SYMMETRY ENERGY FOR NUCLEI BEYOND THE STABILITY VALLEY. V.M. Kolomietz and A.I.Sanzhur

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1 SYMMETRY ENERGY FOR NULEI BEYOND THE STBILITY VLLEY V.M. Kolomietz ad.i.sazhu Istitute fo Nuclea Reseach, 368 Kiev, Ukaie We aly the diect vaiatioal method to deive the equatio of state fo fiite uclei withi the stability valley. The exteded Thomas-Femi aoximatio fo the eegy fuctioal with Skyme foces is used. Usig the letodemous exasio fo the ofile ucleo desities, we have studied the euto coat ad the isosi symmety eegy fo euto ich uclei. Usig equatio of state fo the essue, we deive the egio of siodal istability of fiite uclei ad its deedece o the mass umbe, the asymmety aamete ad the Skyme foce aametes. We suggest the ocedue of deivatio of the isosi symmety eegy fom aalysis of the isosi shift of chemical otetial D l = l - l beyod the beta-stability lie. 1. Itoductio The isosi symmety eegy (ISE) is oe of the key elemets of the uclea equatio of state. May static ad dyamic featues of uclei ae sesitive to isosi degee of feedom. The exasio of the uclea stability valley as well as the existece of the isovecto eigevibatios deed sigificatly o the ISE [1,]. Solutio of some oblems of uclea collisios such as the isoscalig effect, the uclea multifagmetatio accomayig by the emissio of asymmetic clustes, the isosi istability of uclei at high temeatues, etc. deeds o ou kowledge of the uclea equatio of state fo the ISE, amely, its deedece o the aticle desity ad the asymmety aamete X = ( N - Z) /. The elated oblem is the ivestigatio of the ucleo edistibutio withi the suface egio of the ucleus, i aticula, the "euto" coat ad the euto excess fo the uclei fa away fom the beta stability lie.. Eegy fuctioal fo asymmetic uclei We will follow the exteded Thomas-Femi aoximatio (ETF) usig the Skyme-tye foce as the effective ucleo-ucleo iteactio. I geeal, the total eegy E of the ucleus is the fuctioal of the ucleo desities () ad () ò ò E = d e () º d e [ (), ()] (1) tot tot ad deeds o the aticle desity ad its gadiets oly. We oit out that the ETF is the oe ossible ealizatio of geeal Hohebeg-Koh theoem i may body oblem [3]. The ukow values of ad ca be evaluated fom the coditio of equilibium. The equilibium coditio ca be witte as a Lagage vaiatioal oblem. Namely, d( E-l N - l Z) =, ()

2 whee the vaiatio with esect to all ossible small chages of ad is assumed. The Lagage multilies l ad l ae the chemical otetials of the eutos ad the otos, esectively, which ae fixed by the coditio that the umbe of aticles is coseved ò N = d (), Z = d (). (3) The total eegy desity e [, ] i Eq. (1) icludes the kietic eegy desity, e [, ], the tot otetial eegy of NN -iteactio, e [, ], ad the oulomb eegy, e [ ], ot ò ki e [, ] = e [, ] + e [, ] + e [ ]. (4) tot ki ot I the famewok of ETF, both eki[, ] ad e ot[, ] deed o the ucleo desities ad, ad its gadiets. Thei exlicit fom is give i Refs. [4] ad [5]. osideig the asymmetic uclei with X = ( N - Z) / = 1, we will itoduce the ew tial desities, amely the total desity = + + ad the euto excess desity = - -. I this ae, we will ot solve the Eule-Lagage equatio () fo self-cosistet desities q ( q=, ) o ±. Istead, we will follow the diect vaiatioal method (DVM) ad assume the desity ofile fuctio ± to be give by a owe of the Femi fuctio as follows Hee, 1 df() + () = +, f(), -() = -, f() - +, D. (5) d [ ] f( ) = éë1+ ex ( - R) / a ùû, (6) the values +, ad -, ae elated to the bulk desity, R is the uclea adius, a is the diffuseess aamete ad D is the aamete of euto ski. Paamete d i Eq. (6) detemies the behavio of the tial fuctios i the uclea suface egio. The ukow vaiatioal aametes Ra ±,,,,D ad d must be deived fom the vaiatioal icile (). The umbe of vaiatioal aametes fo the tial fuctios (5) is educed due to the estictios of Eq. (3). Below we will assume the letodemous coditio a/ R= 1 which educe the vaiatioal oblem to fou ideedet aametes a +,,,D ad d. We oit out that sigificat advatage of the used DVM is the ossibility to establish the equatio of state fo fiite uclei, i.e., deedece of the eegy e ucleo o the mass umbe, the asymmety aamete X ad the bulk desity +,. The aamete D is elated to the umbe, N D, of eutos i suface egio of the ucleus ("euto coat"). Substitutig Eqs. (5) ad (6) ito Eq. (3) ad usig the letodemous exasio, we obtai fo the euto excess N -d - Z the followig exessio

