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Configuration mixing in 90Zr and the lifetime of the excited O+ state
Volume 41B, number 1
CONFIGURATION
PHYSICS LETTERS
4 September 1972
M I X I N G IN 90 Zr A N D T H E L I F E T I M E O F T H E E X C I T E D 0 + S T A T E * W. J. COURTNEY and H. T. FORTUNE University of Pennsylvania, Philadelphia,Pennsylvania 19104, USA Received 3 July 1972
Inclusion of p 3/2 core excitation in 90Zr removes the previous factor-of-6 discrepancy between measured and calculated lifetime for decay of the first excited 0÷ state. The existence of even nuclei whose first excited states have jTr = 0 + is a rare occurrence. At present only six such nuclei are known; 160, 40Ca, 72Ge, 90Zr, 96Zr and 98Mo. (Recent evidence [1 ] indicates that a seventh nucleide 98Zr may also exhibit this unusual property.) The relative abundance of 0 + first excited states in the mass 90 region has been attributed to mixing of [(2Pl/2) 2 (g9/2) Z-38"2] and (lg9/2) Z'38 proton configurations relative to a Z = 38 core. In particular, the coefficients of these two configurations in the 9°Zr ground state wave function and in the orthogonal 90Zr (0 +, 1.761 MeV state) wave function have been extensively investigated both experimental [ 2 - 9 ] and theoretically [ 1 0 - 1 2 ] . The common fault of the experimental analyses, as discussed by Schreve [6], is the assumption that no additional configuration mixing occurs. The observation [ 7 - 9 ] of lo = 1 strength to known 3 / 2 - states in the 90Zr (r, d) 91Nb reaction makes the validity of this assumption questionable. Indeed, when the monopole matrix element M for the EO transition between the two 9°Zr 0 + states is calculated [10] using the wave functions of those analyses, the value obtained is more than twice that deduced from the measured lifetime of the excited 90Zr state. The failure of the simple model wave functions to reasonably account for the observed lifetime has prompted the analysis presented in this paper. We show that inclusion of proton configurations of the type (2P3/2) -2 (2p 1/2) 2 (1 g9/2) 2, as suggested by the 9°Zr (r, d) results, yields a lifetime in good agreement with experiment. The electric monopole (EO) transition probability * Supported in part by the National Science Foundation.
for K-shell conversion WK has been derived by a number of authors [ 1 3 - 1 5 ] and can be written as
WK = B(Z)F(Z, E, R ) JM 12, where Z is the nuclear charge, E is the transition energy, R is the nuclear radius, B(Z) is a function describing the nuclear charge distribution, F(Z, E, R) is basically the Fermi beta decay function, and M is the matrix element of the monopole operator M, connecting the initial and final states. The monopole operator is usually approximated by A ' (1 + r3)r 2, i=1 where ½ (1 + r3) is the usual proton projection operator and ri is the radial position of the ith nucleon. Using the measured half-life [16] of the 90Zr (0 +, 1.761 MeV) state, tl/2 = 60.3 -+ 2.5 ns, and the measured branching ratios [ 17] for decay by internal conversion from higher atomic orbits [4/K/I41L+M+ N = 7.06 + 0.08 and by pair emission, WK/Wp = 2.08 -+ 0.08, together with a calculation of the functions B(Z)F(Z, E, R), the experimental monopole matrix element IMexp I was obtained. The value thus obtained is IMexp I = 1.60 -+ 0.04 fm 2. It is of interest to note that the extracted value of IMexp I does not depend very strongly on the choice of nuclear radius R = roA 1/3 used in calculating F(Z, E, R). In the present case r 0 = 1.25 fin was used. For the present analysis we define the particle vacuum to consist of a closed core of 38 protons and 50 neutrons. With this basis, allowing only proton excitations, the model waye function for the 90Zr ground state is taken to be
