Chapter'2 Propagation of Acoustic Waves in Crystals

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Chapter.2Propagation of Acoustic Waves in
Crystals

• When applied to a three-dimensional geometry,
  Hookes law takes a relatively complicated form
  because the stiffness components couple two 3x3
  matrices, T c S.
• The nature of the coupling is determined by the
  symmetry properties of the particular crystal
  medium. 21 ? 2
• The threedimensional wave equation is generally
  referred to as the Christoffel equation. It
  admits three solutions, the properties of which
  are determined by the relation of the propagation
  direction to the stiffness matrix.

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Hookes Law in Three Dimensions T c S index
notation
c stiffness matrix s compliance matrix
For i 1, j 1.
.
9x1 matrix
T matrix has 9 components S matrix has 9
components
c s has 81 components (9x9)matrix
9x1 matrix
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Various basic physical principles can reduce the
number and complexity of the stiffness and
compliance matrices.

• From the symmetry of the stress matrix

This reduces the number of independent c values
by 27 to 54.
(6x9 6x6)
Furthermore
The number of independent c values is further
reduced by 18 to a total of 36.
(6x9 6x6)
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or in the matrix form
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The upper left region represents coupling between
longitudinal stresses and strains, and is filled
to all materials. In this region, the diagonal
terms couple stresses and strains in the same
direction the off-diagonal terms (their
presence is dictated by the Poisson ratio) couple
stresses and strains in orthogonal directions.
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The lower right region represents coupling
between shear stresses and shear strains.

• The lower left and upper right regions represent
  coupling
• between shear stresses and longitudinal strains
  and
• longitudinal stresses and shear strains.
• Some of them may or may not be present, depending
  on the
• properties of the material.
• By the same arguments,

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From the definition of energy
Symmetry of the Stiffness and Compliance Matrices
This stiffness matrix is symmetric. Because there
are 30 off-diagonal terms and half of them are
dependent, we conclude that in the most general
case there must be 21 (36 minus 15) independent
terms in the stiffness matrix.
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The Chrisstoffel Equation
or
Differentiating (a) w.r.t. t
a 6x3 matrix
a 3x6 matrix
Index notation
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Now consider a plane wave propagating in a
direction
and
are directional cosine or the projections of the
unit on the three Cartesian axes.
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The l matrix has the same form as the gradient
operator matrix each of its component
represents a propagation direction of the
acoustic wave.
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Likewise, the divergence matrix operator becomes
Christoffel equation in index notation form
a 3x3 matrix Christoffel matrix
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This matrix is 3x3 with elements that depend only
on the propagation direction of the wave
(through the l values) and the stiffness
constants of the crystal.

• Solving the Christoffel matrix involves solving
  an eigenvalue
• problem the eigenvalues are three positive
  numbers that
• are simply the three phase velocities of the
  possible
• propagating waves (one longitudinal two shear
  waves), and
• the three corresponding eigenvectors are the
  particle velocity
• directions, which are defined as the acoustic
  polarization. The
• three polarizations are mutually orthogonal
  (the eigenvectors
• of symmetric matrices are always orthogonal),
  but are not
• necessary parallel to the propagation direction.
  

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Now consider an isotropic crystal in which there
are two independent components of the stiffness
matrix.
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Christoffel matrix
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We now investigate the propagation in xy plane
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This shows that for any direction in the xy plane
there can exist a wave with particle velocity
or polarization that is z-directed.
Pure shear
To have nonzero solution for
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eigenvector
18
They are parallel.
They are perpendicular.
eigenvector
eigenvector
eigenvector
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• There are three velocities corresponding to three
  plane waves. This is true for all crystal
  symmetries.
• The longitudinal mode
• phase velocity
• The shear mode phase velocity
• The phase velocities of the three modes are the
  same for all directions of propagation this is a
  consequence of the isotropy condition.

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Acoustic Propagation in Anisotropic Crystals

• It is customary to speak of seven crystal systems
• (eight if isotropic media are in included).
• Each system represents a general symmetry that is
  further divided into crystal classes, of which
  there
• are 32.
• The acoustically systems are shown in the below
• figure.

