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Kerr effect caused by an electric current in a weakly ionized gas
Volume 15, number 2
OPTICS COMMUNICATIONS
October 1975
KERR E F F E C T CAUSED BY AN ELECTRIC CURRENT IN A WEAKLY IONIZED GAS Siegfried HESS
Institut J~r Theoretische Physik der Universitiit Erlangen-Niirnberg, 1)-8520 Erlangen, Fed. Rep. Germany Received 4 July 1975
The anisotropy of the velocity distribution caused by an electric current in a fluid leads to a molecular alignment via the non-spherical interaction. This alignment implies a birefringence proportional to the square of the applied electric field just as the ordinary Kerr effect. The kinetic theory of this phenomenon is presented for a lorentzian mixture, viz. a gas of few light charged linear molecules and of many heavy optically isotropic particles. The current-induced contribution to the Kerr effect turns out to be much larger than the contribution which stems from the usually considered orienting influence of the electric field on anisotropic molecules.
1. Introduction Application o f a constant electric field to a fluid leads to a difference between the refractive indices in directions parallel and perpendicular to the field. Measurements of this electric birefringence were first reported by Kerr [1,2] one hundred years ago. Two distinct physical mechanisms have been proposed for the explanation of the Kerr effect and corroborated in numerous experiments. These are firstly, as suggested by Voigt [ 3 , 4 ] , an optical anisotropy induced by the electric field in the atoms and molecules (hyperpolarizability) and secondly, as discussed by Langevin [5] and Born [6, 7], the orienting influence of the field on optically anisotropic molecules. It is the purpose of this note to point out that a third mechanism contributes to the Kerr effect if a fluid contains free electric carriers. The physical processes underlying this electric-current induced birefringence are akin to those which occur with other non-equilibrium alignment phenomena such as flow birefringence [2] and birefringence caused by a heat flux [8] or by a diffusion process [9, 10]. The electric-current induced contribution to the Kerr effect should exist in gases, liquids, colloidal and macromolecular solutions. Here, the kinetic theory o f this effect is presented for a gaseous mixture of charged linear molecules (e.g. I-I~2, He~) with mass rn
and number density n, and of optically isotropic particles (e.g. Xe, SF6) with mass m s and number density n s (the subscript "s" refers to scatterer). For simplicity, m "~ m s and n ,¢ n s is assumed. Though this lorentzian mixture is a very special case, it serves to indicate the basic physics underlying the current-induced birefringence and to obtain an estimate o f its order of magnitude. In this note, firstly the relation between the anisotropic part o f the dielectric tensor and the alignment tensor is stated. Then, transport-relaxation equations for the alignment and two other macroscopic variables are derived from a kinetic equation. Finally, the resulting Kerr effect is studied for a stationary situation and its order o f magnitude is estimated.
2. Kinetic equation, transport-relaxation equations A substance is birefringent if the anisotropic (symmetric traceless) part t of its dielectric tensor ~is nonzero. In molecular gases t :/: 0 is due to an alignment of the molecular rotational angular momentum rur. More specifically, for the mixture mentioned above, one has [ 11 ] ~;= e a a = -27rn (oq-ot±) "~l~ca.
(1)
The alignment tensor a is given by 139
Volume 15, number 2
OPTICS COMMUNICATIONS
related to the linearized Waldmann-Snider collision term co(..) by
a =,
~a=N/~c-I(j2-3)-I
JJ~N//~ J - 2 J J ,
(2)
with c = (j2(j2_~)-l)1/2 ~. 1. Here all, a L are the molecular polarizabilities for an electric field parallel and perpendicular to the molecular axis. The brackets ("')0 and (...) refer to averages which have to be evaluated with the equilibrium and the non-equilibrium distribution operators f0 and f of the molecules. The non-equilibrium distribution operator f obeys a kinetic equation, viz. the Waldmann-Snider equation [12]. It can be written as
Of+
-g-i
a
a
e
C . ~r f + m E . -~ f +
coff) 0 =
•
(3)
Here c is the molecular velocity, e is the charge of a molecule and E is the applied electric field. In (3), co(f) stands for the (linearized) Waldmann-Snider collision term; charge transfer processes are disregarded. In the case of the lorentzian mixture considered here, the distribution operator f s of the heavy optically isotropic particles is approximated by an equilibrium distribution. Transport-relaxation equations which allow to relate a and consequently s to the electric field tensor EE can be obtained from the kinetic equation (3) with the help of the moment method [13]. From a comparison of the present problem with the previously developed kinetic theory [11 ] and the first experimental results [14] of the flow birefringence in gases it can be concluded that it should be sufficient to characterize the non-equilibrium state of the gas by three macroscopic variables. In addition to a = (Ca>, these are u = and
e E + coy o = 0 m
a at
V~eE
(4)
" cop(Cp)+ copa'a = 0
~-~ - - v ~-
a__£a + C°ap(Cp) + COaa at
O.
