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A study of root mean square radius series
Volume 81B, number 3, 4
PHYSICS LETTERS
26 February 1979
A STUDY OF ROOT MEAN SQUARE RADIUS SERIES H.A. MAVROMATIS
Michigan State University, East Lansing, MI, USA and American University, Beirut, Lebanon and M.A. JADID
American University, Beirut, Lebanon Received 26 October 1978
Root mean square radius (rms) series are calculated for 4°Ca as a function of b, the oscillator size parameter. Comparisons are made with diagonalization, inversion results and experiment. Higher order terms are shown to reduce the sensitivity of the series to b and, though the series are not convergent, results to third order agree quite well with the "exact " inversion results.
The rms operator for an A nucleon system is given by A
O' =-~1 i ~ (ri_rcm) 2 = 0 - - r £
,
(1)
where the one-body operator 0 = ( l / A ) Z A l r / 2 , and rcm is the center o f mass (cm) vector. If one works with the oscillator basis and adds a cm oscillator potential 1/2Amoo2r2cm to the hamiltonian to localize the cm motion one has an effective hamiltonian H = Hos + V, where the residual interaction A
V=V-1/ernw2(i~=lr2-Ar2)
= V-~moo2AO '.
Both Hos and V depend on the oscillator size parameter b = (7~/mw')ll 2 , while V also depends on the particular interaction V one uses. The present calculation employs b o t h the original Sussex Matrix Elements (SME) given in ref. [1 ] and the saturating SME o f ref. [2]. The former interaction gives reasonable resuits for ground state energies etc. using a correct density imposed b y a suitable choice of b, but does not
give a maximum in the first order binding energy at this density. The latter involves, in addition to the original SME, a Gc term obtained from a hard core potential, where the choice of core radius c = 0.3 is discussed in ref. [2] (c = 0 corresponding to the original SME). To zero order in V the rms operator for the ground states (g) o f closed shell nuclei has the value 0'g(°) =(010 - rcm 2 10) = 000 ,
---~
(2n h + lh + 3/2) - 362
2A' '
(2)
where the sum is over the relevant holes. Thus (0'g (0))1/2 equals approximately 1.72 b for calcium, and depends linearly on the size parameter b. To zero order one gets roughtly the experimental rms radius [3] for this nucleus at b = 2. Higher order terms in the rms series may be obtained directly from the energy series using essentially the "linear replacement m e t h o d " o f refs. [4,5]. This involves substituting V -+ V + aO' and differentiating the resulting energy expressions with respect to a. Thus
273
Volume 81B, number 3, 4 0'g (n)(~) = oE(n+l *~g
PHYSICS LETTERS (3)
)(T + ~0')/~1~=0,
where 0'g(n) is the nth order term in the 0' series and E(n+l) the (n + l)th order term in the energy series. In the present calculation we set up the energy matrix for 4°Ca in the space of 0 and all 2hco excitations i.e. use a 176 dimensional space. Schematically (H) = H)O I
+
V02
~10
g l l - TOO --a
V12
~20
V21
+
V22 - VO0 -
+
0
-G
1
1(~)/
(4)
where a = -2hco, I is the identity matrix and the subscripts 0, 1 and 2 stand for 0p~3h, l p - l h and 2 p - 2 h states, respectively. We also calculate the corresponding matrix of 0,
0ooi+
0
001
010
011
0
0
0 -
000
0
+2
Too +i])O]o ,-2
+ To,(O,] - 00o+,/)
,]-
a-2
- Too+,,+)°+o a-3
(7)
E t (V) = v + ( V ) G ( T > ( T ) ,
(8)
where v, v+ and G - 1 are defined in expression (4). An analogous linked expansion for the 0 operator, the summation of expression (7) to all orders in the space of 0 + 2hco excitations, may be obtained using the linear replacement method. Thus (9)
Using the matrix identity OM/O¢~= -M(OM -1/Ou) X M, the relations OG -1 (V + c~0)0~ = -~', 0v(V + a0)0a = ~, where f and r7 are defined in expression (5) and the fact that 0 and T are hermitian operators one obtains
EL(V ) : TOO + goiTio a-1
+ Voi(Vi] - Voo6i]) V]oa - 2
(6)
+
....
- T00+; ) T 0a -3 +
An inversion technique analogous to the the Q box approach of ref. [4] yields the exact linked diagram expansion result for the binding energy of a closed shell nucleus in the space of 0 + 2hco excitations [6]. This corresponds to summing expression (6) to all orders with the result
OL(V ) = OEL (V + c~O)/O~I u=O.
