Table Of ContentADVANCES IN
ELECTRONICS
Edited by
L. MARTON
National Bureau of Standards, Washington, D. C.
Editorial Board
T. E. Allibone W. B. Nottingham
H. B. G. Casimir E. R. Piore
L. T. DeVore M. Ponte
W. G. Dow A. Rose
A. 0. C. Nier L. P. Smith
VOLUME IV
1952
ACADEMIC PRESS INC., PUBLISHERS
NEW YORK, N. Y.
COPYRIGHT@ 1952 BY ACADEMICPR ESS INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM
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United Kingdom Edition
Published by
ACADEMIC PRESS INC. (LONDON) LTD.
BERKELESYQ UAHROE USEL,O NDOWN . 1
First Printing, 1952
Second Printing, 1964
PRINTED IN THE UNITED STATES OF AMERICA
CONTRIBUTORS TO VOLUME IV
J. S. DONAL,J R., Radio Corporation of America, RCA Laboratories
Division, Princeton, New Jersey
WINFIELDE . FROMMA,i rborne Instruments Laboratory, Inc., Mineola,
New York
H. S. W. MASSEYF, . R. S., Department of Mathematics, University College,
London, England
G. A. MORTONR,a dio Corporation of America, RCA Laboratories Division,
Princeton, New Jersey
M. G. PAWLEYN, ational Bureau of Standards, Corona, California
C. V. L. SMITHO, fice of Naval Research, Washington, D. C.
W. E. TRIEST,I nternational Business Machines Corporation, Pough-
keepsie, New York
ALDERTVA N DER ZIEL, Department of Electrical Engineering, Institute of
Technology, University of Minnesota, Minneapolis, Minnesota
PREFACE
Another volume of “Advances in Electronics” the fourth in the
series, is now presented to the scientific community. It is with deep
satisfaction that members of the Editorial Board note the growing
recognition of these volumes. In fact the book reviews of the past years
have been on the average so favorable that, more than anything else, the
obligation to keep up with the various reviewers’ expectations, has set the
level of this and future volumes. It is sincerely hoped that the present
volume will find as favorable a reception as its predecessors.
L. MARTON
Washington, D.C.
Electron Scattering in Solids
H. S. W. MASSEY, F. R. S.
Department of Mathematics, University College, London, England
CONTENTS
Page
I. Introduction.. ..................................................... 2
11. Elastic Scattering.. ...... ..........
1. Elastic Scattering of Fa
a. Scattering by Free Atoms.. ..................................... 3
b. Relativistic Correction ............ ... 8
c. Validity of Born’s App .......................... 9
d. Comparison of Born’s Approximation with Observation. ............ 10
........ 13
........ 13
b. Effect of Atomic Binding Forces.. ........ .................. 15
III. Inelastic Scattering. ...... .......... ................. 17
1. Inelastic Scattering of s-Born’s Approxi-
mation ......................................................... 18
a. Angular Distribution of the Totality of Inelastically Scattered Elec-
trons.. .... .. ......
b. Total Cross Sections for Inelastic
and Total Ionization ........................................... 23
c. Relativistic Modifications. ....... ..........
