---
abstract: 'This article continues a discussion raised in previous publications (LANL preprint server, nucl-th/0202006 and nucl-th/0202020). I try to convince my opponents that general arguments are not “my case" and may be applied to their model.'
author:
- 'V. Yu. Ponomarev[@byline2]'
title: 'On “the authentic damping mechanism” of the phonon damping model. II [^1]'
---

To remind in brief a discussion which is already distributed over several publications:

A damping mechanism of giant resonances (GR) is well established and represents now basic knowledge in nuclear structure physics. Calculations performed by many groups of authors within different microscopic approaches confirm that a spreading width (due to a coupling of collective modes, phonons, to complex configurations) is the main part of the total GR width in medium and heavy nuclei. In light nuclei, a coupling to continuum (an escape width) also plays an essential role.

A damping mechanism of GRs in a phenomenological phonon damping model (PDM) in its PDM-1 version is different from that (see, an important clarification in [@note0]). A collective phonon fragments within PDM-1 as a result of coupling to simple and not to complex configurations, i.e. only the so-called Landau damping mechanism is accounted for. A coupling strength is a phenomenological model parameter which is adjusted to reproduce the GR width known from experiment. Agreement with data provided by fits within the PDM may be defined as from very good to excellent.

In a recent article [@n9] which raised the present discussion, it has been concluded that these type of fits confirm “the [**authentic**]{} damping mechanism” of the PDM as “the result of coupling between collective phonon and non-collective $p$-$h$ configurations” (i.e. the well established knowledge on the GR properties was put in doubt). This conclusion has been criticized in my article [@m]. It has been argued that this model has the Breit-Wigner (BW) form for the phonon distribution as an [*ad hoc*]{} input and thus, even excellent description of the data available is not surprising. A fruitfulness of an idea to make conclusions from fits in which model parameters are adjusted to described physical observables has been put in doubts.

Although my evaluation of the PDM in [@m] was made for the point of view of general physical grounds, Dang [*et al.*]{} did not agree with me in the forthcoming publication [@nn]. They claim that I consider some specific case (“his case") which cannot be attached to the PDM and all my arguments “[*are either wrong or irrelevant*]{}". I cannot agree with their conclusion and present below additional arguments in a sequence following the paragraphs in [@nn]:

[**2.**]{} For the giant dipole resonance (GDR), the energy scale associated with variations in a coupling matrix between a phonon and uncorrelated $1p1h$ states is of the order of a few hundred keV. The width of the GDR strength function is of the order of a few MeV. So, I do not agree that the condition cited in [@nn] from [@Boh69] is satisfied in the GDR region: why are a few MeV small compared to a few hundred keV?

I know only one PDM-1 article [@n1] in which it is assumed that a phonon interacts 40 times stronger with some specific configurations than with other ones (see more on this article in [**9.**]{} below). In all other PDM-1 papers, we find a single phonon which interacts equally with all $1p1h$ configurations. I do not want to discuss here the PDM fits at non-zero temperature. To keep on reproducing the data in hot nuclear, Dang [*et al.*]{} have to assume for unclear reasons that a phonon prefers to interact with $1p1p$ and $1h1h$ configurations about 10 times stronger than with $1p1h$ configurations. Again, as in the case of cold nuclei, an idea to provide the best fits is preferred to understanding of the physics. I think it is a blind way for theory.

It is true that PDM equations are presented in a general form in many papers by this group with different $V_{q_1 s_1}$. But the point is that they are never used in actual calculations in this form. For this reason, I prefer to discuss what is used in calculations rather than what is written and not used even by the PDM authors themselves.

[**3.**]{} It is very simple to transform Eq. (1) in [@nn] for $m_q^{(2)}$ into Eq. (1) in [@m] for $W_2$, although Dang [*et al.*]{} claim it is impossible. For that, one needs to switch off an additional PDM smearing, i.e., consider the limit $\varepsilon \to 0$. This would bring immediately to the first line of Eq. (2D-14) in [@Boh69]. Eq. (1) in [@m] (for a constant coupling strength) or its general form in [@Boh69]: $$\hspace*{60mm}W_2 = \sum_{a,\alpha} (V_{a \alpha})^2 \hspace*{60mm}
\mbox{(2D-14)}$$ for the second moment $W_2$ is relevant to the PDM as well as to any model which deals with interactive systems.

