**8. The quasistatic macroscopic mixing theory for magnetic permeability**

Although mixing formulas for the effective-medium type of approximations for the dielectric permittivities in the infinite-wavelength (i.e., quasistatic) limit (Lamb et al., 1980), such as the Maxwell Garnett formula (Garnett, 1904), have been popularly applied in the whole spectral range of electromagnetic fields, their magnetic counterpart has seldom been addressed up to this day. The current effort is thus to derive such an equation to approximately predict the final permeability as the result of mixing together several magnetic materials (Chang & Liao, 2011).

Lightwave Refraction and Its Consequences: A Viewpoint of

was the induced dipole moment and *Em*

inclusions, the polarizability became a scalar, such that *Em*

all incorporated molecules were randomly distributed, *Enear*

further approximated that *E EP <sup>m</sup>* / 3

being its polarization density) was *P* / 3

was the average field within the bulk host, *Ep*

However, it was well-known that for isotropic media <sup>0</sup> 1 *P E <sup>r</sup>*

Lorenz formula readily followed ((Lorenz, 1880), (Lorentz, 1880)):

inclusions), and Eq. (35) would have to satisfy (Garnett, 1904):

 

*eff r*

then generalized to the famous Maxwell Garnett mixing formula:

*eff h h*

where *mp*

Here *E* 

further expressed as:

(Garnett, 1904):

*Enear*

*P* 

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 243

at the location of the molecule. Since the treatment was aiming for uniform spherical

molecular location caused by all surrounding concentric spherical shells of the bulk, and

 was due to asymmetry within the inclusion. In those cases of interest where either the structure of the inclusion was regular enough, such as a cubical or spherical particulate, or

being the polarization density associated with a uniformly polarized sphere, and *ε*0 being the permittivity in free space. Hence, given Eq. (32), with the number density of such included molecules denoted as *n*, and *P n <sup>m</sup> <sup>p</sup>* (Cheng, 1989), the polarization density was

relative permittivity (i.e., the electric field at the center of a uniformly polarized sphere (with

 3 1 2

> 0 0

2 

 <sup>3</sup> 2 

 *s h*

*s h sh*

*s s*

 *s*

> 

2 

In those special cases where the permittivity of each tiny included particle was *εs* and the host material was vacuum (*εr* =1), such that *n* = *V*-1 (*V* being the volume of the spherical

Combining Eqs. (35) and (36) gave the effective permittivity (*εeff*) of the final mixture

with *f = nV* being the volume ratio of the embedded tiny particles (0 ≤ *f* ≤ 1) within the final mixture. If, instead of vacuum, the host material was with a permittivity of *εh*, Eq. (37) was

 *s s*

3

00 0

 

3

 

 *r*

 0 0 / 3

 

was the polarizing electric field intensity

*E EE E <sup>m</sup> p near* (33)

<sup>0</sup> (Purcell, 1985), to be elaborated later, with *<sup>P</sup>*

*Pn EP* (34)

<sup>0</sup> ). Then, a relation known as the Lorentz-

*<sup>r</sup> <sup>n</sup>* (35)

*V* (36)

*<sup>f</sup> f* (37)

*<sup>f</sup> f* (38)

 0

0 0

 

was expressed as (Purcell, 1985):

was the electric field at this

became essentially zero. It was

where *εr* stood for the

Fig. 21. Calculated SPR reflectivities for: (a) β-PVDF, (b) poled-β-PVDF.

#### **8.1 A brief review of the derivation leading to the Maxwell Garnett and Bruggeman formulas**

Historically, an isotropic host material was hypothesized to embed with a collection of spherical homogeneous inclusions. With the molecular polarization of a single molecule of such inclusions being denoted *α*, the following relation was established within the linear range (Cheng, 1989):

$$
\bar{p}\_m = \alpha \varepsilon\_0 \bar{E}\_m \tag{32}
$$

Fig. 21. Calculated SPR reflectivities for: (a) β-PVDF, (b) poled-β-PVDF.

**Bruggeman formulas** 

range (Cheng, 1989):

**8.1 A brief review of the derivation leading to the Maxwell Garnett and** 

Historically, an isotropic host material was hypothesized to embed with a collection of spherical homogeneous inclusions. With the molecular polarization of a single molecule of such inclusions being denoted *α*, the following relation was established within the linear

> 0

*m m p E* (32)

where *mp* was the induced dipole moment and *Em* was the polarizing electric field intensity at the location of the molecule. Since the treatment was aiming for uniform spherical inclusions, the polarizability became a scalar, such that *Em* was expressed as (Purcell, 1985):

$$
\bar{E}\_m = \bar{E} + \bar{E}\_p + \bar{E}\_{\text{near}} \tag{33}
$$

Here *E* was the average field within the bulk host, *Ep* was the electric field at this molecular location caused by all surrounding concentric spherical shells of the bulk, and *Enear* was due to asymmetry within the inclusion. In those cases of interest where either the structure of the inclusion was regular enough, such as a cubical or spherical particulate, or all incorporated molecules were randomly distributed, *Enear* became essentially zero. It was further approximated that *E EP <sup>m</sup>* / 3 <sup>0</sup> (Purcell, 1985), to be elaborated later, with *<sup>P</sup>* being the polarization density associated with a uniformly polarized sphere, and *ε*0 being the permittivity in free space. Hence, given Eq. (32), with the number density of such included molecules denoted as *n*, and *P n <sup>m</sup> <sup>p</sup>* (Cheng, 1989), the polarization density was further expressed as:

$$
\bar{P} = n\alpha \varepsilon\_0 \left[ \bar{E} + \bar{P} / \left( 3\varepsilon\_0 \right) \right] \tag{34}
$$

However, it was well-known that for isotropic media <sup>0</sup> 1 *P E <sup>r</sup>* where *εr* stood for the relative permittivity (i.e., the electric field at the center of a uniformly polarized sphere (with *P* being its polarization density) was *P* / 3 <sup>0</sup> ). Then, a relation known as the Lorentz-Lorenz formula readily followed ((Lorenz, 1880), (Lorentz, 1880)):

$$\alpha = \frac{\Im \left( \varepsilon\_r - 1 \right)}{n \left( \varepsilon\_r + 2 \right)} \tag{35}$$

In those special cases where the permittivity of each tiny included particle was *εs* and the host material was vacuum (*εr* =1), such that *n* = *V*-1 (*V* being the volume of the spherical inclusions), and Eq. (35) would have to satisfy (Garnett, 1904):

$$\alpha = \mathfrak{W} \frac{\mathfrak{e}\_s - \mathfrak{e}\_0}{\mathfrak{e}\_s + \mathfrak{e}\_0} \tag{36}$$

