A special case—liquid helium

The special case of liquid helium allows us to present one aspect of direct geometry spectroscopy in particular. Since helium atoms interact so weakly, there are almost no restoring forces in the liquid and only the conservation of momentum plays a significant role in its INS spectrum. The spectrum of a mixture of liquids He and He is shown in Fig. 5.9 [21], as can be seen from the figure it consists of two continuous traces. The first response, with a slope of 5.6, is from the light isotope of helium, He. The second response, of slope 4.2, is from the common isotope of helium, mass four. (The ratio of the slopes is 4.2/5.6 = 3/4.) There are no excitations in the spectrum and the observed response is the result of atomic recoil. 

The conservation of momentum restricts the scattering intensity to lie on a line with a slope related to the mass of the scattering atom. The lines in Fig. 5.9 trace the maximum scattering strength from each isotope and at a given value of energy transfer the response is broad in Q. Substituting the atomic mass for the effective mass in Eq. (2.77)  

The slope is given in the unconventional units, cm (A ) . (Conventionally the momentum transfer in Fig. 5.9 would be linear in Q and the lines of equivalent mass become quadratic curves (see Fig. 9.18).) 

Conservation of momentum is also found in the oscillator responses from molecules but here the simultaneous conservation of energy as well 

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