Of course, to perform this transformation one should use the PDM strength function introduced in Ref. [@o1]: $$S_q(E) = \frac{1}{\pi} 
\frac{\gamma_q(E)}{\left(E-\omega_{q}-P_q(E)\right)^2+\gamma_q^2(E)}~
\label{e1}$$ where $\gamma_q(E)$ is the PDM damping, $P_q(E)$ is the polarization operator (see, e.g., Ref. [@o1] for definitions), and $\omega_{q}$ is a phonon energy, a model parameter. The strength function $S_q(E)$ presents fragmentation properties of a PDM phonon over eigen-states of the PDM Hamiltonian smeared with an additional parameter $\varepsilon$. Parameter $\varepsilon$ appears in $\delta(E) = \varepsilon/[\pi \cdot (E^2+ \varepsilon^2)]$ for $\delta$-functions in $\gamma_q(E)$.

I point this out because the strength function (\[e1\]) has been replaced in the forthcoming PDM articles [@n9; @nn; @n1; @n2; @n3; @n4; @n5; @n6; @n7; @n8; @n10; @n11; @n12; @n13], by its approximate form: $$S_q'(E) = \frac{1}{\pi} 
\frac{\gamma_q(E)}{\left(E-E_{GDR}\right)^2+\gamma_q^2(E)}
\label{e2}$$ where $E_{GDR}$ should be taken as a solution of $$f(E) \equiv E-\omega_{q}-P_q(E)=0~.
\label{e3}$$ Eq. (\[e2\]) has been obtained from Eq. (\[e1\]) by expanding $P_q(E)$ near a solution of Eq. (\[e3\]), $E_{GDR}$, and then extrapolating the properties of this approximation far away from $E_{GDR}$. In the limit $\varepsilon \to 0$, Eq. (\[e3\]) has $N+1$ solutions corresponding to eigen-energies of the PDM Hamiltonian.

[**4.**]{} I never claimed that the BW form for the phonon distribution is assumed within the PDM. But it is indeed an [*ad hoc*]{} input for PDM calculations. I may refer again to [@Boh69] where we read that “[*the Breit-Wigner form for the strength function is an immediate consequence of the assumption of a constant coupling to the other degrees of freedom of the system*]{}". The BW under discussion has nothing to do with definition of the PDM strength function. Indeed, in the limit $\varepsilon \to 0$, $S_q(E)$ turns into a set of infinitely narrow lines while their envelope still remains the BW.

[**5.**]{} I do not agree that the calculation with random values of $E_{\alpha}$ in [@m] “[*no longer corresponds to the PDM*]{}". I have used the PDM Hamiltonian and details on a spectrum and model parameters are only technical details of a calculation. The purpose of my calculation is to demonstrate that “[*the crucial feature of the PDM is the use of realistic single-particle energies*]{}" [@nn] is of marginal importance when a configuration space is not small; everything is determined by the BW discussed above.

$E_0$ in Ref. [@nn]belongs to the Lorentz line in a hypothetical nucleus and not to my PDM fits. Eigen energies in my calculation in Ref. [@m] were obtained from Eq. (\[e3\]) in the limit $\varepsilon \to 0$.

[**6.**]{} I agree that if something “[*is by no mean\[s\] obvious*]{}" it has to be checked. My experience of microscopic calculations tells me that the increase of collectivity tends to the increase of a coupling strength. Of course, it is not necessary that everybody should trust my experience. But then, there are no other alternatives: the one, who puts it in doubt, should check it independently.

