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Statistical estimate of the breathing mode energy
Volume 31B, number 3
PHYSICS LETTERS
STATISTICAL
ESTIMATE
OF
THE
2 February 1970
BREATHING
MODE
ENERGY
K. A. B R U E C K N E R Institute f o r Pure and Applied Physical Sciences, University of California, La Jolla, California, USA and M. J. GIANNONI and R. J . LOMBARD Institut de Physique Nucl$aire, Division de Physique Th$orique *, 91-Orsay, France Received 1 October 1969
Monopole vibrations of spherical nuclei, the so-called breathing modes, are described in terms of harmonic oscillations of the nuclear radius. The energy density formalism is used to estimate the excitation energy of the one phonon state. The p u r p o s e of this p a p e r is to p r e s e n t a s i m p l e way in e v a l u a t i n g the b r e a t h i n g mode e n e r gy. The equations of motion of s p h e r i c a l l y s y m m e t r i c c o m p r e s s i o n a l o s c i l l a t i o n s have b e e n s o l v e d by Wal eck a [1] and Woeste [2] f o r the quantized liquid drop. G e n e r a l l y the s o lu ti o n s a r e much m o r e difficult to r e a c h , s p e c i a l l y f o r s y s t e m s having a diffuse s u r f a c e . The p r o b l e m , howe v e r , r e d u c e s c o n s i d e r a b l y if the s y s t e m can be c h a r a c t e r i z e d by a single quantity. In s p h e r i c a l nuclei the r a d i u s , a c t u a l l y its r . m . s , v a l u e , cons t i t u t e s a n at u ral choice. In such a c a s e the b r e a t h i n g m o d e s can be d e s c r i b e d as a r i s i n g f r o m h a r m o n i c v i b r a t i o n s of the r a d i u s R around its e q u i l i b r i u m value R o. Consequently we have R(t) = Ro(1 + o~(t)) with R 2 = fp(a,r)r
2 d3r
(1)
On the o t h er hand it is p o s s i b l e to e x p r e s s the total e n e r g y of the n u c l e u s as a function of R which can be a p p r o x i m a t e d by a T a y l o r expansion in the neighbourhood of Ro: E ( R ) = E ( R o) + ½ a 2 R 2 ( O 2 E / d R 2 ) R = R o
(2)
T h e r e f o r e the h a m i l t o n i a n f o r h a r m o n i c m o n o pole v i b r a t i o n s is e a s i l y w r i t t e n in t e r m s of the dynamical variable a: 1
H = ½Bo 4 2 + ~C o a
2
w h e r e B o = A M R 2 is the m a s s p a r a m e t e r . The e n e r g y of the o n ° p h o n o n s ta te is then given by * Laboratoire Associ6 au C.N.R.S.
(3)
~w = ~JC~o
(4)
The problem remains to calculate the restoring force parameter Co. The energy density formalism provides a practical tool to do it. As shown in a p r e v i o u s p a p e r [3] the total e n e r g y of a s y s t e m of A n u c l e o n s can be e x p r e s s e d as a functional of the d en si t y E[p(r)] :
f[pe (p)
+ ½epp~ c + 2~MT?(Vp)2]d3r
(5)
where E(p) = C k P 213 + B l p + B 2 P 413 +B3PS/3 r e p r e s e n t s the binding e n e r g y p e r p a r t i c l e in nuc l e a r m a t t e r a s a function of the density. The second t e r m g i v e s the Coulomb e n e r g y and the g r a d i e n t c o r r e c t i o n takes c a r e of the s u r f a c e eff e c t s . The ground state of any s p h e r i c a l nucleus is obtained in m i n i m i z i n g (5) under the condition that A = f p ( r ) d 3 r . T h i s may be done e i t h e r v a r i ationally in c h o o s i n g an a p p r o p r i a t e t r i a l function p(r) o r by s o l v i n g the E u l e r - L a g r a n g e equations a s s o c i a t e d with eq. (5), the f o r m e r leading to a second o r d e r d i f f e r e n t i a l equation f o r p(r). If use is m a d e of t r i a l p a r a m e t r i z e d d e n s i t i e s , the function E ( R ) is e a s i l y r e a c h e d in v a r i i n g the p a r a m e t e r s around t h e i r o p t i m al v al u es. In c a s e of the d i f f e r e n t i a l equation it r e q u i r e s the i n t r o duction of a convenient L a g r a n g e m u l t i p l i e r . H o w e v e r an even s i m p l e r p r o c e d u r e e x i s t s . Since eq. (5) is not valid at low density, it is p o s s i b l e , even su i t ab l e, to i n t r o d u ce a cut-off fitting an exponential tail to the distribution. The function E (R) is thus obtained in so l v i n g the d i f f e r e n t i a l equation f o r d i f f e r e n t initial v a l u e s of the density. Both m et h o d s yield equivalent r e s u l t s . 97