2. Experimental Evidence on Inelastic S
3. Influence of the Solid Binding.. .................................... 28
a. Dynamical Polarization. ...................................... 28
6. Study of Low Energy-Loss Collisions with a Solid.. ............... 30
IV. Multiple Scattering. ........... ................. 32
1. The Boltzmann Equation .......................... 33
2. The Angular Dist ................... 34
a. Momentum Loss Cross Section and Mean Free Path.. . . . . . . . . . . . 35
b. Small-Angle Multiple Scattering. ....................... 36
c. Multiple Scattering Distribution in Terms of Projected Angle of
Scattering ..................................................... 40
d. Mean Values.. ............................................... 41
e. Allowance for Energy Loss.. .................................... 41
f. Experimental Evidence on Multiple Scattering of Fast Electrons in
Foils ......................................................... 42
g. Multiple Scattering of Electrons in Photographic Emulsions. ....... 43
3. Space Distribution of Multiply Scattered Electrons-Absorption of
Electrons in Plates.. .............. ......................... 43
1
2 H. S. W. MASSEY
Page
4. The Diffusion Stage.. .. ...................
V. Energy Loss of Electrons in hrough Solids. ......
1. Stopping by Free Atoms ................................. 49
2. Effect of Atomic Interaction in the Solid State.. ................... 51
3. Attempts to Detect Atomic Interaction Effects.. .................... 55
4. The Range of Electrons in Matter.. ............................... 55
VI. The Mobility of Electrons in Metals, Alloys and Semi-Conductors. ....... 58
1. Scattering by Lattice Vibrations-Resistance of Pure Metals.. ........ 61
2. Scattering by Foreign Atoms-Resistance of Alloys.. ................. 63
3. The Resistance of Semi-Conductors.. ........................... 64
a. Non-degenerate Case .................................... 65
b. Degenerate Case ......................................... 66
References.. ..................................................... 66
I. INTRODUCTION
Many phenomena associated with the scattering of electrons by solids
are of great importance and application in various branches of physics.
Electron diffraction provides a valuable supplementary technique to the
diffraction of x-rays and of neutrons for the exploration of the structure
of solid materials. The electron microscope, now proving such an impor-
tant tool in many fields of research, depends for image formation on small
angle scattering by the specimen. In recent years the photographic
plate has become a most useful medium for the investigation of the prop-
erties of high-energy particles including, as a result of the latest develop-
ments, electrons. For these studies it is essential to have reliable infor-
mation on the rate of energy loss of electrons in the solid material of the
plate as well as of the probability of multiple scattering. The determina-
tion of the range of electrons in a chosen solid material has long been an
important method for measuring the initial energy of the electrons. Sec-
ondary electron emission is another phenomenon which has been put to
use in electron multiplier tubes and in any case has always to be reckoned
with in any apparatus in which electrons impinge on a solid. The elec-
trical resistance of conductors and semi-conductors is another property
of great practical importance which is determined by the probability of
scattering of the conduction electrons within the material. In addition
to all these specifically solid state phenomena a great number of funda-
mental investigations on single scattering of electrons have been carried
out, using of necessity solid scatterers. Included among these have been
the attempts, now partly successful, to detect the polarization of electrons
by double scattering.
It is clearly out of the question in the present review to attempt even a
cursory description of all these aspects of the subject. Instead we shall
discuss a selection only. Secondary electron emission has already been
the subject of a review in this series,' and we shall omit any further dis-
ELECTRON SCATTERING IN SOLIDS 3
cussion of it here, Another major subject we shall exclude is electron
diffraction, as many books on this subject2 already exist.
The first section will be devoted to a discussion of single elastic
scattering of fast electrons which may be treated, apart from superposed
coherent diffraction effects, in much the same way as scattering by single
atoms of the material.
Single inelastic scattering mill form the subject of the second section.
Attention will be directed particularly toward obtaining formulas which
are likely to be useful in application to the electron microscope. The
rather meager information available about the probability of energy
losses due to excitation of the loosely bound electrons in a solid mill also
be reviewed.
The theory of multiple scattering and diffusion of electrons in a solid
scatterer forms the subject of the third section, together with a brief
discussion of experimental evidence. Formulas are obtained which allow
for energy loss in passing through the material, but a detailed discussion
of the determination of the rate of energy loss is reserved for the next
section. This section includes an elementary account of the way in
which the dynamic polarization of the medium influences the energy loss.
The final section is concerned with the consideration of the relative
importance of the different scattering processes which determine the
electrical resistance of metals, alloys, and semi-conductors.
11. ELASTICSC ATTERING
If a fast electron enters a solid the chance that it will undergo an
elastic collision in a small distance 6x mag be calculated to a good approxi-
mation by regarding the atoms of the solid as free. This is particularly
true of collisions in which the electron is scattered through a large angle.
Such collisions involve close approach of the electron and an atomic
nucleus so that the modification of the interaction due to the solid binding
is quite negligible. On the other hand the probability of distant collisions
in which the electron suffers only a very small deviation may be markedly
influenced by the solid binding. Thus in a metal in which the atoms are
ionized and the valence electrons more or less free, the charge distribu-
tion of the latter electrons is very different from that in the free atom, and
this will be reflected in the small angle elastic scattering.
We shall begin by a discussion of the elastic scattering by isolated
atoms and then consider in what way the results are modified by the
solid binding.
1. Elastic Scattering of Fast Electrons
a. Scattering by Free Atoms. The scattering of fast electrons by an
atom may be treated by regarding the atom as a static center of force
which exerts a force of potential energy V(r)o n an electron at distance r
4 H. S. W. MASSEY
from it. Polarization and other distortion effects which are important
when the velocity of the electron is comparable with that of the atomic
electrons may be neglected.
