- Email: [email protected]
Infrared absorption of cube pairs
__El3
I April 1997
2%
OPTICS
COMMUNICATIONS
ELSEVIER
Optics Communications 136 (I 997) 395-398
Infrared absorption of cube pairs R. Ruppin Depurtmmt
ofPhysicsnnd
Applied
Mathemutics,
’ Sorey NRC,
Yuune 81800,
Received 9 September 1996; revised 24 October 1996; accepted I3
November
Isrue 1996
Abstract The infrared absorption spectra of pairs of small cube shaped particles of diatomic ionic solids are calculated. The dependence of the absorption on the separation between the cubes and on the polarization direction of the incident electromagnetic field is investigated. The spectra exhibit a strong dependence on the electromagnetic interaction between the cubes, so that the single cube theory is useless, unless the crystallites are well separated. PACT
78.30;
Keywords:
68.35
Infrared absorption spectroscopy; Microcrystals
1. Introduction In optical experiments on small ionic crystal particles the crystallites are usually irregular in shape. Of the regular shapes, the sphere appears sometimes [l-5], but the cube occurs more often [6-91. A simple experimental technique, in which cube shaped microcrystals tend to predominate, is that of smoke collection on a substrate [IO-161. On the theoretical side the spherical geometry is by far the most often employed, because of the availability of an analytic solution, which enables the interpretation of the infrared absorption peaks in terms of the optical surface phonons of the sphere [ 171. For the cube shape no analytic solution is possible, and it has been tackled (in the non-retarded limit) by multipole expansion methods [8, IS], or by numerically solving a self-consistent equation for the induced surface polarization [ 191. In most experimental samples the microcrystals are not well separated, and sometimes they even tend to form clusters. The electromagnetic interaction between the crystallites can modify the infrared absorption spectra in a profound way. The simplest model for studying the effects of inter-particle interaction on the absorption spectrum is
’ E-mail: [email protected]. 0030.4018/97/$17.00 PIf
Copyright
SOO30-4018(96)00721-3
0
1997
that of a pair of microcrystals. The absorption spectra of a two sphere system have been calculated by various versions of multipole expansion methods [20-271 (which can also be generalized to clusters containing more than two spheres). Another approach, that of using the bispherical coordinate system [28], is restricted to the electrostatic limit (sphere size small in comparison with the wavelength) and to two spheres only. Also, in the case of touching spheres, the tangent spherical coordinate system has to be used instead [29]. For cube pairs most of these methods of calculation are inapplicable. Here we calculate the infrared absorption properties of a pair of cubes by using the discrete dipole approximation [30]. We investigate the dependence of the spectrum on the separation between the cubes and on the orientation of the incident field.
2. Method of calculation
We employ the discrete dipole approximation (DDA), developed by Purcell and Pennypacker [30], in which the microcrystal is replaced by an array of N point dipoles. Each dipole has an oscillating polarization in response to both the incident plane wave and the electric field due to all of the other dipoles in the array. The self-consistent
Elsevier Science B.V. All rights reserved.
R. Ruppin/Optics
396
Communicution.s 136 (1997) 395-398
solution for the dipole moments P,. j= I,. .., N is obtained by solving the N simultaneous complex vector equations
C AjkP”).
(‘1
k#j
Here E,,,,+ is the incident electric field at the position of the jth dipole, and -A,, Pk is the contribution to the electric field at position j due to the dipole at position k: exp( i kr,,) A,,Pk
= ‘;k
3
k2rlrX(rj,XPk)
b 0.2
-
(1 -ikr,k)
+
2 ‘jk
0.0 190
I
I
210
230
250
Frequency(cm-‘) For LY,, the polarizability of each dipole. the simplest choice is the Clausius-Mossotti polarizability
a
3
extinction
cross section
of a NaCl cube of
E-l
(3)
J =GE+I’
where E is the dielectric constant of the crystallite material, and n is the number density of dipoles. Solving the system of 3N simultaneous linear equations (I) yields the polarizations P,, and from these the extinction cross section Q can be computed by
(4) where k = w/c. The extinction cross section will be normalized by dividing by vR2. Here R is the radius of a sphere of equal volume, i.e., R = (3V/47r)‘/“, where V is the volume of the object in question ( a3 for a cube of edge a, 2a3 for a pair of cubes). Efficient algorithms for solving the system (1) have been discussed by Draine [3 I] and Draine and Flatau [32], who have also developed a computer program DDSCAT, which we will employ here for our calculations.
