22.54 Neutron Interactions and Applications (Spring 2004)
Chapter 7 (2/26/04)
Neutron Elastic Scattering - Thermal Motion and Chemical Binding Effects
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References -J. R. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading,
1966), chap 2.
S. Yip, 22.111 Lecture Notes (1975), chap 7.
G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York,
1970), chap 7.
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All cross sections are point functions when it comes to the spatial location of the
interaction. The range of force in nuclear interaction is small compared to neutron
wavelength at any reasonable energy so interaction can be regarded as occurring at a
point as opposed to spread over a region of finite extent.
In this lecture we will focus on understanding the energy dependence of elastic
scattering cross section σ ( E ) , where E is the neutron energy in LCS. While we have
derived the energy distribution in the form of the energy transfer kernel F ( E → E ') in the
preceding lecture, we have thus far not said anything about the energy dependence of
Eq.(6.1). The reason we postponed this discussion until now is that the behavior of σ ( E )
can be more involved than the behavior of F ( E → E ') . For neutrons of thermal energies,
the understanding of σ ( E ) requires considerations of the effects of thermal motion and
chemical binding of the target atom. There is much to be said about these effects, not
only at the level of σ ( E ) , but also at the level of the double differential scattering cross
section d 2σ / dΩdE ' . We will examine only the total cross section here and leave the
discussion of the double differential cross section to later in the term. For a brief
overview of the qualitative energy variations of σ ( E ) , see Lec2 (2003).
As we have noted in Lec3 (2003), at incoming neutron energy E such that the
neutron speed is much greater than the speed of the target nucleus, it is a good
approximation to simplify the kinematics analysis by taking the target nucleus to be
stationary. In the cross section discussion in Lec4, we transform the two-body collision
problem into an effective one-body problem, that of scattering of a particle by a potential
field V(r). Here the vector r is the relative position of the neutron with respect to the
target nucleus. Thus the Schrodinger
equation to be solved is in CMCS and the cross
section subsequently obtained is also in CMCS. We have seen that for low-energy
scattering only the s-wave contribution is needed, in which case the angular differential
cross section is spherically symmetric, and the total cross section is a constant (= 4π a 2 ).
Thermal Motion Effects
In the thermal energy region the neutron energy is comparable to the energy of the
target nuclei which follows a Maxwellian distribution characterized by the target
temperature. Then it is no longer justified to assume the target nucleus is stationary. To
take into account the thermal motions of the target explicitly one should specify the
physical state of the target, such as a crystalline or a liquid target. We will defer dealing
with the dynamics of the target nuclei to a later lecture and consider only the simpler
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situation of elastic scattering of neutron in a gas at thermal equilibrium. In this case the
target nuclei move about in straight-line trajectories with velocities governed by a
Maxwellian distribution at temperature T.
In this discussion it is important to distinguish the cross section that one measures
in the laboratory from the cross section which we have calculated from theory. To be
explicit we denote the former as σ obs and the latter as σ theo . Now σ obs is a function of
the neutron energy in LCS since the observation is made only in the laboratory, whereas
σ theo is a function of the neutron energy in CMCS, or more specifically the relative
energy E in Eq. (4.6). Thus we understand σ obs = σ obs (v) and σ theo = σ theo ( v − V ) , where
v
and V are the neutron and target nucleus velocities in LCS, respectively. The connection
between these two cross sections is
vσ obs (v) = ∫ v − V σ theo ( v − V )P(V )d 3V
(7.1)
where P(V) is Maxwellian distribution of the target nucleus velocity. Eq. (7.1) is the
fundamental statement relating the scattering rates in LCS and CMCS. (Strictly
speaking, for (7.1) to be a scattering rate one should multiply both sides by the density of
target nuclei.)
If the neutron energy is low enough to satisfy the condition of kro < 1, then we
know that σ theo is just a constant, σ theo = 4π a2 ≡ σ so and the integral in (7.1) can be
further reduced. We write the Maxwellian distribution P(V) as
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P(V )d V
 M
=
 2π k T
B





3/2

MV 2  2
V dVd ΩV
 2k T 
B 

exp  −
(7.2)
Inserting σ theo and (7.2) into (7.1), and denoting the observed cross section as
σ obs (v) = σ s (v) , we have
σ s (v) = σ so ∫ vr P(V )d 3V
(7.3)
v
Notice that in (7.3) we are denoting the relative speed as vr = v − V . Since for purpose of
integration over target velocity we can take the z-axis to be along the neutron velocity v,
vr = v 2 + V 2 − 2vV µ , and (7.3) becomes


σ s
(v) = σ so  M 
v  2π kB
T 
Carrying out the
µ
3/ 2
1
∞
∫−1d µ ∫0 dV 2π V
2
vr exp− MV
2 / 2k
BT
(7.4)
-integration one finds
σ s ( E ) = σ so2 ( β 2 +1/ 2)erf ( β ) + (1/ π ) β e− β
β 
2


(7.5)
2
where erf(x) is the error function,
x
erf ( x) = (2/ π )∫ e−t dt
2
(7.6)
0
with limiting behavior


x3 x5
x7
erf ( x) → (2/ π )  x − +
−
+ ... , x <<1
3 5 ⋅ 2! 7 ⋅ 3! 

