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Unformatted text preview: New Physics: Sae Mulli (The Korean Physical Society), Volume 60, Number 2, 2010¸ 2, pp. 113∼118 4 Z DOI: 10.3938/NPSM.60.113 ll? c¤ c¤ ¿ce ˜옹R n£ ê0ý b•-b• {;" ŒŽŒV úŒ V?8 Yrast ô› ;U; 6” m ÅmÅË Þ; s §¨ pÖ c.c X ¢ ŽÃ 8X OMŸk ­ä ýÇ ß]ŒÅ Å Ø Ëö«Ä » c£å 6£ ×> ¨* · ò­ · *.H B <  /<§ ü<  “ @†“ Óo†õ, “… 402-751 Æ tÆ ; (2010¸ 1 13{ ~6, þ7Ã&‘ 2010¸ 2 5{ ~6)  4  Χ  Z 9 ã jáºñ: x r  4  Χ  Z 9 ã  7f ¼2 > â« B”_ »òí Ž£ AŒ Œº Œº þf øøí s ë\" ۗ _” + /d Ä´$` 7 l 0 # ‹Ã-‹Ã Ù\" ìƒì„$ HH H ; r > N ¦ x ••˜ Í Í G © ×F I_ Yrast [> \t\ s /d` &6 %. ƒì„$ ×F I\ &6K‘ (Û æœ tu þp -  B” h i øí G © h : ¼ N¦ x  Í æ œ x r ; 2\ > ú < §Œ  : øøí G  -  OQ — _” t ·) õü q“ # ^ M, ìƒì„$ ×F s\ \t Øìy Z# r §H  ¦ Í Í æ H ær  R ” l¼Ð ¼2 > â« B” ¼2 > ú â«B”  8 »¦ + 4 et 3 ٖ, ۗ _” + /ds ۗ\ _” t · +/d\ qK  Äo “ ½ w ; r > N ; r §H >N É º O ß Q ¨ ŸŒ øøí G © à \. të sô ƒ½\ : # ìƒì„$ ×F I_ Yrast [> \t #b>   – Ç ¦x Í Í æ œ tu þp - Q  G æ G\ > ú º ” ¸ ŒŒ G © ú ¼2 > ú < § ×F _” t · à e. ¢ô yy_ ×F I\ ´ð ۗ\ _” t · õü q“ rH ˜  Ç •• æœ r ; r §H    M Œº G © ¸ ˺ G © ‰ ú ¼2 >  s hÉ hº K ^ :, ‹Ã ×F I ¢ .à ×F I „^\ ´ð ۗ _” õ › &“ >Ã_ ¦ • æœ Hf æœ r ; r  H ` r  ¦ x  H ¤   i £f ª ò í ú “\ s6 % 8€\" Õ ´6$` °. x ¦ H ˜ þ”Q Ùd#: Yrast \t +/d, ۗ\ _”  +/d, ‹Ã-‹ÃÙ_ øƒì„$ ×F I_ Yrast - â«B” ¼2 > â«B” Œº Œºþ Íøí G © >N ; r H >N • • ˜ ì Í æ œ tu þp - [> \t, Ùà ˜ þº Spin-dependent Empirical Formula for the Yrast Excitation Energies of the Unnatural Parity States in Even-even Nuclei Dooyoung Kim · Dongwoo Cha · Jin-Hee Yoon∗ Department of Physics, Inha University, Inchon 402-751 (Dated: Received 13 January 2010, in final form 5 February 2010) We test the validity of the spin-dependent empirical formula for the Yrast excitation energies of the unnatural parity states in even-even nuclei. Compared to those for the natural parity states, the energies are not spaced enough to be differentiated by using the spin-dependent empirical formula. Therefore, we do not find the spin-dependent empirical formula to be more advantageous than the