Today, I will present the first of two lectures on neutron interactions.
I first need to acknowledge that these two lectures were based on lectures presented
previously in Med Phys I by Dr Howell.
1
Before we start, here are some good references on neutrons. Please note that you
will be using the Knoll book in the Radiation Detection course.
2
We know that at present the majority of radiation therapy is delivered using highenergy photons or electrons. In addition, in an increasing number of radiation
centers, protons are being used. So, why do we, as medical physicists, care about
neutron interactions?
There are two important scenarios where we encounter neutrons in radiation
therapy.
First, neutrons themselves can be used for radiation therapy, and there are, in fact,
several dedicated neutron radiotherapy centers in the US.
Second, neutrons can be a source of contamination dose in both high-energy photon
therapy and in proton therapy. This contamination dose is relevant for patients,
personnel, and this is an important component in shielding design.
I will come back to both of these topics in the next lecture, but today we will focus
on neutron interactions, in general.
3
In today’s lecture, I will first discuss general neutron properties and reaction cross
sections. Then, the remainder of the lecture will focus on specific neutron
interactions. This will provide context for the next lecture.
4
Two properties of neutrons are very important to keep in mind.
First of all, neutrons are uncharged particles. Unlike electrons, protons, or other
charged particles, they do not interact via long-range coulombic forces. Rather,
they interact via short-range nuclear forces. Much like photons, they can penetrate
several centimeters of target material without interacting.
The second property is that their interactions are with the nuclei of absorbing
material. They do not interact with orbital electrons. Consequently, neutrons are
considered to be indirectly ionizing radiation.
5
Reaction cross sections are an important concept that you will repeatedly encounter
in the neutron literature because reaction cross sections are used to describe neutron
interaction probabilities. The concept of reaction cross section is analogous to
linear attenuation coefficient, which we use to describe photon interaction
probabilities.
A neutron reaction cross section quantitatively describes the probability of a
particular interaction occurring between a neutron and target material.
6
When the reaction cross section is defined microscopically on a nucleus, it is
denoted by the Greek letter σ and has units of area.
The SI unit for reaction cross section is cm2. However, neutron reaction cross
sections are commonly reported in barns, which is 10-24 cm2.
Reaction cross sections are BOTH energy and interaction type dependent and thus
can be tabulated as a function of energy and interaction type.
7
This graph is an example of a reaction cross section for a specific type of neutron
interaction. In this case, we have an (n,α) reaction between an incident neutron and
a Li-6 nucleus. In this figure, the reaction cross section is shown on the y-axis and
energy is on the x-axis. Note that both axes are logarithmic.
You can clearly see that the probability of this particular interaction is highly
dependent on energy.
8
Reaction cross sections can also be defined macroscopically, as a probability per
unit path length.
The macroscopic cross section is denoted by the capital letter Σ, and is simply
determined by taking the product of the microscopic cross section and the number
of nuclei per unit volume. The units of macroscopic cross section are units of
reciprocal length, continuing the analogy with linear attenuation coefficient.
9
If we have a variety of neutron interactions, then the probability of any type of
neutron interaction can be calculated by taking the sum of the cross sections for all
possible types of neutron interactions.
10
Neutrons are neutral and thus, like photons, undergo exponential attenuation.
In a well-collimated beam, the neutrons that pass through an absorbing material
without interacting can be determined according to the exponential attenuation
equation.
11
Here is an example of a calculation using exponential attenuation. You should be
able to perform such calculations.
In an experiment designed to measure the total cross section of lead for 10 MeV
neutrons, it was found that a 1-cm thick lead absorber attenuated the neutron flux to
84.5% of its initial value. The atomic weight of lead is 207.21, and its density is
11.3 g cm-3. Calculate the total cross section from these data.
12
The solution to this type of problem is relatively straightforward.
We rearrange the attenuation equation and solve for σ.
I0/I is the reciprocal of the attenuation, or 1/0.845, or 1.18.
