cha10 - PHYS 635 Solid State Physics Take home exam 1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PHYS 635 Solid State Physics Take home exam 1 Gregory Eremeev Fall 2004 Submitted: November 8, 2004 Problem 1:Ashcroft & Mermin, Ch.10, p.189, prob.2 a) Lets prove xx = yy = zz = (1) xx =- Z d r * x ( r ) x ( r ) U ( r ) =- Z d r x 2 | ( r ) | 2 U ( r ) =- Z d r y 2 | ( r ) | 2 U ( r ) =- Z d r * y ( r ) y ( r ) U ( r ) = yy (2) Now 0 = xx- yy =- Z d r ( x 2- y 2 ) | ( r ) | 2 U ( r ) (3) If we here make the change of variables: x = x- y ; y = x + y , which is basically rotation in space, then we will get 0 = xx- yy =- Z d r 0 ( x 0 y ) | ( r ) | 2 U ( r ) = xy = 0 (4) b) In the case of a simple cubic Bravais lattice with ij ( R ) negligible for all but the nearest-neighbor R lets calculate xy ( k ). We should take into account 6 neighbors: R = a ( 1 , , 0); a (0 , , 1); a (0 , 1 , 0) (5) xy ( k ) =- e ik x a Z d r xy ( r ) ( ( x- a ) 2 + y 2 + z 2 ) 1 2 U ( r )- e- ik x a Z d r xy ( r ) ( ( x + a ) 2 + y 2 + z 2 ) 1 2 U ( r )- e ik y a Z d r x ( y- a ) ( r ) ( x 2 + ( y- a ) 2 + z 2 ) 1 2 U ( r )- e- ik y a Z d r x ( y + a ) ( r ) ( x 2 + ( y + a ) 2 + z 2 ) 1 2 U ( r )- e ik z a Z d r xy ( r ) ( x 2 + y 2 + ( z- a ) 2 ) 1 2 U ( r )- e- ik z a Z d r xy ( r ) ( x 2 + y 2 + ( z + a ) 2 ) 1 2 U ( r ) = 0 (6) , because under rotation transformation x y, y -x each integral changes sign....
View Full Document

This note was uploaded on 06/07/2010 for the course MECH 1122 taught by Professor Dont during the Spring '10 term at A.T. Still University.

Page1 / 5

cha10 - PHYS 635 Solid State Physics Take home exam 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online