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Polarisation of a dipolar gas by drift
Volume 98, number 4
POLARISATION
CHEMICAL
PHYSICS
1 JuIy 1983
LETTERS
OF A DIPOLAR GAS BY DRIFT
FKh. GEL’MUKHANOV and L.V. IL’ICHOV Institute ofAutomation and Electrometrics. Siberian Branch of the Academy of Science of the USSR. Novosibirsk. USSR Received 31 December 1982; in final form 29 April 1983
Linear molecules without a center of symmetry
are shown to be oriented during their drift with respect to a buffer gas.
1. Introduction
Drift or diffusional motion of one component of a gas or liquid mixture relative to another is possible. For example, it can be a drift of charged particles in a gas discharge, an ordinary diffusion, or a light-induced drift [l]_ In the present paper molecules without a center of symmetry are shown to be oriented during their drift with respect to a buffer gas. The physics of this phenomenon is ultimately simple, and it can be likened to the orientation of a weathercock by the wind. In our case the polar molecules play the role of this weathercock and the buffer gas is the “wind”_ The polar molecules have a dipole moment and an electric field appears with their orientation_
2. Vector of space orientation Space orientation of linear molecules is characterized by a dependence of the density matrix p(n) on a single vector n, which is parallel to the molecular axis. For simplicity we speak only about molecules in X states without spin. Information about the orientation of the molecules is contained in a multipole expansion of ~(12):
where Y,,(n) is a spherical harmonic_ For example, the average value of the vector n is determined by plq _Quantities pKq are simply expressed through the density matrix in the M representation p(JMsIM’). The connection between pKq and p(J&fJ’M’) is established by direct comparison of (1) with the expansion of p(n) with the eigenfunctions of the rotation of a molecule YJM(n) exp(-iEJ r/h): )j’ [(u + I)(=’
pK,(JJ’)
=MG, (-)J’-M’(JMJ’
+ 1)/45r(2~ + 1)] ‘~2(~~‘oIKo>pK4 (Js> ,
-M’(f@p(JMJ’M’)
exp[i(Er
- EJ)tlfi]
_
(2)
The quantity p,,(JJ’) is known as a density matrix in the representation of irreducible tensor operators or in the Kq representation [2] _ In the present paper we are interested only in such quantities as an average value of the vector n and the con--
0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
349
stant electric field E which appears due to the orientation “0
=
P(J)
1 July 1983
CHEMICAL PHYSICS LETTERS
Volume 98. number 4
s
rip(n)) dn = -2
=exp(iwJt)
Re F&J+ 1
Ceqplq(JJ+l),
of the polar molecules
with dipole moment
p:
.
~)]l/zp(~)]
fiw~=E~+l
-
EJ ,
4
E = (dq&,
.
(3)
herical component of a unit vector. m-r,e,istheqthsp here eu = S.S5 X IO- 12CV-r Formula (3) is obtained by means of eq. (1) and the hermitian property of the density matrix. It is evident from (3) that no is determined by the elements of the density matrix which are-J-nondiagonalIn the physics of magnetic properties of atoms and molecules another vector plq(JJ) is known. It is the vector of magnetic orientation [2] _It is proportional to the magnetic moment of the gas and is a pseudovector. To distinguish the true vector pi (J J+l) and the pseudovector plq(JJ) we call p(J) the vector of space orientation. Thus as one could expect, t41e space orientation is connected with the rotational transition J -+ J + 1, or more exactly with the coherence between the rotational levels. plq(JJ+l) f 0. The magnetic orientation is connected with transitions between magnetic levels M -+hl4 1.
