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Classical theory of the dielectric loss for NH3 and ND3
Volume 45A, number 4
PHYSICS LETFERS
8 October 1973
CLASSICAL THEORY OF THE DIELECTRIC LOSS FOR NH3 AND ND3 D.R.A. McMAHON and Ii. McLAUGHLIN Department of Physics, La Trobe University, Bundoora, Victoria, 3083, Australia Received 8 June 1973 A formula for the frequency dependent dielectric loss in dilute gases of NH3 and ND3 describing the deviation of the nonresonant spectrum from the Debye shape and demonstrating the importance of dipole reorientation memory loss effects in lineshift calculations has been developed.
The absorption coefficient a(w)may be expressed in terms of the dipole autocorrelation for N absorbers in the volume V by 2N ~ ~rw ~ = 3ckTV Re J (~i(O).~I (t))~~ exp (—iw t)d r.( 1)
shape parameters are then determined in terms of moments of the following functions; the phase change distribution functions D1 (ij.i) and D2 (i/i) in elastic collisions i ~ i in the zero reorientation and nonzero reorientation regions respectively, the distributions C 1 (~)and C2 (~)for phase changes 0 in the inelastic
The autocorrelation function for a stochastic model may be expressed in terms of an equilibrium distribution function p(X0) and a conditional probability distribution function G(XIX0t) as [1]
collision processes i ~±f representing the same transitions as those taking place in the electromagnetic process and a kernelf(u, u’) for orientations u’-~u where X(t) = cos~u(t) is the angle between the electric field vector of the incident radiation and the molecular dipole moment. The absorption coefficient is given by 2)avI3CkTV] (4) a(w) = [4~,2 N
(Jt(O)~Jt(t))av =ff,i(X).,&(X0)G(X110t)p(Xo)dXdXo
(2) where Xis a set of phase space dynamical variables. G(X X is determined from the stochastic Liouville0 ;t) equation
—
.~
—
0
a G(XIX 0 t) at
L(X)G(XIX0t) = ~(t)&(X—X0).
(3)
The stochastic Liouville formalism permits a general discussion of the simultaneous memory loss contributions to the spectral shape function of a number of different internal degrees of freedom of the absorber and has recently been used in a classical theoretical study of shapes ofthe NHmicrowave and infrared spectral band 3 and ND3 [2]. Generalizing the classical oscifiator analogue for ammonia suggested byofBen-Reuven [3], coll~ion duced reorientations molecular dipole momentinhave been explicitly included into the equations of motion. The intermolecular potential is assumed to consist of roughly two regions, one with both collision induced oscillator phase changes and dipole reorientations and one with phase changes only. The spectral
where w~ = w0
+&
is the shifted mean resonant ab-
sorption frequency for the inversion band. Eq. (4) may be approximated by the Van Vleck-Weisskopf lineshape with a half-width ~ at low pressures, but collapses to the nonresonant shape at high pressures to give 2N<112>avI3CkTV] (5) a(w) = [41rw X
—k--—
where1
+~
[(1
+-}
j~)
~ _~)2 + 21 2 +~
k W2~7 7+7?
t~
—
i~=~(1_62/~2)1I2and 1
+K
(1 +~)(l_~2/~2)1/2.
The spectral shape parameters are defined in terms of the following moments; 343
Volume 45A, number 4
PHYSICS LETTERS
collision theory the measured microwave line shifts of
f D1(~)d~p,~ f Q(0)d0~ f exp (±i~)D1(iP)di~i, ill1) f exp (±iØ)c~(0)d0,
=
(6b)
NH3 with various foreign gas perturbers [4]. Nonresonant cross-sections for foreign gas and ND3 mixtures are known to be of the order of factors 3—10 times larger than the NH3 shift cross-sections [5], so that reorientation memory loss effects may be a dominating consideration in a microwave lineshift theory. Further, as Birnbaum [6] has argued a strong collision theory is required for nonresonant diameters, such a theory is likely to be required for line shifts. The parameter i~is the memory dependent quantity characterizing the deviation of the absorption coefficient from the Debye shape. Formal expressions for such deviations have been given by Gordon [7], BenReuven [3] and Birnbaum [6] but the formalism of Gordon who regarded this deviation as indicative of memory dependent of “persistence” effects is more closely comparable with this work. Eq. (4) gives an excellent fit to the dielectric loss data of Maryott and Kryder for ND3 [8]. However a distribution of nonresonant relaxation rates can also give a good fit in principle, although in practice a calculation fitting both low and high pressure spectral shapes is more difficult. As such, something of an ambiguity exists in the
(6c)
interpretation of the deviations from the Debye shape for NH3 and ND3.
