<doc:document xmlns:doc="http://www.elsevier.com/xml/document/schema" xmlns:dp="http://www.elsevier.com/xml/common/doc-properties/schema" xmlns:cps="http://www.elsevier.com/xml/common/consyn-properties/schema" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:dct="http://purl.org/dc/terms/" xmlns:prism="http://prismstandard.org/namespaces/basic/2.0/" xmlns:oa="http://vtw.elsevier.com/data/ns/properties/OpenAccess-1/" xmlns:cp="http://vtw.elsevier.com/data/ns/properties/Copyright-1/" xmlns:cja="http://www.elsevier.com/xml/cja/schema" xmlns:ja="http://www.elsevier.com/xml/ja/schema" xmlns:bk="http://www.elsevier.com/xml/bk/schema" xmlns:ce="http://www.elsevier.com/xml/common/schema" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:cals="http://www.elsevier.com/xml/common/cals/schema" xmlns:tb="http://www.elsevier.com/xml/common/table/schema" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/schema" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/schema" xmlns:xlink="http://www.w3.org/1999/xlink"><rdf:RDF><rdf:Description rdf:about="http://dx.doi.org/10.1016/j.physletb.2010.08.077"><dct:format>application/xml</dct:format><dct:title>Proton radioactivity at non-collective prolate shape in high spin state of 94Ag</dct:title><dct:creator>Mamta Aggarwal</dct:creator><dct:subject><rdf:Bag><rdf:li>94Ag</rdf:li><rdf:li>One proton radioactivity</rdf:li><rdf:li>Nuclear structure</rdf:li><rdf:li>Structural transitions</rdf:li><rdf:li>Strutinsky shell correction</rdf:li><rdf:li>Non-collective prolate shape</rdf:li></rdf:Bag></dct:subject><dct:description>Physics Letters B 693 (2010) 489-493. doi:10.1016/j.physletb.2010.08.077</dct:description><prism:aggregationType>journal</prism:aggregationType><prism:publicationName>Physics Letters B</prism:publicationName><prism:copyright>Copyright © unknown. Published by Elsevier B.V.</prism:copyright><dct:publisher>Elsevier B.V.</dct:publisher><prism:issn>0370-2693</prism:issn><prism:volume>693</prism:volume><prism:number>4</prism:number><prism:coverDisplayDate>11 October 2010</prism:coverDisplayDate><prism:coverDate>2010-10-11</prism:coverDate><prism:pageRange>489-493</prism:pageRange><prism:startingPage>489</prism:startingPage><prism:endingPage>493</prism:endingPage><prism:doi>10.1016/j.physletb.2010.08.077</prism:doi><prism:url>http://dx.doi.org/10.1016/j.physletb.2010.08.077</prism:url><dct:identifier>doi:10.1016/j.physletb.2010.08.077</dct:identifier><oa:openAccessInformation><oa:openAccessStatus xmlns:cp="http://www.elsevier.com/xml/common/consyn-properties/schema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full</oa:openAccessStatus><oa:openAccessEffective xmlns:cp="http://www.elsevier.com/xml/common/consyn-properties/schema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">2014-01-01T00:14:32Z</oa:openAccessEffective><oa:sponsor xmlns:cp="http://www.elsevier.com/xml/common/consyn-properties/schema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><oa:sponsorName>SCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physics</oa:sponsorName><oa:sponsorType>http://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBody</oa:sponsorType></oa:sponsor><oa:userLicense xmlns:cp="http://www.elsevier.com/xml/common/consyn-properties/schema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">http://creativecommons.org/licenses/by/3.0/</oa:userLicense></oa:openAccessInformation></rdf:Description></rdf:RDF><dp:document-properties><dp:aggregation-type>Journals</dp:aggregation-type><dp:version-number>S300.2</dp:version-number></dp:document-properties><ja:article docsubtype="sco" xml:lang="en" version="5.1"><ja:item-info><ja:jid>PLB</ja:jid><ja:aid>27033</ja:aid><ce:pii>S0370-2693(10)01050-6</ce:pii><ce:doi>10.1016/j.physletb.2010.08.077</ce:doi><ce:copyright type="unknown" year="2010"/><ce:doctopics><ce:doctopic><ce:text>Theory</ce:text></ce:doctopic></ce:doctopics></ja:item-info><ce:floats><ce:figure id="fg0010"><ce:label>Fig. 1</ce:label><ce:caption><ce:simple-para id="sp0010" view="all">Evaluation of equilibrium shape and deformation by minimization of free energy <ce:italic>F</ce:italic> at <ce:italic>T</ce:italic><ce:hsp sp="0.2"/>=<ce:hsp sp="0.2"/>0.78 MeV and <ce:italic>M</ce:italic> = (a) 0, (b) 21<ce:italic>ℏ</ce:italic>, (c) 52<ce:italic>ℏ</ce:italic>. Deformations and shapes obtained by free energy minima are mentioned in each figure.</ce:simple-para></ce:caption><ce:link locator="gr001"/></ce:figure><ce:figure id="fg0020"><ce:label>Fig. 2</ce:label><ce:caption><ce:simple-para id="sp0020" view="all">One proton separation energy <mml:math altimg="si85.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> of <ce:sup loc="pre">94</ce:sup>Ag as a function of spin and temperature with <ce:italic>M</ce:italic><ce:hsp sp="0.2"/>=<ce:hsp sp="0.2"/>0–60<ce:italic>ℏ</ce:italic>, <ce:italic>T</ce:italic> (MeV) = 0.5, 0.6, 0.65, 0.7, 0.78, 1.0 and 1.5.</ce:simple-para></ce:caption><ce:link locator="gr002"/></ce:figure><ce:figure id="fg0030"><ce:label>Fig. 3</ce:label><ce:caption><ce:simple-para id="sp0030" view="all">Occupation probability <mml:math altimg="si94.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup></mml:math> plotted for <ce:sup loc="pre">94</ce:sup>Ag vs. single particle energies <mml:math altimg="si95.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup></mml:math> for protons at <ce:italic>T</ce:italic><ce:hsp sp="0.2"/>=<ce:hsp sp="0.2"/>0.78MeV and different spins <ce:italic>M</ce:italic> = (a) 0, (b) 21<ce:italic>ℏ</ce:italic>, (c) 33<ce:italic>ℏ</ce:italic> and (d) 52<ce:italic>ℏ</ce:italic>. The Fermi level <mml:math altimg="si96.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>5.08</mml:mn><mml:mi>ℏ</mml:mi><mml:mi>ω</mml:mi></mml:math>. Only few levels around the Fermi level are shown here.