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f72 neutron occupancies in the Ca isotopes
Volume 77B, number 2
PHYSICS LETTERS
31 July 1978
f7/2 NEUTRON OCCUPANCIES IN THE Ca ISOTOPES C.F. CLEMENT Theoretical Physics Division, AERE Harwell, Didcot, Oxon 0)(11 ORA, UK
and S.M. PEREZ Physics Department, University o f Cape Town Rondebosch, Cape, South Africa
Received 10 April 1978
Methods of obtaining nucleon occupancies from total energy-weighted sum rules for spectroscopic factors are described and applied to f7/2 neutron transfer data for Ca isotopes. The f7/2 neutron occupancies obtained are consistent with shell model expectations and, for 41Ca and 43Ca, with values previously obtained from an analysis using non-energy-weighted sum rules.
Spin-dependent non-energy weighted sum rules (NEWSR) [1,2] and energy weighted sum rules (EWSR) [ 3 - 5 ] are proving to be a powerful tool in the study of nuclear physics. They have been used to determine spins of final states [6,7], absolute DWBA normalizations and occupancies of valence orbitals [7,8], lightion overlaps [9,10], and effective potential matrix elements [5,11 ]. However, the total energy-weighted sum rule [12,13], to which the EWSR reduce for a spin zero target, has hitherto had little application. We present here an analysis of data for f7/2 neutron stripping and pickup on the isotopes 40Ca,'41Ca, 42Ca, 43Ca, and 44Ca, which is based on this total sum rule. The f7/2 occupancies deduced are close to values expected on the basis of the simple shell model, and reinforce our previous conclusions concerning the validity of the shell model for f7/2 shell nuclei [8]. For the transfer o f a (/t3) nucleon from a target state, r, to stripping states, n, and pickup states, a, the total EWSR may be obtained directly [12], or by summing the partial EWSR [11 ] over Jn, in the form, E+S+ + E - S - = [ / ] E c ( J t 3 ) + V S - ( J t 3 ) ,
(1)
where []] - (2/" + 1), S +, S - are t h e / t 3 hole and particle occupancies respectively, and E+ (] t 3), E - (/. t 3 ) are the energy centroids of the (stripping, pickup) strengths
defined by, E+S + = ~
(2)
(E n - e r) [Jn]C2Sn/[Jr] ,
n
E-S-
= ~ ] (6 r - E a ) C 2 S ~ .
(3)
The contributions from the spectroscopic factors S n and Sa, including the isospin Clebsch-Gordan coef-
ficient C2, are weighted by the separation energies E n - e r and e r - E~. The single particle core energy, E c ( / t 3 ) , represents the potential energy contribution
of all occupied orbits (not necessarily closed) other t h a n / t 3 , and the averaged (/t3)2 interaction is V=
Z~ (2 [ J 1 ] / [ f ] ) V j I 1 T 3 { ( f t 3 ) 2 ) , J1 even
(4)
where Vjr~1/'3 is the potential matrix element for two ]t 3 particles in the state J1, T= 1, T 3. Neglecting 1/A corrections [2], the total NEWSR is s + + s-
(5)
= [/].
Eliminating S + between eqs. (1) and (5) we obtain, (E + - Ec) [1] = S - ( V + E + - E - ) .
(6)
Now E + and E - may be taken from experiment and 145
Volume 77B, number 2
PHYSICS LETTERS
31 July 1978
Table 1 Neutron binding energies a) and f7/2 neutron stripping and pickup spectroscopic factors for the Ca isotopes normalized as discussed in text, together with the corresponding excitation energies of the final states. All energies are in MeV.
Table 2 f7/2 neutron occupancies for the Ca isotopes as a function of the f7/2 proton occupancy, 6p.
