A microscopic approach to the response of $^{\bf 3}$He -$^{\bf 4}$He mixtures

Correlated Basis Function perturbation theory is used to evaluate the zero temperature response $S(q,\omega)$ of $^3$He-$^4$He mixtures for inelastic neutron scattering, at momentum transfers $q$ ranging from $1.1$ to $1.7 \AA^{-1}$. We adopt a Jastrow correlated ground state and a basis of correlated particle-hole and phonon states. We insert correlated one particle-one hole and one-phonon states to compute the second order response. The decay of the one-phonon states into two-phonon states is accounted for in boson-boson approximation. The full response is splitted into three partial components $S_{\alpha \beta}(q,\omega)$, each of them showing a particle-hole bump and a one phonon, delta shaped peak, which stays separated from the multiphonon background. The cross term $S_{34}(q,\omega)$ results to be of comparable importance to $S_{33}(q,\omega)$ in the particle-hole sector and to $S_{44}(q,\omega)$ in the phonon one. Once the one-phonon peak has been convoluted with the experimental broadening, the computed scattering function is in semiquantitative agreement with recent experimental measurements.


INTRODUCTION
Isotopic atomic Helium mixtures are an intriguing case for many-body physicists. There exists a large body of experimental data, concerning mostly static properties (for instance, the chemical potentials and the maximum solubility 1-3 ). Excitation spectra and related quantities, as the zero concentration (x 3 = 0) 3 He effective mass (m * 3 ), have been also measured 4 . Recently, inelastic neutron scattering experiments have been carried out both at low, or intermediate, and high momentum transfers 6,7 . In both regions the measured response presents two generally distinguishable structures, to be ascribed to boson like collective excitations (phonons and rotons) and to Fermi particle-hole ones. However, this apparently simple picture hides a large interplay between the components of the mixtures, each of them probably contributing on comparable foot to both the branches of the response. The reason for this lies in the large correlation effects, which are present in the system because of the strong interatomic potential and of the large density. This are also the motivations why truly microscopic and ab initio studies of Helium mixtures are difficult, and, in the case of the response, practically absent in literature.
Qualitative studies of the response have been done in ref. 8 (using a matrix dispersionrelation representation)and in ref. 9 with a correlated RPA approach (very similar, in spirit, to the phenomenological Polarization Potential method used in ref. 10 . Here we will employ the Correlated Basis Function (CBF) perturbation theory, to embody the above correlation effects directly into the basis functions. CBF has shown to be a powerful tool to succesfully study Helium at zero temperature: the energetics of both pure 4 He and 3 He are well described by sophisticated correlated ground state wave functions, containing explicit two-, threebody, back-flow and spin correlations [11][12][13] ; properties of one 3 He impurity in 4 He, such as chemical potential and effective mass are also quantitatively reproduced by such correlated wave functions 14 . In particular, by using CBF based perturbation theory, with the insertion of up to two correlated independent phonon intermediate states, the impurity effective mass m * 3 turns out to be 2.2m 3 , to be compared with the experimentally measured 2.3m 3 value. The behavior of the 3 He effective mass with the concentration in dilute mixtures has been recently object of some debate. Specific heat measurements 15,16 at finite x 3 do not show appreciable deviations from its x 3 = 0 value. In ref. 6 the Authors have to postulate a much larger value (m * 3 ∼ 2.95m 3 at x 3 = 0.05) in order to reproduce the position of the particlehole response with a Lindhard like function and using a simple Landau-Pomeranchuk (LP) quasiparticle spectrum 17 , This contradiction does not appear if one modifies the LP spectrum ( LP modified, or LPM) as: There are both experimental 6,4 and theoretical 5 indications of a deviation from the simple LP form.
In a CBF based approach, we assume to have an homogeneous mixture of N 3 3 He atoms and N 4 4 He atoms in a volume Ω, with partial densities ρ α=3,4 = N α /Ω, total density ρ = ρ 3 + ρ 4 and concentrations x α = ρ α /ρ. We will keep constant densities, while letting N α and the volume going to infinity. The nonrelativistic Hamiltonian of the mixture is where the interaction is the same for all the different pairs of the mixture.
A realistic, correlated, variational ground state wave function Ψ 0 is obtained by the Jastrow-Feenberg ansatz 18 where φ 0 (N 3 ) is the ground state Fermi gas wave function for the 3 He component and F J , F T and F BF are N-body correlation operators including explicit two-, three-body and backflow dynamical correlations respectively. We will limit our analysis to the case of two-body, state independent (or Jastrow) correlations only (F T = F BF = 1). F J results to be where f (α,β) (r) are two-body correlation functions determined by minimizing the variational ground state energy.