3. Solution of the kinetic equation The initial equation deterntiing
plq(J J+I) = (pIq(J J-f-1 u)> is the Boltzmann equation in the
Kq
representa-
tion [Z]:
(alar + uV)P,,(JJ'U)=S~~(JJ'U) exp[i(EJ-
_!$)t/fz]
,
(4)
where the integration over velocity is designated by the angular brackets (2, u is the molecular velocity. We solve eq. (-I) by perturbation theory. where the perturbation is a collisional integral. We use the equilibrium distribution over M-levels for the zero approximation. Buffer gas is designated by the indes “b” and has no internal structure for simphcity_ Supposing the drift velocity u to be less than the thermal one;, we have the expression for the zero appro_xrmation of the density matrix of polar molecules: p:oq)(JJ’u) = 6JJ,6A06qop(Ju)/(2J+
1)‘i2,
PV$
= riJ0
•t
w~%w~1
WI
(5)
and .rn~ogous expression for the density matrix of buffer gas &,(ub). in (5) p(J) is a density andj(J) is a flow of molecules with the angular momentum J, W(J) is the maxwellian distribution. If the molecule has no time to pass the distance I of space inhomogeneity of the system during the rotational period “w;l (u/hJ 4 I), we can neglect uD in (4). Let us isolate the pdrt containing pKq that is responsible for relaxation of this multipole in the collisional inof the molecules and buffer gas to the rest of the cohitegral S, 4. Then let us put the functions of distributions s1ma1 integral and integrate (4) over velocity. As a result we obtain the equation for the vector of space orientat1lm:
[alar + ~~(4 + ioJ]p(J)
=s(J),
s(J) = C e cs(O)(JJ+l 4
q
lq
u))
_
(6)
The relation constant for the vector of space orientation voQ (Re vo(J) > 0) is of the order of the frequency of collisions which cause transitions between the rotational levels. We do not give an explicit expression for vo(J), because usuahy lvo(J)l Q uJ and only in liquids [v~(J)I > wJ_ Taking into account (5) we obtain the next form of the collisional integral:
350
Volume
98, number
u~~(JJ’J~u)
CHEMICAL
4
=*$,
(-)r-M*
PHYSICS
LETTERS
1 July 1983
- M’ilq)
1
c (-)r-M’(JMf =MM,M
qq(JJ’4J1J&
-M’IIq)f(JM~IJIMlul)f’(J’M’uVIMlul),
(7)
1
where 6,(x) = :6(x) i i/27rx, 1-1is the reduced mass of colliding particles, f(JMuV,M,u,) is the scattering amplitude_ When fio~/~iT~ < 1 the difference between Ey and EJ can be neglected. One integrates over the velocity of the center of mass of colliding particles using (5), takes Er = EJ and under the assumption that the molecules and buffer particles have the same temperature:
+
Jdii,(u, V)ul(JJ’uIJIJlul)
1,
(8)
whereur = [u2 - ( 2 /g)(EJ, - EJ)] l/2, Wg(u) is the maxwellian distribution of the particles with mass I.C,P= CJ@), U1 = uI/uI. We do not designate the lower limit of the integration over U, assuming it to be equal to 0 II2 when E -t > Ep The following equation defmes the “relative velocwhen EJl =GE~md to [(2/P)(Ejl -EJ)] ity” of the drift Y and the “function of distribution”&er J, W(J): yw(Jl)ppb
= {
[i(Jl)pb -&#(JI)~/(~I
i- l)} expKEJ, - E=)/kTl .
If the molecules in different J levels have the same drift velocity r+, one obtains b(J) = u&I)]: v=u,-,
-“b
,
P(JI)IP
WJl) =m
expKE~l
- E~)H”l
,
where +, =jb/pb is the drift velocity of the buffer particles_ One integrates over the direction of u in (8)_ For this one expresses the cross sections ulq which are defined in the system connected to the collisional plane uul using their tensor properties: Qlq
-- Cd?) 4’
44
z,,.
,
(9)
where Dzi(a, P, y) is Wigner’s rotation matrix; (Y,p, 7 are Euler’s angles which connect the laboratory coordinate system to a new one, the z axis is parallel to u and they axis is in the plane of uu1.
351
Using ($$ the optical theorem and Tinvariance, one can obtain for’S(J)(J’ S(J)
= +% V‘i;) P,
I July 1983
CHEBfICAL PHYSICS LE-iTERS
Volume 98,number4
= J f 1, dSL = 2n sin 0 de)
I
YJ)3 g (-1 J’-nf (JMJ’ --Ml 10) T(JMu 1J’Mu)
+ (2rrWiCt) I[W(J’)-
1 -
GO)
The collision frequency p(J) depends on the components of the vectora(JJir &FIJ1ul) = Zqeqolq(JJ’uVIJIul)_ It is evident from (IO) that the collisional frequency y(J) is complex in the general case: fm v(J) J: 0, Re Y(J) JI 0. The physical meaning of cr is as follows: molecules with chaotic orientation are oriented in the directiona
which depends on the relative velocities of particles before and after the collision, u and ul. It is evident that the vector of orientation is zero for molecules with a center of symmetry, because collisional transitions with a change ofJ by an odd integer are forbidden.
