How Do Ashcroft and Mermin Derive the Cyclotron Effective Mass Expression?

[BeauGeste]

could anyone elucidate on how Ashcroft and Mermin get
$$\sqrt{\frac{det M}{M_{zz}}}$$
for the cyclotron effective mass (page 571)?
Is this true for all energy contours or only certain ones? And if that's the case which because they don't seem to specify?
thanks.

 

[drummerguy]

It's from problem 12.2 - essentially it's valid near a maximum or minimum in the band (so you get an approximately quadratic dispersion). I haven't solved the problem yet, but I think the comment on page 568/569 about an ellipsoidal constant energy surface is key (in addition to the fact/assumption that the effective mass tensor is symmetric (this assumption is necessary to solve problem 12.3a)).