Augusthy, A

Rajput, M S

English

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M »S IU jput and  a . A coustiiyDeparl'inent of Phyaics, AUgaih Muslim Un/iversity Aligarh {U.P.), India{Received 19 December 1909—Revised 31 March 1970)E_, transition piobabilitips for 2^ ' —► 0+ giounrl atato tranaitions of ovoii-evon doformoi] niirloi havo Imon ralmlatod u.siiig Dav'pydov and liostovakv (DH) oatimatos. Tho pjodictions of this inotlol have bpfii (’ oinpniud with tho oxponmontal data. From this eoinjianson, it has b(-vn found tliat tho D ll UfltnnatnH aio cloaor to tho oxpon-niontal ratoa. Tho variations of tho I’atios and nonaxialityparamotor ‘yp’ of Davydov and Filippov theory which is a Tnc'asuro o f the degree ol depailuro fium tho axial syinmotiy, have boon studied.Tmluin J. Phys 44, 1^8-203 (1970)E 2 tra n s itio n  p r o b a b ilit ie s  in  e v e n  n u c le iI nteodtjcttonSystcmaticB of transition pi'obabilitios for 2'^ —> 0 ' ground state transitiouR in evon-evon deformed nuclei in ilio mass region 150 ^  A ^  200 and A 230 have been studied by several authiirs. No definite correlation has been found between the experimental and theoretical transition probabilities Several attempts by Alder ei al (1956), Coleman (1957), McGowan & Stelsou (1901), Curio (1962), Kajput & Schgal (1967), Schwarz child (1966) and Rajput (1970) have been made to explain the fastness of these transitions. The various models proposed to explain tho fastness ol'these transitions have confirmed their eolleetivo behaviour. The most generally used single particle model predictions have shown that the experimental transition probabilities are larger by a factor ranging from 30 to 3Q0 in the a])ovc mass region Tn this work wo have studied the variations of the ratios of experimental transition probabilities to those theoretically calculated on tho basis of the single particle, (Blatt & Weiskopf 1952) as well as the noiiaxial collective miKlels (Davydov & Rostovsky 1904), with the nonaxiality parameter of Davydov and Filippov theory (1958). Davydov and Fillippov have suggesf-od a model for deformed nuclei in which they assume that tho rotation of the nucleus takes place without change of fhc intiinsic slate. The oijuililirjum shape of the nucleus is like a triaxial cllipsoiil and is determined by the two parameters /?q and where /?q is the deforma­tion parameter and y  ^ is the nonaxiality parameter. When there is a devia­tion from axial symmetry, it lowers one of the pi'iiicipal moraoiits of inertia and incT'eases the other The nuohnis rotates about the axis with the largest momf'.'iit of inertia.198transition probabilities in even nuclei 199Davydov and Rostovsky (19G4) have treated the problem of collective excitations by taking into account the intet actions of rotations with ft and y  vibra­tions. Tlie excited states generated in rotation duo to the quadriipole vibrations of the nuclear surface are called y  vibrations. When the nucleus passes into tlie excited states, the shape of the nucleus changes. This increase in deformation mcroasos the moment of inertia and hence causes the (sentrifugal stretching. iSiich vibrations are called /?-vibrations.C a l c u l a t i o n  or  T r a n s i t i o n  X^ r o b a b t l i t i e sDavydov ami Rostovsky (1964) have derived tlu‘ expiessions for the E,^  iraiisition probalulitios from one (;c>]lcctiv(^  state i,o tlie othoj' in terms of the usual parameters of DF theory (Dav\ dov & Fi]lipi)ov 19o8)). The reduced transition probability for transitions from tlie ground state to the first 2 i- state of ground state rotational band is given by.0+-> 2 '■) = Wm \2*Sf \2 ■ 3f/ / .. (1)where ... (2) ‘^222^0.. (3)q ^ ••• (1) and Uq — 1 2xJ0"i=JxAi''»cm ... (5)Hero is idle