4682:, since it is assumed that the domains are spatially separated and electrostatic interaction between the chosen inclusions and all other neighbouring inclusions is neglected. The Maxwell Garnett formula, in contrast to Bruggeman formula, ceases to be correct when the inclusions become resonant. In the case of plasmon resonance, the Maxwell Garnett formula is correct only at volume fraction of the inclusions
3603:
3262:
4632:
4931:
482:
3407:
3088:
4461:
5357:
4360:
2360:
5611:
98:
There are many different effective medium approximations, each of them being more or less accurate in distinct conditions. Nevertheless, they all assume that the macroscopic system is homogeneous and, typical of all mean field theories, they fail to predict the properties of a multiphase medium close
2467:
The main approximation is that all the domains are located in an equivalent mean field. Unfortunately, it is not the case close to the percolation threshold where the system is governed by the largest cluster of conductors, which is a fractal, and long-range correlations that are totally absent from
500:
Here the positive sign before the square root must be altered to a negative sign in some cases in order to get the correct imaginary part of effective complex permittivity which is related with electromagnetic wave attenuation. The formula is symmetric with respect to swapping the 'd' and 'm' roles.
152:
of the medium. These parameters are interchangeable in the formulas in a whole range of models due to the wide applicability of the
Laplace equation. The problems that fall outside of this class are mainly in the field of elasticity and hydrodynamics, due to the higher order tensorial character of
6806:
and the effective resistance can be modelled in terms of graph measures and geometrical properties of networks. Assuming, edge length is much less than electrode spacing and edges to be uniformly distributed, the potential can be considered to drop uniformly from one electrode to another. Sheet
156:
EMAs can be discrete models, such as applied to resistor networks, or continuum theories as applied to elasticity or viscosity. However, most of the current theories have difficulty in describing percolating systems. Indeed, among the numerous effective medium approximations, only
Bruggeman's
5066:
The
Maxwell Garnett equation only describes the optical properties of a collection of perfectly spherical nanoparticles. However, the optical properties of nanocomposites are sensitive to the nanoparticles shape distribution. To overcome this limit, Y. Battie et al. have developed the shape
4733:
Maxwell
Garnett Equation describes optical properties of nanocomposites which consist in a collection of perfectly spherical nanoparticles. All these nanoparticles must have the same size. However, due to confinement effect, the optical properties can be influenced by the nanoparticles size
4725:. The applicability of effective medium approximation for dielectric multilayers and metal-dielectric multilayers have been studied, showing that there are certain cases where the effective medium approximation does not hold and one needs to be cautious in application of the theory.
4739:
285:
630:
2137:
1381:
1615:
6778:
6314:
967:
Without any loss of generality, we shall consider the study of the effective conductivity (which can be either dc or ac) for a system made up of spherical multicomponent inclusions with different arbitrary conductivities. Then the
Bruggeman formula takes the form:
4223:
5167:
2201:
1092:
5174:
5477:
2455:
Eq. (1) can also be obtained by requiring the deviation in current to vanish. It has been derived here from the assumption that the inclusions are spherical and it can be modified for shapes with other depolarization factors; leading to Eq. (2).
3598:{\displaystyle \varepsilon _{\mathrm {eff} }\,=\,\varepsilon _{m}\,{\frac {2\delta _{i}(\varepsilon _{i}-\varepsilon _{m})+\varepsilon _{i}+2\varepsilon _{m}}{2\varepsilon _{m}+\varepsilon _{i}-\delta _{i}(\varepsilon _{i}-\varepsilon _{m})}},}
4019:
2948:
5743:
3257:{\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-\varepsilon _{m}}{\varepsilon _{\mathrm {eff} }+2\varepsilon _{m}}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-\varepsilon _{m}}{\varepsilon _{i}+2\varepsilon _{m}}}\right),}
4627:{\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-\varepsilon _{m}}{\varepsilon _{\mathrm {eff} }+2\varepsilon _{m}}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-\varepsilon _{m}}{\varepsilon _{i}+2\varepsilon _{m}}}\right)}
71:
of the materials as a whole. In this sense, effective medium approximations are descriptions of a medium (composite material) based on the properties and the relative fractions of its components and are derived from calculations, and
4194:
3072:. 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.
5067:
distributed effective medium theory (SDEMT). This effective medium theory enables to calculate the effective dielectric function of a nanocomposite which consists in a collection of ellipsoĂŻdal nanoparticles distributed in shape.
7003:
6801:
For a network consisting of a high density of random resistors, an exact solution for each individual element may be impractical or impossible. In such case, a random resistor network can be considered as a two-dimensional
63:. Precise calculation of the many constituent values is nearly impossible. However, theories have been developed that can produce acceptable approximations which in turn describe useful parameters including the effective
509:
6457:
5057:
is the first electric Mie coefficient. This equation reveals that the classical
Maxwell Garnett equation gives a false estimation of the volume fraction nanoparticles when the size distribution cannot be neglected.
4926:{\displaystyle {\frac {(\varepsilon _{\text{eff}}-\varepsilon _{m})}{\varepsilon _{\text{eff}}-2\varepsilon _{m}}}={\frac {3i\lambda ^{3}}{16\pi ^{2}\varepsilon _{m}^{1.5}}}{\frac {f}{R_{m}^{3}}}\int P(R)a_{1}(R)dR}
2027:
1251:
1494:
6663:
918:
6180:
5934:
2724:
5865:
7740:
7229:
Bruggeman, D. A. G. (1935). "Berechnung verschiedener physikalischer
Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen".
981:
477:{\displaystyle \varepsilon _{\mathrm {eff} }={\frac {H_{b}+{\sqrt {H_{b}^{2}+8\varepsilon _{m}\varepsilon _{d}}}}{4}},{\text{ with }}H_{b}=(3c_{d}-1)\varepsilon _{d}+(3c_{m}-1)\varepsilon _{m}.}
749:
4395:
3313:
953:
838:
5983:
3933:
920:. Thus only one picked particle is considered in Bruggeman's approach. The interaction with all the other particles is taken into account only in a mean field approximation described by
2868:
2767:
6097:
5620:
4723:
1660:'s denote the appropriate doublet/triplet of depolarization factors which is governed by the ratios between the axis of the ellipse/ellipsoid. For example: in the case of a circle (
5352:{\displaystyle \beta ={\frac {1}{3}}\iint P(L_{1},L_{2})\sum _{i\mathop {=} 1}^{3}{\frac {\varepsilon _{m}}{\varepsilon _{m}+L_{i}(\varepsilon _{i}-\varepsilon _{m})}}dL_{1}dL_{2}}
5072:
90:
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.
2191:
2013:
959:
if their size is 10 nm or smaller. However, it is unable to describe the size dependence for the resonant frequency of plasmon excitations that are observed in experiments
711:
3070:
7800:
Belyaev, B. A.; Tyurnev, V. V. (2018). "Electrodynamic calculation of effective electromagnetic parameters of a dielectric medium with metallic nanoparticles of a given size".
7645:
Belyaev, B. A.; Tyurnev, V. V. (2018). "Electrodynamic
Calculation of Effective Electromagnetic Parameters of a Dielectric Medium with Metallic Nanoparticles of a Given Size".
6617:
6486:
5412:
4355:{\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-1}{\varepsilon _{\mathrm {eff} }+2}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-1}{\varepsilon _{i}+2}}\right)}
4118:
8029:
4422:
4073:
3367:
3340:
2991:
2825:
2798:
2536:
2509:
803:
776:
221:
194:
8074:
5461:
7984:
669:
150:
6653:
6162:
6062:
4680:
4449:
3394:
2430:
2403:
1975:
1928:
1444:
1236:
1209:
1182:
1155:
6577:
2858:
6932:
6026:
1863:
1822:
1781:
1740:
1699:
8194:
8149:
8104:
8059:
8014:
7969:
7908:
6544:
6517:
2355:{\displaystyle \delta _{1}{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,+\,\delta _{2}{\frac {\sigma _{2}-\sigma _{e}}{\sigma _{2}+2\sigma _{e}}}\,=\,0}
7720:
Levy, O., & Stroud, D. (1997). Maxwell
Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers. Physical Review B, 56(13), 8035.
4214:
4113:
1464:
1417:
127:
6835:
5985:
In this paper
Bruggeman's approach was used, but electromagnetic field for electric-dipole oscillation mode inside the picked particle was computed without applying
6882:
6862:
5055:
5008:
4046:
3022:
1890:
1658:
865:
275:
248:
4981:
2468:
Bruggeman's simple formula. The threshold values are in general not correctly predicted. It is 33% in the EMA, in three dimensions, far from the 16% expected from
5606:{\displaystyle \varepsilon _{\text{eff}}={\frac {1}{4}}\left(H_{\varepsilon }+i{\sqrt {-H_{\varepsilon }^{2}-8\varepsilon _{m}\varepsilon _{d}J(k_{m}a)}}\right),}
157:
symmetrical theory is able to predict a threshold. This characteristic feature of the latter theory puts it in the same category as other mean field theories of
6922:
6902:
6117:
5028:
4952:
4093:
2450:
1948:
1484:
1128:
6334:
6579:. This formula was derived in dipole approximation. Magnetic octupole mode and all other magnetic oscillation modes of odd orders were neglected here. When
3926:
For the derivation of the Maxwell Garnett equation we start with an array of polarizable particles. By using the Lorentz local field concept, we obtain the
59:
the multiple values of the constituents that directly make up the composite material. At the constituent level, the values of the materials vary and are
52:
7375:
7273:
2558:
2472:
and observed in experiments. However, in two dimensions, the EMA gives a threshold of 50% and has been proven to model percolation relatively well.
5763:
8341:
8319:
8297:
8275:
8256:
7705:
7523:
7066:
7444:
Davidson, A.; Tinkham, M. (1976). "Phenomenological equations for the electrical conductivity of microscopically inhomogeneous materials".
4048:
is the number of particles per unit volume. By using elementary electrostatics, we get for a spherical inclusion with dielectric constant
2550:. He proposed his formula to explain colored pictures that are observed in glasses doped with metal nanoparticles. His formula has a form
873:
8075:"Determination of morphological characteristics of metallic nanoparticles based on modified Maxwell-Garnett fitting of optical responses"
7336:
Granqvist, C. G.; Hunderi, O. (1978). "Conductivity of inhomogeneous materials: Effective-medium theory with dipole-dipole interaction".
5874:
7391:
7729:
Liu, Tong, et al. "Microporous Co@ CoO nanoparticles with superior microwave absorption properties." Nanoscale 6.4 (2014): 2447-2454.
8360:
625:{\displaystyle \Delta \Phi =\iint \varepsilon _{r}(\mathbf {r} )E_{n}(\mathbf {r} )ds-\varepsilon _{\mathrm {eff} }\iint E_{0}ds=0,}
8030:"Experimental and theoretical determination of the plasmonic responses and shape distribution of colloidal metallic nanoparticles"
4734:
distribution. As shown by Battie et al., the Maxwell Garnett equation can be generalized to take into account this distribution.
2132:{\displaystyle {\overline {p}}\,\propto \,V\,{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,{\overline {E_{0}}}\,.}
1376:{\displaystyle {\frac {1}{n}}\,\delta \alpha +{\frac {(1-\delta )(\sigma _{m}-\sigma _{e})}{\sigma _{m}+(n-1)\sigma _{e}}}\,=\,0}
7922:
Sukham, J.; Takayama, O., Mahmoodi, M.; Sychev, S., Bogdanov, A.; Hassan Tavassoli, S., Lavrinenko, A. V.; Malureanu R. (2019).
1610:{\displaystyle \alpha \,=\,{\frac {1}{n}}\sum _{j=1}^{n}\,{\frac {\sigma -\sigma _{e}}{\sigma _{e}+L_{j}(\sigma -\sigma _{e})}}}
6489:
68:
7985:"Extended Maxwell-Garnett-Mie formulation applied to size dispersion of metallic nanoparticles embedded in host liquid matrix"
6773:{\displaystyle \mu _{\text{eff}}=\mu _{d}\left(1+{\frac {c_{m}}{10}}{\frac {\omega ^{2}}{c^{2}}}a^{2}\varepsilon _{m}\right).}
4217:
7739:
Jepsen, Peter Uhd; Fischer, Bernd M.; Thoman, Andreas; Helm, Hanspeter; Suh, J. Y.; Lopez, René; Haglund, R. F. Jr. (2006).
