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On the J-dependence of excitation energies in fp-shell nuclei
Volume 104B, number 4
PHYSICS LETTERS
3 September 1981
ON THE J-DEPENDENCE OF EXCITATION ENERGIES IN fp-SHELL NUCLEI R.B.M. MOOY, P.W.M. GLAUDEMANS and A.G.M. van HEES Fysisch Laboratorium, Rijksuniversiteit, Utrecht, The Netherlands Received 24 March 1981 Revised manuscript received 25 June 1981
More than twenty nuclei with A = 52-60 are successfully descried by the shell model. The excitation energies show a systematic dependence on the spin J, i.e. when they are plotted versus the spinJ a parabola-shaped curve results. This shape can be well explained with statistical assumptions as a consequence of the J-dependence of the dimensions of the configuration space and the J-dependence of the T = 1 part of the two-body interaction.
It has been shown [ 1 - 5 ] that many properties of nuclei in the middle of the fp-shell can be understood well with the shell model. Ref. [1] gives a brief summary of early shell-model calculations on some Fe and Ni nuclides. The influence of fT/2"h°le admixtures on properties of Ni nuclides is studied in ref. [2]. Calculations including high-spin states in Fe nuclides are described in refs. [3,4]. An extensive study o f A = 53 nuclei is presented in ref. [5]. In refs. [ 3 - 5 ] some collective aspects are discussed. These investigations made it tempting to look for a possible systematic behavior of some properties. In the present paper the behavior of excitation energies for nuclei in the mass region A = 5 2 - 6 0 is discussed. The calculations are performed in an fp-shell configuration space. The configurations taken into account can be specified by f - n r m , where f denotes the f7/2 orbit and r stands for any of the other fpshell orbits. The numbers n and m represent the minimum number of holes and particles, respectively, needed to describe the low-lying states of a specific nucleus. The numbers n and m are defined with respect to the doubly ma~ic 56Ni core. For instance one hasn, m = 3, 0 for .,3 26Fe27 and n, m = 1, 3 for 58Co31. It has been shown [ 1 - 5 ] that this space is too limited and should be extended by taking into account all configurations with an additional f7/2 hole. Hence we included also the particle-hole excited configurations given by f - n - 1 r m + 1.
From previous investigations with various interactions it became clear [ 1 - 5 ] that renormalized K u o Brown (KB) matrix elements [6] are well suited when the f7/2-hole structure dominates the properties, whereas the schematic SDI matrix elements [7] yield better results when the particles in the other fp-shell orbits strongly determine the calculated observables. The results presented here are obtained with an interaction that contains empirically determined _f2/2 matrix elements. For the two-body matrix elements and the KB values are used, while for the remaining set of matrix elements
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
251
Volume 104B, number 4
PHYSICS LETTERS
3 September 1981
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Volume 104B, number 4
PHYSICS LETTERS
producing other experimental data such as electromagnetic properties [8]. The systematics of various calculated observables suggest the existence of several band-like structures in many of the nuclei considered. The spectra of 18 nuclei are shown in fig. 1. The calculated excitation energies o f the lowest four states for each J are plotted as a function of J. The following features can be observed: (i) A parabola-like shape, to be discussed below, emerges in particular for the states somewhat above the yrast line. (ii) The even-J states for rather low J-values in e v e n - e v e n nuclei have a relatively low excitation energy, since only these states can contain large admixtures of components with the low seniorities v = 0 or 2. These even-J states can be formed b y the breakup of at most one zero-coupled pair and are thus favoured by the nuclear interaction. All states with other J values have seniority o ~> 4. (iii) F o r odd-A nuclei the position of the minimum of the yrast line is determined by the v = 1 state with the unpaired particle in the lowest available singleparticle orbit. (iv) F o r o d d - o d d nuclei the shape of the yrast line for low-spin states is sometimes irregular. The structure of these states is dominated by the simplest v = 2 shell-model configurations with one active proton and one active neutron. The parabola-like shapes, which show up most clearly for states somewhat above the yrast line (fig. 1), can be explained rather well with a simple statistical approach. Let us assume that for given spin value J the shell-model state density, i.e. the number of states per energy interval, is gaussian [9]. It has been shown [10,11] that this assumption is quite well justiffed. In addition, since only a few quantities thus determine the shape, it facilitates a physical understanding of the E x - J plots. The assumption o f a pure gaussian distribution allows a calculation of the position of levels from three quantities, i.e. the dimension o f the configuration space the centroid and the width o f the distribution. The centroid and width can be calculated [9], with the following expressions
o(J)
d(J),
EO(J) =d-~ ~i Hii(S)'
EO(J)
1
(1)
3 September 1981
~i/H2(j)-E2(j),
o 2 ( J ) = d~j)
(1 con'd)
Hij(J)
where denotes the matrix elements of the manyparticle hamiltonian. The energy o f the nth state of spin J denoted by (n = 1 for the yrast state) can be obtained from the gaussian density distribution with the relation [9]
En(J)
( ~
(- [r-Eo(J)]21~]
~(J)_
exp \
d r = n - 1/2.
