SMALL ANGLE SCATTERING OF NEUTRONS

102 Pages·07.031 MB·English

by  WEISS, RICHARD JEROME

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♦G7 Weias, RicM k I Jetreift^. W 3 -
X951 small augl*. scattering of 'neutrons.
.'*45 3pJcJ-3apt‘ lllus. ,diagrs.
Thesis (Lhoio) - PVi.U., Graduate
i School, 1951o
Bibliography: poJ^a
C 81979,,
loXeutrons. 2 .cj.ioctr.lc discharges
through gases. I.Title* 3* Dissertations,
Academic - K.Y.U. - 1951*
Shall list
Xerox University Microfilms, Ann Arbor, Michigan 48106
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
library of
M W YORK RRlVRRSX'pj
UNIVERSITY HEIGHT.-?
SMALL ANGLE SCATTERING OF NEUTRONS
R. J. weiss
August 31y-195Q"
A dissertation in the Department of Physics submitted in partial
fulfillment of the requirements for the degree of Doctor of Philosophy,
at New York University.
TABLE OF CONTENTS
Introduction p i
Theory p 2
Bragg Reflection p 13
Effect of a Conglomerate of Spheres P 15
Particle Shape P 17
Other Applications p 18
Experiment p 20
Refraction p 20
1. Effect of n. p 22
R
2. Effect of p 22
f
3. Effect of p 2A
A. Effect of \ p 25
5. Effect of a. p 26
b
6, Effect of P 30
Diffraction P 32
1. Effect of P 32
A
2. Effect of P 3A
Intermediate Range P 3A
Magnetic Scattering p 36
Summary P 36
Acknowledgements P 37
Appendix A
Appendix B
Appendix C
Appendix D
Bibliography
List of Symbols
4 G 0 G
r
Co s turning point of tiKB wave equation
?,= [ + ‘-
Gr, - I*'-
- phase shift of /£*^‘ partial wave.
T = positional coordinate
Jn = Btssel function
5 = dist ace wave passes from center ->f particle
Q s angle between incident and scattered wave
X
s angle between chord of partial wave and radius of particle
j? = $:$m6 =
p.ayleigh-Oans distribut:' en fu ction =
0" = total scattering cross section per orrticle
= form /'actor
to = mosaic spre-d
rt»)= scat :ering amplitude of powder sample
Cj"(&)- differential scattering cross section for powder sample
V = volume of powder sa sole
\fj - probability function
71
r no. of particles in powder sanple
U) - beam half width after scattering
M)0= 11 " M before "
Co = deviation of wave at surface of particle
rt = number of particles traversed
s coherent cross section
ct, = density
A = Avagadro's no.
s atomic weight
[) =s constant
-2-
F) = Ilaxwellian distribution function
(T - neutron velocity
/I
X - average value of in the i.axwellian distri but Ion above cr'ii leal
wave length.
C - absorpt or coefficient
Jf. - j«ak intensity of bea irons■rdttei throw ;1 powder sample
^ = total reflected intensit,r from 'Mirror (no sample)
■7" = •o°nk " " " 11 ” "
8 - cons'.ant
A = full width at half :-.axi -.urn (m m )
N - number of particles per cc in powder sample
•p - fracti n of beau suffering 3 ’all an pie scatterin'*
dL - der’s.it-- ior rticle
(jL - a.(parent densit• of powder sample
% - t’-ic mess of .powder sample
M- - neutron '?.guetic moment

