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REFLECTION SPECTRA OF NEUTRONS FROM CRYSTALS
M. L. Goldberger
Argonne National Laboratory
SUiqIV OF LLI
This document consists of 9 pages.
Date of Manuscript: February 8, 1945
Date Declassified: June 24, 1947
This document is issued for official use.
Its issuance does not constitute authority
to declassify copies or versions of the
same or similar content and title
and by the same authorss.
Technical Information Division, Oak Ridge Directed Operations
Oak Ridge, Tennessee
UNITED STATES ATOMIC ENERGY COMMISSION
REFLECTION SPECTRA OF NEUTRONS FROM CRYSTALS
By M. Ginsburg, M. L. Goldberger, and F. Seitz
Experiments on the distribution of Bragg reflected neutrons have been performed by Messrs.
W:Sin and Sturm at Argonne and Messrs. Wollan and Borst at Clinton. In each case a collimated beam
:'of neutrons was allowed to impinge upon a crystal and the reflected intensity was measured. In one
!:ase to be described herein, the source of neutrons was a graphite thermal column; in the others,
Radiation taken directly from the pile was used.-We have attempted to explain the observed results
aon the basis of the theory developed by us in reports CP-2419 and CP-2512.
The expression for the reflected intensity from a crystallographic plan characterized by the
generalized Miller indices HHHL is [CP-2419, equation 45]
na k2 2 -kM2mkeT
k2 sin2 e 2mkoT
S/ar sJ iKH. r (2)
aH = aH,HHs = 4nno 4 e
k = 2z/A, A is the neutron wave length, KH = 2 7 (H,bi + H2,b + Hb3), bi is the reciprocal lattice trans-
lational vector, 6 is the Bragg angle, ko is Boltzmann's constant, T is the average neutron tempera-
ture in degrees Kelvin, m is the neutron mass, no is the number of unit cells per unit volume, F) is
the position vector to jth atom in the unit cell. k I K, and 0 are related by the Bragg equation which
can readily be shown to be
KH= KH =2k sin (3)
It is to be noted that the quantity aH is just 4 no times the conventional expression for the X-ray
structure factor in which the atomic scattering power fj has been replaced by + The
question of the ambiguity of algebraic sign entering here will be taken up later.
Equation 1 must be modified to take into account the sensitivity of the detector. BF3 counters are
. generally used in neutron experiments. On the basis of arguments presented in CP-2512, we shall as-
sume that the detector sensitivity for the entire energy spectrum of the neutron beam is proportional
MDDC 1035 [ 1
2 MDDC 1035
to 1/v where v is the neutron velocity. Using the well-known relation v = and combining
unimportant constants, we may write equation I as
C aH| k' -k2mkoT (4)
RH k.k' sins e
where aH = a4 Using equation 3, this may be written as
-o csc2 9
RH =C a4 1H csc3 e (5)
where I = 8mkT
It is important to note in connection with equations 1, 2 and 5 that a different value of aH and KH
must be used for every plane with different indices (H,,H2,H,). Stated otherwise, different values must
be used for every "order" of reflection. There is ambiguity about the meaning of the term "order".
The source of the ambiguity is as follows: It is customary among crystallographers to choose a large
cubic unit cell in preference to the true translational unit cell when treating face-centered cubic
crystals. We shall designate planes in the large cell by (HI,H2,H3), those in the true cell by (K,,K,,K,).
It is found that only half of the values of (H1,H2,H,) can be associated with values of (K1,K,,Ks). Con-
sequently, since reflections are permitted only for planes corresponding to the indices (Ki,K,,K), it
will be found that certain reflections will be forbidden when the (H,,H,,H3) designation is employed.
For example, in reflecting from planes parallel to a cleavage surface in NaCl the only permitted re-
flections correspond to the indices (200), (400), (600), when (HI,H2,H,) are used.
