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Effect of γ-softness on continuum gamma-ray spectra
Volume 150B, number 1,2,3
EFFECT OF 7SOFTNESS
3 January 1985
PHYSICS LETTERS
ON CONTINUUM GAMMA-RAY
SPECTRA
Ikuko HAMAMOTO and Naoki ONISHI ’ Department
of Mathematical Physics, Lund Institute of Technology,
Lund, Sweden
Received 26 September 1984
In the case that a nuclear system has a large fluctuation in the direction of triaxiality, we examine the possible feature expected to appear in the continuum gamma-ray spectra, especially a possibility of the filling in the central valley of the two-dimensional gammaenergy coincidence spectra.
The quantitative information on the yrast spectroscopy of nuclear high-spin states comes from the analysis of discrete gamma-rays emitted from evaporation residues in (HI, xn) reactions. The highest spin (the highest rotational frequency) obtained in this kind of analysis is at present Z 5 45 (Rwrot 5 550 keV). The information on the nuclear structure at still higher spin is contained in the unresolved or “continuum” gamma-ray spectra. Experimental techniques and ways of analyzing data to get nuclear structure information from continuum gamma-ray spectra are being rapidly developed [l] . One of the most popular ways to analyze the data is to plot the two-dimensional gammaenergy coincidence spectra [2] as a function of gamma-ray energies, E1 and E2. The detailed structure of the transition-energy correlation spectra depends on both the nuclei and the way how they are formed. However, the following two points seem to be fairly general: (1) A valley along the diagonal line (namely, El = E2) is clearly seen up to a certain energy (-1 MeV). (2) Above that energy (“1 MeV) the valley tends to be filled and at the same time the location of the first ridges becomes less clear. The filling in the central valley is often interpreted [3] as an indication of the interband transitions (at band crossings), assuming that there are only collective transitions (namely, AZ = 2 E2, transitions). r Permanent address: Institute of Physics, College of General Education, University of Tokyo, Komaba, Meguro, Tokyo, Japan.
6
In the present letter we want to examine the possibility that a part of the filling in the central valley can be used by the correlation between stretched (AZ = 2) transitions and non-stretched (AZ # 2) transitions if the system has a large fluctuation in the direction of triaxiality (i.e. y-variable). (The filling in the central valley is generally expected, when the fluctuation in nuclear shape is considerable so that the nuclear quadrupole deformation and, thus, collective AZ = 2 transitions within “a band” are no more well defined.) In the following we take into account only quadrupole deformation. If the potential energy function has a well-defined minimum at (PO # 0, r. # 0), the low-energy quadrupole degrees of freedom separate into the (/3- and y-) vibrational and the rotational components. The rotational components, namely the motion of the asymmetric rotor and the related E2 transition matrix elements have been discussed, for example, in ref. [4]. The calculation shows that the collective AZ = 2 E2transition matrix elements within bands are much larger than other E2 transition matrix elements. Thus, depending on y- and Z-values, the ridges in the correlation spectra can be smeared out a little, but the filling in the central valley is hardly obtained, in any case in the region of El = E2 - 1 MeV. On the other hand, if the potential-energy function is very flat in /3-and/or y-direction, the rotational components are no more well-separated from the vibrational components. In the following we take a macroscopic Bohr model [5] and we are only concerned 0370-2693/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 150B, number 1,2,3
PHYSICS LETTERS
with the deviation from axial-symmetry by assuming that 13is fixed at some constant value, fl0- Thus, our hamiltonian is written as H - - ~ 0 + V(3'),
(1)
where H 0=T+Tro _
T
t,
]~2
(2) 1
2B132 sin 33'
" 3 ~, "-~ sin 3'~-~-
O
(3)
~/2n2 Tro t =
2Jk,
(4)
Jk(7) = 4B~32 sin2(3' - ~ztk).
(5)
The potential V(7) is a function of cos 33' due to the symmetry argument [5] and, thus, is expanded as V(3') = / ~ Vlel(cos 33').
(6)
First, we consider the V(3') = 0 case, namely the 3'unstable case. It is known that the eigenfunctions of H 0 in (2) have the form of [6]
O ~ ' ( 3 " ) = D C ~n lhn^'" - t c o s 33")ng~K(') ') nja
(7)
with the eigenvalues (~2/2B/j2) 3`(3`+ 3).
(8)
For a given I one has 3 ` = 3 n ,,/ +/a,
(9)
where n v takes non-negative integers, and
la = 1/2,1/2 + 1 .... , I =(1+3)/2,(1+3)/2+1
for even I
..... I
for odd I.
