Review
The Schrodinger equation is used to find the partition function, which is related to thermodynamic quantities. Decouple degrees of freedom to divide the Hamiltonian into multiple components. By the rigid rotor approximation, the rotational and vibrational degrees of freedom are independent. Two other degrees of freedom, electrical and nuclear, are not included in the relation below.
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<br>
<math>\hat H_{\mbox{particle}} = \hat H_{\mbox{trans}} + \hat H_{\mbox{rot}} + \hat H_{\mbox{vib}}</math>
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</center>
Energies are found by a summation. The single-particle partition function is a product of different components. The Boltzman approximation is used at high temperature.
Vibrational spectrum
Solve the Hamiltonian below to find the vibrational spectrum. There are vibrational frequencies, and the force constants say how rigid the vibrations are. Sum over all the states and introduce <math>\theta_v</math>. This term is a measure of how easy it is to excite vibrational modes. A large value is associated with difficulty exciting vibrations.
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<br>
<math>\hat H_{\mbox{vib}} = \frac
\frac
+ \frac
f \zeta^2</math>
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</center>
Rotational spectrum
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Solve a Hamiltonian in spherical coordinates. There is a measure of how easy it is to excite. Consider cases of high temperature and low temperature. Approximate with an integral when the temperature is much greater than <math>\theta_r</math>. Consider a few terms when the temperature is about equal to <math>\theta_r</math>. A value of a vibrational portion of partition function is below. Read in MacQuarrie about symmetric and asymmetric cases.
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<br>
<math>\frac
= q_{\mbox{vib}}</math>
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Additional notes
Decouple and sum energies of states. The term <math>\theta_r</math> can be very small. Evaluate heat capacity from various degrees of freedom. Use limits, such as a case when the temperature is much less than <math>\theta_v</math> and greater than <math>\theta_v</math>. There are two vibrational degrees of freedom and four rotational degrees of freedom 
Heat Capacity
Consider the vibrational portion of heat capacity in the 19th century before quantum mechanics. According to classical theory, the heat capacity is constant as a function of temperature. But measurements at low temperature demonstrated that heat capacity varies at low temperature. Einstein contributed in 1907. There is a quantum mechanical concept of vibrations. Vibrational energy is quantized, and at low temperatures, there is an impact due to the quantization of energy. New energy are not able to be reached.
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Vibrations in a monoatomic solid (crystal)
A first approximation is analogous to the model of vibrations of a diatomic molecule. Below is an expression of energy as a function of displacement.
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<math>U( \zeta) = U(0) + \frac
\frac
\zeta^2</math>
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<math>\frac
= f</math>
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</center>
Consider localized atoms on a surface wherein the potentials are independent of what other atoms are doing. Each particle does not interchange position. Each atom attempts to restore back to equilibrium and all are independent. There is a single-particle Hamiltonian. Solve the Schrodinger equation.
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<br>
<math>\left ( \frac
\frac
+ \frac
f x^2 \right ) \Psi_s = \epsilon_s \Psi_s</math>
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<math>\epsilon_n = \left ( n + \frac
\right ) \hbar \omega</math>
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<math>\omega = \sqrt{ \frac
}</math>
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</center>
The force corresonding to the potential is <math>f</math> and the mass is <math>m</math>. Imagine the particle being restricted by force but not seeing other particles. Use energies to find thermodynamic quantities. Sum over all states to find the single-particle partition function.
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<br>
<math>q_{\mbox{single}} = \sum_
^
e^{-\beta \left ( n + \frac
\right ) \hbar \omega}</math>
<br>
<math>\theta = \frac
</math>
<br>
<math>q_{\mbox{single}} = \frac{e^{\frac
}}{e^{\frac
{T}} - 1}</math>
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<math>Q_{\mbox{system}} = q^N</math>
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</center>
Generalize the result to three dimensions. The partition function is a measure of the number of thermally accessible states. Increasing to three dimensions results in increasing the number of available states by a power of three.