7845:"Experimental demonstration of effective medium approximation breakdown in deeply subwavelength all-dielectric multilayers"
7056:
6309:{\displaystyle \mu _{\text{eff}}={\frac {1}{4}}\left(H_{\mu }+i{\sqrt {-H_{\mu }^{2}-8\mu _{m}\mu _{d}J(k_{m}a)}}\right),}
6519:
is relative complex permeability of the background medium containing small spherical inclusions of relative permeability
2800:
is relative complex permittivity of the background medium containing small spherical inclusions of relative permittivity
2193:. If the average deviation is to vanish, the total polarization summed over the two types of inclusion must vanish. Thus
6803:
5466:
The SDEMT theory was used to extract the shape distribution of nanoparticles from absorption or ellipsometric spectra.
8365:
8208:
Kumar, Ankush; Vidhyadhiraja, N. S.; Kulkarni, G. U . (2017). "Current distribution in conducting nanowire networks".
2543:
2481:
1895:
The most general case to which the Bruggeman approach has been applied involves bianisotropic ellipsoidal inclusions.
672:
7843:
Zhukovsky, S. V.; Andryieuski, A., Takayama, O.; Shkondin, E., Malureanu, R.; Jensen, F., Lavrinenko, A. V. (2015).
7540:
716:
4365:
3927:
3283:
923:
808:
7395:
7188:
Lova, Paola; Megahd, Heba; Stagnaro, Paola; Alloisio, Marina; Patrini, Maddalena; Comoretto, Davide (2020-06-15).
7139:
5986:
5939:
5936:
is wave number. It is supposed here that the time dependence of the electromagnetic field is given by the factor
83:
7299:
2745:
2432:
are respectively the volume fraction of material 1 and 2. This can be easily extended to a system of dimension
82:
are averaged dielectric and magnetic characteristics of a microinhomogeneous medium. They both were derived in
5162:{\displaystyle \varepsilon _{\text{eff}}={\frac {(1-f)\varepsilon _{m}+f\beta \varepsilon _{i}}{1-f+f\beta }}}
8118:
Battie, Y.; Izquierdo-Lorenzo, I., Resano-Garcia, A.; En Naciri, A., Akil, S.; Adam, P.M., Jradi, S. (2016).
7409:
Stroud, D. (1975). "Generalized effective-medium approach to the conductivity of an inhomogeneous material".
6067:
4453:
As the model of Maxwell Garnett is a composition of a matrix medium with inclusions we enhance the equation:
1087:{\displaystyle \sum _{i}\,\delta _{i}\,{\frac {\sigma _{i}-\sigma _{e}}{\sigma _{i}+(n-1)\sigma _{e}}}\,=\,0}
4685:
107:
3025:
2016:
868:
3621:
so long as the denominator does not vanish. A simple MATLAB calculator using this formula is as follows.
2162:
1984:
678:
8188:
8143:
8098:
8053:
8008:
7963:
7902:
7124:
M. Scheller, C. Jansen, M. Koch, "Applications of Effective Medium Theories in the Terahertz Regime" in
7032:
7027:
3042:
3036:
100:
7844:
6582:
6464:
5364:
4400:
4051:
3345:
3318:
2969:
2803:
2776:
2514:
2487:
781:
754:
199:
172:
8217:
8120:"How to determine the morphology of plasmonic nanocrystals without transmission electron microscopy?"
7866:
7809:
7756:
7654:
7602:
7552:
7488:
7453:
7418:
7376:"Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium"
7345:
7238:
7151:
2156:
1399:
This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity
7923:
5417:
4014:{\displaystyle {\frac {\varepsilon -1}{\varepsilon +2}}={\frac {4\pi }{3}}\sum _{j}N_{j}\alpha _{j}}
2994:
2943:{\displaystyle \varepsilon _{\text{eff}}=\varepsilon _{d}+c_{m}{\frac {p_{m}}{\varepsilon _{0}E}},}
2547:
130:
8165:"Ellipsometry of Colloidal solutions: New experimental setup and application to metallic colloids"
7595:
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
651:
135:
7951:
7890:
7856:
7825:
7782:
7678:
6622:
6122:
6031:
4658:
4427:
3372:
2469:
2408:
2381:
1953:
1906:
1903:
The figure illustrates a two-component medium. Consider the cross-hatched volume of conductivity
1422:
1214:
1187:
1160:
1133:
158:
6549:
5738:{\displaystyle H_{\varepsilon }=(2-3c_{m})\varepsilon _{d}-(1-3c_{m})\varepsilon _{m}J(k_{m}a),}
2830:
5992:
1827:
1786:
1745:
1704:
1663:
8337:
8329:
8315:
8307:
8293:
8285:
8271:
8252:
8119:
7943:
7882:
7701:
7670:
7620:
7519:
7254:
7211:
7062:
6522:
6495:
44:
40:
20:
5463:
is the distribution of depolarization factors.f is the volume fraction of the nanoparticles.
4729:
Generalization of the Maxwell Garnett Equation to describe the nanoparticle size distribution
4199:
4098:
1449:
1402:
112:
8225:
8176:
8164:
8131:
8086:
8041:
7996:
7935:
7874:
7817:
7772:
7764:
7662:
7610:
7560:
7496:
7461:
7426:
7387:
7353:
7307:
7246:
7201:
7159:
7098:
7052:
6810:
6867:
6840:
5033:
4986:
4024:
3000:
1868:
1636:
843:
253:
226:
4957:
4655:
In general terms, the Maxwell Garnett EMA is expected to be valid at low volume fractions
4189:{\displaystyle \alpha =\left({\frac {\varepsilon _{i}-1}{\varepsilon _{i}+2}}\right)a^{3}}
2539:
7190:"Strategies for Dielectric Contrast Enhancement in 1D Planar Polymeric Photonic Crystals"
8221:
7870:
7813:
7760:
7658:
7606:
7556:
7492:
7457:
7422:
7349:
7242:
7155:
2452:
that has an arbitrary number of components. All cases can be combined to yield Eq. (1).
6907:
6887:
6102:
5013:
4937:
4078:
3029:
2435:
1978:
1933:
1469:
1130:
that has an arbitrary number of components, the sum is made over all the constituents.
1113:
87:
6028:
is due to the field nonuniformity inside the picked particle. In quasi-static region (
103:
due to the absence of long-range correlations or critical fluctuations in the theory.
8354:
7955:
7829:
7682:
7924:"Investigation of effective media applicability for ultrathin multilayer structures"
7786:
867:
is the normal component of the macroscopic electric field. Formula (4) comes out of
7894:
7878:
2770:
2459:
A more general derivation applicable to bianisotropic materials is also available.
956:
64:
7083:
6998:{\displaystyle R_{sn}\,=\,{\frac {\pi }{2}}{\frac {\rho }{w\,t\,{\sqrt {N_{E}}}}}}
713:
is the component of microscopic electric field normal to the integration surface,
8180:
955:. Formula (3) gives a reasonable resonant curve for plasmon excitations in metal
5030:
are the mean radius and the volume fraction of the nanoparticles, respectively.
60:
48:
7768:
7568:
7541:"Two-dimensional current percolation in nanocrystalline vanadium dioxide films"
7539:
Rozen, John; Lopez, René; Haglund, Richard F. Jr.; Feldman, Leonard C. (2006).
7102:
8135:
7821:
7666:
7500:
7084:"Predictions of effective physical properties of complex multiphase materials"
7674:
7624:
7465:
7430:
7357:
7258:
7250:
7215:
7167:
1950:
and assume it is embedded in a uniform medium with an effective conductivity
7983:
Battie, Y.; Resano-Garcia, A., Chaoui, N.; Zhang, Y., En Naciri, A. (2014).
7115:
T.C. Choy, "Effective Medium Theory", Oxford University Press, (2016) 241 p.
56:
8073:
Battie, Y.; Resano-Garcia, A., En Naciri, A.; Akil, S., Chaoui, N. (2015).
8028:
Resano-Garcia, A.; Battie, Y., En Naciri, A.; Akil, S., Chaoui, N. (2015).
7947:
7886:
7615:
7590:
5474:
A new formula describing size effect was proposed. This formula has a form
7315:
1184:
are respectively the fraction and the conductivity of each component, and
8163:
Battie, Y.; Stchakovsky, M., En Naciri, A.; Akil, S., Chaoui, N. (2017).
7392:
10.1002/(SICI)1098-2760(199707)15:4<263::AID-MOP19>3.0.CO;2-8
7206:
7189:
6452:{\displaystyle H_{\mu }=(2-3c_{m})\mu _{d}-(1-3c_{m})\mu _{m}J(k_{m}a).}
3712:'WARNING: volume portion of inclusion material is out of range!'
7939:
8229:
8090:
8045:
8000:
7777:
7564:
7163:
2484:
approximation, the effective medium consists of a matrix medium with
7311:
913:{\displaystyle \operatorname {div} (\varepsilon _{r}\mathbf {E} )=0}
7861:
8290:
Electromagnetic Fields in Unconventional Materials and Structures
5929:{\displaystyle k_{m}={\sqrt {\varepsilon _{m}\mu _{m}}}\omega /c}
4983:
are the nanoparticle radius and size distribution, respectively.
3629:% constant of a mixture of an inclusion material in a base medium
3626:% This simple MATLAB calculator computes the effective dielectric
751:
is the local relative complex permittivity which takes the value
2719:{\displaystyle \varepsilon _{\text{eff}}=\varepsilon _{d}\left,}
8312:
Introduction to Complex Mediums for Optics and Electromagnetics
1211:
is the effective conductivity of the medium. (The sum over the
7306:. Vol. 40. American Institute of Physics. pp. 2–45.
7140:"Generalized approach to multiphase dielectric mixture theory"
16:
Method of approximating the properties of a composite material
5860:{\displaystyle J(x)=2{\frac {1-x\cot(x)}{x^{2}+x\cot(x)-1}},}
5062:
Generalization to include shape distribution of nanoparticles
3650:% eps_mean: effective dielectric constant of the mixture.
277:, D.A.G. Bruggeman proposed a formula of the following form:
6164:
and formula (5) becomes identical with Bruggeman's formula.
7128:, ed. by K.Y. Kim, Intech, Croatia, Vukovar (2010), p. 231.
6172:
Formula for effective permeability of mixtures has a form
8334:
Electromagnetic Anisotropy and Bianisotropy: A Field Guide
7369:
7367:
3641:% eps_incl: dielectric constant of inclusion material;
7479:
Kirkpatrick, Scott (1973). "Percolation and conduction".
8249:
Selected Papers on Linear Optical Composite Materials
7374:
Weiglhofer, W. S.; Lakhtakia, A.; Michel, B. (1998).
7138:
Tinga, W. R.; Voss, W. A. G.; Blossey, D. F. (1973).
6935:
6910:
6890:
6870:
6843:
6813:
6666:
6625:
6585:
6552:
6525:
6498:
6467:
6337:
6183:
6125:
6105:
6070:
6034:
5995:
5942:
5877:
5766:
5623:
5480:
5420:
5367:
5177:
5075:
5036:
5016:
4989:
4960:
4940:
4742:
4688:
4661:
4464:
4430:
4403:
4368:
4226:
4202:
4121:
4101:
4081:
4054:
4027:
3936:
3644:% vol_incl: volume portion of inclusion material;
3638:% eps_base: dielectric constant of base material;
3410:
3375:
3348:
3321:
3286:
3091:
3045:
3003:
2972:
2871:
2833:
2806:
2779:
2748:
2561:
2517:
2490:
2438:
2411:
2384:
2204:
2165:
2030:
1987:
1956:
1936:
1909:
1871:
1830:
1789:
1748:
1707:
1666:
1639:
1497:
1472:
1452:
1425:
1405:
1254:
1217:
1190:
1163:
1136:
1116:
984:
926:
876:
846:
811:
784:
757:
719:
681:
654:
512:
288:
256:
229:
202:
175:
138:
115:
7058:
Optical Metamaterials: Fundamentals and Applications
4397:
is the effective dielectric constant of the medium,
3868:'WARNING: the effective medium is singular!'
