of the Tb3+

 <sup>c</sup> \*

 <sup>c</sup> \*

 <sup>c</sup> \*

 <sup>c</sup> \*

 <sup>c</sup> \*

 <sup>c</sup> \*

 <sup>c</sup>

 <sup>c</sup>

**5.2.1 Representation analysis of Ia 3 d** 

**5.2 Representation analysis of magnetic structures** 

between the magnetic modes *F*<sup>j</sup>

The adapted magnetic modes for the RE3+{24c} and Fe3+(24d) sublattices with j = 1-3 are

$$\begin{aligned} \mathbf{F\_j} &= \mathbf{S\_j} + \mathbf{S\_{j+3}} + \mathbf{S\_{j+6}} + \mathbf{S\_{j+9}}; \mathbf{G\_j} = \mathbf{S\_j} - \mathbf{S\_{j+3}} + \mathbf{S\_{j+6}} - \mathbf{S\_{j+9}}; \\\\ \mathbf{C\_j} &= \mathbf{S\_j} + \mathbf{S\_{j+3}} - \mathbf{S\_{j+6}} - \mathbf{S\_{j+9}}; \mathbf{A\_j} = \mathbf{S\_j} - \mathbf{S\_{j+3}} - \mathbf{S\_{j+6}} + \mathbf{S\_{j+9}} \end{aligned} \tag{12}$$

For the Fe3+[16a] sublattice, it is necessary to consider two distinct magnetic modes. The first chosen notation is the unprimed magnetic modes for the numbered spins *SSS* p p p8 (p = 1-4)

$$F = \mathbf{S}\_1 + \mathbf{S}\_2 + \mathbf{S}\_3 + \mathbf{S}\_4;\\\mathbf{G} = \mathbf{S}\_1 - \mathbf{S}\_2 + \mathbf{S}\_3 - \mathbf{S}\_4;\\\mathbf{C} = \mathbf{S}\_1 + \mathbf{S}\_2 - \mathbf{S}\_3 - \mathbf{S}\_4;\\\mathbf{A} = \mathbf{S}\_1 - \mathbf{S}\_2 - \mathbf{S}\_3 + \mathbf{S}\_4 \tag{13}$$

The second chosen notation of the magnetic modes is the primed notation (**'**) for the numbered spins *SSS* p' p' p' 8 (p**'** = 5-8)

$$\mathbf{F}' = \mathbf{S}\_5 + \mathbf{S}\_6 + \mathbf{S}\_7 + \mathbf{S}\_8;\\\mathbf{G}' = \mathbf{S}\_5 - \mathbf{S}\_6 + \mathbf{S}\_7 - \mathbf{S}\_8;\\\mathbf{C}' = \mathbf{S}\_5 + \mathbf{S}\_6 - \mathbf{S}\_7 - \mathbf{S}\_8;\\\mathbf{A}' = \mathbf{S}\_5 - \mathbf{S}\_6 - \mathbf{S}\_7 + \mathbf{S}\_8 \tag{14}$$

For the RE3+ ions in the Wyckoff site {24c}, the local axes (*U*,*V*,*W*) in the D2(222) symmetry are identified to the parameters *g*α (α = *x*,*y*,*z*) of the magnetic tensor *g* in the hypothesis of the effective spin Hamiltonian model (Wolf et al., 1962, Wolf, 1964). The particular superstructure lines (222)**\*** and (622)**\*** are not observed in the whole temperature range below *T*N. For example, the associated magnetic structure factors are all equal to zero

$$\left(\mathbf{F\_{M}}^{\text{(a)}}\right)\left(222\right)^{\*} = \mathbf{F} - \mathbf{F'} = 0\\\therefore \mathbf{F\_{M}}^{\text{(d)}}\left(222\right)^{\*} = \mathbf{+i}\left(\mathbf{A\_{1} + \mathbf{A\_{2}} + \mathbf{A\_{3}}\right) = 0}\tag{15}$$

$$\left(F\_{\mathbf{M}}\,^{\left(\mathbf{c}\right)}\left(222\right)^{\*}\right) = \mathrm{-i}\left(\mathbf{A}\_{1} + \mathbf{A}\_{2} + \mathbf{A}\_{3}\right) = \mathbf{0};\\\mathrm{F}\_{\mathbf{M}}^{\left(\mathbf{c}\right)}\left(622\right)^{\*} = \mathrm{+i}\left(\mathbf{A}\_{1} - \mathbf{A}\_{2} - \mathbf{A}\_{3}\right) = \mathbf{0}\tag{16}$$

In these conditions, the spins vectors *S*<sup>j</sup> and *S*j+6 (j = 1-3) of Tb3+{24c} and Fe3+(24d) ions are coupled ferromagnetically and the symmetry operation ( 1 0,0,0) is an inversion center. These significant properties related to magnetic symmetry involve that all the magnetic modes *A*j are absent. This absence will be accompanied by the elimination of the modes *C*j. Consequently, the above crystallographic sites split into six magnetically inequivalent sublattices Cj, Cj and Dj, Dj with (j = 1-3) respectively as it is indicated on Table 1. In each sublattice we have four ions which are equivalent under the symmetry operations: ( 1 0,0,0) and (11/2,1/2,1/2). The sublattices C2 and C3 are related to C1 by a rotation of 120 and 240° around the 3-fold symmetry [111] axis (also for Cj); the same remarks can be made for the sublattices Dj, Dj (j = 1-3). Contrary to previous spin rotation observed at *T*comp (260 5 K) by mössbauer spectroscopy (Hong et al., 2004), no deviation from the colinearity along the easy axis [111] for the spins of the Fe3+[16a] ions is evidenced around *T*comp (244 ± 10 K) (right of Fig. 9). In this *T*-region, this sublattice is described by one ferromagnetic configuration of the cubic magnetic modes *F* and *F***'** with the equality of the left of the equation (15). The magnetic structure factors *F*M(c)(hkl)**\*** calculated for all the observed superstructure lines imply to know if we need both magnetic modes *F*<sup>j</sup> (c) and *G*<sup>j</sup> (c) in the description of the non collinear magnetic structures at low temperature

$$E\_{\rm M}^{\rm (c)}\left(100\right)^{\*} = \sqrt{2}\,\left/\,2\left(\mathbf{G}\_{1} - \mathbf{G}\_{2}\right) + \mathrm{i}\left[\sqrt{2}\,\,/\,2\left(\mathbf{C}\_{1} + \mathbf{C}\_{2}\right) + A\_{3}\right] \tag{17}$$

jjj 3 j 6 j 9 j jj 3 j 6 j 9

*FSS S S G SS S S*

12 3 4 12 34 1234 123 4 *F S S S SG S S S SC S S S S A S S S S* ; – –; ––; – – (13)

567 8 5678 5678 567 8 *F*' *S S S SG S S S SC S S S S A S S S S* ;' – – ;' – – ;' – – (14)

; – –;

(12)

jjj 3 j 6 j 9 j jj 3 j 6 j 9

For the Fe3+[16a] sublattice, it is necessary to consider two distinct magnetic modes. The first chosen notation is the unprimed magnetic modes for the numbered spins

The second chosen notation of the magnetic modes is the primed notation (**'**) for the

For the RE3+ ions in the Wyckoff site {24c}, the local axes (*U*,*V*,*W*) in the D2(222) symmetry are identified to the parameters *g*α (α = *x*,*y*,*z*) of the magnetic tensor *g* in the hypothesis of the effective spin Hamiltonian model (Wolf et al., 1962, Wolf, 1964). The particular superstructure lines (222)**\*** and (622)**\*** are not observed in the whole temperature range

<sup>a</sup> \* \* (d)

c c \* \*

In these conditions, the spins vectors *S*<sup>j</sup> and *S*j+6 (j = 1-3) of Tb3+{24c} and Fe3+(24d) ions are coupled ferromagnetically and the symmetry operation ( 1 0,0,0) is an inversion center. These significant properties related to magnetic symmetry involve that all the magnetic modes *A*j are absent. This absence will be accompanied by the elimination of the modes *C*j. Consequently, the above crystallographic sites split into six magnetically inequivalent sublattices Cj, Cj and Dj, Dj with (j = 1-3) respectively as it is indicated on Table 1. In each sublattice we have four ions which are equivalent under the symmetry operations: ( 1 0,0,0) and (11/2,1/2,1/2). The sublattices C2 and C3 are related to C1 by a rotation of 120 and 240° around the 3-fold symmetry [111] axis (also for Cj); the same remarks can be made for the sublattices Dj, Dj (j = 1-3). Contrary to previous spin rotation observed at *T*comp (260 5 K) by mössbauer spectroscopy (Hong et al., 2004), no deviation from the colinearity along the easy axis [111] for the spins of the Fe3+[16a] ions is evidenced around *T*comp (244 ± 10 K) (right of Fig. 9). In this *T*-region, this sublattice is described by one ferromagnetic configuration of the cubic magnetic modes *F* and *F***'** with the equality of the left of the equation (15). The magnetic structure factors *F*M(c)(hkl)**\*** calculated for all the observed superstructure lines

<sup>M</sup> <sup>M</sup> <sup>123</sup> *F FF F* 222 – ' 0; 222 i *AAA* 0 (15)

(c) and *G*<sup>j</sup>

<sup>M</sup> 1 2 12 3 *F* 100 2 / 2( – ) i[ 2 / 2( ) ] *GG CC A* (17)

(c) in the description of the non

<sup>M</sup> 123 M <sup>123</sup> *F A* 222 –i *A AF A* 0; 622 i – – 0 *A A* (16)

below *T*N. For example, the associated magnetic structure factors are all equal to zero

*C SS S S A SS S S*

– –; – –

The adapted magnetic modes for the RE3+{24c} and Fe3+(24d) sublattices with j = 1-3 are

