Hall Effect constant and Cyclotron frequency

In summary, 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.
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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?
Screenshot_20201030-114753.png
 
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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
 
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Thanks it helps me to understand more.
 
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Related to Hall Effect constant and Cyclotron frequency

1. What is the Hall Effect constant?

The Hall Effect constant, also known as the Hall coefficient, is a physical constant that describes the relationship between the electric field and the induced magnetic field in a conductive material. It is denoted by the symbol RH and is measured in units of meters cubed per coulomb (m3/C).

2. How is the Hall Effect constant related to the Hall Effect?

The Hall Effect constant is directly related to the Hall Effect, which is the phenomenon of a voltage difference being created across a conductive material when a magnetic field is applied perpendicular to the direction of current flow. The Hall Effect constant is used to calculate the magnitude and direction of the induced magnetic field in this situation.

3. What is the significance of the Hall Effect constant in materials science?

The Hall Effect constant is an important parameter in materials science as it can provide information about the type of charge carriers (electrons or holes) and their concentration in a material. It is also used to characterize the electrical properties of semiconductors and determine their mobility and resistivity.

4. What is the Cyclotron frequency?

The Cyclotron frequency, denoted by the symbol fc, is the frequency at which a charged particle in a magnetic field will complete one full circular orbit. It is dependent on the strength of the magnetic field and the mass and charge of the particle.

5. How is the Hall Effect constant related to the Cyclotron frequency?

The Hall Effect constant and the Cyclotron frequency are related through the equation fc = eB/m, where e is the charge of the particle, B is the magnetic field strength, and m is the mass of the particle. This relationship is used to determine the charge-to-mass ratio of a particle in a magnetic field.

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