2
1−
e− x
x π

1
1− 2

2x

+

1⋅ 3
1⋅ 3 ⋅ 5
−
+ ... , x >>1
2 2
2 3
(2x ) (2x )

(7.7)
(7.8)
In (7.5), β 2 = AE / kBT , A being the mass ratio M/m and E = mv2/2 is the neutron energy in
LCS. Using (7.7) and (7.8) we see that in the limit of low neutron energy (in the sense of
small β , or equivalently high temperature),
σ s ( E ) ∝ σ so / v
(7.9)
and in the limit high neutron energy (large β or low temperature) ,
σ s ( E ) → σ so
(7.10)
The two limiting behavior, (7.9) and (7.10), characterize rather well the typical behavior
actually observed for many nuclei, a slowly rising cross section with decreasing energy at
low energies, and a constant cross section at high energies. See, for example, Fig. 5.2 (or
equivalently, Fig. 2.6 in Lec2 (2003)).
One may ask whether the analysis we have just carried out for the total cross
section can be applied also to the energy transfer kernel so that a differential cross section
is obtained that depends on temperature. The answer is that this is indeed fesible.
Without going into the details of the calculations we show the qualitative behavior of the
results in Fig. 1. The temperature-dependent energy transfer kernel, still labeled as
F ( E → E ') , is seen to approach the limiting behavior of (6.12) when E/kBT is large. This
behavior is more clear for A > 1 (right panel) but is seen nonetheless for A = 1
(hydrogen). Recall that when we assume the target nucleus is at rest, there can be no
upscattering of the neutron (E' > E) since the nucleus has no energy to give. Once E/kBT
is finite (as opposed to approaching infinity) one sees a finite probability of upscattering,
the magnitude growing as E → kBT in Fig. 1.
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Fig 1. Distributions in the energy of scattered neutron E' (in unit of initial energy E) for a
gas of target nuclei with two mass ratios, A (= M/m) = 1 and 16, at various ratios of
E/kBT. The scaling shows that one can obtain the same effect by varying either the
energy E or the target temperature T. [Adapted from Bell and Glasstone, Figs. 7.5 and
7.6.]
Chemical Binding Effects - Bound-Atom vs. Free-Atom Cross Sections
To treat chemical binding effects properly one needs to consider the double
differential scattering cross section. However, one can gain some qualitative
understanding by observing that the scattering cross section ought to depend on the
neutron energy when it varies over a range from being small compared to the binding
energy of atoms and molecules to being large compared to this energy. Why should it
matter?
If the neutron energy is small compared to the binding, then the scattering nucleus is
effectively rigidly bound to an object that has the mass of the molecule rather than just
the mass of the nucleus. In the case of water, the difference is a mass of 18 (one oxygen
at mass 16 and two hydrogens at mass 1 each) compared to a mass of 1 for a standalone
hydrogen. Conversely, if the neutron energy is large compared to the binding, then the
fact the scattering proton is bound to a water molecule is of no consequence; in this case,
the scattering mass is just that of the proton. Now it turns out that we can show that the
cross section for neutron scattering by a nucleus is proportional to the square of the
reduced mass,

σ ~ µ 2 =  mM
M

+ m 
2

A 

 A +1 
=
2
(7.11)
For neutron scattering by hydrogen (in water) in the energy range ~ eV and above, the
neutron energy is large compared to the binding energy of the water molecule, the
situation then corresponds to a reduced mass of 0.5. Let us call the cross section in this
case the free-atom cross section, σ free , meaning that it is the cross section in the 'highenergy' region where the chemical binding has no effect. In contrast, in the energy range
below 0.025 eV, the binding energy is now larger than the neutron energy and the
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reduced mass becomes 1 (because the effective mass of the scatterer is 18). We call this
cross section the bound-atom cross section, σ bound as if the proton mass has increased to
18. This argument shows that the free-atom and bound-atom cross sections are related by
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σ bound =  A +1  σ free



A 
(7.12)
Summarizing, we then expect the neutron scattering cross section, which we know has a
value of 20 barns in the eV energy range (the free-atom value), to rise by about a factor
of 4, to 80 barns in the energy region around 0.025 eV. This is the rough explanation of
the observed behavior of neutron scattering in water, shown in Fig. 2.
Fig2. Typical behavior of elastic scattering cross section of a target at energies below ~ 1
eV. The increase of cross section as energy decreases is attributed to chemical binding
effects which may be expressed in terms of the concept of bound-atom cross section
(schematic shown in the left panel, Bm is the binding energy). In the energy range above
~ 1 eV the cross section takes on a constant value known as the free-atom cross section.
The cross section of a H2O molecule with contributions from two hydrogens and an
oxygen is shown in the right panel. [From Lamarsh, Figs. 2-17 and 2-18.]]
There are other characteristic chemical binding effects which we will want to
discuss. When neutron is scattered by a polycrystalline moderator the energy variation of
the elastic scattering cross section can take on the behavior shown in Fig. 3 for the case of
C12. The constant cross section at 5 barns over a wide range from ~ 0.02 eV to 0.1 MeV
is what we have just referred to as the free-atom cross section; it is also the cross section
denoted as σ so earlier in this lecture. We will return later to discuss several interesting
features seen in the low energy range, below 0.02 eV.
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Fig3. Variation of neutron cross section of C12 with neutron energy in a polycrystalline
target showing chemical binding effects at low energies, a constant elastic scattering
contribution over a wide range ( ~0.02 eV - ~0.3 MeV), and a series of broad resonances
with some non-scattering contributions above ~ 2 MeV. [From Lamarsh, Fig. 2-9.]
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