6p
4°Ca
41Ca
42Ca
43Ca
Target
0.00 0.20 0.40 0.20 b)
0.00 0.20 0.40 0.20
1.10+_0.03a) 1.18+_0.03 1.29+_0.05 1.18+-0.03
1.99+0.05 2.05+0.07 2.12+0.09 2.40+_0.09
3.22+_0.12 3.22+0.12 3.24+0.13 4.35+_0.20
Binding Stripping energy .
Pickup
4°Cab) 41CaC)
En
[Jn]C2Sn/[Jr]
E*
-15.63
0.00 2.89
7.79 0.21
2.80 d) I.A.S.
- 8.36
0.00 1.53 1.84 2.42 2.75 3.19 3.25 3.30 3.39 3.65
0.18 0.54 0.04 0.59 1.64 3.45 0.42 0.01 0.01 0.12
0.00 0.98 3.35 0.02 9.30 d) I.A.S.
0.00 1.87 2.97 0.13 7.17 d) I.A.S.
42Cab)
-11.47
0.00 5.90 2.67 0.10
43Ca c)
- 7.93
0.00 1.16 1.88 2.28 2.66 3.04 3.28 4.09
44Ca b)
-11.14
0.33 0.33 0.07 0.21 0.40 1.34 2.24 0.08
0.00 3.49 1.97 0.10 2.97 0.41
C2S~
where A is the mass number. This assumption implies that, for adjacent isotopes A - 1, A, and A + 1, V and E c are essentially constant, and that we can solve the three equations corresponding to eq. (6) to obtain, for example,
0.00 1.52 1.84 2.42 2.75 3.19 3.25 10.5 d,e)
0.68 0.22 0.06 0.19 0.68 1.06 0.11 I.A.S.
0.00 4.00 8.76 d) I.A.S.
a) Neutron binding energy in 45Ca = -7.41 MeV. b) Refs. [15,16]. c) Data used in ref. [8]. d) Isobaric analogue states from refs. [14,16]. See text for values of C2Sc~. e) Estimated position. are independent of absolute magnitudes assumed for spectroscopic factors so that, given any two of E0, V, and S - , the third quantity may be determined from eq. (6) with a similar independence. Another natural approach for a chain of isotopes or isotones is to assume that additional nucleons preferentially populate the orbit it3, so that S~+ 1 = S~ + 1 , 146
a) Errors calculated assuming 0.1 fractional errors in the spectroscopic factors of table 1. b) Values obtained using data from ref. [17] for neutron pickup on 43Ca.
(7)
S~={[/](E~+ 1 - 2E~ + E ~ _ I ) -- ( E f l + l - E ~ _ I
- E~+ 1 +
E~_I))[
(8)
((E~+I - 2E~I+E~_ I) - (E~+I - 2E~ + E~_I) } . We note that no assumption has been made about the detailed structure of the nuclei involved or on the magnitude of ground state-ground state spectroscopic factors, only a less stringent one about the preferential filling of a shell. In our application to the Ca isotopes below, we check on the self-consistencyof the assumption which is supported experimentally by the small and move-or-less constant occupancy of valence orbitals other than the f7/2 neutron [14]. The RHS of eq. (8) depends only on relative spectroscopic factors with fractional errors probably ~ 0.1 [8]. Larger errors may apply where final states are widely separated in energy, as for example here for pikcup to T< and T> final states, and an attempt is made to take this into account. The structure of eq. (8) is such that errors from small missing strength which is systematically undetected should cancel to some extent. We have tested the above methods for obtaining occupancies using the f7/2 neutron stripping and pickup data for Ca isotopes summarized in table 1. The spectroscopic factors for T< states have been normalized to the number of f7/2 holes and particles expected on the basis of a simple shell model. As such they are self-consistent apart from a ~ 3 0 % discrepancy
Volume 77B, number 2
PHYSICS LETTERS
31 July 1978
Table 3 t"7/2 neutron occupancies, S~I, average interaction, V, and core energy, Ec, calculated using the data of taOle 1 with f7/2 proton occupancy, 6p = 0.2. Nuclei
