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Absolute scattering cross-section measurements by a ratio technique
Volume 17, number 1
OPTICS COMMUNICATIONS
April 1976
ABSOLUTE SCATTERING CROSS-SECTION MEASUREMENTS BY A RATIO TECHNIQUE* Leslie A. REITH and Harry L. SW1NNEY Physics Department, City College o f the City University o f New York,
New York 10031, USA Received 5 January 1976
Calculations and measurements are presented which show that absolute scattering cross sections ("Rayleigh ratios") can be determined by measuring the ratio of the depolarized intensity of doubly scattered light to the polarized intensity of singly scattered light. This technique should be useful in determining the susceptibilities of fluids and the molecular weights of macromolecules. Absolute scattering intensity measurements are widely used to investigate the susceptibilities o f liquids and solids, and the molecular weights o f macromolecules. The determination of an absolute scattering cross-section is notoriously difficult, requiring various scattering geometry corrections and measurements of incident and scattered intensities which differ by many orders o f magnitude. Even the cross-section for benzene, which has been frequently measured since benzene is used as a secondary standard in p h o t o m e t r y , has varied widely in careful measurements in recent years (see the discussion in ref. [1 ]). We have developed a ratio technique for measuring absolute scattering cross-sections which largely circumvents the major difficulties encountered in absolute intensity measurements. In our approach the cross-section per unit volume, o (the "Rayleigh ratio"), is determined from measurements o f the ratio o f the depolarized intensity o f doubly scattered light, IvdH , to the polarized intensity o f the singly scattered light, I ~ v . Since the double scattering intensity is proportional to cr2 while the single scattering intensity is proportional to o, the ratio IdH/l~/V will simply be proportional to o. The results o f calculations o f the proportionality factor that relates IdH/ISw to a are presented here for a fluid near the critical point and for a solution o f rodshaped macromolecules; these calculations are based on integral expressions for the scattering intensity derived previously [2, 3]. This paper concludes with a * Research supported by the National Science Foundation.
I
Fig. 1. Scattering geometry used in the present depolarization calculation and experiment. A laser beam, incident along the x-axis and polarized along the z-axis, illuminates a cylindrical volume with diameter d and length 2R. The detector, located on the y-axis at a large distance from the sample, views the paraUelpiped shown, which has dimensions 2R × w × h. The total sample volume is a parallelpiped (not shown) with dimensions 2R × 2R × H. In the present experiment h and z s were varied and the other dimensions were held fixed at the values (in mm): 2R = 6.00, H = 30, w = 0.54, and d = 0.2. discussion of two experimental approaches to the determination ofl~H/I~v, and these two approaches are illustrated with data for xenon. Consider the scattering geometry shown in fig. 1. The parallelpiped sample cell is illuminated by a focused laser beam incident along x, polarized in the z (vertical, V) direction. The detector, located at a large distance away on the y-axis, views a thin rectangular volume Vs of dimensions 2 R (length) X w (width) × h (height). The total sample volume is 2R X 2 R × H , 111
Volume 17, number 1
OPTICS COMMUNICATIONS
April 1976
and the volume illuminated by the laser has a length 2R and a diameter d. The distance from the beam axis to the center of the volume seen by the detector is z sUsing the known form for the polarized single scattering cross-section, we calculate the depolarized intensity arising from two successive scattering events by considering the events to be independent and integrating over the volumes available for each scattering event [2]. This procedure, which assumes that higher order scattering is negligible, is justified by the microscopic scattering theory of Oxtoby and Gelbart [4]. The starting point for our calculation is the cross section per unit volume,
