14 Lattice Vibrations and Thermal Properties 5

Mahavir Singh

epgp books

 

 

 

The objective of the module is to

  • Understand the importance of neutron scattering in order to study phonon interaction in detail.
  • Have an insight of importance of anharmonicity in crystal interactions.
  • Correlate thermal expansion and thermal conduction in context of anhamonic crystal interaction.

   1. Neutron scattering:-

 

The neutrons have a great advantage to be used as a probe in condensed matter physics. Scattering of free neutrons with the matter is used to measure details which are of prime importance in material research. Elastic scattering (neutron diffraction) is used in determining structures whereas inelastic scattering provides valuable information on atomic vibrations and excitations. The interaction of phonons is assumed as scattering collisions.

 

A neutron sees the crystal lattice by interacting with the nuclei of the atoms and exchange energy which is used to measure details of atomic and molecular motions by the use of inelastic neutron scattering (INS). When a crystal gets scattered by a crystal, it results either in absorption or emission of energy of an amount equal to quantum of phonon (hν). The wave vectors before and after the interaction gives information about the phonon energies.

 

Here we study the neutron scattering and phonon interaction dynamics in detail.

 

1.1 Elastic scattering:

 

In elastic scattering of X-rays in a crystal (Bragg’s diffraction) the magnitude of wave vector (or the frequency ) remains unchanged i.e,

 

ω = ώ and

 

?+G = ḱ

 

where k and ḱ are wave vectors of incident and scattered photons, which obey selection rules. G is one of the reciprocal lattice vectors.

 

Also the energy and momenta are conserved i.e,

 

ħω = ħώ              and

 

ħ   + ħG = ħḱ

 

As the frequencies of incident and scattered photons are unchanged the crystal as a whole recoils with momentum -ħG . This process is called N-Process.

 

1.2 Inelastic scattering:

 

But in case the scattering is inelastic, then the process leads to either emission or absorption of a phonon of wave vector K. This phonon will interact with photons, neutrons and electrons as if having a momentum ħk ,however a phonon carries no physical momentum on lattices as the phonon co ordinates (other than k = 0) are linked with the relative co ordinates of the constituents of the lattice. But for the sake of practical purpose, a phonon is considered to have its momentum, more precisely referred to as crystal momentum.

 

The selection rule so obeyed is as under

 

ω = ώ +?

 

here ?  is the frequency of the emitted phonon.

 

And energy and momentum conservation laws become

 

ħω = ħώ + ħ? and

 

ħ? + ħG = ħḱ + ħK

 

The conservation of the selection rules require that

 

K + G = ḱ ± K

 

It is found that the frequency of incident photon and scattered photon is not same i.e. ω ≠ ώ and the momentum is transferred to the whole crystal. This process is known as U-Process.

 

In neutron elastic scattering process a neutron interacts with the crystal by interacting with the nuclei of the atom. It is used for determination of phonon dispersion relation.

 

 

 

 

 

 

 

 

 

 

 

2 Anharmonic Interactions in Crystals

 

The study of lattice dynamics in this chapter has so far been dealt only with potential energies to quadratic terms and the higher terms have been neglected. The lattice vibrations and atomic motions are studied within the harmonic approximations, under which the consequences are that the lattice waves never interact and the wave form never changes or decays with time. To deal with real time phenomenon of thermal expansion, thermal conductivity, temperature dependence of elastic constants and the problem of increase in heat capacity at T > θ ,the anharmonic effects are to be taken into account.

 

Let for an atom in any position obeying Hook’s law, the energy expression is

 

Єr = Єro + A(r-ro)2

 

In such a case, the phonon interaction becomes impossible.

 

One cannot neglect anharmonicity which is a source of coupling between lattice waves and makes scattering of waves possible.

 

Therefore energy expression has to be of the form

 

Єr = Єro + A(r-ro)2 +B(r-ro)3 + _ _ _ +_ _

 

The higher order anharmonic terms makes phonons to collide with perfect crystal by making mean free path of finite dimensions.

 

3 Thermal Expansion in Solids

 

The general answer to why do solids expand on increasing temperature is that an increase in energy results in an increase in equilibrium spacing of atomic bonds.

 

Thermal expansion is measured by the coefficient of linear thermal expansion defined as the increase in length per unit rise in temperature. The thermal expansion in solids is a direct consequence of an harmonicity of the atomic interactions. With the increase in temperature the amplitude of lattice vibrations increases and the equilibrium position shifts as the atoms spend time at greater than original spacing due as the repulsion at short distances is greater than the corresponding attraction at farther distances.

 

In case of a one dimensional solid if the vibrations are considered harmonic , the potential energy will be

 

 

Fig (2) showing the change in equilibrium position of vibrations with respect to energy

 

And it will be a parabola with xo as the equilibrium position, then the harmonic potential is a perfectly parabola function and the mean position does not shift with rise in temperature.

