De Broglie Wavelength

Quantum effects become important if the mean inter-particle distance, $r_d$ , becomes comparable, or less than, the de Broglie wavelength,

$${\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}\equiv \frac{h}{m\,v_t},$$ (1.37)

where $h$ is Planck's constant. According to Equations (1.3) and (1.16), the condition $r_d\ll {\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$ is equivalent to

$$\frac{T^{\,3/2}}{n} \ll \frac{h^3}{(2\,m)^{3/2}}.$$ (1.38)

A plasma that satisfies this condition is said to be degenerate, whereas a plasma that does not is said to be non-degenerate. The behavior of degenerate plasmas is fundamentally different to that of the non-degenerate plasmas discussed in this book (because the former plasmas are governed by quantum mechanics, whereas the latter are governed by classical mechanics). It can be seen that if both species have comparable temperatures then the condition, given in Equation (1.38), for degeneracy is more easily satisfied by the electrons than by the ions. Moreover, it is evident that degenerate plasmas tend to be cold and dense, whereas non-degenerate plasmas are generally hot and diffuse. (See Haas 2011 for a comprehensive discussion of degenerate plasmas.)

It is actually possible for quantum effects to modify collisions in non-degenerate plasmas that satisfy the inequality $r_d\gg{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$ . In fact, the criterion for quantum effects not to modify collisions is $r_c\gg{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$ , where $r_c$ is the mean distance of closest approach during collisions. However, it follows from Equation (1.20) that $r_c\sim r_d/{\mit\Lambda}^{2/3}$ . Hence, the criterion for classical collisions becomes $r_d\gg {\mit\Lambda}^{2/3}\,{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$ . In a weakly coupled plasma, for which ${\mit\Lambda}\gg 1$ , this criterion is harder to satisfy that the criterion, $r_d\gg{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$ , for non-degeneracy.