portionality to h described by eq. (2). Eq. (3) can easily be integrated for P(q) = 1, which corresponds to fluids far from the critical point and to solutions of macromolecules whose size is small compared to the wavelength; we obtaing = ¼7r [2,3]. Thus a measurement ofldvH/I~rv yields directly o O, the angle-independent part of the absolute scattering cross section. The expressions for o 0 for a pure fluid [5], a binary mixture [6], and a dilute solution of macromolecules of molecular weight M [7] are, respectively, o~0 = (rr 2/X4)
(3e/~p)2TkTp2KT ,
(4)
o(q) = ooP(q)
o~ = (rr2/X 4)
OelOc)2rkr(aclau)e,r/p,
(5)
sin2~,
(1)
where the sin2¢ factor describes the angular dependence of Rayleigh scattering (9 is the angle between the incident polarization direction and the propagation vector of the scattered light), P(q) describes departures from the dipole scattering intensity distribution (due, e.g., in a fluid near the critical point, to the long-range correlations), and q = [k 0 - ks[, where k 0 and k s are, respectively, the wave vectors of the incident and scattered light. The quantity of interest is the depolarization ratio ]dR/]~,'V corresponding to the geometry in fig. 1. The depolarized scattering intensity calculation is simplified if (w, h, z s, and d) ~ R, which is not difficult to achieve in practice. The polarized intensity I~rv that we calculate corresponds to a geometry with z s = 0 and h >> d. IV~His also calculated for z s = 0, but IvdH is independent of z s if h ~ R and Zs ~ R. The depolarization ratio calculated for the conditions just described
o~0= (47r2/X4) n 2 (an~tic) 2 Mc/N A ,
(6)
where the symbols have the usual meanings [ 5 - 7 ] , and the expression for o0B is an approximation applicable when scattering arising from pressure and temperature fluctuations can be neglected. Now consider two examples of systems for which P(q) 4= 1 : fluids near the critical point and solutions of large rod-shaped macromolecules. For both pure fluids and binary mixtures near the critical point we have, assuming that the long-range correlations are described 1.0
is [21 t~n 0.1
IdvH/I~rv = g ooh ,
(2)
with
R
R
Zs+½h
g - - f f-R dx -fRdY fzs-½h dzP(q')P(q")x2z2/r 6 , (3) where f = [hP(q)] -1 ; q' and q" are, respectively, the scattering vectors for the first and second scatterings in the double scattering process; and q is the scattering vector for single scattering at 90 °. With the stated assumptions it can be shown that g is independent of the dimensions of the scattering geometry. Thus the only dependence ofldH/I~rv on the geometry is the pro112
0"010.5
1.0
'2.0
5
|0
I
20
50
U Fig. 2. The p r o p o r t i o n a l i t y f a c t o r g in the d e p o l a r i z a t i o n r a t i o e x p r e s s i o n IVH/Ivv d s -gooh, calculated numerically for fluids
near the critical point (o) (see eq. (4.3) in [3]), with u = qG and for a dilute solution of thin rods (A), with u = qL, where q corresponds to 90° scattering. The dashed line is the limit ofg as u --*0.
Volume 17, number 1
OPTICS COMMUNICATIONS
by the O r n s t e i n - Z e r n i k e form,
P(q) = (1 + q2~2)-1 ,
(7)
where ~ is the correlation length. F o r a dilute solution of macromolecules, each with a volume Vm , the angular dependence o f the scattering is described by [7]
P(q)= gm1 f exp(iq'r)dV
,
(8)
Vm
April 1976
both from double scattering and depolarized single scattering; when IVH is measured with Vs positioned below or above the laser beam (as pictured in fig. 1), the detector views only the multiply scattered light. Fig. 3 shows IVH/1VV measurements for (a) z s = 0 and h varied and (b) z s varied and h fixed. In b o t h cases the polarized intensity was measured with z s = 0 and h large (as in the calculations). The depolarization ratio measured with z s= 0 includes a contribution from depolarized single scattering, which is given by the h = 0
which becomes for a thin rod o f length L [7], 4.0 U
P(u)
=
(~u) - 1 f [(sin t)/t] d t - [(sin l u ) / l u ] 2 0
,
(9)
