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Rate coefficient for ion—dipole orbiting collisions
RATE
6 May 1983
CHEMICAL PHYSICS LETTERS
Volume 97, number 1
COEFFICIENT FOR ION-DIPOLE
ORBITING COLLISIONS
D-R. BATES * Harvard-Smithsonian Center for Astrophysics, Harvard University, 60 Garden Street, Gzmbridge, Massachusetts 02138. USA and Center for Earth and Planetary Physics, Harvard University, 29 Oxford Street, Cambridge, Massachusetts 02138. USA Received 16 February
1983
An improved version of the average dipole orientation (ADO) method is used to calculate the rate coefficient for iondipole orbiting (capture) rate coefficients. The results agree well with those obtained by the variational aF.d by the statisti-
cal method.
l_ Introduction
The force on the molecule due to the ion is
Since the pioneering work of Su and Bowers [l] (see also ref. [2]) on the average dipole orientation (ADO) method of calculating the rate coefficient y for ion-dipole orbiting (or capture) collisions several other methods have been develtiped notably the variational [3], the statistical [4] and the perturbed rotational state [5,6] methods. However the ADO method remains attractive because of its conceptual simplicity. Furthermore it is effectively exact in principle because the rotational time period of the molecule is brief compared with the orbital time period. in confirmation of this last, trajectory [3,7] and perturbed stationary-state [6] studies show that the thermal average rate coefficient ;i; does not change appreciably if the moment of inertia I of the molecule is increased from the I + 0 limit (where the ADO method is exact) up to a value far greater than any arising in practice. Let CYbe the polarizability of the neutral molecule, D be its dipole moment and r be its position vector relative to the ion. Put
arc cos(f53
= 0.
-2(&+
0 1983 North-Holland
i- eD cos 8/G $.
(2)
In the I + 0 limit the replacement of cos 8 in (2) by its mean value (cos 0) at separation r does not entail any approximation (unlike the similar replacement in the corresponding expression for the potential which was made in the early version of the ADO method
[11X It is convenient to change the independent variable from r to x = r(kT/eD)‘/2 = r/d
(say),
(3)
T being the temperature. Noting that the work done by an external force in bringing the ion and molecule from an infinite separation to a separation x is the sum of the potential energy kTV(x) and the gain of kTp(x) in the rotational kinetic energy at angle
8= arc cos(cos 8),
(4)
we see from (2) that V(x) = -@x-4
(1)
* Perrnancnt address: Department of Applied Mnthcmntics and Theoretical Physics, Queen’s University of Belfast. Northern Ireland BT 7 INN.
0 009-2614/83/0000-0000/$03.00
F =
- 2 J x
((cos
e)/x3)& - p(x),
(5)
with fl= 2akT/D2.
(6)
19
Volume
97, nuJllher
1
CHEMICAL
PHYSICS
The deteJ-mination of p(x) and then of (cos 8) is the mAin problenl. As well as being a function of the variable indicated. and being parametrically dependent on the values of sever31 entities at infinite separation, p(x) is a function of i-i wherei is a unit vector in the directton of the rotational angular momentum_ I-;thtttg rhts fullyjnto account would be possible but I: would entail extrenmely heavy computing. Fortustately r--i is not of critical importance. Denoting po~cJltl~lk assocJJted withjbeing respectively perpendicuhn to and parallel to i by V,(s) and Vz(x) a simple (.md .~s it proves very satisfactory) physical approxiJu.JtioJi is tu take I-(s) = ; /‘l-,(s)
+ I’,(s)].
(7)
This cmJJJll\ents the need to carry through the laborious tnsk of averaging the r&e coefficient over all orient.utoJJs ofjat iJJliJJite separation. A major adt.~ritdg?e wbicb accrues IS rlial 1’(x) of (7) is paranietric~lly dependent only on k7h the rotational ener,q of thr’ tnadent 111o1csule. Thts enables a closed expres51011for the orhitJng cross section to be derived.
LETTERS
6 May
and that in the librating <‘&z-‘/2[~(@
1983
region
- (1 - mK(nz)J
= f&/2,
0G.X 6x0,
(11)
where K and E are the complete elliptic integrals of +Lhefirst and second kinds. Using (IO), (11) and (8) it is a straightforward task to find 5 as a function ofx for any chosen value of MT the rotational energy at inftite separation_ From the relations (cos 0$ = (J(6$l
cos o1 d(cos $)) -1
j-(6+’
.
d(cosO1)
(12)
e, = [E(_X.e,)] lf2.