 

The mean displacement ,

 

 

But the atomic oscillators differ from perfectly harmonic oscillators and essentially have harmonicity in the potential with comes into play with higher order terms in ‘x’

 

Therefore for  anharmonic potential,

 

 

 

 

 

 

 

 

 

 

 

The above equation suggests a prominent anharmonicity or a departure from harmonic behavior and the curve starts to deviate. The mean positions start shifting with increasing temperature corresponding to thermal expansion in physical behavior as demonstrated in figure (2).

 

For small anharmonicity the first term of the numerator takes the form as under

 

 

 

 

 

 

 

We find that there appears a non zero mean displacement in the presence of the anharmonicity which contributes to thermal expansion.

 

4. Thermal Conductivity

 

As it has been discussed in the previous sections how heat energy gets transmitted through annihilation and creation of phonons in a crystal. An attempt is made to understand the heat conductivity and energy transmission in solids taking phonon factor into account.

 

For a long rod let a steady heat flows along its length with temperature gradient dT/dx. It is assumed that the temperature variation is very small such that the average phonon number is definite. As the neighboring regions have slightly varying temperatures, the phonon number behaves as a function of position. The energy transmitted across unit area per unit time called the flux of thermal energy is given as

 

Eth = – K dT/dx

 

Where K is the coefficient of thermal conductivity in solids.

 

Consider phonon gas analogous to molecular gas then on the basis of kinetic theory of gases,

 

K = 1/3 Cvνλ

 

Cv is the lattice specific heat which is a measure of phonon density.

 

ν is the average phonon velocity and λ is the mean free path of the phonons.

 

Hence the expression for thermal flux reduces to

 

Eth = – 1/3 Cvνλ dT/dx

 

Actually total thermal conductivity receives contribution from conductivity due to electrons and phonons each, hence

 

Ktotal = Kelectron + Kphonon

 

The expression for thermal conductivity due to electrons is deduced and is

 

Kelectron = [π2Nk2τ/3m]T

 

Here N is the number of free electrons per mole and τ is the mean free time of the electron before collision with the positive ion where it gives its entire thermal energy.

 

It is observed that in the expression, Kelectron is almost independent of temperature as the mean free time τ varies as T-1 above Debye’s temperature θD.

 

So the phonons carry most of the heat energy while electrons stay immobile. We have studied in detail the lattice specific heat as given by Debye’s law as

 

Cv = αT3 for T ˂ θD.

 

And Cv = 3Nk for T ˃ θD.

 

Hence the corresponding thermal conductivity expressions are

 

Kphonon = 1/3 νλ αT3 for T ˂ θD

 

Kphonon = 3 νλ Nk for T ˃ θD.

 

The above expressions clearly indicate that thermal conductivity of insulators is constant at high temperatures but is proportional to T3 at lower temperatures and this variation in K arises due to phonon – phonon interaction or as we said anharmonocity. The phonon mean free path λ is found to be inversely proportional to absolute temperature so consequently the expression for thermal conduction due to phonons becomes

 

Kphonon  α T-1

 

So thermal conductivity of an insulator is proportional to T-1  at high temperatures.

you can view video on Lattice Vibrations and Thermal Properties 5

 

Summary:

 

After the completion of this module we are able to understand the following

  • Scattering of neutrons is a probe to study collisions in order to obtain information regarding lattice dynamics through phonon energies.
  • Inelastic scattering and the concept wave vector linked with absorption or emission of phonons is important to determine phonon dispersion relations.
  • Have an insight of importance of anharmonicity in crystal interactions Correlate thermal expansion as a direct consequence of anharmonicity.
  • Visualize thermal conduction in context of anhamonic crystal interaction.

    Value Addition:

    The phonon interaction with photons, electrons, neutrons , magnons and excitons are of prime importance in studying and understanding the physical properties of solids.

 

As in weak coupling cases, scattering caused by phonons puts a limitation on the mean free path of electrons and its importance is well appreciated in understanding the conductivity of metals and mobility of carriers in semiconductors.

 

The electron – phonon interactions have gained significant interest and importance in solid state physics. The transition metals and their compounds are the area of interest in this concern. Interestingly the electron – phonon interactions are thought to be responsible for high temperature superconductivity in compounds like V3Si and NbC.

 

Suggested reading

 

Physics through The 1990’s Condensed Matter Physics , National Academy Press, Washington D.C., 1986

    Glossary:

 

Phonon interaction: it is assumed as a scattering collision during which at a time only a single phonon is absorbed or emitted. Also the energy of the lattice vibrations involved is an integral multiple of ħω.

 

Total energy of phonons: it is actually the thermal energy of the solidbecause at a given temperature the solid is assumed to be completely filled with phonons.

 

Temperature gradient: it is the variation in temperature with respect to distance from the hot end to the cold one.

 

Mean phonon free path:    the average of all the lengths travelled by the phonon between collisions.

 

Excitons : these are electron –hole pairs bound by coulombs electrostatic force that is responsible for transport of energy without transporting net charge.