Nuclear Shell Structure Evolution Theory

Transcription

1 Nulear Shell Struture Evolution Theory Zhengda Wang (1) Xiaobin Wang () Xiaodong Zhang () Xiaohun Wang () (1) Institute of Modern physis Chinese Aademy of SienesLan Zhou P. R. China () Seagate Tehnology 7801 Computer Avenue South Bloomington Minnesota U.S.A () University of Texas M. D. Anderson Caner Center Radiation Physis Department Box 10 Houston Texas U.S.A First draft in Nov 010 1

2 Abstrat The Self-similar-struture shell model (S) omes from the evolution of the onventional shell model () and keeps the energy level of single partile harmoni osillation motion. In single partile motion is the positive harmoni osillation and in S the single partile motion is the negative harmoni osillation. In this paper a nulear evolution equation (NEE) is proposed. NEE desribes the nulear evolution proess from gas state to liquid state and reveals the relations among S and liquid drop model (DM). Based upon S and NEE theory we propose the solution to long-standing problem of nulear shell model single partile s spin-orbit interation energy. We demonstrate that the single partile motion in L normal nulear ground state is the negative harmoni osillation of S [1][][][4] Key words: negative harmoni osillation nulear evolution equation self-similar shell model

3 I. Introdution For nuleus at ground state beause nuleon density saturates at the enter region there is a potential formed at the enter region for the saturated nulear fore. Nulear shell model single partile mean field potential is not the positive harmoni osillator potential in but the 15 negative harmoni osillator potential as in S. For N and O 15 the experimental energy level of the single partile harmoni osillation between 1 and is about: ex hω ( N 15 O 15 ) 7.0MeV [5]. Based upon onventional approah this energy level ould be derived from nulear liquid drop radii R DM : h ω 41A 1/ MeV where 1/ R DM = r A r 1. fm. However for ( N O ) the derived energy 0 0 = level interval ish ω sm ( N 15 O 15 ) 16 MeV muh larger than the experiment ex value. Aording to physis piture the smaller h ω ( N 15 O 15 ) nulei should be formed at a state with radii R(N 15 O 15 ) muh larger than the liquid drop radii R DM (N 15 O 15 ): R(N 15 O 15 ) >> R DM (N 15 O 15 ). The gas state nuleus radius R gas is muh larger than liquid drop radius R DM. For gas state nuleus nulear density is not saturated at the enter region. The nulear fore is not saturated and the single partile mean field potential is well desribed by the positive harmoni ex osillator potential. The experiment observed smaller h ω ( N 15 O 15 )

4 nulei are formed under the gas state ondition aording to piture. ( N 15 O 15 ) nulei originally ome from nulear gas generated by big-bang. ex h ω ( N 15 O 15 ) generated at the gas state is not hanged during nulear evolution liquefation proess. This will be further disussed in later setions. During nulear evolution from gas state to liquid state nuleon density at the enter region inreases. The initial unsaturated nuleon density evolves toward saturation. The nulear shell model single partile mean field potential should be hanged from positive harmoni osillator potential to S negative harmoni osillator potential to desribe nulear evolution proess. Although traditional does not onsider nulear evolution proess it provides the best starting point in the study of nulear shell struture evolution. In the study of nulear evolution the nulear state orresponding to border nuleus is very important. At border nuleus state the shell struture single partile motion hanges from the positive harmoni osillation to the S negative harmoni osillation. The nulear evolution proess before border nuleus is desribed by gas state and the nulear evolution proess after border nuleus is desribed by S liquid state. Border nuleus is the real halo nuleus. In the following we will obtain our onlusions through study of DM S and NEE and verify the theoretial predition on existing experiment measurements. 4

5 II. Self Similar Struture Shell Model (S) and Spin-Orbit Interation Energy Aording to the onventional point of view adding a negative onstant V C to the single partile harmoni osillator potential U os (r) does not hange the mathematial solution of single partile Hamiltonian H 0 sm in. However in S through resaling harmoni osillator Hamiltonian with a negative potential energy V the single partile positive harmoni osillator Hamiltonian is transformed to the negative harmoni osillator Hamiltonian H S 0. This transformation keeps single partile motion energy levels and onfiguration ombinations unhanged. When nuleon density at the enter is not saturated positive harmoni osillation motion is the orret solution for nulear shell struture. When nuleon density saturates at the enter region S negative harmoni osillation motion is the orret solution for nulear shell struture. In the positive harmoni osillation of single partile is desribed by the Hamiltonian H [1] 0. H 0 = H 0 V (1) ( x y z ) h d d d mω H 0 = m dx dy dz () 5