(10)
Since the E2 operator is proportional to the eigenfunction G 10, it has non-zero matrix elements only between the states with 2x3`= 1 and A1 ~< 2. If the shape fluctuation of the relevant states is really large, there is no appreciable difference between the sizes of the E2 transition matrix elements with A1 = 2 and A1 :/= 2. However, for the yrast states with larger/-values the shape with T = 30° is more favoured due to the special form of the expressions (4) and (5). Then, for such states the "collective" E2 transitions with A1 = 2
3 January 1985
become a predominant decay mode. In fact, we have found that while around the yrast states in the region o f / = 2 0 - 3 0 the allowed AI = 2 E2 transition matrix elements are larger by a factor 2 - 3 than allowed A I :~ 2 ones, the difference between the two kinds of matrix elements (i.e. A I = 2 and ~ r 4: 2) becomes smaller as one goes to higher excitation energies above the yrast lines. Now, it is noted that the energy of the transition from the state with (I, X) to the states with (I', X - 1) is equal to (h2/2B/~20)2(3, + 1) irrespective of the angular momentum (I') of the final state. Therefore, though in the whole deexcitation process the number of the non-stretched E2 transitions with particular transition energies could well be appreciable, the two-dimensional gamma-ray energy correlation spectra show the same pattern as that of a perfect rotor. Namely, the intensity is zero along the diagonal, and the ridge structure and the valley structure appear perfectly. For the nuclear shape, which is not completely 3'unstable but very 3'-soft, one may expect both an appreciable amount of the non-stretched E2 transitions and energy spectra which are not so regular as those in (8). Thus, how one can quantitatively expect the filling in the central valley is worthwhile to be examined. We have diagonalized the hamiltonian (1) in terms of the eigenfunctions of H0, G~'/~v in (7). By using the resulting eigenfunctions and energies we calculate the B(E2)-values and the transition probabilities of all possible E2 transitions. Then, by assuming that an entry-line of the gamma-ray cascade lies in the region w i t h I 1 <~I<~I 2 and n 1 ~
Volume 150B, number 1,2,3
PHYSICS LETTERS
3 January 1985
E --- E~+ E2 i n ( z'~'
COUNTS
10,8-< E 512.0
5,6
2000
1000
8,L~E~9,6
3000 13,2<- E <-- 14/+ 2000
1000
o
~ 0
r
0,5
I,O
o
o,5
1,o
r
(EcEz) in
2fl z
(BI3~ J
Fig. 1. Cuts of the contour plot of the estimated two-dimensional gamma-energy correlation spectra across the diagonal in the energy regions indicated. The correlation obtained by using only stretched E2 gamma-rays is shown by the shaded area. See text for the details.
values around the yrast line are shown in fig. 2b. Though the minimum of the potential energy lies at 3' = 0 °, it is seen that at moderately higher spin the energy spectrum shows a clear tendency toward that of a
"/-unstable shape. For reference to the energy scale used, we note that the calculated transition energy from the lowest 2 + state to the ground state is 0.60 in units of 2h2/B[32. In fig. 1 the correlation obtained by
Volume 150B, number 1,2,3
PHYSICS LETTERS
3 January 1985
a 2h---L-z)i V(v) in(~BI3~
0
1'5°
- 5~ -I0
30°
45*
t
b ENERGY in ~,BOo2] J
110 100 90 80 - -
w
- -
70 - -
m
6O
50
- -
" ~
_ _
~
Z
B
- -
w
D
- -
- -
- -
p
m
- m
/,0 - -
D
30
-
_ _
- -
- -
m
n
m
- -
B
m
20
E m
m
w
10
o
_
-o
--
2
t.
6
8
10
12
1~
16
18
20
22
2~
26
>I Fig. 2. (a) The potential Y(~,),which is used in the present numericalexamples. (b) Estimated energyeigenvaluesaround the yrast line. using only stretched E2 gamma-rays is shown by the shaded area. It is seen that the filling in the central valley comes totally from the correlation between the stretched ( A I = 2) and the non-stretched ( A I :/= 2) transitions, the major part of which comes from the A1 = 0 transitions. (In the present model calculation the probability in which two non-stretched transitions occur in a given cascade is pretty small.) When we take a different entry-line, for example 11 = 20,12 = 24
and/or n 1 = 3, n 2 = 8, a very similar pattern of filling in the central valley is obtained as a function o f E 1 + E2, and the only difference which appears is the height of the ridges (namely, the number of the correlations between the stretched transitions). A choice of a different potential V(7), for example, the set ( V 1 = ---6.0, V2 = +3.0) in units of 2h2/B~2 produces a pattern of filling in the central valley which is qualitatively very similar to fig. 1.
