Sum rule approach to decoupled giant dipole state

Volume 251, number 1

PHYSICS LETTERS B

8 November 1990

Sum rule approach to decoupled giant dipole state H. S a g a w a a n d M. H o n m a

Departmentof Physics,Facultyof Science, Universityof Tokyo,Hongo 7-3-1,Bunkyo-ku, Tokyo 113,Japan Received 13 July 1990; revised manuscript received 14 August 1990

We show that the loosely bound neutrons in HLi give rise to a decoupled dipole state at the excitation energy Ex= 1.6 MeV exhausting 8% of the TKR sum rule. On the other hand, there are no such low excited states in 9Li. The decoupled dipole state in ~Li enhances the electromagnetic dissociation cross sections on various targets which explain well the observed cross sections at high bombarding energy Elab/A= 800 MeV.

The giant dipole resonance was first observed in the late '40's and has been o f central interest both experimentally [ 1,2 ] and theoretically [ 3 ]. In a microscopic picture, the giant resonance can be described as a coherent superposition o f particle-hole ( p - h ) excitations which exhausts most of the sum rule value [4]. In nuclei with excess neutrons, low excited dipole states ( L D R ) might decouple from the giant dipole state ( G D R ) while upholding their appreciable transition strengths [5]. Some experimental evidences for the L D R have been reported recently in light [2] and heavy nuclei [6], and they are called "pigmy resonances" or "soft giant resonance". The L D R is expected to have more transition strengths in extremely neutron-rich light nuclei like ~Li since the ratio of neutron to proton numbers is bigger than that o f heavy nuclei; the ratio N / Z is 2.7 in ~lLi, while it is 1.5 in 2°Spb. Moreover, it is expected that the nature o f loosely-bound neutrons will increase further the transition strengths o f the L D R [7 ]. Recently, the electromagnetic dissociation ( E M D ) cross sections o f light neutron-rich nuclei have been obtained from the target dependence o f the interaction cross section [ 8 ]. The E M D cross section is attributed to the break-up process after the excitation of dipole states in the projectile [ 39 ]. The measured E M D cross sections o f light neutron-rich nuclei show an enhancement with respect to the results o f the standard treatment [ 8 ], especially for the high-Z targets. This fact suggests the existence o f low excited

soft giant dipole states [ 10-12 ] in neutron-rich nuclei such as ~lLi. Some microscopic calculations have been done to study the L D R in light neutron-rich nuclei using a large scale shell model [ 7 ], the cluster-orbital shell model [1 1 ] and the r a n d o m phase approximation [ 13 ]. The sum rule approach was also discussed in ref. [ 14 ] in relation with the electromagnetic dissociation cross sections. In this paper, we will study the L D R by using the sum rule approach, especially focusing on the molecular type resonance between the core and the loosely-bound neutrons. Using this method, the average excitation energy and the transition strength can be estimated in relation with the separation energy o f the loosely-bound neutrons in a model independent way. Especially, we are free from the mixture o f the center o f mass excitation which is extremely large in low excited dipole states [ 7 ]. The electromagnetic dissociation cross sections due to the L D R will also be discussed in the cases of medium and high energy heavy-ion collisions with llLi projectile. The kth energy weighted sum for a nucleus with mass number A is defined by

S~k)(E1;A)= ~ (hogn)kl(nlO(E1)lO)[ 2 ,

(1)

n

where I/On is the excitation energy and the dipole operator O(E1 ) is given by

O ( E 1 ) = N / ~ e ~ (½-t~),(r,-R) ,

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

(2)

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PHYSICS LETTERSB

including the center of mass correction. The energy weighted sum S o) is rewritten as S ( * ) ( E 1 ; A ) = I ( 0 I [O*, [H, O ] ] 1 0 )

=½<01 [o*, [T,O]]I0),

(3)

where the second equality holds for a momentum and the isospin independent potential. In this case, the value o f S (~) is evaluated to be S(1)(EI;A) -

9 NZ h z 2 4n A ~m e '

(3')

which is known as the classical T h o m a s - R e i c h e Kuhn sum rule. We will now consider two clusters with nucleon numbers (N~, Z~) and (N2, Z2) in a nucleus (N, Z) (see fig. 1 ). The dipole operator (2) is decomposed as

/5-

ZIA2-Z2A1

Or=44 ~ e

A

(R~ - R 2 ) .

