**4. Spin Hamiltonian**

For a paramagnetic ion, the energy levels are quantized. This energy levels are eigenvalues of the spin Hamiltonian operator which is the representation of total electronic energy for the ion or system. Usually the lowest energy levels states are populated at ordinary temperatures of 300*K* (≈200*cm*-1) and this is the group of levels of the ground state. The ground state is the most of the times the only involved in the resonance experiment, it is to say, the transition are induced between this energy levels of ground state under microwave excitation. The energy of each level depends of the ion properties like electric charge, mass, atomic number, etc. and the energy level depends too to the crystalline field effect and the external magnetic field applied along with appropriate nuclear interactions [9, 12, 15].

The EPR results are interpreted by spin Hamiltonian that describes the system and the interactions mentioned above. The Hamiltonian is given in general by Eq. (2) and Eq. (3) [9–12].

*Paramagnetic Transitions Ions as Structural Modifiers in Ferroelectrics DOI: http://dx.doi.org/10.5772/intechopen.95983*

$$
\hat{H} = \beta \overrightarrow{\mathbf{S}} \bullet \overleftarrow{\mathbf{g}} \bullet \overrightarrow{H} + \overrightarrow{S} \bullet \overrightarrow{A} \bullet \overrightarrow{I} + \overrightarrow{S} \bullet \overrightarrow{D} \bullet \overrightarrow{S} \tag{2}
$$

$$
\hat{H} = \mathbf{g}\beta\overrightarrow{\mathbf{S}}\bullet\overrightarrow{H} + A\overrightarrow{\mathbf{S}}\bullet\overrightarrow{I} + D\left[\mathbf{S}\_x^2 - \frac{\mathbf{1}}{\mathbf{3}}\mathbf{S}(\mathbf{S}+\mathbf{1})\right] + E\left(\mathbf{S}\_x^2 - \mathbf{S}\_y^2\right) \tag{3}
$$

The first term is by Zeeman electronic interaction, the second term is the representation of hyperfine interaction, the third and fourth terms are due to the crystalline field, where *g* \$, *A* \$ and *D* \$ are the spectroscopy, hyperfine interaction and crystalline field third order tensors respectively.

The magnetic interaction is naturally anisotropic, and the tensors are used to describe it, like the magnetic moment for each electron *μ* \$. The anisotropy property for magnetic moment *μ* \$ is measured by spectroscopy factor *g* \$.

The solutions for the Hamiltonian are compared with measures of *g* parameters in the spectrum EPR and other paramagnetic parameters in the system [9–12].

#### **4.1 Zeeman electronic term**

The general expression for Zeeman interaction between external magnetic field *H* ! <sup>0</sup> and the electronic spin *S* ! is given in Eq. (4) and it is rewritten in terms of matrix in Eq. (5) [9–15].

$$
\hat{H}\_{\text{Ze}} = \beta \overrightarrow{\mathbf{S}} \bullet \overleftarrow{\mathbf{g}} \bullet \overrightarrow{H}\_0 \tag{4}
$$

$$
\hat{H}\_{Z\varepsilon} = \beta \begin{pmatrix} \mathbf{S}\_x & \mathbf{S}\_y & \mathbf{S}\_z \end{pmatrix} \begin{pmatrix} \mathbf{g}\_{xx} & \mathbf{g}\_{xy} & \mathbf{g}\_{xx} \\ \mathbf{g}\_{yx} & \mathbf{g}\_{yy} & \mathbf{g}\_{yx} \\ \mathbf{g}\_{xx} & \mathbf{g}\_{xy} & \mathbf{g}\_{xx} \end{pmatrix} \begin{pmatrix} H\_x \\ H\_y \\ H\_x \end{pmatrix} \tag{5}
$$

Where *Hx*, *Hy*, *Hz*, *Sx*, *Sy*, *Sz*, are the three scalar components for external magnetic field *H* ! <sup>0</sup> and *S* ! in a fixed Cartesians coordinates *x*, *y* and *z* in the molecule.