Combining Eqs. (35) and (36) gave the effective permittivity (*εeff*) of the final mixture (Garnett, 1904):

$$
\varepsilon\_{\text{eff}} = \varepsilon\_r \varepsilon\_0 = \varepsilon\_0 + 3f \,\varepsilon\_0 \frac{\varepsilon\_s - \varepsilon\_0}{\varepsilon\_s + 2\varepsilon\_0 - f\left(\varepsilon\_s - \varepsilon\_0\right)}\tag{37}
$$

with *f = nV* being the volume ratio of the embedded tiny particles (0 ≤ *f* ≤ 1) within the final mixture. If, instead of vacuum, the host material was with a permittivity of *εh*, Eq. (37) was then generalized to the famous Maxwell Garnett mixing formula:

$$\mathcal{L}\_{\text{eff}} = \mathcal{E}\_h + \Im f \mathcal{E}\_h \frac{\mathcal{E}\_s - \mathcal{E}\_h}{\mathcal{E}\_s + 2\mathcal{E}\_h - f\left(\mathcal{E}\_s - \mathcal{E}\_h\right)} \tag{38}$$

Lightwave Refraction and Its Consequences: A Viewpoint of

incorporated molecules are randomly distributed, *Bnear*

magnetic dipole moment (*mm*) is:

where *B* 

where 

where

(Cheng, 1989)

(Cheng, 1989)

we obtain:

satisfied by:

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 245

asymmetry in the inclusion. In those cases of interest where either the structure of the included particles is regular enough, such as a cubical or spherical particulate, or all

> 

 

 

with *μr* being the relative permeability. By incorporating Eq. (41) and Eq. (44) into Eq. (43)

<sup>0</sup>

1 3

 *m m*

 <sup>0</sup> 1 

3 1 2 5 

In the special case where the host material is vacuum (*μr* = 1) and the permeability of the spherical particles is *μs*, *n* = *V*-1 (*V* being the volume of a spherical particle), and Eq. (47) is

3

 *r*

0 0

2 5 

*s*

 *s*

 *r r*

 

0

*r*

*M H*

*m*

*m*

> 

 

> 

2

0

 

or

1 1 , *<sup>r</sup>*

*B*

*r*

 0 0 1 1 <sup>0</sup> *Bm rm H nH H m m m m* (44)

*n M M B n* (45)

*B M* (46)

*<sup>r</sup> <sup>n</sup>* (47)

*V* (48)

is the average magnetic flux density within the bulk host and *Bnear*

If the magnetic field intensity at the location of the molecule is denoted *Hm*

*<sup>m</sup>* is the molecular magnetization of the molecule. Because *M*

*<sup>m</sup>* is known as the magnetic susceptibility. Hence, *Bm*

Further, for isotropic magnetized materials (Cheng, 1989):

Substituting Eq. (46) into Eq. (45) gives (Chang & Liao, 2011)

*B BB B <sup>m</sup> c near* (41)

can be taken as zero.

*m H m mm* (42)

*<sup>M</sup> mm mm nH H* (43)

equals *nmm*

can be further expressed as

is due to the

, the induced

, we have

For the view in which the inclusion was no longer treated as a perturbation to the original host material, Bruggeman managed to come up with a more elegant form wherein different ingredients were assumed to be embedded within a host (Bruggeman, 1935). By utilizing Eqs. (35) and (36), he had:

$$\frac{\mathcal{E}\_{\text{eff}} - \mathcal{E}\_0}{\mathcal{E}\_{\text{eff}} + 2\mathcal{E}\_0} = \sum\_i f\_i \frac{\mathcal{E}\_i - \mathcal{E}\_0}{\mathcal{E}\_i + 2\mathcal{E}\_0} \tag{39}$$

where *fi* and *εi* are the volume ratio and permittivity of the *i*-th ingredient.

#### **8.2 The magnetic flux density at the center of a uniformly magnetized sphere**

surface current can be expected to appear on the surface of a uniformly magnetized sphere (wherein *M* is the finalized net anti-responsive magnetization vector, see Fig. 22).

Fig. 22. Situation for calculation of the central magnetic flux density on a uniformly magnetized sphere.

In Fig. 22, *KS* is the induced anti-reactive surface current density (in A/m) on the sphere's surface. By integrating all surface current density on strips of the sphere's surface the magnetic flux density ( *Bc* ) at the center of a uniformly magnetized sphere is obtained to be (Lorrain & Corson, 1970):

$$
\bar{B}\_c = \frac{2\bar{M}\,\mu\_0}{3} \tag{40}
$$

#### **8.3 Magnetic permeabilities formula mixing**

Now, this time consider an isotropic host material embedded with a collection of spherical homogeneous magnetic particles. Given the magnetic flux density at the location of a single molecule of the inclusions being *Bm* , the following relation holds in general:

For the view in which the inclusion was no longer treated as a perturbation to the original host material, Bruggeman managed to come up with a more elegant form wherein different ingredients were assumed to be embedded within a host (Bruggeman, 1935). By utilizing

> 0 0 0 0 2 2

 *eff <sup>i</sup> i*

surface current can be expected to appear on the surface of a uniformly magnetized sphere

is the finalized net anti-responsive magnetization vector, see Fig. 22).

*eff i i*

 

*f* (39)

 

 

where *fi* and *εi* are the volume ratio and permittivity of the *i*-th ingredient.

**8.2 The magnetic flux density at the center of a uniformly magnetized sphere** 

Fig. 22. Situation for calculation of the central magnetic flux density on a uniformly

*KS* is the induced anti-reactive surface current density (in A/m) on the sphere's

) at the center of a uniformly magnetized sphere is obtained to be

, the following relation holds in general:

*<sup>M</sup> <sup>B</sup>* (40)

surface. By integrating all surface current density on strips of the sphere's surface the

<sup>0</sup> 2 3 *c*

Now, this time consider an isotropic host material embedded with a collection of spherical homogeneous magnetic particles. Given the magnetic flux density at the location of a single

Eqs. (35) and (36), he had:

(wherein *M*

magnetized sphere.

magnetic flux density ( *Bc*

(Lorrain & Corson, 1970):

**8.3 Magnetic permeabilities formula mixing** 

molecule of the inclusions being *Bm*

In Fig. 22,

$$
\bar{B}\_m = \bar{B} - \bar{B}\_c + \bar{B}\_{near} \tag{41}
$$

where *B* is the average magnetic flux density within the bulk host and *Bnear* is due to the asymmetry in the inclusion. In those cases of interest where either the structure of the included particles is regular enough, such as a cubical or spherical particulate, or all incorporated molecules are randomly distributed, *Bnear* can be taken as zero.