(5)
(6)
The relaxation coefficients co.. occurring in ( 4 - 6 ) are 140
October 1975
COv = 1( V'CO(I/))0 ,
coik=+(Ci:co(Ck))O ,
with i = a, p, k = a, p and copp -- COp, coaa ------COa"They have the properties w v > 0, COp > 0, coa > 0, coacop > coap copa and co-a = coap" Notice that the non-diagonal relaxation coef~cient coap may either be positive or negative. This quantity is a measure for the collisioninduced coupling between the alignment and anisotropy of the velocity distribution described by (Cp). Both coa and coap vanish for a purely spherical interaction potential. The relaxation coefficients can be written as co = nsOthO.., where n s is recalled as the density of the heavy particles, Uth = (8kT/rrm) 1/2 is a thermal velocity and o.. is an effective cross section. In an ordinary mixture, the co.. depend on both n s and n.
3. C u r r e n t - i n d u c e d b i r e f r i n g e n c e
Next, eqs. ( 4 - 6 ) are discussed for a stationary situation where the time derivatives vanish. Then (4) implies v = (elm) rvE with the relaxation time r v = 6Ov 1 ; the electric conductivity is equal to ne2rv m - I . Inserting the expression for v into (5) one infers from (5,6) 8 = -- N//2 coap e 2
coa mkT r p r v E E
x/~ coap [ el ~ 2
-
2
coa ~kT]
EE,
(7)
with rp =cop 1 ( 1 -
C°apco__ ~ w a cPOap]t - 1
cop-1 .
The mean free path l is defined by
l2 = 7-p7-v 2kTm -1 . Thus a as given by (7) differs from the contribution to the alignment which stems from the orienting influence of the electric field on molecules with the dipole moment eR essentially by the factor coap/coa (l/R) 2" From the experimental data of the flow bireffingence [14] and of the Senftleben-Beenakker effect [14, 15] Icoapl/coa ~ 0.01 to 0.1 can be expected. Thus for R ~ 10 -8 cm and l ~ 10 -5 cm corresponding to a gas
Volume 15, number 2
OPTICS COMMUNICATIONS
at atmospheric pressure and at room temperature, the current-induced contribution to the alignment and consequently to the Kerr effect is about 4 orders of magnitude larger than the contribution which stems from the orienting influence of the electric field. This justifies the neglect of terms which describe the equilibrium alignment in (6) and (8). It should also be mentioned that (7) holds true only if terms of higher than the 2nd power in the electric field can be disregarded. This is the case as long as the magnitude of a (i.e. its largest eigenvalue), is much smaller than about 1. For l~10 -5cm,
E~3X103Vcm
-1,
T~ 300K,
Icoapl/¢oa ~ 10 - 2 ,
e.g., an alignment of the order 10 - 2 is found, and (7) is certainly applicable. Due to (1) and (7), the anisotropic part o f the dielectric tensor can be written as = 2K c E E ,
(8)
where the current induced contribution K c to the Kerr coefficient is K c = 27rn(otll_ot±)
c
C°ap e27"p'rv ~a
mkT
"
(9)
The difference 6v = u l r u ± between the indices of refraction v~l and v± for linearly polarized light with the electric field vectors parallel and perpendicular to E is determined by v 6 v = v(vll-u±) = K c E2 ,
(10)
where u is the refractive index for E = 0. A remark on the expected magnitude o f 6v is in order. For an alignment o f about 10 - 2 which can be achieved under the conditions mentioned above, 16u[ ~ 10 -14 is found for all--a ± ,,~ 10 -24 cm 3 and n -~ 1012 cm - 3 . An effect o f this size should be measurable [14]. Of course, 16ul increases if the number density n o f the charged molecules is raised.
