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Proton occupation numbers in Ge isotopes
Volume 145B, number 1,2
PHYSICS LETTERS
13 September 1984
PROTON OCCUPATION NUMBERS IN Ge ISOTOPES
H.T. FORTUNE, M. CARCHIDI and S. MORDECHAI Physics Department, Universityof Pennsylvania, Philadelphia,PA 19104, USA Received 24 April 1984
Wave functions for ground states in Ge isotopes, derived solely from fits to 2n transfer data, give a good account of the jump in proton occupation numbers between 72Ge and 74Ge, and determine the proton occupation numbers of the basic states.
In a model [1] for the even Ge isotopes that assumes: (1) two-state mixing, i.e. the ground state and an excited 0 + state are linear combinations of basis states ~Ogand Ce; (2) properties Of Cg and ~Pe vary slowly and smoothly with A (mass number of Ge isotopes), with no additional knowledge of the nature of ¢g and ¢e, an analysis [1] of 2n pickup [ 2 - 5 ] and stripping [ 6 - 8 ] , viz. 72,74,76Ge(p, t)7°,72,74Ge and 70,7 2,74Ge(t ' p)7 2,74,76Ge ' data requires that the ground states of 70Ge and 7 2Ge are similar and likewise for those of 74Ge and 76Ge, but that a rather dramatic change has taken place between 72Ge and 74Ge. Of course, this transition is not new, but has been known for a long time. The nature of the abrupt change in structure has been attributed to pure neutron excitations [ 9 - 1 1 ] to pure proton excitations [12], or to a shape transition [ 1 3 - 1 5 ] . The structural change is especially notable in 2n transfer, allowing a quantitative description [I], without any assumptions about detailed structure. Thus, for the expectation value of any operator that is different for tpg and Ce, we expect 70,72Ge(g.s.) to display predominantly the value belonging to ~g and 74,76Ge(g.s.) that oftp e. in the present note we address the question of ground-state proton occupation numbers. This quantity also exhibits an abrupt change between 7 2Ge and 74Ge [16]. The percentages of Cg in the Ge(g.s.) are paremeterized [1] in terms of a single dimensionless parameter R, as depicted in fig. 1. With kinematic depen -~
1.0
.
~
,
0.8 0.6 0.4 0.2 0.0
i 0.8
1.0
R
1.2
1.4
Fig. 1. From ref. [1], a plot of a~l versus R, where a~l is the percentage of ~g in the physical ground state of AGe. dence removed, R 2 is just the ratio of 2n transfer cross sections between ~ et ~ ~1e +2 and ~gA ~ ~9~4+ 2. In ref. [ 1], nothing was assumed about the detag'fled structure of ~pg and Ce. For any observable 0 such that
0g = (~gl0l~g),
0 e = (~Oel0l~e),
(~el0l~g) = 0 ,
we will have 0g s = a20g + 1320e, We now apply this expression to proton occupation numbers in the lf5/2, 2pl/2, 2p3/2 orbitals, and assume the full strength has been observed so that the sum in each isotope is four. In no case is the splitting of l = 1 strength between p 1/ 2 and p 3 / 2 unambiguous, so we lump them together and hence need consider only the 1t"5/2 occupation numbers since Np = 4 - N f . 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 145B, number 1,2
PHYSICS LETTERS
Table 1 lfs/2 proton occupancies in Ge ground states.
A
13 September 1984
I
cO 3.0 r~ LLI m
NA a)
I
CALC.
2.5
76
Z
70 72 74 76
1.17 1.30 2.16 2.37
If we define ng and n e as the number of l f s / 2 protons in tpg and ¢e, respectively, and n A the number o f lf5/2 protons in the physical g.s. of A Ge, then we have
£_) C~ 1.,5 0
72
Z 0 I-- 1.0 0 n-o...
~
"---.~
2- A = 7 0
72
RTO
70
74 -72
n74-~72
0.5 76"74 72-7O
/ 1.0
1.2
R
Note that n e and n g , being properties o f the basis states, should be virtually independent o f A . Experimental occupation numbers [16], normalized to a sum of four, are listed in table 1. Note near equality for 7°Ge and 72Ge, and for 74Ge and 76Ge, as expected. The large jump between 72 and 74 can only mean that ng and n e are quite different. The earlier, simpler model of Vergnes [12] assumed ng = 0, n e = 2. We can, however, use the 2n-transfer analysis to put limits on ng and n e. We note, in passing, that Vergnes' model gives nonsense for 74 and 76, requiring negative a 2. In fact, it is clear that Vergnes' model cannot account for the presence of more than two lf5/2 protons in any Ge g.s., as his basis states contain either zero or two l f s / 2 protons. Using the a 2 values versus R from fig. 1 we have performed a least-squares fit to the experimental occupation numbers, minimizing C76 "( n Aexp
__74
/ 2.0
= aA2 ng +/32ne .
