22.106 Neutron Interactions and Applications (Spring 2005)
Lectures 2-4 (2/3,8,10/05)
Neutron Elastic Scattering: Thermal Motion and Chemical Binding Effects
__________________________________________________________________
Lecture 2 (2003) Neutron Reaction Systematics: Energy Variation of Cross Sections
Chapter 7 (2004) Neutron Elastic Scattering: Thermal Motion and Chemical Binding
Effects
These two lecture notes, written during previous years, are the primary references for our
discussion here; they are still available as standalone notes. For the convenience of the
reader, we also attach them at the end of this Lecture Notes, as Appendices Lec2 (2003)
and Chap7 (2004) without changing any equation or figure numbering.
_____________________________________________________________________
In this set of notes which cover 3 lectures, we review the basic aspects of neutron
scattering having to do with energy variation of the cross section, pulling together notions
that have been discussed in 22.101, as well as additional materials which may not been
introduced before. Our intent is to build a base of understanding of why cross sections
should vary with energy, and what information is contained in these variations. The
reader should bear in mind that there are three kinds of cross sections that will be of
interest to us, the integrated or total cross section for scattering, σ (Ei ) , the energy
differential scattering cross section, dσ / dE f , and the double differential scattering cross
section, d 2σ / dΩdE f . Their relations are best seen from cefinitions that begin with the
double differential cross section. To measure this quantity one requires an incoming
neutron beam of specified energy Ei and momentum (or direction of travel Ω i ), see Fig.
1. The beam is scattered in the sample S and the outgoing beam goes off in an angle θ ,
with energy Ef and direction Ω f . Notice that the scattering angle θ is defined as
cos θ = Ω i ⋅ Ω f . The scattering intensity measured at the detector D is then proportional
to the double differential cross section d 2σ / dΩdE f . Now one writes
1
Fig. 1. Schematic of a scattering experiment to measure the double differential scattering
cross section.
d 2σ / dΩdE f = σ ( Ei ) F ( Ei , E f , Ω)
(2-4.1)
thus defining the double distribution F ( Ei , E f , Ω) . By integrating the double differential
cross section over all angles of scattering one obtains the energy differential cross
section,
dσ / dE f ≡ σ ( Ei ) F ( Ei → E f )
= ∫ dΩ(d 2σ / dΩdE f )
(2-4.2)
thus defining the distribution F ( Ei → E f ) . By integrating the energy differential cross
section one obtains the total cross section,
σ (Ei ) = ∫ dE f F ( Ei → E f )
(2-4.3)
which also follows from the unit normalization of F ( Ei → E f ) . Comparing (2-4.1) and
(2-4.2) one sees the relation between the two F functions. Although we denote them by
the same symbol, the argument of the function should make it clear which function is
being discussed.
2
Since σ (E) appears in all the cross sections, we should first understand its energy
dependence. The energy dependence of the scattering distribution F ( Ei → E f ) has been
previously examined in some detail in 22.101 under the conditions of elastic scattering,
target nucleus at rest, and isotropic scattering in CMCS. We will see how once can relax
the second assumption and thereby study the effects of thermal motion and chemical
binding. The energy variation of F ( Ei , E f , Ω) is more involved. We will defer its
discussion until a later time in this course when we can properly introduce the theory of
inelastic neutron scattering.