022 -- 000
With these two matrices one can obtain any order of the rms series where for all orders beyond zero order 0' and the simpler operator 0 yield identical results. Thus 0g(1)(V) : 0~1)(~) : 2~oiOio/a , where the intermediate states are summed over, and the Voi'S and Oio'S are entries in the matrices (4) and (5), etc. To evaluate only the linked graphs for the rms operator in this space [6] one includes the counterparts of the leading terms in the Rayleigh-Schr6dinger energy series namely:
274
0L(V) = 000 + 2VoiOio a-1
+ 2Voi(0/] - 0006i]) (VlTc - ~ 0 6 j k ) Vko a-3 + ....
I,
;(3)
-=fi00I+ ~
(0) =
Using the linear replacement method the corresponding linked rms series is
+ 2Voi( VO1
26 February 1979
OL(V ) = 2 , ¢ G ( T ) v ( V ) + v + ( T ) G ( T ) ~ G ( T ) v ( T ) .
(lO) Numerical results for the rms radius of Ca are plotted in fig. 1 as a function of b both for c = 0 (unsaturating SME) and c = 3 (saturating SME). One notes from these figures that with the unsaturating SME higher order terms in the rms series do not oppose the collapse of this nucleus and only for the very small rms radius of ~2.6 fm are higher order terms in the rms series small compared to the zero order term. On the other hand while the saturating SME higher orders counteract this tendency (i.e. give a repulsive rms contribution) below b = 1.8 and around b = 1.8 (i.e. a radius of ~3 fm) higher order terms in the rms series are negligible compared to the zero-order term. As
Volume 81B, number 3, 4
26 February 1979
PHYSICS LETTERS
Table 1 Exact, diagonalization and series results for the rms operator in fm 2 . c
b
Zero a)
First
Second
Third
Fourth
Fifth
To Third
To Fifth
Diag.
Exact
0.3 0 0.3 0 0.3 0
2 2 1.8 1.8 1.5 1.5
11.85 11.85 9.60 9.60 6.67 6.67
-1.51 -2.92 -0.07 -l.53 1.51 0.05
-0.32 -0.57 0.07 -0.20 0.43 0.10
-0.21 -0.25 -0.09 -0.18 0.00 -0.12
0.08 0.15 0.12 0.17 0.16 0.23
-0.14 -0.17 -0.17 -0.25 -0.24 -0.43
9.80 8.11 9.51 7.69 8.61 6.70
9.75 8.09 9.46 7.61 8.53 6.50
11.09 10.96 9.57 8.96 7.23 6.78
9.82 8.22 9.56 7.76 8.69 6.73
a) 0~o
=
0oo
3b2/2A.
-
3.5(
* = EXACT
RESULTS
(o)
/~ol . ~
1
C =0
higher orders are included in the rms series the nuclear size b e c o m e s as it should less sensitive to the choice one takes for b. Higher lying excitations m a k e this dep e n d e n c e even less [7]. S o m e exact results (obtained using expressions (10)) are also shown in fig. 1. Table 1 lists these, and the corresponding rms series, as well as diagonalization results for comparison. The rms series (except for b = 1.8, c = 0.3 and b = 1.5, c = 0 where higher orders are small) appear to converge by 3rd order b u t in fact do not. A g r e e m e n t with the exact results is best if one stops while the series is still m o n o t o n i c (i.e. b y third order). Comparison o f the diagonalization and exact linked results shows that the unlinked graphs have no definite sign, and are small only w h e n the higher order rms terms are small.
References
2.501 1.5
I 1.6
i 1.7
1 I.e
I 1,9
I
2.0
b (fro)
Fig. 1. Zeroth, and zeroth plus higher-order terms for the rms operator as a function of b. Exact results are indicated with a star.
[1 ] J.P. Elliot, A.D. Jackson, H.A. Mavromatis, E.A. Sanderson and B. Singh, Nucl. Phys. A121 (1968) 241. [2] E.A. Sanderson, J.P. Elliot, H.A. Mavromatis and B. Singh, Nucl. Phys. A219 (1974) 190. [3] H.A. Bethe, Ann. Rev. Nucl. Sci. 21 (1971) 93. [4] T.T.S. Kuo, S.Y. Lee and K.F. Ratcliff, Nucl. Phys. A176 (1971) 65. [5] I. Lindgren, J. Phys. B7 (1974) 2441. [6] H.A. Mavromatis, Nucl. Phys. A295 (1978) 269. [7] M.A. Jadid and H.A. Mavromatis, to be published.
