F. Pistolesi (ILL).
The hybridisation of the single-particle branch with the two particle
continuum in the region beyond the roton is reconsidered by including
the effect of the interference term between one and two quasiparticle
excitations. Fits to the latest data by B. Fåk and J. Bossy
with our model allow us to extract the dispersion relation of the
final part of the 4He spectrum for momenta between 2.3
Å-1 and 3.6 Å-1 with improved accuracy. In contrast with previous
measurements and data analysis we find that it never crosses the line
of two times the roton energy as expected theoretically. Moreover,
evidence for an attractive interaction between rotons in this range
of momentum is found.
It is well known that the branch of the excitations in 4He flattens out for momentum Q ~ 2.4 - 3.4 Å-1, where one excitation can decay into two excitations (specifically two rotons) fulfilling energy and momentum conservation.
This phenomenon was first predicted by
Pitaevskii who exploited the logarithmic divergence of the two-roton
density of states to find the exact form of the Green functions in a
restricted region of energy around 2 
( is the roton
energy). The explicit value of the parameters entering the Green
functions cannot be inferred from the theory and should be extracted
from the experimental data. In particular a crucial parameter of the
theory is the two-roton interaction potential V4 as it determines
qualitatively the termination of the spectrum. As a matter of facts
for an attractive interaction, due to the effective bi-dimensionality
of the decay process, we always have a bound state
(EB
< 2 ), and in this
case the hybridisation between the one particle branch and the
two-roton excitations gives rise to a one particle branch that
approaches asymptotically EB with a decreasing
weight. For positive V4 , instead, the
spectrum terminates with an essential singularity at a critical value
of momentum Qc at energy 2 . In
both cases we should never observe a sharp peak at energy larger than
2 as for such
energy excitations become unstable towards decay into two
rotons.
The experimental data show clearly that the
hybridisation does take place, but they leave open questions. The
position of the peak at momentum larger than 2.6
Å-1 seems to be above
the 2 line in
contrast with theoretical expectations. The experimental resolution
is not enough to guarantee that the peak is really sharp, so it could
also be a weakly damped peak at energy slightly larger than 2 , but again this would not agree
with theory. Moreover, a satisfying quantitative agreement between
experiment and theory has not been found so far as even the sign of
V4 is
still undetermined.
The validity of Pitaevskii's theory (extended
successively by many authors is restricted to the small region in
energy where the logarithmic singularity dominates, and it cannot be
applied to higher energies. So one encounters difficulties to analyse
data for large value of Q because most of the spectral weight of S(Q,
) is moving to higher energies.
Moreover, so far no theory has taken into account the interplay
between the one- and two-quasiparticle response function that becomes
more and more important for large Q and .
We have thus considered this effect by studying
a simple model where two real numbers 
and 
parametrise
the coupling of neutrons to one and two roton excitations. We thereby
include interference effects in neutron scattering parametrised on
the hybridisation amplitude V3 and the interaction
V4.
The theory depends on the exact roton-roton response function 
() and on the bare one roton Green function. It applies,
in principle, to a larger energy range than the Pitaevskii's one, as
far as one can guess the form of (). In
particular in the narrow region of energy around 2 we can use the explicit expression
for () given by Pitaevskii's theory.
Outside this region Im() should be related to the joint
density of states of two noninteracting rotons but as it is not easy
to have a reliable estimate of the exact one we preferred to extract
also () from the data.
|
|
Figure 1: Data from and best
fit for sets of data with |
We have performed different kinds of fits to
the data by Fåk and Bossy with our expression for S(Q, ). We performed our fits by
assuming no dependence on Q of all the parameters, apart from the
bare dispersion relation which is expected to be the only parameter
strongly dependent on Q. The fits have been done by minimising
globally the 2 for S(Q, ) over
the bi-dimensional region Q and and not by minimising it separately for each value of
momentum. We find that this procedure reduces greatly the error bars
on the fitted parameters and it gives more confidence in the
reliability of the theory.
We studied two different problems. First we
wanted to determine V4 and for this purpose
we performed a fit with a cut-off in energy of 1.7 meV (slightly
larger than 2 ~ 1.5 meV). In
this region the modified Pitaevskii theory should work
quantitatively, as effectively we find. The interaction potential
turns out to be attractive with a V4 ~ -4.7
meVÅ3 and a tiny EB ~ 2 - 1.3 µeV.
Second, we wanted to study the applicability of
the theory to the whole region of energy 0 < < 12 meV and extract from the
data (). This has been possible with a
second fit to the data without any cutoff and minimising the 2 also over the possible
shapes of (). In Fig. 1 we report the result
of the fit compared with the data and the fitted Im () compared with the non-interacting one Im 0().
The peak at 2 is due to the attractive interaction. Note that the
parameter is crucial to obtain a
good fit, also if we have completely free (). The fit with a cutoff in energy (not shown) turns out
to be even more accurate.
The two different fits suggest nearly the same dispersion relation for the quasiparticles as shown in Fig. 2.
|
|
Figure 2: New dispersion
relation for the region |
In conclusion experiments agree with theory
when data are properly analysed. In fact within the resolution of the
data it is not possible to extract the position of the sharp peak
without a suitable analysis. The identification of the energy at
which 2 has a maximum
with the position of the pole in the quasiparticle Green function is
wrong as continuum and discrete contributions are of the same order
of magnitude. The convolution with the experimental resolution leads
to a broad peak at energy slightly higher than 2 that has been previously
interpreted as the quasiparticle energy. Experiments at higher
resolution in this region would be useful to verify the presence of a
sharp peak at the position suggested by our analysis.
Acknowledgement
I am indebted to P. Nozières for many suggestions and discussions. I gratefully acknowledge B. Fåk and J. Bossy for giving me data prior to publication. I also acknowledge B. Fåk, N. Cooper, N. Manini and H. Glyde for useful discussions.