(5)

The kinetic energy referred to the center of mass system is decomposed in a similar way as the dipole operator (4), T= ~ (Pi

•

-

-

P / A ) 2 = T1 + T2 + pz 2rn ~-~,

(6)

where T, is the kinetic energy of the a t h cluster referred to its center-of-mass system and Pr is the conjugate m o m e n t u m of the relative coordinate R r = R l - R 2 ; Pr = (AzPI-A1P2)/A, where g is the reduced mass. The energy-weighted sum rule (3) can be decomposed into three parts: S(*) (El; A) = S ( 1 ) ( E I ; A, ) + S ~l) (El; A2) (7)

+S(~)(EI; rel) ,

O(E1 ) =O1 -[-02 + O r ,

(4)

where O~ ( a = 1, 2) is the operator for the a t h cluster with respect to its center of mass: O a ~-

8 November 1990

E (½-tz),(,'i-Ro),

ei

while Or is the dipole operator for the relative motion between the two clusters,

where

S(I)(EI;

9 (Z1A2-Z2AI) r e l ) = 4n AAIA

2

2 h2

2m

e2

(8)

is the sum rule for the dipole states due to the relative motion of the two clusters [ 15 ]. The non-energy-weighted sum rule is simply given by assuming the perfect decoupling of three kinds of excitations:

S(°)(E1)= ~ I ( n l O ( E 1 ) 1 0 ) l 2 n

valence cluster

=~

I(nllO~(E1)10)l 2+~l(nElOE(E1)10)l 2

nl

n2

+ ~ I(nrlOr(E1)[0)l z nr

= < 0 1 0 ~ o , 10>+ + <01Or*Or [ 0 > ,

Fig. 1. Pictorial representation of two clusters in a neutron-rich nucleus. 18

(9)

where [n~) ( a = 1, 2) are the excited states of the a t h cluster and [nr) are those of the relative motion between the two clusters. The assumption of the perfect decoupling might be justified in the case of a nucleus with loosely-bound clustered neutrons as is discussed in the following. The non-energy weighted sum rule for the cluster vibration is given by

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PHYSICS LETTERS B

S ( ° ) ( E 1 ; rel) = (0t Or*Or J0>

3 e2 ( Z I A 2 - Z 2 A I ) 2 A2

-&

× (0[R 2-2Rt.R2+REI0)

.

(10)

We will discard the cross term R~ "RE in the following discussion. This assumption is reasonable in the case of the loosely-bound valence cluster as will be shown later. Thus, we have to calculate SW)(E1; r e l ) = 3 e 2

(ZIA2-Z2A1)

4n

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9Li, there is no significant difference between two calculations. On the other hand, for ~Li, the value S (°) changes appreciably due to rapid increase of the mean square radius of the lp~//-state for decreasing separation energy, as is shown in table 1. The mean excitation energy of the LDR which is given by the ratio ofS(l)/S ¢o)becomes extremely small in the case of ~lLi, especially with the smaller separation energy / S 2 n = 0.1 MeV. Non-zero contribution for the cross term RI.R 2 comes from the exchange two-body matrix element,

2

A2

2

× ( ( 0 1 R ~ I 0 > + ( 0 1 R 210))

~

iel,j¢2

(ijlRl'R2lji).

(11')

(11)

for the non-energy weighted sum rule. The two terms on the RHS of eq. (11 ) are the fluctuations of the center-of-mass coordinates of the two clusters. We will calculate the sum rules for Li-isotopes, which are considered to be composed of a core cluster and a di-neutron. The S ( 1) ( E 1; rel ) and S (°) ( E 1; rel) values for 9Li and ~lLi are listed in table 1. The value S ~1) (El; rel) exhausts 13% for 9Li and 8% for llLi of the total sum rule value S ~) in eq. ( 3 ' ) . The values S(°) are calculated by using the single-particle wave functions of the lp~//-neutron orbit which are constrained to reproduce the experimental separation energy (see below). Since the center-of-mass motion of the core cluster should be on S-state, the value ( R 2 ) is small, and can be estimated reasonably by using the harmonic oscillator wave functions. We adopt two kinds of separation energies: (a) oneneutron separation energy Sin = 1.0 MeV (4.1 MeV), and (b) half of the two-neutron separation energy ½S2n=0.1 MeV (3.1 MeV) for tlLi (9Li) [16]. For