3315:
is the effective dielectric constant of the medium,
805:
inside the picked dielectric particle and the value
7274:"Silver Nanoparticles: Properties and Applications"
169:For a mixture of two materials with permittivities
106:The properties under consideration are usually the
6997:
6916:
6896:
6876:
6856:
6837:) can be written in terms of edge (wire) density (
6829:
6772:
6647:
6611:
6571:
6538:
6511:
6480:
6451:
6308:
6156:
6111:
6091:
6056:
6020:
5977:
5928:
5859:
5737:
5605:
5455:
5406:
5351:
5161:
5049:
5022:
5002:
4975:
4946:
4925:
4717:
4674:
4626:
4443:
4416:
4389:
4354:
4208:
4188:
4107:
4087:
4067:
4040:
4013:
3597:
3388:
3361:
3334:
3307:
3256:
3064:
3016:
2985:
2942:
2852:
2819:
2792:
2761:
2718:
2530:
2503:
2444:
2424:
2397:
2354:
2185:
2131:
2007:
1969:
1942:
1922:
1884:
1857:
1816:
1775:
1734:
1693:
1652:
1609:
1478:
1458:
1438:
1411:
1375:
1230:
1203:
1176:
1149:
1122:
1086:
947:
912:
859:
832:
797:
770:
743:
705:
663:
624:
476:
269:
242:
215:
188:
144:
121:
8270:(1st ed.). Oxford: Oxford University Press.
7591:"Colours in Metal Glasses and in Metallic Films"
7300:"Electrical conductivity in inhomogeneous media"
7802:Journal of Experimental and Theoretical Physics
7745:thin film observed with terahertz spectroscopy"
7647:Journal of Experimental and Theoretical Physics
1486:dimensional. For randomly oriented inclusions,
3028:of a single inclusion induced by the external
744:{\displaystyle \varepsilon _{r}(\mathbf {r} )}
8336:(1st ed.). Singapore: World Scientific.
7091:Materials Science and Engineering: R: Reports
6797:Effective medium theory for resistor networks
5414:) only depend on the shape of nanoparticles.
4390:{\displaystyle \varepsilon _{\mathrm {eff} }}
3308:{\displaystyle \varepsilon _{\mathrm {eff} }}
948:{\displaystyle \varepsilon _{\mathrm {eff} }}
833:{\displaystyle \varepsilon _{\mathrm {eff} }}
8:
8193:: CS1 maint: multiple names: authors list (
8148:: CS1 maint: multiple names: authors list (
8103:: CS1 maint: multiple names: authors list (
8058:: CS1 maint: multiple names: authors list (
8013:: CS1 maint: multiple names: authors list (
7968:: CS1 maint: multiple names: authors list (
7907:: CS1 maint: multiple names: authors list (
778:inside the picked metal particle, the value
5978:{\displaystyle \mathrm {exp} (-i\omega t).}
3399:The Maxwell Garnett equation is solved by:
1110:In a system of Euclidean spatial dimension
7640:
7638:
7636:
7634:
7293:
7291:
3396:is the volume fraction of the inclusions.
7860:
7776:
7741:"Metal-insulator phase transition in a VO
7614:
7205:
6984:
6978:
6977:
6973:
6964:
6954:
6953:
6949:
6940:
6934:
6909:
6889:
6869:
6848:
6842:
6818:
6812:
6756:
6746:
6734:
6724:
6718:
6707:
6701:
6684:
6671:
6665:
6630:
6624:
6603:
6590:
6584:
6557:
6551:
6530:
6524:
6503:
6497:
6472:
6466:
6434:
6418:
6405:
6380:
6367:
6342:
6336:
6284:
6268:
6258:
6242:
6237:
6228:
6216:
6197:
6188:
6182:
6136:
6124:
6104:
6081:
6077:
6069:
6039:
6033:
6006:
5994:
5943:
5941:
5918:
5907:
5897:
5891:
5882:
5876:
5818:
5785:
5765:
5720:
5704:
5691:
5666:
5653:
5628:
5622:
5581:
5565:
5555:
5539:
5534:
5525:
5513:
5494:
5485:
5479:
5444:
5431:
5419:
5398:
5385:
5372:
5366:
5343:
5330:
5311:
5298:
5285:
5272:
5261:
5255:
5249:
5237:
5233:
5220:
5207:
5184:
5176:
5130:
5111:
5089:
5080:
5074:
5041:
5035:
5015:
4994:
4988:
4959:
4939:
4902:
4875:
4870:
4861:
4852:
4847:
4837:
4822:
4809:
4797:
4781:
4766:
4753:
4743:
4741:
4706:
4693:
4687:
4666:
4660:
4611:
4595:
4583:
4570:
4563:
4553:
4533:
4510:
4509:
4497:
4477:
4476:
4469:
4463:
4451:is the volume fraction of the inclusions.
4435:
4429:
4408:
4402:
4374:
4373:
4367:
4333:
4315:
4308:
4298:
4265:
4264:
4239:
4238:
4231:
4225:
4201:
4180:
4157:
4139:
4132:
4120:
4100:
4080:
4059:
4053:
4032:
4026:
4005:
3995:
3985:
3966:
3937:
3935:
3632:% according to the Maxwell Garnett theory
3580:
3567:
3554:
3541:
3528:
3513:
3497:
3481:
3468:
3455:
3445:
3444:
3438:
3433:
3429:
3416:
3415:
3409:
3380:
3374:
3353:
3347:
3326:
3320:
3292:
3291:
3285:
3238:
3222:
3210:
3197:
3190:
3180:
3160:
3137:
3136:
3124:
3104:
3103:
3096:
3090:
3050:
3044:
3035:. However this equality is good only for
3008:
3002:
2977:
2971:
2925:
2914:
2908:
2902:
2889:
2876:
2870:
2838:
2832:
2811:
2805:
2784:
2778:
2762:{\displaystyle \varepsilon _{\text{eff}}}
2753:
2747:
2696:
2683:
2670:
2657:
2641:
2629:
2616:
2609:
2603:
2579:
2566:
2560:
2522:
2516:
2495:
2489:
2437:
2416:
2410:
2389:
2383:
2348:
2344:
2335:
2319:
2307:
2294:
2287:
2281:
2276:
2272:
2263:
2247:
2235:
2222:
2215:
2209:
2203:
2172:
2166:
2164:
2125:
2114:
2108:
2107:
2098:
2082:
2070:
2057:
2050:
2049:
2045:
2041:
2031:
2029:
2015:then elementary considerations lead to a
1994:
1988:
1986:
1961:
1955:
1935:
1914:
1908:
1876:
1870:
1847:
1835:
1829:
1806:
1794:
1788:
1765:
1753:
1747:
1724:
1712:
1706:
1683:
1671:
1665:
1644:
1638:
1595:
1576:
1563:
1551:
1538:
1537:
1531:
1520:
1506:
1505:
1501:
1496:
1471:
1451:
1430:
1424:
1404:
1369:
1365:
1356:
1328:
1313:
1300:
1275:
1265:
1255:
1253:
1222:
1216:
1195:
1189:
1168:
1162:
1141:
1135:
1115:
1080:
1076:
1067:
1039:
1027:
1014:
1007:
1006:
1000:
995:
989:
983:
932:
931:
925:
896:
890:
875:
851:
845:
817:
816:
810:
789:
783:
762:
756:
733:
724:
718:
695:
686:
680:
653:
601:
581:
580:
559:
550:
538:
529:
511:
465:
446:
427:
408:
389:
380:
363:
353:
337:
332:
326:
317:
310:
294:
293:
287:
261:
255:
234:
228:
207:
201:
180:
174:
137:
114:
7126:Recent Optical and Photonic Technologies
2860:. This formula is based on the equality
2546:, and was named after Garnett's friend,
7044:
6092:{\displaystyle a\leq \mathrm {10\,nm} }
675:flux all over the integration surface,
80:Effective permittivity and permeability
8186:
8141:
8096:
8051:
8006:
7961:
7900:
4718:{\displaystyle \delta _{i}<10^{-5}}
501:This formula is based on the equality
76:. There are two widely used formulae.
7055:; Shalaev, Vladimir (November 2009).
6807:resistance of such a random network (
3080:The Maxwell Garnett equation reads:
1242:Elliptical and ellipsoidal inclusions
7:
6926:
6657:
6174:
5614:
4455:
3401:
3082:
2862:
2552:
2195:
2021:
1488:
1245:
975:
503:
279:
223:with corresponding volume fractions
8314:. Bellingham, WA, USA: SPIE Press.
8251:. Bellingham, WA, USA: SPIE Press.
2995:absolute permittivity of free space
2186:{\displaystyle {\overline {E_{0}}}}
2008:{\displaystyle {\overline {E_{0}}}}
706:{\displaystyle E_{n}(\mathbf {r} )}
7061:. Springer. pp. Chapter 2.4.
6085:
6082:
5950:
5947:
5944:
4517:
4514:
4511:
4484:
4481:
4478:
4381:
4378:
4375:
4272:
4269:
4266:
4246:
4243:
4240:
3423:
3420:
3417:
3299:
3296:
3293:
3144:
3141:
3138:
3111:
3108:
3105:
3065:{\displaystyle \varepsilon _{d}=1}
939:
936:
933:
824:
821:
818:
658:
655:
588:
585:
582:
516:
513:
301:
298:
295:
55:. EMAs or EMTs are developed from
14:
6612:{\displaystyle \mu _{m}=\mu _{d}}
6481:{\displaystyle \mu _{\text{eff}}}
5407:{\displaystyle L_{1},L_{2},L_{3}}
972:Circular and spherical inclusions
8292:. New York: Wiley-Interscience.
8124:Journal of Nanoparticle Research
6655:this formula has a simple form
4417:{\displaystyle \varepsilon _{i}}
4068:{\displaystyle \varepsilon _{i}}
3362:{\displaystyle \varepsilon _{m}}
3335:{\displaystyle \varepsilon _{i}}
2986:{\displaystyle \varepsilon _{0}}
2820:{\displaystyle \varepsilon _{m}}
2793:{\displaystyle \varepsilon _{d}}
2531:{\displaystyle \varepsilon _{i}}
2504:{\displaystyle \varepsilon _{m}}
1930:, take it as a sphere of volume
1446:. The fraction of inclusions is
897:
798:{\displaystyle \varepsilon _{d}}
771:{\displaystyle \varepsilon _{m}}
734:
696:
560:
539:
216:{\displaystyle \varepsilon _{d}}
189:{\displaystyle \varepsilon _{m}}
153:the effective medium constants.
7516:The Physics of Amorphous Solids
6119:this function becomes constant
5871:is the nanoparticle radius and
2463:Modeling of percolating systems
1742:) and in the case of a sphere (
25:effective medium approximations
8310:; Lakhtakia (Ed.), A. (2003).
7879:10.1103/PhysRevLett.115.177402
6443:
6427:
6411:
6389:
6373:
6351:
6293:
6277:
6168:Effective permeability formula
6145:
6129:
6106:
6015:
5999:
5969:
5954:
5842:
5836:
5809:
5803:
5776:
5770:
5729:
5713:
5697:
5675:
5659:
5637:
5590:
5574:
5470:Formula describing size effect
5456:{\displaystyle P(L_{1},L_{2})}
5450:
5424:
5317:
5291:
5226:
5200:
5104:
5092:
4970:
4964:
4914:
4908:
4895:
4889:
4772:
4746:
3586:
3560:
3487:
3461:
2702:
2676:
1601:
1582:
1419:into a matrix of conductivity
1349:
1337:
1319:
1293:
1290:
1278:
1060:
1048:
901:
883:
738:
730:
700:
692:
564:
556:
543:
535:
458:
436:
420:
398:
1:
7298:Landauer, Rolf (April 1978).
6490:relative complex permeability
2771:relative complex permittivity
840:outside the picked particle,
8181:10.1021/acs.langmuir.7b00490
5361:The depolarization factors (
3665:eps_base, eps_incl, vol_incl
2178:
2120:
2036:
2000:
664:{\displaystyle \Delta \Phi }
145:{\displaystyle \varepsilon }
47:modeling that describes the
8247:Lakhtakia, A., ed. (1996).
8034:Journal of Chemical Physics
7989:Journal of Chemical Physics
7700:. Oxford: Clarendon Press.
6648:{\displaystyle k_{m}a\ll 1}
6157:{\displaystyle J(k_{m}a)=1}
6057:{\displaystyle k_{m}a\ll 1}
4675:{\displaystyle \delta _{i}}
4444:{\displaystyle \delta _{i}}
3389:{\displaystyle \delta _{i}}
2425:{\displaystyle \delta _{2}}
2398:{\displaystyle \delta _{1}}
2019:associated with the volume
1970:{\displaystyle \sigma _{e}}
1923:{\displaystyle \sigma _{1}}
1439:{\displaystyle \sigma _{m}}
1231:{\displaystyle \delta _{i}}
1204:{\displaystyle \sigma _{e}}
1177:{\displaystyle \sigma _{i}}
1150:{\displaystyle \delta _{i}}
8382:
8210:Journal of Applied Physics
7769:10.1103/PhysRevB.74.205103
7589:Garnett, J. C. M. (1904).