We need to calculate the atomic density of lead. This is the mass density, 11.3 g cmmultiplied by Avogadro’s number, the number of atoms per mole, divided by the
atomic weight. This gives us an atomic density of 3.29 × 1022 atoms cm-3.
3,
Divide the logarithm of 1.18 by the atomic density, and we obtain a cross section of
5.1 × 10-24 cm2, or 5.1 barn.
13
Another important quantity that is related to reaction cross sections is the mean free
path. This term is a measure of the amount of absorbing material through which
neutrons of a specified energy will travel before interacting. The mean free path is
given by 1 over the (macroscopic) reaction cross section. The macroscopic cross
section is the microscopic cross section multiplied by the atomic density. Thus, a
large reaction cross section corresponds to a small mean free path and, conversely, a
small reaction cross section corresponds to a large mean free path.
In general, slow neutrons have small mean free paths (typically, 1 cm or less)
whereas fast neutrons have large mean free paths (on the order of tens of cm). This
tells us that in general, low-energy neutrons have larger reaction cross sections than
do high-energy neutrons, and consequently, low-energy neutrons have a greater
probability of interaction than do high-energy neutrons.
14
Here is an example of a calculation of mean free path.
In the previous example, the microscopic cross section is 5.1 × 10-24 cm2
and the atomic density is 3.29 × 1022 atoms cm-3, so the mean free path is 1 over
the product of the two quantities, or 0.168 cm.
15
The concepts of the compound nucleus and resonance are frequently encountered
when describing neutron interactions and reaction cross sections.
So, before we proceed with the remainder of the lecture, I would like to briefly
discuss these two topics.
16
Let’s look at the compound nucleus model of an interaction. In a neutron
interaction that follows the compound nucleus model, the incident neutron and
target nucleus fuse together to form a combined system. Then, by successive
nucleon-nucleon collisions within the combined system, the reaction energy
becomes shared among many nucleons.
17
Eventually an equilibrium occurs and the compound nucleus exists in an excited
state. This excited state can exist for on the order of 10-16 – 10-18 seconds.
Excitation is then followed by de-excitation when a single nucleon or group of
nucleons acquires enough energy to escape.
18
An important aspect of the compound nucleus model is that energy and nature of
outgoing particles is determined by properties of the excited compound nucleus and
not by the properties of the colliding particles from which the compound nucleus
was formed.
19
There are many ways in which the compound nucleus can undergo de-excitation.
For example charged particles, neutrons, or gamma rays may be emitted.
However, if the excitation energy is close to the threshold energy for neutron
absorption, the compound nucleus will decay by emitting only γ-rays or by the
competing process of internal conversion of electrons.
We will revisit this topic when we discuss inelastic and non-elastic interactions
because it plays a key role in determining whether a neutron will undergo inelastic
or non-elastic interactions.
20
This is the same example of a reaction cross section that I showed you earlier in the
lecture. However, now, I would like to draw your attention to the peak in the
interaction probability that occurs at approximately 250 keV. This is referred to as a
resonance peak.
In this example, when the incident neutron fuses with the Li-6 nucleus, an excited
Li-7 compound nucleus is formed. The energy of the excited compound nucleus
happens to correspond exactly to a natural excited state (or frequency) of Li-7. The
probability of forming this particular excited compound nucleus dramatically
increases when the energy of the incident neutron is approximately 250 keV and
thus explains the increase in the reaction cross section at or near this energy.
21
This is an example of the neutron reaction cross section for Pu-239. Notice that as
the energy of the incident neutron increases, we observe many large resonance
peaks.
This general pattern in reaction cross sections is frequently observed when high
energy neutrons interact with nuclei of high mass.
22
Because neutron interaction probabilities are energy-dependent, we typically
categorize interactions into a set of specific energy classifications.
Unfortunately, the specific energy ranges are not uniquely defined; the energy
ranges within each classification are somewhat dependent on what reference we use.
