exotic nuclei structure and reaction noyaux exotiques ... - IPN - IN2P3

Figure 3: Ratio of in-medium to free cross section
as function of the relative momentum q for
equilibrium distribution function. We compare our
frequencies and damping rates with the results of
three recent experiments [Altmeyer et al., Phys.
Rev. A 76, 033610 (2007); Wright et al., Phys.
Rev. Lett. 99, 150403 (2007); Riedl et al., Phys.
Rev. A 78, 053609 (2008)]. The main success of
our model is that with the help of the mean field U
we can reproduce the observed upwards shift of
the quadrupole mode in the collisionless regime
compared to the ideal gas result. However, in the
cases where the frequencies and damping rates
are dominated by collisions, the description of the
data becomes worse if one uses the in-medium
cross section instead of the free one: Because of
the enhanced cross section, the relaxation time τ
gets too short. As an example we show the results
for the scissors mode in fig. 4. The data are from
the experiment by Wright et al. [loc. cit.], where
the trap frequencies in the x and y directions were
ω x =2π·830Hz and ω y =2π·415Hz. In the hydrodynamic
(τ→0) and collisionless (τ→∞) limits, the
scissors mode gets undamped and approaches
the limiting frequencies √(ω x 2 +ω y 2 )=2π·944Hz and
ω x +ω y =2π·1245Hz, respectively. As can be seen
from fig. 4, the results with the in-medium cross
Figure 4: Frequency (top) and damping (bottom)
of the scissors mode as a function of temperature.
Figure 5: Quadrupole response of a gas of
10000 atoms with interaction strength k F a = -2
and temperature T = 0.4 ε F .
section (red lines), especially the damping rates Γ,
go too fast to the hydrodynamic limit at lower
temperatures, i.e., the relaxation time is too small.
Numerical solution of the Boltzmann equation
The approximations which are implicitly made in
the method of moments could be a possible
reason for the disagreement between theory and
experiment. This is why we developed a code for
the numerical solution of the Boltzmann equation
based on the pseudoparticle method. The principle
is the same as in the usual codes for the simulation
of heavy-ion collisions, but the practical problems
are different because of the much larger
number of particles in the trapped gas.
As a first application of this code, we studied the
collective quadrupole oscillation of the cloud in a
spherical harmonic trap, without mean field and
with the free cross section. In the hydrodynamic
limit (τ→0), the quadrupole mode becomes
undamped and approaches the frequency √2ω 0 ,
where ω 0 denotes the trap frequency. In the
collisionless limit (τ→∞), the mode is undamped,
too, but its frequency is higher and approaches
2ω 0 . In fig. 5 we show the response obtained from
the pseudoparticle simulation (red), compared
with the method of moments (blue). While the
general behaviour is similar, the simulation gives a
higher frequency than the method of moments. If
we use the form of the response function from the
method of moments, but consider τ as a fitting
parameter, we can very well reproduce the result
of the simulation (purple). The fitted relaxation time
τ obtained in this way is significantly higher (by
~50%) than that obtained from the method of
moments, and a similar increase of τ is found in
the whole range of interaction strengths and
temperatures. It seems that the method of
moments systematically overestimates the effect of
collisions. This is a very encouraging result, since,
as mentioned above, the relaxation time obtained
by the method of moments for realistic parameters
and with the in-medium cross section was too
short in comparison with experimental data.
References:
S. Chiacchiera, T. Lepers, D. Davesne, and M.
Urban, Phys. Rev. A 79, 033613 (2009).
77