spin-independent one. However, a trial to fit all even or odd multipole state Yrast excitation energies yields information on the spin dependence of the Yrast excitation energies, and the fit for all energies is as good as a separate fit for each multipole state energy. PACS numbers: 21.10.Re, 23.20.Lv Keywords: Empirical formula, Spin-dependent empirical formula, Yrast unnatural parity state excitation energy, Valence nucleon numbers ∗ E-mail: jinyoon@inha.ac.kr -113- -114- ÇGt Æ Dü<r hü ô²Óo†t “DÓo”, Volume 60, Number 2, 2010¸ 2 t 4 Z I. "  e] Ø ˜ ¦ H þ í9 ©î < ŒQ þf ª í9 Q Ù_ $|` ½"  X # Ù\" Õ $|s #b ˜  G  í& úRÐ ¯É ” 9 ih > ìŸ÷t ¶(˜ “ _pe {s. %& r H˜ H r H   ¼Ð þ ø¦h 9¾B”É ª /ðh V + º ” ܖ Ù_ 쓄& ||/d“ Õ @³& \ ½ à e ˜ Í  ÓNr  É    [1]. sô ~Z“ Ù_ :&ô $|s Ù_ r& Q ½OÉ þ £ñ í9 þ h Ç Ór ˜ ¤ Ç  ˜    ð 9¾º ªíº ¸ æíº p  Q “ t³(||Ã~ €$à ¢ $à 1)\ # Ó  œ H × x G a   ÷  / Ð ú9¡¼Ð‹ ª í b> ƒ›s & t\ @ô ñ˜\ ·ºÜ–+ Õ $ H Ç& ¦˜ §  Hé¦ É 9 ¶ Òñ+ º ” ½O jB j r |_ "` Æ&½ à e ~Z` ]/ô. þ : H Ó¦ NÇ H‘  ¨ªf Œº Œº þf ø G ƒ½ÕÒ\" ‹Ã-‹Ã Ù\"_ ƒì„í ×F ¨ H• • ˜ Í$ æ œ © I(natural parity multipole state)–_ Yrast [> \ Ð tu þp  - 9¾º t\ ||Ã, €$Ã(valence proton number),  ¦ Ó œ ªíº æ íº ×$Ã(valence neutron number)_ †Ã– ³‰ #  <ºÐ ð&Œ Ê ³ >N¦ ¦  â«B” ¸i +/d` •Ø % [2]. Õ + e²“ ë : ª ' ±úÉ 7¦ Ÿ ˜r HH` x Œ  B” ø Œº G ©   # s /ds ƒì„í ‹Ã ×F I÷ m  N Í$ • æ œr Í Ë øí .º G © ƒì„$ fà ×F I_ Yrast [> \t\• ¸ æœ tu þp -¸ ú ˜ xc h ¨ º ”£ Ði¼ &6| à e6` ˜%Ü9 [3,4], ¢ô # ×F I §¦  ¸ ŒQ G © Ç æœ  BŸh¼Ð h + º ” l9 ú 1 A \ /:&ܖ &6½ à e 1{ô “°` ¹l 0 Nx xÉ H xÇ  ¯¦ Ô Œ ¼2 > ºñ â« B” jߦ  # ۗ\ _”  Ã& + /d` ]î “, s ; r H a > N¦ – ¦•  Œº G .º G ŒŒ h   j \ ‹Ã ×Fõ fà ×F yy\ &6ô õ\ ] æ Ë æ •• xÇ  ¦ i r % [5]. sô ƒ½\ : # ‹Ã-‹Ã Ù\"_   ¨ ŸŒ Œº Œº þf Ç  ¦ x ••˜ øí G ©f ƒì„$ ×F I\" Yrast [> \t || Í æ œ tu þp - 9¾ Ó º Ã, €$Ã, ×$Ãëܖ &]y ["½ à e œ ªíº æ íºß¼Ð hX Oî+ º ” –   É  H œ¦ ß  ª” 1€/3¼  €e` µ)?