4. Concluding remarks In this note, the special case of a lorentzian mixture has been considered in order to demonstrate the basic mechanism for the current-induced birefringence:
October 1975
the electric current gives rise to an anisotropy of the velocity distribution which, via the non-spherical interaction between the particles, leads to an alignment of the molecules. The extension of the present theory to general mixtures, to ionized atomic vapors, as well as the investigation o f the influence of a magnetic field on the birefringence should be straightforward with the help of the methods presented in refs. [ l 0, 16, 17]. For liquids and macromolecular solutions, a theoretical treatment along the lines developed in ref. [9] seems to be feasible. Finally, it seems worth mentioning that a particle flux caused by gravity or by a centrifugal force will, in principle, also induce a birefringence in a fluid.
References [1] J. Kerr, Phil. Mag. (4) 50 (1875) 446; J. Kerr, Phil. Mag. (5) 8 (1879) 85,229. [2] A. Peterlin and H.A. Stuart, Hand- und Jahrbuch der Chem. Phys. 8/I B (Leipzig 1943); G. Szivessy, Handbuch der Physik, eds. H. Geiger and K. Scheel, Vol. 21 (Berlin 1929) 724. [3] W. Voigt, Ann. Phys. (4) 4 (1901) 197. [4] A.D. Buckingham and J.A. Pople, Proc. Phys. Soc. A 68 (1955) 905; A.D. Buckingham and D.A. Dunmur, Trans. Faraday Soc. 64 (1968) 1776. [5] P. Langevin, J. Phys. Radium 7 (1910) 249. [6] M. Born, Ann. Phys. 55 (1918) 177; R. Gans, Ann. Phys. 64 (1921) 481. [7] P. Mazur and B.J. Postma, Physica 25 (1959) 251. [8] S. Hess, Z. Naturforsch. 28a (1973) 1531. [9] S. Hess, Phys. Lett. 45A (1973) 77; S. Hess, Physica 74 (1974) 277. [10] W.E. K6hler and J. Halbritter, Physica 74 (1974) 294; W.E. K/Shler and J. Halbritter, Physica 76 (1974) 224. [11] S. Hess, Phys. Lett. 30A (1969) 239; S. Hess, Springer Tracts in Mod. Phys. 54 (1970) 136; S. Hess, in: The Boltzmann Equation, Theory and Applications, eds. E.G.D. Cohen and W. Thitring, (Springer, Wien 1973); S. Hess, Acta Physica Austriaca Suppl. X (1973) 247; S. Hess, Z. Naturforsch. 29a (1974) 1121. [12] L. Waldmann, Z. Naturforsch. 12a (1957) 660; L. Waldmann, Z, Naturforsch. 13a (1958) 606; R.F. Snider, J. Chem. Phys. 32 (1960) 1051. [13] S. Hess and L. Waldmann, Z. Naturforsch. 21a (1966) 1529; H. Raum and W.E. K6hler, Z. Natttrforsch. 25a (1970) 1178. [14] F. Baas, Phys. Letters 36A (1971) 107; see also J.J.M. Beenakker, in: The Boltzmann Equation, 141
Volume 15, number 2
OPTICS COMMUNICATIONS
Theory and Applications, eds. E.G.D. Cohen and W. Thirring, Springer, Vienna 1973; Acta Physica Austriaca Suppl. X (1973) 267. [15] J.J.M. Beenakker and F.R. McCourt, Ann. Rev. Phys. Chem. 21 (1970) 136.
142
October 1975
[16] W.E. K6hler and S. Hess, Z. Naturforsch. 28a (1973) 1543. [17] A.G. St. Pierre, W.E. K6hler and S. Hess, Z. Naturforsch. 27a (1972) 721.
