Ultrasonic Methods in Solid State Physics

469 Pages·1969·7.775 MB·English

by  Rohn Truell, Charles Elbaum and Bruce B. Chick (Auth.)

Ultrasonic Methods
in Solid State Physics
ROHN TRUELL
DEPARTMENT OF APPLIED MATHEMATICS
BROWN UNIVERSITY
CHARLES ELBAUM
DEPARTMENT OF PHYSICS
BROWN UNIVERSITY
BRUCE B. CHICK
DEPARTMENT OF APPLIED MATHEMATICS
BROWN UNIVERSITY
1969
ACADEMIC PRESS New York and London
COPYRIGHT © 1969, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY
FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS,
WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
Berkeley Square House, London W.l
LIBRARY OF CONGRESS CATALOG CARD NUMBER: 68-18684
PRINTED IN THE UNITED STATES OF AMERICA
The untimely death of Rohn Truell came on
January 10, 1968. Let this book be a tribute to his
pioneering work in the field of ultrasonics.
Preface
It is the hope of the authors that this book will provide the reader with
a large part of the special background necessary to read critically the many
research papers and special articles concerned with the use of ultrasonic
methods in solid state physics. The book is intended to help the person
beginning work in this field. At the same time, we trust it will also be useful
to those actively involved in such work. An attempt has been made to
provide a fairly general and unified treatment suitable for graduate students
and others without extensive experience.
The first chapter is concerned with a classical treatment of wave propaga-
tion in solids.
The second chapter deals with methods and techniques of ultrasonic
pulse echo measurements.
The third chapter treats the physics of ultrasonically measurable prop-
erties of solids.
Some details of a specialized character are dealt with more extensively
in the various appendices.
The authors are obviously indebted to many sources of information and
stimulation. We are especially grateful to Andrew Granato, Akira Hikata,
Humphrey Maris, and George P. Anderson. Other colleagues and many
students have contributed in a variety of ways in the preparation of this
book.
We also express our appreciation to Springer-Verlag for permission to
use certain parts of High-Frequency Ultrasonic Stress Waves in solids by
R. Truell and C. Elbaum, in "Handbuch der Physik," Volume XI, part 2.
ROHN TRUELL
CHARLES ELBAUM
BRUCE B. CHICK
vii
Infroduction
The study of the propagation behavior of high-frequency elastic waves
in solids is now rather well established as an effective means for examining
certain physical properties of materials. It is particularly well adapted
to examining changes in such physical properties while they occur. Recent
developments both in techniques of measurement and in understanding of
the mechanisms of energy loss have brought forth results which clearly
demonstrate the capabilities of ultrasonic methods in the study of fundamental
physical properties of solid materials. This book is devoted to studies
of energy loss and velocity of ultrasonic waves which have a bearing
on present-day problems in solid-state physics. The discussion is particularly
concerned with the type of investigation that can be carried out in the
megacycle range of frequencies from a few megacycles to kilomegacycles ;
it deals almost entirely with short-duration pulse methods rather than with
standing-wave methods.
The term "attenuation" is used throughout to mean energy losses (as
measured by amplitude decay) arising from all causes when ultrasonic waves
are propagated through a solid medium. These "total" losses can be classed
broadly as scattering and absorption arising from the intrinsic physical
character of the solid under study, as well as diffraction, geometrical, and
coupling losses.
It is often important in the study of either scattering or absorption
mechanisms that attenuation and velocity measurements be made at the
same time on the same sample. The measurement of both quantities
provides much more information because a complete description of any
wave propagation can only be given when both attenuation and (phase)
velocity are known. Such measurements permit one to study the influence
on the ultrasonic wave propagation of any property of a solid that is
sufficiently well coupled to the lattice.
xiii
XIV INTRODUCTION
While the content of this book is limited mainly to discussion of phenomena
observed at megacycle frequencies, there are sections, such as that on disloca-
tion damping, where the results are of interest in the lower frequency ranges
as well.
The measurement of attenuation and velocity in solids with ultrasonic
waves constitutes a form of spectroscopy much as that found in other branches
of physics. In contrast with the electromagnetic spectrum, in which the
only pure mode form is transverse, the mechanical spectrum includes as
pure modes both compressional and transverse waves, to say nothing of
the more complex types. The availability of transverse and compressional
waves is, however, frequently an advantage because it permits separation
or sorting out of effects which interact differently with the two types of wave.