Volume 31B, n u m b e r 3
PHYSICS L E T T E R S
T h e p r e s e n t e s t i m a t e i s b a s e d on t h e d i f f e r e n t i a l e q u a t i o n . T h e n u c l e a r m a t t e r p a r t of (5), n a m e l y E(p), i s d e d u c e d f r o m a c a l c u l a t i o n m a d e for variable neutron excess ~ = ( N - Z ) / A in which the Brueckner-Gammel-Thaler potential was u s e d . F o r s a k e of s i m p l i c i t y p r o t o n a n d n e u t r o n d e n s i t i e s a r e a s s u m e d to b e p r o p o r t i o n a l , w h a t is accurate enough for a first estimate. The cutoff i s i n t r o d u c e d a t 10% of t h e c e n t r a l d e n s i t y , t h e r e s u l t s b e i n g r a t h e r i n s e n s i t i v e to t h e c h o i c e of t h i s p o i n t . T h e e n e r g y of t h e b r e a t h i n g m o d e h a s b e e n calculated for few typical nuclei. The results are g i v e n in t a b l e 1. A s e x p e c t e d t h e s e 0 + s t a t e s l i e rather high, somewhat higher than the giant reso n a n c e . Of c o u r s e , t h e r e s u l t s m a y d e p e n d on t h e c h o i c e of t h e p a r a m e t e r 77w h i c h h a s to b e f i t t e d to s o m e e x p e r i m e n t a l d a t a . T h e e n e r g i e s q u o t e d i n t a b l e 1 c o r r e s p o n d to a v a l u e of 77, n a m e l y ~? = = 8, w h i c h y i e l d s c o r r e c t b i n d i n g e n e r g i e s f o r t h e doubly even N = Z nuclei. The calculation has been repeated for three nuclei using a larger v a l u e , ~? = 12, w h i c h i s r e q u i r e d to f i t t h e s u r f a c e e n e r g y c o e f f i c i e n t of t h e s e m i - e m p i r i c a l mass law from the semi-infinite nuclear matter. A s s e e n f r o m t a b l e 2, t h e b r e a t h i n g m o d e e n e r g y is only slightly affected by changes. It w o u l d b e i n t e r e s t i n g to c o m p a r e o u r e s t i m ates with more sofisticated calculations. Oneparticle-one-hole excitations have been considere d b y B l o m q v i s t [4] i n t h e f r a m e w o r k of t h e R . P . A . H i s r e s u l t s a r e s h o w n in t a b l e 2; t h e y a r e l o w e r t h a n o u r s b y a f a c t o r 2. H o w e v e r t h i s d i s c r e p a n c y i s not a s r e l e v a n t a s it c o u l d s e e m , s i n c e t h e r e a r e l a r g e u n c e r t a i n t i e s in Blomqvist's evaluation arising from a renormaliz a t i o n he h a s to i n t r o d u c e f o r t h e e f f e c t i v e m a t r i x e l e m e n t . A t t h i s p o i n t it i s w o r t h w h i l e m e n t i o n i n g a r e s u l t by B r i n k a n d N a s h : in 1 6 0 t h e y have been looking at 0 + states, mixing one-particle-one-hole and two-particle-two-hole excitat i o n s , u s i n g t h e SU3 c l a s s i f i c a t i o n . T h e y f o u n d a n a v e r a g e e x c i t a t i o n e n e r g y of a b o u t 24 M e V , which lies in between our present estimate and the Blomqvist's value. F i n a l l y we w o u l d l i k e to p o i n t out t h a t t h e restoring force parameter C o is nothing but a m e a s u r e of t h e n u c l e a r c o m p r e s s i b i l i t y . F o r v e r y large nuclei, surface effects become negligible
98
2 F e b r u a r y 1970
a n d r e a s o n a b l e e s t i m a t e s of t h e b r e a t h i n g m o d e e n e r g y s h o u l d b e o b t a i n e d in u s i n g n u c l e a r m a t t e r d a t a . A s s u m i n g a n A 113 l a w f o r t h e n u c l e a r r a d i u s we g e t
~i~ ~ ( ] f / r o A t/3) ~
(6)
where r o is deduced from the equilibrium density Table 1 Equilibrium m a s s radius, binding energy and energy of the b r e a t h i n g mode for 77 - 8.
