私立中原大學八十九學年度博士班招生考試命題紙

所組別：

應用物理學系

科目：

固態物理

考試時間：

6月14日第1節

1.

(20%) Consider a linear chain of atoms. Assume only the nearest--neighbors interaction, and use the Lennard--Jones potential for the interaction between pairs $$ U(R) = 4\epsilon \ [ ({\sigma\over R})^{12} -({\sigma\over R})^6 ] $$ (1). Determine the equilibrium distance $R_0$ and the cohesive energy for each atom. (2). Write $R = R_0 + u$, where $u$ is the displacement from the equilibrium position, and expand $U(R)$ to $O(u^2)$. (3). Write down the equation of motion for the s--th atom. Use $u_s$ to denote the displacement of the s--th atom from its equilibrium position. (4). Solve for $\omega(k)$.

2.

(15%) Assume silver is a monovalent metal with a spherical Fermi surface. Use the free electron model to calculate the Fermi energy in electron volts. Silver density is 10.5 g/cm$^3$, and mass number is 107.87. Electron mass is $m = 0.91 \times 10^{-27}$g and the Planck's constant is $h = 6.63 \times 10^{-27}$ erg $\cdot$ s. The experimental value is 5.48 eV.

3.

(15%) In the nearly free electron model, consider a one-dimensional lattice of spacing $a$ with potential $U(x) = - 2 U_0 \cos({{2\pi x}\over{a}})$. (1). Obtain all the non--zero $U_G$'s. (2). Where does the energy band splitting happen and by how much? (3). What are the wavefunctions for the lower and higher band at the splitting respectively?

4.	(10%) Explain the best you can why the crystal momentum $\hbar k$ is not the real momentum.

5.	(10%) Consider a neutron scattering process which absorbs one phonon. Write down and solve the conservation equations.

6.	(30%) Explain the following terms: (1) reciprocal lattice (2) Bloch theorem (3) De Haas--van Alphen effect (4) exciton (5) Raman effect (6) Cooper pair

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