PHYSICS LETTERS
Volume 41B, number 1
(3) The values of a 2,/32 and ,f2 thus obtained were normalized to unity. Two of the three equations necessary to determine the coefficients of the model wave function for the first excited state are provided by the requirements of orthogonality and unit normalization. The third was obtained from the lp = 1 spectroscopic strength observed [4] to the two 0 + states in the 8 9 y ( r , d) reaction. The analysis was somewhat complicated by the suggestion of configurations of the type (2P3/2) -2 (2Pl/2) 1 (lg9/2) 2 necessary to account for the additional l, = 1 strength observed to higher excited states in the 8~y (r, d)90Zr reaction. With this in mind we took as the model wave function for the 89y ground state
190Zr, g.s.) = c¢(2p 1/2) 2 J0) + +/3(lg9/2) 2 IO) + 7(2P3/2)-2(2Pl/2)2(lg9/2) 210) where tx2 +/32 + 72 = 1. The model wave function for the 90Zr (0 +, 1.761 MeV) state is taken to consist of the same configurations, with coefficients a', /3' and 7', where cg 2 +/3'2 + 3,'2 = 1. It is a straightforward exercise to show that the monopole matrix element in this case is given by IMI =
2{aa'(r2)lgp/2 +/3~'(r2)2pl/2
4 September 1972
+T)t'(r2)2p3/2} ,
where (r 2)n0. is the matrix element of r 2 for a proton in the initial and final configurations with quantum numbers nlj. In the present work the (r2)nl/were calculated using bound-state wave functions obtained from a real Woods-Saxon potential well and derivative Woods-Saxon ~pin-orbit potential well. The well depth was adjusted to give the binding energy equal to the observed separation energies of the two states of interest. The parameters chosen were r 0 = 1.25 fm, a = 0.65 fro, and kso = 25. This procedure yielded (r 2) = 22.91 fm 2, 18.72 fm 2 and 19.27 fm 2 for lg9/2, 2Pl/2, and 2P3/2 protons respectively. In previous analyses 3' and 7' were taken to be zero, so that ct =/3' and/3 = -t~'. Typical values obtained from such analyses [5] are a2 = 0.64 and /32 = 0.36 which yield a value of IMJ = 4.02 fm 2, 2.5 times the experimental value. Our analysis including a non-zero 7 was conducted as enumerated below. (1) The value of c~2 was determined from the absolute spectroscopic factor C2S for lg9/2 proton pick-up [5] on 9°Zr. (2) The values of/32 and 72 were determined from the sum of absolute spectroscopic strength (2Jr + l ) X C2S for 2Pl/2 and 2P3/2 proton stripping on 9°Zr respectively [7].
]89y, g.s.) = A(2Pl/2) 1 10) + B(2P3/2) -2 (2Pl/2) 1 (lg9/2) 210), where A 2 + B 2 = 1. (4) An initial value for B was determined by assuming that the lp = 1 strength to 9°Zr states higher than the first excited state was due entirely to 2P3/2 proton stripping. (5) An unnormalized value for A was found using the Pl/2 spectroscopic strength for populating the 9°Zr ground state by (r, d). The coefficients A and B were then normalized. (6) The ratio of spectroscopic strengths to the two 0 ÷ states from the (r, d) reaction then provided the third needed equation. (7) The values of a',/3' and 7' were determined and the monopole matrix element calculated. The results of this analysis are presented in table 1. It is seen that the inclusion of IP3/2 core excitation, which is required by single-proton transfer data, also removes the previous discrepancy between measured and calculated lifetime of the excited 0 ÷ state in 90Zr.