*SSS* p p p8 (p = 1-4)

numbered spins *SSS* p' p' p' 8 (p**'** = 5-8)

imply to know if we need both magnetic modes *F*<sup>j</sup>

collinear magnetic structures at low temperature

 <sup>c</sup> \*

$$\left(\mathbf{F\_{M}}^{(c)}(\text{310})\right)^{\*} = \sqrt{2}\,\left/\,2(\mathbf{-G\_{1}} + \mathbf{G\_{2}}) + \mathrm{i}\left[\sqrt{2}\,\right/\,2(\mathbf{C\_{1}} - \mathbf{C\_{2}}) + \mathbf{A\_{3}}\right] \tag{19}$$

$$F\_{\rm M}^{\rm (c)} \left( 411 \right)^{\*} = \sqrt{2} \left/ \left< 2(\mathbf{G}\_{2} - \mathbf{G}\_{3}) + i \right[ \sqrt{2} \left/ \left< \mathbf{C}\_{2} + \mathbf{C}\_{3} \right> - A\_{1} \right] \right. \tag{20}$$

$$\left(\mathbf{F\_M}^{(c)}(330)\right)^{\*} = \sqrt{2}\left/\left<\mathbf{-G\_1} + \mathbf{G\_2}\right> + i\left[\sqrt{2}\left/\left<\mathbf{C\_1} + \mathbf{C\_2}\right> - A\_3\right]\right.\tag{21}$$

$$\left(\mathbf{F\_{M}}^{(c)}(\text{510})\right)^{\*} = \sqrt{2}\,\left/\,\text{2(-G\_{1}-G\_{2})} + \mathrm{i}\left[\sqrt{2}\,\left/\,\mathbf{2(-C\_{1}+C\_{2}) + A\_{3}}\right.\right] \tag{22}$$

$$\left(F\_{\rm M}\,^{(c)}\right)\left(433\right)^{\*} = \sqrt{2}\,^{\prime}\left(2\left(-\mathcal{G}\_{2} + \mathcal{G}\_{3}\right) + i\left[\sqrt{2}\,^{\prime}\left(2\left(\mathcal{C}\_{2} + \mathcal{C}\_{3}\right) + A\_{1}\right]\right]\tag{23}$$

$$\left[\text{F}\_{\text{M}}^{\text{(c)}}(\text{530})\right]^{\*} = \sqrt{2}\,\left/\,2(\text{-G}\_{1}\text{-G}\_{2}) + \text{i}\left[\sqrt{2}\,\,/\,2(\text{-C}\_{1}\text{-C}\_{2}) - \text{A}\_{3}\right] \tag{24}$$

$$\mathbf{F\_{M}}^{\text{(c)}} \text{(600)}^{\text{\*}} = -\mathbf{F\_{2}} + \mathbf{F\_{3}} - \mathbf{i}\mathbf{A\_{1}} \tag{25}$$

$$\left(\mathbf{F\_{M}}^{\mathrm{(c)}}\right)\left(\mathbf{442}\right)^{\*} = -\mathbf{F\_{2}} + \mathbf{F\_{1}} + \mathbf{i}\mathbf{A\_{3}}\tag{26}$$

From these expressions, one can observe that both magnetic modes *F*<sup>j</sup> (c) and *G*<sup>j</sup> (c) are necessary in the description of the non collinear structures of the Tb3+ ions. The absence of the small superstructure lines (200)**\*** and (600, 442)**\*** above 13 K gives rise to the equality between the magnetic modes *F*<sup>j</sup> (c) (j = 1-3). It must be noted that the superstructure line (510)**\*** appears at the same Bragg peak of the pure nuclear reflection (431) (θ = 30.7°). A magnetic contribution of the RE3+ ions exists for (431). We observe also a magnetic contribution for another reflection (541). It is very difficult to isolate only the magnetic contribution of these two reflections which are represented by their magnetic structure factors

$$\mathrm{F\_M}^{(c)}(431) = \sqrt{2}\left/\left<\mathrm{-G\_2} + \mathrm{G\_3}\right> + i\left[\sqrt{2}\left/\mathrm{2(C\_2 - C\_3) - A\_1}\right.\right] \tag{27}$$

$$\mathbf{F\_{M}}^{(c)}(541) = \sqrt{2}\,/\,2(\mathbf{G\_{1}} + \mathbf{G\_{3}}) + i[\sqrt{2}\,/\,2(-\mathbf{C\_{1}} + \mathbf{C\_{3}}) - \mathbf{A\_{2}}] \tag{28}$$

#### **5.2 Representation analysis of magnetic structures**

Bertaut (Bertaut, 1963, 1968, 1971, 1972) has created a group theory method called "*Representation Analysis*" which has been widely used in the last four decades by Bertaut himself and by other researchers. The essential role is plaid by the "*Basis Vectors of Irreductible Representations*" of the paramagnetic space group Ia 3 d of TbIG and its highest subgroups.

#### **5.2.1 Representation analysis of Ia 3 d**

The representation analysis of the cubic space group Ia d was applied in the past (Bertaut et al., 1970; Tchéou et al., 1970a) in order to determine the spin configurations of the Tb3+

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 217

11 = *A*x + *ЄC*<sup>y</sup> + *Є*<sup>2</sup>*G*<sup>z</sup> 12 = - (*A***'**<sup>y</sup> + *Є*<sup>2</sup>*G***'**x + *ЄC***'**z) 21 = - (*A***'**<sup>y</sup> + *ЄG***'**x + *Є*<sup>2</sup>*C***'**z) 22 = *A*x + *Є*<sup>2</sup>*C*y + *ЄG*<sup>z</sup>

11 = *F*x + *F***'**<sup>x</sup> 12 = *C*z + *C***'**<sup>y</sup> 13 = *G*<sup>y</sup> + *A***'**<sup>z</sup> 21 = *F*<sup>y</sup> + *F***'**<sup>y</sup> 22 = *G*x + *G***'**<sup>z</sup> 23 = *A*z + *C***'**<sup>x</sup> 31 = *F*z + *F***'**<sup>z</sup> 32 = *A*<sup>y</sup> + *A***'**<sup>x</sup> 33 = *C*x + *G***'**<sup>y</sup>

11 = *F*x - *F***'**<sup>x</sup> 12 = *C*z - *C***'**<sup>y</sup> 13 = *G*<sup>y</sup> - *A***'**<sup>z</sup> 21 = *F*<sup>y</sup> - *F***'**<sup>y</sup> 22 = *G*x - *G***'**<sup>z</sup> 23 = *A*z - *C***'**<sup>x</sup> 31 = *F*z - *F***'**<sup>z</sup> 32 = *A*<sup>y</sup> - *A***'**<sup>x</sup> 33 = *C*x - *G***'**<sup>y</sup>

In the high temperature region, the observed spectra which are well interpreted within the ferrimagnetic model of Néel are easily identified to the magnetic modes *F*j, *F* and *F***'**. At *T*comp = 243.5 K the mean exchange field acting on the Tb3+ ions by the iron sublattices is too strong (~ 174 kOe) by comparison with the Tb3+–Tb3+ exchange field (~ 8 kOe) (Lahoubi, 1986); this last coupling beetwen the Tb3+ ions will be not able to decouple at low temperature the two sublattices. This remarkable property excludes permanently the intervention of the three-dimensional irreductible representation (Г5g = T2g) where the magnetic modes of the rare earth sublattice are along the <0 1 1> directions. Using the basis vectors of (Г4g = T1g), we present in Fig. 10 the four cubic models of "*double umbrella*" of type

" II" : f, F G, F G ; *Double umbrella of type S S* 1 7 *S S* 4 10 f, F – G, F – G ; c.p. (29)

1g = A1g 11 = (*A*x + *C*<sup>y</sup> + *G*z) - (*G***'**x + *A***'**<sup>y</sup> + *C***'**z) 2g = A2g 11 = (*A*x + *C*<sup>y</sup> + Gz) + (*G***'**x + *A***'**<sup>y</sup> + *C***'**z)

3g = Eg

4g = T1g

5g = T2g

Table 4. Basis vectors of the Fe3+ in site (16a) of Ia 3 d

II (Lahoubi, 1986) in the irreductible representation (Г4g = T1g) of Ia 3 d

Fig. 10. Four cubic "*double umbrella*'' models of type II for (Г4g = T1g) of Ia 3 d.

ions in TbIG. In the point group *Oh* ten possible irreductible representations Гig and Гiu (i = 1-5) are present and listed usually on quantum mechanics text books (Flury Jr, 1980; Kahan, 1972). The subscripts g and u refer to irreductible representations which are even (gerade) and odd (ungerade) respectively under the inversion I = ( 1 0,0,0). In their original works, two representations (Г4g = T1g) and (Г5g = T2g) were used. However, the coupling between the RE3+ in sites {24} and the two iron ions Fe3+ in sites [16a] and (24d) has not been taken account in their study. The representation analysis of Ia 3 d has been developed later completely by the author (Lahoubi, 1986) in order to choose the common irreductible representation which could be able to represent both the Néel model at high temperature below *T*N and the non collinear magnetic structures observed below 160 K. For both RE3+{24c} and Fe3+(24d) sublattices, we need the even Гig (i = 1-5) which appear only for Fe3+[16a] sublattice. Using the precedent linear combinations of the spins of equations (12), (13) and (14), the sets of magnetic basis vectors of the three sublattices belonging to Гig (i = 1- 5) which are formed by the functions lm (l = 1-dig, dig the dimension of Гig and m fixed) are listed on Tables 2, 3 and 4. The letters A and E are assigned to one and two dimensional representations where (11)\* belongs to the complex conjugate representation with *Є* = {exp(2 πi/3)}. Due to the equation (13), it can be show that (Г5g = T2g) is excluded and only (Г4g = T1g) may be used to describe formally in a first approximation all magnetic structures.