8
= me tic induction
-3-
SMALL ANGLE SCATTERING OF NEUTRONS
Introduction
When a well collimated beam of neutrons or x-rays ( ' minutes of
arc) is passed through finely divided material, the beam is found to di-
1,2
verge. This is attributed to diffraction and refraction occurring at
each individual particle. The theoretical interpretation of the broaden­
ing of the beam has been the subject of much discussionj and a great deal
3
of x-ray work has been done in this field. Two "conflicting” theories
4 5 6
dominated the field - that of Rayleigh-Gans * or Guinier (diffraction
ry
only) and that of Von Nardroff (refraction only). As late as 1949 papers
appeared which refuted one theory or the other on the basis of experimen-
“I p
tal results, even though a thesis by Van de Hulst had appeared in 194-6
which showed that for electromagnetic radiation the Rayleigh-Gans and Von
Nardroff theories were both limiting cases of the correct approach to the
problem. (Unfortunately the thesis was published abroad). The important
consideration, as pointed out by Van de Hulst, is the relative phase change
in traversing the particle as compared to the phase change in traversing
the same path length in vacuo. If this phase change p << / the Rayleigh-
Gans theory is valid while if ^ >>/ the Von Nardroff theory is valid.
A recheck into experimental results has shown that this, indeed, was the
case and only the accidental choice of particle size determined the mag­
nitude of
The present work is theoretically and experimentally extended to the
neutron case; the essential difference being that we deal with a scalar
wave equation (Schrodinger*s equation) for neutrons and a vector wave
equation (Maxwell's equations) for x-rays.
THEORY
We begin by calculating the effect of a single spherical particle
containing N nuclei/cc on a plane wave of neutrons impinging on the par-
\ -«»8
tide. The wave length of the neutrons is A ^ 10 cm. and the particle
A .
diameter afI > 100
g
The Schrodinger equation within the particle is
= - «• <p (2)
where = wave number in free space = and <>c , the Fermi poten­
tial, is given by
« . v ' l K Z - w ' & f * (*,
V
The j/ are the indices corresponding to lattice positions and
y
O" = scattering cross section for the y th nucleus
OL* - absorption 11 ii ii
The + refers to the phase of scattering.
- jJf/ . _"i<w5 t^ _ y -3ff x.
For most nuclei « « //OO cOm. J O~' ^ ^ c>*
We drop terms in O a. leaving
We next expand <<. into a Fourier series in periodicities corresponding
V
to reciprocal lattice vectors •>
-. 2 * « r 2 * P & Kr<*Z> u)
* y
»<e f1 ^
Multiplying through by j£ and integrating over we have
-> -7
«•< ■ v - 2v * [ £ A -A * r' (5)
The Schrodinger equation with a periodic potential has a solution of the
form ' -? rf?
.’*-r
r * * 2 c * * <6)
2
* - -
If we substitute this into the Schrodinger equation as well as the
expression for *c we get
‘-9 ^
z r & t /r) - **]je x 2 2 a*k c«je *
/ tic.r **.
Equating coefficients of £ we have
(8)
When there is no Bragg scattering the wave is little deviated and CQ is
larger than all the C „ . We then have
(p*Co£ (9)
£' » wave number in lattice
and
»/7if (io)
M c«
(n)
« r Z * ( £ f
f ** i i(
Substituting ^ y* (XL- = scattering amplitude
we have
(12)
s j
where the j?% »s are the isotopic abundancesj the i's are the nuclear spins
and the CL and are scattering amplitudes for the spins
of the neutron and nucleus parallel and anti-parallel. CL is termed
the coherent scattering amplitude.
The index of refraction is defined as
771L ' &T
if-'‘-if*-- (13)
(**-/) * ‘ ° -firtti
We let
Rewriting the wave equation in the particle by substituting (9) in (1)
we have , .
^ «<£.
^7 ^ */* = O in the medium
and
V $ +~ -O in free space.
9
The general solution can be given as a sum of partial waves.
00
(15)
Jtco
Due to symmetry we need not consider the azimuthal angle *-f
X* satisfies the differential equation
(16)
t _ f i-# o
d.rl
in the particle and by replacing by -ft we have the equation satis­
fied in free space. The differential scattering cross section is given by9
ais)- 2 ^ * 0(1-2 '^) (17)
^ r |
JtsO
where ^ is the phase shift of the Jt ^ partial wave. To determine
this phase shift we make use of the WKB method.^ The WKB method is a
semi-classical approach which is useful when the fractional change in the
wave number is small in a distance . This is true for neutrons and
*«r
x-rays for which the index of refraction is very close to unity, differing
by S’ = 10“6.
The solutions to (16) are^
I
/ I f? *TT* ^
pVn [ j ^ dT— IC ^ the particle (18)
* V
where is the turning point, is the particle radius and
f a ] V - <* £] (19)
ln ^ree space (20)
cv j>-. (%£] ™