1. Elastic Properties of Solids Topics Discussed in Kittel, Ch
Elastic Properties of Solids Topics Discussed in Kittel, Ch. 3, pages 73-85

2. Consider the propagation of a mechanical wave
Hooke's “Law”
A property of an ideal spring of spring constant k is that it takes twice as much force to stretch the spring twice as far. That is, if it is stretched a distance x, the restoring force is given by F = - kx. The spring is then said to obey
Hooke's “Law”.
An elastic medium is one in which a disturbance can be analyzed in terms of Hooke’s “Law” forces.
Consider the propagation of a mechanical wave
(disturbance) in a solid.
We are interested in the case of very long wavelengths, when the wavelength is much, much larger than the interatomic spacing:
 >> a
so that the solid can be treated as a continuous elastic medium & the fact that there are atoms on a lattice is irrelevant to the wave propagation.

3. A Prototype Hooke’s Law System
A Mass-Spring System
in which a mass m is
attached to an ideal
spring of spring
constant k. That is, the
Simple Harmonic
Oscillator
(SHO)

4. Simple Harmonic Oscillator
Stretch the spring a distance A & release it:
Fig. 1
Fig. 2
Fig. 3
In the absence of friction, the oscillations go on forever.
The Newton’s 2nd Law equation of motion is:
F = ma = m(d2x/dt2) = -kx
Define: (ω0)2  k/m  (d2x/dt2) + (ω0)2 x = 0
A standard 2nd order time dependent differential equation!

5. Simple Harmonic Oscillator
Hooke’s “Law” for a vertical spring (take + x as down):
Static Equilibrium:
∑Fx = 0 = mg - kx0
or x0 = (mg/k)
Newton’s 2nd Law
Equation of Motion:
This is the same as
before, but the
equilibrium position is
x0 instead of x = 0

6. An Elastic Medium is defined to be one in which a disturbance from equilibrium obeys Hooke’s “Law” so that a local deformation is proportional to an applied force.
If the applied force gets too large, Hooke’s “Law” no longer holds. If that happens the medium is no longer elastic. This is called the Elastic Limit.
The Elastic Limit is the point at which permanent deformation occurs, that is, if the force is taken off the medium, it will not return to its original size and shape.

7. Longitudinal Transverse Sound Waves
Sound waves are mechanical waves which propagate through a material medium (solid, liquid, or gas) at a speed which depends on the elastic & inertial properties of the medium. There are 2 types of wave motion for sound waves:
Longitudinal
and
Transverse 7

8. Sound Waves Longitudinal Waves
Because we are considering only long wavelength mechanical waves ( >> a) the presence of atoms is irrelevant & the medium may be treated as continuous.
Longitudinal Waves 8

9. Sound Waves Longitudinal Waves Transverse Waves
Because we are considering only long wavelength mechanical waves ( >> a) the presence of atoms is irrelevant & the medium may be treated as continuous.
Longitudinal Waves
Transverse Waves 9

10. they correspond to the low frequency, long
Sound waves propagate through solids. This tells us that wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The detailed atomic structure is unimportant for these waves & their propagation is governed by the macroscopic elastic properties of the crystal.
So, the reason for discussing sound waves is that
they correspond to the low frequency, long
wavelength limit of the more general lattice
vibrations we have been considering up to now.
At a given frequency and in a given direction in a crystal it is possible to transmit 3 different kinds of sound waves, differing in their direction of polarization and in general also in their velocity.

11. Consider Longitudinal Elastic Wave Propagation in a Solid Bar
Elastic Waves
So, consider sound waves propagating in a solid, when their wavelength is very long, so that the solid may be treated as a continous medium. Such waves are referred to as elastic waves.
Consider Longitudinal Elastic Wave
Propagation in a Solid Bar
At the point x the elastic displacement (or change in length) is U(x) & the strain e is defined as the change in length per unit length.
x x+dx A

12. In general, a Stress S at a point in space is defined as the force per unit area at that point.
x x+dx
Hooke’s “Law” tells us that, at point x & time t in the bar, the stress S produced by an elastic wave propagation is proportional to the strain e. That is:
C  Young’s Modulus

13. Mass  Acceleration = Net Force resulting from stress
To analyze the dynamics of the bar, choose an arbitrary segment of length dx as shown above. Use Newton’s 2nd Law to write for the motion of this segment,
x x+dx A
C  Young’s Modulus
Mass  Acceleration = Net Force resulting from stress

14. k = wave number = (2π/λ), ω = frequency, A = amplitude
Equation of Motion
So, this becomes:
Cancelling common terms in Adx gives:
This is the wave equation a plane
wave solution which gives the
sound velocity vs:
Plane wave solution:
k = wave number = (2π/λ), ω = frequency, A = amplitude

15. At small λ (k → ∞), scattering from discrete atoms occurs.
Unlike the case for the discrete lattice, the dispersion relation ω(k) in this long wavelength limit is the simple equation:
At small λ (k → ∞), scattering from discrete atoms occurs.
At long λ (k → 0), (continuum) no scattering occurs.
When k increases the sound velocity decreases.
As k increases further, the scattering becomes greater since the strength of scattering increases as the wavelength decreases, and the velocity decreases even further. k ω
Continuum
Discrete

16. Speed of Sound C  Bulk Modulus ρ  Mass Density
The speed VL with which a longitudinal elastic wave moves through a medium of density ρ is given by:
C  Bulk Modulus
ρ  Mass Density
The velocity of sound is in general a function of the direction of propagation in crystalline materials.
Solids will sustain the propagation of transverse waves, which travel more slowly than longitudinal waves.
The larger the elastic modulus & the smaller the density, the larger the sound speed is. 16

17. Speed of Sound for Several Common Solids
Structure Type
Nearest Neighbor Distance (A°)
Density ρ
(kg/m3)
Elastic bulk modulus Y
(1010 N/m2)
Calculated Wave Speed
(m/s)
Observed speed of sound
Sodium
B.C.C
3.71
970
0.52
2320
2250
Copper
F.C.C
2.55
8966
13.4
3880
3830
Aluminum
2.86
2700
7.35
5200
5110
Lead
3.49
11340
4.34
1960
1320
Silicon
Diamond
2.35
2330
10.1
6600
9150
Germanium
2.44
5360
7.9
5400
NaCl
Rocksalt
2.82
2170
2.5
3400
4730
Most calculated VL values are in reasonable agreement with measurements. Sound speeds are of the order of 5000 m/s in typical metallic, covalent & ionic solids :