In l l t i , the i index corresponds to the ls~/2-0rbit, while t h e j index means the lp~/2-0rbit. This matrix element is small in 9Li and ~lLi because of the following reasons. The first is that the number of available configurations is very limited, i.e., only one in both nuclei. The second is the small overlap between the 1 s- and lp-wave functions and, on the contrary, the large matrix element ( R 2 ) due to the nature of the loosely-bound valence cluster. Especially, in the case (b) of ~'Li, the cross term ( 1 1 ' ) is calculated to be 0.2 fm 2, while the value ( R 2 ) is 23.1 fm 2. Thus, we can use the formula (11 ) as a reasonable approximation in the case of the loosely-bound valence core. The coupling matrix element between the G D R of the core and the p h - h states of the valence neutrons, which are responsible for the LDR, might be estimated simply by using a short-range interaction

Vlv= Zi
Table 1 Sum rule values for the dipole excitation (LDR) between the two valence neutrons and the core. The mean square radii of the valence neutrons are calculated by imposing two different constraints; (a) the one-neutron separation energy S~,, and (b) the half of the twoneutron separation energy ½S2..

9Li "Li

Separation energy S [MeV]

( r 2) lpt/2 [fm 2 ]

S( ~) [e2MeV fln 2 ]

S (°> [e2fm 2]

h ~ = S ~t )/S (o)

Sl,=4.1 ½S2.=3.1 S I . = 1.0

8.63 9.47 16.8 46.1

4.24 4.24 2.69 2.69

0.501 0.546 0.621 1.666

8.46 7.77 4.33 1.61

½S2,=0.1

[MeV]

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( ( p h - l)J'~= 1- I VIGDR) =/(iv f d r ( ( p h - ' ) [ ~

~zi~(f--I'i)10)

x ( 0 1 Z r~ja(r-rj) lGDR) J

--Kiv f r 2 dr

( - 1x/r~ )jv+j~- 1

(Pll Y1 Hh) (12)

×Rp(r)Rh(r)SPcoR(r) ,

where the transition density ~RGDRcan be obtained from the generalized sum rule method, and is proportional to the derivative of the ground state density [ 17 ]. The coupling matrix element is determined by the overlap of the transition density and the p-h wave functions. The calculated results are shown in table 2, where the coupling strength/(iv is taken to be 370 MeV fm 3. This value reproduces well the excitation energies of GDR in many nuclei. The wave functions of the p-h states are calculated by using the modified H - F potential described before. In general, the coupling matrix elements are very small compared to the diagonal one between the GDR which is given in the last column of table 2. The matrix elements involving (2sl/21pi-/~)J~= 1- are especially small because of the cancellation of the surface and the inner contributions. The quenching of the matrix elements in l~Li is due to the nature of the loosely bound neutrons which prevents a large overlap of the p-h wave function and 8PGDR. The mixing amplitude of GDR and LDR might be determined by the energy difference and the coupling matrix element. The GDR in 7Li is observed at an excitation energy above 15 MeV [ I ] and those in

8 November 1990

other Li-isotopes are expected to be at similar excitation energies [7,13]. The LDR in llLi is very unlikely to mix strongly with the GDR since the coupling matrix element is small (less than 1 MeV) and the energy difference is large (more than 10 MeV). On the other hand, we can expect a significant configuration mixing of the LDR and the GDR in 9Li because of the small energy gap and the larger coupling matrix elements. The EMD cross section is obtained by the integration of the dipole excitation probability P(b) over the impact parameter b [ 9,14 ], OO

0"IMD =

) e(b)2zrb db,

(13)

bo

where the lower limit bo is taken as the sum of the nuclear radii of target and projectile reflecting the strong absorption effect inside nuclei. The excitation probability is written to be P(b) =

