**Superspace Group Approach to the Crystal Structure of Thermoelectric Higher Manganese Silicides MnSi***<sup>γ</sup>*

Yuzuru Miyazaki

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

### **1. Introduction**

230 Neutron Diffraction

Smokotin, É. M., Kvashnini O. P. & Kapitonov, A. M. (1985). *Phys. Status Solidi (a)* Vol. 87, K

Tchéou, F.; Bertaut, E. F.; Delapalme, A.; Sayetat, F. & Fuess, H. (1970a). *Colloque International* 

Wolf, W. P.; Ball, M.; Hutchings, M. T.; Leask, M. J. M. & Wyatt, A. F. G. (1962). *J. Phys. Soc.* 

Wolf, W. P. (1964). Local Anisotropy in Rare earth Garnets, In: *Proceedings of the International* 

Zhang, G.-y.; Wei, M.; Xia, W.-s. & Yang. G. (2009). *J. Magn. Magn. Mater.,*Vol. 321, pp. 3077-

Zvezdin, A. K. (1995). Field Induced Phase Transitions in Ferrimagnets, In: *Handbook of* 

*Magnetic Materials*, K. H. J. Buschow (Ed.), 4, Vol. 9, pp. 405-543, Elsevier Science,

*C.N.R.S.* ''*Les Eléments des Terre Rares''* 180, Vol. II, pp. 313-332 Tchéou, F.; Bertaut, E. F. & Fuess, H. (1970b). *Solid State Commun.*, Vol. 8, pp. 1751-1758 Tchéou, F.; Fuess, H. & Bertaut, E. F. (1970c). *Solid State Commun.*, Vol. 8, pp. 1745-1749

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

*Conference on Magnetism* (ICM) Nottingham, pp. 555-560

*Japan,* Vol. 17, Suppl., B-I, pp. 443-448

pp. 53-56

3079

Amsterdam

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 typical thermal power station.

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

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

al (Miyazaki, 2008) succeeded in determining the detailed modulated structure using this superspace group approach. In this chapter, we describe the method by which the structure

<sup>233</sup> Superspace Group Approach to the Crystal Structure

Suppose we have a crystal consisting of two tetragonal basic unit cells (i.e., subsystems) of [Mn] and [Si] with a common *a*-axis but different *c*-axes, *c*Mn and *c*Si as shown in Fig. 2. If the ratio *c*Mn:*c*Si can be expressed as simple integers, such as 2:1, 3:2, etc., the whole crystal structure can be represented by a three dimensional (3D) unit cell with *c* = *m*×*c*Mn = *n*×*c*Si (*m* and *n* are integers) and we no longer need to assume two basic units. The three HMSs shown in Fig. 1 are examples of such commensurate structures of Mn*m*Si*n*, although rather complicated ones. In contrast, if the *c*Mn:*c*Si ratio is irrational, we then have to assume a unit cell with an infinite length along the *c*-axis. In such cases, the number of atomic sites grows too large and structural analysis becomes practically impossible. The concept of superspace group is the best way to accurately describe such a structure (De Wolff, 1974; Janner & Janssen,

[Mn] subsystem [Si] subsystem

Fig. 2. Two tetragonal subsystems of [Mn] (left) and [Si] (right) with a common *a*-axis but

To apply the superspace group approach, we need to introduce a (3+1)-dimensional unit vectors, **as1**, **as2**, **as3**, and **as4**. Using the 3D unit bases and another **d**, perpendicular to this 3D

⎞

⎛

**a b c d** ⎞

⎟⎟⎠

(1)

⎜⎜⎝

⎟⎟⎠

cMn cSi

of such a composite crystal is determined based on the superspace group formalism.

**2. Superspace group approach**

of Thermoelectric Higher Manganese Silicides MnSi*γ*

a

**2.2 General description of composite crystals**

space, the relationship between these basis sets can be defined as, ⎛

⎜⎜⎝

**as1 as2 as3 as4** ⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

where **k** = (*αβγ*) is a modulation vector to index the electron diffraction patterns. For the HMSs, the modulation exists only in the *c*-axis, i.e., **k** = (00*γ*) with *γ* = *c*Mn/*c*Si. From equation 1, both the **as1**- and **as2**-axes lie in the 3D physical space **R3** and the **as4**-axis is perpendicular to **R3**, as shown in Fig. 3. Similar to the unit bases, fractional coordinates in the (3+1)D space,

b

different *c*-axis lengths, *c*Mn and *c*Si.

c

**2.1 Backgrounds**

1980).

only recognized in a microscopic domain observed using transmission microscopy (Ye & Amelinckx, 1986). In contrast to the existence of several phases, there is only one line compound, Mn11Si19, in the Mn-Si binary phase diagram near the corresponding composition (Okamoto, 1991).

Fig. 1. Crystal structure of commensurate HMSs; Mn4Si7 (*c*∼ 1.75 nm), Mn11Si19 (*c*∼ 4.81 nm) and Mn15Si26 (*c*∼ 6.53 nm).

Based on the observed electron diffraction patterns, Ye and Amelinckx (Ye & Amelinckx, 1986) proposed that the HMS phases are composed of two tetragonal basic units of Mn and Si, with an identical *a*-axis but different *c*-axes. With this idea, a variety of complicated superlattice reflections can be well indexed based on the two basic units distinguished by *c*-axis lengths, *c*Mn and *c*Si. Yamamoto (Yamamoto, 1993) employed the concept of superspace group (De Wolff, 1974) to appropriately describe the detailed crystal structure of HMSs. He proposed that the HMSs belong to a family of "composite crystals", consisting of two tetragonal subsystems of [Mn] and [Si]. By means of neutron diffraction data, Miyazaki et al (Miyazaki, 2008) succeeded in determining the detailed modulated structure using this superspace group approach. In this chapter, we describe the method by which the structure of such a composite crystal is determined based on the superspace group formalism.

### **2. Superspace group approach**

### **2.1 Backgrounds**

2 Will-be-set-by-IN-TECH

only recognized in a microscopic domain observed using transmission microscopy (Ye & Amelinckx, 1986). In contrast to the existence of several phases, there is only one line compound, Mn11Si19, in the Mn-Si binary phase diagram near the corresponding composition

Mn4Si7 Mn11Si19 Mn15Si26

Fig. 1. Crystal structure of commensurate HMSs; Mn4Si7 (*c*∼ 1.75 nm), Mn11Si19 (*c*∼ 4.81

Based on the observed electron diffraction patterns, Ye and Amelinckx (Ye & Amelinckx, 1986) proposed that the HMS phases are composed of two tetragonal basic units of Mn and Si, with an identical *a*-axis but different *c*-axes. With this idea, a variety of complicated superlattice reflections can be well indexed based on the two basic units distinguished by *c*-axis lengths, *c*Mn and *c*Si. Yamamoto (Yamamoto, 1993) employed the concept of superspace group (De Wolff, 1974) to appropriately describe the detailed crystal structure of HMSs. He proposed that the HMSs belong to a family of "composite crystals", consisting of two tetragonal subsystems of [Mn] and [Si]. By means of neutron diffraction data, Miyazaki et

Mn Si

a b

c

nm) and Mn15Si26 (*c*∼ 6.53 nm).