</ce:simple-para></ce:caption><ce:link locator="gr003"/></ce:figure></ce:floats><ja:head><ce:title>Proton radioactivity at non-collective prolate shape in high spin state of <ce:sup loc="pre">94</ce:sup>Ag</ce:title><ce:author-group><ce:author><ce:given-name>Mamta</ce:given-name><ce:surname>Aggarwal</ce:surname><ce:e-address type="email">mamta.a4@gmail.com</ce:e-address></ce:author><ce:affiliation><ce:textfn>UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Kalina Campus, Mumbai 400 098, India</ce:textfn></ce:affiliation></ce:author-group><ce:date-received day="10" month="7" year="2009"/><ce:date-revised day="2" month="6" year="2010"/><ce:date-accepted day="31" month="8" year="2010"/><ce:miscellaneous>Editor: W. Haxton</ce:miscellaneous><ce:abstract class="author"><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para id="sp0040" view="all">We predict proton radioactivity and structural transitions in high spin state of an excited exotic nucleus near proton drip line in a theoretical framework and investigate the nature and the consequences of the structural transitions on separation energy as a function of temperature and spin. It reveals that the rotation of the excited exotic nucleus <ce:sup loc="pre">94</ce:sup>Ag at excitation energies around 6.7 MeV and angular momentum near 21<ce:italic>ℏ</ce:italic> generates a rarely seen prolate non-collective shape and proton separation energy becomes negative which indicates proton radioactivity in agreement with the experimental results of Mukha et al. for <ce:sup loc="pre">94</ce:sup>Ag.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:keywords class="keyword"><ce:section-title>Keywords</ce:section-title><ce:keyword><ce:text><ce:sup loc="pre">94</ce:sup>Ag</ce:text></ce:keyword><ce:keyword><ce:text>One proton radioactivity</ce:text></ce:keyword><ce:keyword><ce:text>Nuclear structure</ce:text></ce:keyword><ce:keyword><ce:text>Structural transitions</ce:text></ce:keyword><ce:keyword><ce:text>Strutinsky shell correction</ce:text></ce:keyword><ce:keyword><ce:text>Non-collective prolate shape</ce:text></ce:keyword></ce:keywords></ja:head><ja:body view="all"><ce:sections><ce:section id="se0010" view="all"><ce:label>1</ce:label><ce:section-title>Introduction</ce:section-title><ce:para id="pr0010" view="all">Study of exotic nuclei is of prime importance nowadays as it helps to know the regions of the nuclear chart unexplored yet and understand many fundamental theories of Nuclear Physics and Astrophysics. A precise knowledge of binding energy, separation energy, deformation, level densities etc. of proton drip line nuclei may help to understand the astrophysical processes <ce:cross-ref refid="br0010">[1]</ce:cross-ref> of nucleosynthesis by rapid proton (rp) capture. The advent of radioactive ion beam and recent experiments on proton radioactivity (1p and 2p) <ce:cross-refs refid="br0020 br0030 br0040">[2–4]</ce:cross-refs> and proton drip line have provided valuable information about the exotic nuclei which give impetus to the theoretical models trusted for the stable nuclei to test their predictions on the unstable nuclei that too in the conditions of high temperature and rotation.</ce:para><ce:para id="pr0020" view="all">Recent observation <ce:cross-refs refid="br0020 br0030">[2,3]</ce:cross-refs> of direct one and two-proton decay of a 21<ce:sup loc="post">+</ce:sup> isomer in proton rich <mml:math altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mi>Z</mml:mi></mml:math> nuclide <ce:sup loc="pre">94</ce:sup>Ag has inspired the present investigation of proton radioactivity from the neutron deficient nucleus <ce:sup loc="pre">94</ce:sup>Ag in the ground and excited state in a theoretical framework. The main objective of this Letter is to study one proton radioactivity from <ce:sup loc="pre">94</ce:sup>Ag, though it also exhibits two-proton radioactivity <ce:cross-refs refid="br0020 br0030">[2,3]</ce:cross-refs> and <ce:italic>β</ce:italic>-decay followed by <ce:italic>γ</ce:italic>-ray or proton emission <ce:cross-refs refid="br0050 br0060 br0070">[5–7]</ce:cross-refs> as well. In fact, <ce:sup loc="pre">94</ce:sup>Ag has been considered <ce:cross-refs refid="br0020 br0030">[2,3]</ce:cross-refs> to be the first unique nucleus to exhibit one and two-proton radioactivity. We use statistical theory <ce:cross-refs refid="br0080 br0090">[8,9]</ce:cross-refs> of hot rotating nucleus combined with the macroscopic–microscopic approach <ce:cross-ref refid="br0100">[10]</ce:cross-ref> and study one proton radioactivity from the spinning hot proton rich nucleus <ce:sup loc="pre">94</ce:sup>Ag. It is conjectured that the hot rotating <ce:sup loc="pre">94</ce:sup>Ag nucleus decays by proton emission at <mml:math altimg="si2.gif" display="inline" overflow="scroll"><mml:mn>21</mml:mn><mml:mi>ℏ</mml:mi><mml:mtext>–</mml:mtext><mml:mn>31</mml:mn><mml:mi>ℏ</mml:mi></mml:math> and 5–7 MeV excitation energy because one proton separation energy of these nuclear states becomes 0 and this indicates proton emission. One of these proton unbound states at 21<ce:italic>ℏ</ce:italic> and <mml:math altimg="si3.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>6.7</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> is in agreement with the experimental observation of I. Mukha et al. <ce:cross-refs refid="br0020 br0030">[2,3]</ce:cross-refs> which shows the reliability of our theoretical formalism for this new domain of exotic nuclei that too in the conditions of high spin and temperature. Since we are not calculating the life time of the state, we do not comment on the isomerism of the nuclear state. One of the most remarkable feature of this work is the observation of a rarely seen prolate non-collective shape phase in excited <ce:sup loc="pre">94</ce:sup>Ag in the angular momentum range <mml:math altimg="si4.gif" display="inline" overflow="scroll"><mml:mn>20</mml:mn><mml:mtext>–</mml:mtext><mml:mn>32</mml:mn><mml:mi>ℏ</mml:mi></mml:math> and its influence on proton radioactivity. Claim of strong deformation with prolate shape by Mukha et al. <ce:cross-ref refid="br0020">[2]</ce:cross-ref> is not confirmed though the shape of the high spin state found by us is also prolate. However, the deformation found by us is not very large as also suggested by Kaneko et al. <ce:cross-ref refid="br0110">[11]</ce:cross-ref> using the Shell Model calculations. The observation of rarely seen prolate non-collective shape allows us to add one more parameter to treat <ce:sup loc="pre">94</ce:sup>Ag as unique <ce:cross-ref refid="br0120">[12]</ce:cross-ref>. Dependence of proton radioactivity on other degrees of freedom like deformation, spin, temperature, and separation energy is also established. <ce:sup loc="pre">94</ce:sup>Ag lies close to proton drip line and is loosely bound with proton separation energy <mml:math altimg="si5.