[**7.**]{} I never claimed that there are some reasons “[*why the values of $f_1$ for $^{40}$Ca and $^{48}$Ca should be the same*]{}" as there are no reasons to keep this parameter fixed along chains of isotopes. As pointed out in [@m], this parameter has no physical meaning. My issue is that one cannot learn anything from agreement with experiment from fits in which a free parameter is adjusted to a described observable. It is important to stress once again:

On the page 4 of the article under discussion [@n9], we find: “[*For double closed-shell nuclei $^{16}$O and $^{40,48}$Ca, where the pairing gap is zero, such a kind of enlargement of configuration space is compensated simply by a renormalization of $f_1$, which reduces its value by $\sim 25\%$ for $^{16}$O, and $\sim 35-37\%$ for $^{40,48}$Ca.*]{}" I read this statement after discussion of open-shell nuclei as a renormalization of $f_1$ in closed-shell nuclei in respect to open-shell nuclei. Or do Dang [*et al.*]{} mean to say that calculations in double-magic nuclei have been performed with pairing as one may conclude now from [@nn]: “[*The results for GDR in $^{16}$O have been obtained already within the enlarged space*]{}"? Obviously, only the authors know whether they renormalized their $f_1$ in calculations along chains or not. Of course, I will take out the statement on the $f_1$ renormalization from $^{16}$O to $^{18}$O if it is not true. But before that, I need some help from the authors as to how a reader should interpret the above cited statements.

[**8.**]{} It is clearly explained in [@m] why comparing the PDM predictions in $^{40}$Ca to the data [@Har00], the strongest $1^-$ state observed should be excluded from consideration (because it has a two-phonon nature and two-phonon states are not included in the PDM model space). Thus, the PDM 0.25% of the TRK EWS corresponds to 0.007% and not to 0.025% from this experiment. It seems to me that Dang [*et al.*]{} try to hide again a huge disagreement by misleading comparison.

The same conclusion, that the PDM is not capable to reproduce “[*the significant experimental difference in the E1 strengths*]{}" $^{40}$Ca/$^{48}$Ca, has been obtained independently by another group of authors [@Har01]. As they write, “[*It is important to note that the parameters of the PDM are adjusted to reproduce the gross structure of the GDR while investigations of $\gamma$-ray strength function models show that the extrapolation of the strength distribution down to energies below the particle threshold leads to unrealistic high dipole strengths and overestimates the experimental data*]{}". Thus, it is not only my point of view that a conclusion in [@n9] (on a quantitative description of pygmy dipole resonance within the PDM) is not justified.

It is not true that the PDM with a structureless phonon has no problems with double counting. If a phonon internal structure is not accounted for, it does not change the physical meaning of the phonon. The PDM configuration space contains a phonon and uncorrelated $1p1h$ configurations. The last ones are also excited from the ground state and each of them has its own $B_{1p1h}(E1)$ value. If $1p1h$ spectrum is rather complete (it is always true in the PDM calculations) these uncorrelated $1p1h$ state alone exhaust about 100% of the TRK EWSR. But the PDM physics is determined only by the phonon strength function and not by its sum with $N$ strength functions of uncorrelated $1p1h$ configurations. This is equivalent to $B_{1p1h}(E1) = 0$ within the PDM.

[**9.**]{} The previous article on pygmy resonances by Dang [*et al.*]{} [@n1] was not a subject of [@m]. But if Dang [*et al.*]{} raise discussion on it in [@nn], I have some comments on it too:

The capability of the phonon damping model (PDM) to describe giant dipole resonance (GDR) damping in neutron-rich nuclei has been tested in [@n1]. To mimic essential differences of double-magic and exotic nuclei, a coupling between a phonon and some $1p1h$ configurations near the Fermi surface has been strongly enhanced in the last ones. As a result, a phonon interacts with these selected $1p1h$ configurations with a strength “[*equal to 41 MeV for oxygen isotopes, 13.856 MeV for calcium isotopes, and 6.928 MeV for tin isotopes*]{}" [@note1]. Let us try to understand how it is possible to stand such an enormous coupling strength which is far away from nuclear structure scales and report an agreement with experiment from this type of calculations. For that, I have repeated the PDM calculations for $^{16}$O and $^{18}$O at zero temperature keeping all the details of Ref. [@n1] and employing realistic $1p1h$ spectrum from Hartree-Fock calculation with SGII Skyrme forces [@note2].