The ambiguity can be resolved by using (H,,H2,H) indices consistently and recognizing that
reflections will always be absent for certain planes. It is convenient, however, to write
H, = nh,
H, = nh,
H, = nh,
where (h,,h,h3) are the Miller indices for the reflecting plane and the integer n, which will be called
the "order" of the (h,,h2,h,) reflection, take successive integer values. Bragg's equation then becomes
nA = 2 d dh ,hsin
where d =h,
hh,,h3 hh h,
in which ao is the length of an edge of the cubic cell. The integer n takes only even values for the (100)
planes. It is readily seen that
MDDC 1035 [3
We shall rewrite 5 to show explicitly the dependence of order:
-a csc2 0
R = C a nK, csc3 e (6)
We shall fix the constant C by the demand that the maximum of the first order reflectivity be unity.
This condition yields
C 3 e 3
la K, ( )(/2
The positions of the maxima are given by
n = sin-an
The final form for the reflectivity which was used in the numerical computations is
nas es/h a- 2 csc 92
Snai;A e IanI anS 6
R a 3 a csc I e
DETERMINATION OF an
It has been pointed out previously that there is an ambiguity of algebraic sign appearing in the
definition of an. This sign cannot be predicted with any degree of accuracy theoretically because of
inadequate knowledge of nuclear energy levels. The only methods at present for determining these
signs are by total reflection_experiments and measurements of the intensities of reflection spectra.
We shall use a combination of these methods in resolving the problems treated in this report. The
spectra to be analyzed are from CaCO3, LiF, and NaCl.
(i) CaCO,.-From total reflection experiments performed at Argonne, it is known that graphite
has an index of refraction for neutrons of less than unity. This implies that the sign preceding the
carbon cross section is negative. The signs for calcium and oxygen alone must be determined. The
I all's for the first five orders taking all possible sign permutations are given in the following
1st 2nd 3rd 4th 5th
(1) 2.971 1.029 2.971 12.971 2.971
(2) 3.353 7.353 3.353 6.647 3.353
(3) 1.029 2.971 1.029 11.029 1.029
(4) 7.353 3.353 7.353 17.353 7.353
The numerical values used were
a Ca = 10 b, aC = 4.8 b, aO = 4.0 b.
The positional coordinates of the atoms in the unit cell may be found in Compton and Allison, X-Rays
in Theory and Experiment, p. 382. The cleavage plane in the unit cell described there is the 211 plane;
the interplanar distance dh is 3.02904 A. Other numerical values used in computing the spectrum from
equation 9 are an2 = 0.08564 n2, for T = 300"K, an2 = 0.06761 nr for T = 380*K.
Referring to the experimental curve shown in Figure 1, we see that rows (2) and (3) in the above
table represent impossible sign permutations, since they predict that the second order reflection
should be larger than the first order. We are left therefore with (1) and (4) which correspond respec-
tively to positive phases for Ca and O, and negative phases for these elements. The reflectivities for
the first five orders for these two possibilities were computed and added. The resulting curves when
compared to the experimental curve indicated that case (1), that of positive phase for Ca and O, is the
This point could be settled by a total reflection experiment with calcite, since the index of refrac-
tion will be determined primarily by the phase of the oxygen atoms in CaCO3, and the phase of calcium
must be the same as that of oxygen. This experiment was performed with the result that calcite is not
totally reflecting; the conclusion is that the phase of the oxygen and hence of calcium is positive, as was
(ii) and (iii) LiF and NaCl.-The determination of Iani for both of these crystals may be discussed
simultaneously because they are both of face-centered cubic structure. We consider reflections from
the 100 or cleavage plane for which lan = 2(1 + cos nn)(f, + f,) where fj = /sJ. We notice that an is
zero when n is an odd integer and takes equal non-vanishing values when n is an even integer. Thus we
do not need to know the scattering cross sections in the cases of LiF and NaCl and an may be absorbed
in the constant C of equation 9.
THE REFLECTION SPECTRA
(i) CaCOs-Thermal column radiation.-The reflectivity computed with positive phases for Ca and
O has been plotted for two temperatures in Figure 1. There is some uncertainty about the neutron
temperature, although it is reasonable to expect that neutrons at the end of the thermal column are at
room temperature. The maxima of the theoretical curve can be made to coincide with the experimental
maximum by choosing a temperature of 340 K; however, this does not explain the important points of
disagreement which will be discussed later.