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.89</mml:mn></mml:math><ce:cross-ref refid="br0130">[13]</ce:cross-ref> in ground state. Hence the threshold excitation energy for particle emission would be much lesser than for a normally bound nucleon.</ce:para></ce:section><ce:section id="se0020" view="all"><ce:label>2</ce:label><ce:section-title>Theoretical formalism</ce:section-title><ce:para id="pr0030" view="all">The excited high spin states of the nucleus are treated using the statistical theory <ce:cross-refs refid="br0080 br0090">[8,9]</ce:cross-refs> with the grand canonical partition function <mml:math altimg="si6.gif" display="inline" overflow="scroll"><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> of the deformed nuclear system of <ce:italic>N</ce:italic> neutrons and <ce:italic>Z</ce:italic> protons<ce:display><ce:formula id="fm0010"><ce:label>(1)</ce:label><mml:math altimg="si7.gif" display="inline" overflow="scroll"><mml:mi>Q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>Z</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:mi mathvariant="normal">exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>E</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>Z</mml:mi></mml:msub><mml:msub><mml:mi>Z</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:msub><mml:mi>N</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>M</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> The Lagrangian multipliers<ce:cross-ref refid="fn0010"><ce:sup loc="post">1</ce:sup></ce:cross-ref><ce:footnote id="fn0010"><ce:label>1</ce:label><ce:note-para>The Lagrangian multipliers <mml:math altimg="si8.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> and <mml:math altimg="si9.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> should not be confused with the deformation parameters (<mml:math altimg="si10.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi></mml:math>). The primes are put just to differentiate them from deformation parameters. The notations <mml:math altimg="si11.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> and <mml:math altimg="si12.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> used in the text from Eq. <ce:cross-ref refid="fm0010">(1)</ce:cross-ref> to Eq. <ce:cross-ref refid="fm0070">(7)</ce:cross-ref> are Lagrangian multipliers, elsewhere in this Letter, <ce:italic>β</ce:italic> and <ce:italic>γ</ce:italic> are used as deformation parameters only.</ce:note-para></ce:footnote><ce:italic>α</ce:italic>, <mml:math altimg="si13.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> and <mml:math altimg="si14.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> project the particle number, total energy and angular momentum of the system in an approximate manner through the saddle point equations at temperature <ce:italic>T</ce:italic> (<mml:math altimg="si15.gif" display="inline" overflow="scroll"><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math>)<ce:display><ce:formula id="fm0020"><ce:label>(2)</ce:label><mml:math altimg="si16.gif" display="inline" overflow="scroll"><mml:mo>−</mml:mo><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mi>Q</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0030"><ce:label>(3)</ce:label><mml:math altimg="si17.gif" display="inline" overflow="scroll"><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mi>Q</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>Z</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0040"><ce:label>(4)</ce:label><mml:math altimg="si18.gif" display="inline" overflow="scroll"><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mi>Q</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0050"><ce:label>(5)</ce:label><mml:math altimg="si19.gif" display="inline" overflow="scroll"><mml:mo>∂</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mi>Q</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">〉</mml:mo></mml:math></ce:formula></ce:display> The corresponding equations in terms of single-particle eigen values for the protons <mml:math altimg="si20.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup></mml:math> with spin projection <mml:math altimg="si21.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup></mml:math> and neutrons <mml:math altimg="si22.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup></mml:math> with spin projection <mml:math altimg="si23.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup></mml:math><ce:cross-ref refid="br0140">[14]</ce:cross-ref> of the deformed oscillator potential of the Nilsson Hamiltonian diagonalized with cylindrical basis states <ce:cross-refs refid="br0150 br0160">[15,16]</ce:cross-refs><ce:display><ce:formula id="fm0060"><ce:label>(6)</ce:label><mml:math altimg="si24.gif" display="inline" overflow="scroll"><mml:mo stretchy="false">〈</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi mathvariant="normal">exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>Z</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msubsup><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0070"><ce:label>(7)</ce:label><mml:math altimg="si25.gif" display="inline" overflow="scroll"><mml:mo stretchy="false">〈</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi mathvariant="normal">exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msubsup><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0080"><ce:label>(8)</ce:label><mml:math altimg="si26.