# How to intuitively interpret the dependence of temperature and density on Debye Length?

I am learning plasma physics through Chen and in the first section it discusses Debye shielding. Essentially, if you place a test charge in a plasma such that it generates a positive potential $$\phi_0$$, it will be shielded by the surrounding electrons which comes out to be (equation 1.17 in Chen)

$$\phi = \phi_0\exp(-|x|/\lambda_D)$$ Where $$\lambda_D = \Bigg(\frac{\epsilon_0KT_e}{ne^2}\Bigg)^{1/2}$$

Is the Debye length which controls the shielding distance of the sheath.

I am confused as to how to intuitively think about the dependence on temperature and density here. Is the correct way to think about the Temperature in the numerator being that if the temperature increases, you'd expect the sheath distance to increase because the higher temperatures you go, the more thermally agitated the particles might be, and they could wiggle out of the potential's grasp, and so the potential is able to access longer distances to get the particles it needs to fully shield its potential?

And the density would act to counteract this wiggling? Although electrons might have the tendency to wiggle out more frequently, if the density is higher then there will be more electrons that can't wiggle out, and thus the sheath would be smaller because the potential doesn't have to reach as far to find electrons that can't wiggle out?

What is the correct way to interpret the temperature in the numerator and density in the denominator?

You can think of the Debye length as the ratio of the rms thermal speed to the electron plasma frequency. This isn't really a loose analogy, it works out to be the correct parameters. The rms thermal speed and plasma frequency are given by: \begin{align} V_{Te}^{2} & = \frac{ k_{B} \ T_{e} }{ m_{e} } \tag{0a} \\ \omega_{pe}^{2} & = \frac{ n_{e} \ e^{2} }{ \varepsilon_{o} \ m_{e} } \tag{0b} \end{align} where $$k_{B}$$ is the Boltzmann constant [J/K], $$e$$ is the fundamental charge [C], $$m_{s}$$ is the mass of species $$s$$ [kg], $$\varepsilon_{o}$$ is the permittivity of free space [F/m], $$n_{s}$$ is the number density of species $$s$$ [$$m^{-3}$$], and $$T_{s}$$ is the temperature of species $$s$$ [K].