energy of 2^  state of ground state rok t^ ioual band, the energy ol 2  ^ stalte oF the y-vibrational band and Eqp is the energj  ^ of 0+ state of the fi- vibrational band. is the deformation parameter The effective values of these parameters have boon calculated from the expeiimentally measured energy levads. The equilibrium deformations used in the calculations were taken from the work of Bes & Szjunanski (1901) corrected by Marshaok & Rasmussen (U)f>Ii). Their values wmro obtained using Nilson levels and including pairing corroliitions and arc m good agreement witli experimental ones Using these r(‘latimis the transition probabilities have been calculated in 20 cases The single liarticle reduced transition probabilities B(E.u 0 '‘ - > 2 )^^ 7:> have been calculated using the relationB{E ,^ 0 2 ■^) e® X 10“ *^“ cm^ (0)The experimental transition probabilities have boon compiled from Nuclear Data (1905). In some cases, the transition probabilities have been calculated Irom measurements where direct measurements are not available.200 M. S. R a jp u t  a n d  A . A iig u sth yR e s u l t s  a n d  D i s c u s s i o nTablo ] gives the values of the parameters used in the calculations. The column 7 gives the values of transiion probabilities calculated using equation 1. We define the factoi'S Fjjjt and Fsp as follow s :For -Fst> —B{E )^ exp ~B(E.^DRB{E.,) expBTEJSPAlder cl al (105G) studied the systematics of E.^  transition lU'obabilities and pointed out that the interactions of the rotational and vibrational bands ai-e responsible for tlu^  cmhancciment of these probabilities. Nathan & Nilson (1965) furtlier emphasized the occurrence of collective vibrations and theirrelation with shell structure. The ratio being always greater than 1,confirms the fact that the first 2^  states are largely collective Several attempts (Alder ct al 1950, Coleman 1957, McGowan & Stelson 1901 and Curie 1962) have been madcj to explain the collective biihaviour of these transitions No definite correlation was found between the experimental data and the propq ?^ed systematics. Rajput & 8ehgal (1967) found that the enhancement factors of the transitions for 2+ to 0+ state decreases gradually with the increase in the value of non-axiality parameter of Davydov & Rilhppov (1958) theory. Davydov & Rostovsky (1964) have reviewed the DF theory and calculated the transition probabilities taking into account fully the collective excitations of all types.Fxperimental values of transition iirobabihties are compared with Davydov & Rostovsky (1964) and single partit;le estimates (Blatt & Weisskopf 1962). The theoretical values calculated using DR theory are closer to the experimental values than the single paiticle values This indicates the superiority of DR estimates over the single particle estimates.Further interesting information is obtained when the factors Ej)ji and Fg plotted against the nonaxiality parameter y,, of DF theory. It w^ as reported caidier by Rajput & Sehgal (1907) that the factor F^p decreases gradually with increase in value of the nonaxiality parameter. Rut in this work wo have ob­served that the factor incjcases slowly with the increase in the value of the nonaxiality parameter. The variations of Fj^ j^  and F^p with nonaxiality para­meter y^  are shown in figui“e 1. From this figure it is clear that the DR estimates are better appj oximations than the 8P estimates; the two factors follow completely different trends. The variation of the factor Fpp with neutron number is shown in figure 2. The factor increases as the magic number is approached and again decreases as we go away from the magic number.E-2, tran sition  probabilities in  even nuclei 20 1Fif?ure 1. Vaiiation oJ and with nunaxiality [larainoiei yuFigure 2. Variation o f  with neutron number N.202 M. S. R a jp u t  and  A . A u g u sth yc3HfsS-xI3Q (M^  COBpq/fees*-■ CJ5 ^M  lO ®  CD COCO d  eo o  -H05 lO «i tH CO U3i - i e o o 5 ( NC5 0 - ^ o oi-(i-t i-l(M «N eO N r-li-tCO CO i-t 05 05 eo GO-*) tJH 05 05 I* lO I> tH CO l> »> »0Tti 1—I lo inN (M (M (MII -H H-l -II Hi -H -H HH HI HH H-l +1 -H H-i H-1 -H “H H-l -H H-l +1 HI «  o  “  ® ® «  «  s  5? 