(7
(2)
d(J)
The dimensions depend strongly o n J . The centroid and width show a weaker dependence on J as can be seen from fig. 2, for typical ex-
EO(J)
~oood t t
(7(J)
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Fig 2. Calculated J-dependence of various quantities for the nuclei 53Cr, S6Fe and SaCo. In the top figure the dimension of the complete configuration spaces (d) and (ten times) the dimension of the spaces without particle-hole excitation (10 d s) are plotted versusJ. The middle figure shows (i) the width a(J) of the distribution of eigenvalues (solid curve, o), (ii) the width ff only the T = 0 part of the interaction is taken into account (dotted curve, T = 0) and (iii) similarly the width for the T = 1 part of the interaction (dashed curve, T = 1). In the bottom figure are shown the eentroid the yrast line obtained from eq. (2) [dotted curve (1)], and the yrast line obtained after diagonalization of the energy matrix [solid curve (2)1. The dashed curve (3) represents the results obtained from eq. (2) with constant E 0 and a(see text). The circles denote the lowest states with at least 50% admixtures of single particle-hole excitations.
EO(J),
253
Volume 104B, number 4
PHYSICS LETTERS
amples of an even-odd, an even-even and an odd-odd nucleus. The separate and combined effects of the dimension, width and centroid on the calculated shape of the yrast line can be judged from a comparison of curves (1), (2) and (3) given in fig. 2. The statistical approach with the assumption of a purely gaussian density distribution yields the yrast line given by curve (1). In particular for higher J-values curve (1), calculated with eqs. (1) and (2), reproduces very well curve (2), i.e. the position of the yrast states obtained from a diagonalization of the hamiltonian. It should be remarked that the dimensions of the configuration space for the particle-hole excited states f - n - 1 rm+l are much larger than those of the much simpler fnrm space, as is illustrated in fig. 2. The values o f d ( J ) , EO(J) and o(J) obtained from eq.'(1) for the combined space are thus almost completely determined by the particle-hole excited configurations. Hence it is not surprising that the yrast states of the latter space can be reproduced well by the present statistical approach. This is illustrated in fig. 2, where the energy of the lowest state with at least 5 0% intensity of the f - n - 1 r m + 1 configurations has been plotted versus J. Note that the very good agreement found for high-spin states follows from the fact that these states can be formed by particle-hole excited configurations only. Since the statistical approach is seen to reproduce well the excitation energies of the lower particle-hole excited states, we can investigate the origin of the Parabola-like shapes of the E x - J plots. The statistical method shows that the J-dependence of the dimension quite strongly determines the observed shape. This follows from a comparison of curves (1) and (3). The latter is obtained from eq. (2) by keeping the values of the width and centroid fLxed. In this case o(J) and EO(J) are evaluated for the spin that yields the maximum dimension d(J). The width o(J) has also an important effect on the shape of the Ex-J plot. The observed J-dependence of the width, which changes from about 3.5 MeV for l o w J to about 2 MeV for high J may be qualitatively understood from general properties of T = 0 and T = 1 two-body matrix elements. We first investigate the effect of the T = 1 part of the matrix elements. When the T = 0 interaction is switched off, the width is determined (apart from 254
3 September 1981
