Layer- and Bulk-Roton Excitations of ${}^4\mathrm{H}\mathrm{e}$ in Porous Media

We examine the energetics of bulk- and layer-roton excitations of ${}^4\mathrm{H}\mathrm{e}$ in various porous medial such as aerogel, Geltech, or Vycor, in order to find out what conclusions can be drawn from experiments on the energetics about the physisorption mechanism. The energy of the layer-roton minimum depends sensitively on the substrate strength, thus providing a mechanism for a direct measurement of this quantity. On the other hand, bulklike roton excitations are largely independent of the interaction between the medium and the helium atoms, but the dependence of their energy on the degree of filling reflects the internal structure of the matrix and can reveal features of ${}^4\mathrm{H}\mathrm{e}$ at negative pressures. While bulklike rotons are very similar to their true bulk counterparts, the layer modes are not in close relation to two-dimensional rotons and should be regarded as a third, completely independent kind of excitation.

Figure 1

The density profiles of the first three layers are shown as a function of the depth $D$ of the substrate potential. The substrate is at $z<0$.

Figure 2

The figure shows the map of the dynamic structure function $S(\mathbf{k},\omega )$ for a ${}^4\mathrm{H}\mathrm{e}$ film for the potential strength $D=24\mathrm{K}$. The two layer rotons, the bulk roton, and the ripplon are indicated by arrows. Notice how the lower energy layer roton has minimum at lowest $k$, meaning it experiences the highest “effective” density. Indeed, the corresponding transition density depicted in Fig. 3 has nodes at the density minima.

Figure 3

The transition densities of the three lowest excitations are shown as a function of $z$ (in Å) for $D=24\mathrm{K}$ and $C_3=1500\mathrm{K}{\AA }^3$, normalized to the same maximum value. For comparison, the density profile of the film is shown as a gray-shaded area.

Figure 4

The figure shows the energy of the 2D roton (long-dashed line) and the energy of the layer roton in a film (solid line) as a function of areal density. The upper horizontal axis shows the corresponding values of the well depth $D$. Also shown is the energy of a two-dimensional roton obtained with shadow wave functions (SWF) [26]. The short-dashed horizontal lines show experimental values of the layer-roton energy on aerogel [6, 10] (7.31 and 7.43 K), Geltech [9] (6.85 K), Vycor [7, 8] (6.38 K), and graphite [27] (6.26 K). Their energies are marked in the right margin.

Figure 5

The figure shows the energy of the bulk roton in a gap of 25 Å width as a function of the two-dimensional coverage and the corresponding average density. The points of zero pressure, the low-density and the high-density spinodal points, and the bulk chemical potential are indicated as points. The dashed line shows the roton energy of a purely 3D calculation.