Paramagnetism

1. Paramagnetism Let us come back to the case of the two-state paramagnet A particular microstate (....) isspecified if the directions of all spins are specified. A macrostate is specified by the total # of dipoles that point “up”, N (the # of dipoles that point “down”, N  = N - N). E E2 = + B an arbitrary choice of zero energy 0 E1 = - B system spontaneously absorbs energy from its environment system spontaneously gives up energy to its environment all dipoles polarized all dipoles polarized

2. Certain systems can achieve negative temperature; that is, their thermodynamic temperature can be expressed as a negative quantity on the kelvin scale. A system with a negative temperature is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. That a system at negative temperature is hotter than any system at positive temperature is paradoxical if absolute temperature is interpreted as an average internal energy of the system. The paradox is resolved by understanding temperature through its more rigorous definition as the tradeoff between energy and entropy, with the reciprocal of the temperature, thermodynamic beta, as the more fundamental quantity. Systems with positive temperature increase in entropy as one adds energy to the system. Systems with negative temperature decrease in entropy as one adds energy to the system. Most familiar systems cannot achieve negative temperatures, because adding energy always increases their entropy. The possibility of decreasing in entropy with increasing energy requires the system to "saturate" in entropy, with the number of high energy states being small. These kinds of systems, bounded by a maximum amount of energy, are generally forbidden classically. Thus, negative temperature is a strictly quantum phenomenon. Some systems, however, have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.

3. Langevin paramagnetic equation

5. Paramagnetism Paramagnetic materials present a magnetization that is proportional to the applied field and reinforces it. This arises from the existence of magnetic dipoles in the material. Bohr magneton The net magnetic moment of an atom is the vector sum of its orbital and spin magnetic moments. (Note that e is negative!) Nuclei also have magnetic moments. At room temperature kT~1/40 eV so Curie’s law