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Title: Mesons in strong magnetic fields: (I) General analyses

Abstract

Here, we study properties of neutral and charged mesons in strong magnetic fields |eB| >> Λ2QCD with ΛQCD being the QCD renormalization scale. Assuming long-range interactions, we examine magnetic-field dependences of various quantities such as the constituent quark mass, chiral condensate, meson spectra, and meson wavefunctions by analyzing the Schwinger–Dyson and Bethe–Salpeter equations. Based on the density of states obtained from these analyses, we extend the hadron resonance gas (HRG) model to investigate thermodynamics at large B. As B increases the meson energy behaves as a slowly growing function of the meson's transverse momenta, and thus a large number of meson states is accommodated in the low energy domain; the density of states at low temperature is proportional to B2. This extended transverse phase space in the infrared regime significantly enhances the HRG pressure at finite temperature, so that the system reaches the percolation or chiral restoration regime at lower temperature compared to the case without a magnetic field; this simple picture would offer a gauge invariant and intuitive explanation of the inverse magnetic catalysis.

Authors:
 [1];  [2];  [3]
+ Show Author Affiliations
  1. Brookhaven National Lab. (BNL), Upton, NY (United States); Nishina Center, RIKEN, Saitama (Japan)
  2. Central China Normal Univ., Wuhan (China); Univ. of Illinois at Urbana-Champaign, Urbana, IL (United States)
  3. Goethe-Univ. Frankfurt, Frankfurt am Main (Germany)
Publication Date:
Research Org.:
Brookhaven National Laboratory (BNL), Upton, NY (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Nuclear Physics (NP)
OSTI Identifier:
1254125
Report Number(s):
BNL-112145-2016-JA
Journal ID: ISSN 0375-9474; R&D Project: PO-3
Grant/Contract Number:  
SC00112704
Resource Type:
Accepted Manuscript
Journal Name:
Nuclear Physics. A
Additional Journal Information:
Journal Volume: 951; Journal Issue: C; Journal ID: ISSN 0375-9474
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Riken BNL Research Center; strong magnetic fields; Meson structure; hadron resonance gas; inverse magnetic catalysis

Citation Formats

Hattori, Koichi, Kojo, Toru, and Su, Nan. Mesons in strong magnetic fields: (I) General analyses. United States: N. p., 2016. Web. doi:10.1016/j.nuclphysa.2016.03.016.
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Hattori, Koichi, Kojo, Toru, & Su, Nan. Mesons in strong magnetic fields: (I) General analyses. United States. https://doi.org/10.1016/j.nuclphysa.2016.03.016
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Hattori, Koichi, Kojo, Toru, and Su, Nan. Mon . "Mesons in strong magnetic fields: (I) General analyses". United States. https://doi.org/10.1016/j.nuclphysa.2016.03.016. https://www.osti.gov/servlets/purl/1254125.
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@article{osti_1254125,
title = {Mesons in strong magnetic fields: (I) General analyses},
author = {Hattori, Koichi and Kojo, Toru and Su, Nan},
abstractNote = {Here, we study properties of neutral and charged mesons in strong magnetic fields |eB| >> Λ2QCD with ΛQCD being the QCD renormalization scale. Assuming long-range interactions, we examine magnetic-field dependences of various quantities such as the constituent quark mass, chiral condensate, meson spectra, and meson wavefunctions by analyzing the Schwinger–Dyson and Bethe–Salpeter equations. Based on the density of states obtained from these analyses, we extend the hadron resonance gas (HRG) model to investigate thermodynamics at large B. As B increases the meson energy behaves as a slowly growing function of the meson's transverse momenta, and thus a large number of meson states is accommodated in the low energy domain; the density of states at low temperature is proportional to B2. This extended transverse phase space in the infrared regime significantly enhances the HRG pressure at finite temperature, so that the system reaches the percolation or chiral restoration regime at lower temperature compared to the case without a magnetic field; this simple picture would offer a gauge invariant and intuitive explanation of the inverse magnetic catalysis.},
doi = {10.1016/j.nuclphysa.2016.03.016},
journal = {Nuclear Physics. A},
number = C,
volume = 951,
place = {United States},
year = {Mon Mar 21 00:00:00 EDT 2016},
month = {Mon Mar 21 00:00:00 EDT 2016}
}
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margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Avancini, Sidney S.; Farias, Ricardo L. S.; Tavares, William R.