(Okamoto, 1991).

Suppose we have a crystal consisting of two tetragonal basic unit cells (i.e., subsystems) of [Mn] and [Si] with a common *a*-axis but different *c*-axes, *c*Mn and *c*Si as shown in Fig. 2. If the ratio *c*Mn:*c*Si can be expressed as simple integers, such as 2:1, 3:2, etc., the whole crystal structure can be represented by a three dimensional (3D) unit cell with *c* = *m*×*c*Mn = *n*×*c*Si (*m* and *n* are integers) and we no longer need to assume two basic units. The three HMSs shown in Fig. 1 are examples of such commensurate structures of Mn*m*Si*n*, although rather complicated ones. In contrast, if the *c*Mn:*c*Si ratio is irrational, we then have to assume a unit cell with an infinite length along the *c*-axis. In such cases, the number of atomic sites grows too large and structural analysis becomes practically impossible. The concept of superspace group is the best way to accurately describe such a structure (De Wolff, 1974; Janner & Janssen, 1980).

Fig. 2. Two tetragonal subsystems of [Mn] (left) and [Si] (right) with a common *a*-axis but different *c*-axis lengths, *c*Mn and *c*Si.

#### **2.2 General description of composite crystals**

To apply the superspace group approach, we need to introduce a (3+1)-dimensional unit vectors, **as1**, **as2**, **as3**, and **as4**. Using the 3D unit bases and another **d**, perpendicular to this 3D space, the relationship between these basis sets can be defined as,

$$
\begin{pmatrix}
\mathbf{a\_{s1}} \\
\mathbf{a\_{s2}} \\
\mathbf{a\_{s3}} \\
\mathbf{a\_{s4}}
\end{pmatrix} = \begin{pmatrix}
1 & 0 & 0 & -\alpha \\
0 & 1 & 0 & -\beta \\
0 & 0 & 1 & -\gamma \\
0 & 0 & 0 & 1
\end{pmatrix} \begin{pmatrix}
\mathbf{a} \\
\mathbf{b} \\
\mathbf{c} \\
\mathbf{d}
\end{pmatrix} \tag{1}
$$

where **k** = (*αβγ*) is a modulation vector to index the electron diffraction patterns. For the HMSs, the modulation exists only in the *c*-axis, i.e., **k** = (00*γ*) with *γ* = *c*Mn/*c*Si. From equation 1, both the **as1**- and **as2**-axes lie in the 3D physical space **R3** and the **as4**-axis is perpendicular to **R3**, as shown in Fig. 3. Similar to the unit bases, fractional coordinates in the (3+1)D space,

classified under the 3D space group of *I*41/*amd*. For the second subsystem [Si], originally

<sup>235</sup> Superspace Group Approach to the Crystal Structure

Si atoms are at the origin and the equivalent positions generated by the *P*4/*nnc* space group. According to Yamamoto (Yamamoto, 1993), the most appropriate (3+1)D superspace group is designated as *P* : *I*41/*amd* : 11*ss* (Mn) *W* : *P*4/*nnc* : *q*1*q*1 (Si), or based on the modulation

Based on the adopted superspace group, the translation parts for the [Mn] subsystem are

The translation parts and the symmetry operation for the [Si] subsystem can be obtained by

Polycrystalline samples were prepared in a tetra-arc-type furnace under an Ar atmosphere using tungsten electrodes and a water-cooled copper hearth. Appropriate amounts of Mn (99.9%) and Si (99.99%) powders were mixed in an alumina mortar and pressed into pellets. The pellets were melted four times, and turned over each time to obtain full homogeneity. As the satellite reflections were observed up to eighth order in the electron diffraction patterns, positional modulation of the atomic sites was introduced taken to the eighth order of cosine

Figure 4 shows the observed, calculated, and difference profiles of the neutron diffraction (ND) data for MnSi*<sup>γ</sup>* at 295 K. The ND data were collected using a Kinken powder diffractometer for high efficiency and high resolution measurements, HERMES (Petricek et al., 2000), at the Institute for Materials Research (IMR), Tohoku University, installed at the JRR-3M reactor of the Japan Atomic Energy Agency (JAEA) at Tokai. A monochromatized incident neutron beam at *λ* = 1.8265 Å was used. The ND data were analyzed with the JANA2000

and sine components of the Fourier terms, *An* and *Bn* (*n*= 0-8), of equation 3.

<sup>√</sup><sup>2</sup> <sup>×</sup> *<sup>c</sup>*Si, the *<sup>a</sup>*-axis length is adjusted to that of [Mn]. The

having dimensions of *a*/

+(0, 0, 0, 0); +(1/2, 1/2, 1/2, 0),

(ii) -*y*, *x*+1/2, *z*+1/4, *t*; (iii) -*x*, *y*, *z*, *t*+1/2; (iv) -*x*, -*y*, *z*, *t*;

(v) -*y*, -*x*+1/2, *z*+1/4, *t*+1/2; (vi) *y*, -*x*+1/2, *z*+1/4, *t*; (vii) *x*, -*y*, *z*, *t*+1/2;

(viii) *y*, *x*+1/2, *z*+1/4, *t*+1/2; (ix) -*x*, -*y*+1/2, -*z*+1/4, -*t*;

(xi) *x*, -*y*+1/2, -*z*+1/4, -*t*+1/2; (xii) *x*, *y*+1/2, -*z*+1/4, -*t*; (xiii) *y*, *x*, -*z*, -*t*+1/2; (xiv) -*y*, *x*, -*z*, -*t*;

(xv) -*x*, *y*+1/2, -*z*+1/4, -*t*+1/2;

(xvi) -*y*, -*x*, -*z*, -*t*+1/2.

**2.4 Structural modulation**

with symmetry operations represented as:

expressed as:

(i) *x*, *y*, *z*, *t*;

(x) *y*, -*x*, -*z*, -*t*;

<sup>√</sup><sup>2</sup> <sup>×</sup> *<sup>a</sup>*/

of Thermoelectric Higher Manganese Silicides MnSi*γ*

vector **k** = (00*γ*) more conveniently as *I*41/*amd*(00*γ*)00*ss*.

simply interchanging the third and the fourth components.

