Biomedical Engineering Reference
In-Depth Information
Fig. 9.8
Normalized energy-loss spectrum for scattering of
neutrons of energy
E
n
from hydrogen.
P
(
Q
)d
Q
is the
probability that the energy loss is between
Q
and
Q
+d
Q
.
Isotropic scattering in the center-of-mass system leads to the
flat spectrum.
2
E
n
cos
θ
sin
θ
.
3)
Equation (9.5)
implies that
d
Q
/d
θ
=
Therefore,
the neutron
energy-loss spectrum (9.10) becomes, simply,
2
sin
θ
cos
θ
d
Q
1
E
n
d
Q
.
P
(
Q
)d
Q
=
=
(9.11)
2
E
n
cos
θ
sin
θ
The (normalized) spectrum for the scattering of neutrons of energy
E
n
by pro-
tons is shown in Fig. 9.8. Because the spectrum is flat and the maximum energy
loss is
Q
max
=
E
n
, the probability that a neutron loses an amount of energy
Q
is
simply the fraction
Q
/
E
n
(independently of where
Q
is located in Fig. 9.8). The
1
2
E
n
)
. This relationship, mentioned earlier
as a rule of thumb for neutron scattering, is exact for isotropic scattering in the
center-of-mass system. The energy-loss spectrum is also flat for isotropic center-of-
mass scattering when the masses are unequal, and
Q
avg
=
1
2
Q
max
(
average energy loss is
Q
avg
=
=
1
2
Q
max
,with
Q
max
given
by Eq. (9.3). The last relation is approximately valid for neutron elastic scattering
by C, N, and O in tissue.
Example
A 2.6-MeV neutron has a collision with hydrogen. (a) What is the probability that it
loses between 0.63 and 0.75 MeV? (b) If the neutron loses 0.75 MeV, at what angle
is it scattered? (c) What is the average energy lost by 2.6-MeV neutrons in collisions
with carbon? (d) In (b), how much energy does the neutron lose in the center-of-mass
system?
Solution
(a) In Fig. 9.8, with
E
n
=
0.12 MeV, it follows that
the probability of a neutron energy loss in the specified interval is
Q
/
E
n
= 0.0462.
2.6 MeV and
Q
=
0.75 - 0.63
=
3 We ignore the negative sign from
differentiating
cos
decreases as
increases. We write all
probabilities as positive definite.
θ
θ
, which indicates that
Q