%Ü9, sô Óo|[_ %½\ @  Q ü¾þ i+ / Ç t Ót É f¸ 7i K"• _ % [6]. H  j Q « B” øøí G þ\ sô â+ /d` ìƒì„$ ×F HH Ç > N¦ Í Í æ œ © I(unnatural parity multipole state)\" Yrast [> f tu þp -¸ h + º ”  / ¨ Ÿ’ \t\• &6½ à e t\ @ô ½ 'K& xÉ H ǃ    [7]. ìƒì„$ ×F I–_ „s „l$ Í Í æ œ øøí G ©Ð  í  H   „s(electric transition)  l$ „s(magnetic  í    transition)\ _K ñ÷9, " Õ ìŸ €    & & f ª í ª r œ Œ Q" 9í / Q> ªQ  ¨ # #‹ {›$` l@ l #§. Õ s ƒ½\  a ¦   Ô øø G ©f £ñ ؀ ìƒì„í ×F I\" 8& Yrast [  Í Í$ æ œ ¤ a t þ u p - ©© ©Ë ¸þ¦ Д¸ ¨ > \t {y Ô½gô —_ ˜e\• Ô½ H œœ ¦ :Ç v`  ¦ ¦ â« B” ª Ÿl¦ h ú Oî¦ ” “, + /ds Õ '1 &X > ¸ [" “ e. > N x` ˜   Í Í æ œ øøí G ©f ìƒì„$ ×F I\" Yrast [> \t  tu þp -  H Í • øí Œº G ©  G ß - ƒì„$ ‹Ã ×F I\ qK ×F ç \t æœ æ– H   O p ¼Ð 9”  ø s \s q5ô ßl– ]e. s ƒì  wÇ t H Í $ Ë  .º G ©ü¸ p f ø „í fà ×F I<• q5 . " ƒì„ æœ w Í Ë í .º G ©f  ú h  â« B”É $ fà ×F I\" sp ¸ &6 + /d“ æœ ˜ xa > Nr Q¼ ñ¸ øøí ©¸ ú h & / #Ö &• ìƒì„$ I\• ¸ &6÷o l@ Í Í œ ˜ x &3 ¼2 > â« B”É Q  7f ÷%. ۗ _” + /d“ #b? s ë\"  ; r > Nr  HH H ¦ :Ç Ÿx¦ ©Ë l Ð øøí ©¸ G Ô½gô '1` ˜s ìƒì„$ I\• ×F H Í Í œ æ  ©O ¼2 > l9 ú¼Ð Oî \ ›\s ۗ\ _”  1{ô “°Ü– [" œa ; r H xÇ  ¯   p+  £ ¼2 > â« B” h Œ s 0½ ¯“t, 7 ۗ _” + /d` &6 # xÉ  ¤ ; r > N¦ x þ ü¾ Ÿl hX ü+ º ”  ú s[ Óo|_ '1` &]y lÕ½ à e t\ · t t Ó x¦  tÉ H ¦ ˜ Ð9  ˜ ô. Ç (a) Even multipoles 10 Excitation Energy Ex (MeV) 1 2 - 4 - 6 - 8 - 10 - (b) Odd multipoles 10 1 1 0 + 3 + 5 + 7 150 + 9 200 + 50 100 250 Mass number A Fig. 1. Measured excitation energies of the lowest unnatural parity states in even-even nuclei. The upper part is for the even multipole states and the lower part is for the odd multipole states. The experimental values are form Ref. [8]. Open circles(dark yellow in color version) are 2− or 1+ , solid circles(blue in color) are 4− or 3+ , open triangles(green in color) are 6− or 5+ , solid triangles(red in color) are 8− or 7+ , open squares(black in color) are 10− or 9+ for even or odd multipole unnatural parity states. II. ­ä ýÇ ß]ŒÅÊ § X“ ŽÃ 8X OMŸkÝ Ž ìÓ Å Ø ËöÄ ÄÞ j £ñ ¸ Œº Œº þf øø s]t 8& —Ž ‹Ã-‹Ã Ù\", ìƒì„ ¤ a H • • ˜ Í Í æœ í G © $ ×F I_ Yrast [> \t Fig. 1 üe tu þp - \ <”  −   < . s Xs' Ref. [8]\" “ s. ŒÃ(2 ∼ H f : ¯ •º r ‹ 10− )ü fÃ(1+ ∼ 9+ )_ ×F I\ @K Œy 0ü < .º Ë  G © / •Œ A< æœ y•   ª9R ”¼ A €\ Õ4 eÜ9, yy_ ×F I  l   •• ŒŒ G © É  æœHr ñ : Q  ñ Ð ð&&Q ” Œº (““ !