3. Numerical results and discussion For the frequency dependent dielectric constant of the microcrystals we employ the usual diatomic ionic crystal expression c( 0) =
Fi g. I. Normalized edge 0.1 km.
Eo - E, E,
+
I - (w/~r)~
- iy( w/or)
’
(5)
where +, and s, are the static and the high frequency dielectric constants, respectively, wr is the transverse optical frequency, and y is a damping constant. In the numerical calculations we will employ the NaCl values [33]: co = 5.934, E, = 2.328, wr = 164 cm-‘, y= 0.02.
First, we calculate the extinction cross section of a single NaCl cube, of edge a = 0.1 km. We started with an array of 6 X 6 X 6 = 216 dipoles, and gradually increased the number of dipoles until the results converged, so that the locations of the main absorption peaks have stabilized to within 0.1 cm-‘. The appropriate number of dipoles was I2 X I2 X I2 = 1278, and the calculated extinction cross section is shown in Fig. 1. Six absorption peaks are discernible in the spectrum, with the two at lowest frequencies being much stronger than the other ones. This is in qualitative agreement with the results of Fuchs [19], although the peak positions computed by him differ somewhat from the more exact results presented here. Next, we perform calculations for a pair of cubes, each of size a = 0.1 km. The cubes are oriented in parallel, with a separation d between the nearest faces (Fig. 2). Two independent polarizations of the external field will be considered, with the applied electric field parallel (E,,) or perpendicular (E,) to the line joining the cube centers. Again, the DDA is used, with each cube represented by 1278 dipoles. The extinction for a separation d = a/2 is shown in Fig. 3. Comparing with the single cube spectrum (Fig. l), it is found that the main effect is the shift of the absorption peaks for the parallel polarization to the low frequency side. This shift increases with decreasing intercube separation, as can be seen from Fig. 4, which was
I n--dC I
El
a
---GFig. 2. Configuration
of two cubes.
R. Ruppin/Optics
136 (1997) 395-398
397
i
t 0.6
Communications
-
1
b 0.4 -
b
_._
190
230
210
250
210
Frequency(cm-‘) Fig. 3. Normalized
extinction
cross
230
250
Frequency(cm-‘)
section
of a pair of NaCl
Fig. 5. Normalized extinction cross section of a pair of NaCl
cubes, with u = 0.1 pm, d= u/2. Full curve - parallel polarization, dashed curve - perpendicular polarization.
cubes, with a = 0.1 pm, d = 0. Full curve - parallel polarization, dashed curve - perpendicular polarization.
calculated for d = a/6. There also occurs a redistribution of the peak strengths, with the lowest frequency peak becoming much stronger than all the other ones. When the
in all calculations presented here.) As an example, Fig. 6 shows the resulting spectrum for d = a/4. in comparison with the isolated cube. (dashed curve of Fig. 61, randomly oriented cube-pairs, of fixed separation d, yield a broadened spectrum, with the peaks becoming less distinct. In actual experimental situations the inter-cube distance is not fixed, so that further averaging, over a distribution of d would be needed. Furthermore, we have only considered the simplest cube-pair configuration, with the two cubes oriented in parallel. In general, averaging over configurations in which one cube is rotated with respect to the other would also be needed. All these averaging procedures would further broaden the spectrum, and wipe out any distinct peaks. It is therefore pointless to try to fit a
two cubes
come
into contact,
forming
a rectangle
of length
in Fig. 5. For the parallel polarization the low frequency peak at 194.4 cm-’ completely dominates the spectrum, while for the perpendicular polarization the changes in the spectrum are relatively small. For randomly oriented cube pairs we have to average over the two polarizations, giving the perpendicular one double weight. (This averaging procedure is exact in the non-retarded limit, when the wavelength is much longer than the dimensions of the two cube system, as is the case 2~.
the spectrum
assumes
the form
shown
11
0.8
0.6
b 0.4
0.4
b 0.2
210
230
Frequency(cm-‘) Fig. 4. Normalized extinction cross section of a pair of NaCl cubes, with a = 0.1 km, d= u/6. Full curve - parallel polarization, dashed curve - perpendicular polarization.