</span> </li> <li> Physical Review D, Vol. 99, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/physrevd.99.056009" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/physrevd.99.056009<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/physrevd.97.034026" target="_blank" rel="noopener noreferrer" class="name">Meson properties in magnetized quark matter<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2018-02-01">February 2018</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Wang, Ziyue; Zhuang, Pengfei</span> </li> <li> Physical Review D, Vol. 97, Issue 3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/physrevd.97.034026" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/physrevd.97.034026<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/physrevd.94.114032" target="_blank" rel="noopener noreferrer" class="name">Electrical conductivity of quark-gluon plasma in strong magnetic fields<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2016-12-01">December 2016</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Hattori, Koichi; Satow, Daisuke</span> </li> <li> Physical Review D, Vol. 94, Issue 11</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/physrevd.94.114032" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/physrevd.94.114032<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/physrevd.96.094009" target="_blank" rel="noopener noreferrer" class="name">Bulk viscosity of quark-gluon plasma in strong magnetic fields<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-11-01">November 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Hattori, Koichi; Huang, Xu-Guang; Rischke, Dirk H.</span> </li> <li> Physical Review D, Vol. 96, Issue 9</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/physrevd.96.094009" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/physrevd.96.094009<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> </div> <div class="pagination-container small"> <a class="pure-button prev page" href="#" rel="prev"><span class="sr-only">Previous Page</span><span class="fa fa-angle-left"></span></a> <ul class="pagination d-inline-block" style="padding-left:.2em;"></ul> <a class="pure-button next page" href="#" rel="next"><span class="sr-only">Next Page</span><span class="fa fa-angle-right"></span></a> </div> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All Cited By</a></li> <li class="small" style="margin-left:.75em; 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float:none;">[ × clear filter / sort ]</a> </div> <input type="submit" id="sort_submit_citations" name="submit" aria-label="submit" style="display: none;"/> </form> </div> </div> </div> </section> <section id="biblio-related" class="tab-content tab-content-sec " data-tab="biblio"> <div class="row"> <div class="col-sm-9 order-sm-9"> <section id="biblio-similar" class="tab-content tab-content-sec active" data-tab="related"> <div class="padding"> <p class="lead text-muted" style="font-size: 18px; margin-top:0px;">Similar Records in DOE PAGES and OSTI.GOV collections:</p> <aside> <ul class="item-list" itemscope itemtype="http://schema.org/ItemList" style="padding-left:0; list-style-type: none;"> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="0" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/952987-condensates-quantum-chromodynamics-cosmological-constant" itemprop="url">Condensates in Quantum Chromodynamics and the Cosmological Constant</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Brodsky, Stanley J</span> ; <span class="author">Shrock, Robert</span> <span class="text-muted pubdata"> - Submitted to Nuclear Physics B</span> </span> </div> <div class="abstract">Casher and Susskind have noted that in the light-front description, spontaneous chiral symmetry breaking in quantum chromodynamics (QCD) is a property of hadronic wavefunctions and not of the vacuum. Here we show from several physical perspectives that, because of color confinement, quark and gluon QCD condensates are associated with the internal dynamics of hadrons. We discuss condensates using condensed matter analogues, the AdS/CFT correspondence, and the Bethe-Salpeter/Dyson-Schwinger approach for bound states. Our analysis is in agreement with the Casher and Susskind model and the explicit demonstration of 'in-hadron' condensates by Roberts et al., using the Bethe-Salpeter/Dyson-Schwinger formalism for QCD bound<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> states. These results imply that QCD condensates give zero contribution to the cosmological constant, since all of the gravitational effects of the in-hadron condensates are already included in the normal contribution from hadron masses.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/servlets/purl/952987" title="Link to document media" target="_blank" rel="noopener" data-ostiid="952987" data-product-type="Journal Article" data-product-subtype="FT" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1501855-intrinsic-transverse-motion-pions-valence-quarks" itemprop="url">Intrinsic Transverse Motion of the Pion’s Valence Quarks</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Shi, Chao</span> ; <span class="author">Cloët, Ian C.</span> <span class="text-muted pubdata"> - Physical Review Letters</span> </span> </div> <div class="abstract">Starting with the solution to the Bethe-Salpeter equation for the pion, in a beyond rainbow-ladder truncation to QCD's Dyson-Schwinger equations, we determine the pion's<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> $$\mathcal{l}$$<sub>z</sub> = 0 and |$$\mathcal{l}$$<sub>z</sub>| = 1 leading Fock-state light-front wave functions (LFWFs) [labeled by $$\psi$$<sub>$$\mathcal{l}$$<sub>z</sub></sub>($$\mathcal{x}$$, $$\mathcal{k}$$$$2\atop{T}$$)]. The leading-twist time-reversal even transverse momentum dependent parton distribution function (TMD) of the pion is then directly obtained using these LFWFs. A key characteristic of the LFWFs, which is driven by dynamical chiral symmetry breaking, is