6 6 ) ( ) ( 1 ) ( ) ( ) ( ) ( 0 0 ϕ θ α ψ ϕ θ α ψ ω ϕ θ α ψ ω ϕ θ α ψ r r H n V r V n r H = = = h h () = 1 ) ( hω n V (4) ) ( (0) ) ( 0 ϕ θ α ψ ϕ θ α ψ r r H = (5) V n = ω h 0) ( (6) h ω α m = (7) In S the negative harmoni osillation of single partile is desribed by the Hamiltonian H 0 S []. ( ) ( ) = = = 0 0 S z y x m z d d y d d x d d m z y x m dz d dy d dx d m H H ω ω h h (8) ω ω S = (9) α α = (10) ) ( (0) ) ( 0 ϕ θ α ψ ϕ θ α ψ r r H S S S S = (11) S V n = = ω h (0) 0) ( (1) Conventional neglets V thus ould not obtain the negative

7 harmoni osillation motion as in S. The energy level differenes in S are the same as those of the However S hω in S is different from hω in. This needs to be pointed out speifially. In S the single partile negative osillation irular frequeny S ω depends upon the single partile onfiguration. It is not a onstant as in. This makes nuleon spatial probability distribution more reasonable. The physis piture of S single partile negative harmoni osillation motion in image spae (irip) is similar to the physial piture of positive harmoni osillation motion in real spae (r p). While the real spae orresponds to positive energy spae the image spae orresponds to negative energy spae. S spin-orbit interation energy s L (S ) 1 1 du ( S ) = m r S ( ω h) n = m h ls ( r) r r s L dr l n ls = h for j = l 1/ (14) ( l 1) n ls = h for j = l 1/ (15) where U S ( r) is the single partile negative harmoni osillator potential: os os is: (1) U S os S m( ω ) r = (16) Above formula s L (S ) without fator is derived diretly from the relativity quantum mehanis. This spin-orbit interation formula has 7

8 been suessfully used in desribing atom shell struture. fator omes from the onsideration of a single nulear partile onsisting of three quarks. Three quarks take part in the spin-orbit interation. One single partile orbit orresponds to three quark orbits and one single partile spin orresponds to three quarks spins. Although there is no reason to believe that quantum mehanis spin-orbit interation formula ould not be used in nulear shell model the spin-orbit interation of single partile motion in nulear shell model has not been solved for long time. There are two reasons that formula in is not suessful. Firstly beause single partile mean field is positive harmoni osillator potential with j = l 1/ has positive value and with j = l 1/ has negative value. The sign obtained from s L ( ) is opposite to the experiment value s L (EXP) for normal nulei. The sign of s L ( ) is the same to the sign of eletron orbit oupling energy in atom shell struture. Seondly the onventional s L ( ) formula does not onsider that a single partile (nuleon) onsists of three quarks. fator is not inluded in the onventional spin-orbit interation formula. Thus the alulated 8

9 ( ) value is muh smaller than that of the experiment value (EXP). In S the single partile mean field is negative harmoni osillator potential. The sign between s L (S ) and s L ( ) is exatly opposite. s L (S ) has the same sign as experiment measurement (EXP) for normal nuleus. Also here we onsider that a single nulear partile onsists of three quarks and three quarks take part in the spin-orbit interation diretly. In nulear shell struture the two single partile states of j = l ±1/ orresponds to the two quark states of j = l ±1/ and the quark state has S degeneray. Also ω in S depends on single partile onfiguration the problem of hanging with orbit angular momentum is solved naturally. In the later setion of alulation and disussions it will be shown that alulated s L (S ) fits very well to experiment measurements s L (EXP). This validates the negative harmoni osillation motion in S. S spin-orbit interation energy s L (S ) an be written as: r s L r S = ( ) ( S ) s L (17) S ( ω ) ( S ) = h m h (18) S energy orretion term L (S ) due to ˆL an be written in a 9