(13)
where e(x. 0) is the rotational energy at (x. 0,) it may further be shown [S] that (cos 0 1) = $ [ 1 + _Y(.Y~<;- 3&p”
= 1 -j&1,
- _&I ] ,
x a-q).
(14)
0
(1%
Finally pi(x) of (5) may now be expressed in terms of known
2. Theory 1 he two ~C~ICJ~IJ~S Jnentioned in section 1 will be ioJ:xJdered sepzrdtrly. The same symbols will be used fog corrcspondJJJg quantities with subscripts 1 or 2 .Jfti\cd .is ,Jppropriste.
Ii the rcJt.JtioJial aJigul.rr JnomeJituJii vector is perprJJdJ~u1.u IO the position vector use may be made of th- .xi~dhdt~s iJiv.iriance [9] ofJp dlf/1/7n in which p Is .I11 3pproprLJte JiioJJieJJtu~Ji and q is the conjugate zo0JdirJaJc. 12cprescJit the instaritaneous rotational energy at tlle orieiit~tion at which the potential energo of (he 1o11-dipole mterxztton is a Jninimunr by x 75 3Jd \\ rite ttt = f $.1-J.
@I
sg = ~l2;,~1/z_
(9)
!t
IIIJ~
be sho\vn
p&r)
quantities:
= Is-1 -
22. PolethaI
A -
thus
(1 -
(cos
0$)x-q.
(16)
V2(_xJ
Only elementary mechanics enter if the rotational angular momentum vector is parallel to the position vector. Two couples act on the rotating molecule (here taken to be diatomic): the couple due to the ion-dipole interaction and the couple due to the centrifugal force acting on the constituent atoms. Equating these gives that the angle 8, which the axis of the dipole makes with ; is such that co.5 82 = l/2r2&.
(17)
Angular momentum is conserved servation equation is
in this case. The con-
h= I2 sin40, [7] that iJJ the rotating
region
(18)
so that (17) becoJnes cos
e2 = sin4B2/2x2X
and the gain in the rotational
(19) energy is kTp2(x)
with
6
CHEMICAL PHYSICS LE-ITERS
Volume 97, number 1
p2(x) = A(cosecQ2 - 1).
(20)
f(x)=&
+$(2
7 X
2.3. Rate coefficient The effective potential tion is kTU(x), where U(x) = (?Jb’/d’)x-2
controlling
+ $[2V,(x)
+2x3-
the radial mo-
+ Iqc)]
,
(21)
dpl(x) dx
(2(cose~)+cos
May
1983
B&-%x
dpd-1
+x3 -
(26)
1-
dx
On multiplying by the velocity of relative motion and integrating over a maxwellian distribution in 77we get
kfi
being the energy of relative motion at infinite separation and b being the impact parameter. At the top of the centrifugal barrier dU(x)/dr
=0
(22)
and g = U(x) + :x diJ(x)/dx
(23)
(in which the terms involving b cancel). We find from (23) that 7j = ;px-4 0
+ $ (2(cos 0,) + cos fqx-2 (
(27) in
which y(h) is the orbiting rate coefficient molecule of rotational energy kTA and
for a
7L = 27r(cue?/~)1~~
(2s)
& being the reduced mass) is the Langevin rate coefficient_ The themlal average for a Boltzmann distribution in the rotational levels is YWJY~
e-XrO)lrL
= 7
(2%
a-
0
(24) from which x(q) may be obtained by inverse interpolation: and we find from (22) that the orbiting cross section is nb’ = (7&Q)
(25)
f(x),
where
Table 1 compares the results of the present calculations with those obtained by Chesnavich et al. 131 using their variational method and those of Celli et al. [4] using their statistical method_ The agreement is good_ Comparison with the perturbed rotational state results of Sakimoto [6] is not straightforward because they are given as cross sections_ When the application of adiabatic invariance to
=ble 1 Thcrnlally nvcmged ratio y(h)/-y~ Dj(2d~T)“~
Adhbatic invarhnce zipproximation [S ]
Present ADO calculations
V;lri.&onnl method [ 3 ]
Statistical method [4] 3)
4
1.739 2.819
1.85 3.10
6 8
4.006
4.40
4213
4.20
5.230
5.70
5.474 8.031 10.61 15.80 21.00
5.45 8.00 10.6 15.8 21.0
_I_2
12 16 24 32
1.827 2.981
1.83 2.97
3) Read from graph.
