What is the Relation Between Wave Vectors in Bloch's Theorem?

[Modey3]

Hello,

I've been self teaching myself solid state physics by reading Ashcroft and Mermin. What confuses me is that for the wave function in the lattice is written as the Fourier series of waves obeying the Born-Von Karman Boundary Conditions while the potenial is written as the Fourier series of waves in the reciprocal lattice. Look on page 137 in Ashcroft and Mermin to see what I mean. Is there any difference in the wave vectors used in both Fourier series. Actually two different summation indices are used (K for the wavefunction and q for the potential) I would think that since a wave is in the reciprocal lattice it should obey the Born-Von-Karman BC's. I'm just not getting a feel for how K and q are related. Thanks for any help.

Modey3

 

[inha]

K is summation over all reciprocal lattice vectors. So it is always of the form n1*b1+n2*b2+n3*b3 where n's are integers and b's are reciprocal lattice vectors. If you look at the previous page the allowed wave vectors are not of the same form but satisfy the peridiocity of the reciprocal lattice.

 

[Modey3]

Thanks for the response. Isn't it true though that a wave vector that satifies the Born-Von-Karman BC's doesn't nessarily satisfy the periodicity of the lattice. For instance consider a simple cubic lattice. The allowed wave vectors that satisfy the periodicity of the lattice are pi/a, 2*pi/a, 3*pi/a etc... The Born-Von-Karman BC's don't restrict the wave vectors to these values. In fact according to the Born-Von-Karman BC you could have a wave vector (0.5)*pi/a which doens't satisfy periodicity. Thanks

Modey3

 

[armandowww]

q's and K's belong to k-space... both of them. But you must be careful when you create a picture. q's are all in first Brillouin zone (remember that its linear size goes like 1/a). K's are equivalent to R's in reciprocal lattice... now you should understand that while Born&Von-Karman have made a restriction to R variability, differently they could not perform the same on K's which are ruled by potential periodicity.
I suggest you to postpone the demonstration and come back to relations like (8.29).
Keep in mind continously a common analogue: the string of a guitar! It's the same, but you have physically to repeat the chord backward and foreward until you got suitable boundary conditions!

 

[Modey3]

Okay, thanks for everyones responses. The only problem I have is that a wave with a wavelength of 2a, 3a, or greater satisfy the Born-Von-Karman, but do not satisfy the periodicity of the lattice since psi(r) doesn't equal psi(r+R) for a wavelength of 3a. In this case R could be equal to a. Thanks for any help.

Modey3