0.0
T-----l 190
210
230
250
Frequency(cm-‘) Fig. 6. Normalized extinction cross section of randomly oriented pairs of NaCl cubes, with o = 0. I km. d = n/4 (full curve). The dashed curve shows the result for a single cube.
398
R: Ruppin/
Optics Communiccrtions 136 (1997) 395-398
spectrum measured on cube shaped microcrystals with the theoretical spectrum calculated for a single cube [8,19], unless the experimental procedure involves special precautions in order to keep the crystallites well separated [ 14,151. Finally, we compare the results for two cubes with analogous calculations for two spheres [20-291. The following features are common to sphere-pairs and cube-pairs: For E,, the spectrum shifts strongly to the low frequency side with decreasing particle separation, until, for touching particles, the spectrum is dominated by one low frequency peak. For E I the positions of the absorption peaks change relatively little with decreasing separation between the crystallites.
References
[i] T.R. Steyer, K.L. Day and D.R. Huffman, Appl. Optics 13 (1974) 1586. [2] C.J. Sema, M. Ocana and J.E. Iglesias, J. Phys. C 20 (1987) 473. [3] M. Ocana, V. Fomes, J.V. Garcia Ramos and C.J. Sema, J. Sol. State Chem. 75 (1988) 364. [4] J.E. Iglesias, M. Ocana and C.J. Sema, Appl. Spectrosc. 44 (1990) 418. [5] C. Pecharromen, T. Gonzalez Carreno and J.E. Iglesias, Appl. Spectrosc. 47 (1993) 1203. [6] J.T. Luxon, D.J. Montgomery and R. Summitt, J. Appl. Phys. 41 (1970) 2303. [7] V.V. Bryksin, Y.M. Gerbstein and D.N. Mirlin, Solid State Commun. 9 (1971) 669. [8] A.P. van Gelder. J. Holvast, J.H.M. Stoelinga and P. Wyder, J. Phys. C 5 ( 1972) 2757.
191 R. Kahn and F. Kneubuhl, Infrared Phys. I6 (1976) 491. [lOI J.T. Luxon, D.J. Montgomery and R. Summitt, Phys. Rev. 188 (1969) 1345. [I II T.P. Martin, Solid State Commun. 9 (I 97 I) 623. I121 K.H. Rieder, M. Ishigame and L. Genzel, Phys. Rev. B 6 ( 1972) 3804. ]131 S. Hayashi, N. Nakamori, J. Hirono and H. Kanamori, J. Phys. Sot. Japan 43 (1977) 2006. ]I41 T.P. Martin and H. Schaber, Phys. Stat. Sol. (b) 81 (1977) K41. ]I51 I.J. Dayawansa and C.F. Bohren, Phys. Stat. Sol. (b) 86 (I 978) K27. [I61 0. Matumura and M. Cho, J. Opt. Sot. Am. 71 ( 198 I) 393. ]I71 R. Ruppin and R. Englman, Rep. Prog. Phys. 33 (I 970) 149. [If31D. Langbein, J. Phys. A 9 (1976) 627. ]I91 R. Fuchs, Phys. Rev. B 1 I (1975) 1732. ]201 J.M. Gerardy and M. Ausloos, Phys. Rev. B 22 (1980) 4950. [21] J.M. Gerardy and M. Ausloos, Surface Sci. 106 (1981) 319. ]221 P. Clippe and M. Ausloos, Phys. Stat. Sol. (b)llO (1982) 21 I. [2?1 F. Clara, Phys. Rev. B 25 (1982) 7875. ]241 F. Claro, Solid State Commun. 49 (1984) 229. ]251 J.M. Gerardy and M. Ausloos, Phys. Rev. B 30 (1984) 2167. ]26] F. Clara, Phys. Rev. B 30 (1984) 4989. ]271 L. Fu and L. Resca, Phys. Rev. B 52 (1995) 10815. ]28] R. Ruppin, J. Phys. Sot. Japan 58 (1989) 1446. 1291 A.V. Paley, A.V. Radchik and G.B. Smith, J. Appl. Phys. 73 (1993) 3446. 1301 E.M. Purcell and C.P. Pennypacker, Astrophys. J. 186 (I 973) 705. 1311 B.T. Draine, Astrophys. J. 333 (1988) 848. 1321 B.T. Draine and P.J. Flatau, J. Opt. Sot. Am. A I I (1994) 1491. ]331 H. Bilz, L. Genzel and M. Happ, Z. Phys. 160 (1960) 535.