OPTICS COMMUNICATIONS
October 1975
KERR E F F E C T CAUSED BY AN ELECTRIC CURRENT IN A WEAKLY IONIZED GAS Siegfried HESS
Institut J~r Theoretische Physik der Universitiit Erlangen-Niirnberg, 1)-8520 Erlangen, Fed. Rep. Germany Received 4 July 1975
The anisotropy of the velocity distribution caused by an electric current in a fluid leads to a molecular alignment via the non-spherical interaction. This alignment implies a birefringence proportional to the square of the applied electric field just as the ordinary Kerr effect. The kinetic theory of this phenomenon is presented for a lorentzian mixture, viz. a gas of few light charged linear molecules and of many heavy optically isotropic particles. The current-induced contribution to the Kerr effect turns out to be much larger than the contribution which stems from the usually considered orienting influence of the electric field on anisotropic molecules.
1. Introduction Application o f a constant electric field to a fluid leads to a difference between the refractive indices in directions parallel and perpendicular to the field. Measurements of this electric birefringence were first reported by Kerr [1,2] one hundred years ago. Two distinct physical mechanisms have been proposed for the explanation of the Kerr effect and corroborated in numerous experiments. These are firstly, as suggested by Voigt [ 3 , 4 ] , an optical anisotropy induced by the electric field in the atoms and molecules (hyperpolarizability) and secondly, as discussed by Langevin [5] and Born [6, 7], the orienting influence of the field on optically anisotropic molecules. It is the purpose of this note to point out that a third mechanism contributes to the Kerr effect if a fluid contains free electric carriers. The physical processes underlying this electric-current induced birefringence are akin to those which occur with other non-equilibrium alignment phenomena such as flow birefringence [2] and birefringence caused by a heat flux [8] or by a diffusion process [9, 10]. The electric-current induced contribution to the Kerr effect should exist in gases, liquids, colloidal and macromolecular solutions. Here, the kinetic theory o f this effect is presented for a gaseous mixture of charged linear molecules (e.g. I-I~2, He~) with mass rn
and number density n, and of optically isotropic particles (e.g. Xe, SF6) with mass m s and number density n s (the subscript "s" refers to scatterer). For simplicity, m "~ m s and n ,¢ n s is assumed. Though this lorentzian mixture is a very special case, it serves to indicate the basic physics underlying the current-induced birefringence and to obtain an estimate o f its order of magnitude. In this note, firstly the relation between the anisotropic part o f the dielectric tensor and the alignment tensor is stated. Then, transport-relaxation equations for the alignment and two other macroscopic variables are derived from a kinetic equation. Finally, the resulting Kerr effect is studied for a stationary situation and its order o f magnitude is estimated.
2. Kinetic equation, transport-relaxation equations A substance is birefringent if the anisotropic (symmetric traceless) part t of its dielectric tensor ~is nonzero. In molecular gases t :/: 0 is due to an alignment of the molecular rotational angular momentum rur. More specifically, for the mixture mentioned above, one has [ 11 ] ~;= e a a = -27rn (oq-ot±) "~l~ca.
(1)
The alignment tensor a is given by 139
Volume 15, number 2
OPTICS COMMUNICATIONS
related to the linearized Waldmann-Snider collision term co(..) by
a =
~a=N/~c-I(j2-3)-I
JJ~N//~ J - 2 J J ,
(2)
with c = (j2(j2_~)-l)1/2 ~. 1. Here all, a L are the molecular polarizabilities for an electric field parallel and perpendicular to the molecular axis. The brackets ("')0 and (...) refer to averages which have to be evaluated with the equilibrium and the non-equilibrium distribution operators f0 and f of the molecules. The non-equilibrium distribution operator f obeys a kinetic equation, viz. the Waldmann-Snider equation [12]. It can be written as
Of+
-g-i
a
a
e
C . ~r f + m E . -~ f +
coff) 0 =
•
(3)
Here c is the molecular velocity, e is the charge of a molecule and E is the applied electric field. In (3), co(f) stands for the (linearized) Waldmann-Snider collision term; charge transfer processes are disregarded. In the case of the lorentzian mixture considered here, the distribution operator f s of the heavy optically isotropic particles is approximated by an equilibrium distribution. Transport-relaxation equations which allow to relate a and consequently s to the electric field tensor EE can be obtained from the kinetic equation (3) with the help of the moment method [13]. From a comparison of the present problem with the previously developed kinetic theory [11 ] and the first experimental results [14] of the flow birefringence in gases it can be concluded that it should be sufficient to characterize the non-equilibrium state of the gas by three macroscopic variables. In addition to a = (Ca>, these are u =
e E + coy o = 0 m
a
V~eE
(4)
" cop(Cp)+ copa'a = 0
~-~ - - v ~-
a__£a + C°ap(Cp) + COaa at
O.
