Solving for A in a Simple Cubic Lattice

[Josh1079]

Homework Statement

1. For a simple cubic lattice with a lattice constant of a, the energy band can be expressed as:E = Acos(k_xa)cos(k_ya)cos(k_za) + B.

(a) Suppose the effective mass for the electron at conduction band is m* = -ħ²/2a², find A.

Homework Equations

The Attempt at a Solution

I know the effective mass equation and how the effective mass tensor is calculated, but I can't really see how does the effective mass appear to be as indicated(m* = -ħ²/2a²).

Thanks!

 

[Dr Transport]

What is the definition of the effective mass? it isn't an extremely complicated expression...

 

[Josh1079]

I think it's m* = ħ²/(∂²E/∂k²).
but how am I going to use this?
there are three components of k, and I'm not sure how to combine the three masses.

Thanks for the reply.

 

[Dr Transport]

you have the expression for E, do the derivatives, the effective mass will be anisotropic...

 

[Josh1079]

Thanks, I've done the derivatives, but the result is a tensor(shouldn't it?).

how should I relate this with the single value provided by the question (m* = -ħ²/2a²)?