The results of my calculations [@code] are shown in Fig. \[f1\]. A difference in $S_q(E)$ for $^{16}$O and $^{18}$O in Fig. \[f1\] is dramatic but not surprising. It is due to the fact that a phonon couples to all $1p1h$ configurations with an equal strength of $F_1=1.025$ MeV (a PDM parameter) in $^{16}$O while a coupling strength for $[1p_{1/2} 2d_{5/2}]_{\nu}$ configuration at $E_{[1p_{1/2} 2d_{5/2}]_{\nu}}=8.2$ MeV in $^{18}$O has been enhanced to $F_1'= 40 F_1 = 41$ MeV following the details of calculations in Ref. [@n1]. As a result, we find the GDR in $^{18}$O between 40 and 60 MeV and about 40% of its strength is pushed to $-20$ MeV. The energy of the ground state is 0 MeV.

Comparing dashed lines in Fig. \[f1\], one may be surprised that $S_q'(E)$ does not feel this enormous matrix element of 41 MeV in $^{18}$O confirming the results in [@n1]. But a very important detail is that $S_q'(E)$ in Fig. \[f1\] has been calculated with $E_{GDR} = 22.5$ MeV from Ref. [@n1]. It has been done in an attempt to reproduce the GDR strength functions published in Fig. 3 of [@n1] which are found in agreement with the data available in $^{16}$O and $^{18}$O. Taking into account that the employed $1p1h$ spectrum might be not exactly the same as in Ref. [@n1], it is possible to conclude that dashed curves in Fig. \[f1\] reproduce the results in Fig. 3a, 3b of Ref. [@n1] on a rather good qualitative level.

The only problem is that it is not possible to obtain $E_{GDR} = 22.5$ MeV in $^{18}$O, reported in Ref. [@n1], with parameters from this article as a solution of Eq. (\[e3\]). To demonstrate this, let us consider a behavior of the function $f(E)$ of Eq. (\[e3\]) in $^{16}$O and $^{18}$O. In $^{16}$O, it has a tendency of a continues increase with fluctuations reflecting $1p1h$ poles smeared by the parameter $\varepsilon$ and crosses $y=0$ line in my calculation at $E_{GDR} = 18.5$ MeV (see, Fig. \[f2\], left). In $^{18}$O (Fig. \[f2\], right), a fluctuation around $E_{[1p_{1/2} 2d_{5/2}]_{\nu}}$ increases enormously because of 41 MeV coupling matrix element corresponding to this pole yielding a spurious solution of Eq. (\[e3\]) at this energy. The physical PDM solutions of Eq. (\[e3\]) in $^{18}$O have $E_{GDR} = -19.7$, 50.6, and 55.0 MeV which are very different from $E_{GDR} = 22.5$ MeV, reported in Ref. [@n1]. $S_q'(E)$ calculated with any them is dramatically different from the one in Fig. \[f1\] (right).

The “$^{18}$O effect" can be obtained even without any calculations. For that, one may neglect a phonon coupling to all $1p1h$ configurations (with a “weak" matrix element $F_1$) except for $[1p_{1/2} 2d_{5/2}]_{\nu}$ configuration. Then, in the limit $\varepsilon \to 0$, Eq. (\[e3\]) transforms into quadratic equation: $$E-\omega_{q}-\frac{(F_1')^2 \cdot n}{E-E_{[1p_{1/2} 2d_{5/2}]_{\nu}}}=0~.
\label{e4}$$ where a factor $n$ accounts partial occupation of $\nu2d_{5/2}$ level in $^{18}$O. Eq. (\[e4\]) yields the PDM eigen states at $-19.0$ and 49.4 MeV with a phonon strength distribution among them as 40% and 60%, respectively.