Table 1. Configurations* and transition matrix element for first two 0÷ states of 9°Zr
c~
/3
3,
a'
#'
V'
Mcalc
Mexp
0.632 0.8
0.626 0.6
0.456 0
0.290 + 0.6
- 0.737 - 0.8
0.611 0
1.85 4.02
1.60 -+0.04 1.60 ± 0.04
* 19°Zr(g.s.)> = a(2Pl/2) 210) +/3(lg9/2) 210> + 7(2P$/2) -2 (2pl/2) 2 (lg9/2) 210) 190Zr(1.76I)> = a'(2p 1/2)21 O> + 13'(Ig9/7.)21 O) + ~"(2P3/2)-2(2p i/2)2 (Ig9/~.)21 O>
Volume 41B, number 1
PHYSICS LETTERS
References [1] B. Fogelberg, Phys. Letters 37B (1971) 372. [2] S. Bjornholm, O.B. Nielsen and R.K. Sheline, Phys. Rev. 115 (1959) 1613. [3] R.B. Day, A.G. Blair and D.D. Armstrong, Phys. Letters 9 (1964) 327. [4] G. Vourvopoulos and J.D. Fox, Phys. Letters 25B (1967) 543; Phys. Rev. 177 (1969) 1558. [5] B.M. Preedom, E. Newman and J.C. I-liebert, Phys. Rev. 166 (1968) 1156. [6] D.C. Schreve, Univ. of Washington annual report (1968) p. 32. [7] M.R. Cares, J.B. Ball and E. Newman, Phys. Rev. 187 (1970) 1682.
4 September 1972
[8] K.T. Knopfle et al., Nucl. Phys. A159 (1970) 642. [9] G. Vourvopoulos et al., in, Nuclear isospon, eds. Anderson et al. (Academic Press, N.Y., 1970) p. 205. [10] B.F. Bayman, A.S. Reiner and R.K. Sheline, Phys. Rev. 115 (1959) 1627. [1 i] I. Talmi and I. Unna, Nucl. Phys. 19 (1960) 225. [12] J. Vervier, Nucl. Phys. 75 (1966) 17. [13] R. Thomas, Phys. Rev. 38 (1940) 714. [14] E.L. Church and J. Weneser, Phys. Rev. 100 (1955) 953; 103 (1956) 1035. [15] A.S. Reiner, Physica 23 (1957) 338. [16] D. Butch et al., Bull. Am. Phys. Soc. 17 (1972) 514 and to be published. [17] M. Nessin, T.H. Kruse and K,E. Eklund, Phys. Rev. 125 (1962) 639.
CONFIGURATION
PHYSICS LETTERS
4 September 1972
M I X I N G IN 90 Zr A N D T H E L I F E T I M E O F T H E E X C I T E D 0 + S T A T E * W. J. COURTNEY and H. T. FORTUNE University of Pennsylvania, Philadelphia,Pennsylvania 19104, USA Received 3 July 1972
Inclusion of p 3/2 core excitation in 90Zr removes the previous factor-of-6 discrepancy between measured and calculated lifetime for decay of the first excited 0÷ state. The existence of even nuclei whose first excited states have jTr = 0 + is a rare occurrence. At present only six such nuclei are known; 160, 40Ca, 72Ge, 90Zr, 96Zr and 98Mo. (Recent evidence [1 ] indicates that a seventh nucleide 98Zr may also exhibit this unusual property.) The relative abundance of 0 + first excited states in the mass 90 region has been attributed to mixing of [(2Pl/2) 2 (g9/2) Z-38"2] and (lg9/2) Z'38 proton configurations relative to a Z = 38 core. In particular, the coefficients of these two configurations in the 9°Zr ground state wave function and in the orthogonal 90Zr (0 +, 1.761 MeV state) wave function have been extensively investigated both experimental [ 2 - 9 ] and theoretically [ 1 0 - 1 2 ] . The common fault of the experimental analyses, as discussed by Schreve [6], is the assumption that no additional configuration mixing occurs. The observation [ 7 - 9 ] of lo = 1 strength to known 3 / 2 - states in the 90Zr (r, d) 91Nb reaction makes the validity of this assumption questionable. Indeed, when the monopole matrix element M for the EO transition between the two 9°Zr 0 + states is calculated [10] using the wave functions of those analyses, the value obtained is more than twice that deduced from the measured lifetime of the excited 90Zr state. The failure of the simple model wave functions to reasonably account for the observed lifetime has prompted the analysis presented in this paper. We show that inclusion of proton configurations of the type (2P3/2) -2 (2p 1/2) 2 (1 g9/2) 2, as suggested by the 9°Zr (r, d) results, yields a lifetime in good agreement with experiment. The electric monopole (EO) transition probability * Supported in part by the National Science Foundation.