where u = qL. The depolarization integral [eq. (3)] has been evaluated by the Monte Carlo technique for the angle-dependent factors given by eqs. (7) and (9), and the result for g is plotted in fig. 2. The curve for fluids in the critical region is applicable except very near the critical point where scattering higher than secondorder contributes significantly; the contribution o f triple and higher order scattering becomes important for some fluids for ( T - Tc)/T c ~ 10 - 3 , but for binary mixtures whose refractive indices are nearly equal the present theory is valid in to ( T - Tc)/T c ~ 10 - 5 . The result for g for rod-shaped macromolecules in solution illustrates that depolarization ratio measurements can be used to determine the molecular weight o f macromolecules if size and shape.are approximately known (from, e.g., electron microscopy). If the molecules are sufficiently small, no information on the shape is required, and g is given by ¼ rr. To illustrate the use of the depolarization ratio technique in determining scattering cross sections we have performed measurements on xenon near the critical point in two ways: with z s = 0 and h varied, and with z s varied and h fixed. In our apparatus h is varied by varying the height o f a slit in front of the detector (see [2]), and z s is varied by translating the collection optics and detector vertically. We measured the polarized scattering intensity 1w with V s centered on the laser beam. Since single scattering can occur only in the illuminated volume, whereas double scattering can occur anywhere in the sample volume, I v v includes contributions from b o t h single and double scattering, but IVdV ~ I ~ r v . When the depolarized intensity is measured with z s = 0, IVH includes contributions arising
3.0 en I0
"~2.0
1.0
1
O0
I
O.S
1.0
h 6.0
1.5
(mm)
( b ) h = 0 . 2 mm
I0
4.0
3,0
I
-1.0
I
i
-0.5
i
0 Zs
0.5
I
1.0
Imml
Fig. 3. The depolarization ratio measured for xenon at T - Tc = 0.302 K, p = Pc: (a) with z s = 0 and h varied, and (b) with z s varied and h = 0.200 mm. The scattering geometry was that shown in fig. 1 and the wavelength of the incident light was 457.9 run-;g = 0.672 for this temperature and laser wavelength. The data have been corrected for the small dependence ofg on h and z s (see eq. (4.8) in [3]); this correction is only a few percent for most of the data and is at the most 17%. 113
Volume 17, number 1
OPTICS COMMUNICATIONS
intercept (see fig. 3a), and a contribution from double scattering, which is linearly proportional to h [see eq. (2)] ; o 0 is simply obtained from the slope of the straight line, which is g o O. The problem with this technique for determining o 0 is that for weakly scattering samples the variation of the depolarization ratio with h is small compared to the intercept. For weakly scattering samples it is preferable to measure the depolarized intensity with z s > h, so the directly illuminated volume is not seen by the detector; then the measured depolarized intensity includes only the contributions from double scattering. Measurements of the depolarization ratio as a function Of Zs, with h fixed, as shown in fig. 3b, include the depolarized single scattering as a peak centered at z s = 0, while the depolarization arising solely from double scattering, indicated by the horizontal dashed line in fig. 3b, is independent of z s. The data in fig. 3 yield o 0 = (3.87-+ 0.20) X 10 - 2 cm - 1 , while the value for cr0 calculated from eq. (4), using independent data tabulated in ref. [8], is 4.6 X 10 - 2 cm - 1 . In conclusion, it has been shown that measurements of the depolarization ratio o f the light scattered by fluids and solutions of macromolecules yield, with the proper choice of scattering geometry, the absolute
114
April 1976
scattering cross-section. Since ratios o f scattering intensities can almost always be measured more accurately and easily than absolute intensities, the depolarization ratio technique should be generally useful in determining scattering cross-sections. Future work could extend this technique to systems for which the single scattering is depolarized and to systems for which triple and higher order scattering is important. We thank A. Bray, R.F. Chang, D. Beysens and P. Calmettes for helpful discussions.
References [ 1 ] E.R. Pike, W.R.M. Pomeroy and J.M. Vaughan, J. Chem. Phys. 62 (1975) 3188. [2] L.A. Reith and H.L. Swinney, Phys. Rev. A12 (1975) 1094 [3] A. Bray and R.F. Chang, Phys. Rev. A12 (1975) 2594. [4] D.W. Oxtoby and W.M. Gelbart, J. Chem. Phys. 60 (1974) 3359. [5] M. Fixman, J. Chem. Phys. 23 (1955) 2074. [6] R.D. Mountain and J.M. Deutch, J. Chem. Phys. 50 (1969) 1103. [7] G. Oster, Chem. Reviews 43 (1946) 319. [8] H.L. Swinney and D.L. Henry, Phys. Rev. A8 (1973) 2586.
