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Evaluation of classical differential cross sections — simplifying methods
Volume
42, number
CHEMICAL
2
EVALUATION
OF CLASSICAL
PHYSICS
DIFFERENTIAL
LETTERS
1 September
CROSS SECTIONS
- SIMPLIFYING
1976
METHODS *
E.A. GISLASON Department of Chemistry, University of iilinois at Chicago Circle. Uticago. IIlinois 60680.
Received
26 Aprd
USA
1976
The general evaluation oi the classical differential cross section I(@, @) requires the determination of the 2 x 2 jacobian a(e,@)/a(b, @), where 0 and 0 are the final polar and azimuthal scattering angles, b is the impact parameter, and Q is the initial azimuthal angle. It is demonstrated here that the cross section can be evaluated from the 1 X 1 jacobian Mlab whenever I(0, @) is independent of 9, even If 8 and Q depend on a. In addition, a method is suggested for avoiding problems associated with “rainbow” singularities where ae/ab = 0.
I_ introduction Ln a classical treatment of the nonreactive scatterof two atoms or molecules interacting through a potential V the final polar deflection angle 0 and final azimuthal angle Q can be determined uniquely from the impact parameter b, the initial azimuthal angle +, and the various molecuiar orientation parameters Q’ [l ] _Thus, we can write ing
6 = 6(B, GJ, a’),
(la)
Q = Q(b, @, a’).
(lb)
Note that 8 2 0. The variables b, +, and a’ are displayed in fig. 1, which shows a coordinate system for the collision of an atom A with a molecule BC. The classical differential cross section at the angles 60, @o is given by [l-3] ‘Coo, @,) = WV-’
jko,
Qo; a’) da’,
z Fig. 1. The origin of the XYZ coordinate system is at the center-of-mass of mo!ecule BC. Atom A is approachmg with an initial relative velocity u parallel to the Z-axis. The X-axis is chosen along some convenient laboratory direction (for CLample, pardllel to the floor of the laboratory). The unperturbed trajectory passes through the XY-plane at point P a distance b from the origin. The angle @ is the azimuthal angle of P about the Z-axis. The polar angle p and the azimuthal angle Q locate atom B at time r = 0.
(2)
(5) where Ast = jd&, 1(eo,~o;52’)sineo * Support
(3) = c$iJ;.,-‘,
(4)
i
from the National acknowIedged.
Science
Foundation
is gratefully
The integrals in eqs. (2) and (3) extend over the full range of the parameters S2’. The sum in eq. (4) is carried out over all pairs (bi, @) which give scattering at 00 and 00 for a particular value of Sz’. An earlier paper [4] used the expression I(t), ; 52’) sin 19~= Ci $1&i-l,
Volume 42, number 2
1 September 1976
CHEMICALPHYSICSLE'ITERS
(7)
in place of eq. (4). While considerably simpler than eq. (4), eq. (6) ignores the dependence on + of 0 and Q and gives the wrong result in some cases. The evaluation of I(e,,
Q. ; S2’) using eq. (4) is nor-
mally quite difficult. Classical trajectories must be carried out for a wide range of b, a, and at’, and the jacobian J must be evaluated numerically. Consequently eqs_ (2)-(S) are normally not used by people carrying out trajectory studies. Rather, a large number of trajectories are determined for a random distribution of b, a, and a’, and a histogram is constructed of the number of collisions which give scattering in the “box” boundzd by 0,&J + AlJ,& and 4 + A& From the histogram I(L?, 4) can be determined [S] _This method completely avoids the problem of evahrating the jacobian. Eq. (4) is normally used when analytic expression for 0(b, +, S2’) and Q(b, ‘P, S2’) can be obtained. This is common in the determination of the small-angle scattering of two molecules using classical perturbation theory [4,6-S] . Even here, however, the evaluation of the jacobian can be quite difficult. By comparison, eq. (6) is much easier to use. Only the single derivative &I/ab must be calculated, and the work involved is similar to the problem of elastic scattering of two structureless particles. In this paper we will show that eq. (6), properly interpreted, is equivalent to eq. (4) whenever 1(0c, Qo) is independent of Go. This allows considerable simplification in the evaluation of the differential cross section. In addition, we will show an alternate method of calculating I((.+, , &,) which promises to further simplify this computation.
dbbS(B -@,)S(&O,), 0
(9)
0
where the functions 8 and Q are described Substitution of eq. (9) into eq. (2) gives
in eq. (1).