160 40Ca 90Zr ll6Sn 140Ce 208pb
- E / A [MeV] 7.72 8.45 8.45 8.20 8.12 7.23
rM [fm] 2.42 3.08 3.92 4.26 4.53 5.15
~o[MeV] 30.5 29.5 22.0 21.0 19.5 16.5
Table 2 Equilibrium m a s s radius, binding energy and b r e a t h i n g mode energy calculated for ~ = 12. F o r comparison the r e s u l t s of Blomqvist a r e also quoted.
16o 40Ca 208pb
rM
- E/A
2.56 3.21 5.26
6.38 7.43 6.65
~o3 30.0 26.5 15.5
(Blomqvist) 15.0 12.0 8.
a n d K t h e c o m p r e s s i b i l i t y of t h e n u c l e a r m a t t e r . For the Brueckner-Gammel-Thaler potential u s e d in t h i s w o r k , we h a v e ]fw ~ 9 7 / A 113 MeV. T h i s r e s u l t i s v e r y s i m i l a r to t h e o n e o b t a i n e d b y W a l e c k a [1]. T w o of u s (M. J . G i a n n o n i a n d R. J . L o m b a r d ) w o u l d l i k e to e x p r e s s t h e i r t h a n k s to P r o f . D. M. Brink for many useful discussions.
References 1. J. D. Walecka. Phys. Rev. 126 (1962) 653 (in which a m o r e complete list of r e f e r e n c e s can been found, as well as in ref. 4). 2. K. Woeste. Z. Physik 133 (1952) 370. 3. K.A B r u e c k n e r , J . R . Buchler. S . J o r n a and R . J . Lombard, Phys. Rev. 171 (1968) 1188; K. A. B r u e c k n e r , J . R . Buchler, R.C. Clark and R. J. Lombard. to be published. 4. J. Blomqvist. Nucl. Phys. A103 (1967) 644. 5. D. M. Brink, G. F. Nash. Nucl. Phys. 40 (1963) 608.
PHYSICS LETTERS
STATISTICAL
ESTIMATE
OF
THE
2 February 1970
BREATHING
MODE
ENERGY
K. A. B R U E C K N E R Institute f o r Pure and Applied Physical Sciences, University of California, La Jolla, California, USA and M. J. GIANNONI and R. J . LOMBARD Institut de Physique Nucl$aire, Division de Physique Th$orique *, 91-Orsay, France Received 1 October 1969
Monopole vibrations of spherical nuclei, the so-called breathing modes, are described in terms of harmonic oscillations of the nuclear radius. The energy density formalism is used to estimate the excitation energy of the one phonon state. The p u r p o s e of this p a p e r is to p r e s e n t a s i m p l e way in e v a l u a t i n g the b r e a t h i n g mode e n e r gy. The equations of motion of s p h e r i c a l l y s y m m e t r i c c o m p r e s s i o n a l o s c i l l a t i o n s have b e e n s o l v e d by Wal eck a [1] and Woeste [2] f o r the quantized liquid drop. G e n e r a l l y the s o lu ti o n s a r e much m o r e difficult to r e a c h , s p e c i a l l y f o r s y s t e m s having a diffuse s u r f a c e . The p r o b l e m , howe v e r , r e d u c e s c o n s i d e r a b l y if the s y s t e m can be c h a r a c t e r i z e d by a single quantity. In s p h e r i c a l nuclei the r a d i u s , a c t u a l l y its r . m . s , v a l u e , cons t i t u t e s a n at u ral choice. In such a c a s e the b r e a t h i n g m o d e s can be d e s c r i b e d as a r i s i n g f r o m h a r m o n i c v i b r a t i o n s of the r a d i u s R around its e q u i l i b r i u m value R o. Consequently we have R(t) = Ro(1 + o~(t)) with R 2 = fp(a,r)r
2 d3r
(1)
On the o t h er hand it is p o s s i b l e to e x p r e s s the total e n e r g y of the n u c l e u s as a function of R which can be a p p r o x i m a t e d by a T a y l o r expansion in the neighbourhood of Ro: E ( R ) = E ( R o) + ½ a 2 R 2 ( O 2 E / d R 2 ) R = R o
(2)
T h e r e f o r e the h a m i l t o n i a n f o r h a r m o n i c m o n o pole v i b r a t i o n s is e a s i l y w r i t t e n in t e r m s of the dynamical variable a: 1
H = ½Bo 4 2 + ~C o a
2
w h e r e B o = A M R 2 is the m a s s p a r a m e t e r . The e n e r g y of the o n ° p h o n o n s ta te is then given by * Laboratoire Associ6 au C.N.R.S.