It is possible to generate a correlated basis through the operator (5), to be used in a CBF perturbation theory (CBFPT). This theory has been succesfully adopted for computing the inclusive response of nuclear matter and heavy nuclei to electron and hadron scattering [19][20][21] , and has shown to be able to provide a semiquantitative agreement with experimental neutron inelastic scattering (nIS) data in pure, liquid atomic 4 He 22 .
In this paper, we will apply CBFPT to compute the nIS response of the mixture, by considering as intermediate states the normalized, correlated 4 He n-phonon states (nPH) |k 1 , .., k n , and 3 He n-particle, m-hole states (np-mh) |p 1 , .., p n , h 1 , .., h m , .
The nPH states are given by where ρ 4 (k) is the 4 He density fluctuaction operator Correlated np-mh states are obtained in a similar way, by applying the correlation operator to the Fermi gas excited states Φ np−mh (N 3 ), We will consider 1-phonon (1PH) and 1p-1h intermediate correlated states, which we will term as One Intermediate Excitation (OIE) states. The response computed at the OIE level will be called variational. In addition, we will also consider the possible decay of 1PH states into 2PH ones, which is essential in giving a physically meaningful 4 He excitation spectrum and provides a quenching of the one-phonon peak. This term will be computed in a boson-boson approximation, i.e. neglecting the 3 He antisymmetry. Such an approach may be justified on the basis of the low 3 He concentration.
1p-1h states may also be coupled to 1PH and 2PH. Such a coupling may be taken into account by a corresponding self-energy insertion. Its analogous in the problem of the single 3 He atom in 4 He is responsible for the impurity large effective mass. To estimate the importance of this effect we will use the on-shell part of the impurity self-energy, again relying on the small value of x 3 .
The plan of the paper is as follows. In section II we will briefly outline the CBFPT for the response of the mixture and the variational calculation will be described in some details. Section III is devoted to the description of the calculation of the coupling with the 2PH states and of the decays into 1PH and 2PH states. Section IV contains results for the response and the comparison with the experimental scattering functions. Moreover, the 4 He and 3 He excitation spectra are presented and discussed. Conclusions are drawn in section V.

CBFPT FOR THE RESPONSE
The Dynamical Structure Function (DSF) S(q, ω) of a 3 He-4 He mixture at T = 0 is given by the imaginary part of the polarization propagator D(q, ω) where and and N = N 3 + N 4 . In eq.(10),Ψ 0 is the exact ground state of H with eigenvalue E 0 .
The total DSF may be expressed in terms of partial αβ DSF, S αβ (q, ω), as S(q, ω) = α,β=3,4 x αβ S αβ (q, ω), The experimentally measured nIS double differential cross section for the mixture directly provides access to the the total scattering functionŜ(q, ω), wich is in turn related to the partial DSFs' by the relation . The where I i is the spin of 3 He i-nucleus.
We will focus, in the remainder, mainly on the calculation of S αβ . To derive a perturbative expansion it is convenient to split H into an unperturbed piece H 0 and an interaction term H 1 , as follows and Here |m are correlated basis states, eigenstates of H 0 . In particular, |0 = |Ψ 0 is not an eigenstate of H and its difference from |Ψ 0 is treated perturbatively. The expansion is obtained by writing where ∆E 0 is the correction to the variational ground state energy E v 0 , and by developing the If the expansion is truncated at the zeroth order, the partial DSF are given by: with ω n = E v n − E v 0 . As stated in the introduction, we will first consider only OIE insertions, i.e. correlated 1PH and 1p-1h intermediate states, defined as:

THE VARIATIONAL RESPONSES
The variational response is given by the sum of two components, where S 1P H αβ (q, ω) has a 1PH intermediate state and S 1p−1h ω k and ε p − ε h are the variational energies of the OIE states considered.
ω k is given by and corresponds to the well known Feynman spectrum 24 . In eq.(26), S 44 (k) is the variational estimate of the 44 component of the Static Structure Function (SSF), S αβ (k), given by: In a similar way, ǫ x=p,h is obtained by where |x is a particle or hole correlated state. We will discuss later the evaluation of ǫ x .
By using the definition of the SSF given in eq. (27), giving, for S 1P H αβ (q, ω), The one-phonon contribution to the variational α-β responses shows a delta-like behavior, whose strenght is Z v αβ (k) = S α4 (k)S β4 (k)/S 44 (k), and it is located at the Feynman phonon energy. We notice that (i) Z v 44 (k) = S 44 (k) and that (ii) the 33 and 44 variational DSF are positive (S 44 (k) being positive), whereas this may not be true for the 34 DSF.
