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Rotational rainbows in inelastic atom—molecule differential cross sections
Volume
62. number 2
ROTATIONAL
CHEMICAL
RAINBOWS
IN INELASTIC
PHYSICS
1 April I979
-RS
ATOM-MOLECULE
DIFFERENTIAL
CROSS SECTIONS *
Joel M. BOWhMN s Department of Ciremistry, Illinois Institute of Technology, Chicago. Illitrois 60616. USA Received 18 November 1978;in fiil
form 19 December 1978
The existence of rotational rainbows in atom-rigid rotor inelastic differential cross sections is shown b> considerins the classical limit of the centrifugal sudden-rotational sudden expression for the scattering amplitude.
Rainbow features in the differential cross section in central field atom-atom scattering are well known [1] and well understood to be due to the long range attraction between atoms. There is at present much interest in the nature of rainbow features in inelastic differential cross sections [2,3]. in this letter we present some formal and computational results which shed light on the nature of these rainbows for rctationally inelastic processes_ The classical limit of the sudden rotation (10s) expression for the scattering amplitude for a transition il = 0 to& has been recently derived by us [4]: it is
a[2t7(1,7)lia7-j2(i.7)
exp [2iQ7)]
where y is the angle between the molecular asis and the scattering position vector, R. The phase shift 7~[(7) is computed for ffled values of this orientation angle. The complete stationary phase limit of (1) can be taken in the usual way to yield the following results. The stationary points satisfy the equations =x(E, 7) = 5 f? 7
0)
(Z$- ;)‘I? r;-, -jz (0) = -
(2#2
k , sin’&
x laiz/a7i,*'~lax/arl,*'~ esp[i9,(7i-li)]_
(4)
The subscript i indicates the ith root to eqs. (2) and (3)_ For the present purpose it is unnecessary to give the complete. more complicated stationary phase ekpression for i;-, _i’j2 (0) *_ inspection of this expressmn indicates that classical rainbows can occur when = 0,
(5)
a.dar=0 _
(6)
or
P,(cos 0)) (1)
a[2tl(z# 7)llar
-
3rd & += (0) is given by
aj2ia7 d7e-iV~-il)Y
=j2 -il
(2)
* Support from the &Conal Science Foundation is gratefully acknowledpd** Alfred P. Sloan Fellow.
The former rainbow corresponds to a rotational transition rninbow and the latter one is just the familiar rainbow in the scattering angle deflection function_ Rainbow phenomena in inelastic scattering amplitudes are known, especially from classical S-matrix theory [S] _The present expression is of interest since the rotational rainbow arises directly from the properties of the very familiar phase shift q(Z. -y) through the rotational deflection function (3). + The full two-dimensional stationary phase expression for fj,-.j., (0) contains a 2 X 2 determinant. and a summatmn over dl roots to eqs. (2) and (3).