Thus, rather than memorizing the exact energy ranges in each energy classification,
I recommend that you use these general classifications to speculate on the
likelihood that a particular interaction type will occur.
23
The National Council on Radiation Protection Report 38 defines three neutron
energy classifications.
The lowest energy classification is that of thermal neutrons. These neutrons are in
thermal equilibrium with the medium that they are in. The average energy of
thermal neutrons is typically below 1 eV, depending on temperature. The most
probable velocity for thermal neutrons is 2200 meters per second at 20.44oC. This
velocity corresponds to an energy of 0.0253 eV. Consequently, we typically use an
energy of around 0.025 eV to characterize thermal neutrons.
24
The next category are the intermediate energy neutrons with energies in the range of
above 1 eV to tens of keV.
And, finally, fast neutrons are those neutrons with energies above those of
intermediate neutrons.
25
As I mentioned, the classification of neutrons by energy is somewhat dependent on
the reference text. Some sources may include an epithermal category, whereas other
sources only include fast and slow (thermal).
Notice that the break point between defining a neutron as thermal or epithermal is
0.5 eV. This value is derived from radiation measurement techniques in which
neutrons above and below 0.5 eV can be distinguished by repeating the
measurements with and without a cadmium cover over the thermal neutron detector.
We will revisit this topic in detail next semester in Radiation Detection.
26
The remainder of the lecture will focus on specific neutron interactions. As we
discuss each interaction type, keep in mind that the probability of a particular
neutron interaction type in a particular absorbing material is highly dependent on
the energy of the incident neutron.
27
This all-to-busy slide provides a broad overview of the various different types of
neutron interactions that can occur. The terminology used in the neutron literature is
often inconsistent and many of the interaction types are referred to by many
different names. On this slide, I have attempted to delineate the most common
nomenclature as well as some of the “alternate” nomenclatures.
While there appear to be numerous possible interactions, all of these interactions
can essentially be divided into two categories, scatter and absorption.
Scatter reactions can be further subdivided into elastic or inelastic processes.
Absorption reactions can be subdivided according to the type of radiation that is
emitted following the absorption.
28
Neutron scatter can be broadly defined as an interaction in which a neutron interacts
with a target nucleus and changes speed and/or direction, but leaves the nucleus of
the absorbing material with the same number of protons and neutrons as it had
before the interaction.
We will look at the differences between elastic and inelastic scatter in just a
moment.
29
Neutron absorption can be broadly defined as an interaction in which a neutron is
absorbed by a target nucleus. A wide range of radiations can be emitted following
an absorption reaction including gamma rays, charged particles, neutral particles,
and fission fragments. The nucleus of the absorbing material has a different number
of protons and/or neutrons from what it had before the interaction.
30
The type of neutron interaction that occurs between an incident neutron and
absorbing material can be generalized according to the energy of the incident
neutron.
For example, fast neutrons are most likely to undergo scatter interactions with
atoms in their environment, with elastic scatter dominating at lower energies and
inelastic scatter dominating at higher energies.
Lower-energy neutrons (that is, thermal or near thermal neutrons) are more likely to
undergo absorption reactions.
31
This slide provides a general overview of the main aspects of the two different types
of neutron scatter.
The key difference between these two scatter mechanisms is whether or not kinetic
energy is conserved. For elastic scatter, kinetic energy is conserved, whereas it is
not conserved for inelastic scatter.
Elastic scatter is more likely to occur with low-Z target materials and with neutrons
having lower incident energies. The maximum amount of energy that can be lost
during elastic scatter is a function of the mass of the target nucleus. In addition,
elastic scatter cross sections are larger in magnitude than inelastic cross sections.
32
Inelastic scatter is more likely to occur with high-Z target materials and with
neutrons having higher incident energies, typically greater than 1 MeV. Large
amounts of energy can be lost during inelastic scatter. However, the amount of
energy lost in a particular inelastic collision is not dependent on the mass of the
target nuclei.
