  Metal tubes plated with nickel-phosphorus are used in many fundamental
physics experiments using ultracold neutrons (UCN) because of their ease of
fabrication. These tubes are usually polished to a average roughness of 25-150
nm. However, there is no scattering model that accurately describes UCN
scattering on such a rough guide surface with a mean-square roughness larger
than 5 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 input of the model. This model
was used to describe UCN transmission experiments conducted at the pulsed UCN
source at 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 nm to 17 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 $p_{L} = 0.039\pm0.003$ 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
nm.