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>〈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>〉</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:mo>+</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup></mml:math></ce:formula></ce:display><ce:display><ce:formula id="fm0090"><ce:label>(9)</ce:label><mml:math altimg="si27.gif" display="inline" overflow="scroll"><mml:mo stretchy="false">〈</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">〉</mml:mo><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:msubsup><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:mo>+</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup><mml:msubsup><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0040" view="all">The excitation energy of the system is found by<ce:display><ce:formula id="fm0100"><ce:label>(10)</ce:label><mml:math altimg="si28.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></ce:formula></ce:display> where <mml:math altimg="si29.gif" display="inline" overflow="scroll"><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> is the ground state energy of the nucleus given by <mml:math altimg="si30.gif" display="inline" overflow="scroll"><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:mo>+</mml:mo><mml:mo>∑</mml:mo><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi><mml:mi>N</mml:mi></mml:msubsup></mml:math>. As illustrated by Moretto <ce:cross-ref refid="br0170">[17]</ce:cross-ref>, the laboratory-fixed <ce:italic>z</ce:italic>-axis can be made to coincide with the body-fixed <ce:italic>z</ce:italic>′-axis and it is possible to identify and substitute <ce:italic>M</ce:italic> for the total angular momentum <ce:italic>I</ce:italic>. In the quantum-mechanical limit, the <ce:italic>z</ce:italic> component <ce:italic>M</ce:italic> of the total angular momentum <mml:math altimg="si31.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>Z</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mi>I</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:math>, where <ce:italic>I</ce:italic> is the total angular momentum.</ce:para><ce:para id="pr0050" view="all">The entropy of the system is found by<ce:display><ce:formula id="fm0110"><ce:label>(11)</ce:label><mml:math altimg="si32.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mo>∑</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi mathvariant="normal">ln</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>]</mml:mo></mml:mrow></mml:math></ce:formula></ce:display></ce:para><ce:para id="pr0060" view="all">To evaluate the separation energy <mml:math altimg="si33.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> of an excited nucleus as a function of temperature and spin, we first calculate the binding energy and the ground state separation energy using the macroscopic–microscopic approach as done earlier <ce:cross-ref refid="br0180">[18]</ce:cross-ref>.<ce:display><ce:formula id="fm0120"><ce:label>(12)</ce:label><mml:math altimg="si34.gif" display="inline" overflow="scroll"><mml:msub><mml:mi mathvariant="italic">BE</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">BE</mml:mi><mml:mi mathvariant="italic">LDM</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>shell</mml:mtext></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></ce:formula></ce:display> where the macroscopic binding energy <mml:math altimg="si35.gif" display="inline" overflow="scroll"><mml:msub><mml:mi mathvariant="italic">BE</mml:mi><mml:mi mathvariant="italic">LDM</mml:mi></mml:msub></mml:math> is obtained from the LDM mass formula of Moller–Nix <ce:cross-ref refid="br0190">[19]</ce:cross-ref> which reproduces the binding energies quite well over a wide range of nuclei. The microscopic effects arising due to nonuniform distribution of nucleons are included through the Strutinsky's shell correction <mml:math altimg="si36.gif" display="inline" overflow="scroll"><mml:mi>δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>shell</mml:mtext></mml:msub></mml:math><ce:cross-ref refid="br0200">[20]</ce:cross-ref> along with the deformation energy <mml:math altimg="si37.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>E</mml:mi><mml:mi mathvariant="italic">def</mml:mi></mml:msub></mml:math> (obtained from the surface and Coulomb effects). Shell correction to energy <mml:math altimg="si38.gif" display="inline" overflow="scroll"><mml:mi>δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>shell</mml:mtext></mml:msub></mml:math> can be written as<ce:display><ce:formula id="fm0130"><ce:label>(13)</ce:label><mml:math altimg="si39.gif" display="inline" overflow="scroll"><mml:mi>δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>shell</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>A</mml:mi></mml:munderover><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:math></ce:formula></ce:display> the first term being the shell model energy in the ground state and the second term is the smoothed energy with the smearing width <mml:math altimg="si40.gif" display="inline" overflow="scroll"><mml:mn>1.2</mml:mn><mml:mi>ℏ</mml:mi><mml:mi>ω</mml:mi></mml:math>. The smearing is done by the Gaussian distribution function (see Ref. <ce:cross-ref refid="br0210">[21]</ce:cross-ref> for details). The difference between the binding energies <mml:math altimg="si41.gif" display="inline" overflow="scroll"><mml:msub><mml:mi mathvariant="italic">BE</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math> of the parent and daughter nuclei gives the corrected ground state separation energy.</ce:para><ce:para id="pr0070" view="all">Total energy <ce:italic>E</ce:italic> (<mml:math altimg="si42.gif" display="inline" overflow="scroll"><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant="italic">BE</mml:mi></mml:math>) of a nucleus at finite temperature <ce:italic>T</ce:italic> and angular momentum <ce:italic>M</ce:italic> is calculated by blending the macroscopic–microscopic approach and statistical theory by summing the ground state energy obtained from the macroscopic–microscopic approach (Eq. <ce:cross-ref refid="fm0120">(12)</ce:cross-ref>) with the total excitation energy <mml:math altimg="si43.