Table 2. Basis vectors of the RE3+ in site {24c} of Ia 3 d


Table 3. Basis vectors of the Fe3+ in site [24d] of Ia 3 d


Table 4. Basis vectors of the Fe3+ in site (16a) of Ia 3 d

ions in TbIG. In the point group *Oh* ten possible irreductible representations Гig and Гiu (i = 1-5) are present and listed usually on quantum mechanics text books (Flury Jr, 1980; Kahan, 1972). The subscripts g and u refer to irreductible representations which are even (gerade) and odd (ungerade) respectively under the inversion I = ( 1 0,0,0). In their original works, two representations (Г4g = T1g) and (Г5g = T2g) were used. However, the coupling between the RE3+ in sites {24} and the two iron ions Fe3+ in sites [16a] and (24d) has not been taken account in their study. The representation analysis of Ia 3 d has been developed later completely by the author (Lahoubi, 1986) in order to choose the common irreductible representation which could be able to represent both the Néel model at high temperature below *T*N and the non collinear magnetic structures observed below 160 K. For both RE3+{24c} and Fe3+(24d) sublattices, we need the even Гig (i = 1-5) which appear only for Fe3+[16a] sublattice. Using the precedent linear combinations of the spins of equations (12), (13) and (14), the sets of magnetic basis vectors of the three sublattices belonging to Гig (i = 1- 5) which are formed by the functions lm (l = 1-dig, dig the dimension of Гig and m fixed) are listed on Tables 2, 3 and 4. The letters A and E are assigned to one and two dimensional representations where (11)\* belongs to the complex conjugate representation with *Є* = {exp(2 πi/3)}. Due to the equation (13), it can be show that (Г5g = T2g) is excluded and only (Г4g = T1g) may be used to describe formally in a first approximation all magnetic structures.

1g = A1g

3g = Eg

4g = T1g

5g = T2g

2g = A2g

3g = Eg

4g = T1g

5g = T2g

2g = A2g 11 = *G*1x + *G*2 y + *G*3z

Table 2. Basis vectors of the RE3+ in site {24c} of Ia 3 d

Table 3. Basis vectors of the Fe3+ in site [24d] of Ia 3 d

1g = A1g 11 = *G*1x + *G*2 y + *G*3z

21 = - *Є*(11)**\***

11 = *G*1x + *ЄG*2 y + *Є*<sup>2</sup>*G*3z

11 = *G*1x + *ЄG*2y + *Є*<sup>2</sup>*G*3z

11 = *F*3x - *F*2x <sup>12</sup> = *G*3y + *G*2z 21 = *F*1y - *F*3y <sup>22</sup> = *G*1z + *G*3x 31 = *F*2z - *F*1z <sup>32</sup> = *G*2x + *G*1y

21 = + *Є*(11)**\***

11 = *F1x* 12 = *F*2x + *F*3x 13 = *G*2z + *G*3y 21 = *F*2y 22 = *F*1y + *F*3y 23 = *G*3x + *G*1z 31 = *F*3z 32 = *F*1z + *F*2z 33 = *G*1y + *G*2x

11 = *F*3x - *F*2x 13 = *G*3y - *G*2z 21 = *F*1y - *F*3y 23 = *G*1z - *G*3x 31 = *F*2z - *F*1z 33 = *G*2x - *G*1y

11 = *F*1x 12 = *F*2x + *F*3x 13 = *G*3y - *G*2z 21 = *F*2y 22 = *F*1y +*F*3y 23 = *G*1z - *G*3x 31 = *F*3z 32 = *F*1z + *F*2z 33 = *G*2x - *G*1y In the high temperature region, the observed spectra which are well interpreted within the ferrimagnetic model of Néel are easily identified to the magnetic modes *F*j, *F* and *F***'**. At *T*comp = 243.5 K the mean exchange field acting on the Tb3+ ions by the iron sublattices is too strong (~ 174 kOe) by comparison with the Tb3+–Tb3+ exchange field (~ 8 kOe) (Lahoubi, 1986); this last coupling beetwen the Tb3+ ions will be not able to decouple at low temperature the two sublattices. This remarkable property excludes permanently the intervention of the three-dimensional irreductible representation (Г5g = T2g) where the magnetic modes of the rare earth sublattice are along the <0 1 1> directions. Using the basis vectors of (Г4g = T1g), we present in Fig. 10 the four cubic models of "*double umbrella*" of type II (Lahoubi, 1986) in the irreductible representation (Г4g = T1g) of Ia 3 d

" II" : f, F G, F G ; *Double umbrella of type S S* 1 7 *S S* 4 10 f, F – G, F – G ; c.p. (29)

Fig. 10. Four cubic "*double umbrella*'' models of type II for (Г4g = T1g) of Ia 3 d.

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 219

±(1/8,0,3/4); c.p. 12f 1

O2- 96h 1 12f 1 OI OII OIII OIV

adapted to describe the magnetic structures of the three magnetic ions. For the RE3+ ions, the preceding magnetically inequivalent sublattices Cj and Cj (j = 1-3) become crystallographic inequivalent sites 6e and 6e; they are described in the rhombohedral axis X, Y, Z by a new linear combination for the ferromagnetic mode *f*j = *S*j + *S*j+6 (j = 1-3). Concerning the iron ions sublattices Dj and D'j of the 24d site, they will be associated to a new 12f site where the basis vectors are described with the ferromagnetic mode *V*j = *S*j + *S*j+6 (j = 1-3). The associated basis vectors of (Г2g = A2g) are presented here on Table 6. A new combination of these basis vectors is proposed and four modified double umbrella models are presented for the six sublattices Cj and Cj on Table 7. In this description, the first part (f*n)* of the moment *m*<sup>j</sup> represents the collinear ferromagnetic mode of the component along the [111] direction.

6e

2

±(X,1/2-X,1/4) X ≈ 3/8; c.p. ±(X,1/2-X,1/4) X ≈ 7/8; c.p.

±(X,Y,Z); c.p.

±(Y+1/2,X+1/2,Z+1/2); c.p. X≈5/8,Y≈3/8,Z≈1/4

> 0,0,0; 1/2,1/2,1/2 0,1/2,1/2; c.p. 1/2,0,0; c.p.

2

3

1

11(I) = *f*1X + *f*2Y + *f*3Z 11(II) = *f*1Y + *f*1Z + *f*2Z + *f*2X + *f*3X + *f*3Y

11(IV) = *f*4X + *f*5Y + *f*6Z 11(V) = *f*4Y + *f*4Z + *f*5Z + *f*5X + *f*6X + *f*6Y

11(I) = *V*1X + *V*2Y + *V*3Z + *V*4X + *V*5Y + *V*6Z 11(II) = *V*2X + *V*3Y + *V*1Z + *V*6X + *V*4Y + *V*5Z 11(III) = *V*3X + *V*1Y + *V*2Z + *V*5X + *V*6Y + *V*4Z

11(I) = (*S*1 + *S*5) X + (*S*1 + *S*5) Y + (*S*1 + *S*5) Z

11(II) = (*S*2 + *S*6) X + (*S*3 + *S*7) Y + (*S*4 + *S*8) Z 11(III) = (*S*3 + *S*8) X + (*S*4 + *S*6) Y + (*S*2 + *S*7) Z 11(IV) = (*S*4 + *S*7) X + (*S*2 + *S*8) Y + (*S*3 + *S*6) Z

6e

2b

6d

Ions Sites Symmetry Positions Sites Symmetry Positions

Ia 3 d R 3 c

±(1/8,0,1/4); c.p.

±(3/8,0,3/4); c.p.

±(3/8,0,1/4); c.p.

0,0,0; 1/4,1/4,1/4 0,1/2,1/2; c.p. 1/4,3/4,3/4; c.p.

Table 5. Correspondence between the positions of the ions in Ia 3 d and R 3 c

 R 3 c Ions Sites Basis vectors

Table 6. Basis vectors of ions in the irreductible representation A2g of R 3 c

RE3+ 24c 222

Fe3+ 24d <sup>4</sup>

Fe3+ 16a 3

RE3+

6e

6e

6d

Fe3+ 12f

Fe3+ 2b

In these four models, the angles and the modulus of the moments *m*1 and *m*'1 are different. They must to respect some requirements: the components of the moments *m*1, *m*'1 along the cubic axis [100] are necessary equals; also for the components of the moments *m*2, *m*'2 and *m*3, *m*'3 respectively along [010] and [001]. The components of the moment *m*1 along the cubic axes [010] and [001] must be equals; also for *m*2 and *m*'2 along [100] and [001]; also for *m*<sup>3</sup> and *m*'3 along [100] and [010]. The first refinements at 4.2 K (with *m*a = *m*d = 5 B) lead to values of the moment *m*'1 (~ 10 B), i.e., above the free Tb3+ ion value (9 B). The reason of this discrepancy is associated to the deviation of the cubic description which imply a rhombohedral distortion observed on powder sample by X-ray diffraction at 6 K (Bertaut et al., 1970, Sayetat, 1974, 1986 and refs. herein) and neutron diffractions at 5 K (Hock et al., 1990) where two subgroups of Ia 3 d have been proposed respectively: R 3 c and R 3 .