A
4°Ca-41Ca-42Ca
41 Ca- 42Ca-43Ca 42Ca-43Ca-a4Ca 4°Ca-a2Ca-4aCa
41 42 43 42
$4
V
Ec
1.18_+0.03 a) 2.05+-0.07 3.22_+0.12 2.45-+0.05
_1.86+_0.13 - 1.41+-0.17 -1.04-+0.10 -1.05_+0.10
-8.47-+0.04 -8.55-+0.04 -8.71-+0.06 -8.79_+0.06
a) Errors calculated assuming 0.1 fractional errors in the spectroscopic factors only. in the magnitudes quoted for the 43Ca-44Ca ground state transition. The f7/2 proton occupancy, 6p, is assumed constant throughout, and the f7/2 neutron pickup strength to T> states is taken to be 6p/(2T + 1), where T is the target isospin, located at the energy of the lowest T> state of the correct parity in the final nucleus. Occupancies calculated from eq. (8) are shown in table 2 for 6p = 0.0 (0.2) 0.4, where, for self-consistency, eq. (7) should be satisfied. The results, which are consistent with those obtained for 41Ca and 43Ca in an independent NEWSR analysis [8], are not very sensitive to 6p. Nonetheless it is gratifying that we have approximate self-consistency for 6p ~ 0.2, in agreement with the value 0.21 obtained in a DWBA analysis o f the 40Ca(p, d)39Ca reaction using the effective binding energy prescription to calculate the overlap function [16], and with the averaged shell-model value of 0.20 [16]. Of interest also are the results of table 2 obtained with 6p = 0.2 and data [17] which include f7/2 pickup contributions to Final states above 3.25 MeV excitation in 42Ca. These contributions, amounting to ~ 2 0 % o f the total strength, may be suspect [8], but, if confirmed, could point to similar undetected
pickup strength for the other Ca isotopes. In any event the results give an indication of the sensitivity o f the occupancies calculated to the input data. For 6p = 0.2, full results for S ~ , V, and E c obtained by solving sets of three equations (6) for the triads o f nuclei shown are given in table 3. The results for 4 0 C a - 4 2 C a - 4 4 C a are of interest in that none o f the data involved were used in our NEWSR analysis [8 ] or partial EWSR analysis [11]. The values o f V a n d E c obtained appear to exhibit systematic trends with A in the opposite direction. In fact this result is spurious, and a further analysis exposes a limitation in the use of the total EWSR in the form (6) to obtain individual values of V and E c. With S - and the experimental input E + and E - fixed, it is apparent that eq. (6) is satisfied for a range of V, Ec, which satisfy S - g + [/'] E c--- c o n s t a n t .
(9)
Sets o f equations (6) still largely retain this ambiguity and we can alter V and E e in the solutions shown in table 3 whilst retaining approximately the same experimental input and value o f S - . The errors quoted for V and E c in table 3 are thus unrealistic. For example f7/2 occupancies, S - , calculated from eq. (6) using the
Table 4 f7/2 neutron occupancies, S-, for Ca isotopes calculated directly from eq. (6) using values for E c and V. Ec (MeV)
V (MeV)
_8.71+0.06 a) -8.66+-0.09 b)
-1.04+0.10 a) -0.9 +-0.3b)
S4OCa
41Ca
42Ca
43Ca
44Ca
0.37+0.05 c)
_ d)
0.33+-0.08
_ d)
2.21+-0.18 2.01+-0.30
3.23+0.71 2.57+-1.10
4.23+-0.20 3.95+-0.42
a) Values taken from 42Ca-4aCa-44Ca of table 3. b) Partial EWSR values. c) Errors calculated assuming independent errors for E c and V only. d) Indeterminate with the errors given. 147