the single-particle energies) by the T = 1 part of the two-body interaction. The absolute values of the diagonal T = 1 two-body matrix elements are large for low J and small for high J as follows from plots [12] of these matrix elements versusJ. A similar behaviour is expected [7] also for off-diagonal twobody matrix elements. The many-particle hamiltonian matrix elementsHii(J) used in eq. (1) can be expressed as a linear combination of two-body matrix elements. The Hi/(J) for high J-values are dominated by contributions from two-body matrix elements of high J-value. Since the latter are small one expects from eq. (1) that also o(J) for T = 1 becomes small for high J-values, which is illustrated in fig. 2. For the T = 0 part of the interaction the situation is more complex. The two-body matrix elements are large for low and for high J-values and small for intermediate J-values [12]. As a result one may expect no clear J-dependence of the width for the T = 0 part of the interaction. This is confirmed by the exactly calculated width o for T = 0, see fig. 2. The combined effects of the T = 1 and T = 0 two-body matrix elements may thus explain the decrease of o(J) for increasing J. The J-dependence of the centroid E 0 is found to be rather small for most nuclei and also depends mainly on the T = 1 part of the interaction. Thus the shape of the E x - J plot depends mainly on the Jdependence of dimensions and widths. It should be remarked that the dimensions of the hamiltonian matrices also depend strongly on the model space employed. For example allowing more f7/2 holes, one would obtain much larger matrices. For a much larger model space than the one used presently the maximum dimensions shift to somewhat higher J-values. Consequently the minima of the parabolas shift to higher J-values. However, changes in widths and centroids as well as possible deviations from a purely gaussian density distribution make it more difficult to draw conclusions about the behaviour of the minima for much larger model spaces. Concluding we can say that the parabola-like shape of the excitation energies as a function of J is mainly the result of the J-dependence of the dimensions of the hamiltonian matrices and the T = 1 part of the residual interaction as follows from a statistical approach. The deviations from these shapes in the low-spin low-energy region can be accounted for by
Volume 104B, number 4
PHYSICS LETTERS
large admixtures in the wave functions of a small number of components with very low seniority. More experimental information on the position of high-spin states is needed to test their presently predicted excitation energies. We thank J.J.M. Verbaarschot, P.J. Brussaard and B.C. Metsch for stimulating discussions on the subject. This work was performed as part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (ZWO).
References
3 September 1981
[2] A.G.M. van Hees, P.W.M. Glaudemans and B.C. Metsch, Z. Phys. A293 (1979) 327. [3] R. Vennink and P.W.M. Glaudemans, Z. Phys. A294 (1979) 241. [4] P.W.M. Glaudemans and R. Vennink, Phys. Lett. 95B (1980) 171. [5] B.C. Metsch and P.W.M. Glaudemans, Nucl. Phys. A352 (1981) 60. [6] T.T.S. Kuo and G.E. Brown, Nucl. Phys. A l l 4 (1968) 241. [7] P.J. Brussaard and P.W.M. Glaudemans, Shell-model applications in nuclear spectroscopy (North-Holland, Amsterdam, 1977). [8] R.B.M. Mooy, to be published. [9] K.F. Ratcliff, Phys. Rev. C3 (1971) 117. [10] J.N. Ginocchio and M.M. Yen, Nucl. Phys. A239 (1975) 365. [11] F.S. Chang and A. Zuker, Nucl. Phys. A198 (1972) 417. [12] J.P. Schiffer, Ann. Phys. 66 (1971) 798.
[ 1] P.W.M. Glaudemans, Inst. Phys. Conf. Set. No. 49 (1979) 11.