*xs*1, *xs*2, *xs*<sup>3</sup> and *xs*<sup>4</sup> can be defined as,

$$
\begin{pmatrix}
\overline{\pi}\_{S1} \\
\overline{\pi}\_{S2} \\
\overline{\pi}\_{S3} \\
\overline{\pi}\_{S4}
\end{pmatrix} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\alpha & \beta & \gamma & 1
\end{pmatrix} \begin{pmatrix}
\overline{\pi} \\
\overline{\nu} \\
\overline{\pi} \\
t
\end{pmatrix} \tag{2}
$$

where the parameter *t*, called as an internal coordinate, represents the distance from **R3** and is related to the fourth coordinate *xs*<sup>4</sup> by *xs*<sup>4</sup> = *γz* + *t*.

Fig. 3. A unit cell in the (3+1)-dimensional superspace and the relationship between the coordinates without modulation, *xsi*, with modulation *xsi* (*i* = 1-4), and internal coordinate *t*. The red curve represents the positional modulation of an atom in subsystem 1. **R3** denotes the 3D real space.

Due to the periodic difference along a particular direction, the composite crystals show positional (*displacive*) modulation of atomic sites. The positional modulation is periodic with an interval of 0 ≤*xs*4≤ 1 as shown with the red curve in Fig. 3. The modulated position of an atom is then mathematically represented with a Fourier series as:

$$\mathbf{x}\_{\rm si} = \overline{\mathbf{x}}\_{\rm si} + \sum\_{n=0}^{m} \left[ A\_{\rm ll} \cos(2\pi n \overline{\mathbf{x}}\_{\rm si4}) + B\_{\rm ll} \sin(2\pi n \overline{\mathbf{x}}\_{\rm si4}) \right],\tag{3}$$

where *i* = 1,2,3.

#### **2.3 Symmetry of HMSs**

To specify the superspace group to the present HMSs, we need to consider an additional axis *c*Si parallel to *c*Mn. The first subsystem [Mn] has a *β*-Sn type arrangement of Mn atoms 4 Will-be-set-by-IN-TECH

⎞

⎛

*x y z t*

⎞

⎟⎟⎠

γ

R3

[*An* cos(2*πnxs*4) + *Bn* sin(2*πnxs*4)], (3)

as3

(2)

⎜⎜⎝

⎟⎟⎠

⎛

⎜⎜⎝

where the parameter *t*, called as an internal coordinate, represents the distance from **R3** and

*x*s3 *x*s3

Fig. 3. A unit cell in the (3+1)-dimensional superspace and the relationship between the coordinates without modulation, *xsi*, with modulation *xsi* (*i* = 1-4), and internal coordinate *t*. The red curve represents the positional modulation of an atom in subsystem 1. **R3** denotes

Due to the periodic difference along a particular direction, the composite crystals show positional (*displacive*) modulation of atomic sites. The positional modulation is periodic with an interval of 0 ≤*xs*4≤ 1 as shown with the red curve in Fig. 3. The modulated position of an

To specify the superspace group to the present HMSs, we need to consider an additional axis *c*Si parallel to *c*Mn. The first subsystem [Mn] has a *β*-Sn type arrangement of Mn atoms

γ*z*

*z*

⎛

*xS*<sup>1</sup> *xS*<sup>2</sup> *xS*<sup>3</sup> *xS*<sup>4</sup> ⎞

⎟⎟⎠ =

⎜⎜⎝

is related to the fourth coordinate *xs*<sup>4</sup> by *xs*<sup>4</sup> = *γz* + *t*.

t

*z*

*x*s4 *x*s4

atom is then mathematically represented with a Fourier series as:

*m* ∑ *n*=0

as4

*xsi* = *xsi* +

the 3D real space.

where *i* = 1,2,3.

**2.3 Symmetry of HMSs**

*xs*1, *xs*2, *xs*<sup>3</sup> and *xs*<sup>4</sup> can be defined as,

classified under the 3D space group of *I*41/*amd*. For the second subsystem [Si], originally having dimensions of *a*/ <sup>√</sup><sup>2</sup> <sup>×</sup> *<sup>a</sup>*/ <sup>√</sup><sup>2</sup> <sup>×</sup> *<sup>c</sup>*Si, the *<sup>a</sup>*-axis length is adjusted to that of [Mn]. The Si atoms are at the origin and the equivalent positions generated by the *P*4/*nnc* space group. According to Yamamoto (Yamamoto, 1993), the most appropriate (3+1)D superspace group is designated as *P* : *I*41/*amd* : 11*ss* (Mn) *W* : *P*4/*nnc* : *q*1*q*1 (Si), or based on the modulation vector **k** = (00*γ*) more conveniently as *I*41/*amd*(00*γ*)00*ss*.

Based on the adopted superspace group, the translation parts for the [Mn] subsystem are expressed as:

+(0, 0, 0, 0); +(1/2, 1/2, 1/2, 0), with symmetry operations represented as: (i) *x*, *y*, *z*, *t*; (ii) -*y*, *x*+1/2, *z*+1/4, *t*; (iii) -*x*, *y*, *z*, *t*+1/2; (iv) -*x*, -*y*, *z*, *t*; (v) -*y*, -*x*+1/2, *z*+1/4, *t*+1/2; (vi) *y*, -*x*+1/2, *z*+1/4, *t*; (vii) *x*, -*y*, *z*, *t*+1/2; (viii) *y*, *x*+1/2, *z*+1/4, *t*+1/2; (ix) -*x*, -*y*+1/2, -*z*+1/4, -*t*; (x) *y*, -*x*, -*z*, -*t*; (xi) *x*, -*y*+1/2, -*z*+1/4, -*t*+1/2; (xii) *x*, *y*+1/2, -*z*+1/4, -*t*; (xiii) *y*, *x*, -*z*, -*t*+1/2; (xiv) -*y*, *x*, -*z*, -*t*; (xv) -*x*, *y*+1/2, -*z*+1/4, -*t*+1/2; (xvi) -*y*, -*x*, -*z*, -*t*+1/2.

The translation parts and the symmetry operation for the [Si] subsystem can be obtained by simply interchanging the third and the fourth components.

### **2.4 Structural modulation**

Polycrystalline samples were prepared in a tetra-arc-type furnace under an Ar atmosphere using tungsten electrodes and a water-cooled copper hearth. Appropriate amounts of Mn (99.9%) and Si (99.99%) powders were mixed in an alumina mortar and pressed into pellets. The pellets were melted four times, and turned over each time to obtain full homogeneity. As the satellite reflections were observed up to eighth order in the electron diffraction patterns, positional modulation of the atomic sites was introduced taken to the eighth order of cosine and sine components of the Fourier terms, *An* and *Bn* (*n*= 0-8), of equation 3.