309
Volume 62, number 2
CHEMICAL
PHYSICS LETTERS
Consider a model atom-rigid rotor system employed previously by McCurdy and Miller [6] and shown by us to one for which the dassical Iimit of the rotational sudden approximation [eq. (l)] is quantitativeIy valid [4] _ The interaction potential is
where
VL_J(R) =
l[(R,/R)‘*
- WZm/R)6J
-
R, = 6-67 au, E = 10-7 cm-I ~ the relative translational energy E IS 150 E, the rotationa constant B is 0.1 E, and the reduced mass is appropriate for He-CO. Let us focus now on locating the rotational rainbows for this system for Aj equal to 2,4.6 and 8_ These rainbows occur at vaIues of C for which there is a coaIescence of roots yi to eq_ (3X Due to the symmetry of the potential about y = z there will be two isolated sets of coalescing roots at the rainbow I [7,8] denoted by Idi_ The coalescence of one of these sets (the one for which rj is Ies~ than K) is shown in fig. I for Aj equaI to 2.4.6 and 8_ First. note that as Aj increases the corresponding coalescence points occur at decreasing values of L This rest& implies that rhe range of V(R. y) effective in promoting transitions is a decreasing function of Aj, as expected. Second, note that the value of yi at which the roots coalesce is 1-I radians for all of the transitions_This is a direct consequence of the sud-
1 April 1979
den approximation we have empIoyed (and for the present example it must be approximately correct)_ It is now a simple matter to predict the scattering angle at which the rotational rainbow feature [due to the classical divergence caused by eq. (6)] should be manifested in the differential cross section for a given Aj transition. We simply read off the angle corresponding to Yj = 1.I radians and the t corresponding to the coalescence of roots (from fig. 1) for a Aj transition_ This is done by inspection of x(y = 1.I, z), plotted in fig 2- Qualitatively, we see that the rotational rainbow angle is an increasing function of Aj; this folIows since l,,., is a decreasing function of Aj, as already noted. Note also that the minimum in x(y = I-1, r) with respect to 2 occurs at a value of 1 which is always greater than ZAi- Thus, in the purely classical picture this rainbow will be unobservable a_ These qualitative conclusions are verified by inspection of the differential cross sections plotted in fig_ 3 for Aj= 2,4,6 and 8, calculated according to the quantaI sudden rotation (10s) approximation [9-I I] _ The maxima observed correspond very weI1 to those predicted from the ciassical rainbow
* In a uniform, i e., Airy, semichssicd analysis the probability for a nj transition decays smoothly to zero for I > lo,and it is possible that the rainbow in the x(7.0 deflection function will manifest itseif as a weak, additional rainbow feature- Classically. this probability is identically zero for I>Cpj-
L2
OS Aj-4 04 1 E 4a
I 52
56
60
Fig- I - CO&SCCXCLXof roots to the stationary phase equation [eq- (3)] as a function of orbital angular momentum. I. for z.%j trmsitions indicated.
310
Fig. 2. Scattering angle deflection function x(1.7) as a function of orbita anguku momentum, l, for 7 = 1.I radkms.
Volume
62.
number 2
CHEMICAL PHYSICS LETTERS
1 April 1979
of inelastic differential cross sections. (2) The scattering angle at which rotational rainbow features are prominent is an increasing function of Aj_ (3) The appearance, location, and sharpness of rotational rainbows are sensitive functions of the mzisotropy of the interaction potential. More extensive discussion of these conclusions as well as additional calculations will be presented in a later publication. I thank Ki Tung Lee for CSI+I~ computations.
out some of the
References [l ] R.B. Bernstein, Adun_ Chem- Pltys_ 10 (1966) 75.
analysis described above. The Aj = 2 differential cross
section shows two rainbow features, one at 0 =z 30” and the other at 0 * 1OO”_The first one corresponds to the classical rotational rainbow discussed above and the second one is a supernumerary rotational rainbow. Based on the above theory we conclude that for atom-molecule systems (1) Rotational rainbows should be general features
[2] WV. Ewes, U. Ross and J.P. Toennies, to be pubhshed. (31 L.D. Thomas, W.P. Kraemer, G.H.F. Diercksen and P. McGuire. Chem. Phys. 27 (1978) 237. [4] J-M. Bowman and K.-T. Lee, Chem. Phls. Letters 60 (1979) 212. [S] W.H. Miller, J_ Chem. Ph)s. 53 (1970) 3578. 161 C-W’.McCurdy and W.H. Miller, J. Chem. Phls. 67 (1977) 463. [7] R-A. Marcus. I. Chem. Phys. 54 (I 971) 3965. IS] WV-H. Miller, J. Chem. Phys. 54 (1971) 5386. [9] D. Secrest. J. Chem. Phys. 63 (1975) 710. [IO]R-T Rck. J. Chem. Phys. 60 (1974) 633. 111] J-M. Boaman and J. Arru&, Chem. Phys. Letters 41 (lS76) 43.