for K-shell conversion WK has been derived by a number of authors [ 1 3 - 1 5 ] and can be written as
WK = B(Z)F(Z, E, R ) JM 12, where Z is the nuclear charge, E is the transition energy, R is the nuclear radius, B(Z) is a function describing the nuclear charge distribution, F(Z, E, R) is basically the Fermi beta decay function, and M is the matrix element of the monopole operator M, connecting the initial and final states. The monopole operator is usually approximated by A ' (1 + r3)r 2, i=1 where ½ (1 + r3) is the usual proton projection operator and ri is the radial position of the ith nucleon. Using the measured half-life [16] of the 90Zr (0 +, 1.761 MeV) state, tl/2 = 60.3 -+ 2.5 ns, and the measured branching ratios [ 17] for decay by internal conversion from higher atomic orbits [4/K/I41L+M+ N = 7.06 + 0.08 and by pair emission, WK/Wp = 2.08 -+ 0.08, together with a calculation of the functions B(Z)F(Z, E, R), the experimental monopole matrix element IMexp I was obtained. The value thus obtained is IMexp I = 1.60 -+ 0.04 fm 2. It is of interest to note that the extracted value of IMexp I does not depend very strongly on the choice of nuclear radius R = roA 1/3 used in calculating F(Z, E, R). In the present case r 0 = 1.25 fin was used. For the present analysis we define the particle vacuum to consist of a closed core of 38 protons and 50 neutrons. With this basis, allowing only proton excitations, the model waye function for the 90Zr ground state is taken to be
PHYSICS LETTERS
Volume 41B, number 1
(3) The values of a 2,/32 and ,f2 thus obtained were normalized to unity. Two of the three equations necessary to determine the coefficients of the model wave function for the first excited state are provided by the requirements of orthogonality and unit normalization. The third was obtained from the lp = 1 spectroscopic strength observed [4] to the two 0 + states in the 8 9 y ( r , d) reaction. The analysis was somewhat complicated by the suggestion of configurations of the type (2P3/2) -2 (2Pl/2) 1 (lg9/2) 2 necessary to account for the additional l, = 1 strength observed to higher excited states in the 8~y (r, d)90Zr reaction. With this in mind we took as the model wave function for the 89y ground state
190Zr, g.s.) = c¢(2p 1/2) 2 J0) + +/3(lg9/2) 2 IO) + 7(2P3/2)-2(2Pl/2)2(lg9/2) 210) where tx2 +/32 + 72 = 1. The model wave function for the 90Zr (0 +, 1.761 MeV) state is taken to consist of the same configurations, with coefficients a', /3' and 7', where cg 2 +/3'2 + 3,'2 = 1. It is a straightforward exercise to show that the monopole matrix element in this case is given by IMI =
2{aa'(r2)lgp/2 +/3~'(r2)2pl/2
4 September 1972
+T)t'(r2)2p3/2} ,
where (r 2)n0. is the matrix element of r 2 for a proton in the initial and final configurations with quantum numbers nlj. In the present work the (r2)nl/were calculated using bound-state wave functions obtained from a real Woods-Saxon potential well and derivative Woods-Saxon ~pin-orbit potential well. The well depth was adjusted to give the binding energy equal to the observed separation energies of the two states of interest. The parameters chosen were r 0 = 1.25 fm, a = 0.65 fro, and kso = 25. This procedure yielded (r 2) = 22.91 fm 2, 18.72 fm 2 and 19.27 fm 2 for lg9/2, 2Pl/2, and 2P3/2 protons respectively. In previous analyses 3' and 7' were taken to be zero, so that ct =/3' and/3 = -t~'. Typical values obtained from such analyses [5] are a2 = 0.64 and /32 = 0.36 which yield a value of IMJ = 4.02 fm 2, 2.5 times the experimental value. Our analysis including a non-zero 7 was conducted as enumerated below. (1) The value of c~2 was determined from the absolute spectroscopic factor C2S for lg9/2 proton pick-up [5] on 9°Zr. (2) The values of/32 and 72 were determined from the sum of absolute spectroscopic strength (2Jr + l ) X C2S for 2Pl/2 and 2P3/2 proton stripping on 9°Zr respectively [7].