#### **5.2.2 Representation analysis of R 3 c**

A detailed description of the representation analysis of the subgroup R 3 c is presented here. According to the earlier precise X-ray diffraction measurements of the rhombohedral distortions carried out on single crystals of the terbium-yttrium iron garnet system Tb Y Fe O x 3 x 5 12 with 0 ≤ x ≤ 3 (hereafter Tb Y IG x 3x ) (Levitin et al., 1983), the choice of the subgroup R 3 c seems more appropriate. This choice will be confirmed later by our high field magnetization measurements and the use of the method of "*the symmetry lowering device*" (Bertaut, 1981) which is connected with representation analysis of Bertaut (Bertaut, 1968, 1971, 1972). From the International Tables (Hahn, 1983), the Bravais lattice of the crystallographic space group <sup>R</sup> <sup>3</sup> c-( <sup>6</sup> D ) No. 167 is defined in the system of the rhombohedral directions 3d [ 111 ], [ 111 ], [ 111 ] with the three fundamental vectors: *A*1, *A*2, *A*3 with the same parameter *A*Rh and an angle αRh # 90° and < 120° respectively along the unit vectors *g*j, j = 1-3). There are related to the cubic axes by unit vectors {*i*, *j*, *k*}

$$\mathbf{g}\_1 = \frac{1}{\sqrt{3}}(-i+j+k); \mathbf{g}\_2 = \frac{1}{\sqrt{3}}(+i-j+k); \mathbf{g}\_3 = \frac{1}{\sqrt{3}}(+i+j-k) \tag{30}$$

$$\mathbf{X} = \mathbf{y} + \mathbf{z}; \mathbf{Y} = \mathbf{x} + \mathbf{z}; \mathbf{Z} = \mathbf{x} + \mathbf{y} \tag{31}$$

The correspondence between the ion positions in the two space groups Ia 3 d and R 3 c is reported on Table 5. For TbIG, the values of the parameters found at 6.75 K (Sayetat, 1974, 1986) are: *A*Rh = 10.7430 Å; αRh = 109°24'40''. The rhombohedral unit cell contains only the half atoms of the cubic unit cell. In order to use the representation analysis to determine the magnetic basis vectors, we choose the following generators of the space group R 3 c: the identity E = (10,0,0), the inversion I = ( 1 0,0,0), a ternary axis 3 = (30,0,0) and a diagonal binary axis 2d = (2 XY 1/2,1/2,1/2). This axis is perpendicular to the glide plane c = I. 2d of the symbol R 3 c. The wave vector being *k* = 0, the six irreductible representations of the space group R 3 c are those of the point group D3d, Гjg and Гju with j = 1-3. The previous three-dimensional irreductible representation (Г4g = T1g) of *Oh* is reduced to: Г2g = A2g + Eg. Only the one-dimensional irreductible representation will be chosen in our study (the twodimensional irreductible representation Eg being complex, she is not considered). It appears that the previous linear combinations of the spin vectors *F, G, C* and *A* are not in reality

In these four models, the angles and the modulus of the moments *m*1 and *m*'1 are different. They must to respect some requirements: the components of the moments *m*1, *m*'1 along the cubic axis [100] are necessary equals; also for the components of the moments *m*2, *m*'2 and *m*3, *m*'3 respectively along [010] and [001]. The components of the moment *m*1 along the cubic axes [010] and [001] must be equals; also for *m*2 and *m*'2 along [100] and [001]; also for *m*<sup>3</sup> and *m*'3 along [100] and [010]. The first refinements at 4.2 K (with *m*a = *m*d = 5 B) lead to values of the moment *m*'1 (~ 10 B), i.e., above the free Tb3+ ion value (9 B). The reason of this discrepancy is associated to the deviation of the cubic description which imply a rhombohedral distortion observed on powder sample by X-ray diffraction at 6 K (Bertaut et al., 1970, Sayetat, 1974, 1986 and refs. herein) and neutron diffractions at 5 K (Hock et al.,

1990) where two subgroups of Ia 3 d have been proposed respectively: R 3 c and R 3 .

A detailed description of the representation analysis of the subgroup R 3 c is presented here. According to the earlier precise X-ray diffraction measurements of the rhombohedral distortions carried out on single crystals of the terbium-yttrium iron garnet system Tb Y Fe O x 3 x 5 12 with 0 ≤ x ≤ 3 (hereafter Tb Y IG x 3x ) (Levitin et al., 1983), the choice of the subgroup R 3 c seems more appropriate. This choice will be confirmed later by our high field magnetization measurements and the use of the method of "*the symmetry lowering device*" (Bertaut, 1981) which is connected with representation analysis of Bertaut (Bertaut, 1968, 1971, 1972). From the International Tables (Hahn, 1983), the Bravais lattice of the crystallographic space group <sup>R</sup> <sup>3</sup> c-( <sup>6</sup> D ) No. 167 is defined in the system of the rhombohedral directions 3d [ 111 ], [ 111 ], [ 111 ] with the three fundamental vectors: *A*1, *A*2, *A*3 with the same parameter *A*Rh and an angle αRh # 90° and < 120° respectively along the unit vectors *g*j, j = 1-3). There are

<sup>123</sup> <sup>111</sup> – ; –; – <sup>333</sup> *<sup>g</sup> <sup>i</sup> <sup>j</sup> <sup>k</sup> <sup>g</sup> <sup>i</sup> <sup>j</sup> <sup>k</sup> <sup>g</sup> <sup>i</sup> <sup>j</sup> <sup>k</sup>* (30)

The correspondence between the ion positions in the two space groups Ia 3 d and R 3 c is reported on Table 5. For TbIG, the values of the parameters found at 6.75 K (Sayetat, 1974, 1986) are: *A*Rh = 10.7430 Å; αRh = 109°24'40''. The rhombohedral unit cell contains only the half atoms of the cubic unit cell. In order to use the representation analysis to determine the magnetic basis vectors, we choose the following generators of the space group R 3 c: the identity E = (10,0,0), the inversion I = ( 1 0,0,0), a ternary axis 3 = (30,0,0) and a diagonal binary axis 2d = (2 XY 1/2,1/2,1/2). This axis is perpendicular to the glide plane c = I. 2d of the symbol R 3 c. The wave vector being *k* = 0, the six irreductible representations of the space group R 3 c are those of the point group D3d, Гjg and Гju with j = 1-3. The previous three-dimensional irreductible representation (Г4g = T1g) of *Oh* is reduced to: Г2g = A2g + Eg. Only the one-dimensional irreductible representation will be chosen in our study (the twodimensional irreductible representation Eg being complex, she is not considered). It appears that the previous linear combinations of the spin vectors *F, G, C* and *A* are not in reality

X y z; Y x z; Z x y (31)

**5.2.2 Representation analysis of R 3 c** 

related to the cubic axes by unit vectors {*i*, *j*, *k*}



adapted to describe the magnetic structures of the three magnetic ions. For the RE3+ ions, the preceding magnetically inequivalent sublattices Cj and Cj (j = 1-3) become crystallographic inequivalent sites 6e and 6e; they are described in the rhombohedral axis X, Y, Z by a new linear combination for the ferromagnetic mode *f*j = *S*j + *S*j+6 (j = 1-3). Concerning the iron ions sublattices Dj and D'j of the 24d site, they will be associated to a new 12f site where the basis vectors are described with the ferromagnetic mode *V*j = *S*j + *S*j+6 (j = 1-3). The associated basis vectors of (Г2g = A2g) are presented here on Table 6. A new combination of these basis vectors is proposed and four modified double umbrella models are presented for the six sublattices Cj and Cj on Table 7. In this description, the first part (f*n)* of the moment *m*<sup>j</sup> represents the collinear ferromagnetic mode of the component along the [111] direction.



Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 221

Fig. 12. a. Local axes of the D2(222) symmetry, b. Novel "*double umbrella*"at 5 K

a b

components *m*1x and *m*'1z above the value (9 B) of the free Tb3+ ion.

**6. Temperature evolution of the double umbrella structure** 

between 6 and 10 % in the 4.2−283 K temperature range.

C1: *m*1 = 8.07 B ; 1 = 32° ; 1 = 180° (32)

 C'1: *m*'1 = 8.90 B ; '1 = 27° ; '1 = 0° (33) where, 1 and 1 are the angles from *p*1 in the (111) plane and = 0.20 B; with (*M*S)<111> = *M*Scal – *M*Smes = 0.03 B/mol. For a better presentation one can show the novel "*double umbrella* "(Fig. 12b) in the local axes of the D2(222) symmetry (Fig. 12a). These results are in good agreement with our recent high magnetic field magnetizations performed at 4.2 K (Lahoubi, 2012) where a third low critical field *H*c0 (Fig. 13) is observed along the <100> direction and added to the previous *H*c1 and *H*c2 (Lahoubi et al., 1984). They confirm unambiguously the presence of the three magnetic glide planes c' of the symbol R 3 c'. The earlier results at 5 K described in the subgroup R 3 (Hock et al., 1990) lead to values of the

The parameters for the sites C1 (*m*1, 1, 1 = 180°) and C1 (*m*1, 1, 1 = 0°) are refined with the same model found at 5 K with the condition 0 (*m*<sup>1</sup> = -*m*<sup>1</sup> ) due to the absence of the superstructure lines (200)**\*** and (600,442)**\*** above 13 K (D1B) (Fig. 4). The magnetic moments *m*a and *m*d of the iron sublattices were not concerned by the refinements for all reflections of the patterns and the observed values found by N.M.R. experiments (Gonano et al., 1967) are used. Good agreement is obtained with the reliability factors *R* varying


Table 7. Four rhombohedral models in A2g of R 3 c and new model for TbIG (Lahoubi, 2012)

