Effective permittivity and permeability

Effective permittivity and permeability are averaged dielectric and magnetic characteristics of a microinhomogeneous medium. They are subject of Effective medium theory.[1] There are two widely used formulae.[2] They both were derived in quasi-static approximation when electric field inside a mixture particle may be considered as homogeneous. So, these formulae can not describe the particle size effect. Many attempts were undertaken to improve these formulae.

Maxwell Garnett's formula

The first formula was proposed by J.C. Maxwell Garnett.[3] Maxwell Garnett was the son of physicist William Garnett, and was named after Garnett's friend, James Clerk Maxwell. He proposed his formula to explain colored pictures that are observed in glasses doped with metal nanoparticles. His formula has a form

${\displaystyle \epsilon _{eff}=\epsilon _{d}\left[1+3c_{m}{\frac {\epsilon _{m}-\epsilon _{d}}{\epsilon _{m}+2\epsilon _{d}-c_{m}(\epsilon _{m}-\epsilon _{d})}}\right],}$ (1)

where ${\displaystyle \epsilon _{eff}}$ is effective relative complex permittivity of the mixture, ${\displaystyle \epsilon _{d}}$ is relative complex permittivity of the background medium containing small spherical inclusions of relative permittivity ${\displaystyle \epsilon _{m}}$ with volume fraction of ${\displaystyle c_{m}<<1}$. This formula is based on the equality

${\displaystyle \epsilon _{eff}=\epsilon _{d}+c_{m}{\frac {p_{m}}{\epsilon _{0}E}},}$ (2)

where ${\displaystyle \epsilon _{0}}$ is the absolute permittivity of free space and ${\displaystyle p_{m}}$ is electric dipole moment of a single inclusion induced by the external electric field E. However this equality is good only for homogeneous medium and ${\displaystyle \epsilon _{d}=1}$. Moreover the formula (1) ignores the interaction between single inclusions. Because of these circumstances, formula (1) gives too narrow and too high resonant curve for plasmon excitations in metal nanoparticles of the mixture.[4]

Bruggeman's formula

The second popular formula was proposed by D.A.G. Bruggeman.[5] His formula has a form

${\displaystyle \epsilon _{eff}={\frac {H_{b}+{\sqrt {H_{b}^{2}+8\epsilon _{m}\epsilon _{d}}}}{4}},}$ ${\displaystyle H_{b}=(2-3c_{m})\epsilon _{d}-(1-3c_{m})\epsilon _{m}.}$ (3)

Here positive sign before the square root must be altered to negative sign in some cases in order to get correct imaginary part of effective complex permittivity which is related with electromagnetic wave attenuation. This formula is based on the equality

${\displaystyle \Delta \Phi =\int \int \epsilon _{r}(\mathbf {r} )E_{n}(\mathbf {r} )ds-\epsilon _{eff}\int \int E_{0},ds=0,}$ (4)

where ${\displaystyle \Delta \Phi }$ is the jump of electric displacement flux all over the integration surface, ${\displaystyle E_{n}(\mathbf {r} )}$ is the component of microscopic electric field normal to the integration surface, ${\displaystyle \epsilon _{r}(\mathbf {r} )}$ is the local relative complex permittivity which takes the value ${\displaystyle \epsilon _{m}}$ inside the picked metal particle, the value ${\displaystyle \epsilon _{d}}$ inside the picked dielectric particle and the value ${\displaystyle \epsilon _{eff}}$ outside the picked particle, ${\displaystyle E_{0}}$ is the normal component of the macroscopic electric field. Formula (4) comes out of Maxwell's equality ${\displaystyle \mathrm {div} (\epsilon _{r}\mathbf {E} )=0}$. Thus only one picked particle is considered in Bruggeman's approach. The interaction with all the other particles is taken into account only in mean field approximation described by ${\displaystyle \epsilon _{eff}}$. Formula (3) gives a reasonable resonant curve for plasmon excitations in metal nanoparticles if their size is 10 nm or smaller. But it is unable to describe the size dependence for the resonant frequency of plasmon excitations that are observed in experiment [6]