The potential energy V(r)o f the atomic field can only be calculated
accurately for atomic hydrogen but for other atoms approximate methods
exist which are accurate enough for many purposes.
For atoms which are not too light the most convenient approximation
is obtained by treating the atom as a statistical assembly of electrons
obeying the Fermi-Dirac statistics. This method, due to Thomas3 and
Fermi4 gives, for a neutral atom,
where Z is the nuclear charge and the length p is given by
p = 0.885aoZ-~, (2)
where a0 is the radius, 0.53 X lo-* cm, of the first Bohr orbit of hydrogen.
The function 4, which represents the effect of the atomic electrons in
screening the nuclear charge, has been tabulated by Bush and Cald~ell.~
+
Various analytical approximations for exist but for many purposes it is
sufficient to write
+(r/p) = e-ar/P (31
where s is a constant of order unity to be determined empirically [see
(30)]. The actual function falls off much more slowly at large distances
but at such distances the statistical theory considerably over-estimates
the field so that (3) probably gives as good an average representation of
the field as may be obtained with any simple formula.
For light atoms the statistical model is not very accurate. In such
cases the Hartree-Fock self-consistent field method may be used. This
has to be determined separately for each atom. Results exist in tabular
form for a number of atoms and ions.6
We first calculate the scattering in the nonrelativistic approximation.
The Schrodinger equation for the motion of an electron of kinetic energy
E( = k2h2/2m)is given by
+
V2+ (k2 - 2mV/h2)+ = 0. (4)
The electron incident in the direction of the unit vector noi s represented
by a plane wave A exp (ik no . r). At a great distance from the scattering
atom the scattered electrons will be represented by an outgoing spherical
wave Ar-1eikrf(0,4).R emembering that the current density of electrons
ELECTRON SCATTERING IN SOLIDS 5
represented by a wave function J. is given by
i€h
j = - (J.* grad # - J. grad #*), (5)
2m
we have that the number of electrons incident per square centimeter per
second is vAA* and that the number scattered into the solid angle dw
about (B,+) is vAA*If(B,+)I2dw. The ratio of these gives the differential
elastic scattering cross section
I(%,+>dw= lf(@)I2dw. (6)
The total elastic cross section Qo is then given by
Thus, if the atom were actually a spherical obstacle of cross section Qo
the number of incident electrons colliding with the sphere per second
would be vAA*Qo exactly as in the above formulation.
To solve the scattering problem it is therefore necessary to obtain a
solution of the equation (4) which, while being a well-behaved function
throughout space, has the- as ymptotic form
+
J. eikno.r r-leikrf(84, ). (8)
Born’s approximation, satisfactory for the discussion of the scattering of
fast electrons by most atoms, may be obtained by treating the scattering
potential ‘v as a small perturbation and regarding the scattering proba-
bility as small. Writing (4) in the form
+
V2# k2# = 2mVJ./h2. (9)
we then substitute for J. on the right-hand side of (9), simply the part
exp (&no . r) representing the incident wave. This gives
+
VY, k2# = 2mVeikno’r/h2, (10)
with the right-hand side a known iunction of r. This equation may be
solved by Green’s theorem to give as a well-behaved solution’
which has the asymptotic form (8) with
where nli s a unit vector in the direction (%,+)m easured with respect to
the direction of incidence as polar axis.
6 H. S. W. MASSEY
Since
Ino - rill = 2 sin +6,
we have
sin 0’ dr’ do’ d+’,
-- - V(r’)s in (2kr’ sin @)r’ dr’. (14)
h2k sin $3
The scattered amplitudef(O,+) may be related to the atom form factor
F for scattering of x rays by writing
where p(rl) is the density of the atomic electrons. On substitution in
(12) and use of the formulas
/om
4R . 1
Ir - rll dT1 = -K2 ’ lim e-ar sin Xrd r = xy (16)
LT+O
we find
p(r’) sin (2 kr’ sin +e)rfd r’,
is the atomic form factor. In this form the term F(0) represents the
shielding effect of the atomic electrons. If it is omitted f(e,+) takes the
well-known form for the Rutherford scattering of electrons by a charge ZE.
If the form (1) is substituted for V(r)i n (14) we have
= (Zp2/ao)G(krs in +O), (19)
where
It follows then that
I(e) = lf(o)121
= (Z2p4/uO2)J(kspin ae),
where
J(Y) = (G(?4)12.