Many times is found that tensor *g* \$ is a symmetric matrix, which could be diagonalized through appropriate transformation [9, 16] *M g*\$*M*-<sup>1</sup> <sup>¼</sup> *<sup>g</sup>* \$ *diagonal*. This transformation corresponds to axes reorientation and the matrix *M* redefine the orientation for new principal axes respecting to previous axes. After to diagonalized the Zeeman's Hamiltonian writes like Eq. (6).

$$
\hat{H}\_{\rm Z\varepsilon} = \beta \left( \mathbf{g}\_{\rm xx} H\_{\rm x} \mathbf{S}\_{\rm x} + \mathbf{g}\_{\gamma\gamma} H\_{\gamma} \mathbf{S}\_{\gamma} + \mathbf{g}\_{\rm xx} H\_{\rm x} \mathbf{S}\_{\rm x} \right) \tag{6}
$$

The *g* \$ components *gxx*, *gyy* and *gzz* measured the contribution of magnetic moment along principal direction *xx*, *yy* and *zz* of magnetic field. There is spherical symmetry for the electron *μ* \$ð Þ*g* when *gxx* ¼ *gyy* ¼ *gzz*.

The Hydrogen atom have spherical symmetry, the spin Hamiltonian have a *g* isotropic factor for an electron and isotropic hyperfine interaction, *A*, between electron and nucleus. In the most molecules these quantities vary with applied magnetic field direction and the spin Hamiltonian is anisotropic [9–16].

#### **4.2 Axial symmetry**

If the *g* \$ tensor is anisotropic, there is the axial symmetry when *gxx* <sup>¼</sup> *gyy* 6¼ *gzz* . Usually the EPR bibliography writes *gxx* ¼ *gyy* ¼ *g*<sup>⊥</sup> and *gzz* ¼ *g*<sup>∥</sup> [9–12].

Suppose that magnetic field *H* ! is applied with *θ* angle respect to *z* axis. Rewriting the components, *H* cos *θ* is parallel to *z* and *H* sin *θ* is parallel to *x*, them the Zeeman *H*^ *Ze* writes like Eq. (7) [9].

$$\hat{H}\_{\rm Z\varepsilon} = \beta H \left( \mathbf{g}\_{\parallel} \mathbf{S}\_{\mathbf{x}} \cos \theta + \mathbf{g}\_{\perp} \mathbf{S}\_{\mathbf{x}} \sin \theta \right) \tag{7}$$

Where *Sx* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ð Þ *S*<sup>þ</sup> þ *S* written in terms of created and annihilated spin operators. With low symmetry and magnetic field random oriented, using the director cosines *l*, *m*, *n* with respect *x*, *y* and *z* axis, then the spin Hamiltonian writes like Eq. (8).

$$
\hat{H}\_{\rm Z\varepsilon} = \beta H \left( \mathbf{g}\_{\rm xx} l \mathbf{S}\_{\rm x} + \mathbf{g}\_{\rm yy} m \mathbf{S}\_{\rm y} + \mathbf{g}\_{\rm xz} n \mathbf{S}\_{\rm x} \right) \tag{8}
$$

This correspond to rhombic symmetry *gxx* 6¼ *gyy* 6¼ *gzz*.

For the axial asymmetry g factor is dependent of angle *<sup>θ</sup>* by *<sup>g</sup>*ð Þ*<sup>θ</sup>* <sup>2</sup> <sup>¼</sup> *<sup>g</sup>*<sup>2</sup> <sup>⊥</sup> sin *θ* þ *g*2 <sup>∥</sup> cos *θ*, for an axial g matrix with *g*<sup>∥</sup> >*g*⊥, the line shape of the corresponding EPR spectrum are drawn in **Figure 3**, assuming a large number of paramagnetic systems with random orientation of their *g* \$ ellipsoids with respect to the static magnetic field *H* ! [9]. This situation is typical for a powder sample that contain all possible direction *g* ellipsoids. For a given magnetic strength H, all spins fulfilling the resonance condition *g*ð Þ¼ *θ hν=βH*, i. e., all spins for which H makes an angle *θ* with the *z* axis of the *g* ellipsoid, contribute to the spectrum and are considered to form a spin packet. The extreme positions (*θ* ¼ 0° and *θ* ¼ 90°) of the powder spectrum are obtained by inserting *g*<sup>∥</sup> and *g*<sup>⊥</sup> into the resonance condition. If *g*<sup>∥</sup> >*g*<sup>⊥</sup> the asymmetry line shape is mainly due to the fact that the number of the spin packets contributing to the spectrum is much larger in the *xy*-plane than the along the *z* axis. If *g*<sup>∥</sup> <*g*<sup>⊥</sup> the asymmetry line shape is mainly due to the fact that the number of the spin packets contributing to the spectrum is much larger along