255
PHYSICS LETTERS
3 September 1981
ON THE J-DEPENDENCE OF EXCITATION ENERGIES IN fp-SHELL NUCLEI R.B.M. MOOY, P.W.M. GLAUDEMANS and A.G.M. van HEES Fysisch Laboratorium, Rijksuniversiteit, Utrecht, The Netherlands Received 24 March 1981 Revised manuscript received 25 June 1981
More than twenty nuclei with A = 52-60 are successfully descried by the shell model. The excitation energies show a systematic dependence on the spin J, i.e. when they are plotted versus the spinJ a parabola-shaped curve results. This shape can be well explained with statistical assumptions as a consequence of the J-dependence of the dimensions of the configuration space and the J-dependence of the T = 1 part of the two-body interaction.
It has been shown [ 1 - 5 ] that many properties of nuclei in the middle of the fp-shell can be understood well with the shell model. Ref. [1] gives a brief summary of early shell-model calculations on some Fe and Ni nuclides. The influence of fT/2"h°le admixtures on properties of Ni nuclides is studied in ref. [2]. Calculations including high-spin states in Fe nuclides are described in refs. [3,4]. An extensive study o f A = 53 nuclei is presented in ref. [5]. In refs. [ 3 - 5 ] some collective aspects are discussed. These investigations made it tempting to look for a possible systematic behavior of some properties. In the present paper the behavior of excitation energies for nuclei in the mass region A = 5 2 - 6 0 is discussed. The calculations are performed in an fp-shell configuration space. The configurations taken into account can be specified by f - n r m , where f denotes the f7/2 orbit and r stands for any of the other fpshell orbits. The numbers n and m represent the minimum number of holes and particles, respectively, needed to describe the low-lying states of a specific nucleus. The numbers n and m are defined with respect to the doubly ma~ic 56Ni core. For instance one hasn, m = 3, 0 for .,3 26Fe27 and n, m = 1, 3 for 58Co31. It has been shown [ 1 - 5 ] that this space is too limited and should be extended by taking into account all configurations with an additional f7/2 hole. Hence we included also the particle-hole excited configurations given by f - n - 1 r m + 1.
From previous investigations with various interactions it became clear [ 1 - 5 ] that renormalized K u o Brown (KB) matrix elements [6] are well suited when the f7/2-hole structure dominates the properties, whereas the schematic SDI matrix elements [7] yield better results when the particles in the other fp-shell orbits strongly determine the calculated observables. The results presented here are obtained with an interaction that contains empirically determined _f2/2 matrix elements. For the two-body matrix elements
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
251
Volume 104B, number 4
PHYSICS LETTERS
3 September 1981
0
e~
t ~r
~(A~I~I)
x3
f
t
[ k X
~(A~IN ) x-~
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x~/~F~.
.
~
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~, o
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.
II
.~
Volume 104B, number 4
PHYSICS LETTERS
producing other experimental data such as electromagnetic properties [8]. The systematics of various calculated observables suggest the existence of several band-like structures in many of the nuclei considered. The spectra of 18 nuclei are shown in fig. 1. The calculated excitation energies o f the lowest four states for each J are plotted as a function of J. The following features can be observed: (i) A parabola-like shape, to be discussed below, emerges in particular for the states somewhat above the yrast line. (ii) The even-J states for rather low J-values in e v e n - e v e n nuclei have a relatively low excitation energy, since only these states can contain large admixtures of components with the low seniorities v = 0 or 2. These even-J states can be formed b y the breakup of at most one zero-coupled pair and are thus favoured by the nuclear interaction. All states with other J values have seniority o ~> 4. (iii) F o r odd-A nuclei the position of the minimum of the yrast line is determined by the v = 1 state with the unpaired particle in the lowest available singleparticle orbit. (iv) F o r o d d - o d d nuclei the shape of the yrast line for low-spin states is sometimes irregular. The structure of these states is dominated by the simplest v = 2 shell-model configurations with one active proton and one active neutron. The parabola-like shapes, which show up most clearly for states somewhat above the yrast line (fig. 1), can be explained rather well with a simple statistical approach. Let us assume that for given spin value J the shell-model state density, i.e. the number of states per energy interval, is gaussian [9]. It has been shown [10,11] that this assumption is quite well justiffed. In addition, since only a few quantities thus determine the shape, it facilitates a physical understanding of the E x - J plots. The assumption o f a pure gaussian distribution allows a calculation of the position of levels from three quantities, i.e. the dimension o f the configuration space the centroid and the width o f the distribution. The centroid and width can be calculated [9], with the following expressions
o(J)
d(J),
EO(J) =d-~ ~i Hii(S)'
EO(J)
1
(1)
3 September 1981
~i/H2(j)-E2(j),
o 2 ( J ) = d~j)
(1 con'd)
Hij(J)
where denotes the matrix elements of the manyparticle hamiltonian. The energy o f the nth state of spin J denoted by (n = 1 for the yrast state) can be obtained from the gaussian density distribution with the relation [9]
En(J)
( ~
(- [r-Eo(J)]21~]
~(J)_
exp \
d r = n - 1/2.