16z~Z2e4B(EI, t )~2 9h2vEb2

× ( K , (~)2+ K°(~)2~ ---~],

(14)

where ~= Exb/fi),v is the adiabaticity parameter, Ex is the excitation energy, v is the velocity of the projectile, 7 is the Lorentz dilatation factor and Kn (~) is the modified Bessel function of order n. The calculated EMD cross sections due to the LDR are tabulated in table 3 for the various targets. The strength distribution T(co) of the LDR is given by the gaussian shape

Table 2 The coupling matrix elements between the GDR and the p-h states. The transition density of GDR in eq. (12) is normalized to exhaust the sum rule value of the core assuming Ex= 15 MeV. The wave functions of the p-h states are calculated by using the modified H - F potential adjusted to reproduce the separation energy of the 1P t/2 orbit.

9Li I ILi

20

Separation energy [MeV]

(2Sl/21pi-/~ I VIGDR) [MeV]

( ld3/21pi-/~ I V[GDR) [MeV]

(GDRI VIGDR) [MeV]

Sl~=4.1 ½S2~=3.1 Sin = 1.0 ½S2n=0.1

0.17 0.11 0.02 -0.02

1.30 1.21 0.93 0.76

10.36 9.44

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Table 3 The E M D cross sections induced by the L D R in HLi projectiles with bombarding energies Elab/A = 40 and 800 MeV. The values in the upper rows are obtained by using the lpi/2 neutron wave function which is calculated with the constraint ofS~n = 1.0 MeV, while the values in the lower rows are obtained from the wave function with the constraint of ~$2,=0.1 MeV. The experimental data of the two neutron removal EMD cross sections are taken from ref. [ 8 ]. Target

Be C A1 Cu Pb

o'~LDD R) (b)

a [ ~ ) ") (b)

El'----~b= 4 0 MeV A

~-~-~ = 800 MeV

0.002 0.012 0.005 0.028 0.019 0.119 0.073 0.525 0.381 3.457

0.001 0.002 0.001 0.005 0.006 0.024 0.027 0.111 0.186 0.810

T(o9) = To exp

2F 2

),

~ - ~ = 800 MeV

0 0.21 +0.04 0.89+0.10

( 15 )

where the normalization To is adjusted to reproduce the sum rule values S (°) and S (~) in table 1. The width parameter F is taken to be 0.5 MeV. The calculated results of EMD cross sections for the medium and high energy projectile Etab/A = 40 and 800 MeV are given in table 3, where the constraint for the lpl/2 wave function is taken to be (a) the one neutron separation energy $1, = 1.0 MeV, and (b) the half of the two neutron separation energy ½S2.= 0.1 MeV. Although the energy weighted sum rule value is not affected by the different constraints, the non-energyweighted value in 11Li is changed much by taking different separation energies and so does the average excitation energy of the LDR. As was discussed in refs. [ 8,10,12 ], the EMD cross section is very sensitive to the excitation energy of the dipole state, especially in the low excitation energy region. This is clearly indicated by the calculated EMD cross sections tr~LDl~ R) , which is several times larger in the case (b) than in the case (a) for every target. Since the LDR is understood to arise from the oscillation between the core