I(do, @o1 sin 0, = (An)-’
JdQ’
Tdb j:
db bS(B-Bo)6(r$-q50). 0 0 (10) We note that evaluation of the multiple integrals in eq. (10) using the Monte-Carlo technique is equivalent to use of the histogram method described earlier. Eq. (10) can be simplified when 1(0,, Go) is independent of Q. _This is no really the case because the orientation parameters X2 are distributed randomly as is @, and the averaging in eq. (10) removes all the $+, dependence. The one exception to this is when a polarizing field is present in the apparatus [8] _ Assumis independent of oo, we obtain ingW(j) =rte,,&J) from eq. (10) 2n I(0,)sin e.
= (2~r)-l s d@oI(BO,Go) sin8, 0
= (2nAR)-1
JdL?’
Using the properties rewrite this as 1(6,)
sin 19~= (2~)-l
Td@ 0
7 0
db b 6(U-8,).
of &-functions
(11)
again [9] , we can
~dOF(Bg,@),
(12)
0 with
2. Theory JZq. (4) can be rewritten
flfI,,G’) formally
as
l(Oa, Go : !2’) sin 0, = cjd@J
db b/J\-‘S(b-bi) S(@-@.$ (8) 0 i 0 Here 6(z) is the delta function. Using the properties of the delta function [9], this can be rewritten as
316
= (Afi)-l[dS2’
~biW/abl,!,,,,, i
i
- (13)
Except for the integration over (b, this result is similar to eqs. (2) and (6). However, eq. (13) shows explicitly that 8 is, in general, a function of @_ It is possible to prove directly for any binary collision involving atoms, linear molecules, and/or nonlinear molecules that F(Bo , @) defied in eq. ( 13) is independent of +. The proof is carried out in the appendix for the coliision of an atom with a linear mole-
Volume 42, number 2
CHEWIC%LPHYSICS
cule- In each case the proof shows that the averaging over SZ2’is sufficient to remove the @ dependence. However, this result is also suggested by the isotropy of space assumed here and by the fact that the precise direction of the X and Y axes (shown in fig. 1) is arbitrary. Consequently, we can write (14)
whereji is given in eq. (7), and it is understood
1
LETFERS
September 1976
It is clear that no singularities can occur in this result. Thus, the ~v~uation of a(@*) from eq. (16) is straightforward. The differential cross section I(@) can then be determined from the property do/d8 = -2~ sin 0 I(0).
that
all calculations are to be carried out at a fuied value of tf, {normally @ = O)_ One problem which can arise when eq. (14) is used
(18)
The function u(0) is also useful in certain total cross section calculations f 101 .
to determine 1(6,) is that for certain values of bi, a, and R’ the jacobian ii is zero. This “rainbow” scattering gives an infinite spike in the cross section which is
3. Discussion
difficult to treat mathematically. By comparison, the cross section I(f?u), which is averaged over all St’, will probably not show any such spike. To avoid this problem, it is convenient to define
As an example of the calculations described in this paper we consider the case of the small-angle scattering of a positive ion by a nonrotating linear molecule with dipole moment D. The potential is given by V(r, SL’)= .&v/+3,
o(P)=
2~~r(~*)sin6~de~. e* Using the argument of the preceding paragraph eq. (11) we can show that o(6*) = (2zr/AG?)ldSZ’
_i: dbbh(&-f?“),
(15) and
(16)
6 where tz(z) is the Heaviside function given by
h(z) = 0 =I
2<0;
(171
z 2 0.