Fermi’s Golden Rules
In quantum mechanics one can calculate reaction cross sections by using a
method known as time-dependent perturbation theory. E. Fermi had used this method to
discuss the systematics of nuclear reaction cross sections [Enrico Fermi, Nuclear Physics,
Lecture Notes compiled by J. Orear, A. H. Rosenfeld, and R. A. Schluter (Univ. Chicago
Press, 1949), revised edition, chap. VIII]. We recall this development very briefly to
show that this is one way to get at the energy dependence of cross sections. For more
detailed discussions the reader should see the Appendix Lec2 (2003), and the references
given therein. Consider the two-body reaction a + A → b + B, where a is the incoming
particle and B the outgoing. Fermi had formulated two calculations, one for the case
where there is no resonance reaction (compound nucleus formation) and the other is for
resonance reaction. For some reason he called the first case Golden Rule No. 2 and the
second case Golden Rule No. 1. For our purposes we are interested only in Rule No. 2
which gives the transition probability per unit time (reaction rate) as
Wab =
2π
H ab
h
2
dn
dE
(2-4.4)
where Hab is the transition matrix element about which we will say very little (because it
gets into the theory of nuclear reactions, a topic that is beyond the scope of our interest),
other than it is the matrix element of an operator that causes the system to go from an
initial state which contains the incoming particle a to a final state that contains the
3
outgoing particle b. The last factor in (2-4.4) is the density of energy levels of the final
state. Typically one estimates what this is using a simple idealization, a particle in a box.
To go from the transition probability per unit time to a cross section, one writes
σ (a, b) = Wab / na v rel
(2-4.5)
where na is the number density of incoming particle a and v rel is the relative speed
between the incoming particle a and the target nucleus A. We can rationalize this
conversion by noting that Wab is the reaction rate, so that dividing by the number of
reacting particles per unit volume and by the relative speed makes it a cross section
(recall our discussion of the meaning of the cross section in Lec1). With the calculation
of the density of states, we find the cross section to be given by
σ (a, b) ≈ U 2 exp(−Ga − Gb ) pb2 / v a vb
(2-4.6)
where U 2 is the nuclear interaction part of the transition, which we will assume is
independent of the energy of particle a, G is the Gamow factor which governs the
penetrability if the particle were charged (the Coulmb interaction part). The remaining
factors in the expression have to do with the kinematics of the reaction. The usefulness
of the Fermi expression is that it separates the cross section into parts that are nuclear or
Coulomb interaction in nature, and factors that are kinematics. We will apply this
decomposition to the case of neutron elastic and inelastic scattering.
Elastic scattering (n,n) -- With incoming and outgoing particles being uncharged, the
transmission factor is unity. Also, with v a = vb , the kinematical factor pb2 / v a vb is a
constant. Eq.(2-4.6) then predicts the cross section to be a constant. Elastic scattering,
strictly speaking, is not a proper reaction.
Inelastic scattering (n,n') -- Since the product nucleus is left in an excited state, this is
an endothermic reaction that requires a threshold value for the neutron energy, E* = -Q ~
4
1 MeV. Velocity of the outgoing neutron is v n ' , with v n2' = (excess energy above
threshold)/(mn/2) ~ (En - E*). Thus,
σ (n, n') ~ v n ' ~ (E n − E*)1/ 2
See Fig. 2-4.1 for a schematic sketch of the characteristic energy dependence of reaction
cross sections. For further discussions, see Lec2 (2003) in the Appendix.
Fig. 2-4.1. Schematic variation of cross section with neutron energy, elastic scattering,
an exothermic reaction, and inelastic scattering.
Thermal Motion Effects in Elastic Scattering Cross Sections
We now focus on only elastic scattering and ask how one can understand the effects of
thermal motions of the target nuclei. We recall that in our previous calculation of
scattering cross section using the phase shift method, the fact that the target nucleus is in
motion never entered into our analysis since we started by reducing the two-body
scattering problem into an effective one-body problem. This being the case, where did
we throw out the effect of target motion? The answer is that thermal motion effects do
not enter into a two-body problem, but they do show up in a many-body problem. To
calculate the cross section that one measures in the laboratory, it is not sufficient just to
calculate the effective one-body problem since one has to average over a distribution of
target velocities. This means that what one calculates in the effective one-body problem
can be related to the measured cross section, but they are NOT the same quantities. The
relation between the two makes it clear where thermal motions can enter into the
problem. (For more background, see Chap7 (2004) in the Appendix.)