(5)
(6)
The relaxation coefficients co.. occurring in ( 4 - 6 ) are 140
October 1975
COv = 1( V'CO(I/))0 ,
coik=+(Ci:co(Ck))O ,
with i = a, p, k = a, p and copp -- COp, coaa ------COa"They have the properties w v > 0, COp > 0, coa > 0, coacop > coap copa and co-a = coap" Notice that the non-diagonal relaxation coef~cient coap may either be positive or negative. This quantity is a measure for the collisioninduced coupling between the alignment and anisotropy of the velocity distribution described by (Cp). Both coa and coap vanish for a purely spherical interaction potential. The relaxation coefficients can be written as co = nsOthO.., where n s is recalled as the density of the heavy particles, Uth = (8kT/rrm) 1/2 is a thermal velocity and o.. is an effective cross section. In an ordinary mixture, the co.. depend on both n s and n.
3. C u r r e n t - i n d u c e d b i r e f r i n g e n c e
Next, eqs. ( 4 - 6 ) are discussed for a stationary situation where the time derivatives vanish. Then (4) implies v = (elm) rvE with the relaxation time r v = 6Ov 1 ; the electric conductivity is equal to ne2rv m - I . Inserting the expression for v into (5) one infers from (5,6) 8 = -- N//2 coap e 2
coa mkT r p r v E E
x/~ coap [ el ~ 2
-
2
coa ~kT]
EE,
(7)
with rp =cop 1 ( 1 -
C°apco__ ~ w a cPOap]t - 1
cop-1 .
The mean free path l is defined by
l2 = 7-p7-v 2kTm -1 . Thus a as given by (7) differs from the contribution to the alignment which stems from the orienting influence of the electric field on molecules with the dipole moment eR essentially by the factor coap/coa (l/R) 2" From the experimental data of the flow bireffingence [14] and of the Senftleben-Beenakker effect [14, 15] Icoapl/coa ~ 0.01 to 0.1 can be expected. Thus for R ~ 10 -8 cm and l ~ 10 -5 cm corresponding to a gas
Volume 15, number 2
OPTICS COMMUNICATIONS
at atmospheric pressure and at room temperature, the current-induced contribution to the alignment and consequently to the Kerr effect is about 4 orders of magnitude larger than the contribution which stems from the orienting influence of the electric field. This justifies the neglect of terms which describe the equilibrium alignment in (6) and (8). It should also be mentioned that (7) holds true only if terms of higher than the 2nd power in the electric field can be disregarded. This is the case as long as the magnitude of a (i.e. its largest eigenvalue), is much smaller than about 1. For l~10 -5cm,
E~3X103Vcm
-1,
T~ 300K,
Icoapl/¢oa ~ 10 - 2 ,
e.g., an alignment of the order 10 - 2 is found, and (7) is certainly applicable. Due to (1) and (7), the anisotropic part o f the dielectric tensor can be written as = 2K c E E ,
(8)
where the current induced contribution K c to the Kerr coefficient is K c = 27rn(otll_ot±)
c
C°ap e27"p'rv ~a
mkT
"
(9)
The difference 6v = u l r u ± between the indices of refraction v~l and v± for linearly polarized light with the electric field vectors parallel and perpendicular to E is determined by v 6 v = v(vll-u±) = K c E2 ,
(10)
where u is the refractive index for E = 0. A remark on the expected magnitude o f 6v is in order. For an alignment o f about 10 - 2 which can be achieved under the conditions mentioned above, 16u[ ~ 10 -14 is found for all--a ± ,,~ 10 -24 cm 3 and n -~ 1012 cm - 3 . An effect o f this size should be measurable [14]. Of course, 16ul increases if the number density n o f the charged molecules is raised.
