107
2.6. Stochastic Coulomb scattering
initial condition s 0 . Separately, the separation s
� in the absence of
interaction is also determined analytically from s 0 . From (2.286)
the trajectory displacement f is thus found in terms of s 0 . This
permits us to perform the integral (2.291). Substituting this into
(2.283), we obtain the general solution for the stochastic Coulomb
interaction, in the vector superposition of two-body interactions.
We are often interested in only the transverse coordinates in some
target plane, or alternatively, the broadening of kinetic energy, for
example. In these cases, the other degrees of freedom, such as the
displacement in the axial coordinate, are superfluous. We need
to integrate over all of the superfluous degrees of freedom. Fortunately, the form of (2.287) makes this particularly simple. Due
to a theorem of Fourier transforms, setting the frequency k equal
to zero is equivalent to integrating over the entire range of the
variable in direct space. By setting the superfluous components of
k equal to zero, we automatically integrate over these degrees of
freedom in the direct space of f. This leaves only those degrees of
freedom we are interested in. In particular, this applies to (2.291),
where the superfluous degrees of freedom integrate to unity.
The general solution for the trajectory displacement is given by
(2.283, 2.291). In general, the integral in (2.291) must be performed numerically. A special case exists for which a closed-form
analytic solution exists, however. This is the case in which all particles are initially at rest in the rest frame of the beam particles.
This is equivalent to an initially monoenergetic beam with zero energy in the rest frame. It follows that the beam is monoenergetic
in the lab frame as well. It is simpler to perform the calculation
in the rest frame, as the magnetic Lorentz force is zero, and the
choice of reference frame does not affect the transverse position.
The initial condition for the particle separation in six dimensions
is
P 0 (s 0 ) = ψ(r 0 ) · δ(p 0 ),
(2.292)
where r 0 is the initial particle spatial separation, and p 0 is initial
momentum difference. The spatial separation distribution ψ(r 0 )
will be determined later. Integrating over all momenta by setting
2.6. Stochastic Coulomb scattering
initial condition s 0 . Separately, the separation s
� in the absence of
interaction is also determined analytically from s 0 . From (2.286)
the trajectory displacement f is thus found in terms of s 0 . This
permits us to perform the integral (2.291). Substituting this into
(2.283), we obtain the general solution for the stochastic Coulomb
interaction, in the vector superposition of two-body interactions.
We are often interested in only the transverse coordinates in some
target plane, or alternatively, the broadening of kinetic energy, for
example. In these cases, the other degrees of freedom, such as the
displacement in the axial coordinate, are superfluous. We need
to integrate over all of the superfluous degrees of freedom. Fortunately, the form of (2.287) makes this particularly simple. Due
to a theorem of Fourier transforms, setting the frequency k equal
to zero is equivalent to integrating over the entire range of the
variable in direct space. By setting the superfluous components of
k equal to zero, we automatically integrate over these degrees of
freedom in the direct space of f. This leaves only those degrees of
freedom we are interested in. In particular, this applies to (2.291),
where the superfluous degrees of freedom integrate to unity.
The general solution for the trajectory displacement is given by
(2.283, 2.291). In general, the integral in (2.291) must be performed numerically. A special case exists for which a closed-form
analytic solution exists, however. This is the case in which all particles are initially at rest in the rest frame of the beam particles.
This is equivalent to an initially monoenergetic beam with zero energy in the rest frame. It follows that the beam is monoenergetic
in the lab frame as well. It is simpler to perform the calculation
in the rest frame, as the magnetic Lorentz force is zero, and the
choice of reference frame does not affect the transverse position.
The initial condition for the particle separation in six dimensions
is
P 0 (s 0 ) = ψ(r 0 ) · δ(p 0 ),
(2.292)
where r 0 is the initial particle spatial separation, and p 0 is initial
momentum difference. The spatial separation distribution ψ(r 0 )
will be determined later. Integrating over all momenta by setting