Figure 4 shows the observed, calculated, and difference profiles of the neutron diffraction (ND) data for MnSi*<sup>γ</sup>* at 295 K. The ND data were collected using a Kinken powder diffractometer for high efficiency and high resolution measurements, HERMES (Petricek et al., 2000), at the Institute for Materials Research (IMR), Tohoku University, installed at the JRR-3M reactor of the Japan Atomic Energy Agency (JAEA) at Tokai. A monochromatized incident neutron beam at *λ* = 1.8265 Å was used. The ND data were analyzed with the JANA2000

symmetry, the number of refinable parameters of the Fourier terms is limited. Only even terms of the sine wave along the *z*-direction, i.e., *B*2*z*, *B*4*z*, *B*6*z*, and *B*8*z*, are allowed for the Mn atoms. In contrast, odd terms of the sine and cosine waves are allowed for both the *x* and *y* components for the Si atoms, along with *B*4*<sup>z</sup>* and *B*8*<sup>z</sup>* terms. The amplitudes of each cosine wave component in the *x*-*y* plane, such as *A*1*<sup>x</sup>* and *A*1*y*, are equal, whereas those of each sine

<sup>237</sup> Superspace Group Approach to the Crystal Structure

*B*<sup>2</sup> 0 0 -0.0142(12) *B*<sup>4</sup> 0 0 0.017(2) *B*<sup>6</sup> 0 00 *B*<sup>8</sup> 0 00

*B*<sup>4</sup> 0 0 -0.0441(19)

Subsystem 1: [Mn] *x yz*

Subsystem 2: [Si] *x yz A*<sup>1</sup> 0.0772(3) = *A*1*<sup>x</sup>* 0 *B*<sup>1</sup> = *A*1*<sup>x</sup>* = -*A*1*<sup>x</sup>* 0 *A*<sup>3</sup> 0.0103(3) = *A*3*<sup>x</sup>* 0 *B*<sup>3</sup> = -*A*3*<sup>x</sup>* = *A*3*<sup>x</sup>* 0

*A*<sup>5</sup> -0.0040(5) = *A*5*<sup>x</sup>* 0 *B*<sup>5</sup> = *A*5*<sup>x</sup>* = -*A*5*<sup>x</sup>* 0 *A*<sup>7</sup> -0.0034(7) = *A*7*<sup>x</sup>* 0 *B*<sup>7</sup> = -*A*7*<sup>x</sup>* = *A*7*<sup>x</sup>* 0 *B*<sup>8</sup> 0 0 0.017(4)

Table 2. Refined positional modulation wave components and anisotropic displacement

In Fig. 5, we show the revealed positional modulations for the *x*, *y*, and *z* coordinates of each atom plotted against *xs*4. The right vertical axes are re-scaled to represent the displacement (in Angstroms) for each atom. All the displacements are periodic in the interval 0 ≤*xs*4≤ 1. For the Mn atoms, positional modulation is only allowed in the *z* direction and the maximum displacement (∼0.12 Å) from *z* = 0 is recognized at *xs*4∼ 0.18, 0.32, 0.68 and 0.82. The displacement of Si atoms along *z* is comparable to that of the Mn atoms and the maximum displacement of (∼0.13 Å) from *z* = 1/4 can be seen at *xs*4∼ 0.03, 0.19, 0.28, etc. Based on the large sine and cosine components, the rotational modulation in the *x* and *y* directions is significant for the Si atoms. Both the modulation waves in *x* and *y* are identical with the phase-shift of Δ*xs*<sup>4</sup> = 1/4. The maximum displacement of ∼0.54 Å from *x* = 1/4, equivalent to Δ*x*∼ 0.1, is realized at *xs*4∼ 0.48 and 0.98, and at *xs*4∼ 0.23 and 0.73 from the *y* = 1/4 position. Similar rotational modulations have been reported for related chimney-ladder compounds, such as (Mo1−*x*Rh*x*)Ge*<sup>γ</sup>* (Rohrer et al., 2000) and (Cr1−*x*Mo*x*)Ge*<sup>γ</sup>* (Rohrer et al., 2001). The deviation from the fundamental position, Δ*x* and Δ*y*, is also ∼0.1 in these compounds, although the *a*-axis lengths of these phases (*a*∼ 5.9 Å) are much larger than that in the present

*U*<sup>11</sup> = *U*<sup>22</sup> = 0.002(2) (Å2) *U*<sup>33</sup> = 0.013(2) (Å2)

*U*<sup>11</sup> = *U*<sup>22</sup> = 0.009(2) (Å2) *U*<sup>33</sup> = 0.023(2) (Å2)

parameters, *Uij*, for MnSi*γ*.

compound.

wave are equal but their signs are opposite.

of Thermoelectric Higher Manganese Silicides MnSi*γ*

software package (Petricek et al., 2000). The bound coherent scattering lengths used for the refinement were -3.730 fm (Mn) and 4.149 fm (Si).

The short vertical lines below the patterns indicate the peak positions of possible Bragg reflections. Small peaks at 2*θ*∼ 32.5◦, 40.5◦, and 53.0◦, derived from the secondary phase MnSi, were excluded in the refinement cycles. The final *R* factors were *R*wp = 9.8% and *R*p = 6.9%, and the lattice parameters were refined to *a* = 5.5242(3) Å, *c*Mn = 4.3665(3) Å, and *c*Si = 2.5202(3) Å. The numbers in parentheses represent the estimated standard deviation of the last significant digit. The resulting *c*-axis ratio was *γ* = 1.7326(1), different from that of any commensurate HMSs (Flieher et al., 1967; Gottlieb et al., 2003; Knott et al., 1967; Schwomma et al., 1963; 1964; Zwilling & Nowotny, 1973) and that for the sample annealed at 1273 K for 168 h from the same batch of the present sample of *γ* = 1.7361(1) (Miyazaki, 2008).

Fig. 4. Observed, calculated, and difference patterns of powder neutron diffraction data for MnSi*<sup>γ</sup>* measured at 295 K. Short vertical lines below the patterns indicate positions of Bragg reflections. The difference between the observed and calculated intensities is shown below the vertical lines. The peaks, denoted as *hkl*0 and *hk*0*m*, are the fundamental reflections derived from the [Mn] and [Si] subsystems, respectively, while the *hklm* peaks are satellite reflections.

Table 1 summarizes the atomic coordinates and equivalent isotropic displacement parameters, *U*eq, for the fundamental structure of MnSi*<sup>γ</sup>* at 295 K.


Table 1. Atomic coordinates and equivalent isotropic atomic displacement parameters, *U*eq, for the fundamental structure of MnSi*<sup>γ</sup>* at 295 K.

