4. Discuss briefly the origin of the differing electronic properties of metals, semiconductors and insulators. Under what circumstances will the nearly-free electron model be useful for describing the band structure of a solid?

A weak periodic potential

W(z) = W_0 cos (2*pi*z)/a

with W_0 > 0, is imposed on a one-dimensional free-electron gas lying along the z direction. Discuss the form of the resulting wavefunctions at the first Brillouin zone boundary. Explain why a gap arises.

A simple model of a two-dimensional band structure can be obtained by summing together two one-dimentional band structures. Consider the energy band

E(k_x, k_y) = U cos (k_x * a) - U cos (k_y * a)

with U > 0, corresponding to a particular two-dimensional crystal with square lattice and period a. Give the wavevectors and energies of the highest and lowest energy states in teh band. Draw labelled constant energy contours in the first Brillouin zone near these particular states. Comment on the form of the constant-energy contours near the centre fo teh first Brillouin zone.

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(a) (i) What are the main assumptions of the free electron model?
(ii) Give an example where the free electron model is in good agreement with experiment.
(iii) Give an example where the free electron model fails.
(b) (i) How does the nearly free electron model improve on the free electron model?
(ii) Describe one of the s

Two slits are illuminated by a 621 nm light. The angle between the zeroth-order bright band at the center of the screen and the fourth-order bright band is 14.4 degrees. If the screen is 150 cm from the double-slit, how far apart is this bright band from the central peak? Answer in units of cm.

2. (a) How does MO theory differ from Valence bond theory?
(b) Explain how the above concepts of MO theory are applied to rationalize the conductivity of :
i. metals
ii. (SN)x [a polymer]
iii. semiconductors

I need some help with these three questions on organic chemistry bonding:
13. Why it is important that any aldehyde used in Witting reaction be free of carboxylic acid impurities?
14. Explain why the C=C stretching mode gives rise to a rather weak IR band in 1-methylcyclohexane, while in its isomer, methylenecyclohexene,

(See attached file for full problem description)
The pipe U-band in the figure attached is connected to a flow system by flexible hoses that transmit no force. The pupse has inner diameter of 3 inches. Water is flowing through the pipe at a rate of 600 gallons/minute. The pressure at point 1 is 5 psig and at point 2 is 3 psig

Consider an aluminum gate-silicon dioxide p type silicon MOS structure with tox=450 angstroms. The silicon doping is Na=2x10^11cm^-3 and the flat-band voltage is Vfb=-1.0V. Determine the fixed oxide charge Q'ss.

The future value of an annuity is A=$32,000. Periodic payments are made quarterly for four years and the annuity earns 8% compounded quarterly. Find the periodic payments.
A. $422.40
B. $1,716.80
C. $2000.00
D. $2,160.00

(a) Based on the linear fit equation y=395.7x + 1.44 (this equation is derived from a graph where stretch of a band in meters is on the x axis and weight is on the y axis), how much coin weight would it take to stretch the band by one meter?
The force constant k of the band is 395.7.
(b) How does above linear equation rel