275
PHYSICS LETTERS
26 February 1979
A STUDY OF ROOT MEAN SQUARE RADIUS SERIES H.A. MAVROMATIS
Michigan State University, East Lansing, MI, USA and American University, Beirut, Lebanon and M.A. JADID
American University, Beirut, Lebanon Received 26 October 1978
Root mean square radius (rms) series are calculated for 4°Ca as a function of b, the oscillator size parameter. Comparisons are made with diagonalization, inversion results and experiment. Higher order terms are shown to reduce the sensitivity of the series to b and, though the series are not convergent, results to third order agree quite well with the "exact " inversion results.
The rms operator for an A nucleon system is given by A
O' =-~1 i ~ (ri_rcm) 2 = 0 - - r £
,
(1)
where the one-body operator 0 = ( l / A ) Z A l r / 2 , and rcm is the center o f mass (cm) vector. If one works with the oscillator basis and adds a cm oscillator potential 1/2Amoo2r2cm to the hamiltonian to localize the cm motion one has an effective hamiltonian H = Hos + V, where the residual interaction A
V=V-1/ernw2(i~=lr2-Ar2)
= V-~moo2AO '.
Both Hos and V depend on the oscillator size parameter b = (7~/mw')ll 2 , while V also depends on the particular interaction V one uses. The present calculation employs b o t h the original Sussex Matrix Elements (SME) given in ref. [1 ] and the saturating SME o f ref. [2]. The former interaction gives reasonable resuits for ground state energies etc. using a correct density imposed b y a suitable choice of b, but does not
give a maximum in the first order binding energy at this density. The latter involves, in addition to the original SME, a Gc term obtained from a hard core potential, where the choice of core radius c = 0.3 is discussed in ref. [2] (c = 0 corresponding to the original SME). To zero order in V the rms operator for the ground states (g) o f closed shell nuclei has the value 0'g(°) =(010 - rcm 2 10) = 000 ,
---~
(2n h + lh + 3/2) - 362
2A' '
(2)
where the sum is over the relevant holes. Thus (0'g (0))1/2 equals approximately 1.72 b for calcium, and depends linearly on the size parameter b. To zero order one gets roughtly the experimental rms radius [3] for this nucleus at b = 2. Higher order terms in the rms series may be obtained directly from the energy series using essentially the "linear replacement m e t h o d " o f refs. [4,5]. This involves substituting V -+ V + aO' and differentiating the resulting energy expressions with respect to a. Thus
273
Volume 81B, number 3, 4 0'g (n)(~) = oE(n+l *~g
PHYSICS LETTERS (3)
)(T + ~0')/~1~=0,
where 0'g(n) is the nth order term in the 0' series and E(n+l) the (n + l)th order term in the energy series. In the present calculation we set up the energy matrix for 4°Ca in the space of 0 and all 2hco excitations i.e. use a 176 dimensional space. Schematically (H) = H)O I
+
V02
~10
g l l - TOO --a
V12
~20
V21
+
V22 - VO0 -
+
0
-G
1
1(~)/
(4)
where a = -2hco, I is the identity matrix and the subscripts 0, 1 and 2 stand for 0p~3h, l p - l h and 2 p - 2 h states, respectively. We also calculate the corresponding matrix of 0,
0ooi+
0
001
010
011
0
0
0 -
000
0
+2
Too +i])O]o ,-2
+ To,(O,] - 00o+,/)
,]-
a-2
- Too+,,+)°+o a-3
(7)
E t (V) = v + ( V ) G ( T > ( T ) ,
(8)
where v, v+ and G - 1 are defined in expression (4). An analogous linked expansion for the 0 operator, the summation of expression (7) to all orders in the space of 0 + 2hco excitations, may be obtained using the linear replacement method. Thus (9)
Using the matrix identity OM/O¢~= -M(OM -1/Ou) X M, the relations OG -1 (V + c~0)0~ = -~', 0v(V + a0)0a = ~, where f and r7 are defined in expression (5) and the fact that 0 and T are hermitian operators one obtains
EL(V ) : TOO + goiTio a-1
+ Voi(Vi] - Voo6i]) V]oa - 2
(6)
+
....
- T00+; ) T 0a -3 +
An inversion technique analogous to the the Q box approach of ref. [4] yields the exact linked diagram expansion result for the binding energy of a closed shell nucleus in the space of 0 + 2hco excitations [6]. This corresponds to summing expression (6) to all orders with the result
OL(V ) = OEL (V + c~O)/O~I u=O.