The second part ( a*g*j) represents the non collinear antiferromagnetic modes of the components of the moment *m*j along the three rhombohedral axes {[ 1 11], [1 1 1], [11 1 ]}. Some requirements from the previous "*cubic description*" are now ignored in the "*rhombohedral description*": the axis [100] and equivalent directions cease to be principal axes. In the four models of non collinear arrangements for the RE3+ ions around the ternary axis [111], the sublattices Cj, Cj are situated in the three glide planes c, c.3 and c.32 of the subgroup R 3 c: each plane containing [111] (unit vector *n*) and one of the rhombohedral directions {[ 1 11], [1 1 1], [11 1 ]} represented by the unit vectors *g*j, j = 1-3. In this condition, we have: c = (*n*, *g*3); c.3 = (*n*, *g*1) and c.32 = (*n*, *g*2) (Fig. 11a). Furthermore, the projection on the plane (111) of the rhombohedral direction [ 1 11] (*g*1) for example, is equivalent to the low symmetry axis [ 2 11] (with the unit vector *p*1); the local axis [0 1 1] (*W*) is also chosen. If one takes into account the smallness superstructure lines (200)**\*** and (600,442)**\*** at 5 K (D1B), new parameters are found for C1 (*m*1, 1, 1) and C1 (*m*1, 1, 1) sites (Lahoubi, 2012) (Fig. 11b). The moments of C1 and C<sup>1</sup> are drawed in the (π) plane which corresponds to glide plane c.3: this plane remains a principal plane of the magnetic tensor *g* of the earlier "*cubic description*". A good reliability factor *R* of the order of 6.7 % is found for a refinement based on all the reflections and the method of validation of the magnetic structures (Wills, 2007 and refs. herein):

Fig. 11. a. The three glide planes of R 3 c, b. New model for C1 and C'1 at 5 K (D1B)

(6e): Cj : *S*j = *S*j+ 6 = f*n* a*g*<sup>j</sup> C1 : *m*1 = *S*1 = *S*7 = (f – a/3)*n* – ( 2 2

(6e): Cj : *S*j = *S*j+ 6 = f *n* a*g*<sup>j</sup> C1 : *m*1 = *S*4 = **S**10 = (f + a/3)*n* + ( 2 2

method of validation of the magnetic structures (Wills, 2007 and refs. herein):

 Fig. 11. a. The three glide planes of R 3 c, b. New model for C1 and C'1 at 5 K (D1B)

a b

Table 7. Four rhombohedral models in A2g of R 3 c and new model for TbIG (Lahoubi, 2012)

The second part ( a*g*j) represents the non collinear antiferromagnetic modes of the components of the moment *m*j along the three rhombohedral axes {[ 1 11], [1 1 1], [11 1 ]}. Some requirements from the previous "*cubic description*" are now ignored in the "*rhombohedral description*": the axis [100] and equivalent directions cease to be principal axes. In the four models of non collinear arrangements for the RE3+ ions around the ternary axis [111], the sublattices Cj, Cj are situated in the three glide planes c, c.3 and c.32 of the subgroup R 3 c: each plane containing [111] (unit vector *n*) and one of the rhombohedral directions {[ 1 11], [1 1 1], [11 1 ]} represented by the unit vectors *g*j, j = 1-3. In this condition, we have: c = (*n*, *g*3); c.3 = (*n*, *g*1) and c.32 = (*n*, *g*2) (Fig. 11a). Furthermore, the projection on the plane (111) of the rhombohedral direction [ 1 11] (*g*1) for example, is equivalent to the low symmetry axis [ 2 11] (with the unit vector *p*1); the local axis [0 1 1] (*W*) is also chosen. If one takes into account the smallness superstructure lines (200)**\*** and (600,442)**\*** at 5 K (D1B), new parameters are found for C1 (*m*1, 1, 1) and C1 (*m*1, 1, 1) sites (Lahoubi, 2012) (Fig. 11b). The moments of C1 and C<sup>1</sup> are drawed in the (π) plane which corresponds to glide plane c.3: this plane remains a principal plane of the magnetic tensor *g* of the earlier "*cubic description*". A good reliability factor *R* of the order of 6.7 % is found for a refinement based on all the reflections and the

New model for TbIG with f f and a a – ( 0 at *T* > 5 K)

<sup>3</sup> )a*p*<sup>1</sup>

<sup>3</sup> )a*p***<sup>1</sup>**

Four A2g models in R 3 c with f f and a a

Fig. 12. a. Local axes of the D2(222) symmetry, b. Novel "*double umbrella*"at 5 K

$$\mathbf{C}\_{1}\text{: } m\_{1} = 8.07\,\upmu\_{B}\text{: } \theta\_{1} = 32^{\circ}\text{: } \phi\_{1} = 180^{\circ}\tag{32}$$

$$\mathbf{C}\_{1}^{\prime}; m\_{1}^{\prime} = 8.90 \,\upmu\_{B}; \theta\_{1}^{\prime} = 27^{\circ}; \,\phi\_{1}^{\prime} = 0^{\circ} \tag{33}$$

where, 1 and 1 are the angles from *p*1 in the (111) plane and = 0.20 B; with (*M*S)<111> = *M*Scal – *M*Smes = 0.03 B/mol. For a better presentation one can show the novel "*double umbrella* "(Fig. 12b) in the local axes of the D2(222) symmetry (Fig. 12a). These results are in good agreement with our recent high magnetic field magnetizations performed at 4.2 K (Lahoubi, 2012) where a third low critical field *H*c0 (Fig. 13) is observed along the <100> direction and added to the previous *H*c1 and *H*c2 (Lahoubi et al., 1984). They confirm unambiguously the presence of the three magnetic glide planes c' of the symbol R 3 c'. The earlier results at 5 K described in the subgroup R 3 (Hock et al., 1990) lead to values of the components *m*1x and *m*'1z above the value (9 B) of the free Tb3+ ion.

#### **6. Temperature evolution of the double umbrella structure**

The parameters for the sites C1 (*m*1, 1, 1 = 180°) and C1 (*m*1, 1, 1 = 0°) are refined with the same model found at 5 K with the condition 0 (*m*<sup>1</sup> = -*m*<sup>1</sup> ) due to the absence of the superstructure lines (200)**\*** and (600,442)**\*** above 13 K (D1B) (Fig. 4). The magnetic moments *m*a and *m*d of the iron sublattices were not concerned by the refinements for all reflections of the patterns and the observed values found by N.M.R. experiments (Gonano et al., 1967) are used. Good agreement is obtained with the reliability factors *R* varying between 6 and 10 % in the 4.2−283 K temperature range.

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 223

Fig. 14. Thermal variations of (*I*M)obs and (*I*M)cal for the reflections (211), (321), (220) and (440)

Fig. 15. Thermal variations of (*I*M)obs and (*I*M)cal for the superstructure lines (310)**\***, (110)**\***,

(411, 330)**\*** and (530, 433)**\***

Fig. 13. *M*T(*H*) versus *H* at 4.2 K along the <111> and <100> directions (Lahoubi, 2012)

The equation (5) in the section 3, permit us to deduce the magnetic intensities *I*M by assuming negligible the *T*-variations of *I*N with a rather good precision for the reflections (hkl) which have a great magnetic contribution such as (211), (321), (521) and (532, 611) where the ratio (*I*M/*I*N)cal is equal respectively to: 200, 20, 4 and 2. Consequently, the high magnetically reflections with a small nuclear contribution are only (211) and (321). The thermal variations of these two magnetic reflections which are responsible of the collinear ferrimagnetic ordering are reported in Fig. 14. They appear at *T*N and present a first increase below 160 K with a second rapid increase at 68 K. Different and complex temperature dependences are observed for the reflections (220) and (440). At first, the values of (*I*M)cal are higher than those observed for (*I*M)obs. These two reflections increase simultaneously between 4.2 and 160 K and present an inflexion point near 68 K. Above 160 K, they tend to a plateau until room temperature after which they decrease progressively and reach zero at *T*N. The possible explanation of this characteristic behavior seems to be related to the magnetic contribution of the irons in the octahedral site [16a] to the total intensity *I*, which is not the case for the reflections (211) and (321). Another good agreement between (*I*M)obs and (*I*M)cal is found in the thermal variations of the superstructure lines (310)**\***, (110)**\***, (411, 330)**\*** and (530, 433)**\*** plotted in Fig. 15 where a rapid variation is observed at 68 K. Two distinct magnetic behaviors separated by the specific temperature 160 K are clearly evidenced.

The refined values of the parameters (*m*j, *m*j) and (θj, θj) listed on Table 8 in the 4.2−283 K temperature range lead to a good agreement between *M*Scal and *M*Smes. During the refinement at 109 ± 2 K, two different results related to the set {a, b} are found and lead to identical values of *M*Scal and reliability factor *R*. The same feature is observed for the set {c, d} at 127 ± 5 K. We observe in the thermal variations of the parameters (*m*j, *m*j) and (θj, θj) plotted in Fig. 16 a broad variation between 54 K and 80 K which disappears beyond 160 K.

Fig. 13. *M*T(*H*) versus *H* at 4.2 K along the <111> and <100> directions (Lahoubi, 2012)

specific temperature 160 K are clearly evidenced.

