09hw - Solutions to problem set 9 (141A Sp07) 1 1. (ISSP...

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Unformatted text preview: Solutions to problem set 9 (141A Sp07) 1 1. (ISSP 7.4). (a) The potential created by the carbon atoms in diamond can be written as U ( ~ r ) = X ~ R ( v ( ~ r- ~ R ) + v ( ~ r- ( ~ R + ~ )) Here, v ( ~ r ) is the potential created at a point ~ r in space by a single carbon atom located at the origin; and { ~ R } are the real space lattice vectors (ie each ~ R points to a unit cell of the crystal). Carbon atoms are placed at ~ R and ~ R + ~ , where ~ = ( a / 4)( x + y + z ), where a be the size of the conventional cubic unit cell. Then U ~ G = 1 vol Z O d 3 r U ( ~ r ) e- i ~ G ~ r where vol is the volume of the primitive unit cell and O is the primitive unit cell located at the origin (the integration is over this unit cell only). Evaluating this integral gives U ~ G = 1 vol Z O d 3 r U ( ~ r ) e- i ~ G ~ r = 1 vol Z O d 3 r e- i ~ G ~ r X ~ R ( v ( ~ r- ~ R ) + v ( ~ r- ( ~ R + ~ )) = X ~ R 1 vol Z O d 3 r e- i ~ G ~ r v ( ~ r- ~ R ) + X ~ R 1 vol Z O d 3 r e- i ~ G ~ r v ( ~ r- ( ~ R + )) = 1 vol...
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09hw - Solutions to problem set 9 (141A Sp07) 1 1. (ISSP...

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