(7
(2)
d(J)
The dimensions depend strongly o n J . The centroid and width show a weaker dependence on J as can be seen from fig. 2, for typical ex-
EO(J)
~oood t t
(7(J)
C r 110001 , ~ , Fe hoooq / 24 2. I ~ / \25 30 / ~ /
~
\
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Fig 2. Calculated J-dependence of various quantities for the nuclei 53Cr, S6Fe and SaCo. In the top figure the dimension of the complete configuration spaces (d) and (ten times) the dimension of the spaces without particle-hole excitation (10 d s) are plotted versusJ. The middle figure shows (i) the width a(J) of the distribution of eigenvalues (solid curve, o), (ii) the width ff only the T = 0 part of the interaction is taken into account (dotted curve, T = 0) and (iii) similarly the width for the T = 1 part of the interaction (dashed curve, T = 1). In the bottom figure are shown the eentroid the yrast line obtained from eq. (2) [dotted curve (1)], and the yrast line obtained after diagonalization of the energy matrix [solid curve (2)1. The dashed curve (3) represents the results obtained from eq. (2) with constant E 0 and a(see text). The circles denote the lowest states with at least 50% admixtures of single particle-hole excitations.
EO(J),
253
Volume 104B, number 4
PHYSICS LETTERS
amples of an even-odd, an even-even and an odd-odd nucleus. The separate and combined effects of the dimension, width and centroid on the calculated shape of the yrast line can be judged from a comparison of curves (1), (2) and (3) given in fig. 2. The statistical approach with the assumption of a purely gaussian density distribution yields the yrast line given by curve (1). In particular for higher J-values curve (1), calculated with eqs. (1) and (2), reproduces very well curve (2), i.e. the position of the yrast states obtained from a diagonalization of the hamiltonian. It should be remarked that the dimensions of the configuration space for the particle-hole excited states f - n - 1 rm+l are much larger than those of the much simpler fnrm space, as is illustrated in fig. 2. The values o f d ( J ) , EO(J) and o(J) obtained from eq.'(1) for the combined space are thus almost completely determined by the particle-hole excited configurations. Hence it is not surprising that the yrast states of the latter space can be reproduced well by the present statistical approach. This is illustrated in fig. 2, where the energy of the lowest state with at least 5 0% intensity of the f - n - 1 r m + 1 configurations has been plotted versus J. Note that the very good agreement found for high-spin states follows from the fact that these states can be formed by particle-hole excited configurations only. Since the statistical approach is seen to reproduce well the excitation energies of the lower particle-hole excited states, we can investigate the origin of the Parabola-like shapes of the E x - J plots. The statistical method shows that the J-dependence of the dimension quite strongly determines the observed shape. This follows from a comparison of curves (1) and (3). The latter is obtained from eq. (2) by keeping the values of the width and centroid fLxed. In this case o(J) and EO(J) are evaluated for the spin that yields the maximum dimension d(J). The width o(J) has also an important effect on the shape of the Ex-J plot. The observed J-dependence of the width, which changes from about 3.5 MeV for l o w J to about 2 MeV for high J may be qualitatively understood from general properties of T = 0 and T = 1 two-body matrix elements. We first investigate the effect of the T = 1 part of the matrix elements. When the T = 0 interaction is switched off, the width is determined (apart from 254