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<br>
<math>Q_{\mbox{system}}{\mbox{3D}} = \left ( Q_{\mbox{system}}{\mbox{3D}} \right )^3</math>
<br>
<math>Q_{\mbox{system}}{\mbox{3D}} = \left ( \frac{e{\frac
}}{e^{\frac
{T}} - 1} \right )^3 </math>
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</center>
Find properties with the partition function. Below is an expression of energy and values considering different temperature limits.
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<br>
<math>E = \frac
+ 3 N k_B T \left ( \frac{ \frac
{T}}{e^{\frac
{T}} - 1} \right )</math>
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<math>\lim_
E = \frac
</math>
<br>
<math>\lim_{T \to \mbox{high}} E = 3 N k_B T + \frac
</math>
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</center>
An expression of heat capacity with expressions considering different limits.
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<br>
<math>C_v = 3 N k_B \left ( \frac
\right )^2 \left ( \frac{ \frac
{T}}{\left ( e^{\frac
{T}} - 1 \right )^2} \right )</math>
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<math>\lim_
= 0</math>
<br>
<math>\lim_{T \to \mbox{high}}
= 3Nk_B</math>
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</center>
The result agrees experimentally to some degree. This helped convince people that quantum mechanics could be true. But atoms in a solid are not expected to behave independently. There is a discrepency between the experimental and predicted values. The value goes to zero to rapidly. Consider the Einstein model. The last relation below demonstrates a material dependence.
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<math>\theta = \theta_E</math>
<br>
<math>\theta = \frac
</math>
<br>
<math>\frac
\sqrt{\frac
{m}}</math>
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</center>
If the value of <math>\theta</math> is large, a large temperature is required to excite vibrational modes. The force constant is most important. Consider what happens if <math>\theta</math> is five and the temperature is five hundred.
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</p>
Consider values of <math>\theta_E</math> with regard to different materials.
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<table cellpadding = 5>
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<tr>
<td>
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<math>\mbox
</math>
</center>
</td>
<td>
<math>\theta_E</math>
</td>
</tr>
<tr>
<td>
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<math>\mbox
</math>
</center>
</td>
<td>
<math>\mbox
</math>
</td>
</tr>
<tr>
<td>
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<math>\mbox
</math>
</center>
</td>
<td>
<math>\mbox
</math>
</td>
</tr>
<tr>
<td>
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<math>\mbox
</math>
</center>
</td>
<td>
<math>\mbox
</math>
</td>
</tr>
<tr>
<td>
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<math>\mbox
</math>
</center>
</td>
<td>
<math>\mbox
</math>
</td>
</tr>
</table>
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</center>
There is a very large range among the different materials. Excited states are reached at different temperatures. Information can be found about phase transitions. Vibrations play a role in temperature driven phase transitions. Consider the relation below.
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<br>
<math>dS = \frac
dT</math>
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<math>\Delta S = \int \frac
dT</math>
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</center>
In the high temperature limit, all the materials approach the same heat capacity. The entropy of a softly bonded material is greater than the entropy of a stiffer material. Consider a free energy curve versus temperature of body-centered cubic and face-centered cubic. There is a more open structure of body-centered cubic, and the force constants are less. There is an energy contribution from vibrations. Consider a pure element with two competing cyrstal phases. The face-centered cubic phase usually displays a higher force constant than body-centered cubic. There is an entropy-driven transition from face-centered cubic to body-centered cubic at higher temperatures. An expression of the Hemholtz free energy and an approximation is below.
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<math>F = E - TS</math>
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<math>F \approx E - T S_{\mbox{vibrational}}</math>
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The Einsten Model helped prove that quantum mechanics could explain things that classical mechanics could not.
Shortcomings of Einstein Model
The heat capacity is related experimentally to the third power of temperature as the temperature approaches zero. There needs to be interactions taken into account. What follows is a sketched solution, which involves a discrete fourier transformation. Atoms do not vibrate independently. Before, the force constant did not depend on whether a neighbor was moving. Atoms feel different potentials based on the movement of neighbors. Wavelike behavior results from the dependence on motion of neighbors. Write the Taylor expansion.