#### **2.5 Details of modulated structure**

Table 2 summarizes the refined Fourier amplitudes for the positional parameters together with the anisotropic displacement parameters of each atom. Due to the superspace group 6 Will-be-set-by-IN-TECH

software package (Petricek et al., 2000). The bound coherent scattering lengths used for the

The short vertical lines below the patterns indicate the peak positions of possible Bragg reflections. Small peaks at 2*θ*∼ 32.5◦, 40.5◦, and 53.0◦, derived from the secondary phase MnSi, were excluded in the refinement cycles. The final *R* factors were *R*wp = 9.8% and *R*p = 6.9%, and the lattice parameters were refined to *a* = 5.5242(3) Å, *c*Mn = 4.3665(3) Å, and *c*Si = 2.5202(3) Å. The numbers in parentheses represent the estimated standard deviation of the last significant digit. The resulting *c*-axis ratio was *γ* = 1.7326(1), different from that of any commensurate HMSs (Flieher et al., 1967; Gottlieb et al., 2003; Knott et al., 1967; Schwomma et al., 1963; 1964; Zwilling & Nowotny, 1973) and that for the sample annealed at 1273 K for

> 20 40 60 80 100 120 140 2θ (deg.)

Fig. 4. Observed, calculated, and difference patterns of powder neutron diffraction data for MnSi*<sup>γ</sup>* measured at 295 K. Short vertical lines below the patterns indicate positions of Bragg reflections. The difference between the observed and calculated intensities is shown below the vertical lines. The peaks, denoted as *hkl*0 and *hk*0*m*, are the fundamental reflections derived from the [Mn] and [Si] subsystems, respectively, while the *hklm* peaks are satellite

Table 1 summarizes the atomic coordinates and equivalent isotropic displacement parameters,

Subsystem 1: [Mn] *xyzU*eq (Å2)

Subsystem 2: [Si] *xyzU*eq (Å2)

Table 1. Atomic coordinates and equivalent isotropic atomic displacement parameters, *U*eq,

Table 2 summarizes the refined Fourier amplitudes for the positional parameters together with the anisotropic displacement parameters of each atom. Due to the superspace group

0 0 0 0.0057(12)

1/4 1/4 1/4 0.0139(12)

3210

3120

4000

2111

5010, 4310

168 h from the same batch of the present sample of *γ* = 1.7361(1) (Miyazaki, 2008).

refinement were -3.730 fm (Mn) and 4.149 fm (Si).

Intensity (a.u.)

reflections.

1010

2000

2111

*U*eq, for the fundamental structure of MnSi*<sup>γ</sup>* at 295 K.

for the fundamental structure of MnSi*<sup>γ</sup>* at 295 K.

**2.5 Details of modulated structure**

1101

2221

1120

symmetry, the number of refinable parameters of the Fourier terms is limited. Only even terms of the sine wave along the *z*-direction, i.e., *B*2*z*, *B*4*z*, *B*6*z*, and *B*8*z*, are allowed for the Mn atoms. In contrast, odd terms of the sine and cosine waves are allowed for both the *x* and *y* components for the Si atoms, along with *B*4*<sup>z</sup>* and *B*8*<sup>z</sup>* terms. The amplitudes of each cosine wave component in the *x*-*y* plane, such as *A*1*<sup>x</sup>* and *A*1*y*, are equal, whereas those of each sine wave are equal but their signs are opposite.


Table 2. Refined positional modulation wave components and anisotropic displacement parameters, *Uij*, for MnSi*γ*.

In Fig. 5, we show the revealed positional modulations for the *x*, *y*, and *z* coordinates of each atom plotted against *xs*4. The right vertical axes are re-scaled to represent the displacement (in Angstroms) for each atom. All the displacements are periodic in the interval 0 ≤*xs*4≤ 1. For the Mn atoms, positional modulation is only allowed in the *z* direction and the maximum displacement (∼0.12 Å) from *z* = 0 is recognized at *xs*4∼ 0.18, 0.32, 0.68 and 0.82. The displacement of Si atoms along *z* is comparable to that of the Mn atoms and the maximum displacement of (∼0.13 Å) from *z* = 1/4 can be seen at *xs*4∼ 0.03, 0.19, 0.28, etc. Based on the large sine and cosine components, the rotational modulation in the *x* and *y* directions is significant for the Si atoms. Both the modulation waves in *x* and *y* are identical with the phase-shift of Δ*xs*<sup>4</sup> = 1/4. The maximum displacement of ∼0.54 Å from *x* = 1/4, equivalent to Δ*x*∼ 0.1, is realized at *xs*4∼ 0.48 and 0.98, and at *xs*4∼ 0.23 and 0.73 from the *y* = 1/4 position. Similar rotational modulations have been reported for related chimney-ladder compounds, such as (Mo1−*x*Rh*x*)Ge*<sup>γ</sup>* (Rohrer et al., 2000) and (Cr1−*x*Mo*x*)Ge*<sup>γ</sup>* (Rohrer et al., 2001). The deviation from the fundamental position, Δ*x* and Δ*y*, is also ∼0.1 in these compounds, although the *a*-axis lengths of these phases (*a*∼ 5.9 Å) are much larger than that in the present compound.

Si Mn

<sup>239</sup> Superspace Group Approach to the Crystal Structure

Fig. 6. Revealed modulated composite structure of MnSi*<sup>γ</sup>* at 295 K. The upper figure is a *c*-axis projection to illustrate the rotational arrangement of the Si atoms. The lower figure depicts the atoms within 4×*c*Mn lengths. The seven squares on the right are slices through

with each curve duplicated due to equidistant Mn-Si bond pairs. At *t* = 0, the Mn atom has eight Si neighbors with *<sup>d</sup>*Mn−Si∼ 2.4 Å, which is the typical Mn-Si bond distance on the basis of the metallic radii of Mn and Si (*r*Si = 1.17 Å) (Inoue et al., 2001). With increasing *t*, the four bonds (two curves) become shorter towards ∼2.2 Å, and the remaining four bonds become longer. In this way, the Mn atom is always bounded to eight Si atoms within a distance of *<*

In principle, the superspace group approach described so far can also be applicable to *all* the reported commensurate structure of HMSs. Towards a uniform treatment of the crystal structure of the present MnSi*<sup>γ</sup>* and other HMSs, we will first convert the 3D coordinates of Mn4Si7 (Gottlieb et al., 2003) to *xs*4. The Mn4Si7 phase can be regarded as a case for which *γ* = 7/4 = 1.75, wherein 4*c*Mn exactly equals 7*c*Si; the 3D unit cell consists of a stacking of four [Mn] subsystems and seven intervening [Si] subsystems, as shown in Fig. 1. Let us consider the [Si] subsystem closest to the origin. The Si atom at Si1 (8*j*) sites with (0.15715, 0.2015, 0.11253) is

**3. Universal treatment of HMSs by means of the superspace description**

*a*

*b*

*<sup>C</sup>*Mn *<sup>C</sup>*Si

*a*

*b*

of Thermoelectric Higher Manganese Silicides MnSi*γ*

*c*

*a*

*c*

*b*

the 1st to 7th layers of the Si subsystem from the origin.

