Abstract

The long-wave optical modes of vibration in an ionic crystal sphere have been determined, including retardation of the Coulomb forces. These modes, which correspond to coupled excitations of phonons and photons, are also known as polaritons. Their frequencies are complex, the imaginary parts arising from both anharmonic and radiative damping; hence they are virtual modes. It is found that the mode frequencies depend on the radius of the sphere only if retardation is included. The absorption and extinction cross sections for spheres of various sizes are calculated as a function of the frequency of the incident light, and it is shown how the structure in the cross sections is related to the properties of the virtual modes. The theory is used to explain the position and width of an optical absorption peak measured in a polyethylene film containing UO2 particles.

© 1968 Optical Society of America

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References

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  1. J. J. Hopfield, Phys. Rev. 112, 1555 (1958).
    [Crossref]
  2. G. Mie, Ann. Physik 25, 377 (1908).
    [Crossref]
  3. P. Debye, Ann. Physik 30, 57 (1909).
    [Crossref]
  4. J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Company, New York, 1941), pp. 392–395, 414–416, 554–573.
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
  6. R. Fuchs and K. L. Kliewer, Phys. Rev. 140, A2076 (1965).
    [Crossref]
  7. H. Fröhlich, Theory of Dielectrics (Oxford University Press, New York, 1949), pp. 149–155.
  8. H. B. Rosenstock, Phys. Rev. 121, 416 (1961).
    [Crossref]
  9. A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961).
    [Crossref]
  10. T. H. K. Barron, Phys. Rev. 123, 1995 (1961).
    [Crossref]
  11. J. Grindlay, Can. J. Phys. 43, 1605 (1965).
    [Crossref]
  12. R. Englman and R. Ruppin, Phys. Rev. Letters 16, 898 (1966).
    [Crossref]
  13. This is true for suitable linear combinations of the functions (2).
  14. K. L. Kliewer and R. Fuchs, Phys. Rev. 150, 573 (1966).
    [Crossref]
  15. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London, 1954), Ch. II, Sec. 10; Ch. VII, Sec. 47.
  16. We are using gaussian units, whereas Stratton uses mks units. Equation (4) is written for a nonmagnetic material (μ=1). The time dependence e−iωt is understood.
  17. “Electric” and “magnetic” modes are often also called “TM” and “TE” modes, respectively.
  18. In LiF the longitudinal optical frequency, defined as the frequency for which ∊=1, is ΩL=ωL/ωT=2.197 in dimensionless units, while the transverse optical frequency is ΩT=1.
  19. The sphere must in practice remain larger than the lattice parameter.
  20. K. L. Kliewer and R. Fuchs, Phys. Rev. 144, 495 (1966). The surface modes for a slab, shown in Fig. 2 of this reference, approach the frequency determined by ∊=−1 as kx → ∞, kx being the component of the wave vector parallel to the slab. In this limit the fields become localized at the surfaces. The condition ∊=−1 also determines the frequency of the surface modes in a semi-infinite medium. Therefore, if the fields fall off fast enough inside the medium, the frequency of the surface modes does not depend on whether the solid is in the form of a sphere, a slab, or a semi-infinite medium.
    [Crossref]
  21. Our expression for the cross sections can be obtained from those given in Sec. 9.26 of Ref. 4 by dividing by πa2 and making the replacements anr → −1/Al and bnr → −1/Bl. Equations (26), (29), and (31) in Sec. 9.26 have the wrong sign, since they give a negative extinction cross section QT.
  22. R. Fuchs, K. L. Kliewer, and W. J. Pardee, Phys. Rev. 150, 589 (1966).
    [Crossref]
  23. J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons, Inc., New York, 1952), Sec. VIII.
  24. There is one exception to the statement that the surface modes do not give rise to visible structure at low frequencies: the unmarked plateau in the extinction cross section QT between the 1LM1 and 2LM1 peaks can be attributed to the 1S surface mode.
  25. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  26. J. D. Axe and G. D. Pettit, Phys. Rev. 151, 676 (1966).
    [Crossref]
  27. M. Tsuyobi, M. Terada, and T. Shimanouchi, J. Chem. Phys. 36, 1301 (1962).
    [Crossref]