(3)
~w = ~JC~o
(4)
The problem remains to calculate the restoring force parameter Co. The energy density formalism provides a practical tool to do it. As shown in a p r e v i o u s p a p e r [3] the total e n e r g y of a s y s t e m of A n u c l e o n s can be e x p r e s s e d as a functional of the d en si t y E[p(r)] :
f[pe (p)
+ ½epp~ c + 2~MT?(Vp)2]d3r
(5)
where E(p) = C k P 213 + B l p + B 2 P 413 +B3PS/3 r e p r e s e n t s the binding e n e r g y p e r p a r t i c l e in nuc l e a r m a t t e r a s a function of the density. The second t e r m g i v e s the Coulomb e n e r g y and the g r a d i e n t c o r r e c t i o n takes c a r e of the s u r f a c e eff e c t s . The ground state of any s p h e r i c a l nucleus is obtained in m i n i m i z i n g (5) under the condition that A = f p ( r ) d 3 r . T h i s may be done e i t h e r v a r i ationally in c h o o s i n g an a p p r o p r i a t e t r i a l function p(r) o r by s o l v i n g the E u l e r - L a g r a n g e equations a s s o c i a t e d with eq. (5), the f o r m e r leading to a second o r d e r d i f f e r e n t i a l equation f o r p(r). If use is m a d e of t r i a l p a r a m e t r i z e d d e n s i t i e s , the function E ( R ) is e a s i l y r e a c h e d in v a r i i n g the p a r a m e t e r s around t h e i r o p t i m al v al u es. In c a s e of the d i f f e r e n t i a l equation it r e q u i r e s the i n t r o duction of a convenient L a g r a n g e m u l t i p l i e r . H o w e v e r an even s i m p l e r p r o c e d u r e e x i s t s . Since eq. (5) is not valid at low density, it is p o s s i b l e , even su i t ab l e, to i n t r o d u ce a cut-off fitting an exponential tail to the distribution. The function E (R) is thus obtained in so l v i n g the d i f f e r e n t i a l equation f o r d i f f e r e n t initial v a l u e s of the density. Both m et h o d s yield equivalent r e s u l t s . 97
Volume 31B, n u m b e r 3
PHYSICS L E T T E R S
T h e p r e s e n t e s t i m a t e i s b a s e d on t h e d i f f e r e n t i a l e q u a t i o n . T h e n u c l e a r m a t t e r p a r t of (5), n a m e l y E(p), i s d e d u c e d f r o m a c a l c u l a t i o n m a d e for variable neutron excess ~ = ( N - Z ) / A in which the Brueckner-Gammel-Thaler potential was u s e d . F o r s a k e of s i m p l i c i t y p r o t o n a n d n e u t r o n d e n s i t i e s a r e a s s u m e d to b e p r o p o r t i o n a l , w h a t is accurate enough for a first estimate. The cutoff i s i n t r o d u c e d a t 10% of t h e c e n t r a l d e n s i t y , t h e r e s u l t s b e i n g r a t h e r i n s e n s i t i v e to t h e c h o i c e of t h i s p o i n t . T h e e n e r g y of t h e b r e a t h i n g m o d e h a s b e e n calculated for few typical nuclei. The results are g i v e n in t a b l e 1. A s e x p e c t e d t h e s e 0 + s t a t e s l i e rather high, somewhat higher than the giant reso n a n c e . Of c o u r s e , t h e r e s u l t s m a y d e p e n d on t h e c h o i c e of t h e p a r a m e t e r 77w h i c h h a s to b e f i t t e d to s o m e e x p e r i m e n t a l d a t a . T h e e n e r g i e s q u o t e d i n t a b l e 1 c o r r e s p o n d to a v a l u e of 77, n a m e l y ~? = = 8, w h i c h y i e l d s c o r r e c t b i n d i n g e n e r g i e s f o r t h e doubly even N = Z nuclei. The calculation has been repeated for three nuclei using a larger v a l u e , ~? = 12, w h i c h i s r e q u i r e d to