that at typical hadronic scales they are broad functions in the light-cone momentum fraction $$\mathcal{x}$$. The LFWFs have a nontrivial ($$\mathcal{x}$$, $$\mathcal{k}$$$$2\atop{T}$$)) dependence and in general do not factorize into separate functions of each variable. For $$\mathcal{k}$$$$2\atop{T}$$ ≲ 1 GeV<sup>2</sup> the $$\mathcal{k}$$$$2\atop{T}$$ dependence of the LFWFs is well described by a Gaussian; however for $$\mathcal{k}$$$$2\atop{T}$$ ≳ 10 GeV<sup>2</sup> these LFWFs behave as $$\psi$$<sub>0</sub>∝ $$\mathcal{x}$$(1 - $$\mathcal{x}$$)/$$\mathcal{k}$$$$2\atop{T}$$ and $$\psi$$<sub>1</sub> ∝ $$\mathcal{x}$$(1 - $$\mathcal{x}$$)/$$\mathcal{k}$$$$4\atop{T}$$ and therefore exhibit the power-law behavior predicted by perturbative QCD. The pion's TMD naturally inherits many features from the LFWFs. With this being said, the TMD evolution of our result is studied using both the <em>b*</em> and $$\zeta$$ prescriptions which allows a qualitative comparison with Drell-Yan data.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 19<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1103/PhysRevLett.122.082301" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1501855" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1103/PhysRevLett.122.082301</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1501855" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1501855" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/166444-electromagnetic-pion-form-factor" itemprop="url">Electromagnetic pion form factor</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Technical Report</small><span class="authors"> <span class="author">Roberts, C. D.</span> <span class="text-muted pubdata"></span> </span> </div> <div class="abstract">A phenomenological Dyson-Schwinger/Bethe-Salpeter equation approach to QCD, formalized in terms of a QCD-based model field theory, the Global Color-symmetry Model (GCM), was used to calculate the generalized impulse approximation contribution to the electromagnetic pion form factor at space-like q{sup 2} on the domain [0,10] GeV{sup 2}. In effective field theories this form factor is sometimes understood as simply being due to Vector Meson Dominance (VMD) but this does not allow for a simple connection with QCD where the VMD contribution is of higher order than that of the quark core. In the GCM the pion is treated as a composite<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> bound state of a confined quark and antiquark interacting via the exchange of colored vector-bosons. A direct study of the quark core contribution is made, using a quark propagator that manifests the large space-like-q{sup 2} properties of QCD, parameterizes the infrared behavior and incorporates confinement. It is shown that the few parameters which characterize the infrared form of the quark propagator may be chosen so as to yield excellent agreement with the available data. In doing this one directly relates experimental observables to properties of QCD at small space-like-q{sup 2}. The incorporation of confinement eliminates endpoint and pinch singularities in the calculation of F{sub {pi}}(q{sup 2}). With asymptotic freedom manifest in the dressed quark propagator the calculation yields q{sup 4}F{sub {pi}}(q{sup 2}) = constant, up to [q{sup 2}]- corrections, for space-like-q{sup 2} {approx_gt} 35 GeV{sup 2}, which indicates that soft, nonperturbative contributions dominate the form factor at presently accessible q{sup 2}. This means that the often-used factorization Ansatz fails in this exclusive process. A paper describing this work was submitted for publication. In addition, these results formed the basis for an invited presentation at a workshop on chiral dynamics and will be published in the proceedings.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.2172/166444" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="166444" data-product-type="Technical Report" data-product-subtype="" >https://doi.org/10.2172/166444</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/servlets/purl/166444" title="Link to document media" target="_blank" rel="noopener" data-ostiid="166444" data-product-type="Technical Report" data-product-subtype="" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="3" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/20695769-light-scalar-mesons-improved-ladder-qcd" itemprop="url">Light scalar mesons in the improved ladder QCD</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Umekawa, Toru</span> ; <span class="author">Naito, Kenichi</span> ; <span class="author">Oka, Makoto</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review. C, Nuclear Physics</span> </span> </div> <div class="abstract">The light scalar meson spectrum is studied using the improved ladder QCD with the U{sub A}(1) breaking Kobayashi-Maskawa-'t Hooft interaction by solving the Schwinger-Dyson and Bethe-Salpeter equations. The dynamically generated momentum-dependent quark mass is large enough in the low momentum region to give rise to the spontaneous breaking of chiral symmetry. Due to the large dynamical quark mass, the scalar mesons become the qq bound states. Since the parameters have been all fixed to reproduce the light pseudoscalar meson masses and the decay constant, there is no free parameter in the calculation of the scalar mesons. We obtain M{sub {sigma}}=667<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> MeV, M{sub a{sub 0}}=942 MeV, and M{sub f{sub 0}}=1336 MeV. They are in good agreement with the observed masses of {sigma}(600), a{sub 0}(980), and f{sub 0}(1370), respectively. We therefore conclude that these states are the members of the light scalar meson nonet. The mass of K{sub 0}{sup *} is obtained between that of a{sub 0} and f{sub 0} and the corresponding state is not observed experimentally. We also find that the strangeness content in the {sigma} meson is about 5%.