CaCO3-Side hole radiation (Figure 2).-The agreement between theory and experiment is quite
good in this case. The average temperature of the neutrons is 400*K, a value which checks quite well
with velocity measurements made by pyrex plate transmission. The discrepancy at the extreme high
energy side will be explained below.
(ii) LiF- Side hole radiation (Figure 3).-This crystal has particular interest since the nuclei
occurring in it possess a spin, in contrast with the case of calcite. The agreement here is again quite
good at the high energy end (except at very high energies) whereas there seem to be fewer low energy
neutrons in the beam distribution than would be predicted by a Maxwellian distribution. The good
agreement in this case indicates that the spin dependence of nuclear forces which has not been taken
GLANCING ANGLE DEGREESI -
Figure 1. Reflection spectrum from calcite (thermal column radiation).
-- THEORETICAL CURVE (T= 400'K)
o 0 EXPERIMENTAL CURVE
0.5 ______ ___
0.6 o, o
0 I IJ I I I I I I
L 40 50
GLANCING ANGLE (DEGREES)
Figure 2. Reflection spectrum from calcite (side hole radiation).
I i I
0 2 4 6 10
6 ] MDDC 1035
into account explicitly in deriving equation 1 does not cause appreciable incoherent scattering in these
(iii) NaC1-Side hole radiation (Figure 5).-The NaCI spectrum was measured by Mr. Wollan at
Clinton, whereas the previous work was done by Messrs. Zinn and Sturm at Argonne. In addition to the
resultant curve we have_ included a breakdown into orders (Figure 4). From this curve we see that to
angles of about 11" the reflected spectrum consists of only first order reflection. The agreement shown
in Figure 5 is extremely good, again indicating that the nuclear spin does not cause appreciable inco-
herent scattering. The temperature at the maximum is 548"K.
DISCUSSION OF RESULTS
The greatest source of error in the theoretical calculations arises from the uncertainty in the
energy distribution of the incident beam. The disagreement between the theoretical and experimental
curves shown in Figure 1 is most pronounced at two places: extremely low energies and energies
slightly less than the most probable as computed from a Maxwellian distribution. The most .reasonable
explanation of these discrepancies is that the energy distribution is not exactly Maxwellian, but is
distorted on the low energy side of the maximum. The experimental results could be explained if we
assumed that there is an excess of extremely cold neutrons and a deficit of those having energies
somewhat less than the most probable value. This would explain the minimum at 25" as well as the
slight rise from 350 to 500. Such a distribution would not be entirely unreasonable since it is known
that very slow neutrons diffuse through great lengths of graphite without suffering many collisions and
hence do not come into thermal equilibrium with the graphite. This effect is sometimes called the crys-
tal effect. The dip in the proposed distribution could be the result of a preferred absorption of neutrons
having an energy near koT. No attempt has been made at the present time to derive a corrected energy
The principal discrepancies in the remaining spectra can be explained as follows: The neutron
distribution at high energies is no longer Maxwellian but is dE/E. This distribution is discussed by
S. M. Dancoff in CP-2171.
SUGGESTION FOR FUTURE EXPERIMENT
The fact that Ianl depends upon f, + f, for reflections from the (100) planes of crystals having the
NaCI structure, whereas an depends upon f, f, for the (111) reflections, indicates that measured
values of the reflectivity from the two types of planes for a given wave length can be used to determine
the ratio f, f,. It is important to note that the quantities required to determine this ratio are not the
absolute reflectivities but the relative reflectivities for a given incident beam. Since the reflectivity of
a crystal is a function of its perfection, it evidently is almost necessary that the same crystal be used
to determine the relative reflectivity from the two types of crystallographic planes. If the elements in
the crystals are monoisotopic, fL and f, are proportional to the square roots of the scattering cross
sections so that this ratio is given at once. Thus if one of the elements possesses isotopes in compara-
ble abundance since then fi is proportional to + gBat where a is summed over all the isotopes
and ga is the fractional abundance of the ath isotope.