21
Volume 97. number 1
CHEMICAL PHYSICS LETTERS
the ion-dipole problem was introduced [8] attention was confmed to encounters in which the rotational angular momentum vector is normal to the orbital plane. These encounters could be treated esactly and the prime objective was their detailed
study. However a thermal average intended to correspond to that of (29) was obtained by a physical approsimation (unrelated to the one made in section 1). As may be seen from table 1 the results are (fortuitously) very close to those of the present ADO calculations_
by a Smithsonian Regents’ Fellowship NASA under Grant NSG-7176.
6
hiay 1963
and in part by
References [ 1] T. 5.1 and M-T. Bowers, J. Chem. Phys. 58 (1973) 3027_ [Z] L. Bass. T. Su. W-G_ Chesnavich and M-T. Bowers, Chem.
Phys. Letters 24 (1975) 119. T. Su and hl_T. Bowers, J. Chem. Phys.
[ 31 W-G. Chesnatkh,
72 (1980) 2641. [4 J F. Celli. G. Weddle and D-P. Ridge, J. Chem. Phys. 73
(1980) 801. 151 II. Takayanagi, J. Phys. Sot. Japan 45 (1978) 976. Acknowledgement I thank Mrs. Norah Scott for skillfully writing the computer program. The work was supported in part
[6] E. Sakimoto. Chem. Phys. 63 (1981) 419_ [ 7) J-V- Dugan and J-R. hiagee, J. Chem. Phys. 47 (1967) 3103. [8] D-R. Bates. Proc. Roy_ Sot. A 384 (1982) 289. [9 J L.D. Landau and E-hi. Lifshitz. hlechanics. Vol. 9 (Pergamon Press, Oxford, 1960).
6 May 1983
CHEMICAL PHYSICS LETTERS
Volume 97, number 1
COEFFICIENT FOR ION-DIPOLE
ORBITING COLLISIONS
D-R. BATES * Harvard-Smithsonian Center for Astrophysics, Harvard University, 60 Garden Street, Gzmbridge, Massachusetts 02138. USA and Center for Earth and Planetary Physics, Harvard University, 29 Oxford Street, Cambridge, Massachusetts 02138. USA Received 16 February
1983
An improved version of the average dipole orientation (ADO) method is used to calculate the rate coefficient for iondipole orbiting (capture) rate coefficients. The results agree well with those obtained by the variational aF.d by the statisti-
cal method.
l_ Introduction
The force on the molecule due to the ion is
Since the pioneering work of Su and Bowers [l] (see also ref. [2]) on the average dipole orientation (ADO) method of calculating the rate coefficient y for ion-dipole orbiting (or capture) collisions several other methods have been develtiped notably the variational [3], the statistical [4] and the perturbed rotational state [5,6] methods. However the ADO method remains attractive because of its conceptual simplicity. Furthermore it is effectively exact in principle because the rotational time period of the molecule is brief compared with the orbital time period. in confirmation of this last, trajectory [3,7] and perturbed stationary-state [6] studies show that the thermal average rate coefficient ;i; does not change appreciably if the moment of inertia I of the molecule is increased from the I + 0 limit (where the ADO method is exact) up to a value far greater than any arising in practice. Let CYbe the polarizability of the neutral molecule, D be its dipole moment and r be its position vector relative to the ion. Put
arc cos(f53
= 0.
-2(&+
0 1983 North-Holland
i- eD cos 8/G $.
