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Typical interferometer trace of a damped ion acoustic wave and its evaluation.

Here are some comments to the damping of ion acoustic waves:

1. The damping manifests itself by the decay of the wave envelope with distance x. The envelope can be approximated by connecting successive maxima and minima as shown in the upper figure (a). In the lower figure (b) the wave amplitude is displayed on a logarithmic scale at every half wavelength. The straight-line approximation shows that the amplitude varies as A(x)=A(0)exp(-x/L_damping), where L_damping=2pi/k_imaginary is defined as the damping length. It is found to be L_damping ~18(lambda/2)/ln100=9.1 mm. The normalized damping is defined by k_imag/k_real. Since the wavelength is related to the real part of the wavenumber, lambda=2pi/k_real, one finds for k_imag/k_real = (lambda/2pi)/L_damping = ln100/18pi=0.081.
   
   
2. There are several mechanisms giving rise to an amplitude decay in the interferometer trace an ion acoustic wave. (1) If it is not a plane wave there is an amplitude loss due to the geometric spread of the wave (energy conservation). (2) Collisional damping arises when the ion-neutral collision frequency is a significant fraction of the wave frequency. (3) Landau damping arises when particles move near the phase velocity of the wave and there are more slower than faster particles, i.e., the slope of the distribution function is negative at v_phase. Both electrons and ions produce Landau damping. Landau damping is strong when the electron temperature is close to the ion temperature. Even at high electron temperatures, ion Landau damping is strong near the ion plasma frequency where the phase velocity decreases to the ion thermal velocity. (4) Finally, if the wave is scattered by other waves (e.g. in a turbulent plasma) the energy spread in frequency gives rise to an energy loss at the injected signal frequency, hence an amplitude decay in the interferometer trace.
   
   
3. In order to distinguish among the various damping mechanisms one can take certain precautions and perform a few basic tests. For example, plane waves are produced close to a plane grid of dimensions large compared to the wavelength. This holds for the present the grid of 5cm diameter at a wavelength of 2.33 mm. Collisional damping can be tested by varying the neutral pressure. The two traces shown in (b) are taken at different neutral pressures but exhibit nearly the same amplitude decay. Thus, the damping cannot be collisional. Landau damping remains since the plasma is not turbulent.
   
   
4. The theory of Landau damping (see Lab manual for a short version) gives an expression for the normalized damping. In the present case of low frequencies, [(omega)/(omega_pi)]²<<1, it is given by L_damping/(lambda) = 0.09 x^-3/2 exp[x/2(1+3/x)], where x=T_e/T_i. For the measured normalized damping length L_damping/(lambda) =1.96 one finds numerically a temperature ratio x=10.1. With an electron temperature of kT_e=1.78 eV, obtained from the phase velocity, one obtains an ion temperature of kT_i=0.15 eV. Thus, ion acoustic wave measurements are a useful diagnostics for obtaining both electron and ion temperatures.
   
   

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