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  8. Chapter#06: Stastical Mechanics of Interacting System

Chapter#06: Stastical Mechanics of Interacting System

Team Quanta presents all possible short questions of Thermal & Statistical Physics’ Chapter#06: Stastical Mechanics of Interacting System.

Q.1: Give an evidence for existence of phonons.

Answer: The experimental evidences that energy of an elastic wave is quantized or for existence of phonon are;

  • The lattice contribution to heat capacity of solids approaches to zero at absolute zero. This can be explained only if lattice vibration is quantized.
  • In inelastic scattering of X-ray photons or neutrons by crystals with momentum and energy changes according to creation or absorption of one or more phonos.

Changes according to creation or absorption of one or more phonons.

Q.2: What do you mean by lattice vibrations?

Answer: Lattice vibration is study of vibrations of atoms about their equilibrium position in a solid. These vibrations are almost entirely responsible for thermal properties namely heat capacity, thermal conductivity, thermal expansion etc of insulators and contribute greater part of heat capacity of metals.

Q.3: For elastic waves what are allowed values of phonon wave vector?

Answer: For elastic waves only those values of phonon wave vector are allowed for which phase  lies between -\pi and +\pi or  lies between -\frac{\pi}{a}  and +\frac{\pi}{a}

Q.4: What is difference between elastic vibrations and electromagnetic waves?

Answer: Electromagnetic waves exhibit only transverse character where as elastic vibrations Have both longitudinal and transverse characters. Elastic waves require material media to propagate while electromagnetic waves can also travel in vacuum.

Q.5: State important characteristics of phonon.

Answer: Characteristics of phonon are given below:

  • Phonon is result of quantization of lattice vibrations.
  • The energy of phonon of frequency  is \hbar\vec{k} .
  • Phonon has no physical momentum, but crystal momentum is .
  • Spin of phonon is zero; hence phonons obey Bose-Einstein statistics.
  • Phonons travel with Speed of sound in solids.
  •  

Q.6: Differentiate between phonon and photon.

Answer: Difference between phonon and photon is given below:

            Photon:-

  • Photon is quantum of electromagnetic radiation.
  • Photon travel with speed of light.
  • Photon can travel in vacuum.

Phonon:-

  • Phonon is quantum of elastic vibrations.
  • Phonon travels with speed of sound in solids.
  • Phonon can not travel in vacuum.

Q.7: What is basic drawback of Einstein model of specific of heat?

Answer: Einstein model was failed to establish the fact C_V\propto T^3 . This failure is due to reason that Einstein assumed that all atomic oscillators vibrate with same frequency. This discrepant was removed by Debye.

Q.8: How did Debye modify Einstein model of specific heat?

Answer:  Debye said that atoms in crystals are coupled together, so they vibrate with number of frequencies rather than a single frequency as Einstein said. Debye considered all possible modes of vibrations limited by number of atoms in crystal to produce the frequency spectrum.

Q.9: What are limitations of Debye model?

Answer: Limitations of Debye model are given below:

  • Debye model is valid for long wavelengths i.e. only low frequencies are active in solid. Therefore this model ignores dispersion of waves.
  • At low temperature Debye  law does not hold in temperature range \le0.1\theta_D. According to Debye model \theta_D=hv/k_B is independent of temperature but actually \theta_D is found to vary with temperature.
  • The number of vibrational modes is assumed to be 3N. As solid is taken to be an elastic continuum it should have an infinite number of frequencies.
  • Debye theory completely ignores the inter-atomic interaction and electronic contribution lo specific heat of lattice

Q.10: Give a comparison of Einstein and Debye models.

Answer: Results of both models tell us that;

  1.  At high temperature both Einstein and Debye model gives same result C_V\ =\ 3R which agrees with Dulong-Petit law.
  2. At low temp. Einstein theory predicts C_V\propto e^{-\frac{\hbar\omega}{k_BT}}  whereas Debye model says that C_V\propto T^3 which is more closely in agreement with experimental results.

Q.11: Dulong-Petit law is found to generally correct at room temperature. Comment this statement by stating failure of Dulong-Petit law.

Answer: Dulong-Petit law is found to generally correct at room temperature and above for elements of atomic weight greater than 40. The following experimental observations show That this law fails under other conditions:

  • Al low temperature, specific heat of all solids is found to approach zero as  in insulators and as  in metals. In superconductors, the drop is faster.
  • A number of low atomic number and high melting point elements like B, Be, C and Si show lower specific heals than predicted by this law. A number of electropositive metals like Na, Cs, Ca and Mg show an increase of specific heat above maximum value 3R with an increase of temperature.
  • In magnetic solids there is a large contribution to specific heat near the temperature at which magnetic moments become ordered.

Q.12: Silver metal obey Dulong-Petit law at room temperature but diamond does not. Explain.

Answer: According to Einstein model, when hv\ll k_BT, \ \ C_v\rightarrow3R ,  .

At room temperature,

    \[k_BT=1.38\times{10}^{-23}\times300=4.14\times{10}^{21}\ J\]

For silver,

    \[hv=6.63\times{10}^{-34}\times4\times{10}^{12}=2.65\times{10}^{-21}\ J\]

For diamond,

    \[hv=6.63\times{10}^{-34}\times2.4\times{10}^{13}=15.9\times{10}^{21}\ J\]

It is clear that for silver hv<k_BT and for diamond hv>k_BT, so much higher temperature than room temperature is required for Dulong-Petit law to be obeyed for diamond.

Q.13: In low temperature range, energy of solid rises with rise in temperature by means of two mechanisms. State these mechanisms.

Answer: The two mechanisms are:

  • The increase in average energy of every normal mode is proportiorial to  due to rise in probability of its excitation
  • The increase in number of normal modes of lattice is proportional to T3, so total energy of excitation is proportional to T4 and hence rise in specific heat is proportional to .

Q.14: Define the following:

  • Heat capacity
  • Lattice specific heat
  • Kg-mol

Answer:

Heat capacity:- The amount of heat necessary to raise the temperature of a system through one degree is called heat capacity.

Lattice specific heat:- The contribution to total specific heat due to transition of vibrating atoms in the crystal lattice to vibrational states of higher energy is known as lattice specific heat.

Kg-mol:- A kg-mol or  of any substance is that mass of substance that contains a specified number of molecules 6.023\times{10}^{26} , called Avogadro number.

Q.15: How many degrees of freedom are there for a diatomic molecule in a two dimensional world?

Answer: In two dimensions, there are two degrees of freedom for translation, one for rotation and two for.

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