1966 (5)

R. Englman and R. Ruppin, Phys. Rev. Letters 16, 898 (1966).
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 150, 573 (1966).
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 144, 495 (1966). The surface modes for a slab, shown in Fig. 2 of this reference, approach the frequency determined by ∊=−1 as kx → ∞, kx being the component of the wave vector parallel to the slab. In this limit the fields become localized at the surfaces. The condition ∊=−1 also determines the frequency of the surface modes in a semi-infinite medium. Therefore, if the fields fall off fast enough inside the medium, the frequency of the surface modes does not depend on whether the solid is in the form of a sphere, a slab, or a semi-infinite medium.
[Crossref]

R. Fuchs, K. L. Kliewer, and W. J. Pardee, Phys. Rev. 150, 589 (1966).
[Crossref]

J. D. Axe and G. D. Pettit, Phys. Rev. 151, 676 (1966).
[Crossref]

1965 (2)

R. Fuchs and K. L. Kliewer, Phys. Rev. 140, A2076 (1965).
[Crossref]

J. Grindlay, Can. J. Phys. 43, 1605 (1965).
[Crossref]

1962 (1)

M. Tsuyobi, M. Terada, and T. Shimanouchi, J. Chem. Phys. 36, 1301 (1962).
[Crossref]

1961 (3)

H. B. Rosenstock, Phys. Rev. 121, 416 (1961).
[Crossref]

A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961).
[Crossref]

T. H. K. Barron, Phys. Rev. 123, 1995 (1961).
[Crossref]

1958 (1)

J. J. Hopfield, Phys. Rev. 112, 1555 (1958).
[Crossref]

1909 (1)

P. Debye, Ann. Physik 30, 57 (1909).
[Crossref]

1908 (1)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

Axe, J. D.

J. D. Axe and G. D. Pettit, Phys. Rev. 151, 676 (1966).
[Crossref]

Barron, T. H. K.

T. H. K. Barron, Phys. Rev. 123, 1995 (1961).
[Crossref]

Blatt, J. M.

J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons, Inc., New York, 1952), Sec. VIII.

Born, M.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London, 1954), Ch. II, Sec. 10; Ch. VII, Sec. 47.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Debye, P.

P. Debye, Ann. Physik 30, 57 (1909).
[Crossref]

Englman, R.

R. Englman and R. Ruppin, Phys. Rev. Letters 16, 898 (1966).
[Crossref]

Fröhlich, H.

H. Fröhlich, Theory of Dielectrics (Oxford University Press, New York, 1949), pp. 149–155.

Fuchs, R.

K. L. Kliewer and R. Fuchs, Phys. Rev. 150, 573 (1966).
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 144, 495 (1966). The surface modes for a slab, shown in Fig. 2 of this reference, approach the frequency determined by ∊=−1 as kx → ∞, kx being the component of the wave vector parallel to the slab. In this limit the fields become localized at the surfaces. The condition ∊=−1 also determines the frequency of the surface modes in a semi-infinite medium. Therefore, if the fields fall off fast enough inside the medium, the frequency of the surface modes does not depend on whether the solid is in the form of a sphere, a slab, or a semi-infinite medium.
[Crossref]

R. Fuchs, K. L. Kliewer, and W. J. Pardee, Phys. Rev. 150, 589 (1966).
[Crossref]

R. Fuchs and K. L. Kliewer, Phys. Rev. 140, A2076 (1965).
[Crossref]

Grindlay, J.

J. Grindlay, Can. J. Phys. 43, 1605 (1965).
[Crossref]

Hopfield, J. J.

J. J. Hopfield, Phys. Rev. 112, 1555 (1958).
[Crossref]

Huang, K.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London, 1954), Ch. II, Sec. 10; Ch. VII, Sec. 47.

Kliewer, K. L.