—~
±iö1
=
+IT
±
=
+1
3o
+
r~=
5 J(u,u’)du’ and (2o —
÷1
+
77)u =
5 u’J(u,u’)du’.
1
The last two moment conditions on J(u, u’ have been chosen to yield a theory equivalent to the Debye theory of slow rotators when the collision integral is expanded out to produce a Fokker-Planck type equation for small reorientations. The total scattering rates for elastic transitions i ~ i and inelastic transitions i ~ f are then 7 = 7~+7~and ~ = + ~2 respectively and the spectral shape parameters are defined by )
(6a) =
1
~(~2 +~2)1/2,
+g
=
8 October 1973
(1 +~2/~2)1/2,
where (6d) with a dummy13pvariable representing any of the pa7 rameters ~1a1, 1X1 and E~1. Eq. (6) show that ~ which governs the rate of collapse from the resonant shape to the nonresonant shape is a memory dependent parameter in that oscillator phase changes and dipole reorientations reduce its value below the maximum possible of This is not surprising in that the collapse is associated with the combined cross-relaxation effect of two mutually exclusive processes i -+ f and f—~i. Eq. (6) also show that reorientations reduce the observed resonant line shifts, a result relevant to the recent unsuccessful attempt to interpret theoretically with Anderson’s weak ~
1~~2[1—o/(3o+~)],
~•
~.
344
References
-
[1]
S.R. De Groot and P. Mazur, Non-equilibrium Thermodynamics, chaps VII and VIII (North-Holland, 1961). [21to D.R.A. McMahon and Ian L. McLaughlin, J. chem. Phys., be published. [3] A. Ben-Reuven, Phys. Rev. Lett.14 (1965) 349; Phys. Rev. 145
(1966)
7; Adv. Atom. Mo!. Phys. 5 (1968) 201.
[4] R.W. Parsons, V.!. Metchmk and I.C. Story, J. Phys. B.
Atom. Molec. Phys. 5
(1972) 1221.
[5] Maryott and Kryder, unpublished. [6] A.A. G. Birnbaum, Phys.S.J. Rev. 150 (1966) 101. [71R.G. Gordon, J. Chem. 45 (1966) 1635. [8] A.A. Maryott and S.J. Kryder, J. Chem. Phys. 46 (1967) 2856.
PHYSICS LETFERS
8 October 1973
CLASSICAL THEORY OF THE DIELECTRIC LOSS FOR NH3 AND ND3 D.R.A. McMAHON and Ii. McLAUGHLIN Department of Physics, La Trobe University, Bundoora, Victoria, 3083, Australia Received 8 June 1973 A formula for the frequency dependent dielectric loss in dilute gases of NH3 and ND3 describing the deviation of the nonresonant spectrum from the Debye shape and demonstrating the importance of dipole reorientation memory loss effects in lineshift calculations has been developed.
The absorption coefficient a(w)may be expressed in terms of the dipole autocorrelation for N absorbers in the volume V by 2N ~ ~rw ~ = 3ckTV Re J (~i(O).~I (t))~~ exp (—iw t)d r.( 1)
shape parameters are then determined in terms of moments of the following functions; the phase change distribution functions D1 (ij.i) and D2 (i/i) in elastic collisions i ~ i in the zero reorientation and nonzero reorientation regions respectively, the distributions C 1 (~)and C2 (~)for phase changes 0 in the inelastic
The autocorrelation function for a stochastic model may be expressed in terms of an equilibrium distribution function p(X0) and a conditional probability distribution function G(XIX0t) as [1]
collision processes i ~±f representing the same transitions as those taking place in the electromagnetic process and a kernelf(u, u’) for orientations u’-~u where X(t) = cos~u(t) is the angle between the electric field vector of the incident radiation and the molecular dipole moment. The absorption coefficient is given by 2)avI3CkTV] (4) a(w) = [4~,2 N
(Jt(O)~Jt(t))av =ff,i(X).,&(X0)G(X110t)p(Xo)dXdXo
(2) where Xis a set of phase space dynamical variables. G(X X is determined from the stochastic Liouville0 ;t) equation
—
.~
—
0
a G(XIX 0 t) at
L(X)G(XIX0t) = ~(t)&(X—X0).