The expression for the particle-hole response S 1p−1h αβ (q, ω) is more involved. A detailed description for a pure Fermi system (specifically, nuclear matter) can be found in 19 and references therein. On the basis of that formalism, the extension to a boson-fermion mixture is straightforward.
In CBF theory, the non diagonal matrix elements ξ α (q; p, h) = Ψ 0 |ρ † α (q)|p, h are computed by a cluster expansion in Mayer like diagrams, and by summing infinite classes of relevant terms. The ξ α are explicitely given by: with (x, y) = (d, e) and g xy,αβ (r) are partial radial distribution functions (RDF). In fact, the total αβ-RDF, g αβ (r), giving the probability of finding a α-type particle 1 at a distance r 12 from a β-type particle 2, is computed, in Fermi Hypernetted Chain (FHNC) 25 , using the correlated g.s. Ψ 0 and it turns out to be written as: The partial RDF are classified, in FHNC theory, according to whether the external particle (1 or 2) is reached by a statistical correlation (i.e. if the particle is involved in an exchange loop, e-vertex), or by a dynamical correlation, (f (α,β) ) 2 − 1, only (d-vertex).
The definitions of the partial RDF, together with the full set of the related FHNC equations, may be found in 25 .
Actually, eq.(29) sums all cluster diagrams factorizable in products of dressed, two-body diagrams. They do not contain only two-body cluster terms, but include, in turn, an infinite number of particles, as they are written in terms of the RDFs, rather than the bare two-body correlations.
Three-body, non factorizable diagrams are also present in the cluster expansion of ξ α , even if they do not appear in eq. (29). However, they have been inserted, following ref. 19 .
where L(k F r) is the FHNC generalization of the exchange Slater function l(k F r) = Again, as D(x) turns out to be positive, S 1p−1h 33,44 (q, ω) are positive, whereas S 1p−1h 34 (q, ω) may be not.

CORRELATED ONE AND TWO PHONON INTERMEDIATE STATES
In this section we will first study the effect on the phonon responses of the insertion of orthogonal, correlated 2PH states: where the 2PH states of eq.(6) have been orthogonalized to the 1PH ones by a Gram-Schmidt procedure.
2PH states influence the partial polarization propagators D αβ (q, ω) via the direct coupling to the ground state and via the decay of 1PH states into 2PH. The coupling to the g.s goes through the matrix element of the 3 He fluctuaction operator, ( notice that ξ 4 (q; k 1 , k 2 ) vanishes because of the Schmidt orthogonalization of the 2PH states), whereas the decay is driven by the non diagonal matrix element of the hamiltonian These CBF matrix elements have been computed in a boson-boson approximation (treating the 3 He as a mass-3 boson) and by adopting the Convolution Approximation (CA) for the three-body distribution functions.
Their explicit expressions are: and a(k; It is convenient, at this point, to introduce the correlated self-energy and the function χ(q; k, ω) given by: If we define the dressed phonon propagator G d (k, ω) as then, the phonon contributions to the polarization propagators can be rearranged as: and The DSF are then obtained by taking the imaginary parts of D αβ . Terms quadratic in ξ 3 (q; k 1 , k 2 ) have not been considered.
The relevant changes introduced by the insertion of the 2PH states in the phonon responses are: 1. the strengths of the delta-like 1PH peaks Z αβ are generally quenched respect to Z v αβ . In the 44 case, we have Analogous corrections occur for Z 34 (k) and Z 33 (k), which are also affected by those parts of the polarization propagators containing ξ 3 (q; k 1 , k 2 ); 2. the 1PH peaks are shifted by the real part of the on-shell self energy, since the 4 He spectrum is modified as 3. a multiphonon tail appears at large ω-values, beyond the position of the 1PH peak, at the momentum transfers here considered.

CBF RESPONSES
In the class of the Jastrow correlated wave functions, the best variational choice is provided by the solution of the Euler equations The resulting equations have been derived, within the FHNC framework, and solved for the 3 He impurity problem 26,27 , for the boson-boson mixture 28 and, lately, for the real fermion-boson case 9 .
Another, often used approach consists in parametrizing the correlation functions and in minimizing the ground state energy with respect to the parameters. This is the choice we have adopted here. Besides that, some of the results we will present have been obtained within the Average Correlation Approximation (ACA). In ACA, the correlation functions are the same for all the types of pairs (f (3,3) = f (3,4) = f (4,4) ) and the differences in the distribution functions (or in the static structure functions) are due only to the different isotope densities and statistics. We will also show that going beyond the ACA does not affect our results.
We have used three types of correlation functions: the time honored, short ranged McMillan form (SR) and two long ranged functionsa (LR and LR1).
The McMillan correlation, in ACA, is given by: The In order to check the accuracy of ACA, we have also used a LR correlation (LR1), formally identical to f LR , but with parameters depending on the type of the correlated pair.