4. Space orientation caused by light-induced drift Let us consider space orientation caused by lent-indu~d drift [I]. The vector of space o~ent~tion of polar molecules is the sum of those of excited (m) and unexcited (n) molecules. For simplicity we consider inefastic collisions J + J1 with a large change of velocity [3]. In this case the flow of molecules exists in the resonance levels only: im(J) = ~~~e~a,(Jo) 7
in(J)
= &‘fJ;1incJb) *
(1 I)
(a) Tizeflmv regirrze. In the cell with open ends the regime of steady-state flow is realised. One can obtain the ratio between the flows in both levels:
&(Jb> = -(J&Ir&)im(io)
3
(12)
where v1 is the frequency of cohisions with the buffer particles which break the flOWii_ For the vector of space orientation one can obtain with regard for (11) and (12):
r),,Q = ~,vw’,(J~)
I
17,(J)= ~,Ol~V,(J($,
(13)
where the frequencies Y,(J) and vn(J) are defied by (10). fb) Z?zeflo~&~s regke_ In a closed cell the total fiow is equal to zero. So, the vector of space orientation is defined by the following expression: PQ
= ]slrrr(JYW~ + 1) - ~ln(J)Wb
+
l)li,(Jdli~,@
It is to be noted that the orientation vectorp(J)
-
in (13) and (14) is not entirely imaginary, be~~e~~(~
(14)
and
q,(J) are complex. me statistical weights in (13) and (14) make it possible to obtain space orientation even in the case ofequal frequencies and cross sections of the orientational collisions using the transitions of P- and Q-branches when JO 352
CHEMICALPHYSICSLETTERS
Volume98, number4
1 July 1983
and Jb are not too large. One can naturally suppose om and un to be approximately proportional to vrn and vn respectively_ In the case of an open cell the differences (;Tmvn - Znr+n) appear in (13) (without statistical weights), and in (14) <;;;, - zn)_ So we conclude that p(J) is greater in the case of the flowless regime than in the case of free flow, when all other parameters are the same.
5. Conclusion Since the scattering amplitudes f(JMW1 M’) are non-zero only for polar molecules (without a center of symmetry) there is a reason to suppose f to be proportional to the dipole moment p (it is in a high-energy limit)_ So v(J) can be roughly estimated as u(J) = Y@/D)~ where Y is an ordinary gas-dynamic collisional frequency. We give an estimate for the relative number of oriented molecules @I/p:
Whenp=lD,v=107 s-‘,‘dJ% 101os--‘, v/u= lo-’ we obtain lpi/p = 10m4. An electric field E = lo- 1 V/cm corresponds to these oriented molecules when p = 1017 cmm3 _ Let us consider the situation when relative motions of the components is caused by ordinary diffusion. The diffusional velocity of the polar molecule is V = (U”/v) VP/P, where Vp is a density gradient of the dipolar gas. The relative number of oriented molecules when VP/~ = 10 cm-l and U * 10s cm s- 1 is @l/p =:v(r?
i- ,$-+7plp
= 10-4
_
Contrary to orientation caused by light radiation, there is no limitation to densities of the components in this case and liquid can be used. An electric field in liquid can be stronger (p = 1020 cme3): E = lo2 V/cm. Consideration of the self-influence of the polarisation gives the coefficient (2 + E)/~E in the expression for E and does not change the order of it. Here E is the dielectric permeability. This field exists only during the diffusion and is sufficiently strong only in the region of large density gradients_
References [l] F.Kh. Gel’mukhanov and A.M. Shalagin, Pis’maZh. Eksp. Tear- Fis. 29 (1979) 773. [JETP Letters 29, No. 12 (1979)];
Zh Eksp. Teor. Fiz. 78 (1980) 1674 [Soviet Phys. JETP 51 (1980) 839]_ [ 21 S.G. Rautian. G-1. Smimov and A-hi. Shalagin, Nelineinye resonansy v spektrakh atomov i molekul [Nonlinear resonances in atomic and molecular spectra ] (Nauka, Novosibirsk, 1979). [3] V-R. htionenko and A.M. Shalagin. Jzv. Akad. Nauk SSSR Ser. Phys. 45 (1981) 995
353
POLARISATION
CHEMICAL
PHYSICS
1 JuIy 1983
LETTERS
OF A DIPOLAR GAS BY DRIFT
FKh. GEL’MUKHANOV and L.V. IL’ICHOV Institute ofAutomation and Electrometrics. Siberian Branch of the Academy of Science of the USSR. Novosibirsk. USSR Received 31 December 1982; in final form 29 April 1983
Linear molecules without a center of symmetry
are shown to be oriented during their drift with respect to a buffer gas.