the *z* axis than in the *xy*-plane. Also, by definition the *giso* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 <sup>3</sup> *g*<sup>2</sup> *xx* þ *g*<sup>2</sup> *yy* þ *g*<sup>2</sup> *zz* <sup>r</sup> h i [9–17].

#### **4.3 Hyperfine interaction term**

The hyperfine interaction is the interaction between decoupled electrons (*S* ! ) and nuclear magnetic moments ( *I* ! ) of several neighboring nucleus. In the EPR spectra the hyperfine split is due to interaction between electronic spin and nuclear spin, it causes splitting Zeeman levels. Each Zeeman level is splitting 2*I* þ 1 times, where *I* is the nuclear spin [12, 16]. In tensorial form the Hamiltonian spin term is given by *S* ! ∙ *A* \$ ∙ *I* ! . Similarly to g-factor, the *A* -hyperfine factor give the magnitude of hyperfine interaction, in general it is an anisotropic tensor. There are two types of hyperfine interaction [9–15].

The first interaction is the classic interaction between *μ* ! *<sup>S</sup>* and *μ* ! *<sup>I</sup>* dipoles separated a *r* ! distance given by Eq. (9) [9, 12].

$$E\_{dip} = \frac{\overrightarrow{\mu}\_S \bullet \overrightarrow{\mu}\_I}{r^3} - \frac{\Im \left( \overrightarrow{\mu}\_S \bullet \overrightarrow{r} \right) \left( \overrightarrow{\mu}\_I \bullet \overrightarrow{r} \right)}{r^5} \tag{9}$$

*Paramagnetic Transitions Ions as Structural Modifiers in Ferroelectrics DOI: http://dx.doi.org/10.5772/intechopen.95983*

**Figure 3.**

*Microwave absorption spectra and their first derivate for axial symmetry (A) g*<sup>∥</sup> > *g*<sup>⊥</sup> *and (B) g*<sup>∥</sup> < *g*⊥*. Both cases show all contributions for all directions contributions of g*ð Þ¼ *θ hν=βH in a powder samples [9–16].*

For correspondence principle, the quantum Hamiltonian for this interaction is given in Eq. (10).

$$\hat{H}\_{\rm dip} = \mathbf{g}\_e \beta\_e \mathbf{g}\_n \beta\_n \left[ \frac{\overrightarrow{\mathbf{S}} \bullet \overrightarrow{I}}{r^3} + \frac{\mathbf{3} \left( \overrightarrow{I} \bullet \overrightarrow{r} \right) \left( \overrightarrow{\mathbf{S}} \bullet \overrightarrow{r} \right)}{r^5} \right] \tag{10}$$

The second interaction is no classic interaction and comes from to the probability different of zero for found an electron in the nuclear region (0<*r*<*a*0), where *a*<sup>0</sup> is the Bohr's radii, i.e., it is proportional to the square electronic function valuated in the nucleus. Fermi proof that this interaction is isotropy and is called contact interaction or Fermi's interactions, given by Eq. (11) [10–17]. Here Ψð Þ 0 is the electronic wave function valued in the nucleus.

$$
\hat{H} = \left(\frac{8\pi}{3}\right)\overrightarrow{\mu}\_S \bullet \overrightarrow{\mu}\_n |\Psi(\mathbf{0})|^2 \tag{11}
$$

If the molecule have one or more neighboring nuclei to the uncoupled magnetic dipolar momentum, it turns out split hyperfine of energy magnetic levels of the decoupled electron (even without external magnetic field applied) due to interaction of each nucleus with the electronic magnetic momentum.