2.8 Å.

Fig. 5. Revealed positional modulations of the Mn and Si atoms plotted as a function of the fourth superspace coordinate, *xs*4. The right vertical axes are re-scaled to represent the displacement for each atom.

Figure 6 shows the revealed modulated structure of MnSi*<sup>γ</sup>* at 295 K. The top left figure illustrates the *c*-axis projection to represent the helical arrangement of the Si atoms. The bottom left figure depicts the atoms within a partial unit cell of length 4×*c*Mn. The seven squares on the right represent slices of the 1st to 7th layers of Si atoms from the origin. The 1st layer of Si atoms corresponds to *xs*<sup>4</sup> = (1/*γ*)(*z* + *t*) = (1/*γ*)×(1/4 - 1/4) = 0, because the modulation vector component is inverted to *γ*−<sup>1</sup> for the [Si] subsystem. As deduced from Fig. 5, the coordinates at *xs*<sup>4</sup> = 0 are around (*x*, *y*) = (0.35, 0.23). Since there is a two-fold axis at (1/2, 1/2) parallel to the *c*-axis, the other Si atom in the 1st layer is located at around (0.65, 0.77). Similarly, the 3rd (*z* = 5/4) and 5th (*z* = 9/4) layers of Si atoms, corresponding to *xs*<sup>4</sup> = 0.576 and 0.152 (≡ 1.152), are located at around (0.16, 0.32), (0.84, 0.68), and (0.32, 0.16), (0.68, 0.84), respectively. By symmetry, the coordinates of even numbered layers can be obtained as (-*x*, *y*) and (*x*, -*y*). The 2nd (*z* = 3/4) and 4th (*z* = 7/4) layers of Si atoms, corresponding to *xs*<sup>4</sup> = 0.288 and 0.864, are located at around (0.19, 0.85), (0.81, 0.15) and (0.33, 0.67), (0.67, 0.33), respectively. The *z* coordinates of Si and Mn atoms in the modulated structure can also be calculated in a similar way.

Figure 7 shows interatomic distances plotted as a function of *t*. The two periodic curves around the distance of 3.0 Å represent the nearest four Mn-Mn distances, for which each curve is duplicated because of the two equidistant bonds. The nearest Mn-Mn distances, ranging from 2.92 Å to 3.01 Å, are relatively longer than that expected from the atomic radius of Mn, *r*Mn = 1.24 Å (Inoue et al., 2001). However, such long distances have been reported in the structure of *α*-Mn (Bradley & Thewlis, 1927), wherein the Mn-Mn distances vary from 2.24 Å to 2.96 Å. In the case of the Mn-Si bonds, each Mn atom is coordinated to eight Si atoms, as deduced from Fig. 6. The four curves around 2.4 Å represent the nearest eight Mn-Si distances, 8 Will-be-set-by-IN-TECH

0.05 Mn x,y <sup>z</sup>

x y


0.2

0.6 0.3



0


0 0.2 0.4 0.6 0.8 1.0

Fig. 5. Revealed positional modulations of the Mn and Si atoms plotted as a function of the fourth superspace coordinate, *xs*4. The right vertical axes are re-scaled to represent the

Figure 6 shows the revealed modulated structure of MnSi*<sup>γ</sup>* at 295 K. The top left figure illustrates the *c*-axis projection to represent the helical arrangement of the Si atoms. The bottom left figure depicts the atoms within a partial unit cell of length 4×*c*Mn. The seven squares on the right represent slices of the 1st to 7th layers of Si atoms from the origin. The 1st layer of Si atoms corresponds to *xs*<sup>4</sup> = (1/*γ*)(*z* + *t*) = (1/*γ*)×(1/4 - 1/4) = 0, because the modulation vector component is inverted to *γ*−<sup>1</sup> for the [Si] subsystem. As deduced from Fig. 5, the coordinates at *xs*<sup>4</sup> = 0 are around (*x*, *y*) = (0.35, 0.23). Since there is a two-fold axis at (1/2, 1/2) parallel to the *c*-axis, the other Si atom in the 1st layer is located at around (0.65, 0.77). Similarly, the 3rd (*z* = 5/4) and 5th (*z* = 9/4) layers of Si atoms, corresponding to *xs*<sup>4</sup> = 0.576 and 0.152 (≡ 1.152), are located at around (0.16, 0.32), (0.84, 0.68), and (0.32, 0.16), (0.68, 0.84), respectively. By symmetry, the coordinates of even numbered layers can be obtained as (-*x*, *y*) and (*x*, -*y*). The 2nd (*z* = 3/4) and 4th (*z* = 7/4) layers of Si atoms, corresponding to *xs*<sup>4</sup> = 0.288 and 0.864, are located at around (0.19, 0.85), (0.81, 0.15) and (0.33, 0.67), (0.67, 0.33), respectively. The *z* coordinates of Si and Mn atoms in the modulated structure can also be

Figure 7 shows interatomic distances plotted as a function of *t*. The two periodic curves around the distance of 3.0 Å represent the nearest four Mn-Mn distances, for which each curve is duplicated because of the two equidistant bonds. The nearest Mn-Mn distances, ranging from 2.92 Å to 3.01 Å, are relatively longer than that expected from the atomic radius of Mn, *r*Mn = 1.24 Å (Inoue et al., 2001). However, such long distances have been reported in the structure of *α*-Mn (Bradley & Thewlis, 1927), wherein the Mn-Mn distances vary from 2.24 Å to 2.96 Å. In the case of the Mn-Si bonds, each Mn atom is coordinated to eight Si atoms, as deduced from Fig. 6. The four curves around 2.4 Å represent the nearest eight Mn-Si distances,

xs4

0.35

Si

z Si


> 0.30 0.25 0.20

position

0.25

0.15

displacement for each atom.

calculated in a similar way.

