Bose-Einstein Condensation in Solid ${}^4\mathrm{He}$

We present neutron scattering measurements of the atomic momentum distribution $n(\mathbf{k})$ in solid helium under a pressure $p=41\mathrm{bar}$ (molar volume $V_m=20.01\pm 0.02{\mathrm{cm}}^3/\mathrm{mol}$) and at temperatures between 80 and 500 mK. The aim is to determine whether there is Bose-Einstein condensation (BEC) below the critical temperature, $T_c=200\mathrm{mK}$, where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the $k=0$ state below $T_c$, we find a condensate fraction of $n_0=(-0.10\pm 1.20)\%$ at $T=80\mathrm{mK}$ and $n_0=(0.08\pm 0.78)\%$ at $T=120\mathrm{mK}$, consistent with zero. The shape of $n(\mathbf{k})$ also does not change on crossing $T_c$ within measurement precision.

Figure 1

Observed $J(Q,y)$ of solid ${}^4\mathrm{He}$ at $V_m=20.01{\mathrm{cm}}^3/\mathrm{mol}$ (open circles) folded with the instrument resolution at momentum transfer $Q=26{\AA }^{-1}$ and temperatures $T=120$ (top) and 300 mK (bottom). The solid lines are fits of the convolution approach (CA) with final state (FS) function taken from liquid ${}^4\mathrm{He}$ (Ref. 13), shape of $n(\mathbf{k})$ from Ref. 25, and width ${\overline{\alpha }}_2$ parameter as the single free fitting parameter. The data peaks below $y=0$ because of FS effects. The dotted line is the MARI instrument resolution function.

Figure 2

The mean square atomic momentum ${\overline{\alpha }}_2=⟨k_Q^2⟩$ along an axis of solid helium at $V_m=20.01{\mathrm{cm}}^3/\mathrm{mol}$ versus temperature obtained from fits to $J(Q,y)$ such as shown in Fig. 4.

Figure 3

Condensate fractions obtained by fitting the CA at $T=80$ and 120 mK with the width of $n(\mathbf{k})({\overline{\alpha }}_2)$ held fixed at its values shown in Fig. 2. We find $n_0=(-0.10\pm 1.20)\%$ at $T=80\mathrm{mK}$ and $n_0=(0.08\pm 0.78)\%$.

Figure 4

Fits of the additive approach (AA ) which includes FS effects and a simple Gaussian to the observed $J(Q,y)$.

Figure 5

Fitting parameters ${\hslash }^2{\mu }_2=(\lambda Q{)}^2{\overline{\alpha }}_2$ and ${\hslash }^3{\mu }_3=(\lambda Q{)}^2{\overline{a}}_3$ in the AA giving ${\overline{\alpha }}_2=1.67{\AA }^{-2}$ and ${\overline{a}}_3/\lambda =6.21{\AA }^{-4}$.