f i t t h e s u r f a c e e n e r g y c o e f f i c i e n t of t h e s e m i - e m p i r i c a l mass law from the semi-infinite nuclear matter. A s s e e n f r o m t a b l e 2, t h e b r e a t h i n g m o d e e n e r g y is only slightly affected by changes. It w o u l d b e i n t e r e s t i n g to c o m p a r e o u r e s t i m ates with more sofisticated calculations. Oneparticle-one-hole excitations have been considere d b y B l o m q v i s t [4] i n t h e f r a m e w o r k of t h e R . P . A . H i s r e s u l t s a r e s h o w n in t a b l e 2; t h e y a r e l o w e r t h a n o u r s b y a f a c t o r 2. H o w e v e r t h i s d i s c r e p a n c y i s not a s r e l e v a n t a s it c o u l d s e e m , s i n c e t h e r e a r e l a r g e u n c e r t a i n t i e s in Blomqvist's evaluation arising from a renormaliz a t i o n he h a s to i n t r o d u c e f o r t h e e f f e c t i v e m a t r i x e l e m e n t . A t t h i s p o i n t it i s w o r t h w h i l e m e n t i o n i n g a r e s u l t by B r i n k a n d N a s h : in 1 6 0 t h e y have been looking at 0 + states, mixing one-particle-one-hole and two-particle-two-hole excitat i o n s , u s i n g t h e SU3 c l a s s i f i c a t i o n . T h e y f o u n d a n a v e r a g e e x c i t a t i o n e n e r g y of a b o u t 24 M e V , which lies in between our present estimate and the Blomqvist's value. F i n a l l y we w o u l d l i k e to p o i n t out t h a t t h e restoring force parameter C o is nothing but a m e a s u r e of t h e n u c l e a r c o m p r e s s i b i l i t y . F o r v e r y large nuclei, surface effects become negligible
98
2 F e b r u a r y 1970
a n d r e a s o n a b l e e s t i m a t e s of t h e b r e a t h i n g m o d e e n e r g y s h o u l d b e o b t a i n e d in u s i n g n u c l e a r m a t t e r d a t a . A s s u m i n g a n A 113 l a w f o r t h e n u c l e a r r a d i u s we g e t
~i~ ~ ( ] f / r o A t/3) ~
(6)
where r o is deduced from the equilibrium density Table 1 Equilibrium m a s s radius, binding energy and energy of the b r e a t h i n g mode for 77 - 8.
160 40Ca 90Zr ll6Sn 140Ce 208pb
- E / A [MeV] 7.72 8.45 8.45 8.20 8.12 7.23
rM [fm] 2.42 3.08 3.92 4.26 4.53 5.15
~o[MeV] 30.5 29.5 22.0 21.0 19.5 16.5
Table 2 Equilibrium m a s s radius, binding energy and b r e a t h i n g mode energy calculated for ~ = 12. F o r comparison the r e s u l t s of Blomqvist a r e also quoted.
16o 40Ca 208pb
rM
- E/A
2.56 3.21 5.26
6.38 7.43 6.65
~o3 30.0 26.5 15.5
(Blomqvist) 15.0 12.0 8.
a n d K t h e c o m p r e s s i b i l i t y of t h e n u c l e a r m a t t e r . For the Brueckner-Gammel-Thaler potential u s e d in t h i s w o r k , we h a v e ]fw ~ 9 7 / A 113 MeV. T h i s r e s u l t i s v e r y s i m i l a r to t h e o n e o b t a i n e d b y W a l e c k a [1]. T w o of u s (M. J . G i a n n o n i a n d R. J . L o m b a r d ) w o u l d l i k e to e x p r e s s t h e i r t h a n k s to P r o f . D. M. Brink for many useful discussions.
References 1. J. D. Walecka. Phys. Rev. 126 (1962) 653 (in which a m o r e complete list of r e f e r e n c e s can been found, as well as in ref. 4). 2. K. Woeste. Z. Physik 133 (1952) 370. 3. K.A B r u e c k n e r , J . R . Buchler. S . J o r n a and R . J . Lombard, Phys. Rev. 171 (1968) 1188; K. A. B r u e c k n e r , J . R . Buchler, R.C. Clark and R. J. Lombard. to be published. 4. J. Blomqvist. Nucl. Phys. A103 (1967) 644. 5. D. M. Brink, G. F. Nash. Nucl. Phys. 40 (1963) 608.