I April 1997
2%
OPTICS
COMMUNICATIONS
ELSEVIER
Optics Communications 136 (I 997) 395-398
Infrared absorption of cube pairs R. Ruppin Depurtmmt
ofPhysicsnnd
Applied
Mathemutics,
’ Sorey NRC,
Yuune 81800,
Received 9 September 1996; revised 24 October 1996; accepted I3
November
Isrue 1996
Abstract The infrared absorption spectra of pairs of small cube shaped particles of diatomic ionic solids are calculated. The dependence of the absorption on the separation between the cubes and on the polarization direction of the incident electromagnetic field is investigated. The spectra exhibit a strong dependence on the electromagnetic interaction between the cubes, so that the single cube theory is useless, unless the crystallites are well separated. PACT
78.30;
Keywords:
68.35
Infrared absorption spectroscopy; Microcrystals
1. Introduction In optical experiments on small ionic crystal particles the crystallites are usually irregular in shape. Of the regular shapes, the sphere appears sometimes [l-5], but the cube occurs more often [6-91. A simple experimental technique, in which cube shaped microcrystals tend to predominate, is that of smoke collection on a substrate [IO-161. On the theoretical side the spherical geometry is by far the most often employed, because of the availability of an analytic solution, which enables the interpretation of the infrared absorption peaks in terms of the optical surface phonons of the sphere [ 171. For the cube shape no analytic solution is possible, and it has been tackled (in the non-retarded limit) by multipole expansion methods [8, IS], or by numerically solving a self-consistent equation for the induced surface polarization [ 191. In most experimental samples the microcrystals are not well separated, and sometimes they even tend to form clusters. The electromagnetic interaction between the crystallites can modify the infrared absorption spectra in a profound way. The simplest model for studying the effects of inter-particle interaction on the absorption spectrum is
’ E-mail: [email protected]. 0030.4018/97/$17.00 PIf
Copyright
SOO30-4018(96)00721-3
0
1997
that of a pair of microcrystals. The absorption spectra of a two sphere system have been calculated by various versions of multipole expansion methods [20-271 (which can also be generalized to clusters containing more than two spheres). Another approach, that of using the bispherical coordinate system [28], is restricted to the electrostatic limit (sphere size small in comparison with the wavelength) and to two spheres only. Also, in the case of touching spheres, the tangent spherical coordinate system has to be used instead [29]. For cube pairs most of these methods of calculation are inapplicable. Here we calculate the infrared absorption properties of a pair of cubes by using the discrete dipole approximation [30]. We investigate the dependence of the spectrum on the separation between the cubes and on the orientation of the incident field.
2. Method of calculation
We employ the discrete dipole approximation (DDA), developed by Purcell and Pennypacker [30], in which the microcrystal is replaced by an array of N point dipoles. Each dipole has an oscillating polarization in response to both the incident plane wave and the electric field due to all of the other dipoles in the array. The self-consistent
Elsevier Science B.V. All rights reserved.
R. Ruppin/Optics
396
Communicution.s 136 (1997) 395-398
solution for the dipole moments P,. j= I,. .., N is obtained by solving the N simultaneous complex vector equations
C AjkP”).
(‘1
k#j
Here E,,,,+ is the incident electric field at the position of the jth dipole, and -A,, Pk is the contribution to the electric field at position j due to the dipole at position k: exp( i kr,,) A,,Pk
= ‘;k
3
k2rlrX(rj,XPk)
b 0.2
-
(1 -ikr,k)
+
2 ‘jk
0.0 190
I
I
210
230
250
Frequency(cm-‘) For LY,, the polarizability of each dipole. the simplest choice is the Clausius-Mossotti polarizability
a
3
extinction
cross section
of a NaCl cube of
E-l
(3)
J =GE+I’
where E is the dielectric constant of the crystallite material, and n is the number density of dipoles. Solving the system of 3N simultaneous linear equations (I) yields the polarizations P,, and from these the extinction cross section Q can be computed by
(4) where k = w/c. The extinction cross section will be normalized by dividing by vR2. Here R is the radius of a sphere of equal volume, i.e., R = (3V/47r)‘/“, where V is the volume of the object in question ( a3 for a cube of edge a, 2a3 for a pair of cubes). Efficient algorithms for solving the system (1) have been discussed by Draine [3 I] and Draine and Flatau [32], who have also developed a computer program DDSCAT, which we will employ here for our calculations.