This ultrasonic spectroscopy of solids has evolved mainly in the last
fifteen to twenty years, and it was natural, as the frequencies began to
approach those of lattice vibrations, that the idea of quanta of vibration
or phonons should be used in conjunction with ultrasonic waves. Any
discussion of the interaction of ultrasonic waves with lattice vibrations or
with defects in a solid follows a pattern very close to that of thermal conduc-
tivity theory. At the same time, the interaction or coupling between ultrasonic
waves and conduction electrons proceeds through the absorption and emission
of phonons. At sufficiently low temperatures there is also energy transfer from
ultrasonic waves to conduction electrons. As a consequence, ultrasonic methods
can also be used to investigate many of the electronic properties of solids.
The range of wavelengths in the ultrasonic spectrum is now rather large.
Assuming that a compressional wave propagating in a solid has a velocity
of 5 x 105 cm/sec, it is seen that with a frequency spectrum from 5 to
50,000 Mc/sec the corresponding wavelength range extends from 1000 to 0.1
micron. There is no reason to regard this as a limit; a wavelength of 0.05
micron (1011 cycles/sec) is still 100 times larger than a lattice spacing in a
crystal.
At the lower end of the frequency range, the low megacycle frequencies,
there are ultrasonic loss mechanisms which bear more directly on the mechan-
ical properties of a solid. The observation of changes in dislocation damping
has a very direct connection with deformation, creep, and stress cycling.
In fact, all of mechanical testing has benefited and should continue to
benefit from information provided by ultrasonic measurements.
It seems quite clear that ultrasonic methods are going to play a larger
and larger part in solid state physics as well as in engineering applications.
1
Propagation of Stress Waves in Solids
1. INTRODUCTION
The propagation behavior of high-frequency stress waves in solids is
determined by the measurement of the attenuation and velocity of the stress
waves as a function of whatever variables are of interest. Such measurements
permit one to study the influence, on the propagation behavior, of any
property of a solid which is sufficiently well coupled to the lattice. Various
types of defects in the lattice are, for example, closely coupled with the
lattice, and a change in the type or density of defects will, in such cases,
change the propagation behavior of a stress wave. Altogether, there are a
dozen or more known "interactions" between stress waves and various
properties of a solid. Electrons are coupled to the lattice and interact with
stress waves in at least two distinct ways. Lattice waves interact with
ultrasonic stress waves, and nuclear quadrupole coupling links the nucleus
with the lattice and hence with ultrasonic stress waves. Radiation-induced
defects of several types have been investigated, and paramagnetic resonance
effects can be influenced by stress waves of the proper frequency.
Before going further with a discussion of such interactions and their
consequences, a review of certain features of elastic wave propagation will be
given.
2. STRESS, STRAIN, AND DISPLACEMENT RELATIONS
Study of the propagation of stress waves [1-5] in an elastic solid yields
information about the velocities of the stress waves and about the attenuation
or energy loss of these waves. The elastic coefficients of a solid may be
obtained from the wave velocities and the density, and these elastic coefficients
1
o 1. PROPAGATION OF STRESS WAVES IN SOLIDS
are in turn connected with binding forces. Deviations from purely elastic
behavior and the connection of such deviations with loss mechanisms and
nonlinear effects will be introduced later.
In an isotropic elastic solid material, which occupies the entire space,
there are two special types of elastic waves that can be propagated through
the material. One type of wave variously called compressional, longitudinal,
or dilatational is one in which the particle motion in a plane wave is along
the direction of propagation. The second type of wave, called transverse,
shear, or distortional, has the particle motion in a plane normal to the
direction of the propagation. In general, however, for anisotropic media
or for bounded isotropic media, waves are neither longitudinal nor transverse ;
the particle displacement has components both along and transverse to the
direction of propagation. When the solid is finite in extent, and hence
bounded by surfaces, surface waves may also be propagated.
Before discussing the equations of motion for a stress wave in an elastic
solid, a brief review of stress-strain relations will be undertaken.
The stress tensor <r. represents the components of force per unit area
i;
acting on an element of area in the solid ; in this discussion, the first subscript
denotes the direction of the normal to the plane on which the stress component
acts, and the second subscript denotes the direction of the stress component.