4. Concluding remarks In this note, the special case of a lorentzian mixture has been considered in order to demonstrate the basic mechanism for the current-induced birefringence:
October 1975
the electric current gives rise to an anisotropy of the velocity distribution which, via the non-spherical interaction between the particles, leads to an alignment of the molecules. The extension of the present theory to general mixtures, to ionized atomic vapors, as well as the investigation o f the influence of a magnetic field on the birefringence should be straightforward with the help of the methods presented in refs. [ l 0, 16, 17]. For liquids and macromolecular solutions, a theoretical treatment along the lines developed in ref. [9] seems to be feasible. Finally, it seems worth mentioning that a particle flux caused by gravity or by a centrifugal force will, in principle, also induce a birefringence in a fluid.
References [1] J. Kerr, Phil. Mag. (4) 50 (1875) 446; J. Kerr, Phil. Mag. (5) 8 (1879) 85,229. [2] A. Peterlin and H.A. Stuart, Hand- und Jahrbuch der Chem. Phys. 8/I B (Leipzig 1943); G. Szivessy, Handbuch der Physik, eds. H. Geiger and K. Scheel, Vol. 21 (Berlin 1929) 724. [3] W. Voigt, Ann. Phys. (4) 4 (1901) 197. [4] A.D. Buckingham and J.A. Pople, Proc. Phys. Soc. A 68 (1955) 905; A.D. Buckingham and D.A. Dunmur, Trans. Faraday Soc. 64 (1968) 1776. [5] P. Langevin, J. Phys. Radium 7 (1910) 249. [6] M. Born, Ann. Phys. 55 (1918) 177; R. Gans, Ann. Phys. 64 (1921) 481. [7] P. Mazur and B.J. Postma, Physica 25 (1959) 251. [8] S. Hess, Z. Naturforsch. 28a (1973) 1531. [9] S. Hess, Phys. Lett. 45A (1973) 77; S. Hess, Physica 74 (1974) 277. [10] W.E. K6hler and J. Halbritter, Physica 74 (1974) 294; W.E. K/Shler and J. Halbritter, Physica 76 (1974) 224. [11] S. Hess, Phys. Lett. 30A (1969) 239; S. Hess, Springer Tracts in Mod. Phys. 54 (1970) 136; S. Hess, in: The Boltzmann Equation, Theory and Applications, eds. E.G.D. Cohen and W. Thitring, (Springer, Wien 1973); S. Hess, Acta Physica Austriaca Suppl. X (1973) 247; S. Hess, Z. Naturforsch. 29a (1974) 1121. [12] L. Waldmann, Z. Naturforsch. 12a (1957) 660; L. Waldmann, Z, Naturforsch. 13a (1958) 606; R.F. Snider, J. Chem. Phys. 32 (1960) 1051. [13] S. Hess and L. Waldmann, Z. Naturforsch. 21a (1966) 1529; H. Raum and W.E. K6hler, Z. Natttrforsch. 25a (1970) 1178. [14] F. Baas, Phys. Letters 36A (1971) 107; see also J.J.M. Beenakker, in: The Boltzmann Equation, 141
Volume 15, number 2
OPTICS COMMUNICATIONS
Theory and Applications, eds. E.G.D. Cohen and W. Thirring, Springer, Vienna 1973; Acta Physica Austriaca Suppl. X (1973) 267. [15] J.J.M. Beenakker and F.R. McCourt, Ann. Rev. Phys. Chem. 21 (1970) 136.
142
October 1975
[16] W.E. K6hler and S. Hess, Z. Naturforsch. 28a (1973) 1543. [17] A.G. St. Pierre, W.E. K6hler and S. Hess, Z. Naturforsch. 27a (1972) 721.