X2__ 1
""
13_
a) From ref. [161. Normalized to give sum of 4.0 for l f s / : and 2p.
nA
--76
74
Z
o
EXR
-- (o~2ng 0.1
+/3~ ne))2 _
'
where we have assumed an uncertainty of 0.1 for the experimental occupancies. Results are plotted in fig. 2. We note that, for any value of R, the model simultaneously fits all four values of n A , within the uncertainties. The dependence of ng and n e on R is given in fig. 3. We observe that even though they are quite different from Vergnes' assumptions, nevertheless n e - ng is still about 2. Plotted also in fig. 2, again versusR, are the calcu-
Fig. 2. Dependence of lf5/: proton occupation numbers on R, for the even Ge isotopes indicated.
lated differences n A - n A _ 2. Uncertainties in the experimental absolute occupation numbers are probably about 0.1. However, the uncertainty in the difference of n A between isotopes is probably considerably smaller. In the figures, we have assumed an uncertainty of 0.04 in the differences and 0.1 in the absolute values. Thus, for any acceptable value of R, which gives x2/N<~ 1 ,the number, ng of l f s / 2 protons in the
I
F
I
I
Ill
I
2.0
Ge
xz t6
,
--1,2f/ N O.8
/
0.8
1.0 R
¢-0) -o 4 C 123
1,2
/
¢- 2 ° -2
I 0.8
i
t 1.0
I
J 1.2
I
I 1.4.
R Fig. 3. The deduced values, versus R, of ng and ne, the lfs/2 proton occupancies of ~Ogand ~Pe, respectively. Inset shows dependence of X: on R.
Volume 145B, number 1,2
PHYSICS LETTERS
basis state ~g, lies in the region 0 . 5 2 - 1 . 2 3 and the number, n e in the basis state ~0e lies in the range 2 . 3 8 - 4 . 1 6 while the difference, n e - ng, is between 1.69 and 2.94. On the average this difference is about 2, consistent with the assumption b y Vergnes, b u t the state ~0g has about one lf5/2 proton in it, while Vergnes assumed zero. Because we believe the mixing between ~Ogand fie is not contained in standard collective models of Ge isotopes, it is the quanties ng and n e (rather than nA) that should be compared to predictions of those models. We acknowledge financial support from the National Science Foundation.
References [1] M. Carchidi, H.T. Fortune, G.S. Stephans and L.C. Bland, submitted to Phys. Rev. C. [2] F. Guilbault et al., Phys. Rev. C16 (1977) 1840. [3] G.C. Ballet al., Nucl. Phys. A231 (1974) 334.
13 September 1984
[4] A.C. Rester, J.B. Ball and R.C. Auble, Nucl. Phys. A346 (1980) 371. [5] C. Lebrun et al., Phys. Rev. C19 (1979) 1224. [6] S. Mordechai, H.T. Fortune, R. Middleton and G. Stephans, Phys. Rev. C19 (1979) 1733. [7] S. LaFrance, S. Mordechai, H.T. Fortune and R. Middleton, Nucl. Phys. A307 (1978) 52. [8] S. Mordechai, H.T. Fortune, R. Middleton and G. Stephans, Phys. Rev. C18 (1979) 2498. [9] R. Fournier, J. Kroon, T.H. Hsu, B. Hird and G.C. Ball, Nucl. Phys. A202 (1973) 1. [10] G.C. Ball, R. Fournier, J. Kroon, T.H. Hsu andB. Hird, Nucl. Phys. A231 (1974) 334. [11] A. Becker, E.A. Bakkum and R. Kamermans, Phys. Lett. ll0B (1982) 199. [12] M.N. Vergnes, Proc. 6th European Physical Society Nuclear Division, Conf. on the Structure of medium heavy nuclei (Rhodes, 1979), Intern. Phys. Conf. Sec. 49 (1980) 25. [13] M.N. Vergnes et al., Phys. Lett. 72B (1978) 447. [ 14] R. Lecomte, M. Irshad, S. Landsberger, P. Paradis and S. Monaro, Phys. Rev. C22 (1980) 1530. [15] R. Lecomte, G. Kajrys, S. Landsberger, P. Paradis and S. Monaro, Phys. Rev. C25 (1982) 2812. [16] G. Rotbard et al., Phys. Rev. C18 (1978) 86.
