105
2.6. Stochastic Coulomb scattering
use the most general possible approach, namely, calculating the
trajectory displacement in six-dimensional phase space. The following analysis closely follows [41], with a few minor changes in
notation. We define the following quantities:
χ 0 = initial coordinate of an individual particle at time zero
s 0 = initial separation of two particles at time zero
s = separation of two particles at time t, interaction present
s
� = separation of two particles at time t, interaction absent,
(2.285)
where all quantities are six-vectors in phase space. The first three
components are position, and last three components are momentum. All quantities are random variables, described by probability
density functions.
We define an individual particle trajectory displacement as the
difference of two-particle separations s with and s
� without interaction as follows:
f = 2
1 (s − s
� ),
(2.286)
where the factor of
1
2
appears, because the individual particle displacement is half the change in particle-particle separation for particles of equal mass. Consistent with the above hypothesis, we apply the Markov formalism, identifying f with a single Markovian
step (2.282). The required Fourier transform for the distribution
of trajectory displacements f is
τ ˜(k) = d
6 f exp(−ik · f) τ (f).
(2.287)
We define a probability density σ 0 (χ 0 ) of an initial single-particle
six-coordinate χ 0 . For example, the beam might be of uniform
spatial density, and monoenergetic. In this case the initial distribution σ 0 (χ 0 ) is a constant spatially, multiplied by a delta function
in momentum, within the beam volume, and zero outside the beam
volume.
2.6. Stochastic Coulomb scattering
use the most general possible approach, namely, calculating the
trajectory displacement in six-dimensional phase space. The following analysis closely follows [41], with a few minor changes in
notation. We define the following quantities:
χ 0 = initial coordinate of an individual particle at time zero
s 0 = initial separation of two particles at time zero
s = separation of two particles at time t, interaction present
s
� = separation of two particles at time t, interaction absent,
(2.285)
where all quantities are six-vectors in phase space. The first three
components are position, and last three components are momentum. All quantities are random variables, described by probability
density functions.
We define an individual particle trajectory displacement as the
difference of two-particle separations s with and s
� without interaction as follows:
f = 2
1 (s − s
� ),
(2.286)
where the factor of
1
2
appears, because the individual particle displacement is half the change in particle-particle separation for particles of equal mass. Consistent with the above hypothesis, we apply the Markov formalism, identifying f with a single Markovian
step (2.282). The required Fourier transform for the distribution
of trajectory displacements f is
τ ˜(k) = d
6 f exp(−ik · f) τ (f).
(2.287)
We define a probability density σ 0 (χ 0 ) of an initial single-particle
six-coordinate χ 0 . For example, the beam might be of uniform
spatial density, and monoenergetic. In this case the initial distribution σ 0 (χ 0 ) is a constant spatially, multiplied by a delta function
in momentum, within the beam volume, and zero outside the beam
volume.