5
The expression we have in mind is
vσ obs (v) = ∫ v − V σ theo ( v − V )P(V )d 3V
(2-4.7)
where v is the neutron velocity in LCS, V the target velocity also in LCS, the subscripts
denote measured and theoretical cross sections respectively, and P is the Maxwellian
distribution of target nucleus velocities. Notice first the connection between measured
and theoretical is made in terms of the product of speed and cross section. If one
multiplies through by the number density of target nuclei, that makes it the scattering
rate. So (2-4.7) in effect is how one relates the observed scattering rate in the laboratory
where the only coordinate system available is LCS to a calculated scattering rate, where
one takes advantage of the two-body nature of the collision and performs the analysis in
CMCS (relative speed). Notice that the experimental cross section is now temperature
dependent while the theoretical cross section is temperature-independent.
If the neutron energy is low enough to satisfy the condition of kro < 1, then we
know from previous studies in 22.101 that σ theo is just a constant, σ theo = 4π a 2 ≡ σ so and
the integral in (2-4.7) can be further reduced. We write the Maxwellian distribution P(V)
as
⎛
M ⎞⎟
⎜ 2π k T ⎟
B ⎠
⎝
3/2
P(V )d 3V = ⎜
⎛
MV 2 ⎞⎟ 2
V dVd ΩV
⎜ 2k T ⎟
B ⎠
⎝
exp ⎜ −
(2-4.8)
Inserting σ theo and (2-4.8) into (2-4.7), 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
v
(2-4.9)
6
Notice that in (2-4.9) 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 (2-4.9) becomes
⎛
⎞
σ s (v) = σ so ⎜⎜ M ⎟⎟
v ⎝ 2π kBT ⎠
Carrying out the
µ
3/ 2
∞
1
∫−1d µ ∫0 dV 2π V
2
vr exp− MV
2 / 2k
BT
(2-4.10)
-integration one finds
σ s ( E ) = σ so2 ⎡⎢( β 2 +1/ 2)erf ( β ) + (1/ π ) β e− β
β ⎣
2⎤
⎥
⎦
(2-4.11)
where erf(x) is the error function,
x
erf ( x) = (2/ π )∫ e−t dt
2
(2-4.12)
0
with limiting behavior
⎛
⎞
x3 x5
x7
erf (x) → (2/ π ) ⎜⎜ x − +
−
+ ...⎟⎟ , x <<1
3
5
⋅
2!
7
⋅
3!
⎝
⎠
⎞
e−x ⎛
1
1⋅ 3
1⋅ 3 ⋅ 5
⎜⎜1 − 2 +
...⎟
→1−
−
+
⎟ , x >>1
(2x 2 ) 2 (2x 2 ) 3
x π ⎝ 2x
⎠
(2-4.13)
2
(2-4.14)
In (2-4.11), β 2 = AE / k B T , A being the mass ratio M/m and E = mv2/2 is the neutron
energy in LCS. Using (2-4.13) and (2-4.14) 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
(2-4.15)
and in the limit of high neutron energy (large β or low temperature) ,
7
σ s ( E ) → σ so
(2-4.16)
The two limiting behavior, (2-4.15) and (2-4.16), 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,
the cross section of hydrogen as sketched below.
Fig. 2-4.2. Elastic scattering cross section of hydrogen in the low-energy region.
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 feasible.
Without going into the details of the calculations we show the qualitative behavior of the
results in Fig. 2-4.3. The temperature-dependent energy transfer kernel, still labeled
as F ( E → E ') , is seen to approach the limiting behavior we know from 22.101 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
up-scattering 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. 2-4.3.
8
Fig 2-4.3. 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
9
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
=⎜
(2-4.17)
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
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
2
σ bound = ⎜ A +1 ⎟ σ free
⎛
⎞
⎝
A ⎠
(2-4.18)
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. 2-4.4.
10
Fig. 2-4.4. 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. 2-4.5 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.
11
Fig. 2-4.5. 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.]