10 onventional format: L L ( S ) D ( S ) Lˆ = (19) D L D L ( S p) ( S n) 1 (1 e 1 (1 e = 14 6n = n hωbp ) ( n ) hωbn ) ( n ) (0) (1) where h ω b orresponds to border nuleus single partile positive harmoni osillation energy level. This an be alulated theoretially by nulear evolution equation (NEE) whih will be disussed in details in the next setion. In summary S single partile motion Hamiltonian S H is: H S S H s L l = 0 ( S ) ( S ) () H S ψ S S ( α r θ ϕ) = ( S ) ψ ( α r θ ϕ) () S S ( 0) s L l = ( S ) ( S ) (4) III. Nulear Evolution Equation (NEE) Nulear evolution equation NEE desribes nulear evolution from gas state to liquid state. It reveals the relation among S and DM. Beause the single partile motion veloity is muh larger than the nulear evolution veloity the probability distribution of single partile motion in nulear struture an be treated quasi-statially. The average of kineti energy is equal to the average of potential energy for single 10

11 partile harmoni osillation and is half of the single partile energy. The summation of interating energies among all nuleons in nuleus is equal to a half of nulear binding energy. Nulear binding energy is given by DM model E DM. Following relation holds between single partile energy and DM energy E DM ( A N) : 1 A [ (0) ] s L l = E DM ( A N) 1 (5) E DM ( A N) / 1/ A 1 1/ = av A as A az A aα Z A apδa (6) av = 15.85MeV as = 18. MeV a = 0.714MeV aα = 9.8MeV a p = 11. MeV and δ = 1 for even even nulei = 0 δ for odd nulei and δ = 1 for odd odd nulei. Equation (5) is the nulear evolution equation NEE(). The 1 / fator in NEE () is ounting the repeated summation of interations among all nuleons. In single partile energy (0) has two parts. The first part is the potential well depth V and the other part is the energy assoiated with positive harmoni osillating motion h ω (eq(6)). As NEE shows due to the onserved energy A dereasing in V results an inreasing in h ω. 11

12 This redues the mean square root radius r of single partile motion and inreases nuleon density: r n h = mω (7) NEE() desribes the nulear shell struture evolution proess in gas state in terms of V and h ω. Conventional neglets V term and ould not desribe nulear evolution proess. and DM desribe nulear ground state the lowest energy level of an isolated system. Nulear evolution proess desribed by NEE() is a result of energy onservation. When V dereases to the double volume energy av in DM V = a v the nuleus reahes border nuleus state. The single partile motion in shell model hanges suddey from the positive harmoni osillation to the S the negative harmoni osillation. Border nuleus is at the end of evolution proess and at the start of S evolution proess. In nulear evolution proess from gas state to liquid state the shell single partile mean field potential transforms from the positive harmoni osillator potential to the S negative harmoni osillator 1

13 potential. The reasons for nulear single partile motion to hange suddey from the positive harmoni osillation to the negative harmoni osillation at border nuleus V = a are as follows: (1) Beause the v nuleon density and nulear fore at nulear enter region start to saturate at border nuleus a negative harmoni osillator potential in S starts to form. () NEE() energy relation between and DM for border nuleus an be found as: 1 av A = V A (8) / 1/ A 1 as A az A aα Z A apδa 1 l = ( n ) hωb The depth V of potential well in orresponds to the negative volume energy ( ) of DM. The summation of all single partile harmoni av osillation energies in orresponds to the positive energy summation in DM (i.e. Coulomb energy plus surfae energy plus symmetri energy). If we insisted to desribe nulear evolution proess after border nuleus as gas state in the energy relation in eq (8) and eq (9) between and DM would be destroyed and the ontradition between and DM would be resulted. () As mentioned in the introdution if nulear 1/ (9) 1

14 evolution proess after border nuleus were gas state the h ω would inrease and the alulated value of h ω would not be onsistent EXP with experiment value h ω ( N O ) << hω ( N O ). (4) As disussed in the previous setion of spin orbit interation energy if nulear evolution proess after border nuleus were still gas state the theoretial values of s L ( ) would not be onsistent with experiment measurements s L (EXP). (5) At border nuleus all the energies are onverted to the S negative harmoni osillating energy the absolute value of summation of all single partile positive harmoni osillation energies is about equal to the absolute value of summation of all negative harmoni osillation energies ( h ) hωb and 1. These five reasons are also S n ) ω ( n the reasons why we believe single partile motion in normal nuleus at ground state is the negative harmoni osillation as in S. In onventional VC is set to be zero. There is no nulear evolution equation NEE() and the energy relation between and DM (eq (8) and eq (9)) an not be obtained. In S liquefation proess the following onlusions an be 14

