STRUCTURE FUNCTIONS GENERATED BY ZERO SOUND EXCITATIONS

Oct 13, 2020, 6:35 PM
1h
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Poster report Section 1. Experimental and theoretical studies of the properties of atomic nuclei. Poster session 1 (part 1)

Speaker

Dr Valentina Sadovnikova (National research center "Kurchatov institute", PNPI)

Description

We study the form of the structure functions connected to the zero sound excitations in the symmetric and asymmetric nuclear matter (ANM). The density response $\Pi(\omega,k)$ (the retarded polarization operator) of ANM to the small external field $V_0(\omega,k)= \tau_{z}e^{i\vec q\vec r -i(\omega+i\eta)t}$ is considered. The structure function $S(\omega,k)$ is defined as $S(\omega,k) = -\frac1\pi{\rm Im}\, \Pi(\omega,k)$ [1].

In [2] the three complex branches of the zero sound excitations in ANM were obtained: $\omega_{si}(k)$, $i=n,p,np$. We calculate these branches as solutions of the dispersion equation $E(\omega,k)=0$. Calculations were made in the framework of RPA with the Landau-Migdal quasiparticle-quasihole isovector interaction $F'(\vec\tau,\vec\tau')$ with $F'=1.0$.

It was shown that in the external field $V_0(\omega,k)$ the total polarization operator is the sum [3]: $\Pi= \Pi^{pp} + \Pi^{nn} - \Pi^{pn} - \Pi^{np}$. Expressions for $\Pi^{\tau,\tau'}$ are obtained from the system of equations $M$ of the type similar to the system for the effective fields in [4]: $\Pi^{pp}= \Pi_0^p(1-\Pi_0^n\,F^{nn})/\det M(\omega,k)\equiv D^{pp}/\det M(\omega,k)$, $\Pi^{np} = \Pi_0^p\,\Pi_0^n\,F^{np}/ \det M(\omega,k) \equiv D^{np}/\det M(\omega,k)$. Changing $p\leftrightarrow n$ we obtain $ \Pi^{nn}$, $\Pi^{pn}$. Dispersion equation for the frequencies of zero sound excitations is $E(\omega,k)\equiv \det M(\omega,k)=0$. So, the branches $\omega_{si}(k)$ are the zeros of $\det M$ and the poles of $\Pi^{\tau,\tau'}$ by construction.

In our approach $S(\omega,k)$ must be considered as a sum over three independent processes: the widths of the different $\omega_{si}(k)$ correspond to the different decays of excitations. The imaginary part of $\omega_{sn}(k)$ describes in nuclei the semidirect decay due to emission of a neutron, reaction $(\gamma,n)$. Decay of $\omega_{sp}(k)$ accompanied by emission of proton. About of $\omega_{snp}(k)$ we can say that one nucleon is emitted and its isospin is not fixed [2]. We rewrite $S(\omega,k) = \Sigma_i S_i(\omega,k)$.

Near the pole at $\omega\approx {\rm Re}(\omega_{si})$ we approximate $(\det M(\omega,k))^{-1}$ $= {R^i(\omega_{si},k)}/{(\omega-\omega_{si})}$ $+ Reg(\omega,k)$. Here $Reg(\omega,k)$ is a smooth function near the pole. This permits us to write
$S(\omega,k)_i= \,-\frac1{\pi} {\rm Im} [\Sigma_{\tau,\tau'}(D^{\tau\tau'}(\omega,k))\,R^i(\omega_{si},k)/(\omega-\omega_{si}) + Reg]$. Then, let define the envelope curve of the pole terms $S^e(\omega,k)= -\frac1\pi \Sigma_{\tau,\tau'} {\rm Im} [D^{\tau,\tau'}(\omega,k) \Sigma_i\,R^i(\omega_{si},k)/ (\omega-\omega_{si})]$.

We demonstrate results for ANM with asymmetry parameter $\beta=0.2$ Fig.1. In the left figure the branches $\omega_{sn}(k)$, $i=n,p,np$ are shown [2]. In the right figure $S^e(\omega,k)$ are presented for $k/p_0=0.6$ and $k/p_0=0.2$ ($p_0=0.268$GeV). For $k/p_0=0.6$ the structure functions for the different processes $S_i(\omega,k)$, $i=n,p,np$ are presented (the numbers $1,2,3$, correspondingly). As it was expected the form of the structure function is decomposed over the contributions of the definite processes, corresponding to $\omega_{si}(k)$. The widths of maxima ($right$) are determined by the imaginary parts of $\omega_{si}$ ($left$).

[1] E.Lipparini, "Modern Many-particle Physics", 2003, World Scientific Publishing Co.
[2] V.A.Sadovnikova, M.A.Sokolov, Bull.Russ.Acad.Sci.Phys., v.80,p.981(2016); eprint 1807.09580.
[3] A.Pastore, D.Davesne, J.Navarro, Phys.Rept.v.563,p.1(2015).
[4] A.B.Migdal, D.F.Zaretsky, A.A.Lushnikov, Nucl.Phys.,v.A66,p.193(1965)

Primary author

Dr Valentina Sadovnikova (National research center "Kurchatov institute", PNPI)

Presentation materials