The equation (5) in the section 3, permit us to deduce the magnetic intensities *I*M by assuming negligible the *T*-variations of *I*N with a rather good precision for the reflections (hkl) which have a great magnetic contribution such as (211), (321), (521) and (532, 611) where the ratio (*I*M/*I*N)cal is equal respectively to: 200, 20, 4 and 2. Consequently, the high magnetically reflections with a small nuclear contribution are only (211) and (321). The thermal variations of these two magnetic reflections which are responsible of the collinear ferrimagnetic ordering are reported in Fig. 14. They appear at *T*N and present a first increase below 160 K with a second rapid increase at 68 K. Different and complex temperature dependences are observed for the reflections (220) and (440). At first, the values of (*I*M)cal are higher than those observed for (*I*M)obs. These two reflections increase simultaneously between 4.2 and 160 K and present an inflexion point near 68 K. Above 160 K, they tend to a plateau until room temperature after which they decrease progressively and reach zero at *T*N. The possible explanation of this characteristic behavior seems to be related to the magnetic contribution of the irons in the octahedral site [16a] to the total intensity *I*, which is not the case for the reflections (211) and (321). Another good agreement between (*I*M)obs and (*I*M)cal is found in the thermal variations of the superstructure lines (310)**\***, (110)**\***, (411, 330)**\*** and (530, 433)**\*** plotted in Fig. 15 where a rapid variation is observed at 68 K. Two distinct magnetic behaviors separated by the

The refined values of the parameters (*m*j, *m*j) and (θj, θj) listed on Table 8 in the 4.2−283 K temperature range lead to a good agreement between *M*Scal and *M*Smes. During the refinement at 109 ± 2 K, two different results related to the set {a, b} are found and lead to identical values of *M*Scal and reliability factor *R*. The same feature is observed for the set {c, d} at 127 ± 5 K. We observe in the thermal variations of the parameters (*m*j, *m*j) and (θj, θj) plotted in Fig. 16 a broad variation between 54 K and 80 K which disappears beyond 160 K.

Fig. 14. Thermal variations of (*I*M)obs and (*I*M)cal for the reflections (211), (321), (220) and (440)

Fig. 15. Thermal variations of (*I*M)obs and (*I*M)cal for the superstructure lines (310)**\***, (110)**\***, (411, 330)**\*** and (530, 433)**\***

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 225

Fig. 16. Thermal variations of the parameters (*m*j, *m*j) and (θj, θj)

K by assuming the free and quenched ion value respectively.

transitions.

and 50–165 K respectively.

The <111> direction ceases to be the easy axis of magnetization and changes below 140 K to the <100> direction up to 4.2 K in Tb0.37Y2.63IG for example with the appearance of the low symmetry phases <uuw> (Lahoubi et al, 2000) in the spontaneous spin reorientation phase

These results are in good agreement with the previous observed rhombohedral distortion below 190 K (Rodić & Guillot, 1990) and 200 K (Sayetat, 1974, 1986). They are also in good agreement with the anomalous behaviors observed previously below 200 K on TbIG single crystals without applied magnetic fields, in the acoustic properties (Kvashnina et al., 1984; Smokotin et al., 1985) and in the elastic constant measurements (Alberts et al., 1988) along the [100], [110] and [111] crystallographic directions, precisely in the temperature ranges, 60–140 K

It seems that the behavior around 160 K has a relation with the previous predicted momentum angular compensation point *T*J (Nelson & Mayer, 1971) localized at 150 and 190

The large magnetodielectric (MD) effects which have been recently revealed on TbIG single crystal at low temperature as well when a very small external magnetic field (*H*ex < 0.2 T) is


Table 8. Values of the parameters *m*j, *m*j, θj and θj in the 4.2−283 K temperature range with a comparison between the calculated *M*Scal and measured *M*Smes magnetizations.

The thermal variations of the parallel (*m*<sup>j</sup> **//**, *m*<sup>j</sup> **//**) and perpendicular (*m*<sup>j</sup> , *m*<sup>j</sup> ) components are also reported in Fig. 17. The double umbrella magnetic structure appears to close slowly around the <111> direction in the three magnetic glide planes c' near 160 K with an abrupt increase between 54 and 68 K. Previous temperature dependences of the calculated non collinear magnetic structure in TbIG (Druzhinina & Shkarubskii, 1988) and the recent neutron scattering on TbIG single crystal, (Louca et al., 2009) are not consistent with these thermal variations. In another recent study of the magnetic and magneto-optical properties of the Tb3+ ions in TbIG and in the mixed system of terbium-yttrium ferrites garnets Tb*x*Y3 *<sup>x</sup>*IG (x < 1) (Zhang et al., 2009) the differences between the two non collinear magnetic structures which exist at low temperature were not taken account in their calculations.

*<sup>T</sup>*(K) *m*1(μB)

4.2 8.18

<sup>5</sup>8.07

<sup>20</sup>7.49

<sup>54</sup>4.79

<sup>68</sup>4.23

<sup>80</sup>3.57

109 ± 2

127 ± 5

208 ± 2

244 ± 10 *m*'1(μB)

8.90

8.99

8.77

6.27

5.21

4.72

2.76 **<sup>a</sup>** 3.79

3.78

2.59 **<sup>c</sup>** 3.12

3.00

2.31

1.74(3) 1.75(7)

1.34(4) 1.35(8)

1.15

The thermal variations of the parallel (*m*<sup>j</sup>

<sup>160</sup>2.30

<sup>283</sup>1.15

θ1(°)

30.79

32.00

30.11

22.57

19.57

16.72

15.04

15.23 11.20

14.94

14.52 13.08

0.11

0.05

0.08

0

Sites C1 ; C<sup>1</sup> *M*Scal *M*Smes (μB/mol)

<sup>θ</sup>'1(°) [111] [110] [100] [111] [110] [100]

28.07 34.64 28.28 20.00 34.53 28.35 20.17

27.00 34.56 28.22 19.95 " " "

25.39 33.21 27.12 19.18 33.34 27.21 19.56

17.03 21.32 17.41 12.31 21.24 17.24 12.80

15.76 17.10 13.97 9.87 17.01 13.44 10.05

12.56 14.25 11.63 8.23 13.97 11.70 8.38

0.00 4.65 3.80 2.68 4.61 3.76 2.66

0.00 1.92 1.56 1.11 1.97 1.36 1.12

0.00 0 0 0 0 0 0

<sup>0</sup>0.66 0.54 0.38 0.79 0.59 0.63

**//**) and perpendicular (*m*<sup>j</sup> , *m*<sup>j</sup> ) components

10.86 9.48 7.74 5.47 8.78 7.20 5.26 2.78 **<sup>b</sup>**

12.30 7.13 5.82 4.12 6.52 5.40 3.88 2.71**<sup>d</sup>**

Table 8. Values of the parameters *m*j, *m*j, θj and θj in the 4.2−283 K temperature range with a

are also reported in Fig. 17. The double umbrella magnetic structure appears to close slowly around the <111> direction in the three magnetic glide planes c' near 160 K with an abrupt increase between 54 and 68 K. Previous temperature dependences of the calculated non collinear magnetic structure in TbIG (Druzhinina & Shkarubskii, 1988) and the recent neutron scattering on TbIG single crystal, (Louca et al., 2009) are not consistent with these thermal variations. In another recent study of the magnetic and magneto-optical properties of the Tb3+ ions in TbIG and in the mixed system of terbium-yttrium ferrites garnets Tb*x*Y3 *<sup>x</sup>*IG (x < 1) (Zhang et al., 2009) the differences between the two non collinear magnetic structures which exist at low temperature were not taken account in their calculations.

**//**, *m*<sup>j</sup>

comparison between the calculated *M*Scal and measured *M*Smes magnetizations.

Fig. 16. Thermal variations of the parameters (*m*j, *m*j) and (θj, θj)

The <111> direction ceases to be the easy axis of magnetization and changes below 140 K to the <100> direction up to 4.2 K in Tb0.37Y2.63IG for example with the appearance of the low symmetry phases <uuw> (Lahoubi et al, 2000) in the spontaneous spin reorientation phase transitions.

These results are in good agreement with the previous observed rhombohedral distortion below 190 K (Rodić & Guillot, 1990) and 200 K (Sayetat, 1974, 1986). They are also in good agreement with the anomalous behaviors observed previously below 200 K on TbIG single crystals without applied magnetic fields, in the acoustic properties (Kvashnina et al., 1984; Smokotin et al., 1985) and in the elastic constant measurements (Alberts et al., 1988) along the [100], [110] and [111] crystallographic directions, precisely in the temperature ranges, 60–140 K and 50–165 K respectively.

It seems that the behavior around 160 K has a relation with the previous predicted momentum angular compensation point *T*J (Nelson & Mayer, 1971) localized at 150 and 190 K by assuming the free and quenched ion value respectively.

The large magnetodielectric (MD) effects which have been recently revealed on TbIG single crystal at low temperature as well when a very small external magnetic field (*H*ex < 0.2 T) is

Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 227

In this chapter, the temperature evolution of the magnetic structure in TbIG is studied by neutron diffraction experiments below *T*N (568 K). The "double umbrella" structure observed at 5 K appears below a specific temperature (~ 160 K) which could be related to the previous predicted *T*J-point situated in the 150–190 K temperature range. The rapid variation of the Tb3+ moments is observed between 54 and 68 K where the predicted value (58 K) of the *T*B-point is located. The magnetic symmetry doesn't change with the temperature and the magnetic space group is R 3 c'. The author is convinced that the symmetry considerations of the Representation Analysis of Bertaut presented in this chapter have demonstrated their usefulness in the determination of the thermal variation of the double umbrella magnetic structure in TbIG below *T*N. It is hoped that these results which are in good agreement with the magnetization measurements will facilitate a better understanding of the possible correlations between the magnetic properties via the double umbrella magnetic structure and the recent ME and MD effects found in this ferrite garnet.

This work is dedicated to the Academician Dr. E F Lewy Bertaut, the father "*on group theoretical techniques in magnetic structure analysis*", who died in 2003. Some of unpublished results presented here constitute a part of my thesis. I am indebted to Dr. M Guillot (LNCMI, Grenoble) for his experimental contribution as supervisor and for providing the samples. We thank Dr. F Tchéou (UJF, Grenoble) for his helpful discussions during the refinement of the neutron diffraction patterns. Many acknowledgements are also addressed to Dr. B Ouladdiaf (ILL, Grenoble) for his assistance during the recent D1B experiments.

Alberts, H. L.; Palmer, S. B. & Patterson, C. (1988). *J. Phys. C: Solid State Phys.,* Vol. 21, pp.