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math> (Eq. <ce:cross-ref refid="fm0100">(10)</ce:cross-ref>) obtained from the statistical theory as<ce:display><ce:formula id="fm0140"><ce:label>(14)</ce:label><mml:math altimg="si44.gif" display="inline" overflow="scroll"><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant="italic">BE</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="italic">BE</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="italic">BE</mml:mi><mml:mi mathvariant="italic">LDM</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>shell</mml:mtext></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></ce:formula></ce:display> Free energy <ce:italic>F</ce:italic> is minimized <ce:cross-ref refid="br0220">[22]</ce:cross-ref> with respect to the Nilsson deformation parameters <ce:italic>β</ce:italic> and <ce:italic>γ</ce:italic> which gives the shape and deformation of the excited nucleus at a fixed <ce:italic>T</ce:italic> and <ce:italic>M</ce:italic>.<ce:display><ce:formula id="fm0150"><ce:label>(15)</ce:label><mml:math altimg="si45.gif" display="inline" overflow="scroll"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></ce:formula></ce:display> The above calculations are performed for the angular deformation parameter <ce:italic>γ</ce:italic> range from <mml:math altimg="si46.gif" display="inline" overflow="scroll"><mml:mo>−</mml:mo><mml:mn>180</mml:mn><mml:mtext>°</mml:mtext></mml:math> (oblate with symmetry axis parallel to the rotation axis) to <mml:math altimg="si47.gif" display="inline" overflow="scroll"><mml:mo>−</mml:mo><mml:mn>120</mml:mn><mml:mtext>°</mml:mtext></mml:math> (prolate with symmetry axis perpendicular to rotation axis) and then to <mml:math altimg="si48.gif" display="inline" overflow="scroll"><mml:mo>−</mml:mo><mml:mn>60</mml:mn><mml:mtext>°</mml:mtext></mml:math> (oblate collective) to ° (prolate non-collective) and the axial deformation parameter <ce:italic>β</ce:italic> range from 0–0.4 in the steps of 0.01.</ce:para><ce:para id="pr0080" view="all">One proton separation energy of the excited nucleus is obtained as<ce:display><ce:formula id="fm0160"><ce:label>(16)</ce:label><mml:math altimg="si49.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant="italic">BE</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant="italic">BE</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>γ</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mi>E</mml:mi><mml:mi>d</mml:mi><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>)</mml:mo></mml:mrow></mml:math></ce:formula></ce:display> where <mml:math altimg="si50.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math> and <mml:math altimg="si51.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>E</mml:mi><mml:mi>d</mml:mi><mml:mo>∗</mml:mo></mml:msubsup></mml:math> are the excitation energies of the parent and daughter nuclei respectively. According to Eq. <ce:cross-ref refid="fm0160">(16)</ce:cross-ref>, the proton separation energy <mml:math altimg="si52.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> (now onwards we would write <mml:math altimg="si53.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> for <mml:math altimg="si54.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>) which is a function of temperature and spin, is the energy required to remove a proton from the excited state of the parent nucleus leaving the daughter nucleus in an excited state. This concept is most useful when we are dealing with highly excited nuclear systems which decay into an excited daughter nucleus similar to what is observed in the experiment by Mukha <ce:cross-refs refid="br0020 br0030">[2,3]</ce:cross-refs>. When a nucleus is excited, the last bound particle occupies higher excited state above the Fermi level and needs much smaller amount of energy to remove it from the nucleus. Depending upon the amount of excitation energy available to the system, separation energy or the energy required to remove the particle from that excited state would be different for different excitation energies. In this way, one can investigate possibility of nucleon emission from the excited nucleus at different excitation energies which we calculate using our input parameters temperature <ce:italic>T</ce:italic> and angular momentum <ce:italic>M</ce:italic>. As the excitation energy increases, the separation energy decreases as shown in Ref. <ce:cross-ref refid="br0100">[10]</ce:cross-ref>. Separation energy of an excited nucleus is always less than the ground state separation energy. Zero or negative proton separation energy indicates the instability of the nucleus against proton radioactivity.</ce:para><ce:para id="pr0090" view="all">In this Letter, we have investigated proton radioactivity for a wide range of excitation energies starting from <mml:math altimg="si55.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> to <mml:math altimg="si56.gif" display="inline" overflow="scroll"><mml:mo>≈</mml:mo><mml:mn>40</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> corresponding to <mml:math altimg="si57.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn><mml:mtext>–</mml:mtext><mml:mn>1.5</mml:mn><mml:mtext> </mml:mtext><mml:mtext>MeV</mml:mtext></mml:math> respectively with the angular momentum range of <mml:math altimg="si58.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mtext>–</mml:mtext><mml:mn>60</mml:mn><mml:mi>ℏ</mml:mi></mml:math>. Through this investigation, we show the dependence of a nuclear state for being proton bound or proton unbound on the temperature, spin and structural effects.</ce:para></ce:section><ce:section id="se0030" view="all"><ce:label>3</ce:label><ce:section-title>Results and discussion</ce:section-title><ce:para id="pr0100" view="all">Minimization of the ground state energy <mml:math altimg="si59.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math> (<mml:math altimg="si60.gif" display="inline" overflow="scroll"><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="italic">BE</mml:mi><mml:mrow><mml:mi>g</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math> obtained from Eq. <ce:cross-ref refid="fm0120">(12)</ce:cross-ref>) with respect to <ce:italic>β</ce:italic> and <ce:italic>γ</ce:italic> gives us the ground state shape and deformation of the proton rich nucleus <ce:sup loc="pre">94</ce:sup>Ag which turns out to be prolate with <mml:math altimg="si61.