PHYSICS 430L
NUCLEAR PHYSICS LABORATORY
Measurement of Neutron Flux
READ:
REPORT:
4.A
Krane
Chapter 12:
Sections: 1-5
Price, W.J.,
Chapter 9:
Section: 9.18 (Most important reference!)
Segre
Chapter 3
(Radiation Detectors)
Krane
Chapter 7
Sections: 2-3 (Radiation detectors)
These exercises are due at 5:00 PM on the day preceding the laboratory class.
The Maxwell-Boltzmann velocity distribution function for the random motion of a "gas" of
neutrons is written as:
3
mv 2
dn
2  m  2  2kT
.
n(v) 
 no 4 v
e
dv
2kT 
This function is normalized such that
 dn

0 dv

dv   n(v)dv  no .
0
In this equation, n(v) is the number of neutrons at velocity v per unit velocity, m is the neutron
mass, k the Boltzmann constant, and T the absolute equilibrium temperature of the neutron "gas"
in Kelvin. The multiplier no is the total number of neutrons of all speeds per unit volume. This
distribution is characterized by a variety of "measures". Beginning with the distribution function
n(v) answer the following questions.
(a) Show that the most probable neutron speed, vp is
vp =
2kT
m .
(b) Show that the mean neutron speed, <v>is
<v> =
8kT
.
m
(c) Show that the mean square speed <v2> is
Staple
2
3kT
<v2> = m
Staple
3
(d) Show that the following ratio is independent of the equilibrium temperature of the neutron
gas:
<v>
vp = 1.128 .
(e) Show that the kinetic energy, E, of a neutron travelling at the most probable speed is
1
E(vp) = 2 mvp2 = kT .
(f) Show that the corresponding distribution function for the kinetic energies of the neutrons at a
temperature T is given by
dn
E
n(E) = dE = 2no
3
kT
E


e kT .
(g) Use the distribution function in (f) to show that the most probable kinetic energy is
1
Ep = 2 kT .
(h) Use the distribution function in (f) to show that the mean kinetic energy is
3
<E> = 2 kT .
Show also that this value of <E> is just the result you would obtain by expressing
1
<E> = 2 m<v2>.
(i) Numerically evaluate the following quantities for neutrons at 20 o C:
vp, <v>,
(j)
<v2> , E(vp), E(<v>), Ep, <E>
If thermal neutron absorption cross sections are proportional to 1/v, show that if the cross
sections are known for neutrons in equilibrium at 20o C at velocities vp, <v>, and <v2> ,
that the corresponding cross sections at these velocities at another temperature T is found as
(T) = (20oC)
293K
.
T
Staple
4
SYNOPSIS:
Our focus this week will be on a means of measuring a neutron radiative capture cross
sections (n + AN  A+1N + ), where N is any chemical species. This will have direct bearing on our
laboratory work and will be illustrative of the significance of cross sections, neutron fluxes, absolute
measurements, and much more nuclear physics. The beauty of this topic is that we will really be a good
approximation to the actual work done by a nuclear physicist in designing an experiment, carrying out
the measurements, and doing a careful and complete analysis.
Neutrons interact only by the short range nuclear force. Therefore, as neutrons move about in an
environment, they experience interactions only when they essentially come in contact with another
strongly interacting object, e.g., another nucleon or nucleus. There are no long range forces. Moreover,
the neutron either gets absorbed (disappears from the reaction) or collides elastically with these objects.
(Quasi-elastic collisions are also possible with neutrons of some energy wherein the struck nucleus is left
in an excited state.) If the neutrons are formed at higher energies, subsequent elastic collisions reduce
their energies to equilibrium values. This collection of neutrons in thermal equilibrium appears to be an
"ideal gas" (satisfying all the criteria for an "ideal gas") and their energy distributions can be treated as
such. This means that the neutrons will have an energy (or velocity) distribution which does not change
in shape (i.e., they are in equilibrium). On collision with each other or with nuclei, these neutrons may
lose or gain energy, but on average the distribution does not change shape. These neutrons are called
thermal neutrons, i.e., neutrons whose energy distribution is characterized by the same distribution
function as a gas of molecules in thermal equilibrium as described above. The only parameter which will
cause the distribution to change is the temperature, T, of the gas, i.e., the temperature of the medium.
Neutrons of thermal energies are especially important and interesting for studying a whole class of low
energy radiative capture reactions.
In the radiative capture process, a neutron has essentially zero kinetic energy. Since it does not need to
surmount the Coulomb barrier, it can become absorbed on a nucleus at zero kinetic energy. When the
free neutron becomes bound to the struck nucleus, the composite nucleus releases its binding energy (the
negative of the neutron separation energy) by emitting a photon (gamma ray). This is understood in
terms of the composite nucleus moving from a state of excitation (too much mass energy) to a lower
energy state, often, but not always, the ground state. This photon is indication that the reaction has taken
place. This reaction, called a radiative capture reaction, is noted as (n,).
The likelihood that such a reaction might occur on a given nucleus is characterized in terms of the
microscopic cross section . For low energy (thermal) neutrons, this cross section varies as 1/v, where v is
the speed of the neutron. After production, the composite nucleus may drop to the ground state or it may
drop to one of its (characteristic) excited states. We characterize the likelihood of producing the different
states in the composite nucleus with different cross sections. These will be important for our studies.
We will be using Lederer's book, as well as online sources as references.