Fig. 6. Revealed modulated composite structure of MnSi*<sup>γ</sup>* at 295 K. The upper figure is a *c*-axis projection to illustrate the rotational arrangement of the Si atoms. The lower figure depicts the atoms within 4×*c*Mn lengths. The seven squares on the right are slices through the 1st to 7th layers of the Si subsystem from the origin.

with each curve duplicated due to equidistant Mn-Si bond pairs. At *t* = 0, the Mn atom has eight Si neighbors with *<sup>d</sup>*Mn−Si∼ 2.4 Å, which is the typical Mn-Si bond distance on the basis of the metallic radii of Mn and Si (*r*Si = 1.17 Å) (Inoue et al., 2001). With increasing *t*, the four bonds (two curves) become shorter towards ∼2.2 Å, and the remaining four bonds become longer. In this way, the Mn atom is always bounded to eight Si atoms within a distance of *<* 2.8 Å.

### **3. Universal treatment of HMSs by means of the superspace description**

In principle, the superspace group approach described so far can also be applicable to *all* the reported commensurate structure of HMSs. Towards a uniform treatment of the crystal structure of the present MnSi*<sup>γ</sup>* and other HMSs, we will first convert the 3D coordinates of Mn4Si7 (Gottlieb et al., 2003) to *xs*4. The Mn4Si7 phase can be regarded as a case for which *γ* = 7/4 = 1.75, wherein 4*c*Mn exactly equals 7*c*Si; the 3D unit cell consists of a stacking of four [Mn] subsystems and seven intervening [Si] subsystems, as shown in Fig. 1. Let us consider the [Si] subsystem closest to the origin. The Si atom at Si1 (8*j*) sites with (0.15715, 0.2015, 0.11253) is

Similarly, the Si atom at Si4 (8*j*) sites with (0.34518, 0.2274, 0.9620) is converted to (0.34518, 0.2274, 27/4+*δ*') with *xs*<sup>4</sup> = 0.714 (≡ 3.714). In this case, 27/4 + *δ*' equals 7×0.9620 = 6.734 in the [Si] subsystem, corresponding to *z* = 0.734 in the 7th [Si] subsystem from the origin. By applying all the symmetry operations to the Mn1-Mn5 and Si1-Si4 sites, the equivalent positions close to Mn (0, 0, 0) and Si (1/4, 1/4, 1/4) in each subsystem can be obtained as

<sup>241</sup> Superspace Group Approach to the Crystal Structure

Based on this structuring, those equivalent positions as a function of *xs*<sup>4</sup> of Mn11Si19 (*γ* = 1.72˙ 7) (Schwomma et al., 1963; 1964), Mn ˙ 15Si26(*γ* = 1.73) (Flieher et al., 1967; Knott et al., ˙ 1967) and Mn27Si47(*γ* = 1.74˙0˙7) (Zwilling & Nowotny, 1973) are converted into the respective ˙ marks as shown in Fig. 8. The solid lines in these panels correspond to those shown in Fig. 5 but the *x* and *y* lines are interchanged to preserve the original (*x*, *y*) coordinates given in the reports (Flieher et al., 1967; Gottlieb et al., 2003; Knott et al., 1967; Schwomma et al., 1963; 1964; Zwilling & Nowotny, 1973). All the *x* and *y* coordinates of the points in the [Si] subsystem are well superposed on the two *universal* lines, suggesting that the helical arrangement of Si atoms is almost identical and independent of the *γ* values in all HMSs. In contrast, the *z* coordinates for the Mn and Si atoms deviate from the periodic solid lines of the present sample. It would be, however, reasonable to consider different shape of curves, i.e., different *Bz* terms, to fit the *z* coordinates for each HMS because the stacking periodicity might be dependent on *γ*.

A superspace group approach is economical in describing an incommensurate compound by using a much reduced number of parameters. The positional modulation of the atoms can be expressed by means of a Fourier expansion of the modulation functions. For HMSs, both the Mn and Si atoms are at special positions in each subsystem and their modulated positions are excellently described by only *eight* parameters. This is much smaller set compared with other HMSs with a 3D description; the number of refinable positional parameters are 13 (Mn4Si7), 41 (Mn11Si19), 26 (Mn15Si26) and 97 (Mn27Si47). These four phases have different 3D symmetries and lattice parameters but by using the superspace group approach, we can treat these phases as an identical compound MnSi*γ*, with slightly different stoichiometric *γ* values. Moreover, this approach affords a uniform treatment of a compound system which can change its stoichiometry incommensurately upon substitution of the elements. For example, a partial substitution of Cr, Fe, and Co is possible at Mn sites and such a solid solution can also be simply represented as (Mn1−*xMx*)Si*<sup>γ</sup>* (*M* = Cr, Fe and Co) (Flieher et al., 1967). Ranging over the various solid solutions, the *γ* value is known to change from 1.65 through 1.76. The next target should be to optimize the TE properties by tuning the electronic structure of HMSs and we hope that these TE materials can be prepared in the present HMS-based compounds in the

Bradley, A. J. & Thewlis, J. (1927). The crystal structure of *α*-manganese. *Proc. Royal Soc.*

De Wolff, P. M. (1974). The pseudo-symmetry of modulated crystal structures. *Acta Cryst.* A30:

Fedorov, M. I. & Zaitsev, V. K. (2006). Thermoelectrics of transition metal silicides, In:

*Thermoelectrics Handbook Macro to Nano*, Rowe, D. M., (Ed.), chap.31, CRC press, Boca

*London, Ser. A* 115: 456-471, 0080-4630.

Raton, 0-8493-2264-2, London.

777-785, 0567-7394.

filled circles shown in Fig. 8.

of Thermoelectric Higher Manganese Silicides MnSi*γ*

**4. Conclusion**

near future.

**5. References**

Fig. 7. Interatomic distances as a function of the internal coordinate *t* for MnSi*<sup>γ</sup>* at 295 K.

converted to (0.15715, 0.2015, *z*+*δ*) in the [Si] subsystem. Since the Si atom is located at the 2nd (*z* = 3/4) layer, we obtain *xs*<sup>4</sup> = (1/*γ*)×(*z* - 1/4) = 0.286 and 3/4 + *δ* = 7×0.11253 = 0.788.

Fig. 8. The converted coordinates of Mn at ∼ (0, 0, 0) and Si at ∼ (1/4, 1/4, 1/4) in each subsystem of HMSs plotted as a function of the superspace coordinate, *xs*4. Solid lines are the data of the present incommensurate MnSi*<sup>γ</sup>* (*γ* = 1.7326(1)).

Similarly, the Si atom at Si4 (8*j*) sites with (0.34518, 0.2274, 0.9620) is converted to (0.34518, 0.2274, 27/4+*δ*') with *xs*<sup>4</sup> = 0.714 (≡ 3.714). In this case, 27/4 + *δ*' equals 7×0.9620 = 6.734 in the [Si] subsystem, corresponding to *z* = 0.734 in the 7th [Si] subsystem from the origin. By applying all the symmetry operations to the Mn1-Mn5 and Si1-Si4 sites, the equivalent positions close to Mn (0, 0, 0) and Si (1/4, 1/4, 1/4) in each subsystem can be obtained as filled circles shown in Fig. 8.