„\ Òõ l )– ³‰÷# e. ‹Ã r   Ho ³  • G ⺠×F_ Ä 2−  ‘ "ܖ, 4−  5s ð "ܖ, æ H¶   é¼Ð Hq Ͷ  Å ø é¼Ð 6−  5s ‘ [—–, 8− “ 5s ð [—–, 10− “ 5s Hq   Å  j¸Ð rq Í É Å ø j¸Ð rq É Å   1¸Ð ª4¼ ‘ W—– Õ§Ü9, fà ×F_ Ä 1+  ‘ "Ü  Ë .º G ⺠æ H¶   é¼ Ð –, 3+  5s ð "ܖ, 5+  5s ‘ [—–, 7+ “ 5 Hq Ͷ  Å ø é¼Ð Hq   Å  j¸Ð rq ÉÅ +  ø j¸Ð s ð [—–, 9 “ 5s ‘ W—– Õ§. „^&ܖ Í rq  É Å  1¸Ð ª4 ‰h¼Ð   Ќ : 9ñ ©Ëí Ð / Ë ˜€` M {&ô ½g$` ˜so l@ l jŽ ¤¦  Ç : ¦ H µH r¦  í Ð ìŸ\ ˜“. •‹˜ Œº •º þf øøí G © ‹Ã-ŒÃ Ù\" ìƒì„$ ×F I_ Yrast [> \t\ @ô ۗ _” +/d· · · – ^¿% 1 Í Í æ œ tu þp - / ¼2 > â«B” Ç ; r >N  x ”ºò p -115-  < h : â« B”É £ ú s Xs \ &6K‘ + /d“ 6õ °. x r > Nr § Ex = αA−γ + βp e−λp Np + βn e−λn Nn . (1) (a) Data 10 Œf  9¾º #l" A ||Ãs9, Np (¢ Nn )“ €$ H Ó ¸ H r É œ ªí º ¸ Ã(¢ ×$Ã)–" jüF\ ”F  €$ H æ íº Ðf þ@Y > ªí ` r H œ º ¸ íº  ªí¨í ¸ í¨í   Ã(¢ ×$Ã) €$½"(¢ ×$½")_ Ì H æ œ H æ º  ŒÉ ú þ Ã × “ °` ×ô. α, γ , βp , βn , λp , λn “ xhܖ æ •r ¯¦ ˜Ç r É A¼Ð & H  t  & þ  â« B”f O ¸º ñ÷ “[s. s + /d\" sX> —¿ > N H 6>_ “\ 6 X, s °[“ yy_ ×F  h   <  úþÉ ŒŒ G © ¦ xH ¯tr •• æœ  / A&Q jÔÐ ñ £ ŒŒ  I\ @K xh÷# [à– &. 7, yy_ ×  a ¤ •• æ G © ¸º h ú ú   F I —¿ 6>_ “°` °. s õ Ref. œH  ¯¦ H  [7]\ üeÜ9, s\ Table 1\ “6 %. s ³\  <”¼   ¦   i  ð x  ¦ Ð ŒŒ G © / ¸º h ú ˜€ yy_ ×F I\ @K —¿ 6>_ “°õ  •• æœ ¯ ª É ú Œ G © > <  Õ\  χ2 °õ y ×F I\ ”F  Xs Ì r¯•æœ r H 2 º Ã(N0 ) üe. Xs  þ™_ χ °` ë7 • <” < jè ú ßḠ H ¯¦ –¤ ¤ Ÿ Ai< Œf   úÉ < ú 2 xh %X, #l" 6ô χ2 °“ Aü °. H xÇ ¯r N0 1 Excitation Energy Ex (MeV) (b) Spin - Indep. 10 1 (c) Spin - Depend. 10 1 1 χ2 = N0 cal exp log Ex (i) − log [Ex (i)] . i=1 2 2 50 4 100 6 150 8 200 10 250 (2) 0 Mass Number A П ú / < ˜:_ χ2 ° @’ Xs \ log °` 2ô sč °“ x ¯ ¯¦ Ç ú [ » úÉ H r G /f¸ -  ºf ºþ  ×F ?\"• \t_ s Ãz\" ÃÑC\ s æ  ˜ Ô : ⺠Øl Mës. sÄ log °` 2 t ·Ü€ \t H ¯¦ ú [ ú¼  - § H ¦ òi AÒÐ A l ÷Q ŒÉ - òif %%` 0Ŗ xhô !s &# “ \t %%\" Çr •r  H  j/Ð  Ò l O A Ð  ]@– õ\ Åt 3ô. sX> xhô õ– a ¦ wÇ  Ç Ô p  t` x 1· þ¦  Œ ß ú ¹ “[ s6 # >í °s Fig. 2(b)ü Fig. –a ¯ < 3(b)\ Õ4 eÜ9, @^&ܖ „^&“ ìŸ\ ¸  ª9R ”¼  /‰h¼Ð ‰h í ú    r ¦ ˜ " O ¦ ” ð Œ  <” [î “ e. ³_ t} ×\ üe “All”_ _  •¦ H   .