If the magnetic field intensity at the location of the molecule is denoted *Hm* , the induced magnetic dipole moment (*mm*) is:

$$
\bar{m}\_m = \kappa\_m \bar{H}\_m \tag{42}
$$

where *<sup>m</sup>* is the molecular magnetization of the molecule. Because *M* equals *nmm* , we have (Cheng, 1989)

$$
\bar{M} = n\kappa\_m \bar{H}\_m = \mathcal{Z}\_n \bar{H}\_m \tag{43}
$$

where *<sup>m</sup>* is known as the magnetic susceptibility. Hence, *Bm* can be further expressed as (Cheng, 1989)

$$
\bar{B}\_{m} = \mu\_{0}\mu\_{r}\bar{H}\_{m} = \mu\_{0}\left(1 + n\kappa\_{m}\right)\bar{H}\_{m} = \mu\_{0}\left(1 + \chi\_{m}\right)\bar{H}\_{m} \tag{44}
$$

with *μr* being the relative permeability. By incorporating Eq. (41) and Eq. (44) into Eq. (43) we obtain:

$$M = \frac{n\kappa\_m}{\mu\_0 \left(1 + n\kappa\_m\right)} \left(B - \frac{2M\,\mu\_0}{3}\right) \tag{45}$$

Further, for isotropic magnetized materials (Cheng, 1989):

$$
\mathcal{M} = \left(\mu\_r - 1\right)H = \frac{\left(\mu\_r - 1\right)B}{\mu\_0 \mu\_r}, \text{ or }
$$

$$
B = \frac{\mu\_0 \mu\_r}{\left(\mu\_r - 1\right)}M\tag{46}
$$

Substituting Eq. (46) into Eq. (45) gives (Chang & Liao, 2011)

$$\kappa\_m = \frac{3}{n} \left( \frac{\mu\_r - 1}{-2\mu\_r + 5} \right) \tag{47}$$

In the special case where the host material is vacuum (*μr* = 1) and the permeability of the spherical particles is *μs*, *n* = *V*-1 (*V* being the volume of a spherical particle), and Eq. (47) is satisfied by:

$$\kappa\_{\kappa} = 3V \frac{\mu\_{\ast} - \mu\_{0}}{-2\mu\_{\ast} + 5\mu\_{0}} \tag{48}$$

Lightwave Refraction and Its Consequences: A Viewpoint of

*<sup>o</sup>* is the orbital part, and

Kohn-Sham orbitals (

where

where

*r*

& Friedman, 1997)) as:

where *A*<sup>0</sup>

where *B*<sup>0</sup>

wave propagating within; *<sup>i</sup>*<sup>0</sup>

electrons in the material of interest. *PM*

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 247

role of these electron wavefunctions is taken by the one-electron spinorbitals, called the

 *os o s* 

*<sup>s</sup>* is the spin part of

 <sup>2</sup> <sup>1</sup> *<sup>i</sup>*

 *r r* 

electronic energy as well as other electronic properties of this *n*-electron system are known to be unique functions of *ρ* ((Shankar, 1994), (Atkins & Friedman, 1997)). Further, the overall wavefunction satisfying the Pauli exclusion principle is often expressed in terms of the

> <sup>123</sup> 1/2 ! det *<sup>n</sup> total os os os os*

Hence, the primitive transition rate of a typical electronic excitation within a material of interest can now be expressed in terms of the Fermi's golden rule ((Shankar, 1994), (Atkins

<sup>2</sup> 0 0 1 0 <sup>0</sup>

*f f i i*

where *H*1 is the first order perturbation to the Hamiltonian of electrons caused by a light

before and after the perturbation (*H*1) takes place, respectively; *ω* is the radian frequency of the propagating light wave; *ħ* is Planck constant divided by 2*π*; and *C* is a proportional

> <sup>0</sup> 2 *<sup>M</sup> <sup>e</sup> H AP*

> > <sup>0</sup> 2 *<sup>M</sup>*

*M*

constitute electronic polarization. Thus substituting Eq. (57) into Eq. (56) will result in *Rtotal*

 *rn r r r r* 

*total os os os os*

*R C H EE* 

*os*

is the local charge probability density function. Then, the exact ground-state

*n*

*i*

 

spinorbital is a product of an orbital wavefunction and a spin wavefunction:

In the density functional theory, these spinorbitals are solutions to the equation:

Slater determinant ((Shankar, 1994), (Atkins & Friedman, 1997)) as:

,

*os* (of *<sup>i</sup>*<sup>0</sup> *Eos* ) and *<sup>f</sup>* <sup>0</sup>

1

proportional to the imaginary part of *εr* (i.e. *εrI*). At the same time (Shankar, 1994),

1

is the magnetic flux density of the injected light wave;

*<sup>e</sup> H B mc*

*f i*

is the vector potential of the injected light wave; *PM*

constant. *H*1 in Eq. (56) can be expressed as (Shankar, 1994):

moment operator of electrons in the material of interest.

*os* ) (see, e.g., (Shankar, 1994) and (Atkins & Friedman, 1997)). A

*os* .

, (54)

 

> 

*os* (of *<sup>f</sup>* <sup>0</sup> *Eos* ) are the electron states (energies)

*mc* (57)

represents electrons' straight-line motion which

(58)

*M*

represents electrons' angular

is the magnetic

, (56)

is the momentum operator of

, (55)

(53)

Combining Eqs. (47) and (48) gives the effective permeability (*μeff*) of the final mixture, i.e. (Chang & Liao, 2011),

$$
\mu\_{\rm eff} = \mu\_r \mu\_o = \mu\_o + 3f\mu\_o \frac{\mu - \mu\_o}{-2\mu + 5\mu\_o + 2f\left(\mu - \mu\_o\right)}\tag{49}
$$

Where *f = nV* is the volume ratio of the embedded particles within the mixture (0 ≤ *f* ≤ 1). In the more general situations where the host is no longer vacuum but of the permeability *μh*, then the more general mixing formula of permeabilities becomes (Chang & Liao, 2011):

$$
\mu\_{\text{eff}} = \mu\_h + \Im f \,\mu\_h \frac{\mu\_s - \mu\_h}{-2\mu\_s + \Im \mu\_h + 2f \left(\mu\_s - \mu\_h\right)} \tag{50}
$$

As with Bruggeman's approach for dielectrics (Bruggeman, 1935), the derived magnetic permeability formula can be generalized to the multi-component form (Chang & Liao, 2011):

$$\frac{\mu\_{\text{eff}} - \mu\_0}{-2\,\mu\_{\text{eff}} + 5\,\mu\_0} = \sum\_{i} f\_i \frac{\mu\_i - \mu\_0}{-2\,\mu\_i + 5\,\mu\_0} \tag{51}$$

where *fi* and *μi* denote the volume ratios and permeabilities of the involved different inclusions, respectively. Or (Chang & Liao, 2011),

$$\frac{\mu\_{\text{ref}} - 1}{-2\,\mu\_{\text{ref}} + \text{5}} = \sum\_{i} f\_{i} \frac{\mu\_{i\text{i}} - 1}{-2\,\mu\_{i\text{i}} + \text{5}} \tag{52}$$

Although the actual mixing procedures can vary widely such that substantial deviations may result between the theoretical and measured values, Eq. (52) should still serve as a valuable guide when designing magnetic materials or composites.