PHYSICS LETTERS
13 September 1984
PROTON OCCUPATION NUMBERS IN Ge ISOTOPES
H.T. FORTUNE, M. CARCHIDI and S. MORDECHAI Physics Department, Universityof Pennsylvania, Philadelphia,PA 19104, USA Received 24 April 1984
Wave functions for ground states in Ge isotopes, derived solely from fits to 2n transfer data, give a good account of the jump in proton occupation numbers between 72Ge and 74Ge, and determine the proton occupation numbers of the basic states.
In a model [1] for the even Ge isotopes that assumes: (1) two-state mixing, i.e. the ground state and an excited 0 + state are linear combinations of basis states ~Ogand Ce; (2) properties Of Cg and ~Pe vary slowly and smoothly with A (mass number of Ge isotopes), with no additional knowledge of the nature of ¢g and ¢e, an analysis [1] of 2n pickup [ 2 - 5 ] and stripping [ 6 - 8 ] , viz. 72,74,76Ge(p, t)7°,72,74Ge and 70,7 2,74Ge(t ' p)7 2,74,76Ge ' data requires that the ground states of 70Ge and 7 2Ge are similar and likewise for those of 74Ge and 76Ge, but that a rather dramatic change has taken place between 72Ge and 74Ge. Of course, this transition is not new, but has been known for a long time. The nature of the abrupt change in structure has been attributed to pure neutron excitations [ 9 - 1 1 ] to pure proton excitations [12], or to a shape transition [ 1 3 - 1 5 ] . The structural change is especially notable in 2n transfer, allowing a quantitative description [I], without any assumptions about detailed structure. Thus, for the expectation value of any operator that is different for tpg and Ce, we expect 70,72Ge(g.s.) to display predominantly the value belonging to ~g and 74,76Ge(g.s.) that oftp e. in the present note we address the question of ground-state proton occupation numbers. This quantity also exhibits an abrupt change between 7 2Ge and 74Ge [16]. The percentages of Cg in the Ge(g.s.) are paremeterized [1] in terms of a single dimensionless parameter R, as depicted in fig. 1. With kinematic depen -~
1.0
.
~
,
0.8 0.6 0.4 0.2 0.0
i 0.8
1.0
R
1.2
1.4
Fig. 1. From ref. [1], a plot of a~l versus R, where a~l is the percentage of ~g in the physical ground state of AGe. dence removed, R 2 is just the ratio of 2n transfer cross sections between ~ et ~ ~1e +2 and ~gA ~ ~9~4+ 2. In ref. [ 1], nothing was assumed about the detag'fled structure of ~pg and Ce. For any observable 0 such that
0g = (~gl0l~g),
0 e = (~Oel0l~e),
(~el0l~g) = 0 ,
we will have 0g s = a20g + 1320e, We now apply this expression to proton occupation numbers in the lf5/2, 2pl/2, 2p3/2 orbitals, and assume the full strength has been observed so that the sum in each isotope is four. In no case is the splitting of l = 1 strength between p 1/ 2 and p 3 / 2 unambiguous, so we lump them together and hence need consider only the 1t"5/2 occupation numbers since Np = 4 - N f . 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 145B, number 1,2
PHYSICS LETTERS
Table 1 lfs/2 proton occupancies in Ge ground states.
A
13 September 1984
I
cO 3.0 r~ LLI m
NA a)
I
CALC.
2.5
76
Z
70 72 74 76
1.17 1.30 2.16 2.37
If we define ng and n e as the number of l f s / 2 protons in tpg and ¢e, respectively, and n A the number o f lf5/2 protons in the physical g.s. of A Ge, then we have
£_) C~ 1.,5 0
72
Z 0 I-- 1.0 0 n-o...
~
"---.~
2- A = 7 0
72
RTO
70
74 -72
n74-~72
0.5 76"74 72-7O
/ 1.0
1.2
R
Note that n e and n g , being properties o f the basis states, should be virtually independent o f A . Experimental occupation numbers [16], normalized to a sum of four, are listed in table 1. Note near equality for 7°Ge and 72Ge, and for 74Ge and 76Ge, as expected. The large jump between 72 and 74 can only mean that ng and n e are quite different. The earlier, simpler model of Vergnes [12] assumed ng = 0, n e = 2. We can, however, use the 2n-transfer analysis to put limits on ng and n e. We note, in passing, that Vergnes' model gives nonsense for 74 and 76, requiring negative a 2. In fact, it is clear that Vergnes' model cannot account for the presence of more than two lf5/2 protons in any Ge g.s., as his basis states contain either zero or two l f s / 2 protons. Using the a 2 values versus R from fig. 1 we have performed a least-squares fit to the experimental occupation numbers, minimizing C76 "( n Aexp
__74
/ 2.0
= aA2 ng +/32ne .