3. Numerical results and discussion For the frequency dependent dielectric constant of the microcrystals we employ the usual diatomic ionic crystal expression c( 0) =
Fi g. I. Normalized edge 0.1 km.
Eo - E, E,
+
I - (w/~r)~
- iy( w/or)
’
(5)
where +, and s, are the static and the high frequency dielectric constants, respectively, wr is the transverse optical frequency, and y is a damping constant. In the numerical calculations we will employ the NaCl values [33]: co = 5.934, E, = 2.328, wr = 164 cm-‘, y= 0.02.
First, we calculate the extinction cross section of a single NaCl cube, of edge a = 0.1 km. We started with an array of 6 X 6 X 6 = 216 dipoles, and gradually increased the number of dipoles until the results converged, so that the locations of the main absorption peaks have stabilized to within 0.1 cm-‘. The appropriate number of dipoles was I2 X I2 X I2 = 1278, and the calculated extinction cross section is shown in Fig. 1. Six absorption peaks are discernible in the spectrum, with the two at lowest frequencies being much stronger than the other ones. This is in qualitative agreement with the results of Fuchs [19], although the peak positions computed by him differ somewhat from the more exact results presented here. Next, we perform calculations for a pair of cubes, each of size a = 0.1 km. The cubes are oriented in parallel, with a separation d between the nearest faces (Fig. 2). Two independent polarizations of the external field will be considered, with the applied electric field parallel (E,,) or perpendicular (E,) to the line joining the cube centers. Again, the DDA is used, with each cube represented by 1278 dipoles. The extinction for a separation d = a/2 is shown in Fig. 3. Comparing with the single cube spectrum (Fig. l), it is found that the main effect is the shift of the absorption peaks for the parallel polarization to the low frequency side. This shift increases with decreasing intercube separation, as can be seen from Fig. 4, which was
I n--dC I
El
a
---GFig. 2. Configuration
of two cubes.
R. Ruppin/Optics
136 (1997) 395-398
397
i
t 0.6
Communications
-
1
b 0.4 -
b
_._
190
230
210
250
210
Frequency(cm-‘) Fig. 3. Normalized
extinction
cross
230
250
Frequency(cm-‘)
section
of a pair of NaCl
Fig. 5. Normalized extinction cross section of a pair of NaCl
cubes, with u = 0.1 pm, d= u/2. Full curve - parallel polarization, dashed curve - perpendicular polarization.
cubes, with a = 0.1 pm, d = 0. Full curve - parallel polarization, dashed curve - perpendicular polarization.
calculated for d = a/6. There also occurs a redistribution of the peak strengths, with the lowest frequency peak becoming much stronger than all the other ones. When the
in all calculations presented here.) As an example, Fig. 6 shows the resulting spectrum for d = a/4. in comparison with the isolated cube. (dashed curve of Fig. 61, randomly oriented cube-pairs, of fixed separation d, yield a broadened spectrum, with the peaks becoming less distinct. In actual experimental situations the inter-cube distance is not fixed, so that further averaging, over a distribution of d would be needed. Furthermore, we have only considered the simplest cube-pair configuration, with the two cubes oriented in parallel. In general, averaging over configurations in which one cube is rotated with respect to the other would also be needed. All these averaging procedures would further broaden the spectrum, and wipe out any distinct peaks. It is therefore pointless to try to fit a
two cubes
come
into contact,
forming
a rectangle
of length
in Fig. 5. For the parallel polarization the low frequency peak at 194.4 cm-’ completely dominates the spectrum, while for the perpendicular polarization the changes in the spectrum are relatively small. For randomly oriented cube pairs we have to average over the two polarizations, giving the perpendicular one double weight. (This averaging procedure is exact in the non-retarded limit, when the wavelength is much longer than the dimensions of the two cube system, as is the case 2~.
the spectrum
assumes
the form
shown
11
0.8
0.6
b 0.4
0.4
b 0.2
210
230
Frequency(cm-‘) Fig. 4. Normalized extinction cross section of a pair of NaCl cubes, with a = 0.1 km, d= u/6. Full curve - parallel polarization, dashed curve - perpendicular polarization.
0.0
T-----l 190
210
230
250
Frequency(cm-‘) Fig. 6. Normalized extinction cross section of randomly oriented pairs of NaCl cubes, with o = 0. I km. d = n/4 (full curve). The dashed curve shows the result for a single cube.