OPTICS COMMUNICATIONS
April 1976
ABSOLUTE SCATTERING CROSS-SECTION MEASUREMENTS BY A RATIO TECHNIQUE* Leslie A. REITH and Harry L. SW1NNEY Physics Department, City College o f the City University o f New York,
New York 10031, USA Received 5 January 1976
Calculations and measurements are presented which show that absolute scattering cross sections ("Rayleigh ratios") can be determined by measuring the ratio of the depolarized intensity of doubly scattered light to the polarized intensity of singly scattered light. This technique should be useful in determining the susceptibilities of fluids and the molecular weights of macromolecules. Absolute scattering intensity measurements are widely used to investigate the susceptibilities o f liquids and solids, and the molecular weights o f macromolecules. The determination of an absolute scattering cross-section is notoriously difficult, requiring various scattering geometry corrections and measurements of incident and scattered intensities which differ by many orders o f magnitude. Even the cross-section for benzene, which has been frequently measured since benzene is used as a secondary standard in p h o t o m e t r y , has varied widely in careful measurements in recent years (see the discussion in ref. [1 ]). We have developed a ratio technique for measuring absolute scattering cross-sections which largely circumvents the major difficulties encountered in absolute intensity measurements. In our approach the cross-section per unit volume, o (the "Rayleigh ratio"), is determined from measurements o f the ratio o f the depolarized intensity o f doubly scattered light, IvdH , to the polarized intensity o f the singly scattered light, I ~ v . Since the double scattering intensity is proportional to cr2 while the single scattering intensity is proportional to o, the ratio IdH/l~/V will simply be proportional to o. The results o f calculations o f the proportionality factor that relates IdH/ISw to a are presented here for a fluid near the critical point and for a solution o f rodshaped macromolecules; these calculations are based on integral expressions for the scattering intensity derived previously [2, 3]. This paper concludes with a * Research supported by the National Science Foundation.
I
Fig. 1. Scattering geometry used in the present depolarization calculation and experiment. A laser beam, incident along the x-axis and polarized along the z-axis, illuminates a cylindrical volume with diameter d and length 2R. The detector, located on the y-axis at a large distance from the sample, views the paraUelpiped shown, which has dimensions 2R × w × h. The total sample volume is a parallelpiped (not shown) with dimensions 2R × 2R × H. In the present experiment h and z s were varied and the other dimensions were held fixed at the values (in mm): 2R = 6.00, H = 30, w = 0.54, and d = 0.2. discussion of two experimental approaches to the determination ofl~H/I~v, and these two approaches are illustrated with data for xenon. Consider the scattering geometry shown in fig. 1. The parallelpiped sample cell is illuminated by a focused laser beam incident along x, polarized in the z (vertical, V) direction. The detector, located at a large distance away on the y-axis, views a thin rectangular volume Vs of dimensions 2 R (length) X w (width) × h (height). The total sample volume is 2R X 2 R × H , 111
Volume 17, number 1
OPTICS COMMUNICATIONS
April 1976
and the volume illuminated by the laser has a length 2R and a diameter d. The distance from the beam axis to the center of the volume seen by the detector is z sUsing the known form for the polarized single scattering cross-section, we calculate the depolarized intensity arising from two successive scattering events by considering the events to be independent and integrating over the volumes available for each scattering event [2]. This procedure, which assumes that higher order scattering is negligible, is justified by the microscopic scattering theory of Oxtoby and Gelbart [4]. The starting point for our calculation is the cross section per unit volume,