In addition, inelastic scatter events have threshold energies that are particular to
each target nucleus and this type of interaction cannot occur if the energy of the
incident neutron is below the threshold energy.
33
As we indicated earlier, elastic scattering is the most likely interaction between
lower energy fast neutrons and low-Z absorbers.
We can describe elastic scatter as a billiard ball type collision. Consequently, a
direct, head-on collision results in a larger energy transfer than an indirect, grazing
collision.
Note, also, that kinetic energy and momentum are conserved in an elastic scatter
event, and a light, low-Z recoil nucleus can cause high LET tracks.
34
This slide shows the equations that describe the kinematics of neutron elastic
scatter.
If we invoke conservation of energy and momentum in the center-of-mass
coordinate system, we can relate ER, the kinetic energy of the recoil nucleus to En,
the kinetic energy of the incident neutron and Θ, the neutron angle of scatter using
the relation shown on the slide. You can work out the kinematics in a problem set.
35
If we now convert to the laboratory-based system in which the target nucleus is at
rest, we obtain the following relation for the kinetic energy of the recoil nucleus,
energy of incident neutron, and recoil nucleus angle of scatter.
36
Recall the following definitions of symbols in these equations:
A is the mass of the target nucleus (in the laboratory system).
En is the kinetic energy of the incident neutron (in the laboratory system).
ER is the kinetic energy of the recoil nucleus (in the laboratory system).
Θ (upper case) is the angle of scatter of the neutron in the center-of-mass
coordinate system.
θ (lower case) is the angle of scatter of the recoil nucleus in the laboratory system.
37
In the equation for the energy given to the recoil nucleus, we see that energy given
to the recoil nucleus is determined by scattering angle of the recoiled neutron.
Now let’s compare the two extreme elastic scattering scenarios; one in which the
incident neutron grazes the target nucleus, and one in which the incident neutron
interacts with the target nucleus in a direct head-on collision.
38
For a grazing angle encounter, the neutron is only slightly deflected, and the recoil
target nucleus is emitted almost perpendicular to the incident neutron.
The cos2 of 90 is zero. Thus, according to the equation, the energy given to the
recoil nucleus is essentially zero.
This demonstrates that for an elastic scatter grazing hit almost no energy goes to
recoil nucleus, regardless of mass of the target nuclei.
39
On the other extreme, for head-on direct collision between an incoming neutron and
a target nucleus, the recoil nucleus is emitted in almost the same direction, θ is
approximately equal to zero.
In the equation for the energy of the recoil nucleus, the cos2 of 0 is 1. Thus, the
equation is simplified and the ratio of the energy given to the recoil nucleus to the
energy of the incident neutron is given by 4A over (1+A)2. This quantity is also
called the fractional energy transfer.
This relationship demonstrates that for a direct head-on encounter the fractional
energy transfer from elastic scatter is highly dependent on the mass of the target
nucleus.
40
This slide shows the fractional neutron energy transfer for direct-hit elastic scatter
for several light nuclei.
These data demonstrate that the maximum fractional energy transfer increases as the
mass of target nuclei decreases. That is, more energy is transferred to the recoil
nucleus and less energy is retained by the scattered neutron for smaller target nuclei.
From these data we can conclude that nuclei with lower mass are more effective on
a “per collision” basis for slowing down neutrons! This is an important concept for
neutron shielding design.
41
In principle, all scattering angles are allowed in neutron elastic scatter. However,
for most target nuclei, forward and backward scattering are somewhat favored.
Thus, the actual energy distribution of recoil nuclei is a continuum between the two
extremes.
42
When neutron inelastic scatter takes place, the neutron is captured by the target
nucleus and is re-emitted along with a gamma ray.
To understand inelastic scatter, we need to invoke the compound nucleus model.