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>120</mml:mn><mml:mtext>°</mml:mtext></mml:math> and deformation <mml:math altimg="si62.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0.12</mml:mn></mml:math>. But once the nucleus is excited, the equilibrium shape of the hot nucleus becomes spherical at <mml:math altimg="si63.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>⩾</mml:mo><mml:mn>0.65</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> as the deformation producing quantum shell effects become ineffective and this hot nuclear system resembles a classical liquid drop. Rotation of this hot sphere generates a prolate non-collective shape at <mml:math altimg="si64.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>20</mml:mn><mml:mi>ℏ</mml:mi></mml:math> with <mml:math altimg="si65.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0.06</mml:mn></mml:math> which has been caused directly by rotation that creates a residual quantum shell effect as shown by Goodman <ce:cross-ref refid="br0230">[23]</ce:cross-ref>. This hot prolate non-collective equilibrium phase had not been anticipated before Ref. <ce:cross-ref refid="br0230">[23]</ce:cross-ref>. This unexpected prolate non-collective phase generated by rotation undergoes the expected transition to the oblate non-collective phase at <mml:math altimg="si66.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>33</mml:mn><mml:mi>ℏ</mml:mi></mml:math>.</ce:para><ce:para id="pr0110" view="all"><ce:cross-ref refid="fg0010">Fig. 1</ce:cross-ref><ce:float-anchor refid="fg0010"/>(a), (b), and (c) plot free energy <ce:italic>F</ce:italic> as a function of <ce:italic>β</ce:italic> and <ce:italic>γ</ce:italic> for <ce:sup loc="pre">94</ce:sup>Ag at <mml:math altimg="si67.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>0.78</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> and <mml:math altimg="si68.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>, 21<ce:italic>ℏ</ce:italic> and 52<ce:italic>ℏ</ce:italic> showing the minimization of F which predicts the shape and deformation of the nucleus. Shape transitions due to changing angular momentum can be seen through <ce:cross-ref refid="fg0010">Fig. 1</ce:cross-ref>(a), (b) and (c) where the shape and deformation of the nucleus are indicated by <ce:italic>β</ce:italic> and <ce:italic>γ</ce:italic> values corresponding to <ce:italic>F</ce:italic> minima for each M. Equilibrium shape of the hot nucleus is spherical at <mml:math altimg="si69.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> (<ce:cross-ref refid="fg0010">Fig. 1</ce:cross-ref>(a)) which undergoes the transition to uncommon prolate non-collective phase at <mml:math altimg="si70.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>20</mml:mn><mml:mi>ℏ</mml:mi></mml:math> via triaxial phase with <ce:italic>γ</ce:italic> ranging from <mml:math altimg="si71.gif" display="inline" overflow="scroll"><mml:mn>40</mml:mn><mml:mtext>°</mml:mtext></mml:math> to 10° and <ce:italic>β</ce:italic> from 0.01 to 0.04 with increasing <ce:italic>M</ce:italic> and then to usual oblate non-collective at <mml:math altimg="si72.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>33</mml:mn><mml:mi>ℏ</mml:mi></mml:math>. The energy minima shifts from <mml:math altimg="si73.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> (spherical) at <mml:math altimg="si74.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> towards triaxial and then to prolate non-collective (<mml:math altimg="si75.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mtext>°</mml:mtext></mml:math>) at <mml:math altimg="si76.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>20</mml:mn><mml:mi>ℏ</mml:mi></mml:math> (<ce:cross-ref refid="fg0010">Fig. 1</ce:cross-ref>(b)). The deformation in the prolate non-collective shape phase is found to be 0.06 not in support of the strong deformation expected by Mukha et al. <ce:cross-ref refid="br0020">[2]</ce:cross-ref> but supports the idea of Refs. <ce:cross-refs refid="br0110 br0240">[11,24]</ce:cross-refs>. With further increasing spin, the energy minima moves towards oblate non-collective shape phase (<mml:math altimg="si77.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>180</mml:mn><mml:mtext>°</mml:mtext></mml:math>) with <mml:math altimg="si78.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn></mml:math> at <mml:math altimg="si79.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>33</mml:mn><mml:mi>ℏ</mml:mi></mml:math> and remains oblate for <mml:math altimg="si80.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>></mml:mo><mml:mn>33</mml:mn><mml:mi>ℏ</mml:mi></mml:math> with a gradual increase in <ce:italic>β</ce:italic> attaining a value 0.14 at <mml:math altimg="si81.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>52</mml:mn><mml:mi>ℏ</mml:mi></mml:math> (<ce:cross-ref refid="fg0010">Fig. 1</ce:cross-ref>(c)). At <mml:math altimg="si82.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>60</mml:mn><mml:mi>ℏ</mml:mi></mml:math>, <ce:italic>β</ce:italic> has much higher value of 0.24. The shape transition from spherical to prolate non-collective phase does not occur at high temperature <mml:math altimg="si83.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> at which the minima goes directly from spherical to oblate non-collective phase at <mml:math altimg="si84.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>34</mml:mn><mml:mi>ℏ</mml:mi></mml:math>.