  Cold neutrons are converted to ultra-cold neutrons (UCN) by the excitation of
a single phonon or multiphonons in superfluid helium. The dynamic scattering
function S(q, omega) of the superfluid helium strongly depends on pressure,
leading to a pressure- dependent differential UCN production rate. A
phenomenological expression for the multiphonon part of the scattering function
s(lambda) describing UCN production has been derived from inelastic neutron
scattering data. When combined with the production rate from single phonon
processes this allows us to calculate the UCN production for any incident
neutron flux. For calculations of the UCN production from single phonon
processes we propose to use the values for S*(SVP) = 0.118(8) and S*(20 bar) =
0.066(6). As an example we will calculate the expected UCN production rate at
the cold neutron beam for fundamental physics PF1b at the Institut Laue
Langevin. We conclude that UCN production in superfluid helium under pressure
is not attractive.
