Hall Effect constant and Cyclotron frequency

[Rzbs]

TL;DR

Hall constant and cyclotron frequency

In the ch1 if solid state physics Mermin & Ashcroft, in the hall effect section these paragraph are about cyclotron frequency, but what the two last terms want to say(the screen shot of the page is attached)? And I can't understand what happens to hall constant in high-field regime?

 

[Lord Jestocost]

At high magnetic fields (high cyclotron frequencies $\omega$) one has to consider effects which can be neglected in theoretical considerations at low magnetic fields. The criterion is whether the lifetime $\tau$ of an electron between collisions is very long compared with the characteristic time, $1/\omega$, to perform a cyclotron orbit in the given magnetic field. Maybe, the following might be of help:

"If $\omega$ is the cyclotron frequency of a representative electron moving in a flux $\vec B$, and $\tau$ is its average relaxation time in the cyclotron orbit, the classical field conditions introduced in Sec. 1 are formally defined for an orbit by:

(1) $\omega \tau \ll 1$ gives the low-field condition. The representative point completes an infinitesimal segment of the total cyclotron orbit during the electron's lifetime between collisions so that topological features of the orbit are not manifested. When all orbits for a given orientation of $\vec B$ are in the low-field condition, the Hall effect is independent of the global features of the Fermi surface and depends only upon the electrons' effective mass, velocity and anisotropic relaxation time over the Fermi surface.

(2) $\omega \tau \approx 1$ gives the intermediate-field condition. Experimentally this is a very important region in which many studies of alloys and less-pure metals have been made.

(3) $\omega \tau \gg 1$ gives the high-field condition. The electron's lifetime between collisions is very long compared with the time necessary to complete the cyclotron orbit. When all orbits for a given $\vec B$ are in the high-field condition the Hall effect depends only upon the topology of the Fermi surface; electron scattering is generally not manifested, except where it affects the topology."

From: “The Hall Effect and Its Applications”, Edited by C. L. Chien and C. R. Westgate

 

[Rzbs]

Thanks it helps me to understand more.