3 4 3 N Z R 4 R +, -» -, + D. (7) 3 3 The fist tem : R o the ight had side of Eq. (7) is due to edistibutio of the euto excess withi the uclea volume while the secod tem NV -,» (8) +, : R is the umbe of eutos withi euto coat N 13 / æ ö 3 ç 4 3 / D ç +, ç 3 è ø» D. (9) Due to the letodemous coditio, the total eegy (1) takes the followig fom of X, -exasio whee ad E ( X ) is the total oulomb eegy. E/ º e = e + e X + e X + E X /, (1) ( ) 1( ) ( ) ( ) e( ) = c + c + c (11) - 13 / - 3 / i i, i, 1 i, Note that we have omitted i Eq. (11) the tem i the total eegy E. The coefficiets ad d. The last tem : -1 which leads to the -ideedet shift c i, j i Eq. (11) ae the fuctios of the aametes a +,,,D : X i Eq. (1) gives ise to the isosi symmety eegy fo stable uclei as well as fo uclei beyod the beta-stability lie. Fo the fixed values of ad X, the basic obe aametes a +,,,D ad d ae evaluated fom the vaiatioal coditios E/ =, = 1,..., 4, (1) whee E/ is take fom Eq. (1) ad we use the followig shot otatios fo the vaiatioal aametes { } = { +,, a,d, d}, = 1,..., 4. The beta-stability lie X = X * ( ) ca be diectly deived fom Eq. (1) by the coditio whee E/ e ( ) - e ( ) X X e ( ) + e ( ) * 1 = Þ ( ) = -, [ ] 3 æ4+, ö 3 / e( ) = e ç è 3 ø log the beta-stability lie, the bidig eegy e aticle is the give by 13 / (13) * * * * * * * E / = e( ) + e1( ) X + e( ) X + E( X )/, (14) whee the ue idex * idicates that the coesodig quatity is detemied by the vaiatioal coditios (1) take fo fixed ad X = X *. Usig Eq. (13) fo a fixed value of, the bidig eegy ca be witte beyod the beta-stability lie as * * I( )( ) ( ) E/ = E / + b, X X - X +D E X /, (15)

4 * * whee D E ( X)/» e ( )( X - X ) ad b (, X) = b ( X) + b ( X) + b ( X). (16) - 13 / - 3 / I I, vol I, suf I, cuv additioal X -deedece i b( X, ) i Eqs. (15) ad (16) occus because, fo each fixed X I i a small viciity of X *, we have to solve the vaiatioal equatios (1) ude the additioal coditios of Eq. (3) ad, cosequetly, the vaiatioal aametes a +,,,D ad d become X - deedet (see also Fig. ). Usig Eq. (15), oe ca establish the imotat elatio fo the chemical otetial l q beyod the beta-stability lie. Namely, fo fixed, we obtai fom Eqs. (15) ad (13) the followig esult E E E * * * * X X éb I X e ù X X N Z Z N X ë û * X, D l = l - l = - = ( - )» 4 (, ) + ( ) ( - ). O the beta-stability lie, oe has fom Eq. (17) that D l =, as it has to be fom the defiitio of the beta-stability lie. (17) 3. Numeical calculatios Fo the fixed mass umbe ad asymmety aamete X, the vaiatio ocedue i Eq. () with esect to all ossible chages of aametes a +,,,D ad d allows us to deive the aametes R ad -, ad the the aticle desity ± of Eq. (5) as well as the bidig eegy E / of Eq. (15). We have efomed calculatios usig the SkM, SLy3b ad SIII foce [5,6,7]. * I Fig. 1 we comae the esults fo the beta stability lie Z = Z ( N) obtaied fom Eq. (13) with the exeimetal data. * Fig.1. Lie of beta stability Z = Z ( N). Solid ad dashed lies ae obtaied fom Eq. (13) fo SkM ad SLy3b sets of Skyme foce aametes; dots ae the exeimetal data.