(3)
The stochastic Liouville formalism permits a general discussion of the simultaneous memory loss contributions to the spectral shape function of a number of different internal degrees of freedom of the absorber and has recently been used in a classical theoretical study of shapes ofthe NHmicrowave and infrared spectral band 3 and ND3 [2]. Generalizing the classical oscifiator analogue for ammonia suggested byofBen-Reuven [3], coll~ion duced reorientations molecular dipole momentinhave been explicitly included into the equations of motion. The intermolecular potential is assumed to consist of roughly two regions, one with both collision induced oscillator phase changes and dipole reorientations and one with phase changes only. The spectral
where w~ = w0
+&
is the shifted mean resonant ab-
sorption frequency for the inversion band. Eq. (4) may be approximated by the Van Vleck-Weisskopf lineshape with a half-width ~ at low pressures, but collapses to the nonresonant shape at high pressures to give 2N<112>avI3CkTV] (5) a(w) = [41rw X
—k--—
where1
+~
[(1
+-}
j~)
~ _~)2 + 21 2 +~
k W2~7 7+7?
t~
—
i~=~(1_62/~2)1I2and 1
+K
(1 +~)(l_~2/~2)1/2.
The spectral shape parameters are defined in terms of the following moments; 343
Volume 45A, number 4
PHYSICS LETTERS
collision theory the measured microwave line shifts of
f D1(~)d~p,~ f Q(0)d0~ f exp (±i~)D1(iP)di~i, ill1) f exp (±iØ)c~(0)d0,
=
(6b)
NH3 with various foreign gas perturbers [4]. Nonresonant cross-sections for foreign gas and ND3 mixtures are known to be of the order of factors 3—10 times larger than the NH3 shift cross-sections [5], so that reorientation memory loss effects may be a dominating consideration in a microwave lineshift theory. Further, as Birnbaum [6] has argued a strong collision theory is required for nonresonant diameters, such a theory is likely to be required for line shifts. The parameter i~is the memory dependent quantity characterizing the deviation of the absorption coefficient from the Debye shape. Formal expressions for such deviations have been given by Gordon [7], BenReuven [3] and Birnbaum [6] but the formalism of Gordon who regarded this deviation as indicative of memory dependent of “persistence” effects is more closely comparable with this work. Eq. (4) gives an excellent fit to the dielectric loss data of Maryott and Kryder for ND3 [8]. However a distribution of nonresonant relaxation rates can also give a good fit in principle, although in practice a calculation fitting both low and high pressure spectral shapes is more difficult. As such, something of an ambiguity exists in the
(6c)
interpretation of the deviations from the Debye shape for NH3 and ND3.
—~
±iö1
=
+IT
±
=
+1
3o
+
r~=
5 J(u,u’)du’ and (2o —
÷1
+
77)u =
5 u’J(u,u’)du’.
1
The last two moment conditions on J(u, u’ have been chosen to yield a theory equivalent to the Debye theory of slow rotators when the collision integral is expanded out to produce a Fokker-Planck type equation for small reorientations. The total scattering rates for elastic transitions i ~ i and inelastic transitions i ~ f are then 7 = 7~+7~and ~ = + ~2 respectively and the spectral shape parameters are defined by )
(6a) =
1
~(~2 +~2)1/2,
+g
=
8 October 1973
(1 +~2/~2)1/2,
where (6d) with a dummy13pvariable representing any of the pa7 rameters ~1a1, 1X1 and E~1. Eq. (6) show that ~ which governs the rate of collapse from the resonant shape to the nonresonant shape is a memory dependent parameter in that oscillator phase changes and dipole reorientations reduce its value below the maximum possible of This is not surprising in that the collapse is associated with the combined cross-relaxation effect of two mutually exclusive processes i -+ f and f—~i. Eq. (6) also show that reorientations reduce the observed resonant line shifts, a result relevant to the recent unsuccessful attempt to interpret theoretically with Anderson’s weak ~
1~~2[1—o/(3o+~)],
~•
~.
344
References
-
[1]
S.R. De Groot and P. Mazur, Non-equilibrium Thermodynamics, chaps VII and VIII (North-Holland, 1961). [21to D.R.A. McMahon and Ian L. McLaughlin, J. chem. Phys., be published. [3] A. Ben-Reuven, Phys. Rev. Lett.14 (1965) 349; Phys. Rev. 145
(1966)
7; Adv. Atom. Mo!. Phys. 5 (1968) 201.
[4] R.W. Parsons, V.!. Metchmk and I.C. Story, J. Phys. B.
Atom. Molec. Phys. 5
(1972) 1221.
[5] Maryott and Kryder, unpublished. [6] A.A. G. Birnbaum, Phys.S.J. Rev. 150 (1966) 101. [71R.G. Gordon, J. Chem. 45 (1966) 1635. [8] A.A. Maryott and S.J. Kryder, J. Chem. Phys. 46 (1967) 2856.