From the figure below (i) Determine the Miller indices for the plane. (ii) Let the length of all the three base vectors of the direct lattice a = 5 nm. Calculate the distance of the lattice planes (346).
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- By determining the temperature at which the magnetic moment vanishes for a two- dimensional Ising system on a square lattice, show analytically that the phase transition temperature is determined by the equation T- 2Ju² ken(1+√2)solve properlyShow that the volume of the first Brillouin zone is 8³/Vc, where Vc is the volume of a crystal primitive cell. (Hint: remember that Wigner-Seitz cell has the same volume as parallelepiped formed by three basis vectors of a primitive cell)
- a Q₁ Given = AX (a-x), A, a are = limit ofxña at tro ~ (x, 0) Constant 1- Find A for normalized (1x₁) 2- If (AV, 0) represents the wave function of aparticle find all bil in infinite square well outside the well N(x,0) find (x, t)Suppose you measure A with eigenvalues A1, A2, and A3 with corresponding eigenvectors |1), 2), and |3), respectively, on wave function ) = a|1) + B|2) + y|3). Assuming degeneracy where A3, what normalized wave function does ) collapse to when normalized measuring A brings a value of 1? O a[1) + y|3) a² +y? a²+y? O alı)+r|3) Va2+y2Write the possible (unnormalized) wave functions for each of the fi rst four excited energy levels for the cubical box.
- Solve the following.Find the optical and acoustical branches of the dispersion relation for a diatomic linear lattice, solid, physicsCalculate and plot the vibrational partition function of CSe2 between 500K and 1000K (with a step of 100K) given the wavenumbers 313 cm- (bend. two modes). 369 cm-1 ( symmetric stretch) and 1302 cm - (asymmetric stretch) assuming that each vibrational mode can be treated as a simple harmonic oscillator. Explain and discuss each step. (marks 10)
- Consider scattering with electrons of 54 eV on a crystal with planes separated by 0.22 nm. How many diffraction peaks should be expected? Justify your answerFind the normalization constant B for the combination 18. As noted in Exercise 8, a linear combination of two wave functions for the same sysstem is also a valid wave function also a valid wave function functions for the same system 2TX = B sin TX +sin L. L. of the wave functions for then = 1 and n = 2 states od %3D particle in a box L wide. [A + CO]At what displacements is the probability density a maximum for a state of a harmonic oscillator with v = 3?