# Effective Mass

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## effective mass

[ə¦fek·tiv ′mas]## Effective Mass

a quantity that has dimensions of a mass and that characterizes the dynamic properties of quasiparticles. For example, the motion of a conduction electron in a crystal under the action of an external force *F* and the forces within the crystal lattice can be described as the motion of a free electron under the action of force *F* alone (Newton’s law) but with an effective mass *m ^{*}* that differs from the mass

*m*of the free electron. This difference reflects the interaction of the conduction electron with the crystal lattice. Effective mass is defined by the relation

where ℰ is the energy and *p* is the quasimomentum of the conduction electron. If the relation ℰ(*p*) is anisotropic (law of dispersion), then the effective mass is a tensor (inverse mass tensor):

This means that, in general, the direction of acceleration of an electron in a crystal lattice is not parallel to the external force **F.** The acceleration may even be directed antiparallel to **F**, which corresponds to a negative value of effective mass. The properties of electrons with negative effective mass differ from those of ordinary particles to such an extent that it has been found to be more convenient to deal with positively charged holes with positive effective mass.

The cyclotron effective mass of electrons and holes is used in studying galvanomagnetic phenomena:

where *S* is the area of the intersection of the constant energy surface ℰ(*p*) by a plane perpendicular to the magnetic field **H.** Methods like cyclotron resonance and the measurement of electron heat capacity are the most important techniques for determining the effective mass of conduction electrons and holes.

In the theory of quantum fluids for the quasiparticles fermions with isotropic law of dispersion, the effective mass is the ratio

(4) *m** = *p*_{0}/*v*_{0}

where *p*_{0} and *v*_{0} are the absolute values of the momentum and velocity of the quasiparticles at a temperature of absolute zero that correspond to the Fermi level. The effective mass of an atom of liquid ^{3}He is *m** = 3.08*m*_{0}, where *m*_{0} is the mass of a free atom of ^{3}He.

The concept of effective mass has been extended to such quasiparticles as phonons, rotons, and excitons. Relation (1) is valid for all these cases.

M. I. KAGANOV