1. Introduction
Drift or diffusional motion of one component of a gas or liquid mixture relative to another is possible. For example, it can be a drift of charged particles in a gas discharge, an ordinary diffusion, or a light-induced drift [l]_ In the present paper molecules without a center of symmetry are shown to be oriented during their drift with respect to a buffer gas. The physics of this phenomenon is ultimately simple, and it can be likened to the orientation of a weathercock by the wind. In our case the polar molecules play the role of this weathercock and the buffer gas is the “wind”_ The polar molecules have a dipole moment and an electric field appears with their orientation_
2. Vector of space orientation Space orientation of linear molecules is characterized by a dependence of the density matrix p(n) on a single vector n, which is parallel to the molecular axis. For simplicity we speak only about molecules in X states without spin. Information about the orientation of the molecules is contained in a multipole expansion of ~(12):
where Y,,(n) is a spherical harmonic_ For example, the average value of the vector n is determined by plq _Quantities pKq are simply expressed through the density matrix in the M representation p(JMsIM’). The connection between pKq and p(J&fJ’M’) is established by direct comparison of (1) with the expansion of p(n) with the eigenfunctions of the rotation of a molecule YJM(n) exp(-iEJ r/h): )j’ [(u + I)(=’
pK,(JJ’)
=MG, (-)J’-M’(JMJ’
+ 1)/45r(2~ + 1)] ‘~2(~~‘oIKo>pK4 (Js> ,
-M’(f@p(JMJ’M’)
exp[i(Er
- EJ)tlfi]
_
(2)
The quantity p,,(JJ’) is known as a density matrix in the representation of irreducible tensor operators or in the Kq representation [2] _ In the present paper we are interested only in such quantities as an average value of the vector n and the con--
0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
349
stant electric field E which appears due to the orientation “0
=
P(J)
1 July 1983
CHEMICAL PHYSICS LETTERS
Volume 98. number 4
s
rip(n)) dn = -2
=exp(iwJt)
Re F&J+ 1
Ceqplq(JJ+l),
of the polar molecules
with dipole moment
p:
.
~)]l/zp(~)]
fiw~=E~+l
-
EJ ,
4
E = (dq&,
.
(3)
herical component of a unit vector. m-r,e,istheqthsp here eu = S.S5 X IO- 12CV-r Formula (3) is obtained by means of eq. (1) and the hermitian property of the density matrix. It is evident from (3) that no is determined by the elements of the density matrix which are-J-nondiagonalIn the physics of magnetic properties of atoms and molecules another vector plq(JJ) is known. It is the vector of magnetic orientation [2] _It is proportional to the magnetic moment of the gas and is a pseudovector. To distinguish the true vector pi (J J+l) and the pseudovector plq(JJ) we call p(J) the vector of space orientation. Thus as one could expect, t41e space orientation is connected with the rotational transition J -+ J + 1, or more exactly with the coherence between the rotational levels. plq(JJ+l) f 0. The magnetic orientation is connected with transitions between magnetic levels M -+hl4 1.