Volume 77B, number 2
PHYSICS LETTERS
4 2 C a - 4 3 C a - 4 4 C a values o f V and E c of table 3 are shown in table 4 and can be seen to be consistent with the values of table 3. The occupancies o f table 3, with slightly larger errors, are consistent with constant values of V and E c and the condition (7), and thus with the basic assumptions made in their derivation from experimental data. Another example of the direct use o f eq. (6) in calculating S - given V and E c is provided by values, V = - 0 . 9 -+0.3 MeV and E c = - 8 . 6 6 -+0.09 MeV, obtained from a partial EWSR analysis [11 ] redone including transfer data for 41Ca and with the single particle core energy treated as a parameter. The resulting occupancies shown in table 4 are consistent with the other values derived here. In conclusion we have investigated the use o f the total EWSR in analysing spectroscopic data for a series o f isotopes. Whilst it is difficult to obtain reliable values of the single particle core energy and mean interaction in a shell because o f the ambiguity (9), the methods proposed show considerablepromise in the extraction of absolute shell model occupancies. The f7/2 occupancies obtained for 41Ca and 43Ca are consistent with those obtained from an independent NEWSR analysis which are within about 10% of shell model values. The occupancies are again all close to simple shell model values. It would seem most unlikely that such results would be obtained if a large fraction
148
31 July 1978
of the total single particle strength were missing from the experimentally surveyed energy region as was once suggested [18].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
J.B. French, Phys. Lett. 13 (1964) 249. C.F. Clement, Nucl. Phys. A213 (1973) 469. R.K. Bansal and J.B. French, Phys. Lett. 19 (1965) 223. C.F. Clement, AERE TP.615 (1975). C.F. Clement, Univ. of Maryland Technical Report ORO 51268 (1977). C.F. Clement and S.M. Perez, Nucl. Phys. A213 (1973) 510. C.F. Clement and S.M. Perez, J. Phys. A7 (1974) 1464. C.F. Clement and S.M. Perez, Nucl. Phys. A284 (1977) 469. N.S. Chant, N.S. Wall, C.F. Clement, S.M. Perez and J.N. Craig, Phys. Rev. C15 (1977) 53. A. Moalem, Nucl. Phys. A289 (1977) 45. C.F. Clement and S.M. Perez, Phys. Lett. 62B (1976) 381, and unpublished. C.F. Clement, Phys. Lett. 28B (1969) 398. M. Baranger, Nucl. Phys. A149 (1970) 225. P.M. Endt and C. van der Leun, Nucl. Phys. A214 (1973) 1 G. Brown, A. Denning and J.G.B. Haigh, Nucl. Phys. A214 (1974) 267. P. Martin, M. Buenerd, Y. Dupont and M. Chabre, Nucl. Phys. A185 (1972) 465. A. Jamshidi and W.P. Alford, Phys. Rev. C8 (1973) 1782. G.E. Brown, Commun. Nucl. Part. Phys. 3 (1969) 136.
PHYSICS LETTERS
31 July 1978
f7/2 NEUTRON OCCUPANCIES IN THE Ca ISOTOPES C.F. CLEMENT Theoretical Physics Division, AERE Harwell, Didcot, Oxon 0)(11 ORA, UK
and S.M. PEREZ Physics Department, University o f Cape Town Rondebosch, Cape, South Africa
Received 10 April 1978
Methods of obtaining nucleon occupancies from total energy-weighted sum rules for spectroscopic factors are described and applied to f7/2 neutron transfer data for Ca isotopes. The f7/2 neutron occupancies obtained are consistent with shell model expectations and, for 41Ca and 43Ca, with values previously obtained from an analysis using non-energy-weighted sum rules.