311
62. number 2
ROTATIONAL
CHEMICAL
RAINBOWS
IN INELASTIC
PHYSICS
1 April I979
-RS
ATOM-MOLECULE
DIFFERENTIAL
CROSS SECTIONS *
Joel M. BOWhMN s Department of Ciremistry, Illinois Institute of Technology, Chicago. Illitrois 60616. USA Received 18 November 1978;in fiil
form 19 December 1978
The existence of rotational rainbows in atom-rigid rotor inelastic differential cross sections is shown b> considerins the classical limit of the centrifugal sudden-rotational sudden expression for the scattering amplitude.
Rainbow features in the differential cross section in central field atom-atom scattering are well known [1] and well understood to be due to the long range attraction between atoms. There is at present much interest in the nature of rainbow features in inelastic differential cross sections [2,3]. in this letter we present some formal and computational results which shed light on the nature of these rainbows for rctationally inelastic processes_ The classical limit of the sudden rotation (10s) expression for the scattering amplitude for a transition il = 0 to& has been recently derived by us [4]: it is
a[2t7(1,7)lia7-j2(i.7)
exp [2iQ7)]
where y is the angle between the molecular asis and the scattering position vector, R. The phase shift 7~[(7) is computed for ffled values of this orientation angle. The complete stationary phase limit of (1) can be taken in the usual way to yield the following results. The stationary points satisfy the equations =x(E, 7) = 5 f? 7
0)
(Z$- ;)‘I? r;-, -jz (0) = -
(2#2
k , sin’&
x laiz/a7i,*'~lax/arl,*'~ esp[i9,(7i-li)]_
(4)
The subscript i indicates the ith root to eqs. (2) and (3)_ For the present purpose it is unnecessary to give the complete. more complicated stationary phase ekpression for i;-, _i’j2 (0) *_ inspection of this expressmn indicates that classical rainbows can occur when = 0,
(5)
a.dar=0 _
(6)
or
P,(cos 0)) (1)
a[2tl(z# 7)llar
-
3rd & += (0) is given by
aj2ia7 d7e-iV~-il)Y
=j2 -il
(2)
* Support from the &Conal Science Foundation is gratefully acknowledpd** Alfred P. Sloan Fellow.
The former rainbow corresponds to a rotational transition rninbow and the latter one is just the familiar rainbow in the scattering angle deflection function_ Rainbow phenomena in inelastic scattering amplitudes are known, especially from classical S-matrix theory [S] _The present expression is of interest since the rotational rainbow arises directly from the properties of the very familiar phase shift q(Z. -y) through the rotational deflection function (3). + The full two-dimensional stationary phase expression for fj,-.j., (0) contains a 2 X 2 determinant. and a summatmn over dl roots to eqs. (2) and (3).