Formula describing size effect

A new formula describing size effect was proposed.[4] This formula has a form

${\displaystyle \epsilon _{eff}={\frac {H_{\epsilon }+i{\sqrt {-H_{\epsilon }^{2}-8\epsilon _{m}\epsilon _{d}J(k_{m}a)}}}{4}},}$

${\displaystyle H_{\epsilon }=(2-3c_{m})\epsilon _{d}-(1-3c_{m})\epsilon _{m}J(k_{m}a),}$ (5)

${\displaystyle J(x)=2{\frac {1-x\mathrm {cot} (x)}{x^{2}+x\mathrm {cot} (x)-1}}}$,

where a is the nanoparticle radius and ${\displaystyle k_{m}={\sqrt {\epsilon _{m}\mu _{m}}}\omega /c}$ is wave number. It is supposed here that the time dependence of the electromagnetic field is given by the factor ${\displaystyle \mathrm {exp} (-i\omega t).}$ In this paper Bruggeman's approach was used, but electromagnetic field for electric-dipole oscillation mode inside the picked particle was computed without applying quasi-static approximation. Thus the function ${\displaystyle J(k_{m}a)}$ is due to the field nonuniformity inside the picked particle. In quasi-static region ${\displaystyle (k_{m}a<<1}$, i.e. ${\displaystyle a}$ ≤ 10 nm for Ag${\displaystyle )}$ this function becomes constant ${\displaystyle J(k_{m}a)=1}$ and formula (5) becomes identical with Bruggeman's formula (3).

Effective permeability formula

Formula for effective permeability of mixtures has a form [4]

${\displaystyle \mu _{eff}={\frac {H_{\mu }+i{\sqrt {-H_{\mu }^{2}-8\mu _{m}\mu _{d}J(k_{m}a)}}}{4}},}$ (6)

${\displaystyle H_{\mu }=(2-3c_{m})\mu _{d}-(1-3c_{m})\mu _{m}J(k_{m}a).}$

Here ${\displaystyle \mu _{eff}}$ is effective relative complex permeability of the mixture, ${\displaystyle \mu _{d}}$ is relative complex permeability of the background medium containing small spherical inclusions of relative permeability ${\displaystyle \mu _{m}}$ with volume fraction of ${\displaystyle c_{m}<<1}$. This formula was derived in dipole approximation. Magnetic octupole mode and all other magnetic oscillation modes of odd orders were neglected here. When ${\displaystyle \mu _{m}=\mu _{d}}$ and ${\displaystyle k_{m}a<<1}$ this formula has a simple form [4]

${\displaystyle \mu _{eff}=\mu _{d}\left(1+{\frac {c_{m}}{10}}{\frac {\omega ^{2}}{c^{2}}}a^{2}\epsilon _{m}\right).}$ (7)

References

1. T.C. Choy, "Effective Medium Theory", Oxford University Press, (2016) 241 p.
2. M. Scheller, C. Jansen, M. Koch, "Applications of Effective Medium Theories in the Terahertz Regime" in Recent Optical and Photonic Technologies, ed. by K.Y. Kim, Intech, Croatia, Vukovar (2010), p. 231.
3. Garnett, J. C. M. (1904). "Colours in Metal Glasses and in Metallic Films". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 203 (359–371): 385–420. doi:10.1098/rsta.1904.0024. ISSN 1364-503X.
4. Belyaev, B. A.; Tyurnev, V. V. (2018). "Electrodynamic Calculation of Effective Electromagnetic Parameters of a Dielectric Medium with Metallic Nanoparticles of a Given Size". Journal of Experimental and Theoretical Physics. 127 (4): 608–619. doi:10.1134/S1063776118100114. ISSN 1063-7761.
5. Bruggeman, D. A. G. (1935). "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen". Annalen der Physik. 416 (7): 636–664. doi:10.1002/andp.19354160705. ISSN 0003-3804.
6. S.J. Oldenburg. "Silver Nanoparticles: Properties and Applications". Sigma Aldrich. Retrieved 17 May 2019.