12
Appendix Lec2 (2003)
22.54 Neutron Interactions and Applications (Spring 2003)
Lecture 2 (2/11/03)
Neutron Reaction Systematics -- Energy Variations of Cross Sections, Nuclear Data
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References --
Enrico Fermi, Nuclear Physics, Lecture Notes compiled by J. Orear, A. H. Rosenfeld,
and R. A. Schluter (Univ. Chicago Press, 1949), revised edition, chap. VIII.
Marmier and Sheldon, Physics of Nuclei and Particles (Academic Press, New York,
1969), vol. I, pp. 64-95.
S. Pearlstein, "Evaluated Nuclear Data Files", in Advances in Nuclear Science and
Technology (Academic Press, New York, 1975), vol. 8.
__________________________________________________________________
In this lecture we will survey the energy variations of nuclear reaction cross
sections, making use of simple results from perturbation theory in quantum mechanics,
Fermi's Golden Rules. This is to provide some appreciation of the variety of cross
sections that have found useful applications in nuclear science and technology. We will
also give a brief description of nuclear data compilation and evaluation, the importance of
which cannot be overemphasized since no serious nuclear calculations can be performed
without this basic information.
We consider the reaction a + A → b + B, where a is the incoming particle
(neutron), A the target nucleus, b the outgoing particle, and B the product nucleus. First
we are not concerned with rersonance reaction which is usually treated as a two-step
process, involving the formation and subsequent decay of the compound nucleus. This
will be taken up later in this lecture. We define the Q-value for the reaction as
Q ≡ [(ma + M A ) − (mb + M B )]c 2
(2.1)
where m, M are atomic masses, and c is the speed of light. For Q > 0, the reaction is
exothermic, with energy being given off. For Q < 0, the reaction is endothermic, with
energy being absorbed. From conservation of total energy, Q also can be written in terms
of the kinetic energy of the particles and nuclei,
Q = Tb + TB − (Ta + TA )
(2.2)
Notice that despite the appearance of (2.2) which may make one think that Q is
dependent on the coordinate system used to measure the kinetic energies (laboratory vs.
center-of-mass), Q is independent of the reference frame as shown by (2.1). In most
cases, exception is in neutron thermalization discussions, we can ignore the motion of the
13
target nucleus and set TA = 0, a considerable simplification. In the following we will
switch notation and use E instead of T for the kinetic energy. For the neturon the kinetic
energy is then E (or En).
In quantum mechanics the calculation of cross section σ ( a, b) is usually
calculated by applying time-dependent perturbation theory to the reaction A(a,b)B,
obtaining the general expressions known as Fermi's Golden Rules. There are two Golden
Rules; for our discussion we are concerned with Golden Rule no. 2, which deals with socalled first-order transitions and gives the expression for the transition rate (number of
transitions per unit time between initial state a and final state b of the particle) wab ,
wab =
2π
h
H ab
2
dn
dE
(2.3)
where H ab is the matrix element of the perturbation H causing the transition from a to b,
and dn/dE is the density of states, the number of states per unit final energy of the
particle. For a particle in vacuum (free particle) with momentum p, we can readily
calculate the density of states. Let the vacuum have a finite volume Ω (we take the
system volume to be finite for purpose of simple normalization of the wave function), the
number of states with momentum in dp about p is
dn = 4π p 2 dpΩ /(2π h)3
(2.4)
If the outgoing particle b and the product nucleus B have spins Ib and IB, respectively,
then we need to include in Eq.(2.4) a statistical factor, (2Ib+1)(2IB+1) for the number of
spin orientations one can have in the final state. For an outgoing particle with momentum
pb ,
dE = vb dpb
(2.5)
Strictly speaking, vb and pb should be the velocity and momentum of the final state (b+B)
in the center of mass coordinate system (frame of reference), rather than of the outgoing
particle b. The difference is small when the product nucleus is heavy compared to the
mass of the particle. We can rewrite (2.1) by combining all (2.4), (2.5) and the statistical
factor,
wab =
1 pb2
π h vb
4
Ω H ab (2I b + 1)(2I B + 1)
2
(2.6)
To go from the rate of transition to the cross section, we simply write
wab = na vrelσ ( a, b)
(2.7)
14
which essentially introduces and therefore defines the cross section. Compare this
argument with Eq.(1.3) in Lec 1. In (2.7), na is the density of incoming particle and vrel
is the relative velocity in the initial state (a+A). Again, if the target nucleus is massive
compared to the incoming particle, then to a good approximation vrel is given by va, the
velocity of the incoming particle in CMCS. Combining (2.6) and (2.7) we obtain
σ ( a, b) =
1
π h4
ΩH ab
2
pb2
va vb
(2I b + 1)(2I B + 1)
(2.8)
This formula is useful for deducing the qualitative variation of the cross section with
energy for a number of typical reactions, without knowing much about the really
complicated part of the calculation, the matrix element of the perturbation H between
initial and final states.