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1103/PhysRevC.70.055205" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="20695769" data-product-type="Journal Article" data-product-subtype="" >https://doi.org/10.1103/PhysRevC.70.055205</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/21175469-low-energy-qcd-from-effective-quark-quark-interaction" itemprop="url">Low energy QCD from an effective quark-quark interaction</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Meissner, Thomas</span> ; <span class="author">Frank, Michael</span> <span class="text-muted pubdata"> - AIP Conference Proceedings</span> </span> </div> <div class="abstract">We consider a model truncation of QCD which is based on an effective quark-quark interaction. The truncation allows for a phenomenological description in a framework which maintains the global symmetries of QCD and permits a 1/N{sub c} expansion. The applied truncation leads to the Schwinger-Dyson equation for the quark self energy in the rainbow approximation, which is solved numerically for a given model form of the gluon 2 point function D(q{sup 2}). Meson bound states appear as solutions of the homogeneous ladder Bethe-Salpeter equation. This approach allows for a detailed and systematic investigation of nonperturbative phenomena at low and intermediate<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> energies. A systematic chiral low energy expansion is performed leading to a model prediction of all the chiral coefficients (Gasser Leutwyler coefficients). We demonstrate how the U{sub A}(1) anomaly and the splitting between {eta} and {eta}{sup '} can arise in this approach. It turns out that a necessary condition is a 1/q{sup 4} infrared singularity for the gluon 2 point function. Within the truncation a general technique for calculating nonperturbative quark and gluonic vacuum condensates can be developed. We demonstrate this in case of the mixed condensate <q-barG{sigma}q>. Final results for this condensate as well as <q-barq> are presented.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc doi-link " href="https://doi.org/10.1063/1.54275" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="21175469" data-product-type="Journal Article" data-product-subtype="" >https://doi.org/10.1063/1.54275</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9;"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator"/> <br/> <div class="col text-center mt-3"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.osti.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="col text-center small" style="margin-top: 0.5em;margin-bottom:2.0rem;"> <div class="row justify-content-center" style="color:white"> <div class="pure-menu pure-menu-horizontal" style='white-space:normal'> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="https://www.osti.gov/disclaim" class="pure-menu-link" target="_blank" ref="noopener noreferrer"><span class="fa fa-institution"></span> Website Policies <span class="d-none d-sm-inline d-print-none" style="color:#737373;">/ Important Links</span></a></li> <li class="pure-menu-item" style='float:none;'><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span>Contact Us</a></li> <li class="d-block d-md-none mb-1"></li> <li class="pure-menu-item" style='float:none;'><a target="_blank" title="Vulnerability Disclosure Program" class="pure-menu-link" href="https://doe.responsibledisclosure.com/hc/en-us" rel="noopener noreferrer">Vulnerability Disclosure Program</a></li> <li class="d-block d-lg-none mb-1"></li> <li class="pure-menu-item" style="float:none;"><a href="https://www.facebook.com/ostigov" target="_blank" class="pure-menu-link social ext fa fa-facebook" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Facebook</span></a></li> <li class="pure-menu-item" style="float:none;"><a href="https://twitter.com/OSTIgov" target="_blank" class="pure-menu-link social ext fa fa-twitter" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Twitter</span></a></li> <li class="pure-menu-item" style="float:none;"><a href="https://www.youtube.com/user/ostigov" target="_blank" class="pure-menu-link social ext fa fa-youtube-play" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Youtube</span></a></li> </ul> </div> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.240327.0205.css" rel="stylesheet"> <script src="/pages/js/pages.240327.0205.js"></script><noscript></noscript> <script defer src="/pages/js/pages.biblio.240327.0205.js"></script><noscript></noscript> <script defer src="/pages/js/lity.js"></script><noscript></noscript> <script async type="text/javascript" src="/pages/js/Universal-Federated-Analytics-Min.js?agency=DOE" id="_fed_an_ua_tag"></script><noscript></noscript> </body> <!-- DOE PAGES v.240327.0205 --> </html>