It becomes clear that agreement with experiment for $^{18}$O (and accordingly for other neutron-rich nuclei) reported in Ref. [@n1] has been obtained by making use of the approximate PDM strength function $S_q'(E)$ and the GDR energy which is not a solution of Eq. (\[e3\]) as announced. Correct PDM strength function with parameters from Ref. [@n1] for $^{18}$O is presented by solid curve in right part of Fig. \[f1\].

[**1 and 10.**]{} I have examined the PDM from the point of view of general physical grounds. My arguments and conclusions are presented in [@m] and above. I think a reader may independently conclude whether general rules are not for this model (as the claims of Dang [*et al.*]{} in [@nn] may be understood) and whether one learns any physics from the PDM fits in Refs. [@n9; @n1; @o1; @n2; @n3; @n4; @n5; @n6; @n7; @n8; @n10; @n11; @n12; @n13; @n14; @n15; @n16] although agreement with experiment is always reported by the authors.

[99]{}

Permanent address: Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia.

V. Yu. Ponomarev, LANL preprint server, nucl-th/0202006v1, 1 Feb. 2002.

N. Dinh Dang [*et al.*]{}, LANL preprint server, nucl-th/0202020v1, 7 Feb. 2002.

In many articles by Dang [*et al.*]{}, we find that a coupling to complex configurations is effectively accounted for within the PDM-1 in a model parameter $f$ responsible for coupling to simple configurations (a possibility to provide similar fits within the PDM-1 and PDM-2, which phenomenologically accounts for coupling to complex configurations, is used as an only proof). As pointed out in Refs. [@m; @mm], this is a misleading statement and nothing else.

Indeed, from a theoretical point of view, higher-order effects may be claimed to be effectively included in a free parameter of lower-order graphs only if higher-order diagrams can be transformed into the lower-order diagrams with a renormalized vertex. The PDM diagrams are published in Ref. [@n2]: in Fig. 1a for a coupling to simple and in Figs. 1b-e for a coupling to complex configurations.) Since none of the diagrams in Figs. 1b-e can be transformed into the diagram in Fig. 1a with a renormalized vertex, the statement that high-order processes can be effectively accounted for by lower-order processes does not sound as theoretically proved.

It is also not possible to agree that two fits performed within two different, in principle, approaches (PDM-1 and PDM-2) may be used as a theoretical proof that one model automatically includes all ingredients of another model. It is important to notice that the PDM strength parameter $f$ has to be reduced by about two order of magnitude in the transition from the PDM-1 to PDM-2. This takes place because a configuration space of the PDM-2 is much larger as compared to the one of the PDM-1 and there is only one purpose of PDM fits, namely to reproduce the data.

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An interaction strength of 41 MeV is not a misprint. In Ref. [@n1] we find it again as $F_1'=40F_1$ where $F_1=1.025$ MeV for $^{16}$O in table 1. In Ref. [@nn], it is mentioned as “[*the parameter $f_1$ in Ref. \[5\] of \[4\] was increased significantly near the Fermi surface*]{}". To avoid misunderstanding, parameters $F_1$ and $f_1$ are the same, only different notations are used in various PDM articles.

I thank Dr. G. Coló for $1p1h$ spectrum.

Calculations have been performed with a Fortran code pdm.f of 42 lines. It is available from the author. Calculation time of both $S_q(E)$ and $S_q'(E)$ is about 1 sec. on 166 MHz PC.

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[^1]: I cannot not neither confirm nor disprove that an article [@m] was rejected by PRC as a reader learns from [@nn]. It is common practice that a manuscript submission to any scientific journal remains a confidential information of the author(s) until the manuscript is accepted for publication. In PRC, it is protected by a secret accession code. It seems to me at least not ethical to publish openly on Web a confidential information of another person(s) as Mr. N. Dinh Dang did in [@nn]. An editorial decision may be published only with a formal permission from the Editors. I doubt that Mr. N. Dinh Dang has it. I also interpret this fact as an attempt to influence a reader by non-scientific arguments in a scientific discussion.