We now assume that there are two parts to the transition matrix element, one
involving nuclear interactions among the nucleons and the other pertains to electrostatic
(Coulomb) interactions between the incoming or outgoing particle, if it were a charged
particle, and the nucleus (either the target or the product as the case may be). We further
assume that the nuclear interactions, however complicated, may be taken to be a constant
for the purpose of estimating the energy variation of σ ( a, b) . What is left in Hab is the
Coulomb interaction. The effect of this can be estimated by using the model of a charged
particle tunneling through a Coulomb barrier. (Recall from 22.101 that such a model was
introduced to calculate the decay constant for α − decay.) For a positively charged
particle, charge z with velocity v, to penetrate a nucleus, charge Z, the transmission factor
is exp(-G/2), where [Fermi's Notes, p. 143]
G/2 ≈
π Zze 2
hv
(2.9)
where G is called the Gamow factor. Accordingly, if the incoming and outgoing particles
are both charged, we will write
2
H ab ≈ U exp( −Ga − Gb )
2
(2.10)
where U is the nuclear interaction part, which we will take to be energy independent.
Now we are ready to examine the variation with energy of incoming particle of
various neutron reactions at low energies, say eV range. Eq.(2.8) shows that there are
two parts which can vary with energy, the kinematical factor, pb2 / va vb , and the
transmission factors, if present. We can distinguish four types of neutron interactions of
interest.
15
1. Elastic scattering (n,n)
With incoming and outgoing particles being uncharged, the transmission factor is unity.
Also, with va = vb, the kinematical factor pb2 / va vb is a constant. Eq.(3.8) then predicts
the cross section to be a constant, as shown in Fig. 3-1(a). Notice that elastic scattering
is, strictly speaking, not a proper reaction.
2. Charged-Particle Emission (exothermic)
Since Q is typically a few MeV, while the incoming neutron energy is of order eV, pb2 / va vb ≈ 1/ va .
2
H ab ≈ U exp(−Gb ) ≈ constant. So σ ≈ 1/ vn ; this is the '1/v' behavior
2
often seen in reactions like ( n, α ) , ( n, γ ) , etc.
3. Inelastic scattering (n,n')
Since the product nucleus is left in an excited state, this is an endothermic reaction that
requires a threshold value for the neutron energy, E* = -Q ~ 1 MeV. Velocity of the
outgoing neutron is vn', with vn2' = (excess energy above threshold)/(mn/2) ~ (En - E*).
Thus,
σ ( n, n ') ∝ vn ' ~ (En − E * )1/ 2
An example here is O16 (n,α )C 13 , with Q = -2.215 MeV.
4. Charged-Particle Emission (endothermic)
This reaction is like the endothermic reaction of inelastic neutron scattering, except there
is now a Coulomb factor,
σ ( n, b) ~ e − G (En − E * )1/ 2
b
with Gb ~ 1/vb.
16
Fig. 2-1. Schematic energy variations of cross sections, the first four correspond to
neutron elastic scattering, neutron-induced reaction (exothermic), neutron inelastic
scattering, and neutron-induced reaction (endothermic). The last two reactions are
charged particle and uncharged particle, and charged particles in and out, both reactions
being exothermic.
