22.54 Neutron Interactions and Applications (Spring 2004)
Chapter 7 (2/26/04)
Neutron Elastic Scattering - Thermal Motion and Chemical Binding Effects
___________________________________________________________________
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 ( ) , where E is the neutron energy in LCS. While we have derived the energy distribution in the form of the energy transfer kernel
( → 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
F E → E ')
σ ( )
. For neutrons of thermal energies,
requires considerations of the effects of thermal motion and can be more involved than the behavior of
( the understanding of σ ( ) chemical binding of the target atom. There is much to be said about these effects, not only at the level of section d 2 σ / Ω '
σ ( ) , but also at the level of the double differential scattering cross
. 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 ( ) , 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 (= 4 π 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
and σ theo
= σ theo
( ) , where v and V are the neutron and target nucleus velocities in LCS, respectively. The connection between these two cross sections is v σ obs
( ) = ∫ − σ theo
( ) ( ) (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 kr o know that σ is just a constant, σ = 4 π a 2 σ so
< 1, then we
≡ and the integral in (7.1) can be further reduced. We write the Maxwellian distribution P(V) as
( ) =




2 π
M




3/2 exp




−
MV
2 k T
2




Ω
V
(7.2)
Inserting
σ obs
σ theo v = σ s v
and (7.2) into (7.1), and denoting the observed cross section as
, we have
σ s
=
σ v so ∫ r
( ) (7.3)
Notice that in (7.3) we are denoting the relative speed as v r
= v 2 + V 2 − 2 vV µ , and (7.3) becomes
3/2 v r
= − . Since for purpose of integration over target velocity we can take the z-axis to be along the neutron velocity v,
σ s
=
σ v so




2 π
M




∫ 1
− 1 d µ ∫
0
∞ dV 2 π V v r exp
− MV 2 /2 k T (7.4)
Carrying out the µ -integration one finds
σ s
=
σ
β so
2



( β 2 + 1/ 2) erf β + π β e − β 2



(7.5)
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where erf(x) is the error function,
= π ) x
0
∫ t 2 (7.6) with limiting behavior
( ) → (2/ π )



 x x 3 x
⋅
5
− x
⋅
7
−
3
+
5 2! 7 3!
+ ...




, x <<1 (7.7)
1 − e − x 2 x π




1 −
2
1 x 2
+
1 3 x
−
⋅ ⋅ x
+ ...




, x >>1 (7.8)
In (7.5), β 2 = /
, A being the mass ratio M/m and E = mv 2 /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
∝ σ so
/ v (7.9) and in the limit high neutron energy (large β or low temperature) ,
σ s
→ 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
( → E ')
, is seen to approach the limiting behavior of (6.12) when E/k
B
T 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/k
B
T is finite (as opposed to approaching infinity) one sees a finite probability of upscattering, the magnitude growing as E → k
B
T 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/k
B
T. 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,
2 =


 mM
+


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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σ s 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
σ bound
=



A + 1
A



2
σ free
(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.
200
σ bound =
( ) 2
σ free
150
100
σ t
0
σ free
50
2 σ t (H) + σ t
(O)
1
E
B m
0
0.001
0.01
0.1
1
Neutron energy, e
V
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, B m
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 H
2
O molecule with contributions from two hydrogens and an oxygen is shown in the right panel.
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
C 12 . 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.
10
5

10
5
1.0
0.5
0.0001
10
5 o T=1020 K
Cross section
Constant to 0.01 M e
V
0.001
Bragg cutoff
0.01
Neutron energy, e
V
0.1
1.0
0.5
σ t =
σ s
Broad resonances
σ s
σ t
1
0.1
0.01
0.1
1 10 100
Neutron energy, M e
V
Fig3.
Variation of neutron cross section of C 12 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.
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