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<math>U( \zeta_1, \zeta_2, \zeta_3, ....) = U(0) + \sum_
^N \left ( \frac
\right ) \zeta_i + \frac
\sum_
^N \sum_
^N \frac
\zeta_i \zeta_j + ...</math>
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</center>
Neglext the term <math>U(0)</math>. The first derivative of <math>U</math> is equal to 0, the sum is a harmonic approximation. There are different force constants.
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<br>
<math>U( \zeta_1, \zeta_2, \zeta_3, ....) = \frac
\sum_
f_
\zeta_i \zeta_j</math>
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</center>
There are cross-terms, and the Schrodinger equation is difficult to solve as a result. Introduce normal coordinates and change variables. There is a straight-forward treatment to remove the cross-terms in the potential. Do in general case with infinite number of neighbors. A change in variables (discrete Fourier transform) decouples the Hamiltonian into single-particle Hamiltonian. The Hamiltonian decouple into a sum of independent quasi-particles, or phonons. All contributions of phonons add up. The energy of the system, <math>E_
</math>, is the summation of the energies of independent quasi-particles. A general expression is below.
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<br>
<math>\zeta_j = \frac
{\sqrt{N}} \sum_k C_k e^
</math>
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There is a periodic boundary condition.
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<br>
<math>e^
= e^
</math>
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<math>k
= k_n</math>
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<math>k = \frac
</math>
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<math>k = \frac
</math>
<br>
<math> 0 < n < N </math>
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There are a finite number of atoms, and not all wavenumbers are allowed. The discrete <math>k</math> comes from the finite number of atoms. A graph is below, and the region between <math>- \pi / a</math> and <math>\pi / a</math> is considered. There are is zone boundary phonon at the boundary. Consider the cases wherein <math>\lambda</math> is equal to <math>2a</math> and where <math>\lambda</math> approaches infinity. In the limit where <math>\lambda</math> approaches infinity, there is collective motion in one direction. Each <math>k</math> corresponds to one particular collective movement of the atoms. Write the general transform. The summation is over phonons, which are quasi-particles.
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<br>
<math>E_
= \sum_{\mbox
k_n} \left ( n_k + \frac
\right ) \hbar \omega_k</math>
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<math>\omega(k) = \mbox
</math>
<br>
<math>k = \frac
</math>
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</center>
There is a summation over all phonons. There are <math>n_k</math> phonons in the same state. The term specifies how many phonons are in each state. The frequency depends on the wave number. Atoms feel different potentials. Similiar to Einsten, sum over all <math>k_N</math> and get different <math>\omega(k)</math>. Below is a relation of the partition function.
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<math>Q = \sum_{\mbox{states}} e^{- \beta} \left ( \sum n l + \frac
\right ) \hbar \omega (l)</math>
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</center>
The dispersion relation depends on the model. Decouple after the <math>n^
</math> nearest neighbor. In the expression below, the term <math>n_1</math> determines how many phonons are in a particular state. The temperature determines how many are excited.
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<br>
<math>\left ( n_1 + \frac
\right ) \hbar \omega \left ( \frac
\right )</math>
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</center>
Find <math>\omega (l)</math> to get all of thermodynamics. The term is dependent on the approximation of the model.
Einstein
The masses move independently and there is only one frequency.
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Debye
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First-Nearest Neighbor
Consider first-nearest neighbors in a monoatomic linear chain.
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Increasing the number of neighbors results in more branches. There are symmetry laws. Sum over frequencies and look at every mode. Count number to get frequency.
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</p>
Sum over modes instead of <math>\omega (k)</math>. Sum over frequencies. The term <math>g(\nu)</math> is the phonon density of states. It describes how many states there are within a particular frequency. Many modes give the same frequency.
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</p>
Model with Debye and Einstein. Calculate the number of modes and sum over frequencies.
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Exam
Read the first six chapters in MacQuarrie and chapter eleven. Any material in MacQuarrie could be included. Consider the solution part and Clausius Clapeyron.