309
Volume 62, number 2
CHEMICAL
PHYSICS LETTERS
Consider a model atom-rigid rotor system employed previously by McCurdy and Miller [6] and shown by us to one for which the dassical Iimit of the rotational sudden approximation [eq. (l)] is quantitativeIy valid [4] _ The interaction potential is
where
VL_J(R) =
l[(R,/R)‘*
- WZm/R)6J
-
R, = 6-67 au, E = 10-7 cm-I ~ the relative translational energy E IS 150 E, the rotationa constant B is 0.1 E, and the reduced mass is appropriate for He-CO. Let us focus now on locating the rotational rainbows for this system for Aj equal to 2,4.6 and 8_ These rainbows occur at vaIues of C for which there is a coaIescence of roots yi to eq_ (3X Due to the symmetry of the potential about y = z there will be two isolated sets of coalescing roots at the rainbow I [7,8] denoted by Idi_ The coalescence of one of these sets (the one for which rj is Ies~ than K) is shown in fig. I for Aj equaI to 2.4.6 and 8_ First. note that as Aj increases the corresponding coalescence points occur at decreasing values of L This rest& implies that rhe range of V(R. y) effective in promoting transitions is a decreasing function of Aj, as expected. Second, note that the value of yi at which the roots coalesce is 1-I radians for all of the transitions_This is a direct consequence of the sud-
1 April 1979
den approximation we have empIoyed (and for the present example it must be approximately correct)_ It is now a simple matter to predict the scattering angle at which the rotational rainbow feature [due to the classical divergence caused by eq. (6)] should be manifested in the differential cross section for a given Aj transition. We simply read off the angle corresponding to Yj = 1.I radians and the t corresponding to the coalescence of roots (from fig. 1) for a Aj transition_ This is done by inspection of x(y = 1.I, z), plotted in fig 2- Qualitatively, we see that the rotational rainbow angle is an increasing function of Aj; this folIows since l,,., is a decreasing function of Aj, as already noted. Note also that the minimum in x(y = I-1, r) with respect to 2 occurs at a value of 1 which is always greater than ZAi- Thus, in the purely classical picture this rainbow will be unobservable a_ These qualitative conclusions are verified by inspection of the differential cross sections plotted in fig_ 3 for Aj= 2,4,6 and 8, calculated according to the quantaI sudden rotation (10s) approximation [9-I I] _ The maxima observed correspond very weI1 to those predicted from the ciassical rainbow
* In a uniform, i e., Airy, semichssicd analysis the probability for a nj transition decays smoothly to zero for I > lo,and it is possible that the rainbow in the x(7.0 deflection function will manifest itseif as a weak, additional rainbow feature- Classically. this probability is identically zero for I>Cpj-
L2
OS Aj-4 04 1 E 4a
I 52
56
60
Fig- I - CO&SCCXCLXof roots to the stationary phase equation [eq- (3)] as a function of orbital angular momentum. I. for z.%j trmsitions indicated.
310
Fig. 2. Scattering angle deflection function x(1.7) as a function of orbita anguku momentum, l, for 7 = 1.I radkms.
Volume
62.
number 2
CHEMICAL PHYSICS LETTERS
1 April 1979
of inelastic differential cross sections. (2) The scattering angle at which rotational rainbow features are prominent is an increasing function of Aj_ (3) The appearance, location, and sharpness of rotational rainbows are sensitive functions of the mzisotropy of the interaction potential. More extensive discussion of these conclusions as well as additional calculations will be presented in a later publication. I thank Ki Tung Lee for CSI+I~ computations.
out some of the
References [l ] R.B. Bernstein, Adun_ Chem- Pltys_ 10 (1966) 75.
analysis described above. The Aj = 2 differential cross
section shows two rainbow features, one at 0 =z 30” and the other at 0 * 1OO”_The first one corresponds to the classical rotational rainbow discussed above and the second one is a supernumerary rotational rainbow. Based on the above theory we conclude that for atom-molecule systems (1) Rotational rainbows should be general features
[2] WV. Ewes, U. Ross and J.P. Toennies, to be pubhshed. (31 L.D. Thomas, W.P. Kraemer, G.H.F. Diercksen and P. McGuire. Chem. Phys. 27 (1978) 237. [4] J-M. Bowman and K.-T. Lee, Chem. Phls. Letters 60 (1979) 212. [S] W.H. Miller, J_ Chem. Ph)s. 53 (1970) 3578. 161 C-W’.McCurdy and W.H. Miller, J. Chem. Phls. 67 (1977) 463. [7] R-A. Marcus. I. Chem. Phys. 54 (I 971) 3965. IS] WV-H. Miller, J. Chem. Phys. 54 (1971) 5386. [9] D. Secrest. J. Chem. Phys. 63 (1975) 710. [IO]R-T Rck. J. Chem. Phys. 60 (1974) 633. 111] J-M. Boaman and J. Arru&, Chem. Phys. Letters 41 (lS76) 43.
311