</ce:para><ce:para id="pr0120" view="all">The influence of shape transitions on the separation energy of an excited nuclear system is shown in <ce:cross-ref refid="fg0020">Fig. 2</ce:cross-ref><ce:float-anchor refid="fg0020"/> where we have plotted one proton separation energy as a function of temperature and spin obtained from Eq. <ce:cross-ref refid="fm0160">(16)</ce:cross-ref>. Spin degree of freedom inherently involves deformation and structural or shape changes and we know that the separation energy is a function of these changes as illustrated by Faber et al. <ce:cross-ref refid="br0250">[25]</ce:cross-ref>, Rajasekaran et al. <ce:cross-ref refid="br0260">[26]</ce:cross-ref> and our earlier work <ce:cross-refs refid="br0100 br0180 br0270">[10,18,27]</ce:cross-refs>, which is evident in <ce:cross-ref refid="fg0020">Fig. 2</ce:cross-ref> where the proton separation energy is decreasing with increasing <ce:italic>T</ce:italic> but starts showing fluctuations when the angular momentum degree of freedom is incorporated which sets in the structural changes with it. The spin states with <mml:math altimg="si86.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>21</mml:mn><mml:mi>ℏ</mml:mi><mml:mtext>–</mml:mtext><mml:mn>31</mml:mn><mml:mi>ℏ</mml:mi></mml:math> at low temperature <mml:math altimg="si87.gif" display="inline" overflow="scroll"><mml:mo>≈</mml:mo><mml:mn>0.78</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> have zero or negative separation energy. These spin states are built up mostly by rotational states and contribution from the thermal excitation energy is kept to be very small (<mml:math altimg="si88.gif" display="inline" overflow="scroll"><mml:mo>≈</mml:mo><mml:mn>1</mml:mn><mml:mtext>–</mml:mtext><mml:mn>1.5</mml:mn><mml:mtext> </mml:mtext><mml:mtext>MeV</mml:mtext></mml:math>) in order to bring these states as close as possible to the experimentally observed proton decaying state <ce:cross-refs refid="br0020 br0030">[2,3]</ce:cross-refs> with <mml:math altimg="si89.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>6.7</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> and 21<ce:italic>ℏ</ce:italic> which seems to be built up mostly of the rotational modes. Since we are investigating the proton emission from various excited high spin states using a thermodynamical approach in which <ce:italic>T</ce:italic> is an essential component, we keep <ce:italic>T</ce:italic> to its minimum possible value so that the majority of excitation energy is due to rotational degree of freedom. This ratio of thermal and rotational degree of freedom used in our calculations enables us to observe one proton radioactivity (see <ce:cross-ref refid="fg0020">Fig. 2</ce:cross-ref>) from <ce:sup loc="pre">94</ce:sup>Ag at 6.7 MeV excitation energy and 21<ce:italic>ℏ</ce:italic> angular momentum, analogous to the proton emitting state observed in Ref. <ce:cross-ref refid="br0020">[2]</ce:cross-ref>. At <ce:italic>M</ce:italic> around 21<ce:italic>ℏ</ce:italic> and nearby states at low temperatures <mml:math altimg="si90.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn><mml:mtext>–</mml:mtext><mml:mn>0.78</mml:mn><mml:mtext> </mml:mtext><mml:mtext>MeV</mml:mtext></mml:math>, one proton separation energy is zero or very close to 0 which means that the quantum shell effects at angular momentum around 21<ce:italic>ℏ</ce:italic> drive the nuclear shape towards prolate non-collective which favors proton emission as also pointed out by I. Mukha et al. <ce:cross-ref refid="br0020">[2]</ce:cross-ref> that the prolate (cigar-like) shape facilitates the emission of proton. At high temperatures (<mml:math altimg="si91.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math> and 1.5 MeV) and corresponding high excitation energies, many more states become proton radioactive for a larger range of angular momentum values irrespective of shape. Infact at <mml:math altimg="si92.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn><mml:mtext> MeV</mml:mtext></mml:math>, all the states are proton decaying irrespective of angular momentum and shape because excitation energy available to the system is much higher than that spent in rotation even for very high spin states. Therefore, the role of angular momentum is more prominent at lower temperature at which the shape transition to uncommon prolate non-collective shape phase and proton emission is observed whereas the shape transition to usual oblate non-collective shape phase does not indicate proton emission. Hence we suggest strong dependence of proton emission on the shape transitions due to spin at low temperature which is one of the main focus of the present work. It would be interesting to study the effects of angular momentum and shape transitions on proton emission at <mml:math altimg="si93.gif" display="inline" overflow="scroll"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> with some suitable microscopic calculations.</ce:para><ce:para id="pr0130" view="all">The proton unbound states are seen in the deformed prolate non-collective shape phase (<mml:math altimg="si97.gif" display="inline" overflow="scroll"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mtext>°</mml:mtext></mml:math>, <mml:math altimg="si98.gif" display="inline" overflow="scroll"><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0.02</mml:mn><mml:mtext>–</mml:mtext><mml:mn>0.06</mml:mn></mml:math>) at <mml:math altimg="si99.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mtext>–</mml:mtext><mml:mn>7</mml:mn><mml:mtext> </mml:mtext><mml:mtext>MeV</mml:mtext></mml:math> and <mml:math altimg="si100.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>21</mml:mn><mml:mi>ℏ</mml:mi><mml:mtext>–</mml:mtext><mml:mn>31</mml:mn><mml:mi>ℏ</mml:mi></mml:math>. At other excitation energies and <ce:italic>M</ce:italic> values for all other shapes, the system is bound though weakly only. This shows that the prolate non-collective shape favours proton radioactivity whereas other shapes like spherical or triaxial (at 0 or low <ce:italic>M</ce:italic> values) and oblate non-collective (at <mml:math altimg="si101.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>></mml:mo><mml:mn>33</mml:mn><mml:mi>ℏ</mml:mi></mml:math>) even with larger deformations (<ce:italic>β</ce:italic> upto 0.24) indicate a proton bound state with <mml:math altimg="si102.