8 November 1990

and the center of the two neutrons, the value 0"~LDD R) should be compared with the cross section of the two neutron removal channel tr~fi~]). On the other hand, the cross section due to the G D R is responsible for the remainder of the observed EMD. The enhancement of the EMD cross sections is particularly remarkable for Pb target in both the medium and high energy reactions. In table 3 we also compare the calculated values of the EMD cross sections with the experimental values. Our results of case (b) reproduce well the anomalous enhancement of the 0"~42Dn) values of heavy targets, especially that of Pb. Since we calculate the wave functions in the independent particle model, there is an ambiguity in the selection of the separation energy. We need a more sophisticated wave function which takes into account many-body correlation in order to determine the precise radial form of dineutrons. The shell model [ 12 ] and core-cluster shell model calculations [ 11 ] show also appreciable transition strength in the low energy region below 3 MeV. The shell model dipole strengths give smaller EMD cross sections than the present calculation, while the corecluster calculations show the same enhancement of the EMD cross section as ours because of the strong dipole strength at Ex = 0.7 MeV. In summary, the sum rule approach has been applied to study the excitation energy and the transition strength of the low energy dipole resonance (LDR) in Li-isotopes, which arises from the oscillation of the core and the center of the di-neutron. Our calculation shows that the LDR appears at very low energies [Ex = 1.6 MeV in case (b) ] in l lLi and might continue to have a large portion of the classical sum rule value ( ~ 8%) even when the residual interaction is switched on. On the other hand, in 9Li the LDR has an excitation energy close to the G D R and might be absorbed into the G D R since the repulsive p - h interaction is strong enough to mix these two states. The EMD cross section induced by the LDR is calculated in the medium energy and the high energy projectiles (Etab/A = 40 MeV and 800 MeV). The calculated results are compared with available experimental data of the two-neutron removal EMD cross sections and show a fine agreement.

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We t h a n k W. B e n t z for his critical r e a d i n g o f the m a n u s c r i p t a n d K. Y a z a k i for e n l i g h t e n i n g discussions. T h i s w o r k is s u p p o r t e d financially b y the G r a n t i n - A i d for G e n e r a l S c i e n t i f i c R e s e a r c h (No. 6 3 5 4 0 2 0 6 a n d N o . 0 2 4 0 2 0 0 5 ) by t h e M i n i s t r y o f Edu c a t i o n , S c i e n c e a n d Culture.

References [ 1 ] Proc. on Giant multipole resonance topical Conf., ed. F.E. Bertrand (Oak Ridge, CA, 1979 ). [2] R.E. PyweU, B.L. Berman, J.G. Woodworth, J.W. Jury, K.G. McNeil and M.N. Thompson, Phys. Rev. C 32 ( 1985 ) 384. [ 3 ] See for example, A. van der Woude, Prog. Part. Nucl. Phys., Vol. 18 (1987) 217. [4] A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. II (Benjamine, New York, 1975), Ch. 6. [5 ] M. Harvey and F.C. Khanna, Nucl. Phys. A 221 (1974) 77; R. Mohan, M. Danos and L.C. 'Biedenharn, Phys. Rev. C 3 (1971) 1740; Y. Suzuki, K. Ikeda and H. Sato, Prog. Theor. Phys. 83 (1990) 180.

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[6] M. Igashira, H. Kitazawa, M. Shimizu, H. Komano and N. Yamamuro, Nucl. Phys. A 457 (1986) 301. [7] T. Hoshino, H. Sagawa and A. Arima, preprint (1990), to be published. [8]T. Kobayashi, S. Shimoura, I. Tanihata, K. Katori, K. Matsuta, T. Minamisono, K. Sugimoto, W. MUller, D.L. Olson, T.J.M. Symons and H. Wieman, Phys. Lett. B 232 (1989) 51. [9] C.A. Bertulani and G. Baur, Phys. Rep. 163 (1988) 300. [ 10] P.G. Hansen and B. Jonson, Europhys. Lett. 4 (1987) 409. [I1]Y. Suzuki and Y. Tosaka, Nucl. Phys. A (1990), to be published. [ 12] M. Honma and H. Sagawa, Prog. Theor. Phys. (1990), in press. [ 13 ] G.F. Bertsch and J. Foxwell, Phys. Rev. C 41 (1990) 1300. [ 14] G. Baur, Proc. Intern. Symp. on Heavy ion physics and nuclear astrophysical problems (eds. S. Kubono, M. Ishihara and T. Nomura, Tokyo, 1988) p. 225; G. Baur and C.A. Bertulani, Nucl. Phys. A 482 (1988) 313c. [ 15 ] Y. Alhassid, M. Gai and G.F. Bertsch, Phys. Rev. Lett. 49 (1982) 1482. [ 16 ] A.H. Wapstra and G. Audi, Nucl. Phys. A 432 ( 1985 ) 1. [ 17] T. Suzuki and H. Sagawa, Prog. Theor. Phys. 65 (1981) 565.