15 obtained: (1) As V dereases the single partile S h ω in S inreases (eq(4) and eq(9)) and the single partile radius dereases (eq(7)). () In S liquefation proess further dereasing V will not hange the orders and intervals between single partile energy levels S (0) left at nuleus border. () In S liquefation proess all the negative energy assoiated with V dereasing are onverted to negative harmoni osillating energies. The onservation of energy for nuleus system requires nulear temperature inreasing to anel the negative energy V dereasing: T = V a ) (0) ( v where the border nuleus temperature is defined to be zero T b =0. The temperature inreasing energy omes from the liberation energy in liquefation proess. The dereasing of V in the liquefation proess results the inreasing of S h ω and T. This ontinues until the nuleus is fully liquefied to form normal nuleus at ground state. The nulear evolution equation NEE (S) orresponding to S liquefation proess is: 1 S [ (0) ] s L l T = EDM ( A N) l= 1 A (1) 15

16 There is the relation between and S: S ( 0) T = (0 ) ( 0 ) is (0) at border nuleus. In the S liquefation proess the energy relation between S and DM is: 1 av A = ( V T ) A () / 1/ A 1 1/ as A az A aα Z A apδa 1 l = ( n hωb ( S ) ( S ) The energy relation between S and DM in the S proess is similar to the energy relation between and DM at border nuleus. The energy relationships eq() and eq() between S and DM show that S and DM are unified. The S is a quantum liquid drop model (QDM). In S proess the nulear temperature T reflets the disorder motion of nuleons in nuleus. This provides insight to understand the physis piture of DM. IV. Calulations and Disussions () We an test nulear evolution equation NEE and S formula through analyzing experiment measurements of single partile levels. For N 15 and O 15 the experiment data of single partile osillation levels are [5] : ex 15 1 ex 15 5 ex 15 1 ( N ) = 0MeV 5 ( N ) = 5.704MeV ( N ) = 7.011MeV 11 16

17 ex 15 1 ex 15 5 ex 15 1 ( O ) = 0MeV 5 ( O ) = 5.409MeV ( O ) = 6.79MeV ex 15 ex 15 1 h ω( N ) = ( N ) 1 ( N ) = 7.MeV 15 ex 15 ex 15 1 h ω( O ) = ( O ) 1 ( O ) = 6.79MeV For N 15 and O 15 h ω b an be alulated by NEE() at border nuleus (V =-a v ) h ω ( N ) = 7. MeV and h ω ( O ) = 7. MeV. The alulated b b 5 ( 15 O 15 ex h N ) is lose to experiment measurement values h ω ( N 15 O 15 ) : ω b ex hω ( N O ) hω ( N O ). This is an important validation to NEE( b S). As pointed out in the introdution onventional predits h ω ( N 15 O 15 ) about two times bigger than experiment measurements ex h ω ( N 15 O 15 ). Single partile spin-orbit interation is a diret test to determine whether single partile motion in normal nuleus is positive harmoni osillation or S negative harmoni osillation. For N ( N ex) = 0.81MeV s L 15 ( N ex) = 1.MeV an be obtained from 5 ex 15 ex 15 5 energy differene ( N ) 5 ( N ). Aording to s L (S ) formula in this paper S h ω = 1.0MeV and root mean square radius r = r =. 1 fm an be derived. Root mean square radius 5 17 r and ex ex 5 are little larger than R DM =.96fm. Beause ( ) and 5 ( ) are single partile exited states at d shell (whih is one shell above p shell) r 5

18 it is reasonable that alulated root mean square radius is a little bit larger than liquid drop radius. For O 15 s L 15 ( O ex) = 0.6MeV 15 S ( O ex) = 0.9MeV h ω = 11.8MeV and r = r =. 57 fm (S ) formula is also very effetive. Table (1 A) shows the evolution proess of O 15 from gas state to liquid state i. e. from to S in nulear shell struture. From V C =-0MeV to V C =-1.7MeV the O 15 nuleus is in gas state and its evolution follows NEE(). At border nuleus (V C =-a v =-1.7MeV) the shell struture single partile motion transforms suddey from positive harmoni osillation to S negative harmoni osillation. 5 5 Border nuleus energy is h ω = 7. MeV and radius is R b = 4. 5 fm. O 15 b 5 border nuleus radius is muh larger than O 15 liquid drop radius R DM =. 96 fm. O 15 border nuleus is the real halo nuleus. When nulear single partile motion hanges at border nuleus state nuleus radius inreases and the ( S ) of single partile orbit P also transforms suddey from negative value to positive value. From V C =-1.7MeV to V C =-48.MeV O 15 nuleus radius shrinks following S evolution liquefation proess. At V C =-48.MeV O 15 nuleus radius equals liquid drop radius and nuleus are in full liquid state and normal O 15 nuleus is formed. O 15 nuleus liquefation proess is from border nuleus to normal nuleus. The temperature inreases from 0 to 16.6MeV. The root mean square radius for outer shell shrinks from 4.47fm to.96fm. In the 18

