Body Plethysmography

Phenomena

P. Schucka∗, M. Beyerb, G. R¨opkeb, W. Schadowc, and A. Schnelld aInstitut de Science Nucl´eaires, Universit´e Joseph Fourier, CNRS-IN2P3 53, Avenue des Martyrs, F-38026 Grenoble Cedex, France bFachbereich Physik, Universit¨at Rostock, D-18051 Rostock, Germany cTRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., V6T 2A3, Canada Abstract. Residual interactions in many particle systems lead to strong cor- relations. A multitude of spectacular phenomenae in many particle systems are connected to correlation effects in such systems, e.g. pairing, superconductiv- ity, superfluidity, Bose-Einstein condensation etc. Here we focus on few-body bound states in a many-body surrounding.

Introduction

Correlated many particle systems such as, e.g. nuclear matter or strongly cou- pled plasmas, have a complicated dynamical behavior. Only few areas of the density-temperature phase diagram used to characterize the state of the sys- tem in thermal equilibrium may be described in the approximation of nonin- teracting quasiparticles. The dynamics of the quasiparticle is determined by the mean field of the other particles and therefore some prominent features like self-energy corrections and Pauli blocking (Bose enhancement) are sufficiently taken into account . However due to sizable residual interactions many inter- esting and exciting phenomena, such as clustering, formation of condensates, and phase transitions occur. The description of the dynamics of such systems is particularly difficult since the quasiparticle approach reaches its limits.

A large number of these phenomena such as, e.g. two-particle bound state formation, can already be accounted for by explicitly introducing two-body correlations into the formalism. This may be achieved in the frame work of the Green function method and leads to effective two-body equations that include medium effects in a consistent way.

Plasma?

Figure 1.

Diagram Of Nuclear Matter. N0 Denotes

nuclear matter density n0 = 0.17 fm−3.

N0

Figure 2.

Sity . Tc Denotes The Critical Tempera-

ture. The density temperature planes of nuclear matter are shown in Figs. 1 and 2. The phase diagram of nuclear matter turns out to be very rich. In particular the superfluid phase reflecting strong pairing is relevant for the structure of neutron stars . At lower densities bound states occur. This part of the phase diagram may be accessed in the laboratory through heavy ion collisions at intermediate energies. The description of these phenomena in the framework spelled out here constitues a wide and exciting area for an application of few- body methods.

Besides the pairing of nucleons, correlations with more particles are also important. Based on the suitably modified AGS (Alt-Grassberger-Sandhas ) formalism the reaction cross section for break-up (Nd →NNN), relevant e.g.

for the formation of deuterons in a heavy-ion reaction, has been given in . As a first application the life time of deuteron fluctuations in nuclear matter was calculated in . In we have given an exploratory calculation of the four-body condensate in strongly coupled fermionic systems.

Theory

The formalism to derive few-body Green functions within the Dyson equation approach or cluster mean field approximation at finite temperatures and den- sities has been given elsewhere . Here we give some of the basic results and focus on bound states.

(1)

where a1(t) etc. denotes the Heisenberg operator of the particle with quan-

3

tum numbers s1, k1, etc. for spin, momentum etc. In the equal time for- malism we may introduce creation and annihilation operators Aα(t), where

Aα = A1, A†

1a2, a1a2a3, etc. for one-particle, hole-particle, three-particle opera- tors. We describe the propagation of these clusters via the Green’s function at T = 0. However, generalization to T̸ = 0 is straight forward using the Matsub- ara technique . The definition of the Green’s function reads

(2)

where averaging is due to the density operator ρ0. For an open system in thermodynamical equilibrium the extremum condition for the entropy leads to the following expression for the density operator of a grand canonical quantum

E−Β(H−Μ N)

Tr{e−β(H−µ N)}.

(3)

The temperature (1/β = T ) and the chemical potential µ are the corresponding

¯T−T′

γβ .

We Use Nαβ = ⟨[Aα, A†

β]±⟩in the following, where the upper (lower) sign reflect Fermi (Bose) type operators. The mass matrix is given by

−I⟨T [Aα, H](T), [H, A†

β](t′)⟩irr.