Volume 150B, number 1,2,3
PHYSICS LETTERS
The values of angular momentum I in the present calculation should not be directly compared with experimental Z-values, since a number of important degrees of freedom are missing in our model calculation and, especially, we have not included aligned angular momentum of particles. A given deexcitation process in the present calculation such as the one shown in fig. 1 is accompanied with a large number of particle configurations. If we assume that the aligned particles behave as spectators, then, around yrast states, we may just add the total amount of the aligned angular momentum of particles in each configuration to our present Z-values, in order to obtain experimental Z-values which can be compared. Though there has been no clear-cut evidence for the deviation of nuclear shape from axial-symmetry around the ground-states of medium or heavy nuclei, the deviation is generally expected at high spin. There have been publications [7] in which the result of the deviation in discrete gamma-ray yrast-spectroscopy is discussed in connection with available experimental data. The present calculation is clearly too simple to simulate the nuclear system at high spin well above the yrast line, but it shows one of the consequences of the deviation of nuclear shape from axial-symmetry, which are expected to appear in the analysis of the continuum gamma-rays. In order to obtain the filling in the central valley, we need: (1) the nuclear shape is very r-soft but not completely y-unstable, (2) the transition energies have to be larger than certain values, and (3) the deexcitation process goes
10
3 January 1985
through states which he appreciably higher than the yrast line. We note that the contribution of the nonstretched gamma-rays to the filling in the central valley in the correlation spectra can be checked, in principle, by measuring the angular correlation. One of the authors (I.H.) is grateful to Ben R. Mottelson for discussions. References [l] See, e.g., R.M. Diamond and F.S. Stephens, Annu. Rev. Nucl. Part. Sci. 30 (1980) 85 ; B. Herskind, Proc. Intern. Conf. on Nuclear physics, vol. 2, eds. P. Blasi and R.A. Ricci (Tipografia Compositori, Bologna, 1983) p. 117. [ 21 See, e.g., 0. Andersen, J.D. Garrett, G.B. Hagemann, B. Herskind, D.L. Hillis and L.L. Riedinger, Phys. Rev. 43 (1979) 687; M.A. Deleplanque, Phys. Ser. 24 (1981) 158. t31 R.A. Sorensen, Proc. 1982 INS Intern. Symp. on Dynamics of nuclear collective motion, ed. K. Ogawa and K. Tanabe (Mt. Fuji, 1982). [41 A. Bohr and B.R. Mottelson, Nuclear structure, vol. II (Benjamin, New York, 1975). [51 A. Bohr, Dan. Mat. Fys. Medd. 26 (1952) no. 14. [61 N. Onishi, Genshikaku Kenkyu 17 (1972) 151 [in Japanese ] ; N. Onishi, I. Hamamoto, S. Aberg and A. Ikeda, to be published. [71 E.g., I. Hamamoto and B.R. Mottelson, Phys. Lett. 132B (1983) 7; R. Bengtsson, H. Frisk, F.R. May and J.A. Pinston, Nucl. Phys. A415 (1984) 189.
EFFECT OF 7SOFTNESS
3 January 1985
PHYSICS LETTERS
ON CONTINUUM GAMMA-RAY
SPECTRA
Ikuko HAMAMOTO and Naoki ONISHI ’ Department
of Mathematical Physics, Lund Institute of Technology,
Lund, Sweden
Received 26 September 1984
In the case that a nuclear system has a large fluctuation in the direction of triaxiality, we examine the possible feature expected to appear in the continuum gamma-ray spectra, especially a possibility of the filling in the central valley of the two-dimensional gammaenergy coincidence spectra.
The quantitative information on the yrast spectroscopy of nuclear high-spin states comes from the analysis of discrete gamma-rays emitted from evaporation residues in (HI, xn) reactions. The highest spin (the highest rotational frequency) obtained in this kind of analysis is at present Z 5 45 (Rwrot 5 550 keV). The information on the nuclear structure at still higher spin is contained in the unresolved or “continuum” gamma-ray spectra. Experimental techniques and ways of analyzing data to get nuclear structure information from continuum gamma-ray spectra are being rapidly developed [l] . One of the most popular ways to analyze the data is to plot the two-dimensional gammaenergy coincidence spectra [2] as a function of gamma-ray energies, E1 and E2. The detailed structure of the transition-energy correlation spectra depends on both the nuclei and the way how they are formed. However, the following two points seem to be fairly general: (1) A valley along the diagonal line (namely, El = E2) is clearly seen up to a certain energy (-1 MeV). (2) Above that energy (“1 MeV) the valley tends to be filled and at the same time the location of the first ridges becomes less clear. The filling in the central valley is often interpreted [3] as an indication of the interband transitions (at band crossings), assuming that there are only collective transitions (namely, AZ = 2 E2, transitions). r Permanent address: Institute of Physics, College of General Education, University of Tokyo, Komaba, Meguro, Tokyo, Japan.