Bertaut, E. F.; Forrat, F.; Herpin, A. & Mériel, P. (1956b). *C. R. Acad. Sci.*, *Paris,* Vol. 243, pp.

Bertaut, E. F. (1963). Spin Configurations of Ionic Structures: Theory and Practice, In: *Treatise on Magnetism*, Suhl & Rado, Vol. III, Chap. 4, pp. 149-209, Acad. Press, New York

Bertaut, E. F.; Sayetat, F. & Tchéou, F. (1970). *Solid State Commun.,* Vol. 8, No.4, pp. 239-245

Bacon, G. E. (1975). *Neutron Diffraction*, (Third Edition), Clarendon Press, Oxford, UK

Bertaut, E. F. & Forrat, F. (1956a). *C. R. Acad. Sci.*, *Paris,* Vol. 242, pp. 382-384

Bertaut, E. F. (1971). *J. Phys., Colloque C1*, Suppl. No. 2-3, t. 32, pp. C1. 462-470

Bertaut, E. F. (1997). *J. Phys. IV France* 7, *Colloque C1*, Suppl. J. Phys III, pp. C1.11-26 Blume, M.; Freeman, A. J. & Watson, R. E. (1962). *J. Chem. Phys.,* Vol. 37, pp. 1245-1253 Bonnet, M. (1976). *Thesis, Doct. of Sciences in Physics*, Grenoble University, France

**7. Conclusion** 

**8. Acknowledgements** 

**9. References** 

271-275

898-901

Bacon, G. E. (1972). *Acta. Cryst. A,* Vol. 28, pp. 357-358

Belov, K. P. (1996). *Phys. Usp.,* Vol. 39, No. 6, pp. 623-634

Bertaut, E. F. (1968). *Acta Cryst. A,* Vol. 24, No.1, pp. 217-231

Bertaut, E. F. (1981). *J. Magn. Magn. Mater.,*Vol. 24, pp. 267-278

Bertaut, E. F. (1972). *Ann. Phys.,* t. 7, pp. 203-232

Fig. 17. Thermal variations of the parameters (*m*<sup>j</sup> **//**, *m*<sup>j</sup> ) and (*m*<sup>j</sup> **//**, *m*<sup>j</sup> )

applied (Hur et al., 2005) could be combined with the previous huge spontaneous magnetostriction measurements (Sayetat, 1974, 1986; Guillot et al., 1980) where a peak near 70 K has been observed in the thermal variations of *λ*ε, 2(Tb3+)exp/*λ*ε, 2(Tb3+)cal, the ratio of the experimental values of the magnetostriction constant to the theoretical values derived from the one ion model. It correspond to the abrupt change in the long-range magnetic order in the Tb3+ sublattice near the previous predicted low-temperature point *T*B = 58 K (Belov, 1996 and refs. herein) which is situated between 54 and 68 K in this study. More recently, some magnetoelectric (ME) and MD effects in weak and high external magnetic fields (*H*ex up to 2 T and *H*ex up to 10 T respectively) have been reported (Kang et al., 2010). A possible coupling between the magnetic exchange and the ligand-field excitations which occurs at a specific temperature situated between 60 and 80 K has been discovered without external magnetic field (*H*ex = 0) with two distinct behaviors above and below another characteristic temperature (~ 150 K). All results confirm that Landau's theory of second order phase transitions does not apply to TbIG in the 5 K–*T*N temperature range without applied external magnetic field and the magnetic space group is R 3 c' (Bertaut, 1997; Lahoubi et al., 1997).

### **7. Conclusion**

226 Neutron Diffraction

**//**, *m*<sup>j</sup> ) and (*m*<sup>j</sup>

applied (Hur et al., 2005) could be combined with the previous huge spontaneous magnetostriction measurements (Sayetat, 1974, 1986; Guillot et al., 1980) where a peak near 70 K has been observed in the thermal variations of *λ*ε, 2(Tb3+)exp/*λ*ε, 2(Tb3+)cal, the ratio of the experimental values of the magnetostriction constant to the theoretical values derived from the one ion model. It correspond to the abrupt change in the long-range magnetic order in the Tb3+ sublattice near the previous predicted low-temperature point *T*B = 58 K (Belov, 1996 and refs. herein) which is situated between 54 and 68 K in this study. More recently, some magnetoelectric (ME) and MD effects in weak and high external magnetic fields (*H*ex up to 2 T and *H*ex up to 10 T respectively) have been reported (Kang et al., 2010). A possible coupling between the magnetic exchange and the ligand-field excitations which occurs at a specific temperature situated between 60 and 80 K has been discovered without external magnetic field (*H*ex = 0) with two distinct behaviors above and below another characteristic temperature (~ 150 K). All results confirm that Landau's theory of second order phase transitions does not apply to TbIG in the 5 K–*T*N temperature range without applied external magnetic field and

**//**, *m*<sup>j</sup> )

Fig. 17. Thermal variations of the parameters (*m*<sup>j</sup>

the magnetic space group is R 3 c' (Bertaut, 1997; Lahoubi et al., 1997).

In this chapter, the temperature evolution of the magnetic structure in TbIG is studied by neutron diffraction experiments below *T*N (568 K). The "double umbrella" structure observed at 5 K appears below a specific temperature (~ 160 K) which could be related to the previous predicted *T*J-point situated in the 150–190 K temperature range. The rapid variation of the Tb3+ moments is observed between 54 and 68 K where the predicted value (58 K) of the *T*B-point is located. The magnetic symmetry doesn't change with the temperature and the magnetic space group is R 3 c'. The author is convinced that the symmetry considerations of the Representation Analysis of Bertaut presented in this chapter have demonstrated their usefulness in the determination of the thermal variation of the double umbrella magnetic structure in TbIG below *T*N. It is hoped that these results which are in good agreement with the magnetization measurements will facilitate a better understanding of the possible correlations between the magnetic properties via the double umbrella magnetic structure and the recent ME and MD effects found in this ferrite garnet.

### **8. Acknowledgements**

This work is dedicated to the Academician Dr. E F Lewy Bertaut, the father "*on group theoretical techniques in magnetic structure analysis*", who died in 2003. Some of unpublished results presented here constitute a part of my thesis. I am indebted to Dr. M Guillot (LNCMI, Grenoble) for his experimental contribution as supervisor and for providing the samples. We thank Dr. F Tchéou (UJF, Grenoble) for his helpful discussions during the refinement of the neutron diffraction patterns. Many acknowledgements are also addressed to Dr. B Ouladdiaf (ILL, Grenoble) for his assistance during the recent D1B experiments.

### **9. References**


Temperature Evolution of the Double Umbrella Magnetic Structure in Terbium Iron Garnet 229

Kang, T. D.; Standard, E.; Ahn, K. H.; Sirenko, A. A.; Carr, G. L.; Park, S.; Choi, Y. J.;

Kazei, Z. A.; Kolmakova, N.P.; Novak, P. & Sokolov, V. I. (1991). Magnetic Properties of

Kvashnini, O. P.; Kapitonov, A. M.; Smokotin, É. M. & Titova, A. G. (1984). *Sov. Phys. Solid* 

Lahoubi, M.; Guillot, M.; Marchand, A.; Tchéou, F. & Roudaut, E. (1984). *I.E.E.E. Trans. on* 

Lahoubi, M.; Guillot, M.; Marchand, A.; Tchéou, F. & Le Gall, H. (1985). High Magnetic-

(Ed.), pp. 275-282, ISSN 0730-9546, San Francisco, USA, Oct.31-Nov.2, 1984 Lahoubi, M. (1986). *Thesis*, *Doct. of Sciences in Physics*, Grenoble University, France, pp.1-245 Lahoubi, M.; Fillion, G. & Tchéou, F. (1997). *J. Phys. IV France* 7, *Colloque C1*, Suppl. J. Phys

Lahoubi, M.; Younsi, W.; Soltani, M.-L.; Voiron, J. & Schmitt, D. (2009). *J. Phys.: Conf. Ser.,*

Lahoubi, M.; Younsi, W.; Soltani, M.-L. & Ouladdiaf, B. (2010). *J. Phys.: Conf. Ser.,* Vol. 200,

Lahoubi, M. (2012). Symmetry Analysis of the Magnetic Structures in TbIG and Tb:YIG at

Levitin, R. Z.; Markosyan, A. S. & Orlov, V. N. (1983). *Sov. Phys. Solid State,* Vol. 25, (No. 6),

Nathans, R.; Shull, C. G.; Shirane, G. & Andresen, A. (1959). *J. Phys. Chem. Solids*, Vol. 10,

Pauthenet, R. (1958b). *Thesis Doct. of Sciences in Physics*, Grenoble University, France, pp.1-39

Roudaut, E. (1983). Evolution of Position-Sensitive Detectors for Neutron Diffraction

Experiments From 1966 to 1982 in the Nuclear Centre of Grenoble, In: *Position-Sensitive Detection & Thermal Neutrons*, pp. 294-301, Academic Press Inc., ISBN: 0-12-

Louca, D.; Kamazawa, K & Proffen, T. (2009). *Phys*. *Rev*. *B*, Vol. 80, 214406(6 pages)

Nelson T. J. & Mayer D.C. (1971). *I.E.E.E. Trans. on Magn.,* Vol. 7, (No. 3), pp. 616-617 Nekvasil, V. & Veltrusky, I. (1990). *J. Magn. Magn. Mater.,* Vol. 86, pp. 315-325

Pickart, S. J.; Halperin, H. A. & Clark, A. E. (1970). *J. Appl. Phys*., Vol. 41, pp.1192-1193

Sayetat, F. (1974). *Thesis, Doct. of Sciences in Physics*, Grenoble University, France

