Surface roughness effect on ultracold neutron interaction with a wall and implications for computer simulations

We review the diffuse scattering and the loss coefficient in ultracold neutron reflection from slightly rough surfaces, report a surprising reduction in loss coefficient due to roughness, and discuss the possibility of transition from quantum treatment to ray optics. The results are used in a computer simulation of neutron storage in a recent neutron lifetime experiment that reported a large discrepancy of neutron lifetime with the current particle data value. Our partial reanalysis suggests the possibility of systematic effects that were not included in this publication.

Figure 1

For elastic wall interaction of UCN with wave number $k$ we show the projections onto the surface plane of incident (${\mathbf{k}}_i$), mirror-reflected (${\mathbf{k}}_r$), and diffusely scattered (${\mathbf{k}}_s$) wave vectors. The in-plane wave vector transfer q is the component of ${\mathbf{k}}_s-{\mathbf{k}}_r$ parallel to the surface. The vectors are bracketed to indicate projection onto the $\mathit{xy}$ plane. For elastic scattering the end points of q are confined to the circle of radius $k$.

Figure 2

The total diffusely scattered intensity $p_{D0}$ [from (18) for the $K_0$ model] is compared to approximation (24) which is valid for fairly smooth roughness ($\alpha \ll 1$). For the parameters used the agreement is excellent.

Figure 3

Angle dependence of roughness correction to the wall loss coefficient, $\Delta \mu ({\theta }_i)/{\mu }_0({\theta }_i)$. The numerical result from (20) for the $K_0$ model is compared to approximation (25). The agreement is very good except for ${\theta }_i$ near ${90}^{°}$ (glancing incidence). In this plot the scale changes from $n=3$ (expanded scale) for ${\theta }_i<{70}^{°}$ to $n=1$ (direct scale) for ${\theta }_i>{70}^{°}$.

Figure 4

Mean value of roughness correction, $\langle \Delta \mu ({\theta }_i)\rangle$, for isotropic UCN flux, referred to the flat-wall average $\langle {\mu }_0({\theta }_i)\rangle$ given in (27). For the parameters given, the numerical result for both models are compared to the semianalytical approximation (28). Both for the Gauss and the $K_0$ model, the agreement is reasonably good, with maximal deviations of $~20\%$ over a wide range of parameters.

Figure 5

Schematic of the apparatus used in Ref. [3]. UCN from a guide tube 1 pass through a large entrance or exit channel to a cylindrical or “quasispherical” storage chamber 8 that can be rotated about a horizontal axis. After a short or long holding time, UCN in five spectral ranges are, successively, discharged through the same channel to the detector 12. The system 14 is used to coat the walls with a special low-temperature oil.

Figure 6

Simulated count rates, in 20-s bins, versus cycle time $t$, starting from “monitoring” for 300 s where the trap is at hold position ${30}^{°}$ titled away from vertical. The data are normalized to ${10}^6$ runs (one run for each UCN started at $t=0$). Data are shown for both short and long storage and for both the Gauss and $K_0$ models for a fairly smooth surface with similar parameters, as shown in the inset. They correspond to $\alpha =7.1\times {10}^{-4}$ for the Gauss model, and $\alpha =2.4\times {10}^{-4}$ for $K_0$. An example of steeper roughness ($\alpha =0.7$) is shown for comparison. We plot the simulated counts N plus one to allow a distinction between $N=0$ and $N=1$ on the log scale. In the simulations of Fig. 6 (only) we used a constant angular speed of ${4.3}^{°}$/s for all trap rotations.

Figure 7

Comparison of measured (Serebrov et al., 2008: [3]) and simulated time spectra for the first counting period following short storage. For clarity, (b) shows an expanded view of the initial 60 s of the full 150-s interval shown in (a). Some of the simulation curves of (b) are also included in (a). For a discussion see the text.

Figure 8

Experimental (Serebrov et al. 2008: [3]) and simulated data are compared for the first 300 s of storage, i.e., from $t=300$ s to $t=600$ s. The curves are labeled in the inset. The initial decay, with channel time constant 30 s, for the experimental points (curve 1) and for the simulation in Ref. [3] for isotropic diffusivity $0.1\%$ of surface reflection (curve 2), is due to the last stage of UCN drainage from the wide “entrance or exit channel.” The faster initial decrease seen in our simulation data for the trap only (the remaining curves in Fig. 8), occurs because the storage time interval includes the initial 7 s of trap rotation from ${30}^{°}$ to $0^{°}$ [3].

Figure 9

Simulation results for the mean value $\langle \gamma \rangle =\langle \mu \rangle /\eta$, for the five spectral ranges 1 to 5 of Ref. [3], plotted for short storage (300 s, solid lines) and for long storage (2000 s, dashed lines). For the $K_0$ model, the abscissa shows an example of “intermediary roughness” with $b=2$ nm, $w=4$ nm, thus $\alpha =0.7$. The ordinate values are for $b=0.5$ nm, $w=3$ nm, thus $\alpha =2.4\times {10}^{-4}$. The seemingly “erratic” behavior in either case may be explained by mixing between the five spectral intervals and loss due to “spill-over’ during the storage time as discussed in Sec. 8. The statistical errors for the $\langle \gamma \rangle$ values are insignificantly small.

Figure 10

The “cooling” effect during long storage versus short storage. Instead of a small systematic drop $\langle \Delta h\rangle$ of mean spectral height $\langle h\rangle$ as a function of time, expected since the wall loss increases with UCN energy, the simulation shows fairly large and “erratic” values of $\langle \Delta h\rangle$ for the five spectral ranges. Although there is cooling for the full spectrum, the lowest-energy intervals show negative values of $\langle \Delta h\rangle$ (“heating”). The simulation data are for the $K_0$ model with fairly “soft roughness” ($\alpha =2.4\times {10}^{-4}$) and are explained by interspectral mixing and by “spill-over” (see Sec. 8). The $\langle \langle \gamma \rangle \rangle$ and $\langle \langle h\rangle \rangle$ values are averages over the spectral ranges and over the two storage times (300 s and 2000 s).