# Final-state effects on superfluid 4He in the deep inelastic regime.

TL;DR: It is shown that Gersch-Rodriguez theory produces results as accurate as those coming from other more recent FSE theories.

Abstract: A study of final-state effects (FSE) on the dynamic structure function of superfluid {sup 4}He in the Gersch-Rodriguez formalism is presented The main ingredients needed in the calculation are the momentum distribution and the semidiagonal two-body density matrix The influence of these ground-state quantities on the FSE is analyzed A variational form of {rho}{sub 2} is used, even though simpler forms turn out to give accurate results if properly chosen Comparison to the experimental response at high momentum transfer is performed The predicted response is quite sensitive to slight variations on the value of the condensate fraction, the best agreement with experiment being obtained with {ital n}{sub 0}=0082 Sum rules of the FSE broadening function are also derived and commented Finally, it is shown that Gersch-Rodriguez theory produces results as accurate as those coming from other more recent FSE theories {copyright} {ital 1996 The American Physical Society}

## Summary (2 min read)

### I. INTRODUCTION

- The dynamics of the sample is entirely contained in S(q,), the dynamic structure factor, which is the Fourier transform of the density-density correlation function.
- The delta function in Eq. ͑2͒ takes care of the momentum and energy conservation in the scattering event between the neutron and a single atom.
- Assuming S(q,)ϭS IA (q,), the momentum distribution n(k) can be extracted from Eq. ͑2͒ by simple differentiation.

### II. GERSCH-RODRIGUEZ THEORY OF FSE

- The nth order cumulant accounts for the correlations among the struck atom and clusters of n particles in the background.
- In the high momentum transfer limit, those terms with nϭ1 carry the most significant corrections.
- At this level, the FSE broadening function can be expressed as a function of the interatomic potential and the one-and twobody density matrices.

### III. NUMERICAL RESULTS

- The authors present results for the FSE correcting function R(q,Y ) and the response function J(q,Y ) calculated in the framework of the Gersch-Rodriguez formalism.
- The variational minimization has been performed for the HFDHE2 Aziz potential 24 at the experimental equilibrium density (ϭ0.365 Ϫ3 ; ϭ2.556 Å͒.
- The ground-state description obtained with this wave function is in good agreement with recent Green's function Monte Carlo calculations.
- 25, 26 The discussion is separated in two parts, the first one being devoted to the study of both R(q,Y ) and J(q,Y ) and their comparison to experimental data, and the second one to the analysis of the dependence of these functions on the different approximations used in the variational description of the ground-state wave function.

### V. COMPARISON WITH OTHER FSE THEORIES

- FSE theories can be classified in different groups depending on the way they incorporate the corrections to the IA.
- Examples of theories belonging to the first class are those of Silver 12 or Carraro and Koonin.
- Theoretical arguments brought them to fix its value to r 0 ϭ2.5 Å. 10 With this prescription, Gersch and Rodriguez predicted a J(q,Y ) that visibly overestimates the measured strength of the response around its maximum.
- The authors have compared their results for R(q,Y ) and J(q,Y ) with those obtained by Silver 12 and Carraro and Koonin.
- The FSE function R(q,Y ) is slightly different in the three theories, though both the height and width of the central peak are quite similar.

### VI. SUMMARY AND CONCLUSIONS

- He have been studied in the framework of the Gersch-Rodriguez theory using a realistic description of the ground state of the liquid.
- Two quantities describing the ground state of the system are needed.
- The latter is much less affected by FSE, although the effects are nonnegligible.
- The authors have verified that Gersch-Rodriguez theory gives accurate results when proper forms for the one-and two-body density matrices are used.
- A variational 2 obtained in the HNC framework accurately reproduces the experimental response at high q's.

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