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math> though there is no taboo to other deexcitation modes which may exist and the nucleus may decay via other exit channels. In this work, we neither comment if the concerned excited state are isomeric nor we talk about the other decay modes. Our only emphasis is to explore the possibilities of proton radioactivity using the criteria of <mml:math altimg="si103.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>⩽</mml:mo><mml:mn>0</mml:mn></mml:math> which occurs at certain excitations and shapes only.</ce:para><ce:para id="pr0140" view="all">To understand the proton decay in prolate shape phase but not in oblate shape phase at high spin states, we plot occupation probability <mml:math altimg="si104.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math> vs. single particle energy eigen values corresponding to each ± m state at fixed <ce:italic>T</ce:italic> and <ce:italic>M</ce:italic> in <ce:cross-ref refid="fg0030">Fig. 3</ce:cross-ref><ce:float-anchor refid="fg0030"/>. The proton which is farthest from the Fermi level is most likely to escape as it is most weakly bound. Occupation probability shows the rearrangement of particles near the Fermi level. All the occupied ±<ce:italic>m</ce:italic> states will sum up to give <ce:italic>M</ce:italic> (see Eq. <ce:cross-ref refid="fm0090">(9)</ce:cross-ref>). Normally, <mml:math altimg="si105.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup></mml:math> should be 1 upto the Fermi level and 0 beyond it. In <ce:cross-ref refid="fg0030">Fig. 3</ce:cross-ref> (a), <mml:math altimg="si106.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>, and the curve is smooth but <mml:math altimg="si107.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:math> and <mml:math altimg="si108.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>Z</mml:mi></mml:msubsup><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math> for few levels below and above the Fermi level respectively because many higher levels are occupied due to excitation energy. When angular momentum (<ce:italic>M</ce:italic>) increases, large fluctuations appear near the Fermi level which are smeared out at high temperatures as shown in our earlier work <ce:cross-ref refid="br0100">[10]</ce:cross-ref>. At <mml:math altimg="si109.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>21</mml:mn><mml:mi>ℏ</mml:mi></mml:math>, the fluctuations in occupation probability (<ce:cross-ref refid="fg0030">Fig. 3</ce:cross-ref>(b)) are large and stretched far below and above the Fermi level much more than that at higher angular momentum <mml:math altimg="si110.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>33</mml:mn><mml:mi>ℏ</mml:mi></mml:math>, 52<ce:italic>ℏ</ce:italic> states and even at higher deformation (<ce:cross-ref refid="fg0030">Fig. 3</ce:cross-ref>(c) and (d)) with oblate shape. This shows that in the proton decaying state with <mml:math altimg="si111.gif" display="inline" overflow="scroll"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>6.7</mml:mn><mml:mtext> MeV</mml:mtext></mml:math> and <mml:math altimg="si112.gif" display="inline" overflow="scroll"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>21</mml:mn><mml:mi>ℏ</mml:mi></mml:math>, the excited particles have occupied much higher levels above the Fermi level than those at other angular momentum states due to which it is very much likely that the outer most nucleon with higher occupation probability in prolate shape is very weakly bound with a very low or zero separation energy and hence is easily knocked out of the nucleus. This shows that above the Fermi level, farther is the level occupied with higher occupation probability, lower is the separation energy. This analysis indicates that the occupation in farthest levels above <mml:math altimg="si113.gif" display="inline" overflow="scroll"><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>F</mml:mi></mml:msub></mml:math> in prolate non-collective shape favours proton emission.</ce:para></ce:section><ce:section id="se0040" view="all"><ce:label>4</ce:label><ce:section-title>Conclusion</ce:section-title><ce:para id="pr0150" view="all">To conclude, we report one-proton radioactivity from the excited high spin state of <ce:sup loc="pre">94</ce:sup>Ag at excitation energy 5–7 MeV and angular momentum <mml:math altimg="si114.gif" display="inline" overflow="scroll"><mml:mn>21</mml:mn><mml:mi>ℏ</mml:mi><mml:mtext>–</mml:mtext><mml:mn>31</mml:mn><mml:mi>ℏ</mml:mi></mml:math> in a theoretical framework by combining statistical theory and macroscopic–microscopic approach carefully in a way that it proves a simple yet an effective tool in predicting proton radioactivity and establishing the important role played by the structural properties like shape and deformation on the separation energy and proton decay in excited high spin states which is not shown by any other work so far. Structural transition to uncommon prolate non-collective shape influences the proton decay which is an important feature of this work. This formalism can be used to make useful and meaningful predictions which would be very helpful in planning an experiment to study proton radioactivity by providing an excitation energy and angular momentum range in which proton decay could be expected via proton separation energy becoming ⩽0 criteria. A lot more experimental and theoretical work is required for a better insight into this problem.</ce:para></ce:section></ce:sections><ce:acknowledgment><ce:section-title>Acknowledgements</ce:section-title><ce:para id="pr0160" view="all">Financial support from <ce:grant-sponsor xlink:type="simple" xlink:role="http://www.elsevier.com/xml/linking-roles/grant-sponsor">The Department of Science and Technology (DST)</ce:grant-sponsor>, <ce:grant-sponsor xlink:type="simple" xlink:role="http://www.elsevier.com/xml/linking-roles/grant-sponsor">Government of India</ce:grant-sponsor>, under the WOS-A Scheme is acknowledged. Useful discussions with Dr. S. Kailas, and Dr. S. 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