PH-201(Physics-II) B.Tech – II Year Autumn: 2013 – 14

Solid State PhysicsFree Electron Theory, Electrical Conductivity, Band structure, Magnetic Properties, Superconductivity

1. Calculate the energy difference between the nx = ny = nz = 1 level and next higher energy level for free electrons in a solid cube of side 10 mm. [Ans. 1.13 x 10-14eV]

2. The Fermi energy of sodium is 3.0 eV. Calculate the energy difference between the neighbouring energy levels in the vicinity of the Fermi energy if the highest occupied state in a cubical box of side 1.2268 x 10-8 m is given by nx = ny = nz. Ans. 0.098 eV.

3. Calculate the number of energy states available for the electrons in a cubical box of side 1 cm lying below energy of 1 eV. [Ans. 4.5 x 1021]

4. What should be the energy of the free electrons in a metal in the quantum state for which the sum of the squares of the quantum numbers nx, ny, nz is 19? Take the size of the solid to be 10mm x 10mm x 10mm. What are the quantum numbers of the next higher state? [ Ans. 1.14 x 10-32 J]

5. Assuming free electron model for a two-dimensional solid, obtain the expression for the density of electronic states. Show that the average energy of electrons in this solid at T= 0 K is 0.5 EF.

Ans. N(E) = 2 π (2m/h2)

6. Calculate the density of states of electron in 1D, 2D and 3D system at energy of 0.1eV. Also make

qualitative plots of density of states versus energy for all the cases.

7. Copper has f.c.c. structure with a = 3.6 Å. Assume each copper atom contributes one electron for conduction. Find (i) concentration of the electrons (ii) Fermi energy Ef (iii) Fermi temperature (iv) Fermi velocity, (v) Fermi wavevector, and (vi) electrical conductivity if mean free path is 200 times the atomic radius. Also find the Fermi momentum and corresponding deBroglie wavelength of the electrons. (Given density ρ=8.96 gm/cm3 and atomic wt. =63.5 of copper). [Ans. 8.57x1028/m3; 7.1 eV; 8.25x104 K; 1.58x106 m/s; 1.36x1010/m;]

8. The visible light is allowed to fall on a semiconductor crystal having the energy gap of 1.5 eV. Discuss the transparency of the semiconductor to this light.

9. In GaAs, when a conduction band electron recombines with hole in the valence band, the wavelength of light emitted is 0.87μm. Find the band gap.

10. Calculate the mean free path of potassium, if its Fermi energy is eV 1.2 and the electrical conductivity is 1-17 ohm 10 5.1 −× m . [Ans. nm2.33 ]

11. Find the maximum radius R of the sphere that can just fit into the void at the body center of the FCC structure formed by the atoms of radius r. [Ans. R = 0.414 r ]

12. A rod of p-type Ge 10 mm long and 1 mm diameter has a resistance of 100 Ohm at 0 K. Calculate

the concentration of impurity in it given that µ p = 0.19 m2/Vs. [Ans. 4.17x1021/m3]

13. A material has a mean collision time of 10 s. Calculate the drift velocity acquired by the electrons in a field of 10 V/m. Ans. 176 m/s.

14. Calculate the diamagnetic susceptibility of germanium. [ ρ = .,./3.5 ccgm average electronic radius = 0.4 Å, atomic weight = 72 and atomic number=32 . [Ans. 31034.1 −× ]

15. A paramagnetic substance has 1028 atom/m3. The magnetic moment of each atom is 1.8x10-23

ampere meter2. Calculate the paramagnetic susceptibility at 300K. [Ans: 3.28 X 10-4]16. Using the effective number of Bohr magnetons as 0.606 per atom for FCC Ni (a = 3.52 Å),

calculate the saturation magnetization for it at 0K. Ans. 5.15 x 105 A/m.

P.T.O17. The Curie temperature of iron is 1043K. Assume that iron atoms, when in the metallic form, have

moments of two Bohr magneton per atom. Iron has body-centered cubic structure with lattice parameter, a =0.286 nm. Calculate (a) the saturation magnetization (b) the curie constant (c) the mean field constant (d) the magnitude of the internal field.

[Ans: (a) 15.8 X105Am-1 (b) 0.66 (c) 1580 (d) 3135 Tesla]

18. The critical temperature cT for mercury with isotopic mass .185.4 is 5.199 K Calculate the critical temperature when its isotopic mass changes to .4.203 [Ans. K139.4 ]

19. Calculate the critical current, which can flow through a long thin superconducting wire of Al of

diameter m410− . The critical magnetic field for Al is ./109.7 3 mamp×

[Ans. 24.806A ]20. A niobium (Nb) wire has a radius of 3.00 mm and is at a temperature 4.5 K. What is the maximum current the wire can carry and remain superconducting? For Nb, the critical temperature is 9.26 K and critical magnetic field at 0 K is 0.199 T. [Ans. 2.28 x 103 A]

21. For GaV3 the critical fields are mampmamp /102.4 and /104.1 55 ×× for KK 13 and 14 . Calculate the transition temperature and Critical fields at KK 2.4 and 0 .

[Ans. mAmAK /9.18,/7.20..5.14 ]

22. Estimate the energy gap for niobium at T = 0 K, and find the maximum photon wavelength needed to break the Cooper pair. Tc of niobium is 9.25 K. Experimental value of band gap is 3.05 meV.

Ans. 2.82 meV, 4.39 x 10-4m