From equation 28.4, the semi classical equation of motion is given by,
The oscillatory solution to the above equation is,
Substitute the above relation in the equation
Taking the component of the magnetic field in the z direction,
Here, the cyclotron effective mass is .
It is given by the relation
The dispersion relations along the coordinate directions are given as,
The motion is periodic. So, the solution is of the form .
So, putting in the above equations,
There exists non trivial solution when the determinant formed by the above equations vanished. Thus,
Now, the physical acceptable solution is given by the equation.
As the effective mass tensor is diagonal, the dispersion relation can be written as,
Now, the projection of the magnetic field along the principal axes is specified by the direction cosines , and . Thus,
Now, the equation of motions are given as,
Giving solution of the form of in the above equations,
Now, the determinant of the above equation vanished for non-trivial solutions. Thus,
Now, the acceptable solution is given by,
Thus, the cyclotron effective mass is given by,
Therefore, the cyclotron effective mass is related as:
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