º ¸ Œº G ¸ < / ú p fà ¢ ‹Ã ×F_ —Ž Xs \ @ô χ2 ° HË H• æ H ǯ ¦   £ Œ < Ъ j ú + ` _pô. 7, y Xs'_ –Õ _ ]Y` 8 ½ô Ç ¤• L¦ x ËÇ ê ‰ < ºÐ  ú Ê, „^ Xs ÌÖ è °s.  H¯ j O ŒŒ G  É ú ú s] sX> yy_ ×F\   “°` °  •• æ r  ¯¦  H •  / Œº .º G ‰ BŸ¼Ð h + º  @’ ‹Ã fà ×F „^\ /:ܖ &6½ Ã Ë æ Nx xÉ H  ¯¦ Ô ” ú 1Ð  þÉ G > e “°` ¹˜. s “[“ ×F\ _”K  tr æ r   9, s\ ۗ _” +/ds ô. ۗ _”  ¼2 > â«B”  ¼2 > ¦ ; r >N Ç ;r >N â«B”f ” +/d\" d (1)\" α @’\ α0 J a \, γ @’\ H f /  ¦  /  √ c 0 ¦ xÇ    γ0 J \ 6ô. ¢ô λp(n) @’\ λp(n) / J \ 6ô ¦ xÇ    ¸ Ç /  £ . 7, ¤ Ex = α0 J a A−γ0 J + βp e−λp Np / c 0 Fig. 2. Yrast excitation energies of the lowest even unnatural parity states in even-even nuclei. The upper part (a) is the experimental data, same figure as the part (a) of Fig. 1. The middle part (b) is the fit with the spinindependent empirical formula, and the lowest part (c) is the fit with the spin-dependent empirical formula. Symbol representations are same as those of Fig. 1. 0.035–, fà ×F_ Ä 0.042\" 0.074Ð ›Fm Ð .º G âº Ë æ f – ¸K”  x £ i ªQ Œ 7 %. Õ #„y 0.1˜• “ ÃÐ" Xs   и ŒÉ º–f < •r  hX úÒ¦ ”¦ + º ”  ªË¼Ð \ &]y ´Æ“ e“ ½ à e. s\ Õaܖ ¦    É  ¦> · ¯   s Fig. 2(c)ü Fig. 3(c)s. Fig. 2ü 3“ p <  <É r Table 1õ Table 2\ üe °[` s6 Œ yy_   <” úþ  # ŒŒ H ¯t¦ x •• ˜ þ / ßÇ  j9 A « < Ù\ @K >í õs. ]{ 0\ z+ Xs s –ô   H ´> 9, îX ۗ\ _” t · +/dܖ xh r < ¼2 > ú â«B”¼Ð A ; r §H >N Ç   ô õs9, ]{ A ۗ _” +/d` 6ô j9  ¼2 > â«B”    ; r >N¦ xÇ   ¸ õs. ¢ô Fig. 2 ‹Ã ìƒì„$ I\ @ Ç H • Í Í œ  Œº øøí © / Ç   ô õs9, Fig. 3“ fà ìƒì„$ I\ @ô r Ë Í Í œ É .º øøí © / Ç    º ªË <ü   q§Ð õs. s ¿ Õa_ îX< A €` “K˜€ > r ¦   ‰h¼Ð ŒŒ G © AfÒ  „^&ܖ yy_ ×F I_ 0\"Â' At •• æœ  f ú¦ ú ü&¦ ”£ ˜ º ” _ í"  t·“ ¸ lÕ÷“ e6` ú à e. H  §˜t §¦ ·  − − + + – ß ét 2 ü 4 , 1 ü 3  +Ÿ 1 ˜s  ¿ Xs < < '÷ p Ð s º < ¶w   g”Q ¨ îX ú ‰h  _ 5e# ½ìs "S t ·. „^&“ H  r ‰ §   ¸ª ¦ : ¼2 > â« B” ©/h¼Ð Œ  —€`  M, ۗ _” + /ds @&ܖ y  œ¦ ^ ; r > N œ  • æ œt G ©þ  - ú OQ” æ ü ×F I[ s\ \t °_ Z#f` ìy lÕ ¯  ¦ Ør t √ J + βn e−λn Nn / 0 √ J (3)  ¸  Af  & úÉ A ñ” s. ¢ô s xh\" 6÷ χ2 °“ 0_ &_dõ Ç x H ¯r  x l9 ¸ 1{  ì—_ N0  „^ Xs'_ ÌÃs. s õ r H  ‰ < º     Table 2\ üe.  <”  Table 1õ 2\ q“ # ˜€, {é Õ Ä´$`    §Œ Ð 9ß ª »òí Ž ¦  – ¦  2 xÉ £+ º ” 7½ à e χ °s ‹Ã ×F_ Ä 0.027\" H ¯ ú Œº G ⺠• æ f -116- ÇGt Æ Dü<r hü ô²Óo†t “DÓo”, Volume 60, Number 2, 2010¸ 2 t 4 Z Table 1. Values of six parameters in Eq.(1) for the Yrast excitation energies Ex of the even and odd multipole unnatural parity states. We quoted these parameter values from Ref. [7]. The last rows in two pannels show the overall χ2 value and total number of data. π J1 2− 1 4− 1 6− 1 8− 1 10− 1 All 1+ 1 3+ 1 5+ 1 7+ 1 9+ 1 All α (MeV) 48.27 75.04 107.98 277.43 238.48 47.13 49.45 87.00 139.02 172.81 γ 0.73 0.81 0.83 1.00 0.90 0.67 0.76 0.83 0.88 0.86 βp (βn ) (MeV) 1.09(1.59) 1.00(1.27) 0.77(1.40) 0.90(1.50) 1.24(1.76) 0.54(0.99) 1.17(1.49) 1.05(1.25) 1.19(1.48) 1.09(1.61) λp (λn ) 0.19(0.31) 0.17(0.24) 0.19(0.28) 0.15(0.20) 0.44(0.25) 0.76(0.50) 0.58(0.32) 0.40(0.24) 0.28(0.24) 0.34(0.46) χ2 0.058 0.027 0.019 0.017 0.012 0.027 0.079 0.051 0.028 0.021 0.019 0.042 N0 246 253 248 230 199 1176 251 236 250 184 159 1080 Table 2. Values of 8 parameters in Eq.(3) for even and odd multipole states. J Even Odd α0 (MeV) 38.48 53.65 a 0.558 0.156 γ0 0.719 0.738 c 0.07 0.018 βp (βn ) (MeV) 0.909(1.529) 1.055(1.47) λp (λn ) 0.464(0.535) 0.868(0.556) χ2 0.035 0.074 N0 1176 1197  l¦ ”  &©É ˺ G © h t 3 “ e. s ‰ .à ×F I\ &6 w  ³œr f æœ x Ç  f á 8 º¼QR Ð ô õ\" 7  ¿×4 ˜“. §    á 8 j úRÐ AŒ ŒŒ  s õ\ 7  [y ¶(˜l 0 # yy_ “  ¦§ ˜ ••  ¯¦ ú °` Fig. 4\ – Õ˜€. #l" 5s ‘ l   Ð ª9Ќ Œf Å  ñ ¤ q H ¼2 > ú B” h  ú¦ Å ۗ\ _” t · /d\ &6ô “°s“, 5s ; r §H N xÇ  ¯ q Í ø ñ ª /£& ¼2 > B” h   ð l  Õ\ @6÷ ۗ _” /d` &6ô “ H x H ; r N¦ xÇ  ¯ ú ¸ ¶ ñ Œº G / ú °s. ¢ô, " l  ‹Ã ×F\ @ô “°s Çé H• æ Ç ¯ 9, W— l  fà ×F\ @ô “°s. s Õ 1¸ ñ .º G / ú  ª HË æ Ç ¯ >¦  Ë Ð /‰h¼Ð  j@ þÉ ¼2 > a` ˜€ @^&ܖ α\ ]üô “[“ ۗ _”  ¦ Ç  tr ; r N B”f > ú B”f >  O /d\" _” t · /d\" Z s \. r §H N   ªQ  hÉ úf ú 9 ú ” Õ ፠&“ J °\" ¸ {u  J °s &f\ H r ¯ H˜ ¯    ú OQ £ Œº G ⺠ s ´s Z#”. :y ‹Ã ×F_ Ä §  ¤• æ − − 8  10 _ Ä\ Õ 9, fà ×F_ Ä 7+ õ   ⺁ ªQ Ë .