5 The esults of calculatios of symmety eegy aametes b I, vol, b I, suf ad b I, cuv i Eq. (16) ae show i Table 1. Foce X * b I, vol, MeV b I, suf, MeV b I, cuv, MeV SkM SLy3b SIII Table 1. The esults of the ETF calculatio of the equilibium asymmety aamete X * ad the isosi symmety eegies b I, vol, b I, suf ad b I, cuv i Eq. (16) fo the ucleus with mass umbe = 8 fo thee sets of Skyme foces SIII, SkM ad SLy3b fom Refs. [5,6,7]. Table 1 gives the coefficiets i the viciity of the beta-stability lie. Fo the fixed mass umbe, the values of b I, vol, b I, suf ad b I, cuv i Eq. (16) ae slightly deedet o the asymmety aamete X. This effect is illustated i Fig. fo = 8. Fig.. Deedece of values b I, vol, b I, suf, b I, cuv ad bi Eq. (16) o the euto excess N - Z fo the uclei with fixed mass umbe = 8. The calculatio was efomed with SkM foces. I Fig. 3 we have lotted the X-deedece of the isotoic shift of the chemical otetial, D l = l - l, fo fixed mass umbe = 8. s it see fom Fig. 3, thee is the obvious coelatio betwee calculatio ad exeimetal data. Thus, the quatity exeimetal detemiatio of the value of symmety coefficiet b I fom Eq. (16). D l ca be used fo the

6 Fig. 3. Deedece of isotoic shift of the chemical otetial D l = l - l o the asymmety aamete X fo fixed = 8. Solid ad dashed lies ae obtaied fom Eq. (17) fo SkM ad SLy3b sets of Skyme foce aametes; dots ae the exeimetal data fom [8]. I geeal, the total eegy (1) ca be used to evaluate E = E[, ] beyod the equilibium oit =. I aticula, oe ca evaluate the equatio of state, i.e., the deedece of the essue ( eq) ± ± P o the bulk desity +,. We will deive the essue P as [9] + - E E/ P=- =. V +, X, +, X, (18) Fig. 4. The equatio of state fo the uclei 9 Z, 8 Pb ad uclea matte. The calculatio efomed fo SkM foces. The aea o the left had side of dashed stight lies is the siodal istability egio fo the uclea matte ad the ucleus 8 Pb.

