Diffuse scattering model of ultracold neutrons on wavy surfaces

Metal tubes plated with nickel-phosphorus are used in many fundamental physics experiments that use ultracold neutrons (UCN) because of their ease of fabrication. These tubes are usually polished to an average roughness of $25\text{–}150\mathrm{nm}$. However, there is no scattering model that accurately describes UCN scattering on such a rough guide surface with a mean-square roughness greater than $5\mathrm{nm}$. We, therefore, developed a scattering model for UCN in which scattering from random surface waviness with a size larger than the UCN wavelength is described by a microfacet Bidirectional Reflectance Distribution Function model (mf-BRDF model), and scattering from smaller structures by the Lambert's cosine law (Lambert model). For the surface waviness, we used the statistical distribution of surface slope measured by an atomic force microscope on a sample piece of guide tube as an input of the model. This model was used to describe UCN transmission experiments conducted at the pulsed UCN source in J-PARC. In these experiments, a UCN beam collimated to a divergence angle smaller than $\pm 6^{\circ }$ was directed into a guide tube with a mean-square roughness of 6.4 to $17\mathrm{nm}$ at an oblique angle, and the UCN transport performance and its time-of-flight distribution were measured while changing the angle of incidence. The mf-BRDF model combined with the Lambert model with scattering probability $p_L=0.039\pm 0.003$ reproduced the experimental results well. We have thus established a procedure to evaluate the characteristics of UCN guide tubes with a surface roughness of approximately $10\mathrm{nm}$.

Figure 1

Differences in UCN trajectories (left panel) and TOF shapes (right panel) depending on the reflection models. They were calculated by our Monte Carlo simulation. In these calculations, ${10}^6$ parallel UCNs with a diameter of $5\mathrm{mm}$ and a velocity of $5\mathrm{m}/\mathrm{s}$ are injected into a cylindrical guide tube with a length of $1\mathrm{m}$ at an upward angle of ${30}^{\circ }$. The time distribution of the incident UCNs consists of a rectangular pulse with a width of $10\mathrm{ms}$. The value of $p_L$ in the Lambert model and that of $\alpha$ in the mf-BRDF model, which is explained in Sec. 3a, are both set to 0.03. The left panel shows the integration of the three-dimensional UCN trajectories projecting onto the plane of the paper surface. The right panel shows the TOF of the incident UCNs until they reach the downstream end of the guide tube.

Figure 2

Setup of the UCN transmission experiment (top view). The figure shows the setup with the sample guide tube attached at an angle of ${30}^{\circ }$ and the detector position at an angle of $0^{\circ }$. The upper right inset shows an enlarged view of the connection between the guide tube and the UCN detector, and the lower inset shows the setup that changes when the installation angle of the guide tube is changed. The symbols in the figure mean the following: (a) a square collimator with internal dimensions of $45\mathrm{mm}\times 45\mathrm{mm}$ and a length of $61\mathrm{mm}$, coated with ${\mathrm{Gd}}_2{\mathrm{O}}_3$ compound; (b) a square collimator with internal dimensions of $45\mathrm{mm}\times 45\mathrm{mm}$ made of polyethylene plate; (c) a cylindrical collimator with an internal diameter of $50\mathrm{mm}$ and a length of $155\mathrm{mm}$ made of polyethylene; (d) NiC mirror sputtered on a Si wafer; (e) vacuum chamber; (f) $10\text{−}\mu \mathrm{m}$-thick copper film; (g) angled flange; ($L_1$) distance from the point of UCN production in the Doppler shifter to the center of the NiC mirror, which is $250\mathrm{mm}$; ($L_2$) distance from the center of the NiC mirror to the center of the sample guide tube inlet, which is $155\mathrm{mm}$; ($L_3$) distance from the center of the sample guide tube inlet to the exit of the cylindrical collimator, which is $43\mathrm{mm}$.

Figure 3

(a) TOF spectra of transmitted UCN measured without the sample guide tube, and with the tube at different installation angles. (b) Velocity distributions converted from (a) by using TOF distances.

Figure 4

Measured rate of transmitted neutrons with no guide (a red square) and with sample guide installed at different angles (black dots), compared to simulations with no diffuse reflection (blue triangles), pure Lambert scattering with a probability of $p_L=0.081$ (orange circles), and mf-BRDF scattering combined with Lambert scattering with a probability of $p_L=0.039$ (pink squares). Due to gravity and incidence angle, UCN has to undergo between 1 and 7 specular reflections to reach the detector, reducing the transmission compared to the no-diffusion simulation.

Figure 5

The off-specular scattering of incident rays due to surface roughness in the mf-BRDF model [32, 34].

Figure 6

Comparison between the mf-BRDF model and a macroscopic waviness model described by Steyerl (Eq. (26) in Ref. [19]). They were calculated for $\alpha =0.001$, ${\theta }_i={89}^{\circ }$, ${\varphi }_i={180}^{\circ }$, and ${\varphi }_o=0^{\circ }$. In the mf-BRDF model calculation, $F$ was set to 1, but $G$ was calculated exactly. The curve for the mf-BRDF model is normalized to have an integral value of 1, and the curve for the Steyerl's model is normalized to the same value.

Figure 7

(a) Image of the surface of a guide tube sample measured by AFM. (b) Image obtained by frequency filtering to remove structures with a period of less than $128\mathrm{nm}$ from image (a).

Figure 8

The polar angle distribution of the normal vector created from Fig. 7 and the fit curve using Eq. (3).

Figure 9

Variations of $\alpha$ with respect to ${\lambda }_c$ and the fitted power function. The datasets used for the fit are shown in color and with open brackets.

Figure 10

Comparison of velocity distribution shapes between experiment and simulation: simulations with the Lambert model alone with $p_L=0.081$ (top), simulations combining the mf-BRDF model with the Lambert model with $p_L=0.039$ (bottom). Each figure is accompanied by the ${\chi }^2/\mathrm{ndf}$ values computed from their respective measured and simulated data sets.

Figure 11

Reduction in the transport efficiency of UCN at a velocity of $5\mathrm{m}/\mathrm{s}$ with various diffuse scattering models as the length of the guide tube is extended. In panel (a), the UCN returned to the inlet is assumed to be completely lost (open inlet). The statistical errors are smaller than those of the markers. In panel (b), the calculation is performed under the condition that the UCN returned to the inlet is reflected by an NiP mirror (closed inlet). Refer to the text for details of the calculation settings.