43
According to this model, the incident neutron collides with the target nucleus and
they fuse together to form a combined system.
Then by successive nucleon-nucleon collisions within the combined system, the
reaction energy becomes shared among many nucleons.
44
Eventually equilibrium occurs and the compound nucleus exists in an excited state.
Excitation is followed by de-excitation in which the excitation energy is emitted as
a gamma photon. This gamma can have substantial energy, typically on the order of
millions of electron volts.
De-excitation is also accompanied by the emission of a neutron. The neutron that is
emitted from the combined nucleus may not necessarily be the same neutron that
went into the system, but the total number of protons and neutrons remains the same
as before the inelastic scatter collision.
45
Inelastic scatter is a threshold phenomenon.
The threshold energy for inelastic scatter is dependent on the target nucleus. In
general, the threshold energy decreases with increasing atomic mass. For example,
the threshold for hydrogen is essentially infinite and inelastic scatter cannot occur.
The threshold for oxygen and uranium are approximately 6 MeV and 1 MeV,
respectively.
46
Above the threshold energy, the probability of inelastic scatter or the inelastic
scatter cross section for a particular material increases with increasing energy.
The cross section is less than 1 barn for low-energy neutrons, and approaches the
physical cross section of the target nucleus at high energies. That is, inelastic
scatter is the dominant interaction mechanism at higher energies.
47
Another category of neutron interactions is sometimes referred to as a non-elastic
process.
This type of interaction is similar to inelastic scatter in that the process follows a
compound nucleus model and that there is a recoil neutron.
However, non-elastic processes differ from inelastic scatter because instead of
emitting γ-rays, additional secondary particles can be emitted (in addition to
scattered neutron).
48
That is, after the interaction, the target nucleus has a different number of protons
and neutrons from what it had before the interaction.
Non-elastic processes also differ from absorption because the incident neutron is not
absorbed and a scattered neutron is emitted as a product of the reaction.
Non-elastic processes are also sometimes referred to as non-elastic scatter.
49
Both non-elastic and inelastic scatter follow a compound nucleus model.
Whether the compound nucleus will de-excite via non-elastic or inelastic scatter is
determined by the energy of the incident neutron.
50
if the energy of the incident neutron is very close to the threshold energy, deexcitation is more likely to occur by emission of gamma rays rather than by
additional particle emissions. That is, inelastic scatter is favored.
Conversely, if the incident neutron energy is much greater than the threshold energy,
de-excitation is more likely to occur by emission of additional particles i.e. nonelastic scatter is favored.
51
Now, we will switch gears and briefly discuss neutron absorption reactions.
Low energy neutrons (thermal or near thermal) are more likely to undergo
absorption reactions rather than scatter.
In this energy range, the absorption cross-section of many nuclei has been found to
be inversely proportional to the square root of the energy of the neutron.
52
This figure is an example of an absorption reaction cross section for an incident
slow neutron and a boron nucleus. In this figure, reaction cross section is shown on
the y-axis and energy on the x-axis. You can clearly see that the probability for
slow neutron absorption follows the one-over-v law. Note that both axes are
logarithmic.
53
This table shows thermal neutron absorption cross sections for several different
nuclei. The values reported in the table are given for neutrons of energy 0.025 eV.
We earlier said that this is in the range of energies for thermal neutrons.
Note the very high thermal neutron absorption cross section for cadmium. One can
take advantage of this high absorption probability in neutron detection techniques.
Also note the high thermal neutron absorption cross section for boron-10. This high
absorption probability is capitalized upon in boron neutron capture therapy, which
we will discuss in the next lecture.
54
The cross section for any other neutron within the validity of the 1/v law can then be
calculated using the following equation:
55
Neutron activation is the production of a radioactive isotope by absorption of a
neutron. Activation reactions almost always follow absorption reactions.
These radioactive isotopes subsequently decay by various mechanisms by emitting
gamma rays, and various particles such as alpha particles or protons.