An example of B(b) for particular is shown in fig. 2. In this case
values of fit’ and Q,
(19)
where r is a vector from the center-of-mass of the molecule BC (see fig_ I) to the ion A. Classical perturbation theory then gives [4,73 B = (eD/Eob2) sin 0,
(2Oa)
tg=2*-a!,
(2Ob)
where E0 is the relative energy of collision. The differential cross section can be computed from either eq. (2) or eq. (14). Note that for fixed values of p and
Q!there are two pairs of (b, Q) which give scattering at particular angles 9, and GOin eq. (2). By comparison, for fixed values of (3,Q, and a, there is only one
vahre of b which gives scattering at a particular angle 00 in eq. (14). The differential cross section is the same in either case because the jacobians J and i differ by a factor of two. The integral cross section o(t3 *), computed from eq. (IQ, is
0fe*) = (2/4)eD/EofP.
(21)
Using eq. (18) we then obtain fmpact
Parameter
I(O) =
b
Hypothetic& deflection angle curve plotted a&nst impact parameter. The five impact parameters bt, bal bs, and bJ give scattering at e = e*. Fig. 2.
b4.
sin 6-l f?-2 * (wDJ~E,)~-~,
a result which has been obtained
by others [4,7]. The exampIe described here is simple because 6 is
317
Volume 42, number 2
CHEMICAL.PHYSICS LETTERS
a monotonic function of b and is independent of Qr, More ~~rn~~~onIy~B will depend oo % 2nd may welI havt3ode or mare maxima and minima as the impact parameter is varied. One example of this is the scattering of two rotating dipoles [gf , and another is the scatterins of an ion by a nonrotatinl: dipolar molecule with the ion-induced dipole potential included [7,11]_ In both cases 8 depends on Cpin a nontrivial way and has minima and maxima which generate rainbow “spikes” in the differential cross section. The techniques developed in this paper should be very useful here.
1 September 1976
A co&ion
with p~t~cuiar values of (ZJ,& S,o1, and _ terirtg at a particuk~r deff cction angle 0. If the angles @ and 01are then increased by an equal amount (keeping b, p, and (IIfixed), the same deflection angle 0 will be obtained because, except for a common rotation about the Z-axis, it is the identical colhsion. Thus, B depends only on the variables (b, p, 6, w-9) rather that on the complete set (b, p, 6. Q, 43). Consequently, we replace the variables 0: and Cp @) wit!give scat
by y = ~-4, and (f>.Then the evaluation of the integrals in eq. (13) wili in&de an average over y, and all the ch dependence will be removed. We conclude that F(@Q,CD)is independent of Q, in this particular
case. Appendix.
Lack of Cpdependence
atom-hear
mofecule colfision
in F(l+, , ch) for References
Evaluation of the function F(Qo.d;), defined in eq. (131, requires the determination of the polar deflection angle 19as a function of b, 41, and R’. Fig. 1 shows the appropriate coordinate system for the collision of an atom or atomic ion with a linear molecule. In ad-
dition to the angles 0 and LY,which give the orientation of the molecule at time t = 0, it is necessary to specify an additional angle 6 which locates the initial plane of rotation of the molecule. This angle is the dihedral angle between the initial plane ofmofecular rotation and the plane &rough the Z-axis making an angle a! with the X-axis 14,8] _The an& 6 is shown in fig. 2 of ref. [4]. Inspection of that figure shows that ir is possible to vary (Ywhile keeping 6 fixed, but this mvolves a change in the orientation of the initial
plane of molecular
318
rotation.
R.J. C!ros%, J. Chem. Pi~ys. 50 (1969) 1036. R. Felton, J. Ross and R.J. Cross, J. Chem. Phys. 50 (1969) 1038. H. van Dop and A. Tip, P~YSWXI 61 (1972) 607. R.J. Cross, J. Chem. Phys. 46 (1967) 609. J.C. Poianyi and J.L. Schreiber, in: Physical chemistn an advanced treatise, Vol. 6A, ed. IV. Jost (Academic Press, New York, 1974) p_ 383. M.A. Wartell and R J. Cross, J. Cham. Phys. 54 (1971) 4519. W.R. Gentry, X Chem. Phys. 60 (1974) 2547. E.A. Gislason and D-R. Iferschbach, 3. C&cm. Phys. 64 (1976) 2133. B. Friedman, Principles and techniques of applied mathematics (Wiley, New York, 1956) p. 292. FE. Budenholzer, J.J. Galante, E.A. Gislason and A.D. Jorgensen, Chem. Phys. Letters 33 (197.5) 245. EA. Gislason, to be published.