Spin-dependent non-energy weighted sum rules (NEWSR) [1,2] and energy weighted sum rules (EWSR) [ 3 - 5 ] are proving to be a powerful tool in the study of nuclear physics. They have been used to determine spins of final states [6,7], absolute DWBA normalizations and occupancies of valence orbitals [7,8], lightion overlaps [9,10], and effective potential matrix elements [5,11 ]. However, the total energy-weighted sum rule [12,13], to which the EWSR reduce for a spin zero target, has hitherto had little application. We present here an analysis of data for f7/2 neutron stripping and pickup on the isotopes 40Ca,'41Ca, 42Ca, 43Ca, and 44Ca, which is based on this total sum rule. The f7/2 occupancies deduced are close to values expected on the basis of the simple shell model, and reinforce our previous conclusions concerning the validity of the shell model for f7/2 shell nuclei [8]. For the transfer o f a (/t3) nucleon from a target state, r, to stripping states, n, and pickup states, a, the total EWSR may be obtained directly [12], or by summing the partial EWSR [11 ] over Jn, in the form, E+S+ + E - S - = [ / ] E c ( J t 3 ) + V S - ( J t 3 ) ,
(1)
where []] - (2/" + 1), S +, S - are t h e / t 3 hole and particle occupancies respectively, and E+ (] t 3), E - (/. t 3 ) are the energy centroids of the (stripping, pickup) strengths
defined by, E+S + = ~
(2)
(E n - e r) [Jn]C2Sn/[Jr] ,
n
E-S-
= ~ ] (6 r - E a ) C 2 S ~ .
(3)
The contributions from the spectroscopic factors S n and Sa, including the isospin Clebsch-Gordan coef-
ficient C2, are weighted by the separation energies E n - e r and e r - E~. The single particle core energy, E c ( / t 3 ) , represents the potential energy contribution
of all occupied orbits (not necessarily closed) other t h a n / t 3 , and the averaged (/t3)2 interaction is V=
Z~ (2 [ J 1 ] / [ f ] ) V j I 1 T 3 { ( f t 3 ) 2 ) , J1 even
(4)
where Vjr~1/'3 is the potential matrix element for two ]t 3 particles in the state J1, T= 1, T 3. Neglecting 1/A corrections [2], the total NEWSR is s + + s-
(5)
= [/].
Eliminating S + between eqs. (1) and (5) we obtain, (E + - Ec) [1] = S - ( V + E + - E - ) .
(6)
Now E + and E - may be taken from experiment and 145
Volume 77B, number 2
PHYSICS LETTERS
31 July 1978
Table 1 Neutron binding energies a) and f7/2 neutron stripping and pickup spectroscopic factors for the Ca isotopes normalized as discussed in text, together with the corresponding excitation energies of the final states. All energies are in MeV.
Table 2 f7/2 neutron occupancies for the Ca isotopes as a function of the f7/2 proton occupancy, 6p.
6p
4°Ca
41Ca
42Ca
43Ca
Target
0.00 0.20 0.40 0.20 b)
0.00 0.20 0.40 0.20
1.10+_0.03a) 1.18+_0.03 1.29+_0.05 1.18+-0.03
1.99+0.05 2.05+0.07 2.12+0.09 2.40+_0.09
3.22+_0.12 3.22+0.12 3.24+0.13 4.35+_0.20
Binding Stripping energy .
Pickup
4°Cab) 41CaC)
En
[Jn]C2Sn/[Jr]
E*
-15.63
0.00 2.89
7.79 0.21
2.80 d) I.A.S.
- 8.36
0.00 1.53 1.84 2.42 2.75 3.19 3.25 3.30 3.39 3.65
0.18 0.54 0.04 0.59 1.64 3.45 0.42 0.01 0.01 0.12
0.00 0.98 3.35 0.02 9.30 d) I.A.S.
0.00 1.87 2.97 0.13 7.17 d) I.A.S.
42Cab)
-11.47
0.00 5.90 2.67 0.10
43Ca c)
- 7.93
0.00 1.16 1.88 2.28 2.66 3.04 3.28 4.09
44Ca b)
-11.14
0.33 0.33 0.07 0.21 0.40 1.34 2.24 0.08
0.00 3.49 1.97 0.10 2.97 0.41
C2S~
where A is the mass number. This assumption implies that, for adjacent isotopes A - 1, A, and A + 1, V and E c are essentially constant, and that we can solve the three equations corresponding to eq. (6) to obtain, for example,
0.00 1.52 1.84 2.42 2.75 3.19 3.25 10.5 d,e)
0.68 0.22 0.06 0.19 0.68 1.06 0.11 I.A.S.
