The reflection of low energy phonons at the free surface of liquid 4 He. I. Theory

The scattering of a low energy phonon at the free surface of liquid4He at finite temperature is discussed in relation to the reflection measurements in the accompanying paper (II). If not specularly reflected, the phonon may be inelastically scattered, with the emission or absorption of a single ripplon, or it may be absorbed with the emission of two ripplons. The inelastic scattering produces deviations from ideal specular reflection that depend logarithmically on the angular sensitivity of the measurement. For the experiment described in II, the predicted deviations due to scattering and absorption are roughly equal and too small (∼5×10−4) to be measured. In addition, there should be a small broadening (less than 1 ° in angle) of the reflected image of the phonon source due to phonon decay in the bulk liquid. This was calculated from the curvature of the phonon spectrum measured by Rugar and Foster. Phonon decay also determines the distribution of the incident phonon beam with respect to energy. From the known decay rate, the average incident phonon energy in our experiment is calculated to be ∼0.5 K. We also discuss the attenuation of surface second sound due to the inelastic scattering and absorption of thermal phonons at the surface. We find that two ripplon absorption is the dominant effect. Below ∼0.7 K, the attenuation due to phonons is probably just small enough for pure ripplon surface sound to exist in a narrow range of low frequencies. To show this, we have recalculated the ripplon lifetime using the 3-ripplon interaction as recently revised by Rocheet al. The results for the ripplon lifetime are displayed in a simple scaling format.