#### **9. A practical method to secure magnetic permeability in optical regimes**

As widely known, electronic polarization is involved in the absorption of electromagnetic wave within materials and this mechanism is represented by permittivity even in light-wave frequencies. However, refractive index which describes light absorption and reflection is calculated in terms of permittivity and permeability (accounting for magnetization). But permeability spectra in light frequencies are hardly available for most materials. In contrast with it, permeabilities for various materials are fairly well documented at microwave frequencies (see, for example, (Goldman, 1999) and (Jorgensen, 1995)). It is noted that more often than not an electronic permittivity spectrum (*εr*(*ω*)) is secured by measuring its corresponding refractive index (*N*(*ω*)) while bluntly assuming its relative permeability (*μr*) to be unity. Obviously, this approach is one-sided and inappropriate. In particular, as we are entering the nanotech era, many new possibilities should emerge and surprise us with their novel optical permeabilities, for example, the originally nonmagnetic manganese crystal can be made ferromagnetic once its lattice constant is varied (Hummel, 1985). In this section, a method is proposed such that reliable optical permeability values can be obtained numerically.

The magnetic permeability of a specific material emerges fundamentally from wavefunctions of its electrons. In the popular density functional theory (DFT) approach, the

Combining Eqs. (47) and (48) gives the effective permeability (*μeff*) of the final mixture, i.e.

Where *f = nV* is the volume ratio of the embedded particles within the mixture (0 ≤ *f* ≤ 1). In the more general situations where the host is no longer vacuum but of the permeability *μh*, then the more general mixing formula of permeabilities becomes (Chang & Liao, 2011):

As with Bruggeman's approach for dielectrics (Bruggeman, 1935), the derived magnetic permeability formula can be generalized to the multi-component form (Chang & Liao, 2011):

where *fi* and *μi* denote the volume ratios and permeabilities of the involved different

 *reff ri i reff i ri*

Although the actual mixing procedures can vary widely such that substantial deviations may result between the theoretical and measured values, Eq. (52) should still serve as a

As widely known, electronic polarization is involved in the absorption of electromagnetic wave within materials and this mechanism is represented by permittivity even in light-wave frequencies. However, refractive index which describes light absorption and reflection is calculated in terms of permittivity and permeability (accounting for magnetization). But permeability spectra in light frequencies are hardly available for most materials. In contrast with it, permeabilities for various materials are fairly well documented at microwave frequencies (see, for example, (Goldman, 1999) and (Jorgensen, 1995)). It is noted that more often than not an electronic permittivity spectrum (*εr*(*ω*)) is secured by measuring its corresponding refractive index (*N*(*ω*)) while bluntly assuming its relative permeability (*μr*) to be unity. Obviously, this approach is one-sided and inappropriate. In particular, as we are entering the nanotech era, many new possibilities should emerge and surprise us with their novel optical permeabilities, for example, the originally nonmagnetic manganese crystal can be made ferromagnetic once its lattice constant is varied (Hummel, 1985). In this section, a method is proposed such that reliable optical permeability values can be obtained numerically. The magnetic permeability of a specific material emerges fundamentally from wavefunctions of its electrons. In the popular density functional theory (DFT) approach, the

**9. A practical method to secure magnetic permeability in optical regimes** 

00 0

 

*eff <sup>i</sup>*

 

 

inclusions, respectively. Or (Chang & Liao, 2011),

*eff h h*

 

valuable guide when designing magnetic materials or composites.

3

 

*<sup>f</sup> f* (50)

*f* (51)

*f* (52)

0

 

*s h sh*

*eff r <sup>f</sup> <sup>f</sup>* (49)

25 2 

 <sup>3</sup> 252 

 *s h*

0 0 0 0 2 5 25

*i eff i i*

1 1 2 5 25

 

> 

 

0 0

 

(Chang & Liao, 2011),

role of these electron wavefunctions is taken by the one-electron spinorbitals, called the Kohn-Sham orbitals ( *os* ) (see, e.g., (Shankar, 1994) and (Atkins & Friedman, 1997)). A spinorbital is a product of an orbital wavefunction and a spin wavefunction:

$$\left| \left| \boldsymbol{\nu}\_{\boldsymbol{\alpha}} \right> \right| = \left| \left| \boldsymbol{\nu}\_{\boldsymbol{\alpha}} \right> \otimes \left| \boldsymbol{\nu}\_{\boldsymbol{s}} \right> \right. \tag{53}$$

where *<sup>o</sup>* is the orbital part, and *<sup>s</sup>* is the spin part of *os* .

In the density functional theory, these spinorbitals are solutions to the equation:

$$\rho\left(\bar{r}\right) = \sum\_{i=1}^{n} \left| \nu\_{\alpha\_{i}}\left(\bar{r}\right) \right|^{2} \,\, \, \, \tag{54}$$

where *r* is the local charge probability density function. Then, the exact ground-state electronic energy as well as other electronic properties of this *n*-electron system are known to be unique functions of *ρ* ((Shankar, 1994), (Atkins & Friedman, 1997)). Further, the overall wavefunction satisfying the Pauli exclusion principle is often expressed in terms of the Slater determinant ((Shankar, 1994), (Atkins & Friedman, 1997)) as:

$$\left|\boldsymbol{\nu}\_{\boldsymbol{\alpha}\_{\text{ball}}}\left(\boldsymbol{\bar{r}}\right) = \left(\boldsymbol{n}!\right)^{-1/2} \det\left|\boldsymbol{\nu}\_{\boldsymbol{\alpha}\_{1}}\left(\boldsymbol{\bar{r}}\right) \cdot \boldsymbol{\nu}\_{\boldsymbol{\alpha}\_{2}}\left(\boldsymbol{\bar{r}}\right) \cdot \boldsymbol{\nu}\_{\boldsymbol{\alpha}\_{3}}\left(\boldsymbol{\bar{r}}\right) \cdots \boldsymbol{\nu}\_{\boldsymbol{\alpha}\_{n}}\left(\boldsymbol{\bar{r}}\right)\right|,\tag{55}$$