3. Solution of the kinetic equation The initial equation deterntiing
plq(J J+I) = (pIq(J J-f-1 u)> is the Boltzmann equation in the
Kq
representa-
tion [Z]:
(alar + uV)P,,(JJ'U)=S~~(JJ'U) exp[i(EJ-
_!$)t/fz]
,
(4)
where the integration over velocity is designated by the angular brackets (2, u is the molecular velocity. We solve eq. (-I) by perturbation theory. where the perturbation is a collisional integral. We use the equilibrium distribution over M-levels for the zero approximation. Buffer gas is designated by the indes “b” and has no internal structure for simphcity_ Supposing the drift velocity u to be less than the thermal one;, we have the expression for the zero appro_xrmation of the density matrix of polar molecules: p:oq)(JJ’u) = 6JJ,6A06qop(Ju)/(2J+
1)‘i2,
PV$
= riJ0
•t
w~%w~1
WI
(5)
and .rn~ogous expression for the density matrix of buffer gas &,(ub). in (5) p(J) is a density andj(J) is a flow of molecules with the angular momentum J, W(J) is the maxwellian distribution. If the molecule has no time to pass the distance I of space inhomogeneity of the system during the rotational period “w;l (u/hJ 4 I), we can neglect uD in (4). Let us isolate the pdrt containing pKq that is responsible for relaxation of this multipole in the collisional inof the molecules and buffer gas to the rest of the cohitegral S, 4. Then let us put the functions of distributions s1ma1 integral and integrate (4) over velocity. As a result we obtain the equation for the vector of space orientat1lm:
[alar + ~~(4 + ioJ]p(J)
=s(J),
s(J) = C e cs(O)(JJ+l 4
q
lq
u))
_
(6)
The relation constant for the vector of space orientation voQ (Re vo(J) > 0) is of the order of the frequency of collisions which cause transitions between the rotational levels. We do not give an explicit expression for vo(J), because usuahy lvo(J)l Q uJ and only in liquids [v~(J)I > wJ_ Taking into account (5) we obtain the next form of the collisional integral:
350
Volume
98, number
u~~(JJ’J~u)
CHEMICAL
4
=*$,
(-)r-M*
PHYSICS
LETTERS
1 July 1983
- M’ilq)
1
c (-)r-M’(JMf =MM,M
qq(JJ’4J1J&
-M’IIq)f(JM~IJIMlul)f’(J’M’uVIMlul),
(7)
1
where 6,(x) = :6(x) i i/27rx, 1-1is the reduced mass of colliding particles, f(JMuV,M,u,) is the scattering amplitude_ When fio~/~iT~ < 1 the difference between Ey and EJ can be neglected. One integrates over the velocity of the center of mass of colliding particles using (5), takes Er = EJ and under the assumption that the molecules and buffer particles have the same temperature:
+
Jdii,(u, V)ul(JJ’uIJIJlul)
1,
(8)
whereur = [u2 - ( 2 /g)(EJ, - EJ)] l/2, Wg(u) is the maxwellian distribution of the particles with mass I.C,P= CJ@), U1 = uI/uI. We do not designate the lower limit of the integration over U, assuming it to be equal to 0 II2 when E -t > Ep The following equation defmes the “relative velocwhen EJl =GE~md to [(2/P)(Ejl -EJ)] ity” of the drift Y and the “function of distribution”&er J, W(J): yw(Jl)ppb
= {
[i(Jl)pb -&#(JI)~/(~I
i- l)} expKEJ, - E=)/kTl .
If the molecules in different J levels have the same drift velocity r+, one obtains b(J) = u&I)]: v=u,-,
-“b
,
P(JI)IP
WJl) =m
expKE~l
- E~)H”l
,
where +, =jb/pb is the drift velocity of the buffer particles_ One integrates over the direction of u in (8)_ For this one expresses the cross sections ulq which are defined in the system connected to the collisional plane uul using their tensor properties: Qlq
-- Cd?) 4’
44
z,,.
,
(9)
where Dzi(a, P, y) is Wigner’s rotation matrix; (Y,p, 7 are Euler’s angles which connect the laboratory coordinate system to a new one, the z axis is parallel to u and they axis is in the plane of uu1.
351
Using ($$ the optical theorem and Tinvariance, one can obtain for’S(J)(J’ S(J)
= +% V‘i;) P,
I July 1983
CHEBfICAL PHYSICS LE-iTERS
Volume 98,number4
= J f 1, dSL = 2n sin 0 de)
I
YJ)3 g (-1 J’-nf (JMJ’ --Ml 10) T(JMu 1J’Mu)
+ (2rrWiCt) I[W(J’)-
1 -
GO)
The collision frequency p(J) depends on the components of the vectora(JJir &FIJ1ul) = Zqeqolq(JJ’uVIJIul)_ It is evident from (IO) that the collisional frequency y(J) is complex in the general case: fm v(J) J: 0, Re Y(J) JI 0. The physical meaning of cr is as follows: molecules with chaotic orientation are oriented in the directiona
which depends on the relative velocities of particles before and after the collision, u and ul. It is evident that the vector of orientation is zero for molecules with a center of symmetry, because collisional transitions with a change ofJ by an odd integer are forbidden.