7380:Microw. Opt. Technol. Lett
7304:AIP Conference Proceedings
7103:10.1016/j.mser.2008.07.001
6572:{\displaystyle c_{m}\ll 1}
5987:quasi-static approximation
3928:Clausius-Mossotti relation
2853:{\displaystyle c_{m}\ll 1}
2159:produces a deviation from
1981:far from the inclusion is
84:quasi-static approximation
8136:10.1007/s11051-016-3533-8
7822:10.1134/S1063776118100114
7667:10.1134/S1063776118100114
7501:10.1103/RevModPhys.45.574
6021:{\displaystyle J(k_{m}a)}
4218:Clausius Mosotti equation
2542:was the son of physicist
1858:{\displaystyle L_{3}=1/3}
1817:{\displaystyle L_{2}=1/3}
1776:{\displaystyle L_{1}=1/3}
1735:{\displaystyle L_{2}=1/2}
1694:{\displaystyle L_{1}=1/2}
8361:Condensed matter physics
8332:; Lakhtakia, A. (2010).
7514:Zallen, Richard (1998).
7466:10.1103/PhysRevB.13.3261
7431:10.1103/PhysRevB.12.3368
7358:10.1103/PhysRevB.18.1554
7251:10.1002/andp.19354160705
7082:Wang, M; Pan, N (2008).
6546:with volume fraction of
6539:{\displaystyle \mu _{m}}
6512:{\displaystyle \mu _{d}}
3623:
2827:with volume fraction of
2476:Maxwell Garnett equation
8268:Effective Medium Theory
8079:Applied Physics Letters
7849:Physical Review Letters
7698:Effective Medium Theory
6924:) of edges (wires) as:
4209:{\displaystyle \alpha }
4108:{\displaystyle \alpha }
3342:of the inclusions, and
1459:{\displaystyle \delta }
1412:{\displaystyle \sigma }
122:{\displaystyle \sigma }
74:effective medium theory
33:effective medium theory
7696:Choy, Tuck C. (1999).
7616:10.1098/rsta.1904.0024
7518:. Wiley-Interscience.
6999:
6918:
6898:
6878:
6858:
6831:
6830:{\displaystyle R_{sn}}
6774:
6649:
6613:
6573:
6540:
6513:
6482:
6453:
6310:
6158:
6113:
6093:
6058:
6022:
5979:
5930:
5861:
5739:
5607:
5457:
5408:
5353:
5254:
5163:
5051:
5024:
5004:
4977:
4948:
4927:
4719:
4676:
4628:
4445:
4418:
4391:
4356:
4210:
4190:
4109:
4089:
4069:
4042:
4015:
3599:
3390:
3363:
3336:
3309:
3258:
3066:
3026:electric dipole moment
3018:
2987:
2944:
2854:
2821:
2794:
2763:
2720:
2532:
2505:
2446:
2426:
2399:
2356:
2187:
2133:
2009:
1971:
1944:
1924:
1886:
1859:
1818:
1777:
1736:
1695:
1654:
1611:
1536:
1480:
1460:
1440:
1413:
1377:
1232:
1205:
1178:
1151:
1124:
1088:
949:
914:
861:
834:
799:
772:
745:
707:
665:
626:
478:
271:
244:
217:
190:
146:
123:
7033:Percolation threshold
7028:Constitutive equation
7000:
6919:
6899:
6879:
6877:{\displaystyle \rho }
6859:
6857:{\displaystyle N_{E}}
6832:
6775:
6650:
6614:
6574:
6541:
6514:
6483:
6454:
6311:
6159:
6114:
6094:
6059:
6023:
5980:
5931:
5862:
5740:
5608:
5458:
5409:
5354:
5229:
5164:
5052:
5050:{\displaystyle a_{1}}
5025:
5005:
5003:{\displaystyle R_{m}}
4978:
4949:
4928:
4720:
4677:
4629:
4446:
4419:
4392:
4357:
4211:
4191:
4110:
4090:
4070:
4043:
4041:{\displaystyle N_{j}}
4016:
3660:MaxwellGarnettFormula
3600:
3391:
3364:
3337:
3310:
3259:
3067:
3019:
3017:{\displaystyle p_{m}}
2988:
2945:
2855:
2822:
2795:
2764:
2721:
2533:
2506:
2447:
2427:
2400:
2357:
2188:
2134:
2010:
1972:
1945:
1925:
1887:
1885:{\displaystyle L_{j}}
1865:). (The sum over the
1860:
1819:
1778:
1737:
1696:
1655:
1653:{\displaystyle L_{j}}
1612:
1516:
1481:
1461:
1441:
1414:
1378:
1233:
1206:
1179:
1152:
1125:
1089:
950:
915:
862:
860:{\displaystyle E_{0}}
835:
800:
773:
746:
708:
673:electric displacement
666:
627:
479:
272:
270:{\displaystyle c_{d}}
245:
243:{\displaystyle c_{m}}
218:
191:
164:
147:
124:
101:percolation threshold
7601:(359–371): 385–420.
6933:
6908:
6888:
6868:
6841:
6811:
6664:
6623:
6583:
6550:
6523:
6496:
6465:
6335:
6181:
6123:
6103:
6068:
6032:
5993:
5989:. Thus the function
5940:
5875:
5764:
5621:
5478:
5418:
5365:
5175:
5073:
5034:
5014:
4987:
4976:{\displaystyle P(R)}
4958:
4938:
4740:
4686:
4659:
4462:
4428:
4401:
4366:
4224:
4200:
4119:
4099:
4079:
4052:
4025:
3934:
3408:
3373:
3346:
3319:
3284:
3089:
3043:
3001:
2970:
2869:
2831:
2804:
2777:
2746:
2559:
2515:
2511:and inclusions with
2488:
2436:
2409:
2382:
2202:
2163:
2028:
1985:
1954:
1934:
1907:
1869:
1828:
1787:
1746:
1705:
1664:
1637:
1495:
1470:
1450:
1423:
1403:
1252:
1215:
1188:
1161:
1134:
1114:
982:
924:
874:
844:
809:
782:
755:
717:
679:
652:
510:
286:
254:
227:
200:
173:
136:
113:
8286:Lakhtakia (Ed.), A.
8266:Tuck, Choy (1999).
8222:2017JAP...122d5101K
7934:(26): 12582–12588.
7871:2015PhRvL.115q7402Z
7814:2018JETP..127..608B
7761:2006PhRvB..74t5103J
7659:2018JETP..127..608B
7607:1904RSPTA.203..385G
7557:2006ApPhL..88h1902R
7493:1973RvMP...45..574K
7458:1976PhRvB..13.3261D
7423:1975PhRvB..12.3368S
7350:1978PhRvB..18.1554G
7243:1935AnP...416..636B
7207:10.3390/app10124122
7156:1973JAP....44.3897T
7087:(Free PDF download)
6247:
5544:
4880:
4857:
4424:of the inclusions;
3859:small_number_cutoff
3670:small_number_cutoff
2548:James Clerk Maxwell
342:
131:dielectric constant
53:composite materials
8366:Physical chemistry
7940:10.1039/C9NR02471A
7231:Annalen der Physik
6995:
6914:
6894:
6874:
6854:
6827:
6770:
6645:
6609:
6569:
6536:
6509:
6478:
6449:
6306:
6233:
6154:
6109:
6089:
6054:
6018:
5975:
5926:
5857:
5735:
5603:
5530:
5453:
5404:
5349:
5159:
5047:
5020:
5000:
4973:
4944:
4923:
4866:
4843:
4715:
4672:
4624:
4441:
4414:
4387:
4352:
4206:
4186:
4105:
4085:
4065:
4038:
4011:
3990:
3595:
3386:
3359:
3332:
3305:
3254:
3062:
3037:homogeneous medium
3014:
2983:
2940:
2850:
2817:
2790:
2759:
2716:
2528:
2501:
2470:percolation theory
2442:
2422:
2395:
2352:
2183:
2129:
2005:
1967:
1940:
1920:
1882:
1855:
1814:
1773:
1732:
1691:
1650:
1607:
1476:
1466:and the system is
1456:
1436:
1409:
1373:
1228:
1201:
1174:
1147:
1120:
1084:
994:
945:
910:
869:Maxwell's equality
857:
830:
795:
768:
741:
703:
661:
622:
474:
328:
267:
240:
213:
186:
159:critical phenomena
142:
119:
8343:978-981-4289-61-0
8321:978-0-8194-4947-4
8299:978-0-471-36356-9
8277:978-0-19-851892-1
8258:978-0-8194-2152-4
8230:10.1063/1.4985792
8091:10.1063/1.4932638
8046:10.1063/1.4916917
8001:10.1063/1.4862995
7707:978-0-19-851892-1
7565:10.1063/1.2175490
7525:978-0-471-29941-7
7164:10.1063/1.1662868
7068:978-1-4419-1150-6
7019:
7018:
6993:
6990:
6962:
6917:{\displaystyle t}
6904:) and thickness (
6897:{\displaystyle w}
6794:
6793:
6740:
6716:
6674:
6475:
6330:
6329:
6296:
6205:
6191:
6112:{\displaystyle )}
5913:
5852:
5759:
5758:
5593:
5502:
5488:
5321:
5192:
5157:
5083:
5023:{\displaystyle f}
4947:{\displaystyle R}
4881:
4859:
4804:
4784:
4756:
4648:
4647:
4618:
4540:
4346:
4285:
4170:
4095:a polarisability
4088:{\displaystyle a}
3981:
3979:
3961:
3619:
3618:
3590:
3278:
3277:
3245:
3167:
2964:
2963:
2935:
2879:
2756:
2740:
2739:
2706:
2569:
2445:{\displaystyle n}
2376:
2375:
2342:
2270:
2181:
2153:
2152:
2123:
2105:
2039:
2003:
1943:{\displaystyle V}
1631:
1630:
1605:
1514:
1479:{\displaystyle n}
1397:
1396:
1363:
1263:
1123:{\displaystyle n}
1108:
1107:
1074:
985:
646:
645:
498:
497:
383:
375:
369:
165:Bruggeman's model
21:materials science
8373:
8347:
8325:
8308:Weiglhofer (Ed.)
8303:
8281:
8262:
8234:
8233:
8205:
8199:
8198:
8192:
8184:
8160:
8154:
8153:
8147:
8139:
8115:
8109:
8108:
8102:
8094:
8070:
8064:
8063:
8057:
8049:
8025:
8019:
8018:
8012:
8004:
7980:
7974:
7973:
7967:
7959:
7919:
7913:
7912:
7906:
7898:
7864:
7840:
7834:
7833:
7797:
7791:
7790:
7780:
7736:
7730:
7727:
7721:
7718:
7712:
7711:
7693:
7687:
7686:
7642:
7629:
7628:
7618:
7586:
7580:
7579:
7577:
7576:
7567:. Archived from
7545:Appl. Phys. Lett
7536:
7530:
7529:
7511:
7505:
7504:
7476:
7470:
7469:
7452:(8): 3261–3267.
7441:
7435:
7434:
7417:(8): 3368–3373.
7406:
7400:
7399:
7394:. Archived from
7371:
7362:
7361:
7344:(4): 1554–1561.
7333:
7327:
7326:
7324:
7323:
7314:. Archived from
7295:
7286:
7285:
7283:
7281:
7272:S.J. Oldenburg.
