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4.7. Inelastic scattering of a particle by a target atom
one example, a massive incident particle transfers energy to cause
excitation or ionization of the electrons of a target atom. In a second example, the passage of a fast charged particle causes instantaneous polarization of a dielectric medium. The central problem
is to calculate the scattering cross section in each case.
We assume an incident particle of mass m and charge ze, where z is
an integer. The target is a single atom with atomic number Z. In a
single collision the incident particle transfers a small fraction of its
energy to the target atom, initially in its ground state. As a result,
the atom is excited to a higher energy state. In principle this can
include ionization of the atom. The scattered particle then exits
to a final free-particle state with reduced energy and momentum.
The following analysis closely follows the classic paper of Bethe
[5], which is based on the same perturbation-theoretical approach
described above. The reader is referred to Egerton [24], who places
the material in the context of the considerable body of subsequent
work by others.
According to the foregoing analysis, calculation of the differential
scattering cross section σ is reduced to finding the appropriate
matrix element i|H ˆ 1 |j between the initial and final states of the
system. In this case, the system consists of a scattering particle
and a single target atom. The origin of coordinates coincides with
the nucleus of the target atom. The instantaneous position of the
scattering particle we denote by x, and the instantaneous positions
of the Z atomic electrons we denote by (x 1 , . . . , x Z ).
The initial and final eigenfunctions, respectively, can be represented by
u 0 (x; x 1 , . . . x Z ) = √
1 e
ik 0 ·x U 0 (x 1 , . . . , x Z )
V
1 ik·x U n
u n (x; x 1 , . . . x Z ) = √ e
(x 1 , . . . , x Z ), (4.154)
V
where the subscripts 0 and n refer to the ground state and the nth
excited states of the atom, respectively. The quantities U 0 and U n
4.7. Inelastic scattering of a particle by a target atom
one example, a massive incident particle transfers energy to cause
excitation or ionization of the electrons of a target atom. In a second example, the passage of a fast charged particle causes instantaneous polarization of a dielectric medium. The central problem
is to calculate the scattering cross section in each case.
We assume an incident particle of mass m and charge ze, where z is
an integer. The target is a single atom with atomic number Z. In a
single collision the incident particle transfers a small fraction of its
energy to the target atom, initially in its ground state. As a result,
the atom is excited to a higher energy state. In principle this can
include ionization of the atom. The scattered particle then exits
to a final free-particle state with reduced energy and momentum.
The following analysis closely follows the classic paper of Bethe
[5], which is based on the same perturbation-theoretical approach
described above. The reader is referred to Egerton [24], who places
the material in the context of the considerable body of subsequent
work by others.
According to the foregoing analysis, calculation of the differential
scattering cross section σ is reduced to finding the appropriate
matrix element i|H ˆ 1 |j between the initial and final states of the
system. In this case, the system consists of a scattering particle
and a single target atom. The origin of coordinates coincides with
the nucleus of the target atom. The instantaneous position of the
scattering particle we denote by x, and the instantaneous positions
of the Z atomic electrons we denote by (x 1 , . . . , x Z ).
The initial and final eigenfunctions, respectively, can be represented by
u 0 (x; x 1 , . . . x Z ) = √
1 e
ik 0 ·x U 0 (x 1 , . . . , x Z )
V
1 ik·x U n
u n (x; x 1 , . . . x Z ) = √ e
(x 1 , . . . , x Z ), (4.154)
V
where the subscripts 0 and n refer to the ground state and the nth
excited states of the atom, respectively. The quantities U 0 and U n