Hence, the primitive transition rate of a typical electronic excitation within a material of interest can now be expressed in terms of the Fermi's golden rule ((Shankar, 1994), (Atkins & Friedman, 1997)) as:

$$\mathcal{R}\_{\text{total}} = \mathbb{C} \cdot \sum\_{f,i} \left| \left\langle \boldsymbol{\nu}\_{\text{as}}^{f0} \left| \boldsymbol{H}^{1} \left| \boldsymbol{\nu}\_{\text{as}}^{i0} \right| \right|^{2} \delta \left( \boldsymbol{E}\_{\text{as}}^{f0} - \boldsymbol{E}\_{\text{as}}^{i0} - \hbar \boldsymbol{\alpha} \right) \right\rangle \tag{56}$$

where *H*1 is the first order perturbation to the Hamiltonian of electrons caused by a light wave propagating within; *<sup>i</sup>*<sup>0</sup> *os* (of *<sup>i</sup>*<sup>0</sup> *Eos* ) and *<sup>f</sup>* <sup>0</sup> *os* (of *<sup>f</sup>* <sup>0</sup> *Eos* ) are the electron states (energies) before and after the perturbation (*H*1) takes place, respectively; *ω* is the radian frequency of the propagating light wave; *ħ* is Planck constant divided by 2*π*; and *C* is a proportional constant. *H*1 in Eq. (56) can be expressed as (Shankar, 1994):

$$H^1 = \frac{e}{2mc} \bar{A}\_0 \cdot \bar{P}\_{\text{M}} \tag{57}$$

where *A*<sup>0</sup> is the vector potential of the injected light wave; *PM* is the momentum operator of electrons in the material of interest. *PM* represents electrons' straight-line motion which constitute electronic polarization. Thus substituting Eq. (57) into Eq. (56) will result in *Rtotal* proportional to the imaginary part of *εr* (i.e. *εrI*). At the same time (Shankar, 1994),

$$H^1 = \frac{e}{2mc} \bar{B}\_0 \cdot \bar{\mu}\_M \tag{58}$$

where *B*<sup>0</sup> is the magnetic flux density of the injected light wave; *M* is the magnetic moment operator of electrons in the material of interest. *M* represents electrons' angular

Lightwave Refraction and Its Consequences: A Viewpoint of

*<sup>f</sup>*0 and *Es*

 

, spin down:

*SS S*

 *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> 

1 0

01 0 1 0 , , 22 2 1 0 0 01 *xy z*

*i* 

*<sup>s</sup>* both spin up such that

0 0

*f i*

turned off (as in the nonmagnetic case).

calculation of *R's* is now in order.

axis (*Sz*) (i.e., eigenvectors of *<sup>f</sup>* <sup>0</sup>

the two possible states of *<sup>f</sup>* <sup>0</sup>

1 0

 

bracketed term in Eq. (64), we first have:

quantities *<sup>f</sup>* <sup>0</sup>

 *<sup>M</sup> S* ( *<sup>S</sup>*

2 *ge mc*

where

1994):

are:

Spin up:

1. *<sup>f</sup>* <sup>0</sup> 

*s*

combinations of *<sup>f</sup>* <sup>0</sup>

 *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> 

 *<sup>s</sup>* , *<sup>i</sup>*<sup>0</sup> *<sup>s</sup>* , *Es*

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 249

It is a common practice in the first-principle quantum mechanical calculations (such as by the computer codes CASTEP and DMol3 ((Clark et al., 2005), (Delley, 1990)) that *R'o* is normally set equal to *εrI* simulated with the "spin polarized" option in these kind of codes

Substituting Eq. (58) into Eq. (64) gives *R's*. And then the product of *R'o* and *R's*, *R'total*, is proportional to *μrI* (the scaling factor is defined as *CM*). With *μr* being linear and causal, there is an exact one-to-one correspondence between its real and imaginary parts as prescribed by the Kramers-Kronig relation (see, e.g., (Landau & Lifshitz, 1960)). Thus, the real part of *μ<sup>r</sup>* (i.e., *μrR* ) is readily available once *μrI* is numerically obtained. To make all this happen, the

Within the formulation of *R's* spectrum (see its unsmoothed form in Eq. (64)), physical

2005), (Delley, 1990)) simulations by selecting the "spin polarized" option. In evaluating the

1

and *g* = 2 (Shankar, 1994), such that 0 000 2 2 *x x y y z z*

Cartesian coordinate. Further, it is known that the spin angular momentum around the z

+*ħ*/2 (spin up) or -*ħ*/2 (spin down), with *ħ* being the Planck's constant divided by 2*π*. Hence,

Hence, the value of the bracketed term *ts* of Eq. (66) varies according to the four possible

<sup>0</sup> <sup>000</sup> 2 2

*e B <sup>B</sup> <sup>i</sup> B e B t mc i mc* 

01 0 1 0 1

*s s s xx yy zz e e <sup>t</sup> S B <sup>j</sup> m SB SB SB jm*

0 0 0

2 22 2 4 10 0 0 1 0 *y x z*

 

 *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> 

0 1

(Shankar, 1994).

 

*<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup>

*s*

*i*

*<sup>s</sup>* , namely:

*mc mc*

 

00 0 1 0 0 0

 

2 2 *ff f ii i ss s s M s s s <sup>e</sup> t H <sup>B</sup> S B mc*

*R RR total o s* , (65)

*<sup>i</sup>*0 would emerge from CASTEP or DMol3 ((Clark et al.,

 , (66)

*e e S B SB SB SB*

*<sup>s</sup>* possess only two possible eigenvalues: either

*<sup>s</sup>* expressed on the eigenbasis of *Sz* are (Shankar,

, while the spin angular momenta *Sx*, *Sy* and *Sz*

0

*z z*

 in

0 0

*mc mc*

 

being the spin angular momentum), with the gyromagnetic ratio

momentum which constitutes magnetization. Thus substituting Eq. (58) into Eq. (56) will result in *Rtotal* proportional to the imaginary part of *μr* (i.e. *μrI*).

Because

$$\bar{P}\_{\mathcal{M}} \left| \boldsymbol{\nu}\_{s} \right\rangle = 0 \tag{59}$$

Thus according to Eq. (57), the spin part of *os* and *εrI* are irrelative. But *<sup>s</sup>* contains angular momentum and therefore must be involved in *μrI*. From Eq. (56) and Eq. (58), we can obtain *μrI* as follows:

Applying a spin-orbit decomposition (see Eqs. (53) and (55)) on the kernel of the transition rate expression in Eq. (56) suggests the convenience of defining parameters as follows:

$$
\left\langle \boldsymbol{\nu}\_{\boldsymbol{\alpha}}^{\prime 0} \right| \boldsymbol{H}^{1} \left| \boldsymbol{\nu}\_{\boldsymbol{\alpha}}^{\prime 0} \right\rangle = \left( \left\langle \boldsymbol{\nu}\_{\boldsymbol{\circ}}^{\prime 0} \right| \otimes \left\langle \boldsymbol{\nu}\_{\boldsymbol{s}}^{\prime 0} \right| \right) \cdot \boldsymbol{H}^{1} \cdot \left( \left| \boldsymbol{\nu}\_{\boldsymbol{s}}^{\prime 0} \right\rangle \otimes \left| \boldsymbol{\nu}\_{\boldsymbol{s}}^{\prime 0} \right\rangle \right) \equiv \boldsymbol{t}\_{\text{total}} \tag{60a}
$$

