Basic Theoretical Physics A Concise Overview

402 53 Applications I: Fermions, Bosons, Condensation Phenomena Similar behavior, U ∝ V · T 4 (the Stefan-Boltzmann law ) is observed fora photon gas, i.e., in the context of black-body radiation; however this is validat all temperatures, essentially since for photons (in contrast to phonons) thevalue of N is not defined. Generally we can state:a) The low-temperature contribution of phonons, i.e., of sound-wave quanta,thus corresponds essentially to that of light-wave quanta, photons; thevelocity of light is replaced by an effective sound-wave velocity, consideringthe fact that light-waves are always transverse, whereas in addition to thetwo transverse sound-wave modes there is also a longitudinal sound-wavemode.b) In contrast, the high-temperature phonon contribution yields Dulong andPetits’s law; i.e., for kB T ωDebyeone obtains the exact result: U (T, V, N ) = 3N kBT .This result is independent of the material properties of the system con-sidered: once more essentially universal behavior, as is common in ther-modynamics. In the same way one can show that magnons in ferromagnets yield a low-temperature contribution to the internal energy ∝ V · T 5 2which corresponds to a low-temperature contribution to the heat capacity ∝ T 3 . 2This results from the quadratic dispersion relation, ω(k) ∝ k2, for magnonsin ferromagnets. In contrast, as already mentioned, magnons in antiferromag-nets have a linear dispersion relation, ω(k) ∝ k, similar to phonons. Thus inantiferromagnets the low-temperature magnon contribution to the specificheat is ∝ T 3 as for phonons. But by application of a strong magnetic fieldthe magnon contribution can be suppressed.– In an earlier section, 53.1, we saw that electrons in a metal produce a con- tribution to the heat capacity C which is proportional to the temperature T . For sufficiently low T this contribution always dominates over all other contributions. However, a linear contribution, C ∝ T , is not character- istic for metals but it also occurs in glasses below ∼ 1 K. However in glasses this linear term is not due to the electrons but to so-called two- level “tunneling states” of local atomic aggregates. More details cannot be given here.