7269:
7263:
7262:
7226:
7220:
7219:
7209:
7194:Applied Sciences
7185:
7179:
7178:
7176:
7175:
7166:. Archived from
7135:
7129:
7122:
7116:
7113:
7107:
7106:
7088:
7079:
7073:
7072:
7049:
7013:
7004:
7002:
7001:
6996:
6994:
6992:
6991:
6989:
6988:
6979:
6965:
6963:
6955:
6948:
6947:
6927:
6923:
6921:
6920:
6915:
6903:
6901:
6900:
6895:
6883:
6881:
6880:
6875:
6864:), resistivity (
6863:
6861:
6860:
6855:
6853:
6852:
6836:
6834:
6833:
6828:
6826:
6825:
6788:
6779:
6777:
6776:
6771:
6766:
6762:
6761:
6760:
6751:
6750:
6741:
6739:
6738:
6729:
6728:
6719:
6717:
6712:
6711:
6702:
6689:
6688:
6676:
6675:
6672:
6658:
6654:
6652:
6651:
6646:
6635:
6634:
6618:
6616:
6615:
6610:
6608:
6607:
6595:
6594:
6578:
6576:
6575:
6570:
6562:
6561:
6545:
6543:
6542:
6537:
6535:
6534:
6518:
6516:
6515:
6510:
6508:
6507:
6492:of the mixture,
6487:
6485:
6484:
6479:
6477:
6476:
6473:
6458:
6456:
6455:
6450:
6439:
6438:
6423:
6422:
6410:
6409:
6385:
6384:
6372:
6371:
6347:
6346:
6324:
6315:
6313:
6312:
6307:
6302:
6298:
6297:
6289:
6288:
6273:
6272:
6263:
6262:
6246:
6241:
6229:
6221:
6220:
6206:
6198:
6193:
6192:
6189:
6175:
6163:
6161:
6160:
6155:
6141:
6140:
6118:
6116:
6115:
6110:
6098:
6096:
6095:
6090:
6088:
6063:
6061:
6060:
6055:
6044:
6043:
6027:
6025:
6024:
6019:
6011:
6010:
5984:
5982:
5981:
5976:
5953:
5935:
5933:
5932:
5927:
5922:
5914:
5912:
5911:
5902:
5901:
5892:
5887:
5886:
5870:
5866:
5864:
5863:
5858:
5853:
5851:
5823:
5822:
5812:
5786:
5753:
5744:
5742:
5741:
5736:
5725:
5724:
5709:
5708:
5696:
5695:
5671:
5670:
5658:
5657:
5633:
5632:
5615:
5612:
5610:
5609:
5604:
5599:
5595:
5594:
5586:
5585:
5570:
5569:
5560:
5559:
5543:
5538:
5526:
5518:
5517:
5503:
5495:
5490:
5489:
5486:
5462:
5460:
5459:
5454:
5449:
5448:
5436:
5435:
5413:
5411:
5410:
5405:
5403:
5402:
5390:
5389:
5377:
5376:
5358:
5356:
5355:
5350:
5348:
5347:
5335:
5334:
5322:
5320:
5316:
5315:
5303:
5302:
5290:
5289:
5277:
5276:
5266:
5265:
5256:
5253:
5248:
5241:
5225:
5224:
5212:
5211:
5193:
5185:
5168:
5166:
5165:
5160:
5158:
5156:
5136:
5135:
5134:
5116:
5115:
5090:
5085:
5084:
5081:
5056:
5054:
5053:
5048:
5046:
5045:
5029:
5027:
5026:
5021:
5009:
5007:
5006:
5001:
4999:
4998:
4982:
4980:
4979:
4974:
4953:
4951:
4950:
4945:
4932:
4930:
4929:
4924:
4907:
4906:
4882:
4879:
4874:
4862:
4860:
4858:
4856:
4851:
4842:
4841:
4828:
4827:
4826:
4810:
4805:
4803:
4802:
4801:
4786:
4785:
4782:
4775:
4771:
4770:
4758:
4757:
4754:
4744:
4724:
4722:
4721:
4716:
4714:
4713:
4698:
4697:
4681:
4679:
4678:
4673:
4671:
4670:
4642:
4633:
4631:
4630:
4625:
4623:
4619:
4617:
4616:
4615:
4600:
4599:
4589:
4588:
4587:
4575:
4574:
4564:
4558:
4557:
4545:
4541:
4539:
4538:
4537:
4522:
4521:
4520:
4503:
4502:
4501:
4489:
4488:
4487:
4470:
4456:
4450:
4448:
4447:
4442:
4440:
4439:
4423:
4421:
4420:
4415:
4413:
4412:
4396:
4394:
4393:
4388:
4386:
4385:
4384:
4361:
4359:
4358:
4353:
4351:
4347:
4345:
4338:
4337:
4327:
4320:
4319:
4309:
4303:
4302:
4290:
4286:
4284:
4277:
4276:
4275:
4258:
4251:
4250:
4249:
4232:
4215:
4213:
4212:
4207:
4195:
4193:
4192:
4187:
4185:
4184:
4175:
4171:
4169:
4162:
4161:
4151:
4144:
4143:
4133:
4114:
4112:
4111:
4106:
4094:
4092:
4091:
4086:
4074:
4072:
4071:
4066:
4064:
4063:
4047:
4045:
4044:
4039:
4037:
4036:
4020:
4018:
4017:
4012:
4010:
4009:
4000:
3999:
3989:
3980:
3975:
3967:
3962:
3960:
3949:
3938:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3664:
3661:
3658:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3613:
3604:
3602:
3601:
3596:
3591:
3589:
3585:
3584:
3572:
3571:
3559:
3558:
3546:
3545:
3533:
3532:
3519:
3518:
3517:
3502:
3501:
3486:
3485:
3473:
3472:
3460:
3459:
3446:
3443:
3442:
3428:
3427:
3426:
3402:
3395:
3393:
3392:
3387:
3385:
3384:
3368:
3366:
3365:
3360:
3358:
3357:
3341:
3339:
3338:
3333:
3331:
3330:
3314:
3312:
3311:
3306:
3304:
3303:
3302:
3272:
3263:
3261:
3260:
3255:
3250:
3246:
3244:
3243:
3242:
3227:
3226:
3216:
3215:
3214:
3202:
3201:
3191:
3185:
3184:
3172:
3168:
3166:
3165:
3164:
3149:
3148:
3147:
3130:
3129:
3128:
3116:
3115:
3114:
3097:
3083:
3071:
3069:
3068:
3063:
3055:
3054:
3034:
3023:
3021:
3020:
3015:
3013:
3012:
2992:
2990:
2989:
2984:
2982:
2981:
2958:
2949:
2947:
2946:
2941:
2936:
2934:
2930:
2929:
2919:
2918:
2909:
2907:
2906:
2894:
2893:
2881:
2880:
2877:
2863:
2859:
2857:
2856:
2851:
2843:
2842:
2826:
2824:
2823:
2818:
2816:
2815:
2799:
2797:
2796:
2791:
2789:
2788:
2773:of the mixture,
2768:
2766:
2765:
2760:
2758:
2757:
2754:
2734:
2725:
2723:
2722:
2717:
2712:
2708:
2707:
2705:
2701:
2700:
2688:
2687:
2675:
2674:
2662:
2661:
2646:
2645:
2635:
2634:
2633:
2621:
2620:
2610:
2608:
2607:
2584:
2583:
2571:
2570:
2567:
2553:
2537:
2535:
2534:
2529:
2527:
2526:
2510:
2508:
2507:
2502:
2500:
2499:
2451:
2449:
2448:
2443:
2431:
2429:
2428:
2423:
2421:
2420:
2404:
2402:
2401:
2396:
2394:
2393:
2370:
2361:
2359:
2358:
2353:
2343:
2341:
2340:
2339:
2324:
2323:
2313:
2312:
2311:
2299:
2298:
2288:
2286:
2285:
2271:
2269:
2268:
2267:
2252:
2251:
2241:
2240:
2239:
2227:
2226:
2216:
2214:
2213:
2196:
2192:
2190:
2189:
2184:
2182:
2177:
2176:
2167:
2147:
2138:
2136:
2135:
2130:
2124:
2119:
2118:
2109:
2106:
2104:
2103:
2102:
2087:
2086:
2076:
2075:
2074:
2062:
2061:
2051:
2040:
2032:
2022:
2014:
2012:
2011:
2006:
2004:
1999:
1998:
1989:
1976:
1974:
1973:
1968:
1966:
1965:
1949:
1947:
1946:
1941:
1929:
1927:
1926:
1921:
1919:
1918:
1891:
1889:
1888:
1883:
1881:
1880:
1864:
1862:
1861:
1856:
1851:
1840:
1839:
1823:
1821:
1820:
1815:
1810:
1799:
1798:
1782:
1780:
1779:
1774:
1769:
1758:
1757:
1741:
1739:
1738:
1733:
1728:
1717:
1716:
1700:
1698:
1697:
1692:
1687:
1676:
1675:
1659:
1657:
1656:
1651:
1649:
1648:
1625:
1616:
1614:
1613:
1608:
1606:
1604:
1600:
1599:
1581:
1580:
1568:
1567:
1557:
1556:
1555:
1539:
1535:
1530:
1515:
1507:
1489:
1485:
1483:
1482:
1477:
1465:
1463:
1462:
1457:
1445:
1443:
1442:
1437:
1435:
1434:
1418:
1416:
1415:
1410:
1391:
1382:
1380:
1379:
1374:
1364:
1362:
1361:
1360:
1333:
1332:
1322:
1318:
1317:
1305:
1304:
1276:
1264:
1256:
1246:
1237:
1235:
1234:
1229:
1227:
1226:
1210:
1208:
1207:
1202:
1200:
1199:
1183:
1181:
1180:
1175:
1173:
1172:
1156:
1154:
1153:
1148:
1146:
1145:
1129:
1127:
1126:
1121:
1102:
1093:
1091:
1090:
1085:
1075:
1073:
1072:
1071:
1044:
1043:
1033:
1032:
1031:
1019:
1018:
1008:
1005:
1004:
993:
976:
954:
952:
951:
946:
944:
943:
942:
919:
917:
916:
911:
900:
895:
894:
866:
864:
863:
858:
856:
855:
839:
837:
836:
831:
829:
828:
827:
804:
802:
801:
796:
794:
793:
777:
775:
774:
769:
767:
766:
750:
748:
747:
742:
737:
729:
728:
712:
710:
709:
704:
699:
691:
690:
670:
668:
667:
662:
640:
631:
629:
628:
623:
606:
605:
593:
592:
591:
563:
555:
554:
542:
534:
533:
504:
492:
483:
481:
480:
475:
470:
469:
451:
450:
432:
431:
413:
412:
394:
393:
384:
382: with
381:
376:
371:
370:
368:
367:
358:
357:
341:
336:
327:
322:
321:
311:
306:
305:
304:
280:
276:
274:
273:
268:
266:
265:
249:
247:
246:
241:
239:
238:
222:
220:
219:
214:
212:
211:
195:
193:
192:
187:
185:
184:
151:
149:
148:
143:
128:
126:
125:
120:
8381:
8380:
8376:
8375:
8374:
8372:
8371:
8370:
8351:
8350:
8344:
8328:
8322:
8306:
8300:
8284:
8278:
8265:
8259:
8246:
8243:
8241:Further reading
8238:
8237:
8207:
8206:
8202:
8185:
8162:
8161:
8157:
8140:
8117:
8116:
8112:
8095:
8072:
8071:
8067:
8050:
8027:
8026:
8022:
8005:
7982:
7981:
7977:
7960:
7921:
7920:
7916:
7899:
7842:
7841:
7837:
7799:
7798:
7794:
7744:
7738:
7737:
7733:
7728:
7724:
7719:
7715:
7708:
7695:
7694:
7690:
7644:
7643:
7632:
7588:
7587:
7583:
7574:
7572:
7538:
7537:
7533:
7526:
7513:
7512:
7508:
7478:
7477:
7473:
7443:
7442:
7438:
7408:
7407:
7403:
7373:
7372:
7365:
7335:
7334:
7330:
7321:
7319:
7312:10.1063/1.31150
7297:
7296:
7289:
7279:
7277:
7276:. Sigma Aldrich
7271:
7270:
7266:
7228:
7227:
7223:
7187:
7186:
7182:
7173:
7171:
7137:
7136:
7132:
7123:
7119:
7114:
7110:
7086:
7081:
7080:
7076:
7069:
7051:
7050:
7046:
7041:
7024:
7011:
6980:
6969:
6936:
6931:
6930:
6906:
6905:
6886:
6885:
6866:
6865:
6844:
6839:
6838:
6814:
6809:
6808:
6799:
6786:
6752:
6742:
6730:
6720:
6703:
6694:
6690:
6680:
6667:
6662:
6661:
6626:
6621:
6620:
6599:
6586:
6581:
6580:
6553:
6548:
6547:
6526:
6521:
6520:
6499:
6494:
6493:
6468:
6463:
6462:
6430:
6414:
6401:
6376:
6363:
6338:
6333:
6332:
6322:
6280:
6264:
6254:
6212:
6211:
6207:
6184:
6179:
6178:
6170:
6132:
6121:
6120:
6101:
6100:
6066:
6065:
6035:
6030:
6029:
6002:
5991:
5990:
5938:
5937:
5903:
5893:
5878:
5873:
5872:
5868:
5814:
5813:
5787:
5762:
5761:
5751:
5716:
5700:
5687:
5662:
5649:
5624:
5619:
5618:
5577:
5561:
5551:
5509:
5508:
5504:
5481:
5476:
5475:
5472:
5440:
5427:
5416:
5415:
5394:
5381:
5368:
5363:
5362:
5339:
5326:
5307:
5294:
5281:
5268:
5267:
5257:
5216:
5203:
5173:
5172:
5137:
5126:
5107:
5091:
5076:
5071:
5070:
5064:
5037:
5032:
5031:
5012:
5011:
4990:
4985:
4984:
4956:
4955:
4936:
4935:
4898:
4833:
4829:
4818:
4811:
4793:
4777:
4776:
4762:
4749:
4745:
4738:
4737:
4731:
4702:
4689:
4684:
4683:
4662:
4657:
4656:
4653:
4640:
4607:
4591:
4590:
4579:
4566:
4565:
4559:
4549:
4529:
4505:
4504:
4493:
4472:
4471:
4465:
4460:
4459:
4452:
4431:
4426:
4425:
4404:
4399:
4398:
4369:
4364:
4363:
4329:
4328:
4311:
4310:
4304:
4294:
4260:
4259:
4234:
4233:
4227:
4222:
4221:
4198:
4197:
4176:
4153:
4152:
4135:
4134:
4128:
4117:
4116:
4097:
4096:
4077:
4076:
4055:
4050:
4049:
4028:
4023:
4022:
4001:
3991:
3968:
3950:
3939:
3932:
3931:
3924:
3919:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3662:
3659:
3656:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3611:
3576:
3563:
3550:
3537:
3524:
3520:
3509:
3493:
3477:
3464:
3451:
3447:
3434:
3411:
3406:
3405:
3376:
3371:
3370:
3369:of the matrix;
3349:
3344:
3343:
3322:
3317:
3316:
3287:
3282:
3281:
3270:
3234:
3218:
3217:
3206:
3193:
3192:
3186:
3176:
3156:
3132:
3131:
3120:
3099:
3098:
3092:
3087:
3086:
3078:
3046:
3041:
3040:
3032:
3004:
2999:
2998:
2973:
2968:
2967:
2956:
2921:
2920:
2910:
2898:
2885:
2872:
2867:
2866:
2834:
2829:
2828:
2807:
2802:
2801:
2780:
2775:
2774:
2749:
2744:
2743:
2732:
2692:
2679:
2666:
2653:
2637:
2636:
2625:
2612:
2611:
2599:
2589:
2585:
2575:
2562:
2557:
2556:
2544:William Garnett
2540:Maxwell Garnett
2518:
2513:
2512:
2491:
2486:
2485:
2482:Maxwell Garnett
2478:
2465:
2434:
2433:
2412:
2407:
2406:
2385:
2380:
2379:
2368:
2331:
2315:
2314:
2303:
2290:
2289:
2277:
2259:
2243:
2242:
2231:
2218:
2217:
2205:
2200:
2199:
2168:
2161:
2160:
2145:
2110:
2094:
2078:
2077:
2066:
2053:
2052:
2026:
2025:
1990:
1983:
1982:
1957:
1952:
1951:
1932:
1931:
1910:
1905:
1904:
1901:
1872:
1867:
1866:
1831:
1826:
1825:
1790:
1785:
1784:
1749:
1744:
1743:
1708:
1703:
1702:
1667:
1662:
1661:
1640:
1635:
1634:
1623:
1591:
1572:
1559:
1558:
1547:
1540:
1493:
1492:
1468:
1467:
1448:
1447:
1426:
1421:
1420:
1401:
1400:
1389:
1352:
1324:
1323:
1309:
1296:
1277:
1250:
1249:
1244:
1218:
1213:
1212:
1191:
1186:
1185:
1164:
1159:
1158:
1137:
1132:
1131:
1112:
1111:
1100:
1063:
1035:
1034:
1023:
1010:
1009:
996:
980:
979:
974:
965:
927:
922:
921:
886:
872:
871:
847:
842:
841:
812:
807:
806:
785:
780:
779:
758:
753:
752:
720:
715:
714:
682:
677:
676:
671:is the jump of
650:
649:
638:
597:
576:
546:
525:
508:
507:
490:
461:
442:
423:
404:
385:
359:
349:
313:
312:
289:
284:
283:
257:
252:
251:
230:
225:
224:
203:
198:
197:
176:
171:
170:
167:
134:
133:
111:
110:
96:
17:
12:
11:
5:
8379:
8377:
8369:
8368:
8363:
8353:
8352:
8349:
8348:
8342:
8326:
8320:
8304:
8298:
8282:
8276:
8263:
8257:
8242:
8239:
8236:
8235:
8200:
8155:
8110:
8065:
8020:
7975:
7914:
7855:(17): 177402.
7835:
7808:(4): 608–619.
7792:
7755:(20): 205103.
7742:
7731:
7722:
7713:
7706:
7688:
7653:(4): 608–619.
7630:
7581:
7531:
7524:
7506:
7487:(4): 574–588.
7481:Rev. Mod. Phys
7471:
7436:
7401:
7398:on 2013-01-05.
7386:(4): 263–266.
7363:
7328:
7287:
7264:
7237:(7): 636–664.
7221:
7180:
7130:
7117:
7108:
7074:
7067:
7043:
7042:
7040:
7037:
7036:
7035:
7030:
7023:
7020:
7017:
7016:
7007:
7005:
6987:
6983:
6976:
6972:
6968:
6961:
6958:
6952:
6946:
6943:
6939:
6913:
6893:
6873:
6851:
6847:
6824:
6821:
6817:
6798:
6795:
6792:
6791:
6782:
6780:
6769:
6765:
6759:
6755:
6749:
6745:
6737:
6733:
6727:
6723:
6715:
6710:
6706:
6700:
6697:
6693:
6687:
6683:
6679:
6670:
6644:
6641:
6638:
6633:
6629:
6606:
6602:
6598:
6593:
6589:
6568:
6565:
6560:
6556:
6533:
6529:
6506:
6502:
6471:
6448:
6445:
6442:
6437:
6433:
6429:
6426:
6421:
6417:
6413:
6408:
6404:
6400:
6397:
6394:
6391:
6388:
6383:
6379:
6375:
6370:
6366:
6362:
6359:
6356:
6353:
6350:
6345:
6341:
6328:
6327:
6318:
6316:
6305:
6301:
6295:
6292:
6287:
6283:
6279:
6276:
6271:
6267:
6261:
6257:
6253:
6250:
6245:
6240:
6236:
6232:
6227:
6224:
6219:
6215:
6210:
6204:
6201:
6196:
6187:
6169:
6166:
6153:
6150:
6147:
6144:
6139:
6135:
6131:
6128:
6108:
6087:
6084:
6080:
6076:
6073:
6053:
6050:
6047:
6042:
6038:
6017:
6014:
6009:
6005:
6001:
5998:
5974:
5971:
5968:
5965:
5962:
5959:
5956:
5952:
5949:
5946:
5925:
5921:
5917:
5910:
5906:
5900:
5896:
5890:
5885:
5881:
5856:
5850:
5847:
5844:
5841:
5838:
5835:
5832:
5829:
5826:
5821:
5817:
5811:
5808:
5805:
5802:
5799:
5796:
5793:
5790:
5784:
5781:
5778:
5775:
5772:
5769:
5757:
5756:
5747:
5745:
5734:
5731:
5728:
5723:
5719:
5715:
5712:
5707:
5703:
5699:
5694:
5690:
5686:
5683:
5680:
5677:
5674:
5669:
5665:
5661:
5656:
5652:
5648:
5645:
5642:
5639:
5636:
5631:
5627:
5602:
5598:
5592:
5589:
5584:
5580:
5576:
5573:
5568:
5564:
5558:
5554:
5550:
5547:
5542:
5537:
5533:
5529:
5524:
5521:
5516:
5512:
5507:
5501:
5498:
5493:
5484:
5471:
5468:
5452:
5447:
5443:
5439:
5434:
5430:
5426:
5423:
5401:
5397:
5393:
5388:
5384:
5380:
5375:
5371:
5346:
5342:
5338:
5333:
5329:
5325:
5319:
5314:
5310:
5306:
5301:
5297:
5293:
5288:
5284:
5280:
5275:
5271:
5264:
5260:
5252:
5247:
5244:
5240:
5236:
5232:
5228:
5223:
5219:
5215:
5210:
5206:
5202:
5199:
5196:
5191:
5188:
5183:
5180:
5155:
5152:
5149:
5146:
5143:
5140:
5133:
5129:
5125:
5122:
5119:
5114:
5110:
5106:
5103:
5100:
5097:
5094:
5088:
5079:
5063:
5060:
5044:
5040:
5019:
4997:
4993:
4972:
4969:
4966:
4963:
4943:
4922:
4919:
4916:
4913:
4910:
4905:
4901:
4897:
4894:
4891:
4888:
4885:
4878:
4873:
4869:
4865:
4855:
4850:
4846:
4840:
4836:
4832:
4825:
4821:
4817:
4814:
4808:
4800:
4796:
4792:
4789:
4780:
4774:
4769:
4765:
4761:
4752:
4748:
4730:
4727:
4712:
4709:
4705:
4701:
4696:
4692:
4669:
4665:
4652:
4649:
4646:
4645:
4636:
4634:
4622:
4614:
4610:
4606:
4603:
4598:
4594:
4586:
4582:
4578:
4573:
4569:
4562:
4556:
4552:
4548:
4544:
4536:
4532:
4528:
4525:
4519:
4516:
4513:
4508:
4500:
4496:
4492:
4486:
4483:
4480:
4475:
4468:
4438:
4434:
4411:
4407:
4383:
4380:
4377:
4372:
4350:
4344:
4341:
4336:
4332:
4326:
4323:
4318:
4314:
4307:
4301:
4297:
4293:
4289:
4283:
4280:
4274:
4271:
4268:
4263:
4257:
4254:
4248:
4245:
4242:
4237:
4230:
4205:
4196:If we combine
4183:
4179:
4174:
4168:
4165:
4160:
4156:
4150:
4147:
4142:
4138:
4131:
4127:
4124:
4104:
4084:
4062:
4058:
4035:
4031:
4008:
4004:
3998:
3994:
3988:
3984:
3978:
3974:
3971:
3965:
3959:
3956:
3953:
3948:
3945:
3942:
3923:
3920:
3624:
3617:
3616:
3607:
3605:
3594:
3588:
3583:
3579:
3575:
3570:
3566:
3562:
3557:
3553:
3549:
3544:
3540:
3536:
3531:
3527:
3523:
3516:
3512:
3508:
3505:
3500:
3496:
3492:
3489:
3484:
3480:
3476:
3471:
3467:
3463:
3458:
3454:
3450:
3441:
3437:
3432:
3425:
3422:
3419:
3414:
3383:
3379:
3356:
3352:
3329:
3325:
3301:
3298:
3295:
3290:
3276:
3275:
3266:
3264:
3253:
3249:
3241:
3237:
3233:
3230:
3225:
3221:
3213:
3209:
3205:
3200:
3196:
3189:
3183:
3179:
3175:
3171:
3163:
3159:
3155:
3152:
3146:
3143:
3140:
3135:
3127:
3123:
3119:
3113:
3110:
3107:
3102:
3095:
3077:
3074:
3061:
3058:
3053:
3049:
3030:electric field
3011:
3007:
2980:
2976:
2962:
2961:
2952:
2950:
2939:
2933:
2928:
2924:
2917:
2913:
2905:
2901:
2897:
2892:
2888:
2884:
2875:
2849:
2846:
2841:
2837:
2814:
2810:
2787:
2783:
2752:
2738:
2737:
2728:
2726:
2715:
2711:
2704:
2699:
2695:
2691:
2686:
2682:
2678:
2673:
2669:
2665:
2660:
2656:
2652:
2649:
2644:
2640:
2632:
2628:
2624:
2619:
2615:
2606:
2602:
2598:
2595:
2592:
2588:
2582:
2578:
2574:
2565:
2525:
2521:
2498:
2494:
2477:
2474:
2464:
2461:
2441:
2419:
2415:
2392:
2388:
2374:
2373:
2364:
2362:
2351:
2347:
2338:
2334:
2330:
2327:
2322:
2318:
2310:
2306:
2302:
2297:
2293:
2284:
2280:
2275:
2266:
2262:
2258:
2255:
2250:
2246:
2238:
2234:
2230:
2225:
2221:
2212:
2208:
2180:
2175:
2171:
2151:
2150:
2141:
2139:
2128:
2122:
2117:
2113:
2101:
2097:
2093:
2090:
2085:
2081:
2073:
2069:
2065:
2060:
2056:
2048:
2044:
2038:
2035:
2002:
1997:
1993:
1979:electric field
1964:
1960:
1939:
1917:
1913:
1900:
1897:
1892:'s is unity.)