$$\left\langle \left. \nu\_{\boldsymbol{\nu}}^{\prime \prime 0} \right| H^{1} \left| \psi\_{\boldsymbol{\nu}}^{\prime 0} \right\rangle \equiv \mathfrak{t}\_{\boldsymbol{\nu}} \right. \tag{60b}$$

$$
\left\langle \boldsymbol{\nu}\_{s}^{\prime {}^{0}} \left| \boldsymbol{H}^{1} \left| \boldsymbol{\nu}\_{s}^{\prime {}^{0}} \right\rangle \equiv \mathbf{t}\_{s} \right. \tag{60c}
$$

where through explicit matrix operations, it can be shown that

$$\mathbf{t}\_{\text{total}} = \mathbf{t}\_o \mathbf{t}\_s \tag{61}$$

As a result, Eq. (56) can be rewritten as:

$$\boldsymbol{R}\_{\text{total}} = \mathbf{C} \cdot \sum\_{f,i} \mathbf{t}\_{\alpha} \cdot \delta \left( \mathbf{E}\_{\alpha}^{\prime 0} - \mathbf{E}\_{\alpha}^{\prime 0} - \hbar \alpha \boldsymbol{\nu} \right) = \mathbf{C} \cdot \sum\_{f,i} \mathbf{t}\_{\alpha} \cdot \delta \left( \mathbf{E}\_{\alpha}^{\prime 0} - \mathbf{E}\_{\alpha}^{\prime 0} - \hbar \alpha \boldsymbol{\nu} \right) = \mathbf{R}\_{\text{o}} \cdot \mathbf{R}\_{\text{s}} \tag{62}$$

where

$$R\_o = \mathbb{C}\_o \sum\_{f,i} \left| \left\langle \mathbb{W}\_o^{f0} \left| H^1 \left| \mathbb{W}\_o^{i0} \right\rangle \right|^2 \delta \left( E\_o^{f0} - E\_o^{i0} - \hbar o \right) \right\rangle \tag{63}$$

$$R\_s = \mathbb{C}\_s \sum\_{f,i} \left| \left\langle \boldsymbol{\nu}\_s^{\prime 0} \left| \boldsymbol{H}^1 \left| \boldsymbol{\nu}\_s^{\prime 0} \right\rangle \right|^2 \delta \left( \boldsymbol{E}\_s^{\prime 0} - \boldsymbol{E}\_s^{\prime 0} - \hbar \boldsymbol{\alpha} \right) \right. \tag{64} \right. \tag{64}$$

with *Co* and *Cs* being constant coefficients.

As aforementioned, despite being termed "electronic transition rate" to account for the lightwave absorption within a material, the primitive transition rate *Rtotal* is in essence a series of delta functions situated at varying frequencies (or energies, see Eq. (56)), and thus is too spikey to be real. In fact, this spikey nature results from our attempt to describe the dynamics of multitudes of electrons by a limited number of Kohn-Sham orbitals. Therefore, *Rtotal* ought to be "smoothed" prior to being converted to a realistic absorption spectrum. Here, Gaussian functions are adopted to replace all delta functions. The smoothed *Rtotal*, *Ro* and *Rs* are now denoted as *R'total*, *R'o* and *R's*, respectively, and thus from Eq. (62) we have:

momentum which constitutes magnetization. Thus substituting Eq. (58) into Eq. (56) will

0 *PM s* 

angular momentum and therefore must be involved in *μrI*. From Eq. (56) and Eq. (58), we

Applying a spin-orbit decomposition (see Eqs. (53) and (55)) on the kernel of the transition rate expression in Eq. (56) suggests the convenience of defining parameters as follows:

*os H Ht os <sup>o</sup> <sup>s</sup> <sup>o</sup> s total*

 

*<sup>f</sup>* <sup>0</sup> 1 0*<sup>i</sup> f f* 0 0 10 0 *i i*

*f* 0 1 0*i*

*f* 0 1 0*i*

 

 

0 0 0 0

<sup>2</sup> 0 0 1 0 <sup>0</sup>

<sup>2</sup> 0 0 1 0 <sup>0</sup>

*f f i i total os os os o s os os o s*

*f f i i*

*f f i i*

As aforementioned, despite being termed "electronic transition rate" to account for the lightwave absorption within a material, the primitive transition rate *Rtotal* is in essence a series of delta functions situated at varying frequencies (or energies, see Eq. (56)), and thus is too spikey to be real. In fact, this spikey nature results from our attempt to describe the dynamics of multitudes of electrons by a limited number of Kohn-Sham orbitals. Therefore, *Rtotal* ought to be "smoothed" prior to being converted to a realistic absorption spectrum. Here, Gaussian functions are adopted to replace all delta functions. The smoothed *Rtotal*, *Ro* and *Rs* are now denoted as *R'total*, *R'o* and *R's*, respectively, and thus from Eq. (62) we have:

*R C t E E C tt E E R R*

*oo o o o o*

*ss s s s s*

*RC H E E*

*RC H E E*

 

(59)

*o oo H t* , (60b)

*s ss H t* , (60c)

*total o s t tt* , (61)

, (62)

> 

 

, (63)

, (64)

, (60a)

*<sup>s</sup>* contains

*os* and *εrI* are irrelative. But

result in *Rtotal* proportional to the imaginary part of *μr* (i.e. *μrI*).

Thus according to Eq. (57), the spin part of

As a result, Eq. (56) can be rewritten as:

with *Co* and *Cs* being constant coefficients.

 

where through explicit matrix operations, it can be shown that

 

, ,

*f i f i*

,

,

*f i*

*f i*

can obtain *μrI* as follows:

Because

where

*R RR total o s* , (65)

It is a common practice in the first-principle quantum mechanical calculations (such as by the computer codes CASTEP and DMol3 ((Clark et al., 2005), (Delley, 1990)) that *R'o* is normally set equal to *εrI* simulated with the "spin polarized" option in these kind of codes turned off (as in the nonmagnetic case).

Substituting Eq. (58) into Eq. (64) gives *R's*. And then the product of *R'o* and *R's*, *R'total*, is proportional to *μrI* (the scaling factor is defined as *CM*). With *μr* being linear and causal, there is an exact one-to-one correspondence between its real and imaginary parts as prescribed by the Kramers-Kronig relation (see, e.g., (Landau & Lifshitz, 1960)). Thus, the real part of *μ<sup>r</sup>* (i.e., *μrR* ) is readily available once *μrI* is numerically obtained. To make all this happen, the calculation of *R's* is now in order.