K. L. Kliewer and R. Fuchs, Phys. Rev. 144, 495 (1966). The surface modes for a slab, shown in Fig. 2 of this reference, approach the frequency determined by ∊=−1 as kx → ∞, kx being the component of the wave vector parallel to the slab. In this limit the fields become localized at the surfaces. The condition ∊=−1 also determines the frequency of the surface modes in a semi-infinite medium. Therefore, if the fields fall off fast enough inside the medium, the frequency of the surface modes does not depend on whether the solid is in the form of a sphere, a slab, or a semi-infinite medium.
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 150, 573 (1966).
[Crossref]

R. Fuchs, K. L. Kliewer, and W. J. Pardee, Phys. Rev. 150, 589 (1966).
[Crossref]

R. Fuchs and K. L. Kliewer, Phys. Rev. 140, A2076 (1965).
[Crossref]

Maradudin, A. A.

A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961).
[Crossref]

Mie, G.

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

Pardee, W. J.

R. Fuchs, K. L. Kliewer, and W. J. Pardee, Phys. Rev. 150, 589 (1966).
[Crossref]

Pettit, G. D.

J. D. Axe and G. D. Pettit, Phys. Rev. 151, 676 (1966).
[Crossref]

Rosenstock, H. B.

H. B. Rosenstock, Phys. Rev. 121, 416 (1961).
[Crossref]

Ruppin, R.

R. Englman and R. Ruppin, Phys. Rev. Letters 16, 898 (1966).
[Crossref]

Shimanouchi, T.

M. Tsuyobi, M. Terada, and T. Shimanouchi, J. Chem. Phys. 36, 1301 (1962).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Company, New York, 1941), pp. 392–395, 414–416, 554–573.

Terada, M.

M. Tsuyobi, M. Terada, and T. Shimanouchi, J. Chem. Phys. 36, 1301 (1962).
[Crossref]

Tsuyobi, M.

M. Tsuyobi, M. Terada, and T. Shimanouchi, J. Chem. Phys. 36, 1301 (1962).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

Weiss, G. H.

A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961).
[Crossref]

Weisskopf, V. F.

J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons, Inc., New York, 1952), Sec. VIII.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Ann. Physik (2)

G. Mie, Ann. Physik 25, 377 (1908).
[Crossref]

P. Debye, Ann. Physik 30, 57 (1909).
[Crossref]

Can. J. Phys. (1)

J. Grindlay, Can. J. Phys. 43, 1605 (1965).
[Crossref]

J. Chem. Phys. (1)

M. Tsuyobi, M. Terada, and T. Shimanouchi, J. Chem. Phys. 36, 1301 (1962).
[Crossref]

Phys. Rev. (9)

J. J. Hopfield, Phys. Rev. 112, 1555 (1958).
[Crossref]

J. D. Axe and G. D. Pettit, Phys. Rev. 151, 676 (1966).
[Crossref]

R. Fuchs, K. L. Kliewer, and W. J. Pardee, Phys. Rev. 150, 589 (1966).
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 144, 495 (1966). The surface modes for a slab, shown in Fig. 2 of this reference, approach the frequency determined by ∊=−1 as kx → ∞, kx being the component of the wave vector parallel to the slab. In this limit the fields become localized at the surfaces. The condition ∊=−1 also determines the frequency of the surface modes in a semi-infinite medium. Therefore, if the fields fall off fast enough inside the medium, the frequency of the surface modes does not depend on whether the solid is in the form of a sphere, a slab, or a semi-infinite medium.
[Crossref]

K. L. Kliewer and R. Fuchs, Phys. Rev. 150, 573 (1966).
[Crossref]

R. Fuchs and K. L. Kliewer, Phys. Rev. 140, A2076 (1965).
[Crossref]

H. B. Rosenstock, Phys. Rev. 121, 416 (1961).
[Crossref]

A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961).
[Crossref]

T. H. K. Barron, Phys. Rev. 123, 1995 (1961).
[Crossref]

Phys. Rev. Letters (1)

R. Englman and R. Ruppin, Phys. Rev. Letters 16, 898 (1966).
[Crossref]

Other (13)

This is true for suitable linear combinations of the functions (2).