1879:
1875:
1854:
1850:
1846:
1843:
1838:
1834:
1813:
1809:
1805:
1802:
1797:
1793:
1772:
1768:
1764:
1761:
1756:
1752:
1731:
1727:
1723:
1720:
1715:
1711:
1690:
1686:
1682:
1679:
1674:
1670:
1647:
1643:
1629:
1628:
1619:
1617:
1603:
1598:
1594:
1590:
1587:
1584:
1579:
1575:
1571:
1566:
1562:
1554:
1550:
1546:
1543:
1534:
1529:
1526:
1523:
1519:
1513:
1510:
1504:
1500:
1475:
1455:
1433:
1429:
1408:
1395:
1394:
1385:
1383:
1372:
1368:
1359:
1355:
1351:
1348:
1345:
1342:
1339:
1336:
1331:
1327:
1321:
1316:
1312:
1308:
1303:
1299:
1295:
1292:
1289:
1286:
1283:
1280:
1274:
1271:
1268:
1262:
1259:
1243:
1240:
1238:'s is unity.)
1225:
1221:
1198:
1194:
1171:
1167:
1144:
1140:
1119:
1106:
1105:
1096:
1094:
1083:
1079:
1070:
1066:
1062:
1059:
1056:
1053:
1050:
1047:
1042:
1038:
1030:
1026:
1022:
1017:
1013:
1003:
999:
992:
988:
973:
970:
964:
961:
941:
938:
935:
930:
909:
906:
903:
899:
893:
889:
885:
882:
879:
854:
850:
826:
823:
820:
815:
792:
788:
765:
761:
740:
736:
732:
727:
723:
702:
698:
694:
689:
685:
660:
657:
644:
643:
634:
632:
621:
618:
615:
612:
609:
604:
600:
596:
590:
587:
584:
579:
575:
572:
569:
566:
562:
558:
553:
549:
545:
541:
537:
532:
528:
524:
521:
518:
515:
496:
495:
486:
484:
473:
468:
464:
460:
457:
454:
449:
445:
441:
438:
435:
430:
426:
422:
419:
416:
411:
407:
403:
400:
397:
392:
388:
379:
374:
366:
362:
356:
352:
348:
345:
340:
335:
331:
325:
320:
316:
309:
303:
300:
297:
292:
264:
260:
237:
233:
210:
206:
183:
179:
166:
163:
141:
118:
95:
92:
88:electric field
51:properties of
15:
13:
10:
9:
6:
4:
3:
2:
8378:
8367:
8364:
8362:
8359:
8358:
8356:
8345:
8339:
8335:
8331:
8330:Mackay, T. G.
8327:
8323:
8317:
8313:
8309:
8305:
8301:
8295:
8291:
8287:
8283:
8279:
8273:
8269:
8264:
8260:
8254:
8250:
8245:
8244:
8240:
8231:
8227:
8223:
8219:
8216:(4): 045101.
8215:
8211:
8204:
8201:
8196:
8190:
8182:
8178:
8175:: 7425–7434.
8174:
8170:
8166:
8159:
8156:
8151:
8145:
8137:
8133:
8129:
8125:
8121:
8114:
8111:
8106:
8100:
8092:
8088:
8084:
8080:
8076:
8069:
8066:
8061:
8055:
8047:
8043:
8039:
8035:
8031:
8024:
8021:
8016:
8010:
8002:
7998:
7994:
7990:
7986:
7979:
7976:
7971:
7965:
7957:
7953:
7949:
7945:
7941:
7937:
7933:
7929:
7925:
7918:
7915:
7910:
7904:
7896:
7892:
7888:
7884:
7880:
7876:
7872:
7868:
7863:
7858:
7854:
7850:
7846:
7839:
7836:
7831:
7827:
7823:
7819:
7815:
7811:
7807:
7803:
7796:
7793:
7788:
7784:
7779:
7774:
7770:
7766:
7762:
7758:
7754:
7750:
7746:
7735:
7732:
7726:
7723:
7717:
7714:
7709:
7703:
7699:
7692:
7689:
7684:
7680:
7676:
7672:
7668:
7664:
7660:
7656:
7652:
7648:
7641:
7639:
7637:
7635:
7631:
7626:
7622:
7617:
7612:
7608:
7604:
7600:
7596:
7592:
7585:
7582:
7571:on 2012-07-12
7570:
7566:
7562:
7558:
7554:
7551:(8): 081902.
7550:
7546:
7542:
7535:
7532:
7527:
7521:
7517:
7510:
7507:
7502:
7498:
7494:
7490:
7486:
7482:
7475:
7472:
7467:
7463:
7459:
7455:
7451:
7447:
7440:
7437:
7432:
7428:
7424:
7420:
7416:
7412:
7405:
7402:
7397:
7393:
7389:
7385:
7381:
7377:
7370:
7368:
7364:
7359:
7355:
7351:
7347:
7343:
7339:
7332:
7329:
7318:on 2012-07-10
7317:
7313:
7309:
7305:
7301:
7294:
7292:
7288:
7275:
7268:
7265:
7260:
7256:
7252:
7248:
7244:
7240:
7236:
7233:(in German).
7232:
7225:
7222:
7217:
7213:
7208:
7203:
7199:
7195:
7191:
7184:
7181:
7170:on 2012-07-16
7169:
7165:
7161:
7157:
7153:
7149:
7145:
7144:J. Appl. Phys
7141:
7134:
7131:
7127:
7121:
7118:
7112:
7109:
7104:
7100:
7096:
7092:
7085:
7078:
7075:
7070:
7064:
7060:
7059:
7054:
7048:
7045:
7038:
7034:
7031:
7029:
7026:
7025:
7021:
7015:
7008:
7006:
6985:
6981:
6974:
6970:
6966:
6959:
6956:
6950:
6944:
6941:
6937:
6929:
6928:
6925:
6911:
6891:
6871:
6849:
6845:
6822:
6819:
6815:
6805:
6796:
6790:
6783:
6781:
6767:
6763:
6757:
6753:
6747:
6743:
6735:
6731:
6725:
6721:
6713:
6708:
6704:
6698:
6695:
6691:
6685:
6681:
6677:
6668:
6660:
6659:
6656:
6642:
6639:
6636:
6631:
6627:
6604:
6600:
6596:
6591:
6587:
6566:
6563:
6558:
6554:
6531:
6527:
6504:
6500:
6491:
6488:is effective
6469:
6459:
6446:
6440:
6435:
6431:
6424:
6419:
6415:
6406:
6402:
6398:
6395:
6392:
6386:
6381:
6377:
6368:
6364:
6360:
6357:
6354:
6348:
6343:
6339:
6326:
6319:
6317:
6303:
6299:
6290:
6285:
6281:
6274:
6269:
6265:
6259:
6255:
6251:
6248:
6243:
6238:
6234:
6230:
6225:
6222:
6217:
6213:
6208:
6202:
6199:
6194:
6185:
6177:
6176:
6173:
6167:
6165:
6151:
6148:
6142:
6137:
6133:
6126:
6078:
6074:
6071:
6051:
6048:
6045:
6040:
6036:
6012:
6007:
6003:
5996:
5988:
5972:
5966:
5963:
5960:
5957:
5923:
5919:
5915:
5908:
5904:
5898:
5894:
5888:
5883:
5879:
5854:
5848:
5845:
5839:
5833:
5830:
5827:
5824:
5819:
5815:
5806:
5800:
5797:
5794:
5791:
5788:
5782:
5779:
5773:
5767:
5755:
5748:
5746:
5732:
5726:
5721:
5717:
5710:
5705:
5701:
5692:
5688:
5684:
5681:
5678:
5672:
5667:
5663:
5654:
5650:
5646:
5643:
5640:
5634:
5629:
5625:
5617:
5616:
5613:
5600:
5596:
5587:
5582:
5578:
5571:
5566:
5562:
5556:
5552:
5548:
5545:
5540:
5535:
5531:
5527:
5522:
5519:
5514:
5510:
5505:
5499:
5496:
5491:
5482:
5469:
5467:
5464:
5445:
5441:
5437:
5432:
5428:
5421:
5399:
5395:
5391:
5386:
5382:
5378:
5373:
5369:
5359:
5344:
5340:
5336:
5331:
5327:
5323:
5312:
5308:
5304:
5299:
5295:
5286:
5282:
5278:
5273:
5269:
5262:
5258:
5250:
5245:
5242:
5238:
5234:
5230:
5221:
5217:
5213:
5208:
5204:
5197:
5194:
5189:
5186:
5181:
5178:
5169:
5153:
5150:
5147:
5144:
5141:
5138:
5131:
5127:
5123:
5120:
5117:
5112:
5108:
5101:
5098:
5095:
5086:
5077:
5068:
5061:
5059:
5042:
5038:
5017:
4995:
4991:
4967:
4961:
4941:
4933:
4920:
4917:
4911:
4903:
4899:
4892:
4886:
4883:
4876:
4871:
4867:
4863:
4853:
4848:
4844:
4838:
4834:
4830:
4823:
4819:
4815:
4812:
4806:
4798:
4794:
4790:
4787:
4778:
4767:
4763:
4759:
4750:
4735:
4728:
4726:
4710:
4707:
4703:
4699:
4694:
4690:
4667:
4663:
4650:
4644:
4637:
4635:
4620:
4612:
4608:
4604:
4601:
4596:
4592:
4584:
4580:
4576:
4571:
4567:
4560:
4554:
4550:
4546:
4542:
4534:
4530:
4526:
4523:
4506:
4498:
4494:
4490:
4473:
4466:
4458:
4457:
4454:
4436:
4432:
4409:
4405:
4370:
4348:
4342:
4339:
4334:
4330:
4324:
4321:
4316:
4312:
4305:
4299:
4295:
4291:
4287:
4281:
4278:
4261:
4255:
4252:
4235:
4228:
4219:
4203:
4181:
4177:
4172:
4166:
4163:
4158:
4154:
4148:
4145:
4140:
4136:
4129:
4125:
4122:
4102:
4082:
4075:and a radius
4060:
4056:
4033:
4029:
4006:
4002:
3996:
3992:
3986:
3982:
3976:
3972:
3969:
3963:
3957:
3954:
3951:
3946:
3943:
3940:
3929:
3921:
3622:
3615:
3608:
3606:
3592:
3581:
3577:
3573:
3568:
3564:
3555:
3551:
3547:
3542:
3538:
3534:
3529:
3525:
3521:
3514:
3510:
3506:
3503:
3498:
3494:
3490:
3482:
3478:
3474:
3469:
3465:
3456:
3452:
3448:
3439:
3435:
3430:
3412:
3404:
3403:
3400:
3397:
3381:
3377:
3354:
3350:
3327:
3323:
3288:
3274:
3267:
3265:
3251:
3247:
3239:
3235:
3231:
3228:
3223:
3219:
3211:
3207:
3203:
3198:
3194:
3187:
3181:
3177:
3173:
3169:
3161:
3157:
3153:
3150:
3133:
3125:
3121:
3117:
3100:
3093:
3085:
3084:
3081:
3075:
3073:
3059:
3056:
3051:
3047:
3038:
3031:
3027:
3009:
3005:
2996:
2978:
2974:
2960:
2953:
2951:
2937:
2931:
2926:
2922:
2915:
2911:
2903:
2899:
2895:
2890:
2886:
2882:
2873:
2865:
2864:
2861:
2847:
2844:
2839:
2835:
2812:
2808:
2785:
2781:
2772:
2769:is effective
2750:
2736:
2729:
2727:
2713:
2709:
2697:
2693:
2689:
2684:
2680:
2671:
2667:
2663:
2658:
2654:
2650:
2647:
2642:
2638:
2630:
2626:
2622:
2617:
2613:
2604:
2600:
2596:
2593:
2590:
2586:
2580:
2576:
2572:
2563:
2555:
2554:
2551:
2549:
2545:
2541:
2523:
2519:
2496:
2492:
2483:
2475:
2473:
2471:
2462:
2460:
2457:
2453:
2439:
2417:
2413:
2390:
2386:
2372:
2365:
2363:
2349:
2345:
2336:
2332:
2328:
2325:
2320:
2316:
2308:
2304:
2300:
2295:
2291:
2282:
2278:
2273:
2264:
2260:
2256:
2253:
2248:
2244:
2236:
2232:
2228:
2223:
2219:
2210:
2206:
2198:
2197:
2194:
2173:
2169:
2158:
2149:
2142:
2140:
2126:
2115:
2111:
2099:
2095:
2091:
2088:
2083:
2079:
2071:
2067:
2063:
2058:
2054:
2046:
2042:
2033:
2024:
2023:
2020:
2018:
2017:dipole moment
1995:
1991:
1980:
1962:
1958:
1937:
1915:
1911:
1898:
1896:
1893:
1877:
1873:
1852:
1848:
1844:
1841:
1836:
1832:
1811:
1807:
1803:
1800:
1795:
1791:
1770:
1766:
1762:
1759:
1754:
1750:
1729:
1725:
1721:
1718:
1713:
1709:
1688:
1684:
1680:
1677:
1672:
1668:
1645:
1641:
1627:
1620:
1618:
1596:
1592:
1588:
1585:
1577:
1573:
1569:
1564:
1560:
1552:
1548:
1544:
1541:
1532:
1527:
1524:
1521:
1517:
1511:
1508:
1502:
1498:
1491:
1490:
1487:
1473:
1453:
1431:
1427:
1406:
1393:
1386:
1384:
1370:
1366:
1357:
1353:
1346:
1343:
1340:
1334:
1329:
1325:
1314:
1310:
1306:
1301:
1297:
1287:
1284:
1281:
1272:
1269:
1266:
1260:
1257:
1248:
1247:
1241:
1239:
1223:
1219:
1196:
1192:
1169:
1165:
1142:
1138:
1117:
1104:
1097:
1095:
1081:
1077:
1068:
1064:
1057:
1054:
1051:
1045:
1040:
1036:
1028:
1024:
1020:
1015:
1011:
1001:
997:
990:
986:
978:
977:
971:
969:
962:
960:
958:
957:nanoparticles
928:
907:
904:
891:
887:
880:
877:
870:
852:
848:
813:
790:
786:
763:
759:
725:
721:
687:
683:
674:
642:
635:
633:
619:
616:
613:
610:
607:
602:
598:
594:
577:
573:
570:
567:
551:
547:
530:
526:
522:
519:
506:
505:
502:
494:
487:
485:
471:
466:
462:
455:
452:
447:
443:
439:
433:
428:
424:
417:
414:
409:
405:
401:
395:
390:
386:
377:
372:
364:
360:
354:
350:
346:
343:
338:
333:
329:
323:
318:
314:
307:
290:
282:
281:
278:
262:
258:
235:
231:
208:
204:
181:
177:
162:
160:
154:
139:
132:
116:
109:
104:
102:
93:
91:
89:
85:
81:
77:
75:
70:
66:
62:
61:inhomogeneous
58:
54:
50:
46:
42:
39:) pertain to
38:
34:
30:
26:
22:
8333:
8311:
8289:
8267:
8248:
8213:
8209:
8203:
8189:cite journal
8172:
8168:
8158:
8144:cite journal
8127:
8123:
8113:
8099:cite journal
8082:
8078:
8068:
8054:cite journal
8037:
8033:
8023:
8009:cite journal
7992:
7988:
7978:
7964:cite journal
7931:
7927:
7917:
7903:cite journal
7852:
7848:
7838:
7805:
7801:
7795:
7752:
7749:Phys. Rev. B
7748:
7734:
7725:
7716:
7697:
7691:
7650:
7646:
7598:
7594:
7584:
7573:. Retrieved
7569:the original
7548:
7544:
7534:
7515:
7509:
7484:
7480:
7474:
7449:
7446:Phys. Rev. B
7445:
7439:
7414:
7411:Phys. Rev. B
7410:
7404:
7396:the original
7383:
7379:
7341:
7338:Phys. Rev. B
7337:
7331:
7320:. Retrieved
7316:the original
7303:
7278:. Retrieved
7267:
7234:
7230:
7224:
7200:(12): 4122.
7197:
7193:
7183:
7172:. Retrieved
7168:the original
7147:
7143:
7133:
7125:
7120:
7111:
7094:
7090:
7077:
7057:
7053:Wenshan, Cai
7047:
7009:
6800:
6784:
6460:
6331:
6320:
6171:
5760:
5749:
5473:
5465:
5360:
5170:
5069:
5065:
4934:
4736:
4732:
4654:
4638:
3925:
3620:
3609:
3398:
3279:
3268:
3079:
2965:
2954:
2741:
2730:
2479:
2466:
2458:
2454:
2377:
2366:
2157:polarization
2154:
2143:
1902:
1894:
1632:
1621:
1398:
1387:
1109:
1098:
966:
647:
636:
499:
488:
168:
155:
108:conductivity
105:
97:
94:Applications
79:
78:
73:
69:permeability
65:permittivity
36:
32:
28:
24:
18:
7150:(9): 3897.
3907:factor_down
3850:factor_down
3787:factor_down
49:macroscopic
45:theoretical
8355:Categories
8085:: 143104.
8040:: 134108.
7995:: 044705.
7862:1506.08078
7778:2440/36406
7575:2019-04-24
7322:2010-02-07
7174:2019-04-24
7039:References
6884:), width (
4220:, we get:
3922:Derivation
1899:Derivation
1633:where the
41:analytical
7956:195326315
7928:Nanoscale
7830:125250487
7683:125250487
7675:1063-7761
7625:1364-503X
7259:0003-3804
7216:2076-3417
6967:ρ
6957:π
6872:ρ
6754:ε
6722:ω
6682:μ
6669:μ
6640:≪
6601:μ
6588:μ
6564:≪
6528:μ
6501:μ
6470:μ
6416:μ
6396:−
6387:−
6378:μ
6358:−
6344:μ
6266:μ
6256:μ
6249:−
6239:μ
6231:−
6218:μ
6186:μ
6075:≤
6049:≪
5964:ω
5958:−
5916:ω
5905:μ
5895:ε
5846:−
5834:
5801:
5792:−
5702:ε
5682:−
5673:−
5664:ε
5644:−
5630:ε
5563:ε
5553:ε
5546:−
5536:ε
5528:−
5515:ε
5483:ε
5309:ε
5305:−
5296:ε
5270:ε
5259:ε
5243:
5231:∑
5195:∬
5179:β
5154:β
5142:−
5128:ε
5124:β
5109:ε
5099:−
5078:ε
4884:∫
4845:ε
4835:π
4820:λ
4795:ε
4788:−
4779:ε
4764:ε
4760:−
4751:ε
4708:−
4691:δ
4664:δ
4609:ε
4593:ε
4581:ε
4577:−
4568:ε
4551:δ
4531:ε
4507:ε
4495:ε
4491:−
4474:ε
4433:δ
4406:ε
4371:ε
4331:ε
4322:−
4313:ε
4296:δ
4262:ε
4253:−
4236:ε
4216:with the
4204:α
4155:ε
4146:−
4137:ε
4123:α
4103:α
4057:ε
4003:α
3983:∑
3973:π
3952:ε
3944:−
3941:ε
3901:factor_up
3721:factor_up
3647:% OUTPUT:
3635:% INPUTS:
3578:ε
3574:−
3565:ε
3552:δ
3548:−
3539:ε
3526:ε
3511:ε
3495:ε
3479:ε
3475:−
3466:ε
3453:δ
3436:ε
3413:ε
3378:δ
3351:ε
3324:ε
3289:ε
3236:ε
3220:ε
3208:ε
3204:−
3195:ε
3178:δ
3158:ε
3134:ε
3122:ε
3118:−
3101:ε
3048:ε
2975:ε
2923:ε
2887:ε
2874:ε
2845:≪
2809:ε
2782:ε
2751:ε
2694:ε
2690:−
2681:ε
2664:−
2655:ε
2639:ε
2627:ε
2623:−
2614:ε
2577:ε
2564:ε
2520:ε
2493:ε
2414:δ
2387:δ
2333:σ
2317:σ
2305:σ
2301:−
2292:σ
2279:δ
2261:σ
2245:σ
2233:σ
2229:−
2220:σ
2207:δ
2179:¯
2121:¯
2096:σ
2080:σ
2068:σ
2064:−
2055:σ
2043:∝
2037:¯
2001:¯
1977:. If the
1959:σ
1912:σ
1593:σ
1589:−
1586:σ
1561:σ
1549:σ
1545:−
1542:σ
1518:∑
1499:α
1454:δ
1428:σ
1407:σ
1354:σ
1344:−
1326:σ
1311:σ
1307:−
1298:σ
1288:δ
1285:−
1270:α
1267:δ
1220:δ
1193:σ
1166:σ
1139:δ
1065:σ
1055:−
1037:σ
1025:σ
1021:−
1012:σ
998:δ
987:∑
929:ε
888:ε
881:
814:ε
787:ε
760:ε
722:ε
659:Φ
656:Δ
595:∬
578:ε
574:−
527:ε
523:∬
517:Φ
514:Δ
463:ε
453:−
425:ε
415:−
361:ε
351:ε
291:ε
205:ε
178:ε
140:ε
117:σ
86:when the
57:averaging
8288:(2000).
8169:Langmuir
8130:: 1–13.
7948:31231735
7887:26551143
7787:28476406
7097:: 1–30.
7022:See also
4651:Validity
3895:eps_base
3889:eps_mean
3874:eps_mean
3835:eps_incl
3826:vol_incl
3811:eps_base
3802:vol_incl
3781:eps_incl
3772:vol_incl
3751:eps_base
3742:vol_incl
3697:vol_incl
3685:vol_incl
3655:eps_mean
3653:function
963:Formulas
8218:Bibcode
7895:4018894
7867:Bibcode
7810:Bibcode
7757:Bibcode
7655:Bibcode
7603:Bibcode
7553:Bibcode
7489:Bibcode
7454:Bibcode
7419:Bibcode
7346:Bibcode
7239:Bibcode
7152:Bibcode
6064:, i.e.
3076:Formula
2993:is the
2480:In the
129:or the
99:to the
8340:
8318:
8296:
8274:
8255:
7954:
7946:
7893:
7885:
7828:
7785:
7704:
7681:
7673:
7623:
7522:
7280:17 May
7257:
7214:
7065:
6099:for Ag
5867:where
4362:Where
4021:Where
3280:where
2966:where
2742:where
2378:where
648:where
7952:S2CID
7891:S2CID
7857:arXiv
7826:S2CID
7783:S2CID
7679:S2CID
6804:graph
6461:Here
5171:with
2155:This
31:) or
8338:ISBN
8316:ISBN
8294:ISBN
8272:ISBN
8253:ISBN
8195:link
8150:link
8105:link
8060:link
8015:link
7970:link
7944:PMID
7909:link
7883:PMID
7702:ISBN
7671:ISSN
7621:ISSN
7520:ISBN
7282:2019
7255:ISSN
7212:ISSN
7063:ISBN
6619:and
5010:and
4954:and
4700:<
3886:else
3862:disp
3856:<
3706:disp
3700:>
3688:<
3676:1e-6
3039:and
2997:and
2405:and
1157:and
250:and
196:and
67:and
8226:doi
8214:122
8177:doi
8132:doi
8087:doi
8083:107
8042:doi
8038:142
7997:doi
7993:140
7936:doi
7875:doi
7853:115
7818:doi
7806:127
7773:hdl
7765:doi
7663:doi
7651:127
7611:doi
7599:203
7561:doi
7497:doi
7462:doi
7427:doi
7388:doi
7354:doi
7308:doi
7247:doi
7235:416
7202:doi
7160:doi
7099:doi
6673:eff
6474:eff
6190:eff
5831:cot
5798:cot
5487:eff
5082:eff
4854:1.5
4783:eff
4755:eff
3916:end
3913:end
3844:abs
3718:end
3024:is
2878:eff
2755:eff
2568:eff
878:div
43:or
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29:EMA
19:In
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8224:.
8212:.
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6768:.
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