19 alulation ( A N) E DM values in NEE (S) are obtained from experiment O 15 binding energy. Table (1B) shows the evolution proess of L 11 i. S negative harmoni osillation in nulear evolution proess 11 forms the neutron halo of L i at ground state. Large asymmetry energy of L i stops full liquefation proess in L i and neutron halo is formed as ground state. NEE(S) gives the formation of neutron halo for 11 L i nuleus. Fig(1) shows O 15 single partile harmoni osillation energy levels (0) in evolution proess. dereasing of V C results the inreasing of h ω and broadening of osillating energy level differene. Fig() shows O 15 single partile negative harmoni osillation of in S liquefation proess. The energy level differene S (0) is equal to h ω b As V C dereases in S liquefation proess the energy level differene of S (0) does not hange. IN S evolution from border nuleus to normal nuleus the summation of oulomb energy surfae energy and asymmetry energy is onserved. For light nuleus suh as O 15 and N 15 the energy level differene of h ω b remain about the same. This is due to the fat that energy orretion terms s L (S ) and L (S ) are small. The ex experiment measurements of hω ( N O ) hω ( N O ) is a ex onformation for NEE(S). h ω ( N O ) << hω ( N O ) points to the problem of onventional. The physis piture of onventional b Rdm 19

20 is orret in gas evolution proess but is inorret in S liquefation proess. The total evolution proess to form normal nuleus at ground state is desribed by NEE( S). For normal nuleus at ground state the density of single partile level in S is larger than the density of single partile level in and the energy of outside single partile in is larger than the energy of outside single partile in S. The V C derease in S is larger than the V C derease in so that for nulear system S system is more stable than system aording to the priniple of potential energy minimum. The temperature onept in S is interesting beause it illustrates that nulear motion inludes both deterministi part and stohasti part. S fully inorporate DM piture and the nuleus generated by S is more stable than nuleus in form energy minimization point of view. For heavy nuleus beause nulear fore is independent on harge the V C of NEE ( M) should be the same for for neutron and proton. However in evolution proess proton h ω (P) is larger than neutron h ω (N) due to the large oulomb energy in heavy nuleus. For the same single partile onfiguration h ω ( P) > hω ( N) so that the mean square root radius r (P) of proton is smaller than r (N) of neutron r ( P) < r ( N) in evolution proess. Oy after rossing border nuleus proton h ω S (P) is smaller than neutron h ω S (N) : S S S S h ω ( P) < hω ( N) and r ( P) > r ( N) for the same single partile 0

21 onfiguration. This is a very important result. For heavy nuleus the neutron number N is muh larger than the proton number Z N>>Z and the onfiguration state of neutron is higher than the onfiguration states of proton. Aording to the neutron distribution radius would be muh bigger than proton distribution radius. This is inonsistent with liquid drop model. Oy after S evolution proess proton distribution radius R DM (P) equals neutron distribution radius R DM (N). This reprodues the physis piture of liquid drop model: R DM (P)= R DM (N) = R DM. In DM the hanging of binding energy E DM (AN) with ( AN) is smooth but the hanging of s L (S ) with (AN) is flutuating. E DM (AN) an be orreted by s L (S ) : s C C L EDM ( A N) = EDM ( A N) ( S ) (4) The alulated results of s L (S ) show that the orretion of s L (S ) in E DM (AN) orresponds to the shell orretion of E DM (AN). In the alulation of s L L (S ) beause s ( j = L ±1/ ) of inside partile orbits anel eah other oy outside partile orbits needs to be onsidered. Aording to DM for normal nuleus R DM (P) = R DM (N)= R DM the mean square root radius of outside single partile orbits (neutron and proton) are equal to R DM r ( n) = r ( p) = R. S S DM S ω ( n p) of outside single partile orbits an be derived and 1