Low Temperature, In: *J. Phys.: Conf. Ser.,* Vol. 340, 012068(10 pages) *Proceedings of the 5th European Conf. on Neutron Scattering*, Prague, Czech Republic, July 17-22, 2011

Lahoubi, M.; Kihal, A. & Fillion, G. (2000). *Physica B*, Vol. 284-288, pp. 1503-1504

(No.1), 014414 (7 pages)

edition, Berlin Heidelberg

III, pp. C1. 291-292

082018(4 pages)

pp. 1074-1075

(No. 2-3), pp. 138-146

186180-5, London, UK

Néel, L. (1948). *Ann. Phys.* Paris, Vol. 3, pp. 137-198

Pauthenet, R. (1958a). *Ann. Phys*., Paris, Vol. 3, pp. 424

Pearson, R. F. (1965). *Proc. Phys. Soc.,*Vol. 86, pp. 1055-1066

Sayetat, F. (1986). *J. Magn. Magn. Mater.,* Vol. 58, pp. 334-346 Shull, C. G. & Smart, J. S. (1949). *Phys. Rev*. Vol. 76, pp. 1256-1257

Rodić, D. & Guillot, M. (1990). *J. Magn. Magn. Mater.,* Vol. 86, pp. 7-12

Vol. 150, 042108(4 pages)

*State,* Vol. 26, (No.8), pp. 1458-1459

*Magn.,* Vol. MAG-20, (No. 5), pp. 1518-1520

Ramazanoglu, M.; Kiryukhin, V. & Cheong, S.-W. (2010). *Phys. Rev. B,* Vol. 82,

Non-Metallic Inorganic Compounds Based on Transition Elements, In: *Landolt-Börnstein Group III*, H. P. J. Wijn (Ed.), Vol. 27, ISBN 3-540-53963-8, Springer Verlag

Field Magnetization in Terbium Iron Garnet (TbIG), In: *Advances in Ceramics, Proceedings of the Fourth Inter. Conf. on Ferrites (ICF4)*, Vol. 15, Part I, Wang F. F. Y.


Bonnet, M.; Delapalme, A.; Fuess, H. & Becker, P. (1979). *J. Phys. Chem. Solids*, Vol. 40, (No.

Chatterji, T. (2006). Editor, *Neutron Scattering from Magnetic Materials*, Elsevier B.V., pp. 1-559 Convert, P.; Fruchart, D.; Roudaut, E. & Wolfers, P. (1983). 12 Years of Life with Bananas

Ferey, G.; De Pape, R.; Le Blanc, M. & Pannetier, J. (1986). *Rev. Chim. Miner.,* Vol. 23, pp.

Flury Jr, R. L. (1980). *Symmetry Groups: Theory and Chemical Applications*, Prentice-Hall, Inc.,

Fuess, H.; Bassi, G.; Bonnet, M. & Delapalme, A. (1976). *Solid State Commun.*, Vol. 18, (No. 5)

Geller, S. (1978). Crystal and Static Magnetic Properties of Garnets, In: *Proc. of the* 

Guillot, M. & du Tremolet de Lacheisserie, E. (1980). *Z. Phys. B- Condensed Matter,* Vol. 39,

Guillot, M.; Marchand, A.; Tchéou, F & Feldmann, P (1982). *J. Appl. Phys.,* Vol. 53, (No.3), pp.

Guillot, M.; Tchéou, F.; Marchand, A. & Feldmann, P. (1983).*J. Magn. Magn. Mater.,*Vol. 31-

Guillot, M.; Tchéou, F.; Marchand, A. & Feldmann, P. (1984). *Z. Phys. B - Condensed Matter,* 

Guillot, M. (1994). Magnetic Properties of Ferrites, In: *Materials Science and Technology: A* 

Hahn, Th. (1983). (Ed.) Space Group Symmetry, Vol. A, In: *International Tables for* 

Hock, R.; Fuess, H.; Vogt, T & Bonnet, M. (1990). *J. of Solid State Chem.,*Vol. 84, (No. 1) pp.39-

Hock, R.; Fuess, H.; Vogt, T & Bonnet, M. (1991). *Z. Phys. B-Cond. Matter,* Vol. 82, pp. 283-294 Hong, Y. J.; Kum, J.S.; Shim, I. B. & Kim, C. S. (2004). *I.E.E.E. Trans. on Magn*., Vol. 40, (No.

Hur, N.; Park, S.; Guha, S.; Borissov, A.; Kiryukhin, V. & Cheong, S.-W. (2005). *Appl*. *Phys*.

Kahan, T. (1972). *Théorie des Groupes en Physique Classique et Quantique*, tome 3, Dunod, Paris

*Comprehensive Treatment,* Cahn, Haasen & Kramer, (editors), Vol. 3B, Electronic and Magnetic Properties of Metals and Ceramics, Part II, VCH Publishers Inc., Chapter 8, K. H. J. Buschow (Ed.), pp. 7–92, VCH Verlagsgesellschaft mbH, VCH Publishers

*Crystallography*, Published by the IUCr, Dordrecht Reidel Boston, London, kluwer

Geller, S. & Gilleo, M. A. (1957b). *J. Phys. Chem. Solids*, Vol. 3, (No.1-2), pp. 30-36 Gonano, R.; Hunt, E. & Meyer, H. (1967). *Phys. Rev.,* Vol. 156, pp. 521-533

*International School of Physics "Enrico Fermi", 1977, Course LXX, Physics of magnetic Garnets*, A. Paoletti (Ed.), pp. 1-55, North-Holland Publishing Co, Amsterdam

(Curved One-Dimensional Neutron PSDs), In: *Position-Sensitive Detection & Thermal Neutrons*, pp. 302-309, Academic Press Inc., ISBN: 0-12-186180-5, London, UK Druzhinina, R. F. & Shkarubskii, V.V. (1988). *Sov. Phys. Solid State,* Vol. 30, (No. 2), pp. 342-

11), pp. 863-876

343

474-484

pp. 557-562

pp. 109-114

2719-2721

51

34, pp. 631-632

*V*ol. 56, pp. 29-39

4), pp. 2808-2810

Inc. Weinheim, New York, USA

academic publisher's edition

Herpin, A. (1968). *Théorie du Magnétisme*, Puf, Paris, France

*Lett.,* Vol. 87, (No.4), 042901(3 pages)

Herpin, A.; Koehler, W. & Meriel, P. (1960). *C. R. Acad. Sci.*, Vol. 251, p. 1359

Englewood Cliffs, New-Jersey 07632

Geller, S. & Gilleo, M. A. (1957a). *Acta. Cryst.*, Vol. 10, p. 239


**0**

**11**

*Japan*

Yuzuru Miyazaki

**Superspace Group Approach to the Crystal**

*Department of Applied Physics, Graduate School of Engineering, Tohoku University*

Thermoelectric (TE) materials directly convert waste heat into electricity based on the Seebeck effect. This process itself yields no extra gas, noise, or vibration, and it is thus recognized as a clean power generator for next decades. Currently, large amounts of waste heat, ranging from *<*100 ◦C (PCs, TVs, etc.) to ∼1000 ◦C (power stations, incinerators, etc.), are emitted into the environment, but some of which can be recovered as electricity by simply placing TE materials on the waste heat sources. Of the waste heat, exhaust gases from automobiles account for a total energy of 460 Pcal (4.6 <sup>×</sup>1017 cal) per year in Japan (Terasaki, 2003) and if we can recover 10% of this energy, the total generated electricity would equate to that of a

TE generators usually consist in series of ∼100-pair p-n junctions of TE materials. The performance of TE materials is commonly evaluated by the "figure-of-merit" using the Seebeck coefficient *S*, electrical conductivity *σ*, and thermal conductivity *κ* as *Z* = *S*2*σ*/*κ*. We also use the dimensionless figure-of-merit *ZT* (*T* the absolute temperature) and the "power factor" given by *S*2*σ*. For good characteristics in a TE material, a large *S* and *σ* as well as a small *κ* are necessary although all three parameters are dependent on carrier concentration and hence are correlated. The best TE materials are to be found in doped-semiconductors as the Seebeck coefficient is significantly smaller in conventional metals. A *ZT* value larger than unity is regarded as a measure of practical application because it roughly corresponds to a thermal-to-electric conversion efficiency of *η*∼10 %. However, *η>*10 % can be achieved at

Since the 1960's, higher manganese silicides (HMSs) have been extensively studied as potential p-type thermoelectric materials both in Russia and Japan (Nikitin et al., 1969; Nishida, 1972). The compounds exhibit *ZT* = 0.3-0.7 at around 800 K (Fedorov & Zaitsev, 2006), but different structure formulae, e.g., Mn4Si7 (Gottlieb et al., 2003), Mn11Si19 (Schwomma et al., 1963; 1964), Mn15Si26 (Flieher et al., 1967; Knott et al., 1967), Mn27Si47(Zwilling & Nowotny, 1973), were proposed as HMS phases. Figure 1 shows the crystal structures (Momma & Izumi, 2008) of the first three of these; all three have great resemblance apart from the *c*-axis length. Until recently, controversy existed as to whether the compounds were an identical phase or a series of phases with different structures. The existence of Mn7Si12, Mn19Si33 and Mn39Si68 was also reported but these phases were

**1. Introduction**

typical thermal power station.

higher operating temperatures above 600 K even if *ZT<* 1.

**Structure of Thermoelectric Higher**

**Manganese Silicides MnSi***<sup>γ</sup>*


Watson, R. E. & Fremann, A. J. (1961). *Acta. Cryst.*, Vol. 14 pp. 27-37

Wills, A. S. (2007). *Zeitschrift f ür Kristallographie,* Suppl., Vol. 26, pp. 53-58