The 44 correlation function is the same as above, whereas the parameters of the 43 and 33 ones have been obtained by minimizing the energy of the pure 4 He with one and two 3 He impurities, respectively 25 .
Key ingredients in the CBF theory of the response in Helium mixtures are the radial distribution functions g αβ (r) and the static structure functions S αβ (k). Figs. (1) and (2) show these quantities in a 4.7% mixture, at a total density ρ = 0.02160Å −3 , for the f LR1 (r) correlation, in FHNC/0 approximation (i.e. we have neglected the elementary diagrams 25 ).
The results for the SSF, with the f SR (r), differ mainly in the region of low-k values, in agreement with the previous discussion.
Table (1) shows the variational strenghts Z v αβ (k) of the one-phonon response for the same mixture and compares the results obtained with the SR and LR correlation functions at four momentum values, from q = 1.1 to 1.7Å −1 . The positions of the variational delta peaks, ω k , are also given. It has to be noticed that the Feynman spectrum overestimates the experimental data by at least 10K both in the maxon and roton regions. Table (2) provides the same quantities after the insertion of the 2PH states. However, because of the low 3 He density, it turns out to be extremely close to the free Fermi gas spectrum  Even from the CA calculation, a deviation from the LP behavior clearly appears . The estimated CBF value of γ in CA turns out to be γ(CA) = 0.052Å 2 . We stress once more that we expect SA to provide a better description of the 3 He spectrum behavior, as it correctly takes into account the core property of the system, requiring that the three-particle distribution functions vanish when any interparticle distance is lower than the radius of the repulsive core of the potential. Table (3) shows the CBF values of the m n,αβ (q) sum rules of the DSF's, for n = 0, 1, defined as: The exact DSF's satisfy the f −sum rules and for n = 0 one has m 0,αβ (q) = S αβ (q).
The table gives also the variational values of the SSF's of eq.27 and the f −sum rules. To evaluate the total scattering functionŜ(q, ω), the DSF must be multiplied by the elementary cross sections and the concentrations of the species. In fig.s(6a,b) we give the partial CBF scattering functions (PSF): andŜ 33 (q, ω) = The LR1 correlation has been used. At q = 1.1Å −1 , both the position and the strength of the phonon branch are well described by our calculation. When approaching the roton minimun region, the agreement worsens and we overestimate the experimental data. As discussed previously, we expect that the use of SA will improve the CBF description.
An analogous analysis can be performed fort the 1p − 1h sector. The use of the CBF-CA spectrum slightly misses the location of the bump, well described in turn by a LPM parametrization, which is essentially a fit to the experimental data. We recall that the relevant difference between the LPM and the CBF-CA energies lies in the γ-parameter value, smaller by a factor ∼ 0.4 in the latter case. A simple, quadratic LP parametrization with m * 3 = 2.3m 3 seems to be ruled out.
The 3 He scattering functionŜ 3 (q, ω), defined aŝ and the functionS 3 (q, ω), given bȳ in the 1p − 1h sector, are given in fig.(8). The figures contains also the convolution of The microscopic quasiparticle 3 He spectrum clearly shows a deviation from the simple LP form. The spectrum has been actually computed for the single impurity problem, but we do not believe that its evaluation in the low concentration mixture will dramatically change our findings. In particular, a deviation from LP was advocated in ref. 6 to explain the experimental 1p − 1h response, in contrast with a possible large change of the 3 He effective mass in mixture (from m * 3 = 2.3m 3 at x 3 = 0 to m * 3 = 2.9m 3 at x 3 = 4.7%). The CBF spectrum still does not reproduce fully quantitatively the data, and a more accurate calculation is needed.
The 4 He excitation spectra in the phonon-roton branch of the pure system and the mixture at SVP have been compared. The shift between the two excitations appears to be due to the change in density. CBF gives a good description of the maxon region, but overestimates the roton, even if it gives an almost correct q-value for the change of sign of the shift .
The CBF scattering function at low momenta gives a reasonable description of the scattering data (both for the position and strength). The agreement worsens as q increases. The peaks are located at too a large energy and their strength is overestimated. We believe that the reason of this lies in the approximations made to compute the decay of 1PH states into 2PH and in the lack of higher intermediate states, which become more and more important as the momentum increases. In particular, the 1p − 1h sector does not include two probably relevant contributions: the decays of 1p − 1h states into (1) 2p − 2h and (2) 1P H states.
The former adds large energy tails to the 1p − 1h bump reducing its strength, and the latter is known to be responsible for a large part of the 3 He effective mass. Our CBF calculation includes the real part of the 1p − 1h into 1P H decay but does not consider its imaginary part.