In addition, we can consider exothermic reactions with incoming charged
particles. If the outgoing particle is uncharged, then with Ea << Q, pb2 / va vb ~1/va, pb ~
constant, we have
σ ( a, b) ~
1
v
a
e
−G
a
An example would be the inverse reaction to that already mentioned, C 13 (α , n)O16 , Q =
2.215 MeV.
If the outgoing particle is also charged, as in (α , p) , pb2 / va vb ~1/va, and
σ ( a, b) ~
1
v
a
−( Ga +Gb )
e
Both cases are also shown in Fig. 2-1.
Resonance Reactions
The energy variations we have discussed are qualitative and smooth behavior;
they are not intended to apply to resonances which are sharp features of the energy
dependence of the cross sections. To describe resonances one can apply Golden Rule no.
1 which curiously deals with second-order transitions. Instead of going from initial to
final state directly in a transition, as in the case of Golden Rule no. 2, we consider a twostep process where the incoming particle interacts with the target nucleus to form a
compound nucleus which exists for a finite period and then decays to the final state
consisting of an outgoing particle and the product nucleus,
a+ A→C →b+B
(2.11)
17
Without going into details, we will simply give the results for two neutron resonances
which are quite commonly encountered, elastic scattering ( n, n) and radiative capture
( n, γ ) . The cross sections for these two reactions have the form of so-called Breit-Wigner
resonances.
For elastic scattering,
σ ( n, n) = π D f o
Γ n2
2
(E − Eγ ) 2 + Γ 2 / 4
+ 4π DΓ n af o
E − Eγ
(E − Eγ ) 2 + Γ 2 / 4
+σ p
(2.12)
where D = 2π / k , k being the neutron wave number, E = h 2 k 2 / 2m , a is the scattering
length which appears in the definition of the potential scattering cross section σ p = 4π a 2
(we will come back to discuss what we mean by potential scattering in more detail), and
f o = (2J + 1) / 2(2 I + 1)
is the statistical factor for spin orientations, I being the spin of the
target and J is the total spin, I ± 1/ 2 . The other quantities in (2.12) are the resonance
energy Eγ , and the neutron and total resonance width, Γ n and Γ = Γ n + Γγ + ... , with Γγ
being the radiation width; these are typically known as resonance parameters, they are
part of the nuclear data and are tabulated for a given target nucleus. Eq.(2.12) shows
there are three contributions to elastic scattering, the first term is the resonance
contribution, called resonance elastic scattering, the second term is an interference term
between resonance scattering and potential scattering, the latter being the ordinary
scattering in the absence of any resonance. The third term is the potential scattering
which is typically taken to be a constant, depending only on the scattering length, a basic
property of the nucleus (more discussion will be given in the lectures on cross section
calculation and neutron-proton scattering).
To see the cross section variation with neutron energy in elastic scattering, we
note that D ~ v or
E
, Γn ~
E
, Γγ ~ constant. Imagine there is a resonance at energy
Eγ , for neutron energy much less or much greater than Eγ the first and second terms do
not contribute, so the cross section is a constant. In the vicinity of Eγ the cross section
has a variation shown in Fig. 2-2. Notice that the interference effect is destructive at
energies just below the resonance, and constructive at energies above. This is a
18
characteristic feature of resonance reaction which sometimes are observed in actual cross
section measurements.
Fig. 2-2. Schematic energy variation of neutron elastic scattering showing interference
effects between resonant and potential scattering.
For radiative capture the cross section has the pure Breit-Wigner form,
σ ( n, γ ) = π D 2 f o
Γ n Γγ
(E − Eγ ) 2 + Γ 2 / 4
(2.13)
One can see that well below the resonance the cross section behaves like 1/v, and the full
width at half maximum of the resonace peak is Γ , as shown in Fig. 2-3.