(2)
In the I + 0 limit the replacement of cos 8 in (2) by its mean value (cos 0) at separation r does not entail any approximation (unlike the similar replacement in the corresponding expression for the potential which was made in the early version of the ADO method
[11X It is convenient to change the independent variable from r to x = r(kT/eD)‘/2 = r/d
(say),
(3)
T being the temperature. Noting that the work done by an external force in bringing the ion and molecule from an infinite separation to a separation x is the sum of the potential energy kTV(x) and the gain of kTp(x) in the rotational kinetic energy at angle
8= arc cos(cos 8),
(4)
we see from (2) that V(x) = -@x-4
(1)
* Perrnancnt address: Department of Applied Mnthcmntics and Theoretical Physics, Queen’s University of Belfast. Northern Ireland BT 7 INN.
0 009-2614/83/0000-0000/$03.00
F =
- 2 J x
((cos
e)/x3)& - p(x),
(5)
with fl= 2akT/D2.
(6)
19
Volume
97, nuJllher
1
CHEMICAL
PHYSICS
The deteJ-mination of p(x) and then of (cos 8) is the mAin problenl. As well as being a function of the variable indicated. and being parametrically dependent on the values of sever31 entities at infinite separation, p(x) is a function of i-i wherei is a unit vector in the directton of the rotational angular momentum_ I-;thtttg rhts fullyjnto account would be possible but I: would entail extrenmely heavy computing. Fortustately r--i is not of critical importance. Denoting po~cJltl~lk assocJJted withjbeing respectively perpendicuhn to and parallel to i by V,(s) and Vz(x) a simple (.md .~s it proves very satisfactory) physical approxiJu.JtioJi is tu take I-(s) = ; /‘l-,(s)
+ I’,(s)].
(7)
This cmJJJll\ents the need to carry through the laborious tnsk of averaging the r&e coefficient over all orient.utoJJs ofjat iJJliJJite separation. A major adt.~ritdg?e wbicb accrues IS rlial 1’(x) of (7) is paranietric~lly dependent only on k7h the rotational ener,q of thr’ tnadent 111o1csule. Thts enables a closed expres51011for the orhitJng cross section to be derived.
LETTERS
6 May
and that in the librating <‘&z-‘/2[~(@
1983
region
- (1 - mK(nz)J
= f&/2,
0G.X 6x0,
(11)
where K and E are the complete elliptic integrals of +Lhefirst and second kinds. Using (IO), (11) and (8) it is a straightforward task to find 5 as a function ofx for any chosen value of MT the rotational energy at inftite separation_ From the relations (cos 0$ = (J(6$l
cos o1 d(cos $)) -1
j-(6+’
.
d(cosO1)
(12)
e, = [E(_X.e,)] lf2.
(13)
where e(x. 0) is the rotational energy at (x. 0,) it may further be shown [S] that (cos 0 1) = $ [ 1 + _Y(.Y~<;- 3&p”
= 1 -j&1,
- _&I ] ,
x a-q).
(14)
0
(1%
Finally pi(x) of (5) may now be expressed in terms of known
2. Theory 1 he two ~C~ICJ~IJ~S Jnentioned in section 1 will be ioJ:xJdered sepzrdtrly. The same symbols will be used fog corrcspondJJJg quantities with subscripts 1 or 2 .Jfti\cd .is ,Jppropriste.
Ii the rcJt.JtioJial aJigul.rr JnomeJituJii vector is perprJJdJ~u1.u IO the position vector use may be made of th- .xi~dhdt~s iJiv.iriance [9] ofJp dlf/1/7n in which p Is .I11 3pproprLJte JiioJJieJJtu~Ji and q is the conjugate zo0JdirJaJc. 12cprescJit the instaritaneous rotational energy at tlle orieiit~tion at which the potential energo of (he 1o11-dipole mterxztton is a Jninimunr by x 75 3Jd \\ rite ttt = f $.1-J.
@I
sg = ~l2;,~1/z_
(9)
!t
IIIJ~
be sho\vn
p&r)
quantities:
= Is-1 -
22. PolethaI
A -
thus
(1 -
(cos
0$)x-q.
(16)
V2(_xJ
Only elementary mechanics enter if the rotational angular momentum vector is parallel to the position vector. Two couples act on the rotating molecule (here taken to be diatomic): the couple due to the ion-dipole interaction and the couple due to the centrifugal force acting on the constituent atoms. Equating these gives that the angle 8, which the axis of the dipole makes with ; is such that co.5 82 = l/2r2&.