When the conditions are favorable the hyperfine interaction could be measured like in the case of the hyperfine interaction splitting of an octahedral manganese(II) complex with spin *S* ¼ 5*=*2 and nuclear spin *I* ¼ 5*=*2 and the corresponding EPR spectrum are shown in the Drago's book at Figure 13.10 [9].

In the simplest case, the energy levels of a system with one unpaired electron and one nucleus with *I* ¼ 1*=*2 are show in **Figure 4(A)** with sufficiently high fixed magnetic field *H* ! . The dashed line would be the transition corresponding to Δ*W* ¼ *hν* ¼ *gβH* in the absence of hyperfine (*A*0), like is show in **Figure 2**. The solid lines marked Δ*W*<sup>1</sup> and Δ*W*<sup>2</sup> correspond to the allowed EPR transitions with the hyperfine coupling operative. To first order, *<sup>h</sup><sup>ν</sup>* <sup>¼</sup> *<sup>g</sup>β<sup>H</sup>* <sup>1</sup> <sup>2</sup> *A*0, where *A*<sup>0</sup> is the isotropic hyperfine coupling constant. In **Figure 4(B)** are shown the hyperfine splitting as a function of an applied magnetic field. The dashed line corresponds to the transitions in the case of *A*<sup>0</sup> ¼ 0. The solid lines Δ*W*<sup>1</sup> and Δ*W*<sup>2</sup> refer to transitions induced by a constant microwave quantum *hν* of the same energy as for the transition Δ*W*. Here the resonant-field values corresponding to these two transitions are, to first order, given by *<sup>H</sup>* <sup>¼</sup> *<sup>h</sup>ν=g<sup>β</sup>* <sup>1</sup>*=*<sup>2</sup> *ge=<sup>g</sup> a*<sup>0</sup> (in mT) is the hyperfine spplitting constant given approximately by *H*Δ*W*<sup>1</sup> - *H*Δ*W*<sup>2</sup> . Note that these diagrams are specifics to a nucleus with positive *gn* and *A*<sup>0</sup> values, such as hydrogen atom [9, 12]. The **Figure 4 (C)** shows the typical EPR spectrum and the *g value* approximate positions for the case described in (A) and (B) with *a*<sup>0</sup> hyperfine splitting, and additionally shows the hyperfine splitting for and axial symmetry EPR signal with *A*<sup>∥</sup> and *A*<sup>⊥</sup> splitting factor than indicating how hyperfine interactions is affecting the energy levels transitions [11, 12].

#### **Figure 4.**

*(A) At a sufficiently high fixed magnetic field H* ! *. The dashed line would be the transition corresponding to* **Δ***W* ¼ *hν* ¼ *gβH in the absence of hyperfine (A***0***). The solid lines marked* **Δ***W***<sup>1</sup>** *and* **Δ***W***<sup>2</sup>** *correspond to the allowed EPR transitions with the hyperfine coupling operative. To first order,* **<sup>Δ</sup>***<sup>W</sup>* <sup>¼</sup> *<sup>h</sup><sup>ν</sup>* <sup>¼</sup> *<sup>g</sup>β<sup>H</sup>* **<sup>1</sup> <sup>2</sup>** *A***0***, where A***<sup>0</sup>** *is the isotropic hyperfine coupling constant. (B) As a function of an applied magnetic field. The dashed line corresponds to the transitions in the hypothetical case of A***<sup>0</sup>** ¼ **0***. The solid lines* **Δ***W***<sup>1</sup>** *and* **Δ***W***<sup>2</sup>** *refer to transitions induced by a constant microwave quantum hν of the same energy as for the transition* **Δ***W . Here the resonant-field values corresponding to these two transitions are, to first order, given by H* ¼ *hν=gβ* **<sup>1</sup>***=***<sup>2</sup>** *ge=<sup>g</sup> a***<sup>0</sup>** *(in mT) is the hyperfine spplitting constant. (C) Shows the typical EPR spectrum and the <sup>g</sup>* - *value approximate positions for the case described in (A) and (B) with a***<sup>0</sup>** *hyperfine splitting, and additionally shows the hyperfine splitting for and axial symmetry EPR signal with A***<sup>∥</sup>** *and A***<sup>⊥</sup>** *splitting factor than indicating how hyperfine interactions is affecting the energy levels transitions [11, 12].*