Fig. 2-3. Schematic energy variation of a neutron capture resonance, showing a
characteristic 1/v behavior below the resonance and a full width at half maximum of Γ .
To conclude our brief survey of neutron cross sections we show the actual cross
sections for a few nuclei which are of definite interest to nuclear engineering students.
The first example is U235, an isotope of uranium which is fissile (capable of undergoing
fission when bombarded by thermal neutrons).
19
.
Fig. 2-4. Various cross sections of the uranium isotope 235, fission (n,f), radiative
capture ( n, γ ) , elastic scattering (n,n), inelastic scattering (n,n'), and a stripping reaction
(n,2n).
In Fig. 2-4 one can pick out some of the characteristic features discussed above,
energy-independent behavior for potential scattering, 1/v variation for capture and fission
below the resonances, and threshold behavior for inelastic scattering. The student should
also pay attention to the magnitudes of the cross sections and the wide energy range
covered. In the thermal region, around 0.025 eV, the energy dependence is rather simple,
all monotonic variations. Starting at about 1 eV and extending up to about 1 KeV, there
are many sharp resonances. Above 10 KeV the cross sections return to smooth behavior.
In the context of using uranium as the fuel for a nuclear reactor, we can see the reason for
building thermal reactors - to take advantage of the large fission cross sections. Since
neutrons from fission are emitted in the MeV range, one also has the problem of slowing
down the neutrons past the resonance region where there is a high probability of capture
down to the thermal range in order to continue the chain reaction. Thus, the optimum
design of a nuclear system, whether it is a nuclear reactor, a particle accelerator, or
anything else, often comes down to a matter of materials properties, or in this case the
cross sections.
When one compares the cross sections of U235 with those of another isotope U238,
shown in Fig. 2-5, several features can be noted. The absence of thermal fission reaction
in U238 results in rather low cross section values in the thermal energy region. In the
resonance region, the characteristic interference behavior in elastic scattering is even
more pronounced than that sketched in Fig. 2-2. In the Mev region, fast fission is seen to
be a threshold process. U238 is a fertile nucleus in the sense that when it captures a
neutron, it becomes U239 which undergoes two β -decays to reach Pu239 which is fissile.
20
Fig. 2-5. Cross sections of uranium isotope 238.
At the opposite extreme of a light nucleus we can consider the cross section of the
proton, the nucleus of hydrogen atom, shown in Fig. 2-6. The most notable feature here
is the essentially constant value of the cross section at 20 barns over an extended energy
from about 1 eV to well beyond 10 KeV. Given that the thermal absorption cross section
is 0.3 barns, the entire 20 barns can be attributed to potential scattering. An interesting
question is since the proton is the smallest possible nucleus, why should its scattering
cross section be so large, about a factor of 2 or more compared to all the other nuclei.
This clearly suggests that in the case of neutron scattering by hydrogen, the cross section
is not determined only by the size of the nucleus. As we will see later, the answer lies in
the contribution to the cross section from the coupling of the neutron and proton spins
(each has spin 1/2). This explanation is fully consistent with our knowledge of nuclear
interactions as being spin-dependent
Fig. 2-6. Neutron scattering cross section of hydrogen in the low-energy region.
In contrast to the hydrogen cross section, the cross section of carbon, shown in
Fig. 2-7, shows features which arise from the physical state of target atoms. In this case
the target is a crystalline sample. One sees a sharp edge in the cross section variation
which arises from Bragg scattering, a constructive interference effect. Below the edge,
the cross section behaves like 1/v and is temperature-sensitive, higher magnitude at
higher temperature. Beyond about 0.1 eV the cross section reaches a constant at about 5
barns, the same potential scattering behavior as mentioned above. We will come back to
discuss these features in more detail later.
21
Fig. 2-7. Neutron scattering cross section of a polycrystal of natural carbon in the lowenergy region.
Nuclear Data Compilation and Evaluation
As the last part of this lecture we take up the important topic of how information
on nuclear data is gathered, analyzed, and made available to users on a world-wide basis.