X2__ 1
""
13_
a) From ref. [161. Normalized to give sum of 4.0 for l f s / : and 2p.
nA
--76
74
Z
o
EXR
-- (o~2ng 0.1
+/3~ ne))2 _
'
where we have assumed an uncertainty of 0.1 for the experimental occupancies. Results are plotted in fig. 2. We note that, for any value of R, the model simultaneously fits all four values of n A , within the uncertainties. The dependence of ng and n e on R is given in fig. 3. We observe that even though they are quite different from Vergnes' assumptions, nevertheless n e - ng is still about 2. Plotted also in fig. 2, again versusR, are the calcu-
Fig. 2. Dependence of lf5/: proton occupation numbers on R, for the even Ge isotopes indicated.
lated differences n A - n A _ 2. Uncertainties in the experimental absolute occupation numbers are probably about 0.1. However, the uncertainty in the difference of n A between isotopes is probably considerably smaller. In the figures, we have assumed an uncertainty of 0.04 in the differences and 0.1 in the absolute values. Thus, for any acceptable value of R, which gives x2/N<~ 1 ,the number, ng of l f s / 2 protons in the
I
F
I
I
Ill
I
2.0
Ge
xz t6
,
--1,2f/ N O.8
/
0.8
1.0 R
¢-0) -o 4 C 123
1,2
/
¢- 2 ° -2
I 0.8
i
t 1.0
I
J 1.2
I
I 1.4.
R Fig. 3. The deduced values, versus R, of ng and ne, the lfs/2 proton occupancies of ~Ogand ~Pe, respectively. Inset shows dependence of X: on R.
Volume 145B, number 1,2
PHYSICS LETTERS
basis state ~g, lies in the region 0 . 5 2 - 1 . 2 3 and the number, n e in the basis state ~0e lies in the range 2 . 3 8 - 4 . 1 6 while the difference, n e - ng, is between 1.69 and 2.94. On the average this difference is about 2, consistent with the assumption b y Vergnes, b u t the state ~0g has about one lf5/2 proton in it, while Vergnes assumed zero. Because we believe the mixing between ~Ogand fie is not contained in standard collective models of Ge isotopes, it is the quanties ng and n e (rather than nA) that should be compared to predictions of those models. We acknowledge financial support from the National Science Foundation.
References [1] M. Carchidi, H.T. Fortune, G.S. Stephans and L.C. Bland, submitted to Phys. Rev. C. [2] F. Guilbault et al., Phys. Rev. C16 (1977) 1840. [3] G.C. Ballet al., Nucl. Phys. A231 (1974) 334.
13 September 1984
[4] A.C. Rester, J.B. Ball and R.C. Auble, Nucl. Phys. A346 (1980) 371. [5] C. Lebrun et al., Phys. Rev. C19 (1979) 1224. [6] S. Mordechai, H.T. Fortune, R. Middleton and G. Stephans, Phys. Rev. C19 (1979) 1733. [7] S. LaFrance, S. Mordechai, H.T. Fortune and R. Middleton, Nucl. Phys. A307 (1978) 52. [8] S. Mordechai, H.T. Fortune, R. Middleton and G. Stephans, Phys. Rev. C18 (1979) 2498. [9] R. Fournier, J. Kroon, T.H. Hsu, B. Hird and G.C. Ball, Nucl. Phys. A202 (1973) 1. [10] G.C. Ball, R. Fournier, J. Kroon, T.H. Hsu andB. Hird, Nucl. Phys. A231 (1974) 334. [11] A. Becker, E.A. Bakkum and R. Kamermans, Phys. Lett. ll0B (1982) 199. [12] M.N. Vergnes, Proc. 6th European Physical Society Nuclear Division, Conf. on the Structure of medium heavy nuclei (Rhodes, 1979), Intern. Phys. Conf. Sec. 49 (1980) 25. [13] M.N. Vergnes et al., Phys. Lett. 72B (1978) 447. [ 14] R. Lecomte, M. Irshad, S. Landsberger, P. Paradis and S. Monaro, Phys. Rev. C22 (1980) 1530. [15] R. Lecomte, G. Kajrys, S. Landsberger, P. Paradis and S. Monaro, Phys. Rev. C25 (1982) 2812. [16] G. Rotbard et al., Phys. Rev. C18 (1978) 86.