portionality to h described by eq. (2). Eq. (3) can easily be integrated for P(q) = 1, which corresponds to fluids far from the critical point and to solutions of macromolecules whose size is small compared to the wavelength; we obtaing = ¼7r [2,3]. Thus a measurement ofldvH/I~rv yields directly o O, the angle-independent part of the absolute scattering cross section. The expressions for o 0 for a pure fluid [5], a binary mixture [6], and a dilute solution of macromolecules of molecular weight M [7] are, respectively, o~0 = (rr 2/X4)
(3e/~p)2TkTp2KT ,
(4)
o(q) = ooP(q)
o~ = (rr2/X 4)
OelOc)2rkr(aclau)e,r/p,
(5)
sin2~,
(1)
where the sin2¢ factor describes the angular dependence of Rayleigh scattering (9 is the angle between the incident polarization direction and the propagation vector of the scattered light), P(q) describes departures from the dipole scattering intensity distribution (due, e.g., in a fluid near the critical point, to the long-range correlations), and q = [k 0 - ks[, where k 0 and k s are, respectively, the wave vectors of the incident and scattered light. The quantity of interest is the depolarization ratio ]dR/]~,'V corresponding to the geometry in fig. 1. The depolarized scattering intensity calculation is simplified if (w, h, z s, and d) ~ R, which is not difficult to achieve in practice. The polarized intensity I~rv that we calculate corresponds to a geometry with z s = 0 and h >> d. IV~His also calculated for z s = 0, but IvdH is independent of z s if h ~ R and Zs ~ R. The depolarization ratio calculated for the conditions just described
o~0= (47r2/X4) n 2 (an~tic) 2 Mc/N A ,
(6)
where the symbols have the usual meanings [ 5 - 7 ] , and the expression for o0B is an approximation applicable when scattering arising from pressure and temperature fluctuations can be neglected. Now consider two examples of systems for which P(q) 4= 1 : fluids near the critical point and solutions of large rod-shaped macromolecules. For both pure fluids and binary mixtures near the critical point we have, assuming that the long-range correlations are described 1.0
is [21 t~n 0.1
IdvH/I~rv = g ooh ,
(2)
with
R
R
Zs+½h
g - - f f-R dx -fRdY fzs-½h dzP(q')P(q")x2z2/r 6 , (3) where f = [hP(q)] -1 ; q' and q" are, respectively, the scattering vectors for the first and second scatterings in the double scattering process; and q is the scattering vector for single scattering at 90 °. With the stated assumptions it can be shown that g is independent of the dimensions of the scattering geometry. Thus the only dependence ofldH/I~rv on the geometry is the pro112
0"010.5
1.0
'2.0
5
|0
I
20
50
U Fig. 2. The p r o p o r t i o n a l i t y f a c t o r g in the d e p o l a r i z a t i o n r a t i o e x p r e s s i o n IVH/Ivv d s -gooh, calculated numerically for fluids
near the critical point (o) (see eq. (4.3) in [3]), with u = qG and for a dilute solution of thin rods (A), with u = qL, where q corresponds to 90° scattering. The dashed line is the limit ofg as u --*0.
Volume 17, number 1
OPTICS COMMUNICATIONS
by the O r n s t e i n - Z e r n i k e form,
P(q) = (1 + q2~2)-1 ,
(7)
where ~ is the correlation length. F o r a dilute solution of macromolecules, each with a volume Vm , the angular dependence o f the scattering is described by [7]
P(q)= gm1 f exp(iq'r)dV
,
(8)
Vm
April 1976
both from double scattering and depolarized single scattering; when IVH is measured with Vs positioned below or above the laser beam (as pictured in fig. 1), the detector views only the multiply scattered light. Fig. 3 shows IVH/1VV measurements for (a) z s = 0 and h varied and (b) z s varied and h fixed. In b o t h cases the polarized intensity was measured with z s = 0 and h large (as in the calculations). The depolarization ratio measured with z s= 0 includes a contribution from depolarized single scattering, which is given by the h = 0
which becomes for a thin rod o f length L [7], 4.0 U
P(u)
=
(~u) - 1 f [(sin t)/t] d t - [(sin l u ) / l u ] 2 0
,
(9)