The principles involved in the foregoing method of determining the relative scattering cross section
of the elements in a diffracting crystal are obviously not restricted to crystals having an NaCI structure,
but can be used whenever different crystallographic planes involve the fi in independent ways.
If the flux of the incident beam, the dimensions of the beam, the dimensions of counter chambers,
and the counter efficiency, and the magnitudes of the scattering cross sections for a given crystal were
known, it should be possible to predict from equation 1 the actual number of counts recorded by the.
counters for every sign permutation in lanl and hence one would theoretically be able to determine the
signs from a single measurement.
s-*-Ist. p u-
4.0 ---- THEORETICAL CURVE(T=400"K)
0.- ------ "- --------
04 --1- -----------------------
o 1 I
0 2 .6 8 0 12 14 16 0 22 24 P" R O2 3 40 4 46 48 50 s2 54 56 56 60 62 64 66 68 70 22 74 76
GLANCING ANGLE (DEGREES) -
Figure 3. Reflection spectrum from LIF (side hole radiation).
0,9 I 1 I I/ \ I------------------------------
F1 / 2
SI I .------ I
0 2 4 6 8 40
I 40 50
GLANCING ANGLE (DEGREES)
Figure 4. Reflection spectrum from NaCI (resolution of orders).
8 I MDDC 1035
.0 -- THEORETICAL CURVEIT-54.S K
Saoo EXPERIMENTAL CURVE
I 0.4 _,_ _.
0 10 20 30 40 0o 60 70 so80
GLANCING ANGLE (DEGREES) -
Figure 5. Rerlection spectrum from NaCI (side hole radiation).
0.- I \
GLANCING ANGLE iIN DEGREES)
Figure 1. Reflkrlion spectrum from calkte (resolution of orders; side hole radiation).
(1) The recent experiments at Argonne and Clinton on the reflection spectra from crystals are
discussed. A theoretical treatment based on reports CP-2419 and CP-2512 is given.
(2) The reflected intensity from a crystallographic plane characterized by the Miller indices
(hh,hs) for a given order, n, is given by
-a 2 csc2 0
Rn = C an nKI csc Oe n
where K, = a 2= 2 ,
dh n 8mkoT
an i 2 Tn(h,xj + hyj + hzj)
a = + (b) e
where C is an arbitrary constant, dh is the interplanar spacing, A is Dirac's constant, m is the
neutron mass, ko is Boltzman's constant, T is the average neutron temperature, asj (b) is the
scattering cross section of the jth atom in barns, and (xj,yj, zj) are the coordinates of the jth atom in
the unit cell.
(3) The question of the ambiguity of algebraic sign appearing in the definition of an is discussed
and it is shown how by combining the results of reflection spectra and total reflection data this
ambiguity may be resolved. This problem is discussed in detail for CaCO,.
(4) The effect of nuclear spin in producing incoherent scattering is considered. The good agree-
ment between the theoretical and experimental curves in the cases of LiF and NaCl indicates that this
effect is negligible since the derivation of the expression for Rn neglects spin effects.
(5) On the basis of the reflection spectrum from calcite obtained with radiation from a graphite
thermal column an energy distribution for the neutron beam'has been proposed. It is believed that the
energy distribution is not exactly Maxwellian, but is distorted on the low energy side of the maximum.
The experimental results could be explained if we assumed that there is an excess of very cold neu-
trons and deficit in those having energies somewhat less than the most probable.
The energy distribution in the neutron beam coming directly from the pile is Maxwellian over the
entire range with the exception of a high energy tail. This is explained by the fact that the high energy
distribution is given by dE/E where E is the neutron energy.
(6) An experimental method for determining the scattering cross sections of certain elements by
the use of reflection spectra data is proposed. This method is feasible whenever the an's for different
planes involve the scattering cross sections in independent ways.
UNIVERSITY OF FLORIDA
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