22 2 2 2S !> s  ^CO L' O  CO 05 Tt< cq I-I CD CO•-H rH (N (N fisto 00 (N m CO © 00 CM © o O (£) ©©© © ©i> CO © © CM © i " in(M CQ CA CM (N CM © CO (M T f CO CM CO CM eoH I H I H-i H I H I HH 41 H I©^H I c+l H I 4 ) H i 41 41 41 -H 41 41 sTtl001 lo CO -HI o CO © CM Ih (M CM CO 00 © t " 00 § © in om CO Til CO 4 4 lO Til © © © © 00 uo dp-i 4 CO »o CO u i Til ©< N ( M o o e o > 0 ( N ^ e o « D>n ri t~ o iM <5^i - C , - l t ,  l ' . C O * « C » 5( M r - l , _ ( CO OO (M00 CO C5 i.'to  o  05 in m 101—)i—i(-(i—I r H O O O O O o ^ O O i —l<M<M o o o lo >n o inCO (M m fN CO IMo CO o inf   I  I-H (M O o o O-H HH H-l HH HI -H i l  II -H II Hi -H Hi Hi 4 i Hi -4^  -II Hi HI 41 -fi 41 -HO^Q0C0C0OG0c0t ' -05l0C0 10»n”1'I^C0C‘1 ~ ~  ~  ^ ~ ^THococDONt~-^inco. -HcDr ' in®5oiOr- (  co-^coTti ioininin' ^Ttt ’»tco(M(N'^<^' —It"o o o o o o o o o © Q O O O O O O O O O O ® 0O CO M o 2 2 ® o o i n o o o o o N c > o o  coo'^iioooC'5 0' ' i i05ioioe>|S*5oi o c o ‘n i - i o i c o i 3 o c o i M c n i - i r H f - H o o c o c o T t i ' « * i " - i 4 c o ( r f l ' ^ oOl o OOOOpO, - . « M, - . ( M^05  00 05^ 0©>rtl|"C' l ' "ia)iNFH005(»05 o  O  O  O  ®  p-^1 00 <N l"co- tQOmoo505. - ( (Mt>, ' (Moo^(MrHi4o- l incoiooico^f f^^ I-Hi-H fH>-I i-li-l|^pii4.(MC>]0)2 S l I S S S 5 2 w S o o p - i i o Q t ) - H i n . - i i o ) O i n ' H < ' ^ ^ iiHOi-HCiOO OOO» - Hp Hi - i , Hi - i COCOr HOOO©Op oo d o o o d d o o d o d d o d d d d d o © d ® oH! ! 3“ 2 ® m S S “ p i N< i i i g D- # ( p g g ]I I  s s s  ^a & I: I  a I s  s '£^2 t r c L f i s i tw u  p T o b d h ' i / l i t ie s  i n  e v B n  n u c l e iA o k n  o w l b d o e m e n tmThe authors are very thankful to Dr. M. L. Schgal for helpful discussions and encouragement. They are also thankful to Prof. Rais Ahmed for his interest in this work, and to the Council of Scientific and Industrial Research, iSfeAv Delhi, ibr financial assistance.R e f e r e n c e sAKlcir K., Bhor A.. Huns Moifcelyon B .  &  Winthor A. lU.'iO Ifa v . M o d .  P h y s .  28 432.Hus D. Jl. ifc SzyinauHki Z. J9(i] N u c lm r  P k y t,n H  28, 42.Jilatt J. M «& Woiaakopf V. F’. 1952 T h c o r v tk a l  N u c le a r  r h y .k c s  John Wiloy Rons, NowYoikColomau C F. 19.57 N u c le a r  P in /.'iics 7, 4SS.Ciii'io W M 1!U)2 N u c le a r  P l iy s ir / i  32, C74Jlawchiv A S & K()4tf)vkfiy V. R. ItUM N u c lc a i  P h y s ic s  60, 529 |)avy(-lo^  A iS it Filippov 0  F 1958 N u c le a r  P h y s i c s  8, 237 (h'ldziuH L. 19f)2 P h y'^ics le t te r s  2, 88I'lmk F ( t ,  I’utMk B . J .&  Miholioli .J. . 1960 P h y  R ev . 141, 1200.Mai.iluilck F K, & BusmunHeu J (). 1903 N u c le w  P h y s i c s  43, 438 ]\U't5o\Miu F. K  &  RteLsiiii P H ]9(iJ P h i/s .  R co . 132, 1274Nalihaii O <& JSlilson R. G. 19G5 o i- f i-y -r a y  tS p ecIro sco p y . Jklilorl by K Siegbnhii. N. H. Publ. Co., Aiubtoiclain.I^ucleur D a ta  1905. Acadomic PinsiH. Now Yovk.KajfiiiL M. R &  Rohgtt) M. L 1967 J  P h y  f io c . J a p a n  23, 917 Kajpai M R. 1970 N u o v o  (J im c n to  66A, 343..SolwuiizchiJa A. 1966 P h y s .  P c i \  141, 1214.

E2-Transition Probabilities in Even Nuclei

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/2860/1/E2%20transition%20probabilities%20in%20even%20nuclei_M%20S%20Rajput.pdf

E2-Transition Probabilities in Even Nuclei

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