0.00 4.00 8.76 d) I.A.S.
a) Neutron binding energy in 45Ca = -7.41 MeV. b) Refs. [15,16]. c) Data used in ref. [8]. d) Isobaric analogue states from refs. [14,16]. See text for values of C2Sc~. e) Estimated position. are independent of absolute magnitudes assumed for spectroscopic factors so that, given any two of E0, V, and S - , the third quantity may be determined from eq. (6) with a similar independence. Another natural approach for a chain of isotopes or isotones is to assume that additional nucleons preferentially populate the orbit it3, so that S~+ 1 = S~ + 1 , 146
a) Errors calculated assuming 0.1 fractional errors in the spectroscopic factors of table 1. b) Values obtained using data from ref. [17] for neutron pickup on 43Ca.
(7)
S~={[/](E~+ 1 - 2E~ + E ~ _ I ) -- ( E f l + l - E ~ _ I
- E~+ 1 +
E~_I))[
(8)
((E~+I - 2E~I+E~_ I) - (E~+I - 2E~ + E~_I) } . We note that no assumption has been made about the detailed structure of the nuclei involved or on the magnitude of ground state-ground state spectroscopic factors, only a less stringent one about the preferential filling of a shell. In our application to the Ca isotopes below, we check on the self-consistencyof the assumption which is supported experimentally by the small and move-or-less constant occupancy of valence orbitals other than the f7/2 neutron [14]. The RHS of eq. (8) depends only on relative spectroscopic factors with fractional errors probably ~ 0.1 [8]. Larger errors may apply where final states are widely separated in energy, as for example here for pikcup to T< and T> final states, and an attempt is made to take this into account. The structure of eq. (8) is such that errors from small missing strength which is systematically undetected should cancel to some extent. We have tested the above methods for obtaining occupancies using the f7/2 neutron stripping and pickup data for Ca isotopes summarized in table 1. The spectroscopic factors for T< states have been normalized to the number of f7/2 holes and particles expected on the basis of a simple shell model. As such they are self-consistent apart from a ~ 3 0 % discrepancy
Volume 77B, number 2
PHYSICS LETTERS
31 July 1978
Table 3 t"7/2 neutron occupancies, S~I, average interaction, V, and core energy, Ec, calculated using the data of taOle 1 with f7/2 proton occupancy, 6p = 0.2. Nuclei
A
4°Ca-41Ca-42Ca
41 Ca- 42Ca-43Ca 42Ca-43Ca-a4Ca 4°Ca-a2Ca-4aCa
41 42 43 42
$4
V
Ec
1.18_+0.03 a) 2.05+-0.07 3.22_+0.12 2.45-+0.05
_1.86+_0.13 - 1.41+-0.17 -1.04-+0.10 -1.05_+0.10
-8.47-+0.04 -8.55-+0.04 -8.71-+0.06 -8.79_+0.06
a) Errors calculated assuming 0.1 fractional errors in the spectroscopic factors only. in the magnitudes quoted for the 43Ca-44Ca ground state transition. The f7/2 proton occupancy, 6p, is assumed constant throughout, and the f7/2 neutron pickup strength to T> states is taken to be 6p/(2T + 1), where T is the target isospin, located at the energy of the lowest T> state of the correct parity in the final nucleus. Occupancies calculated from eq. (8) are shown in table 2 for 6p = 0.0 (0.2) 0.4, where, for self-consistency, eq. (7) should be satisfied. The results, which are consistent with those obtained for 41Ca and 43Ca in an independent NEWSR analysis [8], are not very sensitive to 6p. Nonetheless it is gratifying that we have approximate self-consistency for 6p ~ 0.2, in agreement with the value 0.21 obtained in a DWBA analysis o f the 40Ca(p, d)39Ca reaction using the effective binding energy prescription to calculate the overlap function [16], and with the averaged shell-model value of 0.20 [16]. Of interest also are the results of table 2 obtained with 6p = 0.2 and data [17] which include f7/2 pickup contributions to Final states above 3.25 MeV excitation in 42Ca. These contributions, amounting to ~ 2 0 % o f the total strength, may be suspect [8], but, if confirmed, could point to similar undetected