398
R: Ruppin/
Optics Communiccrtions 136 (1997) 395-398
spectrum measured on cube shaped microcrystals with the theoretical spectrum calculated for a single cube [8,19], unless the experimental procedure involves special precautions in order to keep the crystallites well separated [ 14,151. Finally, we compare the results for two cubes with analogous calculations for two spheres [20-291. The following features are common to sphere-pairs and cube-pairs: For E,, the spectrum shifts strongly to the low frequency side with decreasing particle separation, until, for touching particles, the spectrum is dominated by one low frequency peak. For E I the positions of the absorption peaks change relatively little with decreasing separation between the crystallites.
References
[i] T.R. Steyer, K.L. Day and D.R. Huffman, Appl. Optics 13 (1974) 1586. [2] C.J. Sema, M. Ocana and J.E. Iglesias, J. Phys. C 20 (1987) 473. [3] M. Ocana, V. Fomes, J.V. Garcia Ramos and C.J. Sema, J. Sol. State Chem. 75 (1988) 364. [4] J.E. Iglesias, M. Ocana and C.J. Sema, Appl. Spectrosc. 44 (1990) 418. [5] C. Pecharromen, T. Gonzalez Carreno and J.E. Iglesias, Appl. Spectrosc. 47 (1993) 1203. [6] J.T. Luxon, D.J. Montgomery and R. Summitt, J. Appl. Phys. 41 (1970) 2303. [7] V.V. Bryksin, Y.M. Gerbstein and D.N. Mirlin, Solid State Commun. 9 (1971) 669. [8] A.P. van Gelder. J. Holvast, J.H.M. Stoelinga and P. Wyder, J. Phys. C 5 ( 1972) 2757.
191 R. Kahn and F. Kneubuhl, Infrared Phys. I6 (1976) 491. [lOI J.T. Luxon, D.J. Montgomery and R. Summitt, Phys. Rev. 188 (1969) 1345. [I II T.P. Martin, Solid State Commun. 9 (I 97 I) 623. I121 K.H. Rieder, M. Ishigame and L. Genzel, Phys. Rev. B 6 ( 1972) 3804. ]131 S. Hayashi, N. Nakamori, J. Hirono and H. Kanamori, J. Phys. Sot. Japan 43 (1977) 2006. ]I41 T.P. Martin and H. Schaber, Phys. Stat. Sol. (b) 81 (1977) K41. ]I51 I.J. Dayawansa and C.F. Bohren, Phys. Stat. Sol. (b) 86 (I 978) K27. [I61 0. Matumura and M. Cho, J. Opt. Sot. Am. 71 ( 198 I) 393. ]I71 R. Ruppin and R. Englman, Rep. Prog. Phys. 33 (I 970) 149. [If31D. Langbein, J. Phys. A 9 (1976) 627. ]I91 R. Fuchs, Phys. Rev. B 1 I (1975) 1732. ]201 J.M. Gerardy and M. Ausloos, Phys. Rev. B 22 (1980) 4950. [21] J.M. Gerardy and M. Ausloos, Surface Sci. 106 (1981) 319. ]221 P. Clippe and M. Ausloos, Phys. Stat. Sol. (b)llO (1982) 21 I. [2?1 F. Clara, Phys. Rev. B 25 (1982) 7875. ]241 F. Claro, Solid State Commun. 49 (1984) 229. ]251 J.M. Gerardy and M. Ausloos, Phys. Rev. B 30 (1984) 2167. ]26] F. Clara, Phys. Rev. B 30 (1984) 4989. ]271 L. Fu and L. Resca, Phys. Rev. B 52 (1995) 10815. ]28] R. Ruppin, J. Phys. Sot. Japan 58 (1989) 1446. 1291 A.V. Paley, A.V. Radchik and G.B. Smith, J. Appl. Phys. 73 (1993) 3446. 1301 E.M. Purcell and C.P. Pennypacker, Astrophys. J. 186 (I 973) 705. 1311 B.T. Draine, Astrophys. J. 333 (1988) 848. 1321 B.T. Draine and P.J. Flatau, J. Opt. Sot. Am. A I I (1994) 1491. ]331 H. Bilz, L. Genzel and M. Happ, Z. Phys. 160 (1960) 535.