Given the importance of having accurate knowledge of cross sections and other data in
nuclear calculations, it is not surprising that there is a whole industry on nuclear data
technology that has been long established and continues to be active in maintaining and
improving the database. The student should beware that our discussion is based on
source material dated around the mid 70's; it is quite possible that significant changes
have since taken place. While detailed information may have changed over the years, the
role of nuclear data in nuclear system analysis and design remains as vital as ever.
Compilation and evaluation of nuclear data are carried out at many national
laboratories and research centers all over the world. In the U.S. the focal point of this
activity has been the National Neutron Cross Section Center at the Brookhaven National
Laboratory. Perhaps the most important contribution of this Center is a library of
evaluated nuclear data, known as the Evaluated Nuclear Data File (ENDF), which was
developed for the storage and retrieval of nuclear data needed for neutronic and photonic
calculations. There are two libraries. ENDF/A is a collection of useful evaluated data
sets. ENDF/B contains the reference data sets recommended by the Cross Section
Evaluation Working Group. At any given time there is only evaluated data set for a
particular material. Most users are concerned only with the B version.
Evaluation of nuclear data means the assignment of the most credible value after
consideration of all the pertinent information. The evaluation is supported by
documentation giving a description of how the value was determined and an estimate of
its uncertainty. As of 1975 the information contained in ENDF consists of: Resonance
22
parameters, cross section tables, angular distributions, energy distributions, double
differential data in angle and energy, scattering law data, and fission parameters.
Four primary centers with responsibilities for collecting and disseminating
nuclear data information have been established to serve the world-wide community. The
Neutron Cross Section Center, Brookhaven National Laboratory, serves U.S. and Canada.
The Neutron Data Compilation Center, Saclay, France, serves Western Europe and Japan.
The Nuclear Data Center, Obninsk, USSR, serves the Soviet Union. The Nuclear Data
Section, IAEA, serves essentially the rest of the world.
For a brief look at the contents of ENDF/B library, we note that ENDF/B-II
contains 3 basic types of data stored on 13 magnetic tapes.
1. Scattering law data for 12 moderator materials. For example, S (α , β ) for a series of
temperatures, 10 temperatures between 296oK and 2000oK in the case of graphite.
2. Neutron cross sections for 78 fissile, fertile, structural, and other materials, each being
an element or an isotope -- total and any significant partial cross sections, reactions
producing outgoing neutrons, angular and energy distributions, radioactive decay chain
data, fission product yield, ν ( E ) .
3. Photon interaction cross sections for 87 elements from Z = 1 to 94 -- photon cross
sections, angular and energy distributions of secondary photons.
Over the years a number of active compilation groups have issued library files: KEDAK
from Karlsruhe, Germany, JAERI (Japan), UKNDL from England, and various libraries
from U.S. national laboratories such as Oak Ridge, Los Alamos, and Lawrence
Livermore.
23
Appendix Chap 7 (2004)
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.
____________________________________________________________________
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).
24
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
&&
equation to be solved is in CMCS and the cross
target nucleus. Thus the Schrodinger
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
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
25
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
3
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 kBT ⎠
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)
26
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
−
+ ...⎟⎟ , x <<1
erf (x) → (2/ π ) ⎜⎜ x − +
3 5 ⋅ 2! 7 ⋅ 3! ⎠
⎝
e− x
x π
2
1−
In (7.5),
β 2 = AE / k BT
⎛
1
⎜1− 2
⎜
2x
⎝
+
⎞
1⋅ 3
1⋅ 3 ⋅ 5
−
+ ...⎟⎟ , x >>1
(2x 2 )2 (2x 2 )3
⎠
(7.7)
(7.8)
, 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 feasible.
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
27
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.
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?
28
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
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
⎛
=⎜
⎝
2
A +1 ⎞
⎟ σ
A ⎠ 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.
29
Fig. 2. 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.
30
Fig. 3. 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.]
31