where u = qL. The depolarization integral [eq. (3)] has been evaluated by the Monte Carlo technique for the angle-dependent factors given by eqs. (7) and (9), and the result for g is plotted in fig. 2. The curve for fluids in the critical region is applicable except very near the critical point where scattering higher than secondorder contributes significantly; the contribution o f triple and higher order scattering becomes important for some fluids for ( T - Tc)/T c ~ 10 - 3 , but for binary mixtures whose refractive indices are nearly equal the present theory is valid in to ( T - Tc)/T c ~ 10 - 5 . The result for g for rod-shaped macromolecules in solution illustrates that depolarization ratio measurements can be used to determine the molecular weight o f macromolecules if size and shape.are approximately known (from, e.g., electron microscopy). If the molecules are sufficiently small, no information on the shape is required, and g is given by ¼ rr. To illustrate the use of the depolarization ratio technique in determining scattering cross sections we have performed measurements on xenon near the critical point in two ways: with z s = 0 and h varied, and with z s varied and h fixed. In our apparatus h is varied by varying the height o f a slit in front of the detector (see [2]), and z s is varied by translating the collection optics and detector vertically. We measured the polarized scattering intensity 1w with V s centered on the laser beam. Since single scattering can occur only in the illuminated volume, whereas double scattering can occur anywhere in the sample volume, I v v includes contributions from b o t h single and double scattering, but IVdV ~ I ~ r v . When the depolarized intensity is measured with z s = 0, IVH includes contributions arising
3.0 en I0
"~2.0
1.0
1
O0
I
O.S
1.0
h 6.0
1.5
(mm)
( b ) h = 0 . 2 mm
I0
4.0
3,0
I
-1.0
I
i
-0.5
i
0 Zs
0.5
I
1.0
Imml
Fig. 3. The depolarization ratio measured for xenon at T - Tc = 0.302 K, p = Pc: (a) with z s = 0 and h varied, and (b) with z s varied and h = 0.200 mm. The scattering geometry was that shown in fig. 1 and the wavelength of the incident light was 457.9 run-;g = 0.672 for this temperature and laser wavelength. The data have been corrected for the small dependence ofg on h and z s (see eq. (4.8) in [3]); this correction is only a few percent for most of the data and is at the most 17%. 113
Volume 17, number 1
OPTICS COMMUNICATIONS
intercept (see fig. 3a), and a contribution from double scattering, which is linearly proportional to h [see eq. (2)] ; o 0 is simply obtained from the slope of the straight line, which is g o O. The problem with this technique for determining o 0 is that for weakly scattering samples the variation of the depolarization ratio with h is small compared to the intercept. For weakly scattering samples it is preferable to measure the depolarized intensity with z s > h, so the directly illuminated volume is not seen by the detector; then the measured depolarized intensity includes only the contributions from double scattering. Measurements of the depolarization ratio as a function Of Zs, with h fixed, as shown in fig. 3b, include the depolarized single scattering as a peak centered at z s = 0, while the depolarization arising solely from double scattering, indicated by the horizontal dashed line in fig. 3b, is independent of z s. The data in fig. 3 yield o 0 = (3.87-+ 0.20) X 10 - 2 cm - 1 , while the value for cr0 calculated from eq. (4), using independent data tabulated in ref. [8], is 4.6 X 10 - 2 cm - 1 . In conclusion, it has been shown that measurements of the depolarization ratio o f the light scattered by fluids and solutions of macromolecules yield, with the proper choice of scattering geometry, the absolute
114
April 1976
scattering cross-section. Since ratios o f scattering intensities can almost always be measured more accurately and easily than absolute intensities, the depolarization ratio technique should be generally useful in determining scattering cross-sections. Future work could extend this technique to systems for which the single scattering is depolarized and to systems for which triple and higher order scattering is important. We thank A. Bray, R.F. Chang, D. Beysens and P. Calmettes for helpful discussions.
References [ 1 ] E.R. Pike, W.R.M. Pomeroy and J.M. Vaughan, J. Chem. Phys. 62 (1975) 3188. [2] L.A. Reith and H.L. Swinney, Phys. Rev. A12 (1975) 1094 [3] A. Bray and R.F. Chang, Phys. Rev. A12 (1975) 2594. [4] D.W. Oxtoby and W.M. Gelbart, J. Chem. Phys. 60 (1974) 3359. [5] M. Fixman, J. Chem. Phys. 23 (1955) 2074. [6] R.D. Mountain and J.M. Deutch, J. Chem. Phys. 50 (1969) 1103. [7] G. Oster, Chem. Reviews 43 (1946) 319. [8] H.L. Swinney and D.L. Henry, Phys. Rev. A8 (1973) 2586.