Supplemental material for
Electron Compton-Like Quasi-Elastic Scattering from H2, D2 and HD
R. A. Bonham(*)
Dept. Biological, Chemical, And Physical Sciences
Illinois Institute of Technology
3101 South Dearborn Street, Chicago, Il 60616 USA
G. Cooper and A. P. Hitchcock
Brockhouse Institute for Materials Research
McMaster University, Hamilton, ON, Canada L8S 4M1
ElectronComptonLikeQuasi-supplement.doc Last changed: 23 Jan 2008 (aph)
This document contains further information relevant to the subject of quasi-elastic scattering in
molecules, but which is not central to the main thrust of the paper.
Contents:
1. Sum rules for quasi-elastic scattering in diatomic molecules
2. Evaluation of the nuclear-nuclear interference scattering form factor
3. Translational Averaging – case of molecular beams
4. Description of a Mathematica script used to calculate the quasi-elastic electron scattering cross
sections as a function of energy loss. (the actual script is included as a separate Supplemental
file)
1. Sum rules for quasi-elastic scattering in diatomic molecules
The energy loss of the center of gravity, E cg , of the quasi-elastic peak is given by
N
Z q2
Sel 0,K 
2
E cg 
K 
Sel 1,K 
q1 2M q
N N

 N 2
TRV
TRV
/ Z q  2  Z q Z p on
cos[K  Rqp ] on

q1
 q1 pq





(S-1)
which simplifies for homo-nuclear diatomic molecules if the small inter-nuclear interference

term in the denominator is neglected as
E cg 

K2
2M1
where M1 is the mass of one of the two nuclei. If the two nuclei have the same charge but different
masses, then
E cg 

K2
K2
1  K 2
K 2  K 2

 


4 M1 4 M 2 2 2M1 2M 2  4
(S-2)
where  is the reduced mass. If the two peaks expected in the energy loss spectrum are well
 each term can be identified with the E cg of each separate peak (i.e. E cg 1 
separated then

K2
)
2M1
since S 0,K  is a linear function of the energy loss and thus must be the average of the E cg of
each separate peak. Note also that the twopeak areas are identical if Z1 Z2 .


2. Evaluation of the nuclear-nuclear interference scattering form factor

The matrix element that arises in the S(-1,K) sum rule can be easily evaluated provided that the
Born-Oppenheimer approximation is valid which allows the ground state wave function to be
written as the product of translational, rotational, and vibration wave functions. Because the
scattering operator in this case does not depend on the position of the center of mass the
translational average reduces to one. Since the sum over the quantum number M, specifying the
spatial degeneracy, of the product of spherical harmonic wave functions is independent of all
orientation angles (Brink and Satchler, page 146, 2nd Eq. from top of page, [1] ) the rotational
average reduces to
o e
iK R pq
1  J  kTe J (J 1) M
sin[ KR(Q)]
e
YJ ,  V e iKR cos V YJM ,    V Q
V Q


N J  0 M J
KR(Q)
B
o 
where N is a normalization constant and

R(Q)  Re 
1
 e
Q
with the vibrational wave function for harmonic motion written as

V (Q) 
1
2v v! 
H v Qe

Q2
2
.
The remaining vibration average can be expressed as


V Q


 e (v 1/ 2)
sin[ KR(Q)]
1
e kT
V Q   v
KR(Q)
N v 0 2 v! 

 dQH Qe
2
v

2
Q 2
sin[ KR(Q)]
KR(Q)
where the sum over all vibration states can be written as [2] :


 e (v 1/ 2)
1
e
H v2 QeQ 

v
N v 0 2 v! 
kT
2
tanh(
e
e
) 2
2kT eQ tanh( 2kT ).

Using this result and expanding the sin term in the operator as


sin[ K(Re  Q)] sin[ KRe ]cos[KQ]  cos[KRe ]sin[ KQ]  Q
 
 ...
1
K(Re  Q)
KRe

 Re

1
with  

1



 dye
the integrations can be carried out analytically by use of the result
 e
2
y 2

K 


cos[K y]  e 

2
where

2
1

.
  tanh[  e ]
e
2kT

The latter term is associated with the root mean square amplitude of vibration, , defined as

1
2 e tanh[
e
2kT
]

which for hydrogen has the value 0.12 a.u.. The final result can be written as

o e
iKR pq
o 
sin[ K(Re 
KRe
2
Re
)]
e

K2
2
2
which is a well known result used in molecular structure determination by gas phase electron

diffraction.
3
3. Translational Averaging – case of molecular beams
Note that the broadening of the vibrational and rotational spectral peaks due to translational
motion depends on how the experiment is carried out. If a perfectly collimated molecular beam
were employed as the target and if the plane determined by the incident electron beam and the
entrance slit of the energy analyzer, a plane containing the momentum transfer K, was
perpendicular to it, then the result of the translational averaging would be simply  D since
K  P  0 . In this case the ro-vibrational peaks would be shifted to higher energy loss by the
same amount as before, K 2 / 2 M t , but the peak widths due to translational
 and thermal motion

would be reduced to their natural spectroscopic line widths.
4. Mathematica script to calculate quasi-elastic scattering electron cross sections as a
function of energy loss.
The constants of the molecule of interest are set at the beginning of the script (the program
comments describe them in detail), while the rest of the program area (the Mathematica “cells”)
contains the calculations and results. Mass, rotational and vibrational constants for the molecule
and several fundamental constants are required to be set before running the program. For the
calculational results reported in the present work,  =1, jmax is set to 30 and all the ro-vibrational
interaction and centrifugal distortion constants are set to zero, in accord with the theory outlined
in the text of the paper.
The program is a second Supplemental Material file, distributed with permission from Wolfram.
Note that a Mathematica license is required to execute the program. Updates and alternate
versions can be obtained by emailing rabonham2000@yahoo.com
References
1. D. M. Brink and G. R. Satchler, Angular Momentum, (Clarendon Press, Oxford, 1968)
2. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol 1 (McGraw-Hill Ryerson,
1953) p. 786. Note that the + sign in the square root in the denominator should be a minus
sign.
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