3 September 1981
the single-particle energies) by the T = 1 part of the two-body interaction. The absolute values of the diagonal T = 1 two-body matrix elements are large for low J and small for high J as follows from plots [12] of these matrix elements versusJ. A similar behaviour is expected [7] also for off-diagonal twobody matrix elements. The many-particle hamiltonian matrix elementsHii(J) used in eq. (1) can be expressed as a linear combination of two-body matrix elements. The Hi/(J) for high J-values are dominated by contributions from two-body matrix elements of high J-value. Since the latter are small one expects from eq. (1) that also o(J) for T = 1 becomes small for high J-values, which is illustrated in fig. 2. For the T = 0 part of the interaction the situation is more complex. The two-body matrix elements are large for low and for high J-values and small for intermediate J-values [12]. As a result one may expect no clear J-dependence of the width for the T = 0 part of the interaction. This is confirmed by the exactly calculated width o for T = 0, see fig. 2. The combined effects of the T = 1 and T = 0 two-body matrix elements may thus explain the decrease of o(J) for increasing J. The J-dependence of the centroid E 0 is found to be rather small for most nuclei and also depends mainly on the T = 1 part of the interaction. Thus the shape of the E x - J plot depends mainly on the Jdependence of dimensions and widths. It should be remarked that the dimensions of the hamiltonian matrices also depend strongly on the model space employed. For example allowing more f7/2 holes, one would obtain much larger matrices. For a much larger model space than the one used presently the maximum dimensions shift to somewhat higher J-values. Consequently the minima of the parabolas shift to higher J-values. However, changes in widths and centroids as well as possible deviations from a purely gaussian density distribution make it more difficult to draw conclusions about the behaviour of the minima for much larger model spaces. Concluding we can say that the parabola-like shape of the excitation energies as a function of J is mainly the result of the J-dependence of the dimensions of the hamiltonian matrices and the T = 1 part of the residual interaction as follows from a statistical approach. The deviations from these shapes in the low-spin low-energy region can be accounted for by
Volume 104B, number 4
PHYSICS LETTERS
large admixtures in the wave functions of a small number of components with very low seniority. More experimental information on the position of high-spin states is needed to test their presently predicted excitation energies. We thank J.J.M. Verbaarschot, P.J. Brussaard and B.C. Metsch for stimulating discussions on the subject. This work was performed as part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (ZWO).
References
3 September 1981
[2] A.G.M. van Hees, P.W.M. Glaudemans and B.C. Metsch, Z. Phys. A293 (1979) 327. [3] R. Vennink and P.W.M. Glaudemans, Z. Phys. A294 (1979) 241. [4] P.W.M. Glaudemans and R. Vennink, Phys. Lett. 95B (1980) 171. [5] B.C. Metsch and P.W.M. Glaudemans, Nucl. Phys. A352 (1981) 60. [6] T.T.S. Kuo and G.E. Brown, Nucl. Phys. A l l 4 (1968) 241. [7] P.J. Brussaard and P.W.M. Glaudemans, Shell-model applications in nuclear spectroscopy (North-Holland, Amsterdam, 1977). [8] R.B.M. Mooy, to be published. [9] K.F. Ratcliff, Phys. Rev. C3 (1971) 117. [10] J.N. Ginocchio and M.M. Yen, Nucl. Phys. A239 (1975) 365. [11] F.S. Chang and A. Zuker, Nucl. Phys. A198 (1972) 417. [12] J.P. Schiffer, Ann. Phys. 66 (1971) 798.
[ 1] P.W.M. Glaudemans, Inst. Phys. Conf. Set. No. 49 (1979) 11.
255