º G ⺠ æ + _ ⺠ªQ 1 ⺠¸º ¼2 > B” ; r N 9  Ä Õ . W Ä —¿ ۗ _” /d\ f ú ¼; > ú ⺠úÐ /‰ "_ α°s ۗ\ _” t · Ä_ ᰘ @^ ¯ 2 r §H ¯  h¼Ð Œ  - ú ‰h¼Ð A Ð  &ܖ . ፠\t °` „^&ܖ 0 A– s • H ¯¦   x l  i+  £ ª ¼ X/ú aº 1 rv %½` ô. 7, Õ ßl_ ]@°` ýtÄ H ɦ Ç ¤  ¯¦  t 9, Qt “[“ þüy ,| ?\"_ \   þÉ j@Œ 9 /f o  tr •  ¦ ü f ú  o l  G lÕô. " α°s J _ \ 3 pu€ ×F tÇ ¯  w æ   OQ l ÷¦  Ð ªË s Øìy Z#tt 3 > &“, s – Õa\ ær  w H > f ˜p ‰ < A Ÿ Q¼ ¸þ¼ " Ðs1 „^ Xs _ 0A ;s ×#׍ —_Ü w ¤¦ Hv Ð ß –  è. Ref. [9]_ õü q“K˜€, ƒì„$ –  < §Ð øí   Í  ⺠Œº G © - _ Ä ‹Ã ×F I_ \t E2 ü E10 _ \ • æœ H <   t 10C\" ß> 100Ct s  X qK f ¼ H    <  H Í Í$ øø  ⺁ ªßp ¼   ú ìƒì„í_ Ä\ Õë ß> s t · H –u § H   ú úRÐ ª ò ”h¼– ¼ º . α_ °` ¶(˜€ Õ ´õ\ fX&ÜРւ à ¯¦ ˜  ¦  ´  ” Œº øí ⺠ú f Œf e. ‹Ã ƒì„$_ Ä α °s 68\" rK" • Í ¯ • 2300 t &tX s Á 300C s\ K{ô.  <  º9 H H  © © œ œÇ ªQ Œº øø  ⺠ª úÉ f Œ Õ ‹Ã ìƒì„í_ Ä Õ °“ 48\" r • Í Í$ ¯r • f K" 240t– Ôõ 5C &•\ Ôõ . " ۗ Ð   ñ¸  f ¼2 ¦ ¦ ; >¸   ú¦  ú _”• _ 0\ ¾“ s °s 0.5 &•_ a°\ ì rH š ¯ ñ¸ ú ø ¯Í √ √ ò&Q ” £ %÷# e. 7, J a ∼ J sٖ &K 10 ∼ 3C &  ¤ ¼Ð ® o ñ ¸ òõ1 Ò l :  Ëà G •_ ´µ\ Ål 3 l Mës. s .º ×F Ú w H Hf æ f¸ ª/Ð < \"• Õ@–  X, 0.156_ a°s _   H  ú p  ¯ H J _ _” ´õ _ \.  > ò  O r  III. + ÇØ s]  7f øøí G ©\f s ë\" ìƒì„$ ×F I" Yrast HH H Í Í æ œ tu þp - ¼2 > â«B” Œ Œº  [> \t\ ۗ _” +/d\  # ‹Ã  ¦ ; r >N H • æ G .º G © ŒŒ h i øø ×Fõ fà ×F I yy\ &6 %. ìƒì Ë æ œ •• x  Í Í  æ œ H • í G © Œº G .º G ¸º „$ ×F I ‹Ã ×Fõ fà ×F —¿, æ Ë æ øí ©  £ Œº ø © ƒì„$ I\ qK, :y ‹Ã ƒì„í I\ Í œ ¤• Í$ œ •‹˜ Œº •º þf øøí G © ‹Ã-ŒÃ Ù\" ìƒì„$ ×F I_ Yrast [> \t\ @ô ۗ _” +/d· · · – ^¿% 1 Í Í æ œ tu þp - / ¼2 > â«B” Ç ; r >N  x ”ºò p -117- 300 (a) Data 10 250 200 even even J odd odd J α γ 2 150 1 1 100 Excitation Energy Ex (MeV) 50 (b) Spin - Indep. 10 2 βp βn 2 1 1 1 (c) Spin - Depend. 10 0.8 λp λn 0.8 0.6 0.6 0.4 0.4 1 0.2 0.2 2 0 + 4 + 6 + 8 150 + 10 200 + 50 100 250 0 2 4 6 8 10 0 2 4 6 8 10 Mass Number A Multipolarity J Fig. 3. Yrast excitation energies of the lowest even unnatural parity states in even-even nuclei. (a)Measured Yrast excitation energies, (b) the fit with the spinindependent empirical formula, and (c) the fit with spindependent formula. Symbol representations are same as those of Fig. 1. Fig. 4. Six