22 s L (S ) an be alulated without introduing any additional parameter. Table () is alulated s L (S ) values nulei for ten nulei. In order to investigate the effet of Most of seleted nulei are magi nulei on E DM (AN) most of nulei are seleted from magi nulei. The alulation results show that s L (S ) orretion is the shell orretion. s L (S ) formula is indeed effetive. Based upon h s L l (S ) and (S ) alulation the ω b S distribution of single partile level ( N P) for neutron and proton an S be obtained: T s L l = ( S ) ( S ). Fig () is the single b S distribution of single partile level ( N P) in S. In S (N) of Fig() there is no single partile level between two levels of j=l ± exept n= shell.in S (P) of Fig() there is also not single partile level between two levels of j=l ± exept n= and n= shells. Fig.(4) shows the dependene of total binding energy (E DM (AN) - E exp (AN)) upon neutron number N. It is also s L (S ) versus N urve. The flutuating s L (S ) versus N urve shows that the (S ) of j=l state inreases binding energy to form magi nulei with N= And then the s L (S ) of j=l - state dereases binding energy. Fig. () is onsistent with Fig. (4). The nulear shell struture depends on nulear evolution history. The magi nulei with

23 NP= 8 and 0 formed in evolution proess and The magi nulei with NP= and 184 formed in S liquefation proess.in the onventional the nulear shell struture does no depend on historial proess of nuleus formation so the onventional is not perfet. The single partile onfigurations in and S are same. The deformation shell model orresponding to S and ranking shell model orresponding to S an be easily obtained by introduing the nulear deformation and nulear olletive rotation [4] [6] [7] to S. It is a nature step to reonstrut more omplete theory of nulear shell struture based upon S and NEE in urrent manusript. Finally let us disuss the relationship between nulear evolution and the universe expansion [8 ]. The universe ame from Big Bang and the universe was expanding rapidly. About min after Big Bang the temperature and the nuleon density dropped steadily. When the nuleon density in the universe was muh smaller than one of normal nuleus the short range strong nulear fore attrats nuleons together to form nuleon gas groups. These nuleon gas groups were separated from

24 eah other due to the universe expansion. The nuleon gas groups went through the evolution proess from gas state to liquid state (from proess to S proess) and formed different nulei in the universe. The universe ontinued expanding and the eletromagneti fore ombined nulei and eletrons to form atoms and moleules. Then Gravity attrated matter together to form galaxies stars and planets. Aknowledgement: It is pleasure for us to aknowledge some useful disussions with professor Shuwei Xu. 4

25 Table (1 A) evolution proess of O 15 V(Mev) T (Mev) (Mev) (Mev) (Mev) / (Mev) / r 11(fm) / r oo(fm) / =-av=-1.7mev RDM(O 15 )=.96fm Eexp(O 15 )=11.0MeV 5

26 Table (1 B) evolution proess of V(Mev) T (Mev) (Mev) (Mev) (Mev) / (Mev) / r 11(fm) / r oo(fm) / =-av=-1.7mev Eexp( )=45.54MeV 6

27 Table ()alulation values of ) for nuleis ZX N 9U 18 8Pb 16 7Hf 11 58Ce 9 54Xe 8 58Ce 7 40Zr 58 8Sr 50 6Fe 8 0Ca

28 Referenes (1)M.G.Mayer and J.H.D.JensenElementary Theory of Nulear shell straturewileynew York1955. ()Zheng-Da Wangxiao-Dong Zhangxiao-xhun Wangxiao-Bin WangZphys1996A56() 55-58;Commun.Theor phys.7 (1997)PP <high energy physis and Nulear physis >Vol. No.4 apr ()V.S.Shirley and G.M.lederer (ed)nulear Wallet Gards Lawrene Berkeley Laboratory1979. (4)A.Bohr and B.R.Mottelson Nulear struture VolⅠVolⅡ(1975). (5)F.A.Jzenberg-Selave Nulear physisa5i (6)Sven G sta Nilsson Mat.Fys.Medd.Dan.Vid.Selsk. 9.no.16(1955)(ed.1960) (7)R. Bengtsson Y.S.hen J.Y.Zhang and S berg Nul.phys.A405(198)1. (8)David Halliday and Robert Resnik <Fumdamntals of physis> Third Edition Extended Volume Two 1988.P s 8

29 9

30 0

31 1

32