42, number
CHEMICAL
2
EVALUATION
OF CLASSICAL
PHYSICS
DIFFERENTIAL
LETTERS
1 September
CROSS SECTIONS
- SIMPLIFYING
1976
METHODS *
E.A. GISLASON Department of Chemistry, University of iilinois at Chicago Circle. Uticago. IIlinois 60680.
Received
26 Aprd
USA
1976
The general evaluation oi the classical differential cross section I(@, @) requires the determination of the 2 x 2 jacobian a(e,@)/a(b, @), where 0 and 0 are the final polar and azimuthal scattering angles, b is the impact parameter, and Q is the initial azimuthal angle. It is demonstrated here that the cross section can be evaluated from the 1 X 1 jacobian Mlab whenever I(0, @) is independent of 9, even If 8 and Q depend on a. In addition, a method is suggested for avoiding problems associated with “rainbow” singularities where ae/ab = 0.
I_ introduction Ln a classical treatment of the nonreactive scatterof two atoms or molecules interacting through a potential V the final polar deflection angle 0 and final azimuthal angle Q can be determined uniquely from the impact parameter b, the initial azimuthal angle +, and the various molecuiar orientation parameters Q’ [l ] _Thus, we can write ing
6 = 6(B, GJ, a’),
(la)
Q = Q(b, @, a’).
(lb)
Note that 8 2 0. The variables b, +, and a’ are displayed in fig. 1, which shows a coordinate system for the collision of an atom A with a molecule BC. The classical differential cross section at the angles 60, @o is given by [l-3] ‘Coo, @,) = WV-’
jko,
Qo; a’) da’,
z Fig. 1. The origin of the XYZ coordinate system is at the center-of-mass of mo!ecule BC. Atom A is approachmg with an initial relative velocity u parallel to the Z-axis. The X-axis is chosen along some convenient laboratory direction (for CLample, pardllel to the floor of the laboratory). The unperturbed trajectory passes through the XY-plane at point P a distance b from the origin. The angle @ is the azimuthal angle of P about the Z-axis. The polar angle p and the azimuthal angle Q locate atom B at time r = 0.
(2)
(5) where Ast = jd&, 1(eo,~o;52’)sineo * Support
(3) = c$iJ;.,-‘,
(4)
i
from the National acknowIedged.
Science
Foundation
is gratefully
The integrals in eqs. (2) and (3) extend over the full range of the parameters S2’. The sum in eq. (4) is carried out over all pairs (bi, @) which give scattering at 00 and 00 for a particular value of Sz’. An earlier paper [4] used the expression I(t), ; 52’) sin 19~= Ci $1&i-l,
Volume 42, number 2
1 September 1976
CHEMICALPHYSICSLE'ITERS
(7)
in place of eq. (4). While considerably simpler than eq. (4), eq. (6) ignores the dependence on + of 0 and Q and gives the wrong result in some cases. The evaluation of I(e,,
Q. ; S2’) using eq. (4) is nor-
mally quite difficult. Classical trajectories must be carried out for a wide range of b, a, and at’, and the jacobian J must be evaluated numerically. Consequently eqs_ (2)-(S) are normally not used by people carrying out trajectory studies. Rather, a large number of trajectories are determined for a random distribution of b, a, and a’, and a histogram is constructed of the number of collisions which give scattering in the “box” boundzd by 0,&J + AlJ,& and 4 + A& From the histogram I(L?, 4) can be determined [S] _This method completely avoids the problem of evahrating the jacobian. Eq. (4) is normally used when analytic expression for 0(b, +, S2’) and Q(b, ‘P, S2’) can be obtained. This is common in the determination of the small-angle scattering of two molecules using classical perturbation theory [4,6-S] . Even here, however, the evaluation of the jacobian can be quite difficult. By comparison, eq. (6) is much easier to use. Only the single derivative &I/ab must be calculated, and the work involved is similar to the problem of elastic scattering of two structureless particles. In this paper we will show that eq. (6), properly interpreted, is equivalent to eq. (4) whenever 1(0c, Qo) is independent of Go. This allows considerable simplification in the evaluation of the differential cross section. In addition, we will show an alternate method of calculating I((.+, , &,) which promises to further simplify this computation.
dbbS(B -@,)S(&O,), 0
(9)
0
where the functions 8 and Q are described Substitution of eq. (9) into eq. (2) gives
in eq. (1).