Within the formulation of *R's* spectrum (see its unsmoothed form in Eq. (64)), physical quantities *<sup>f</sup>* <sup>0</sup> *<sup>s</sup>* , *<sup>i</sup>*<sup>0</sup> *<sup>s</sup>* , *Es <sup>f</sup>*0 and *Es <sup>i</sup>*0 would emerge from CASTEP or DMol3 ((Clark et al., 2005), (Delley, 1990)) simulations by selecting the "spin polarized" option. In evaluating the bracketed term in Eq. (64), we first have:

$$\mathbf{t}\_s \equiv \left\langle \boldsymbol{\nu}\_s^{\prime \, 0} \right| H^1 \left| \boldsymbol{\nu}\_s^{\prime \, 0} \right\rangle = \left\langle \boldsymbol{\nu}\_s^{\prime \, 0} \right| \frac{1}{2} \, \bar{\boldsymbol{\mu}}\_M \cdot \bar{\mathbf{B}}\_0 \left| \boldsymbol{\nu}\_s^{\prime \, 0} \right\rangle = \left\langle \boldsymbol{\nu}\_s^{\prime \, 0} \right| \frac{\mathcal{E}}{2mc} \, \bar{\mathbf{S}} \cdot \bar{\mathbf{B}}\_0 \left| \boldsymbol{\nu}\_s^{\prime \, 0} \right\rangle \, \tag{66}$$

where *<sup>M</sup> S* ( *<sup>S</sup>* being the spin angular momentum), with the gyromagnetic ratio 2 *ge mc* and *g* = 2 (Shankar, 1994), such that 0 000 2 2 *x x y y z z e e S B SB SB SB mc mc* in Cartesian coordinate. Further, it is known that the spin angular momentum around the z axis (*Sz*) (i.e., eigenvectors of *<sup>f</sup>* <sup>0</sup> *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> *<sup>s</sup>* possess only two possible eigenvalues: either +*ħ*/2 (spin up) or -*ħ*/2 (spin down), with *ħ* being the Planck's constant divided by 2*π*. Hence, the two possible states of *<sup>f</sup>* <sup>0</sup> *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> *<sup>s</sup>* expressed on the eigenbasis of *Sz* are (Shankar, 1994):

$$\begin{aligned} \text{Spin up: } \left| \boldsymbol{\nu}\_{\boldsymbol{s}} \right\rangle & \leftrightarrow \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \text{ spin down: } \left| \boldsymbol{\nu}\_{\boldsymbol{s}} \right\rangle \leftrightarrow \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \text{ while the spin angular momenta } S\_{\boldsymbol{s}}, S\_{\boldsymbol{y}} \text{ and } S\_{\boldsymbol{z}};\\ \text{area: } S\_{\boldsymbol{s}} & \leftrightarrow \frac{\hbar}{2} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad S\_{\boldsymbol{y}} \leftrightarrow \frac{\hbar}{2} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \text{ (Sharkar, 1994).} \end{aligned}$$

Hence, the value of the bracketed term *ts* of Eq. (66) varies according to the four possible combinations of *<sup>f</sup>* <sup>0</sup> *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> *<sup>s</sup>* , namely:

1. *<sup>f</sup>* <sup>0</sup> *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> *<sup>s</sup>* both spin up such that

$$t\_s \equiv \left\langle \nu\_s^{\prime/0} \middle| \frac{e}{2mc} \bar{S} \cdot \bar{B}\_o \middle| \nu\_s^{\prime 0} \right\rangle = \frac{e}{2mc} \left\langle j^\prime m^\prime \middle| S\_x B\_{0x} + S\_y B\_{0y} + S\_z B\_{0z} \middle| j m \right\rangle$$

$$\leftrightarrow \frac{e}{2mc} \begin{bmatrix} 1 & 0 \end{bmatrix} \bigg[ \frac{\hbar B\_{0x}}{2} \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} + \frac{\hbar B\_{0y}}{2} \begin{bmatrix} 0 & -i\\ i & 0 \end{bmatrix} + \frac{\hbar B\_{0z}}{2} \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix} \bigg] \begin{bmatrix} 1\\ 0 \end{bmatrix} = \frac{e\hbar}{4mc} B\_{0z} \equiv t\_z \equiv t\_z$$

Lightwave Refraction and Its Consequences: A Viewpoint of

developed, can now be fully explored for the first time.

frequencies (Lide & Frederikse, 1994).

Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 251

As mentioned, in those cases where the "spin polarized" option in, e.g., CASTEP is turned off, the resultant *εrI* spectra actually gives *R'o* in Eq. (65). With both *R's* and *R'o* being revealed, the optical *μr* spectrum of interest can be secured within a proportional constant *CM*, and then the Kramers-Kronig relation. Finally, this universal constant *CM* is uncovered by comparing the erected *μrI* spectrum with existing data covering from low to lightwave

The calculated refractive index spectrum of iron crystal from using the proposed approach is shown in Fig. 23. A comparison with the known refractive index spectrum of iron crystal exposed by a linearly polarized light (Lide & Frederikse, 1994) (see Fig. 24) reveals that both figures are relatively close in features. Accordingly, the universal proportional constant *CM* obtained from comparing the two *NI* curves is about 13.7. The completed iron *μr* spectrum is provided in Fig. 25 after applying *R'o* = *εrI*, Eq. (66) and then the Kramers-Kronig relation. Therefore, the optical *μr* and refractive index spectra of all materials, including those to be

Fig. 23. The calculated relative refractive index spectrum of the Fe crystal (*CM* = 1).

Fig. 24. The refractive index spectrum of Fe crystal in CRC Handbook.

Incidentally, to further challenge the proposed procedure, we erroneously leave the "spin polarized" action of CASTEP on when simulating the copper crystal, a knowingly


It is obvious that the values of |*ts*|2 for cases 1 and 4 are identical, i.e., |*tz*|2 ≡ *Tz*; and those for cases 2 and 3 are the same and are |*txy*|2 ≡ *Txy*.

To simplify the *R's* calculation without loss of generality, the magnetic flux density vector of a linearly polarized electromagnetic wave is oriented to parallel to the z axis, i.e., *B B* 0 0 *<sup>z</sup>* ( 0 0 <sup>0</sup> *B B x y* ). As a result, only *Tz* is relevant in the *R's* calculation. Further, as implied by its representation, *Tz* is invariant in value regardless of what the initial and final energy levels (*Es <sup>f</sup>*0 and *Es <sup>i</sup>*0) are in a transition, so long as Fermi's golden rule is satisfied. This property greatly facilitates the calculation of *R's*, in that *R's* now simply becomes proportional to the number of identical-spin transition electron pairs. Within each transition pair there is an energy difference of *ħω* while being irradiated by a linearly polarized light of frequency *ω*.