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1. Isotropic An arbitrary rotation of the
stiffness matrix leaves the mechanical
properties unchanged. The most important
member of this class is fused (amorphous)
quartz. The stiffness matrix
contains two independent components .
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2. Cubic The x-, y-, and z-axis are
equivalent, and each is a four fold axis of
symmetry. This means that a rotation
leaves the properties of the crystals
unchanged. Acoustically important members
include the semiconductors (gallium
arsenide, gallium phosphide) and
silicon, germanium, and bismuth germanium
oxide (BGO). The stiffness matrix contains
three independent components. There
are five classes in cubic system.
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3. Hexagonal There is sixfold symmetry around
the z-axis. Important acoustic members
include cadmium sulfide and zinc oxide. The
stiffness matrix contains five independent
components and there are seven classes in
the system.
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4. Trigonal There is three fold symmetry
around the z-axis i.e., a rotation of 120
about the z-axis leaves the mechanical
properties unchanged. Important acoustic
members include sapphire ( ), lithium
niobate ( ), lithium tantalate (
), and crystal quartz ( ). There
are five classes in the system, and the
stiffness matrix contains either six or seven
independent components, depending on the
particular class.
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5. Tetragonal There are two equivalent axes (
the x- and y-axes) separated by . There
is fourfold symmetry about the z-axes.
Important members include paratellurite (
), rutile ( ). There are seven
classes, and the stiffness matrix possesses
six or seven independent components.
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6. Orthorhombic There are three major axes,
none of which are equal, each axis possesses
twofold symmetry. Important examples include
barium sodium niobate (
), lithium gallate ( ). This system
contains three classes, and the stiffness
matrix has nine independent components for
all classes.
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Cubic Symmetry

• It has three equivalent axes, we can replace each
• in turn without affecting the material
  properties.
• if x y , y z , z x ,then c is
  unchanged.

21 unknowns
original
transformed
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The final form of the stiffness matrix is
There are three independent components.
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1. Wave propagating alone a major axis, the x
axis
(longitudinal mode)
(pure shear)
2.Wave propagating in the xy plane at an angle of
to the x-axis.
pure shear
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eigenvector
Parallel to the propagating direction.
eigenvector
Perpendicular to the propagating direction.
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Wave propagating in the xy plane in a
directionneither along one of the principal axes
nor at .Because is zero, there is a
pure shear mode polarized in the z direction, as
before.The eigenvalue solution is quite complex.
The mode corresponding to is no longer
precisely parallel to the propagation direction,
and the shear mode (corresponding to ) is no
longer precisely perpendicular. Indeed, the
deviation of these angles depends on the
difference
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Because the angle of deviation form the pure mode
directions vanishes for propagation along the x-
or y-axes as well as at , it is reasonable
to assume that it is a maximum near , and
this is usually the case.One other direction of
interest is the (1,1,1) or body diagonal.
Direct substitution into the Christoffel
equation shows that all modes are pure and that
there is a shear degeneracy.
longitudinal mode
shear mode
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Tetragonal Symmetry
A number of important materials in the class are
invariant under the symmetry operation called
around the z-axis The operation consists of the
rotation around the z-axis and then
inversion through it.
transformed
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After comparison with original stiffness matrix
The two new elements in the lower left and upper
right corners represent coupling between shear
stiffness and longitudinal strains, and vice
versa. These components are present in the
classes , 4 and , but not in
classes 422 and , and thus are not
essential for the symmetry.
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The operation 2 ( a notation about the
z-axis ) eliminates the stiffness components
and . In the following discussion, these
components will be omitted. Thus, the tetragonal
stiffness matrix possesses either six or seven
independent components.
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Wave propagating in the x direction
(longitudinal)
(shear)
(shear)
Propagating in the y direction.
( longitudinal )
( shear )
( shear )
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Propagating in the z direction
(longitudinal)
(shear)
(shear)
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In the xy plane
pure shear mode in z direction
(longitudinal mode)
(shear mode)
Orthorhombic Symmetry
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The form of the stiffness matrix can also be
determined using the symmetry operation 2
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For the principle axes
x - axes
y - axes
z - axes
Problems