pickup strength for the other Ca isotopes. In any event the results give an indication of the sensitivity o f the occupancies calculated to the input data. For 6p = 0.2, full results for S ~ , V, and E c obtained by solving sets of three equations (6) for the triads o f nuclei shown are given in table 3. The results for 4 0 C a - 4 2 C a - 4 4 C a are of interest in that none o f the data involved were used in our NEWSR analysis [8 ] or partial EWSR analysis [11]. The values o f V a n d E c obtained appear to exhibit systematic trends with A in the opposite direction. In fact this result is spurious, and a further analysis exposes a limitation in the use of the total EWSR in the form (6) to obtain individual values of V and E c. With S - and the experimental input E + and E - fixed, it is apparent that eq. (6) is satisfied for a range of V, Ec, which satisfy S - g + [/'] E c--- c o n s t a n t .
(9)
Sets o f equations (6) still largely retain this ambiguity and we can alter V and E e in the solutions shown in table 3 whilst retaining approximately the same experimental input and value o f S - . The errors quoted for V and E c in table 3 are thus unrealistic. For example f7/2 occupancies, S - , calculated from eq. (6) using the
Table 4 f7/2 neutron occupancies, S-, for Ca isotopes calculated directly from eq. (6) using values for E c and V. Ec (MeV)
V (MeV)
_8.71+0.06 a) -8.66+-0.09 b)
-1.04+0.10 a) -0.9 +-0.3b)
S4OCa
41Ca
42Ca
43Ca
44Ca
0.37+0.05 c)
_ d)
0.33+-0.08
_ d)
2.21+-0.18 2.01+-0.30
3.23+0.71 2.57+-1.10
4.23+-0.20 3.95+-0.42
a) Values taken from 42Ca-4aCa-44Ca of table 3. b) Partial EWSR values. c) Errors calculated assuming independent errors for E c and V only. d) Indeterminate with the errors given. 147
Volume 77B, number 2
PHYSICS LETTERS
4 2 C a - 4 3 C a - 4 4 C a values o f V and E c of table 3 are shown in table 4 and can be seen to be consistent with the values of table 3. The occupancies o f table 3, with slightly larger errors, are consistent with constant values of V and E c and the condition (7), and thus with the basic assumptions made in their derivation from experimental data. Another example of the direct use o f eq. (6) in calculating S - given V and E c is provided by values, V = - 0 . 9 -+0.3 MeV and E c = - 8 . 6 6 -+0.09 MeV, obtained from a partial EWSR analysis [11 ] redone including transfer data for 41Ca and with the single particle core energy treated as a parameter. The resulting occupancies shown in table 4 are consistent with the other values derived here. In conclusion we have investigated the use o f the total EWSR in analysing spectroscopic data for a series o f isotopes. Whilst it is difficult to obtain reliable values of the single particle core energy and mean interaction in a shell because o f the ambiguity (9), the methods proposed show considerablepromise in the extraction of absolute shell model occupancies. The f7/2 occupancies obtained for 41Ca and 43Ca are consistent with those obtained from an independent NEWSR analysis which are within about 10% of shell model values. The occupancies are again all close to simple shell model values. It would seem most unlikely that such results would be obtained if a large fraction
148
31 July 1978
of the total single particle strength were missing from the experimentally surveyed energy region as was once suggested [18].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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