Some examples include :
14N(n,p)14C
10B(n,α)7Li
113Cd(n,γ)114C
56
Byproducts of activation can be both good and bad because these byproducts of
activation can have substantial energy.
These byproducts can be measured. This technique is one of the methods exploited
for neutron detection.
These high-energy byproducts can also pose a substantial radiation hazard and must
be considered in neutron shielding design.
57
We will conclude today’s lecture by identifying the most common interactions of
neutrons with tissue.
Let me repeat a statement I made earlier this lecture: The type of interaction and
the amount of dose deposited in the body is strongly dependent on the neutron
energy and the nature of the absorbing material.
Absorbing material in the human body is primarily hydrogen, carbon, nitrogen, and
oxygen.
58
We know that neutrons are indirectly ionizing radiations, but the secondary particles
produced by neutron interactions are densely ionizing particles, such as protons,
alpha particles, and nuclear fragments. These are high-LET particles that then
deposit radiation dose in tissue.
59
For example, high-energy neutrons may interact with carbon and oxygen via nonelastic processes and result in the release of charged alpha particles. These alpha
particles then deliver dose to tissue.
60
Here are two examples:
A neutron can interact with a carbon nucleus, which has six protons and six
neutrons, resulting in the production of three alpha particles. A neutron can also
interact with an oxygen nucleus, which has eight protons and eight neutrons,
resulting in the production of four alpha particles. In either case, it is the alpha
particles that deposit the radiation dose.
61
Intermediate energy neutrons primarily interact with hydrogen nuclei via elastic
scatter.
Elastic scatter with hydrogen nuclei is the dominant mechanism of energy transfer
in soft tissue for 3 reasons:
First of all, hydrogen is the most abundant atom in tissue. Recall that the human
body is mainly water, and there are twice as many hydrogen atoms as there are
oxygen atoms in water.
Second, a hydrogen nucleus, that is, a proton, has almost the same mass as that of a
neutron. The greatest amount of energy transfer will occur when the target has the
same mass as that of the incident neutron.
Finally, hydrogen has a large elastic scatter cross-section for neutrons.
62
Now, as we go down in energy to thermal neutrons, we find that absorption is the
dominant interaction mechanism for thermal neutrons in tissue.
Absorption is followed by activation, and the activation decay products deliver dose
to tissue.
63
Here are some interactions of thermal neutrons in tissue:
The major component of dose from thermal neutrons is a consequence of the (n,p)
reaction with nitrogen. Of the 0.62 MeV of energy transferred in this interaction,
0.04 MeV is given to the recoil carbon nucleus, which deposits its energy locally,
and 0.58 MeV is given to the proton. This proton has a range of about 10-6 m, so it,
too, deposits energy locally. This interaction is the dominant energy transfer
mechanism in the body for thermal and epithermal neutrons. Because the energy is
deposited locally, the kerma is equal to the dose.
64
Another thermal neutron interaction that we need to consider is the (n,γ) reaction
with hydrogen. In this interaction, almost all the 2.2 MeV energy is given to the
gamma with a small amount of energy given to the recoil deuteron. The gamma
deposits its energy some distance away from the interaction, so the kerma is no
longer equal to the dose.
65
To summarize the interactions of neutrons with tissue, we note that the amount of
dose deposited in the body is strongly dependent on neutron energy.
Fast neutrons primarily interact with carbon and oxygen via non-elastic processes
and result in the release of charged α-particles, (n,n’3α) and (n,n’4α). These αparticles then deposit dose to tissue.
Intermediate energy neutrons primarily interact with hydrogen nuclei via elastic
scatter. The recoil proton then deposits dose in tissue.
Absorption is the dominant interaction mechanism for thermal neutrons in tissue
and is followed by activation. The major component of dose from thermal neutrons
is a consequence of the 14N(n,p)14C, which results in the emission of a 0.58 MeV
proton.
66