I(do, @o1 sin 0, = (An)-’
JdQ’
Tdb j:
db bS(B-Bo)6(r$-q50). 0 0 (10) We note that evaluation of the multiple integrals in eq. (10) using the Monte-Carlo technique is equivalent to use of the histogram method described earlier. Eq. (10) can be simplified when 1(0,, Go) is independent of Q. _This is no really the case because the orientation parameters X2 are distributed randomly as is @, and the averaging in eq. (10) removes all the $+, dependence. The one exception to this is when a polarizing field is present in the apparatus [8] _ Assumis independent of oo, we obtain ingW(j) =rte,,&J) from eq. (10) 2n I(0,)sin e.
= (2~r)-l s d@oI(BO,Go) sin8, 0
= (2nAR)-1
JdL?’
Using the properties rewrite this as 1(6,)
sin 19~= (2~)-l
Td@ 0
7 0
db b 6(U-8,).
of &-functions
(11)
again [9] , we can
~dOF(Bg,@),
(12)
0 with
2. Theory JZq. (4) can be rewritten
flfI,,G’) formally
as
l(Oa, Go : !2’) sin 0, = cjd@J
db b/J\-‘S(b-bi) S(@-@.$ (8) 0 i 0 Here 6(z) is the delta function. Using the properties of the delta function [9], this can be rewritten as
316
= (Afi)-l[dS2’
~biW/abl,!,,,,, i
i
- (13)
Except for the integration over (b, this result is similar to eqs. (2) and (6). However, eq. (13) shows explicitly that 8 is, in general, a function of @_ It is possible to prove directly for any binary collision involving atoms, linear molecules, and/or nonlinear molecules that F(Bo , @) defied in eq. ( 13) is independent of +. The proof is carried out in the appendix for the coliision of an atom with a linear mole-
Volume 42, number 2
CHEWIC%LPHYSICS
cule- In each case the proof shows that the averaging over SZ2’is sufficient to remove the @ dependence. However, this result is also suggested by the isotropy of space assumed here and by the fact that the precise direction of the X and Y axes (shown in fig. 1) is arbitrary. Consequently, we can write (14)
whereji is given in eq. (7), and it is understood
1
LETFERS
September 1976
It is clear that no singularities can occur in this result. Thus, the ~v~uation of a(@*) from eq. (16) is straightforward. The differential cross section I(@) can then be determined from the property do/d8 = -2~ sin 0 I(0).
that
all calculations are to be carried out at a fuied value of tf, {normally @ = O)_ One problem which can arise when eq. (14) is used
(18)
The function u(0) is also useful in certain total cross section calculations f 101 .
to determine 1(6,) is that for certain values of bi, a, and R’ the jacobian ii is zero. This “rainbow” scattering gives an infinite spike in the cross section which is
3. Discussion
difficult to treat mathematically. By comparison, the cross section I(f?u), which is averaged over all St’, will probably not show any such spike. To avoid this problem, it is convenient to define
As an example of the calculations described in this paper we consider the case of the small-angle scattering of a positive ion by a nonrotating linear molecule with dipole moment D. The potential is given by V(r, SL’)= .&v/+3,
o(P)=
2~~r(~*)sin6~de~. e* Using the argument of the preceding paragraph eq. (11) we can show that o(6*) = (2zr/AG?)ldSZ’
_i: dbbh(&-f?“),
(15) and
(16)
6 where tz(z) is the Heaviside function given by
h(z) = 0 =I
2<0;
(171
z 2 0.
