Biomedical Engineering Reference
In-Depth Information
Fig. 9.6
Schematic representation of elastic scattering of a
neutron by a proton. The initial neutron energy
E
n
is given in
terms of the proton recoil energy
Q
and angle
θ
by Eq. (9.5).
9.6
Neutron-Proton Scattering Energy-Loss Spectrum
Like Eq. (8.19) for Compton scattering, Eq. (9.5) for neutron-proton scattering is
purely kinematic in nature, reflecting (nonrelativistically) the conservation of ki-
netic energy and momentum. It provides no information about the probability that
the neutron is scattered in the direction
θ
. Experimentally, for neutron energies up
to about 10 MeV, it is observed that neutron-proton scattering is isotropic in the
center-of-mass coordinate system. That is, the neutron (as well as the proton) is
scattered with equal likelihood in any direction in three dimensions in this coordi-
nate system. One can translate this experimental finding into the probability den-
sity for having a proton recoil at an angle
θ
as seen in the laboratory system. Equa-
tion (9.5) can then be used to compute the probability density
P
(
Q
)
for
Q
,which
is then the neutron energy-loss spectrum. As we now show, isotropic scattering in
the center-of-mass system results in a flat energy-loss spectrum for neutron-proton
scattering in the laboratory system.
The collision is represented in the laboratory system by Fig. 9.5. In (a), with the
two masses equal
(
M
=
m
)
, the center of mass is located midway between the neu-
tron and proton and moves toward the right with constant speed
2
V
.Thecenterof
mass crosses the collision point at the instant of collision and continues moving to-
ward the right with speed
2
V
thereafter. (Its motion is unchanged by interaction of
the particles.) Figure 9.7(a) shows the locations of the collision point
O
, the center
of mass
C
, the neutron
N
and proton
P
at unit time after the collision. At this time,
the scattered neutron and proton have displacements equal numerically to
V
and
v
relative to
O
. (We use the same symbols in this discussion as in Figs. 9.5 and 9.6.)
Also, the center of mass
C
bisects the line
NP
at a displacement
1
2
V
from
O
.The
scattering angle of the proton is
θ
in the laboratory system and
ω
in the center-of-
mass system. Before collision, the neutron and proton approach each other from
opposite directions with speeds
2
V
in the center-of-mass system. Since momentum
is conserved, the particles are scattered “back-to-back” with equal speeds. Because
no kinetic energy is lost, the speeds of the neutron and proton in the center-of-mass