(17)
Angular momentum is conserved servation equation is
in this case. The con-
h= I2 sin40, [7] that iJJ the rotating
region
(18)
so that (17) becoJnes cos
e2 = sin4B2/2x2X
and the gain in the rotational
(19) energy is kTp2(x)
with
6
CHEMICAL PHYSICS LE-ITERS
Volume 97, number 1
p2(x) = A(cosecQ2 - 1).
(20)
f(x)=&
+$(2
7 X
2.3. Rate coefficient The effective potential tion is kTU(x), where U(x) = (?Jb’/d’)x-2
controlling
+ $[2V,(x)
+2x3-
the radial mo-
+ Iqc)]
,
(21)
dpl(x) dx
(2(cose~)+cos
May
1983
B&-%x
dpd-1
+x3 -
(26)
1-
dx
On multiplying by the velocity of relative motion and integrating over a maxwellian distribution in 77we get
kfi
being the energy of relative motion at infinite separation and b being the impact parameter. At the top of the centrifugal barrier dU(x)/dr
=0
(22)
and g = U(x) + :x diJ(x)/dx
(23)
(in which the terms involving b cancel). We find from (23) that 7j = ;px-4 0
+ $ (2(cos 0,) + cos fqx-2 (
(27) in
which y(h) is the orbiting rate coefficient molecule of rotational energy kTA and
for a
7L = 27r(cue?/~)1~~
(2s)
& being the reduced mass) is the Langevin rate coefficient_ The themlal average for a Boltzmann distribution in the rotational levels is YWJY~
e-XrO)lrL
= 7
(2%
a-
0
(24) from which x(q) may be obtained by inverse interpolation: and we find from (22) that the orbiting cross section is nb’ = (7&Q)
(25)
f(x),
where
Table 1 compares the results of the present calculations with those obtained by Chesnavich et al. 131 using their variational method and those of Celli et al. [4] using their statistical method_ The agreement is good_ Comparison with the perturbed rotational state results of Sakimoto [6] is not straightforward because they are given as cross sections_ When the application of adiabatic invariance to
=ble 1 Thcrnlally nvcmged ratio y(h)/-y~ Dj(2d~T)“~
Adhbatic invarhnce zipproximation [S ]
Present ADO calculations
V;lri.&onnl method [ 3 ]
Statistical method [4] 3)
4
1.739 2.819
1.85 3.10
6 8
4.006
4.40
4213
4.20
5.230
5.70
5.474 8.031 10.61 15.80 21.00
5.45 8.00 10.6 15.8 21.0
_I_2
12 16 24 32
1.827 2.981
1.83 2.97
3) Read from graph.
21
Volume 97. number 1
CHEMICAL PHYSICS LETTERS
the ion-dipole problem was introduced [8] attention was confmed to encounters in which the rotational angular momentum vector is normal to the orbital plane. These encounters could be treated esactly and the prime objective was their detailed
study. However a thermal average intended to correspond to that of (29) was obtained by a physical approsimation (unrelated to the one made in section 1). As may be seen from table 1 the results are (fortuitously) very close to those of the present ADO calculations_
by a Smithsonian Regents’ Fellowship NASA under Grant NSG-7176.
6
hiay 1963
and in part by
References [ 1] T. 5.1 and M-T. Bowers, J. Chem. Phys. 58 (1973) 3027_ [Z] L. Bass. T. Su. W-G_ Chesnavich and M-T. Bowers, Chem.
Phys. Letters 24 (1975) 119. T. Su and hl_T. Bowers, J. Chem. Phys.
[ 31 W-G. Chesnatkh,
72 (1980) 2641. [4 J F. Celli. G. Weddle and D-P. Ridge, J. Chem. Phys. 73
(1980) 801. 151 II. Takayanagi, J. Phys. Sot. Japan 45 (1978) 976. Acknowledgement I thank Mrs. Norah Scott for skillfully writing the computer program. The work was supported in part
[6] E. Sakimoto. Chem. Phys. 63 (1981) 419_ [ 7) J-V- Dugan and J-R. hiagee, J. Chem. Phys. 47 (1967) 3103. [8] D-R. Bates. Proc. Roy_ Sot. A 384 (1982) 289. [9 J L.D. Landau and E-hi. Lifshitz. hlechanics. Vol. 9 (Pergamon Press, Oxford, 1960).