An example of B(b) for particular is shown in fig. 2. In this case
values of fit’ and Q,
(19)
where r is a vector from the center-of-mass of the molecule BC (see fig_ I) to the ion A. Classical perturbation theory then gives [4,73 B = (eD/Eob2) sin 0,
(2Oa)
tg=2*-a!,
(2Ob)
where E0 is the relative energy of collision. The differential cross section can be computed from either eq. (2) or eq. (14). Note that for fixed values of p and
Q!there are two pairs of (b, Q) which give scattering at particular angles 9, and GOin eq. (2). By comparison, for fixed values of (3,Q, and a, there is only one
vahre of b which gives scattering at a particular angle 00 in eq. (14). The differential cross section is the same in either case because the jacobians J and i differ by a factor of two. The integral cross section o(t3 *), computed from eq. (IQ, is
0fe*) = (2/4)eD/EofP.
(21)
Using eq. (18) we then obtain fmpact
Parameter
I(O) =
b
Hypothetic& deflection angle curve plotted a&nst impact parameter. The five impact parameters bt, bal bs, and bJ give scattering at e = e*. Fig. 2.
b4.
sin 6-l f?-2 * (wDJ~E,)~-~,
a result which has been obtained
by others [4,7]. The exampIe described here is simple because 6 is
317
Volume 42, number 2
CHEMICAL.PHYSICS LETTERS
a monotonic function of b and is independent of Qr, More ~~rn~~~onIy~B will depend oo % 2nd may welI havt3ode or mare maxima and minima as the impact parameter is varied. One example of this is the scattering of two rotating dipoles [gf , and another is the scatterins of an ion by a nonrotatinl: dipolar molecule with the ion-induced dipole potential included [7,11]_ In both cases 8 depends on Cpin a nontrivial way and has minima and maxima which generate rainbow “spikes” in the differential cross section. The techniques developed in this paper should be very useful here.
1 September 1976
A co&ion
with p~t~cuiar values of (ZJ,& S,o1, and _ terirtg at a particuk~r deff cction angle 0. If the angles @ and 01are then increased by an equal amount (keeping b, p, and (IIfixed), the same deflection angle 0 will be obtained because, except for a common rotation about the Z-axis, it is the identical colhsion. Thus, B depends only on the variables (b, p, 6, w-9) rather that on the complete set (b, p, 6. Q, 43). Consequently, we replace the variables 0: and Cp @) wit!give scat
by y = ~-4, and (f>.Then the evaluation of the integrals in eq. (13) wili in&de an average over y, and all the ch dependence will be removed. We conclude that F(@Q,CD)is independent of Q, in this particular
case. Appendix.
Lack of Cpdependence
atom-hear
mofecule colfision
in F(l+, , ch) for References
Evaluation of the function F(Qo.d;), defined in eq. (131, requires the determination of the polar deflection angle 19as a function of b, 41, and R’. Fig. 1 shows the appropriate coordinate system for the collision of an atom or atomic ion with a linear molecule. In ad-
dition to the angles 0 and LY,which give the orientation of the molecule at time t = 0, it is necessary to specify an additional angle 6 which locates the initial plane of rotation of the molecule. This angle is the dihedral angle between the initial plane ofmofecular rotation and the plane &rough the Z-axis making an angle a! with the X-axis 14,8] _The an& 6 is shown in fig. 2 of ref. [4]. Inspection of that figure shows that ir is possible to vary (Ywhile keeping 6 fixed, but this mvolves a change in the orientation of the initial
plane of molecular
318
rotation.
R.J. C!ros%, J. Chem. Pi~ys. 50 (1969) 1036. R. Felton, J. Ross and R.J. Cross, J. Chem. Phys. 50 (1969) 1038. H. van Dop and A. Tip, P~YSWXI 61 (1972) 607. R.J. Cross, J. Chem. Phys. 46 (1967) 609. J.C. Poianyi and J.L. Schreiber, in: Physical chemistn an advanced treatise, Vol. 6A, ed. IV. Jost (Academic Press, New York, 1974) p_ 383. M.A. Wartell and R J. Cross, J. Cham. Phys. 54 (1971) 4519. W.R. Gentry, X Chem. Phys. 60 (1974) 2547. E.A. Gislason and D-R. Iferschbach, 3. C&cm. Phys. 64 (1976) 2133. B. Friedman, Principles and techniques of applied mathematics (Wiley, New York, 1956) p. 292. FE. Budenholzer, J.J. Galante, E.A. Gislason and A.D. Jorgensen, Chem. Phys. Letters 33 (197.5) 245. EA. Gislason, to be published.