parameters(open symbols) in Eq. (1) and the corresponding parameters(filled symbols) in Eq. (3). Circles(blue in color version) are for even multipoles and squares(red in color version) are for odd multipoles. p cý k P8 ò >  - ߏ 9” â¾ Ð ¼2  qK, \t çs ]e †` ˜“. ۗ _ – t H Ó¦  ; r > Nr > â« B”É  >¸Ð  OQ” ü< ” + /d“ J a _ _”•– s Z#f` lÕ X, r  ¦ t H  3Q ºú  Œº< .º G ⺠ŒŒ %#t tð(a)s ‹Ãü fà ×F_ Ä yy H ¯ • Ë æ •• 0.5558õ 0.156ܖ" Õ °s BÄ . s ƒì„  ¼Ðf ª ú º Œ  ø ¯ • H Í æœ í G © h   3Q ºú  Œº $ ×F I\ &6ô õ %#” tð(a)s ‹Ã xÇ    ¯ • < .º G ⺠ŒŒ ü fà ×F_ Ä yy 2.26õ 1.05\ qK “ Ã Ë æ ••    ŒÉ º •r Ðf ø   ¼2>¸  O£ u–", ƒì„í\ qK ۗ_”• _ \6` Í$ ;r §¦  f øø G ©f _pô. " ìƒì„í ×F I\" Yrast Ç Í Í$ æ œ tu þp - ü < ŒŒ G h  [> \t\ lÕ  X yy_ ×F\ &6 ¦ t H H •• æ x H > N  â« B” ¼; > â«B”Ð - OQ  + /ds ۗ _” +/d˜ \t_ Z# 2 r >N  ¦ ” 8 ú ü¦ ’ ªQ G ©  f`  ¸ lÕô“ . Õ ×F I  ˜ tÇ x æœ   úRÐ   ½O¼Ðf  ¸  s_ ›>\ ¶(˜ ô t ~Zܖ" s r• _ a ¦˜ HÇ Ó H  ”¦ + º ”¼ p e“ ½ à eÜ9, s–Â' ìƒì„$ ×  É  ÐÒ øøí  Í Í æ G © F I_ Yrast [> \t\ &f ¯™–" ۗ œ tu þp - ñ± ¹èÐf ¼2 ¦  H ; r É ª  ¼ l¦ : º ” “ Õ _p ßt 3 “ t` à e. w r ¦  [1] C. F. Weizsacker, Z. Physik, 96, 431 (1935); H. A. Bethe and R. F. Bacher, Rev. Mod. Phys., 8 193, (1936); E. Feenberg, Rev. Mod. Phys., 19 239, (1947). [2] E. Ha and D. Cha, Phys. Rev. C75, 057304 (2007). [3] J.-H. Yoon, E. Ha and D. Cha, J. Phys. G:Nucl. Part. Phys. 34, 2545 (2007). [4] G. Jin, J.-H. Yoon and D. Cha, Nucl. Phys. A 812, 58 (2008). [5] G. Jin, D. Cha and J.-H. Yoon, J. Kor. Phys. Soc. 53, 3483 (2008). [6] G. Jin and J.-H. Yoon, Sae Mulli 56, 496 (2008).  ¨ /<§ é¼Ð ºŸ&þ s ƒ½ “ @†“_ t"ܖ Ã'÷3_m.  H Æ ¶  %v p cŠU YwÔ Øô -118- ÇGt Æ Dü<r hü ô²Óo†t “DÓo”, Volume 60, Number 2, 2010¸ 2 t 4 Z [7] D. Kim, J.-H. Yoon and D.o Cha, to be published in J. Kor. Phys. Soc. (2010). [8] ENSDF database, offered by the National Nuclear Data Center at the Brookhaven National Laboratory, is available on the World Wide Web at http:// www.nndc.bnl.gov/endsf/. [9] G. Jin, J.-H. Yoon and D. Cha, J. Phys. G:Nucl. Part. Phys. 35, 1 (2008). ...
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