022 -- 000
With these two matrices one can obtain any order of the rms series where for all orders beyond zero order 0' and the simpler operator 0 yield identical results. Thus 0g(1)(V) : 0~1)(~) : 2~oiOio/a , where the intermediate states are summed over, and the Voi'S and Oio'S are entries in the matrices (4) and (5), etc. To evaluate only the linked graphs for the rms operator in this space [6] one includes the counterparts of the leading terms in the Rayleigh-Schr6dinger energy series namely:
274
0L(V) = 000 + 2VoiOio a-1
+ 2Voi(0/] - 0006i]) (VlTc - ~ 0 6 j k ) Vko a-3 + ....
I,
;(3)
-=fi00I+ ~
(0) =
Using the linear replacement method the corresponding linked rms series is
+ 2Voi( VO1
26 February 1979
OL(V ) = 2 , ¢ G ( T ) v ( V ) + v + ( T ) G ( T ) ~ G ( T ) v ( T ) .
(lO) Numerical results for the rms radius of Ca are plotted in fig. 1 as a function of b both for c = 0 (unsaturating SME) and c = 3 (saturating SME). One notes from these figures that with the unsaturating SME higher order terms in the rms series do not oppose the collapse of this nucleus and only for the very small rms radius of ~2.6 fm are higher order terms in the rms series small compared to the zero order term. On the other hand while the saturating SME higher orders counteract this tendency (i.e. give a repulsive rms contribution) below b = 1.8 and around b = 1.8 (i.e. a radius of ~3 fm) higher order terms in the rms series are negligible compared to the zero-order term. As
Volume 81B, number 3, 4
26 February 1979
PHYSICS LETTERS
Table 1 Exact, diagonalization and series results for the rms operator in fm 2 . c
b
Zero a)
First
Second
Third
Fourth
Fifth
To Third
To Fifth
Diag.
Exact
0.3 0 0.3 0 0.3 0
2 2 1.8 1.8 1.5 1.5
11.85 11.85 9.60 9.60 6.67 6.67
-1.51 -2.92 -0.07 -l.53 1.51 0.05
-0.32 -0.57 0.07 -0.20 0.43 0.10
-0.21 -0.25 -0.09 -0.18 0.00 -0.12
0.08 0.15 0.12 0.17 0.16 0.23
-0.14 -0.17 -0.17 -0.25 -0.24 -0.43
9.80 8.11 9.51 7.69 8.61 6.70
9.75 8.09 9.46 7.61 8.53 6.50
11.09 10.96 9.57 8.96 7.23 6.78
9.82 8.22 9.56 7.76 8.69 6.73
a) 0~o
=
0oo
3b2/2A.
-
3.5(
* = EXACT
RESULTS
(o)
/~ol . ~
1
C =0
higher orders are included in the rms series the nuclear size b e c o m e s as it should less sensitive to the choice one takes for b. Higher lying excitations m a k e this dep e n d e n c e even less [7]. S o m e exact results (obtained using expressions (10)) are also shown in fig. 1. Table 1 lists these, and the corresponding rms series, as well as diagonalization results for comparison. The rms series (except for b = 1.8, c = 0.3 and b = 1.5, c = 0 where higher orders are small) appear to converge by 3rd order b u t in fact do not. A g r e e m e n t with the exact results is best if one stops while the series is still m o n o t o n i c (i.e. b y third order). Comparison o f the diagonalization and exact linked results shows that the unlinked graphs have no definite sign, and are small only w h e n the higher order rms terms are small.
References
2.501 1.5
I 1.6
i 1.7
1 I.e
I 1,9
I
2.0
b (fro)
Fig. 1. Zeroth, and zeroth plus higher-order terms for the rms operator as a function of b. Exact results are indicated with a star.
[1 ] J.P. Elliot, A.D. Jackson, H.A. Mavromatis, E.A. Sanderson and B. Singh, Nucl. Phys. A121 (1968) 241. [2] E.A. Sanderson, J.P. Elliot, H.A. Mavromatis and B. Singh, Nucl. Phys. A219 (1974) 190. [3] H.A. Bethe, Ann. Rev. Nucl. Sci. 21 (1971) 93. [4] T.T.S. Kuo, S.Y. Lee and K.F. Ratcliff, Nucl. Phys. A176 (1971) 65. [5] I. Lindgren, J. Phys. B7 (1974) 2441. [6] H.A. Mavromatis, Nucl. Phys. A295 (1978) 269. [7] M.A. Jadid and H.A. Mavromatis, to be published.
275