]89y, g.s.) = A(2Pl/2) 1 10) + B(2P3/2) -2 (2Pl/2) 1 (lg9/2) 210), where A 2 + B 2 = 1. (4) An initial value for B was determined by assuming that the lp = 1 strength to 9°Zr states higher than the first excited state was due entirely to 2P3/2 proton stripping. (5) An unnormalized value for A was found using the Pl/2 spectroscopic strength for populating the 9°Zr ground state by (r, d). The coefficients A and B were then normalized. (6) The ratio of spectroscopic strengths to the two 0 ÷ states from the (r, d) reaction then provided the third needed equation. (7) The values of a',/3' and 7' were determined and the monopole matrix element calculated. The results of this analysis are presented in table 1. It is seen that the inclusion of IP3/2 core excitation, which is required by single-proton transfer data, also removes the previous discrepancy between measured and calculated lifetime of the excited 0 ÷ state in 90Zr.
Table 1. Configurations* and transition matrix element for first two 0÷ states of 9°Zr
c~
/3
3,
a'
#'
V'
Mcalc
Mexp
0.632 0.8
0.626 0.6
0.456 0
0.290 + 0.6
- 0.737 - 0.8
0.611 0
1.85 4.02
1.60 -+0.04 1.60 ± 0.04
* 19°Zr(g.s.)> = a(2Pl/2) 210) +/3(lg9/2) 210> + 7(2P$/2) -2 (2pl/2) 2 (lg9/2) 210) 190Zr(1.76I)> = a'(2p 1/2)21 O> + 13'(Ig9/7.)21 O) + ~"(2P3/2)-2(2p i/2)2 (Ig9/~.)21 O>
Volume 41B, number 1
PHYSICS LETTERS
References [1] B. Fogelberg, Phys. Letters 37B (1971) 372. [2] S. Bjornholm, O.B. Nielsen and R.K. Sheline, Phys. Rev. 115 (1959) 1613. [3] R.B. Day, A.G. Blair and D.D. Armstrong, Phys. Letters 9 (1964) 327. [4] G. Vourvopoulos and J.D. Fox, Phys. Letters 25B (1967) 543; Phys. Rev. 177 (1969) 1558. [5] B.M. Preedom, E. Newman and J.C. I-liebert, Phys. Rev. 166 (1968) 1156. [6] D.C. Schreve, Univ. of Washington annual report (1968) p. 32. [7] M.R. Cares, J.B. Ball and E. Newman, Phys. Rev. 187 (1970) 1682.
4 September 1972
[8] K.T. Knopfle et al., Nucl. Phys. A159 (1970) 642. [9] G. Vourvopoulos et al., in, Nuclear isospon, eds. Anderson et al. (Academic Press, N.Y., 1970) p. 205. [10] B.F. Bayman, A.S. Reiner and R.K. Sheline, Phys. Rev. 115 (1959) 1627. [1 i] I. Talmi and I. Unna, Nucl. Phys. 19 (1960) 225. [12] J. Vervier, Nucl. Phys. 75 (1966) 17. [13] R. Thomas, Phys. Rev. 38 (1940) 714. [14] E.L. Church and J. Weneser, Phys. Rev. 100 (1955) 953; 103 (1956) 1035. [15] A.S. Reiner, Physica 23 (1957) 338. [16] D. Butch et al., Bull. Am. Phys. Soc. 17 (1972) 514 and to be published. [17] M. Nessin, T.H. Kruse and K,E. Eklund, Phys. Rev. 125 (1962) 639.