Because of symmetry conditions arising from the assumed absence of torques,
the stress tensor is symmetric, cr. = a and it makes no difference if the
i; jif
indices are interchanged.
The strain tensor is defined in terms of the coordinates of a point x(x)
{
and x'(^) before and after deformation with displacement sfo):
s = x' — x.
The coordinates x\ of the displaced point are functions of the coordinates
x of the initial position, hence s = s(^).
{
In a body under deformation, the distance between any two points
changes. By comparing the distances between two points, close together,
before and after deformation, it may be shown* that the squares of the
* This is readily seen by using x' = S(XJ) + x(xi). Then
/ dsk
dxk = dxk + —dxj,
OXj
/ dsk
ôxk = àxk + — δχι,
2. STRESS, STRAIN, AND DISPLACEMENT RELATIONS 3
distances differ by a quantity involving 2ε which defines the deformation
7
in the neighborhood of the point in question. ε is called the strain tensor:
7
_ 1 (dsi dsj ds ds\
k k
£jl - 2\d^ + ΊΪ + Ι^ΎχΊ) ' [ ' }
The strain tensor (2.1) is symmetric, and it contains terms of several
types which may be distinguished from one another as follows. Where
and, where / = 1, 1 — 2,
ll(dsA i (dsA i asidSi i ds*ds* i ds* ds*V
ε =
12 2 \\dx ) x^xj dx dx dx dx dx dx \
2 x 2 1 2 x 2
With small deformations and derivatives that are sufficiently small, so
that their squares are negligible, one can use the simplified forms of ε , as
7
is usually done. The second-order terms are neglected, and
8 + sè<s +si,) M
"=ife S) " '
where the comma notation for differentiation is used. The infinitesimal
strains ε^ defined by (2.1) have a simple geometrical meaning; thus, e (/ = /)
;7
is the change in length per unit length of a straight line segment originally
parallel to the x- axis. ε (j Φ I) is twice the change in an angle whose sides
;7
were originally parallel to the Xj and x axes.
l
It is evident, for example, that
£n = 2(si,i + si,i) = «1,1 » ^13 = i(si,3 + «8, l), etc.
The displacement s^, s , s) may be considered to be made up of three
2 3
parts,
S = S + 8' + S",
0
and, by forming the inner or dot product one finds,
, , I dsi dsj ds ds
k k
dx ôx - dx ôx = dXjôxiX— + — + — —
k k k k
\oxj οχι oxj dxi
t
= dxjôxi (2εji),
which defines the strain tensor εμ.
4 1. PROPAGATION OF STRESS WAVES IN SOLIDS
where s is a rigid body translation, s' is a rigid body rotation, and s" is the
0
local deformation. Only the local deformation is of interest for the study of
interactions between stress waves and defect properties of the lattice.
Consequently, we consider only the last term s", and, assuming Eq. (2.2) is
an appropriate approximation, the components of s" may be written as
51 = £H^l i £ΐ2λ2 i £13x3>
s2 — £21^1 ~Γ ε22"^2 i £23^3 »
s3 = e31^1 "Γ ε32^2 i £33% ·
The elastic coefficients are defined in terms of a linear stress-strain relation :
Oik = CikjiEji, iyk= 1,2, 3, (2.3)
^11 = Cllllell + C1112£12 + C1113£13
+ CH21£21 + C1122£22 + C1123e23
+ C1131e31 + C1132£32 + C1133£33 »
or one may prefer to see this in matrix form:
nill *Ί112 ^1113
<Tn = C1121 C1122 C1123 -12 fc22
C1131 C1132 C1133 ε13 ε23
and so on for als, σ21, σ22, (X23, σ31, σ32, ÖT33 , and σ12.
There are nine Eqs. (2.3) with nine coefficients each or 81 coefficients
altogether.
This expression is the most general linear stress-strain law, yielding zero
stress for zero strain. It follows from the symmetry of the stress and strain
tensors (no body moments) that
Cikjl = Ckijl = Ciklj ·
These symmetry conditions reduce the number of coefficients from 81 to
36. Therefore, the most general linear stress-strain relation will depend on
36 coefficients. The additional condition that there exist an elastic potential
(i.e., that the medium be elastic) amounts to having the strain energy be a
function of state and independent of the path by which the state is reached ;
this imposes the further symmetry relation
Oikjl = Cjlik ·