Based on this structuring, those equivalent positions as a function of *xs*<sup>4</sup> of Mn11Si19 (*γ* = 1.72˙ 7) (Schwomma et al., 1963; 1964), Mn ˙ 15Si26(*γ* = 1.73) (Flieher et al., 1967; Knott et al., ˙ 1967) and Mn27Si47(*γ* = 1.74˙0˙7) (Zwilling & Nowotny, 1973) are converted into the respective ˙ marks as shown in Fig. 8. The solid lines in these panels correspond to those shown in Fig. 5 but the *x* and *y* lines are interchanged to preserve the original (*x*, *y*) coordinates given in the reports (Flieher et al., 1967; Gottlieb et al., 2003; Knott et al., 1967; Schwomma et al., 1963; 1964; Zwilling & Nowotny, 1973). All the *x* and *y* coordinates of the points in the [Si] subsystem are well superposed on the two *universal* lines, suggesting that the helical arrangement of Si atoms is almost identical and independent of the *γ* values in all HMSs. In contrast, the *z* coordinates for the Mn and Si atoms deviate from the periodic solid lines of the present sample. It would be, however, reasonable to consider different shape of curves, i.e., different *Bz* terms, to fit the *z* coordinates for each HMS because the stacking periodicity might be dependent on *γ*.

### **4. Conclusion**

10 Will-be-set-by-IN-TECH

0 0.2 0.4 0.6 0.8 1.0

*Mn-Si*

*Mn-Mn*

t

Mn11Si19 Mn15Si26 Mn27Si47

Mn4Si7

0 0.2 0.4 0.6 0.8 1.0

xs4

Fig. 8. The converted coordinates of Mn at ∼ (0, 0, 0) and Si at ∼ (1/4, 1/4, 1/4) in each subsystem of HMSs plotted as a function of the superspace coordinate, *xs*4. Solid lines are the

y x

Fig. 7. Interatomic distances as a function of the internal coordinate *t* for MnSi*<sup>γ</sup>* at 295 K.

converted to (0.15715, 0.2015, *z*+*δ*) in the [Si] subsystem. Since the Si atom is located at the 2nd (*z* = 3/4) layer, we obtain *xs*<sup>4</sup> = (1/*γ*)×(*z* - 1/4) = 0.286 and 3/4 + *δ* = 7×0.11253 = 0.788.

3.6

3.2

2.8

distances (Å)

2.4

2.0


> 0.30 0.25

> 0.35

0.25

0.15

data of the present incommensurate MnSi*<sup>γ</sup>* (*γ* = 1.7326(1)).

0.20 z

Si

z

Mn

Si

position

A superspace group approach is economical in describing an incommensurate compound by using a much reduced number of parameters. The positional modulation of the atoms can be expressed by means of a Fourier expansion of the modulation functions. For HMSs, both the Mn and Si atoms are at special positions in each subsystem and their modulated positions are excellently described by only *eight* parameters. This is much smaller set compared with other HMSs with a 3D description; the number of refinable positional parameters are 13 (Mn4Si7), 41 (Mn11Si19), 26 (Mn15Si26) and 97 (Mn27Si47). These four phases have different 3D symmetries and lattice parameters but by using the superspace group approach, we can treat these phases as an identical compound MnSi*γ*, with slightly different stoichiometric *γ* values. Moreover, this approach affords a uniform treatment of a compound system which can change its stoichiometry incommensurately upon substitution of the elements. For example, a partial substitution of Cr, Fe, and Co is possible at Mn sites and such a solid solution can also be simply represented as (Mn1−*xMx*)Si*<sup>γ</sup>* (*M* = Cr, Fe and Co) (Flieher et al., 1967). Ranging over the various solid solutions, the *γ* value is known to change from 1.65 through 1.76. The next target should be to optimize the TE properties by tuning the electronic structure of HMSs and we hope that these TE materials can be prepared in the present HMS-based compounds in the near future.

### **5. References**


**12** 

*Japan* 

**Neutron Diffraction** 

Nobuo Niimura

**The pH Dependence of Protonation States of** 

*Frontier Research Center for Applied Atomic Sciences, Ibaraki University, Shirakata* 

The charges of various amino-acid side chains depend on the pH. For example, at a high pH (low acidity conditions), carboxylic acids tend to be negatively charged (deprotonated), and amines tend to be uncharged (unprotonated). At a low pH (high acidity), the opposite is true. The pH at which exactly half of any ionized amino acid is charged in solution is called the *pKa* of that amino acid. These *pKa* values of such ionizable amino-acid side chains are tabulated in standard textbooks. However, the protonation state of a given amino-acid side chain in a protein cannot be estimated from standard *pKa* values measured from isolated amino acids in solution, because inside a protein it may vary significantly depending on the local environment. The electrically charged states of the amino-acid residues are very important in understanding the physiological function of the protein, the interaction

between ligands and proteins, molecular recognition, structural stability, and so on.

Neutron Protein Crystallography (NPC) is the unique method to provide the definite protonation states of the amino acid residue in proteins because neutron can identify not only hydrogen atoms but also protons. (N.Niimura et al. 2004,2006, N.Niimura & R. Bau

Consider one example: Figure 1 shows a typical example of the different protonation states of histidines in met-myoglobin at pH 6.8. To create this figure, the Neutron Protein Crystallography (NPC) of met-myoglobin was carried out at 1.5 Å resolution (A. Ostermann et al. 2005). Met-myoglobin contains 12 histidines. The measured *pKa* value of the histidine amino acid in aqueous solution is about 6.0, so the histidines should be neutral at pH 6.8 aqueous solution. However, four kinds of ionization states are observed by NPC in the imidazole ring of histidines as shown in Figure 1. Two histidines are doubly protonated on both Nδ1 and N <sup>ε</sup>2 (in red in Figure 1, with an average B-factors of 14.9 Å2), three are singly protonated (neutral) on Nδ1 (in yellow, with an average B-factors of 4.5 Å2), and four are singly protonated (neutral) on Nε2 (in blue, with an average B-factors of 16.5 Å2). In the three remaining histidines, the protonation state is not clear because of the disordered state of deuterium atoms, the B-factors of which are rather large (in pink in Figure 1, with an average B-factor of 22.9 Å2). Clearly, histidines inside the protein do not always follow the

**1. Introduction** 

2008, N.Niimura & A. Podjarny 2011)

*pKa* value in solution.

**Polar Amino Acid Residues Determined by** 

