Développement et optimisation d'un système de polarisation de ...

tel-00726870, version 1 - 31 Aug 2012
32 1.2. Ultracold neutrons
Interaction with a set of nuclei
UCN have a large wavelength compared to the atomic radius (50 to 130 nm compared to 0.1
nm). Therefore, during a scattering process, a UCN interacts with hundreds of nuclei. The mean
potential, called the Fermi potential, is given in Eq. (1.7). A critical velocity can be defined
from this potential:
vc = 2VF /mn
(1.14)
Table 1.2 shows several common materials with their Fermi potential and associated critical
velocity.
Table 1.2: Fermi potentials, critical velocities and losses factor (Sec. 1.2.2) of materials used in UCN
physics.
Material VF [neV] vc [m/s] W [neV]
Al 54 3.2 1.1 · 10 −3
Be 252 6.9 1.3 · 10 −3
C (graphite) 195 6.1 1 · 10 −3
C (diamond) 306 7.7 3.1 · 10 −5
Cu 168 5.7 2.5 · 10 −2
DPS 162 5.6 3.8 · 10 −2
Fe 210 6.3 1.9 · 10 −2
Ni (natural) 252 6.9 3.2 · 10 −2
58 Ni 335 8 3 · 10 −2
Quartz 90 4.1 3.6 · 10 −4
Stainless steel 185 5.9 1.5 · 10 −2
The arrival of a neutron on a surface may be described as the arrival of a neutron on a
potential step. The neutron energy can be divided in two parts: one perpendicular to the
surface (E⊥), and one parallel to it (E). Classically, the neutron is transmitted if E⊥ > Ec,
and reflected if E⊥ < Ec. In a quantum approach, it is not the case anymore. Let us consider a
one dimensional problem: the potential step can be described as:
0 x < 0
V (x) =
(1.15)
V x > 0 with V > 0
Figure 1.5 shows this potential and the three waves.
If the UCN energy is written E⊥, then the UCN wave function is:
ψ =
Ae ikxx + rAe −ikx x < 0
tAe iktx x > 0
(1.16)
2mn
¯h 2
2mn
E⊥, kt = ¯h 2 (E⊥ − V ), rA being the amplitude of the reflected wave and
with kx =
tA the amplitude of the transmitted wave. Using the continuity conditions, the probability of
reflection R = r2 and transmission T = kt
t2 are calculated:
kx