In first-principle quantum mechanical simulation codes, such as the CASTEP and DMol3, a finite number of *Es <sup>f</sup>*0 and *Es <sup>i</sup>*0 are generated to approximate the transitions of a multitude of electrons within a material of interest. In other words, each output energy value actually stands for a narrow continuous band of states centered at this specific value. Therefore, to simulate more closely to the reality, all delta functions of *Es <sup>f</sup>*0 and *Es <sup>i</sup>*0 are replaced with Gaussian functions prior to being added up into continuous density spectra, for both the "all spin-up" (case 1) and "all spin-down" (case 4) states, respectively. Since the magnetic properties are manifested by unpaired spins, and on the same energy level a spin-up is neutralized by a spin-down (Pauli's exclusion principle), the net spin density spectrum is thus settled by subtracting that of the spin-downs from that of the spin-ups.

With all inner work laid out, the detailed procedure for evaluating *R's* is outlined as follows:


It is obvious that the values of |*ts*|2 for cases 1 and 4 are identical, i.e., |*tz*|2 ≡ *Tz*; and those

To simplify the *R's* calculation without loss of generality, the magnetic flux density vector of a linearly polarized electromagnetic wave is oriented to parallel to the z axis, i.e., *B B* 0 0 *<sup>z</sup>*

( 0 0 <sup>0</sup> *B B x y* ). As a result, only *Tz* is relevant in the *R's* calculation. Further, as implied by its representation, *Tz* is invariant in value regardless of what the initial and final energy levels

greatly facilitates the calculation of *R's*, in that *R's* now simply becomes proportional to the number of identical-spin transition electron pairs. Within each transition pair there is an energy difference of *ħω* while being irradiated by a linearly polarized light of frequency *ω*. In first-principle quantum mechanical simulation codes, such as the CASTEP and DMol3, a

electrons within a material of interest. In other words, each output energy value actually stands for a narrow continuous band of states centered at this specific value. Therefore, to

Gaussian functions prior to being added up into continuous density spectra, for both the "all spin-up" (case 1) and "all spin-down" (case 4) states, respectively. Since the magnetic properties are manifested by unpaired spins, and on the same energy level a spin-up is neutralized by a spin-down (Pauli's exclusion principle), the net spin density spectrum is

With all inner work laid out, the detailed procedure for evaluating *R's* is outlined as follows: 1. Subtract the density spectrum of the spin-down states from that of the spin-ups to result in the net spin density spectrum. The positive part of it is the net spin-up density spectrum, and the absolute value of the other part is the net spin-down density

2. Randomly sample the net spin-up density spectrum at each energy of interest and denote the sampled energy value *Ei* if it is lower than the Fermi level, otherwise, name

3. Define *Pi,j*, as the product of *ni* and *nj*, where *ni* and *nj* are the density-of-states at *Ei* and *Ej*, respectively. Namely, *Pi,j* is proportional to the number densities associated with a transition pairs of net spin-up electrons linked by an energy difference of *Ei,j* ≡ (*Ej* - *Ei*). 4. Calculate *Pi,j*'s and *Ei,j*'s for each (*Ei*, *Ej*) pair to get the net spin-up *Pi,j* vs. *Ei,j* collection. 5. Obtain the *Pi,j* vs. *Ei,j* collection for the net spin-down states in a similar fashion.

6. Then, obtain the union of the *Pi,j* vs. *Ei,j* collection of the net spin-downs and that of the

7. Replace all *Pi,j* delta peaks by Gaussian functions to arrive at the desired continuous

*<sup>i</sup>*0) are in a transition, so long as Fermi's golden rule is satisfied. This property

*<sup>i</sup>*0 are generated to approximate the transitions of a multitude of

*<sup>f</sup>*0 and *Es*

*<sup>s</sup>* both spin down such that 0 <sup>4</sup> *<sup>s</sup> z z*

*<sup>s</sup>* spin down such that 0 0 <sup>4</sup> *s x <sup>y</sup> xy*

*<sup>s</sup>* spin up such that \*

*<sup>e</sup> t B iB t mc* .

0 0 4 *s x <sup>y</sup> xy <sup>e</sup> t B iB t*

*<sup>i</sup>*0 are replaced with

 .

*mc*

*<sup>e</sup> t Bt mc* .

2. *<sup>f</sup>* <sup>0</sup> 

3. *<sup>f</sup>* <sup>0</sup> 

4. *<sup>f</sup>* <sup>0</sup> 

(*Es*

*<sup>f</sup>*0 and *Es*

finite number of *Es*

spectrum.

spectrum of *R's*.

it *Ej*.

*<sup>s</sup>* spin up, and *<sup>i</sup>*<sup>0</sup>

 *<sup>s</sup>* and *<sup>i</sup>*<sup>0</sup> 

*<sup>s</sup>* spin down, and *<sup>i</sup>*<sup>0</sup>

for cases 2 and 3 are the same and are |*txy*|2 ≡ *Txy*.

*<sup>f</sup>*0 and *Es*

simulate more closely to the reality, all delta functions of *Es*

net spin-ups to result in the total *Pi,j*-*Ei,j* collection.

thus settled by subtracting that of the spin-downs from that of the spin-ups.

As mentioned, in those cases where the "spin polarized" option in, e.g., CASTEP is turned off, the resultant *εrI* spectra actually gives *R'o* in Eq. (65). With both *R's* and *R'o* being revealed, the optical *μr* spectrum of interest can be secured within a proportional constant *CM*, and then the Kramers-Kronig relation. Finally, this universal constant *CM* is uncovered by comparing the erected *μrI* spectrum with existing data covering from low to lightwave frequencies (Lide & Frederikse, 1994).

The calculated refractive index spectrum of iron crystal from using the proposed approach is shown in Fig. 23. A comparison with the known refractive index spectrum of iron crystal exposed by a linearly polarized light (Lide & Frederikse, 1994) (see Fig. 24) reveals that both figures are relatively close in features. Accordingly, the universal proportional constant *CM* obtained from comparing the two *NI* curves is about 13.7. The completed iron *μr* spectrum is provided in Fig. 25 after applying *R'o* = *εrI*, Eq. (66) and then the Kramers-Kronig relation. Therefore, the optical *μr* and refractive index spectra of all materials, including those to be developed, can now be fully explored for the first time.

Fig. 23. The calculated relative refractive index spectrum of the Fe crystal (*CM* = 1).

Fig. 24. The refractive index spectrum of Fe crystal in CRC Handbook.

Incidentally, to further challenge the proposed procedure, we erroneously leave the "spin polarized" action of CASTEP on when simulating the copper crystal, a knowingly

Lightwave Refraction and Its Consequences: A Viewpoint of

membrane double layers.

**11. References** 

(Delley, 1990)

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York. (Haus & Melcher, 1989)

York. (Jorgensen, 1995)

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Microscopic Quantum Scatterings by Electric and Magnetic Dipoles 253

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Garland Science, New York. (Alberts et al., 2007)

nonmagnetic material. It turned out that the numerically converged spin density was a minimal 1.35 x 10-4 *μB*/atom, showing that the proposed procedure is robust against erroneous initial conditions.

Fig. 25. The optical *μr* spectrum of Fe crystal obtained from the proposed procedure.