(7)

For Aα = a1, i.e. the one-particle case, neglecting the term Mirr (represent- ing all retarded irreducible diagrams) leads to the self-energy shift in mean field (e.g. Hartree-Fock) approximation. For a given potential this constitutes a selfconsistent problem to determine µ and β, to be solved by iteration

2

V2(12)f2.

(8)

For T →0 this function approaches a step function at the respective Fermi momentum.

3

Two-body correlations, bound states and Cooper pairing Nuclear matter cannot be considered as a system of independent quasi par- ticles but for a large part it is correlated and, e.g. in the superfluid phase as demonstrated in Fig. 2 . The basis to treat correlated densities is provided

Mentum Between The Deuteron And The

medium.

Temperature (T = 10 Mev) And Density

n = n0/25 . by a generalization of the Beth-Uhlenbeck approach . The nuclear density

As

bound/scattering states. In first iteration these correlations may be treated on the basis of residual interactions between the quasi particles. The exact two particle equations to be solved are known as Bethe-Goldstone or Feynman- Galitski equations depending on some details . In ladder approximation the

(11)

where the dual wave functions ˜Ψ ν are defined by Ψ ν = (1 −f1 −f2) ˜Ψ ν. Intro- ducing the two-body t-matrix in a standard fashion both bound and scattering states have been solved . In first approximation V2 of Eq. (10) corresponds to the bare force. However, at higher densities the force may be substantially screened.

As an example for an in-medium bound state the deuteron binding energy is shown in Fig. 3. The Mott density is defined through the condition Ed = 0. Note that the Fermi functions f1 for particle 1 etc. depend on the relative momentum P between the deuteron and the medium.

The deuteron in the medium also has a finite life time. To second order Born approximation the life time can be extracted from the square of the

D

D

Figure 5. Decay of deuteron in nuclear matter induced by surrounding nucleons.

D Is Given By The Incoherent Decay

of the deuteron into 3p-1h states. At zero temperature one evaluates the 3p-1h

=

Tr1Tr2Tr3Tr4{θ(H1 −εF )θ(H2 −εF )θ(H3 −εF )θ(εF −H4)}

∝

(E −2εF)3 + . . .

(13)

Since the normalization of the deuteron wave function brings a factor (E −

2Εf )−1 We Have Γ ↓

d ∝(E −2εF )2. At finite temperatures T ≪εF we get as in Fermi liquid theory for zero sound a T 2 behavior for Γ ↓

D . In A More Elaborated

theory one has to consider rescattering of the incoming deuteron with the nucleon(s) of the medium. Summing up all the respective diagrams leads to a generic three-body in-medium scattering matrix, that has been derived in [6, 7]. As a consequence the reaction rate depends strongly on the surrounding medium. The numerical result for the life-time of the deuteron is shown in Fig. 4, where we in addition compare the use of the isolated to the in-medium break-up cross section at T = 10 MeV and a nuclear density of n = 0.007 fm−3 ≃n0/25 [7, 10]. The medium dependent elementary cross section leads to shorter fluctuation times. As expected the difference is vanishing for larger momenta. In turn this effect leads to faster chemical relaxation times that is e.g. relevant for the notion of freeze out in heavy ion collisions.

The appearance of a sharp pole at E = 2µ in the respective Green function related to Eq. (10) corresponds to the Thouless criterion for the onset of super- fluidity for np Cooper pairs. Depending on the channels considered, it allows us

T For The Respective Isospin Sin-

glet and triplet channels, as a function of the chemical potential. This is shown

T . Using The Canonical Relation ∆Kf = 1.76Tc

one realizes that the np gap has a maximum value of ≃10 MeV. This is more than 3 times larger than the nn gap! Such high values may have an influence in expanding nuclear matter as produced in central heavy ion collisions.

An interesting extension of Eq. (10) is to write this equation down for the case where we have a condensate of pairs, i.e. a superfluid state. For pairs at

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