Our expression for the cross sections can be obtained from those given in Sec. 9.26 of Ref. 4 by dividing by πa2 and making the replacements anr → −1/Al and bnr → −1/Bl. Equations (26), (29), and (31) in Sec. 9.26 have the wrong sign, since they give a negative extinction cross section QT.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London, 1954), Ch. II, Sec. 10; Ch. VII, Sec. 47.

We are using gaussian units, whereas Stratton uses mks units. Equation (4) is written for a nonmagnetic material (μ=1). The time dependence e−iωt is understood.

“Electric” and “magnetic” modes are often also called “TM” and “TE” modes, respectively.

In LiF the longitudinal optical frequency, defined as the frequency for which ∊=1, is ΩL=ωL/ωT=2.197 in dimensionless units, while the transverse optical frequency is ΩT=1.

The sphere must in practice remain larger than the lattice parameter.

H. Fröhlich, Theory of Dielectrics (Oxford University Press, New York, 1949), pp. 149–155.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Company, New York, 1941), pp. 392–395, 414–416, 554–573.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley & Sons, Inc., New York, 1952), Sec. VIII.

There is one exception to the statement that the surface modes do not give rise to visible structure at low frequencies: the unmarked plateau in the extinction cross section QT between the 1LM1 and 2LM1 peaks can be attributed to the 1S surface mode.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

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Figures (10)

Fig. 1
Fig. 1

Complex frequencies of low-frequency modes with l=1 as functions of sphere radius. The modes are labeled “E” and “M” for “electric” and “magnetic,” respectively. The numerical subscript labels modes of the same type and with a given value of l in order of increasing frequency Ω′.

Fig. 2
Fig. 2

Complex frequencies of low-frequency modes with l=2 as functions of sphere radius.

Fig. 3
Fig. 3

Complex frequencies of high-frequency modes with l=1 as functions of sphere radius.

Fig. 4
Fig. 4

Complex frequencies of high-frequency modes with l=2 as functions of sphere radius.

Fig. 5
Fig. 5

Complex frequencies of surface (S) modes as functions of sphere radius. The number that occurs in the mode label denotes the value of l.

Fig. 6
Fig. 6

Values of ρ1′ for several modes with l=1, as a function of sphere radius. When four symbols appear in a mode label they denote: the value of l; H (high-frequency) or L (low-frequency); E (electric) or M (magnetic); a subscript that has the same meaning as in Figs. 14.

Fig. 7
Fig. 7

Absorption and extinction cross sections (QA and QT) as functions of frequency for sphere radius W=0.1.

Fig. 8
Fig. 8

Absorption and extinction cross sections as functions of frequency for sphere radius W=1.0.

Fig. 9
Fig. 9

Absorption and extinction cross sections as functions of frequency for sphere radius W=10.0.

Fig. 10
Fig. 10

Comparison of the theoretical extinction cross section of UO2 spheres of radius W′=0.7 in polyethylene with experiment. The frequency scale is dimensionless; the transverse optical frequency of UO2 occurs at ΩT=1 and the longitudinal optical frequency, at ΩL=1.97.

Tables (2)

Tables Icon

Table I The asymptotic limits of ρ1′. The three lowest numerical values of ρ1′ for l=1 are given.

Tables Icon

Table II Frequencies of virtual modes and cross-section maxima.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