4. Space orientation caused by light-induced drift Let us consider space orientation caused by lent-indu~d drift [I]. The vector of space o~ent~tion of polar molecules is the sum of those of excited (m) and unexcited (n) molecules. For simplicity we consider inefastic collisions J + J1 with a large change of velocity [3]. In this case the flow of molecules exists in the resonance levels only: im(J) = ~~~e~a,(Jo) 7
in(J)
= &‘fJ;1incJb) *
(1 I)
(a) Tizeflmv regirrze. In the cell with open ends the regime of steady-state flow is realised. One can obtain the ratio between the flows in both levels:
&(Jb> = -(J&Ir&)im(io)
3
(12)
where v1 is the frequency of cohisions with the buffer particles which break the flOWii_ For the vector of space orientation one can obtain with regard for (11) and (12):
r),,Q = ~,vw’,(J~)
I
17,(J)= ~,Ol~V,(J($,
(13)
where the frequencies Y,(J) and vn(J) are defied by (10). fb) Z?zeflo~&~s regke_ In a closed cell the total fiow is equal to zero. So, the vector of space orientation is defined by the following expression: PQ
= ]slrrr(JYW~ + 1) - ~ln(J)Wb
+
l)li,(Jdli~,@
It is to be noted that the orientation vectorp(J)
-
in (13) and (14) is not entirely imaginary, be~~e~~(~
(14)
and
q,(J) are complex. me statistical weights in (13) and (14) make it possible to obtain space orientation even in the case ofequal frequencies and cross sections of the orientational collisions using the transitions of P- and Q-branches when JO 352
CHEMICALPHYSICSLETTERS
Volume98, number4
1 July 1983
and Jb are not too large. One can naturally suppose om and un to be approximately proportional to vrn and vn respectively_ In the case of an open cell the differences (;Tmvn - Znr+n) appear in (13) (without statistical weights), and in (14) <;;;, - zn)_ So we conclude that p(J) is greater in the case of the flowless regime than in the case of free flow, when all other parameters are the same.
5. Conclusion Since the scattering amplitudes f(JMW1 M’) are non-zero only for polar molecules (without a center of symmetry) there is a reason to suppose f to be proportional to the dipole moment p (it is in a high-energy limit)_ So v(J) can be roughly estimated as u(J) = Y@/D)~ where Y is an ordinary gas-dynamic collisional frequency. We give an estimate for the relative number of oriented molecules @I/p:
Whenp=lD,v=107 s-‘,‘dJ% 101os--‘, v/u= lo-’ we obtain lpi/p = 10m4. An electric field E = lo- 1 V/cm corresponds to these oriented molecules when p = 1017 cmm3 _ Let us consider the situation when relative motions of the components is caused by ordinary diffusion. The diffusional velocity of the polar molecule is V = (U”/v) VP/P, where Vp is a density gradient of the dipolar gas. The relative number of oriented molecules when VP/~ = 10 cm-l and U * 10s cm s- 1 is @l/p =:v(r?
i- ,$-+7plp
= 10-4
_
Contrary to orientation caused by light radiation, there is no limitation to densities of the components in this case and liquid can be used. An electric field in liquid can be stronger (p = 1020 cme3): E = lo2 V/cm. Consideration of the self-influence of the polarisation gives the coefficient (2 + E)/~E in the expression for E and does not change the order of it. Here E is the dielectric permeability. This field exists only during the diffusion and is sufficiently strong only in the region of large density gradients_
References [l] F.Kh. Gel’mukhanov and A.M. Shalagin, Pis’maZh. Eksp. Tear- Fis. 29 (1979) 773. [JETP Letters 29, No. 12 (1979)];
Zh Eksp. Teor. Fiz. 78 (1980) 1674 [Soviet Phys. JETP 51 (1980) 839]_ [ 21 S.G. Rautian. G-1. Smimov and A-hi. Shalagin, Nelineinye resonansy v spektrakh atomov i molekul [Nonlinear resonances in atomic and molecular spectra ] (Nauka, Novosibirsk, 1979). [3] V-R. htionenko and A.M. Shalagin. Jzv. Akad. Nauk SSSR Ser. Phys. 45 (1981) 995
353