7 Note that the eegy E/ i Eq. (18) must be miimized with esect to a,d ad d fo each fixed +,. I Fig. 4 we have lotted the equatio of state P P +, = ( ) fo two uclei with = 9 ad = 8. I ageemet with Eq. (18), the equilibium coditio fo the goud state of the ucleus at = meas that ( eq) +, +, ( ) ( eq) P +, =. I Fig. 4, the miimum of the essue P( +, ) is located at = ¹. The ucleus is ustable withi the siodal istability egio ( cit ) +, +, <, whee ( cit) +, +, the icomessibility coefficiet K = 9 P( +, ) / +, is egative, K <. I Table, we show the atio / fo the uclea matte ad the fiite uclei fo diffeet Skyme foces. ( cit ) ( eq) +, +, Foce ucl. matte 8 Pb 1 S 9 Z SkM SLy3b SIII ( cit ) ( eq) Table. The esults of the calculatio of the atio / fo thee diffeet foces used i Table 1. +, +, weak -deedece of the citical desity +, ad, cosequetly, of the siodal istability ( cit) egio is maily caused by both the suface tesio ad the oulomb foce which act i oosite diectios. The iteestig oit is that the citical desity ( cit) +, fo the uclea matte exceeds the oe fo fiite uclei. This is due to the fact that the gadiet tems i Eq (4) give the effect o the suface ad lead (without the oulomb iteactio) to a icease i the ucleo desity i the cete of the ucleus ovidig a additioal stabilisatio of fiite uclei with esect to the bulk desity vaiatio. 4. Summay ad oclusios Statig fom the eegy fuctioal of the exteded Thomas-Femi aoximatio ad assumig the effective Skyme-like foces, we have studied the ISE ad the coesodig equatio of state, amely, deedece of the ISE o the mass umbe, the asymmety aamete X ad the isoscala bulk desity +,. Usig the letodemous exasio fo the ofile ucleo desities () ad (), we have established the -deedece fo the uclea chaacteistics elated to the isosi degee of feedom. simle elatio (7) has bee obtaied fo the edistibutio of the euto excess N - Z withi the ucleus. advatage of the diect vaiatioal method, used i this ae, is the ossibility to deive the equatio of state fo fiite uclei, amely, deedece of the bidig eegy E( ±, ) /, the essue P( ±, ), etc. o the bulk desity ±,. The miimum of the essue P( +, ) is located at ( cit ) +, +, = ¹ which deives the egio of the siodal

8 istability < fo fiite uclei. We oited out that the egio of siodal istability is ( cit) +, +, slightly sesitive to the mass umbe ad aeas at the aticle desity +,» 6., ( cit ) sat, NM whee sat, NM is the satuatio desity fo the uclea matte. The citical desity +, is sesitive to the asymmety aamete X ad the aametes of Skyme foces. We have established the elatio (17) betwee the isosi shift of chemical otetial D l = l - l ad the isosi symmety eegy beyod the beta-stability lie. This elatio allowed us to evaluate the symmety eegy ideedetly o the stadad deivatio fom the mass fomula. Moeove such kid of cosideatio efomed fo diffeet mass umbe, ca be used to deive the "exeimetal" values of the -deedet volume, suface ad cuvatue tems i isosi symmety eegy. We have evaluated the cotibutios to the symmety eegy ad cuvatue, b I fom the volume, ( cit) b I b I, vol, suface, b I, suf, b I, cuv, tems. osideig these values beyod the beta-stability lie, see Fig., we have oted that all of them deedet slightly o the asymmety aamete X. Refeeces 1. Rig P., Schuck P. The Nuclea May-Body Poblem. Beli: Sige-Velag, Myes W.D., Swiatecki W.J. veage Nuclea Poeties //. of. Phys Vol. 55. P Koh W., Sham L. J. Self-osistet Equatios Icludig Exchage ad oelatio Effects // Phys. Rev Vol. 14. P Kolomietz V.M., Sazhu.I. Bulk ad suface symmety eegy fo the uclei fa fom the valley of stability // Nuclea Physics ad tomic Eegy 7. No. (). P Back M., Guet., Håkasso H.-B. Selfcosistet semiclassical descitio of aveage uclea oeties a lik betwee micoscoic ad macoscoic models // Phys. Re Vol. 13. P Liu K.-F., Luo H., Ma Z., She Q. Skyme-Ladau aametizatio of effective iteactios (II). // Nucl. Phys Vol P habaat E., Boche P., Haesel P., Meye J., Schaeffe R. Skyme aametizatio fom subuclea to euto sta desities // Nucl. Phys Vol. 67. P Mati M.J. Nuclea Data Sheets fo = 8 // Nucl. Data Sheets 7. Vol. 18. P Ladau L.D., Lifshitz E.M. Statistical Physics Oxfod: Pegamo Pess, 1958.