ω 2 = [ ( l + 1 ) ω T 2 + l ω L 2 ] / ( 2 l + 1 ) ,
f ( r ) = [ r l Y l m ( θ , φ ) ] ,
( ω ) = + 0 - 1 - ω 2 / ω T 2 - i γ ω / ω T ,
× ( × F ) - ( ω 2 / c 2 ) F = 0.
ψ l m ( r ) = z l ( k r ) Y l m ( θ , φ ) ,
2 ψ + k 2 ψ = 0.
L = ψ
M = L × r
N = k - 1 × M .
M = k - 1 × N .
h l ( 1 ) ( ρ 2 ) d d ρ 1 [ ρ 1 j l ( ρ 1 ) ] - j l ( ρ 1 ) d d ρ 2 [ ρ 2 h l ( 1 ) ( ρ 2 ) ] = 0 ,
h l ( 1 ) ( ρ 2 ) d d ρ 1 [ ρ 1 j l ( ρ 1 ) ] - j l ( ρ 1 ) d d ρ 2 [ ρ 2 h l ( 1 ) ( ρ 2 ) ] = 0.
= - ( l + 1 ) / l .
ψ l m ( r ) = j l ( κ r ) Y l m ( θ , φ ) .
E r = ( / r ) j l ( κ r ) Y l m ( θ , φ ) , E θ = 1 r j l ( κ r ) θ Y l m ( θ , φ ) , E φ = i m r sin θ j l ( κ r ) Y l m ( θ , φ ) ,
E R ( ρ 1 R ) - 1 j l ( ρ 1 R ) Y l m ( θ , φ )
tan ρ 1 = { - i l odd , - i / l even ,
tan ρ 1 = { - i / l odd , - i l even ,
Q A = [ 1 / ( W Ω ) 2 ] l = 1 ( 2 l + 1 ) [ ( A l + A l * - 2 ) / A l A l * + ( B l + B l * - 2 ) / B l B l * ] ,
A l = h l ( 1 ) ( ρ 2 ) ( d / d ρ 1 ) [ ρ 1 j l ( ρ 1 ) ] - j l ( ρ 1 ) ( d / d ρ 2 ) [ ρ 2 h l ( 1 ) ( ρ 2 ) ] j l ( ρ 2 ) ( d / d ρ 1 ) [ ρ 1 j l ( ρ 1 ) ] - j l ( ρ 1 ) ( d / d ρ 2 ) [ ρ 2 j l ( ρ 2 ) ] ,
B l = h l ( 1 ) ( ρ 2 ) ( d / d ρ 1 ) [ ρ 1 j l ( ρ 1 ) ] - j l ( ρ 1 ) ( d / d ρ 2 ) [ ρ 2 h l ( 1 ) ( ρ 2 ) ] j l ( ρ 2 ) ( d / d ρ 1 ) [ ρ 1 j l ( ρ 1 ) ] - j l ( ρ 1 ) ( d / d ρ 2 ) [ ρ 2 j l ( ρ 2 ) ] .
Q S = 1 ( W Ω ) 2 l = 1 ( 2 l + 1 ) [ 2 A l A l * + 2 B l B l * ]
Q T = Q A + Q S = [ 1 / ( W Ω ) 2 ] l = 1 ( 2 l + 1 ) × [ ( A l + A l * ) / A l A l * + ( B l + B l * ) / B l B l * ] .
A l = 1 - i u l , B l = 1 - i v l ,
v l = - j l ( ρ 1 ) d d ρ 2 [ ρ 2 n l ( ρ 2 ) ] - n l ( ρ 2 ) d d ρ 1 [ ρ 1 j l ( ρ 1 ) ] j l ( ρ 1 ) d d ρ 2 [ ρ 2 j l ( ρ 2 ) ] - j l ( ρ 2 ) d d ρ 1 [ ρ 1 j l ( ρ 1 ) ] ;
u l = - ( Ω - ζ ) η + i ( ζ - η ) η ,
Q A 2 ( W Ω ) - 2 Σ ( 2 l + 1 ) ( ζ - η ) η / D ,
Q S 2 ( W Ω ) - 2 Σ ( 2 l + 1 ) ( η ) 2 / D ,
Q T 2 ( W Ω ) - 2 Σ ( 2 l + 1 ) ζ η / D ,
D = ( Ω - ζ ) 2 + ( ζ ) 2 .
l = 1 [ i = 1 { X l L M i + X l H M i + X l L E i } + i = 0 X l H E i + X l S ] ,
Q T = 2 - 4 y - 1 sin y + 4 y - 2 ( 1 - cos y ) ,
( Q A ) max = 2 W - 2 ( 2 l + 1 ) ( ζ - η ) η ( ζ ζ ) - 2
( Q T ) max = 2 W - 2 ( 2 l + 1 ) η ( ζ ) - 2 ( ζ ) - 1 .