6
In the present letter we want to examine the possibility that a part of the filling in the central valley can be used by the correlation between stretched (AZ = 2) transitions and non-stretched (AZ # 2) transitions if the system has a large fluctuation in the direction of triaxiality (i.e. y-variable). (The filling in the central valley is generally expected, when the fluctuation in nuclear shape is considerable so that the nuclear quadrupole deformation and, thus, collective AZ = 2 transitions within “a band” are no more well defined.) In the following we take into account only quadrupole deformation. If the potential energy function has a well-defined minimum at (PO # 0, r. # 0), the low-energy quadrupole degrees of freedom separate into the (/3- and y-) vibrational and the rotational components. The rotational components, namely the motion of the asymmetric rotor and the related E2 transition matrix elements have been discussed, for example, in ref. [4]. The calculation shows that the collective AZ = 2 E2transition matrix elements within bands are much larger than other E2 transition matrix elements. Thus, depending on y- and Z-values, the ridges in the correlation spectra can be smeared out a little, but the filling in the central valley is hardly obtained, in any case in the region of El = E2 - 1 MeV. On the other hand, if the potential-energy function is very flat in /3-and/or y-direction, the rotational components are no more well-separated from the vibrational components. In the following we take a macroscopic Bohr model [5] and we are only concerned 0370-2693/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 150B, number 1,2,3
PHYSICS LETTERS
with the deviation from axial-symmetry by assuming that 13is fixed at some constant value, fl0- Thus, our hamiltonian is written as H - - ~ 0 + V(3'),
(1)
where H 0=T+Tro _
T
t,
]~2
(2) 1
2B132 sin 33'
" 3 ~, "-~ sin 3'~-~-
O
(3)
~/2n2 Tro t =
2Jk,
(4)
Jk(7) = 4B~32 sin2(3' - ~ztk).
(5)
The potential V(7) is a function of cos 33' due to the symmetry argument [5] and, thus, is expanded as V(3') = / ~ Vlel(cos 33').
(6)
First, we consider the V(3') = 0 case, namely the 3'unstable case. It is known that the eigenfunctions of H 0 in (2) have the form of [6]
O ~ ' ( 3 " ) = D C ~n lhn^'" - t c o s 33")ng~K(') ') nja
(7)
with the eigenvalues (~2/2B/j2) 3`(3`+ 3).
(8)
For a given I one has 3 ` = 3 n ,,/ +/a,
(9)
where n v takes non-negative integers, and
la = 1/2,1/2 + 1 .... , I =(1+3)/2,(1+3)/2+1
for even I
..... I
for odd I.
(10)
Since the E2 operator is proportional to the eigenfunction G 10, it has non-zero matrix elements only between the states with 2x3`= 1 and A1 ~< 2. If the shape fluctuation of the relevant states is really large, there is no appreciable difference between the sizes of the E2 transition matrix elements with A1 = 2 and A1 :/= 2. However, for the yrast states with larger/-values the shape with T = 30° is more favoured due to the special form of the expressions (4) and (5). Then, for such states the "collective" E2 transitions with A1 = 2
3 January 1985
become a predominant decay mode. In fact, we have found that while around the yrast states in the region o f / = 2 0 - 3 0 the allowed AI = 2 E2 transition matrix elements are larger by a factor 2 - 3 than allowed A I :~ 2 ones, the difference between the two kinds of matrix elements (i.e. A I = 2 and ~ r 4: 2) becomes smaller as one goes to higher excitation energies above the yrast lines. Now, it is noted that the energy of the transition from the state with (I, X) to the states with (I', X - 1) is equal to (h2/2B/~20)2(3, + 1) irrespective of the angular momentum (I') of the final state. Therefore, though in the whole deexcitation process the number of the non-stretched E2 transitions with particular transition energies could well be appreciable, the two-dimensional gamma-ray energy correlation spectra show the same pattern as that of a perfect rotor. Namely, the intensity is zero along the diagonal, and the ridge structure and the valley structure appear perfectly. For the nuclear shape, which is not completely 3'unstable but very 3'-soft, one may expect both an appreciable amount of the non-stretched E2 transitions and energy spectra which are not so regular as those in (8). Thus, how one can quantitatively expect the filling in the central valley is worthwhile to be examined. We have diagonalized the hamiltonian (1) in terms of the eigenfunctions of H0, G~'/~v in (7). By using the resulting eigenfunctions and energies we calculate the B(E2)-values and the transition probabilities of all possible E2 transitions. Then, by assuming that an entry-line of the gamma-ray cascade lies in the region w i t h I 1 <~I<~I 2 and n 1 ~
Volume 150B, number 1,2,3
PHYSICS LETTERS
3 January 1985
E --- E~+ E2 i n ( z'~'
COUNTS
10,8-< E 512.0
5,6
2000
1000
8,L~E~9,6
3000 13,2<- E <-- 14/+ 2000
1000
o
~ 0
r
0,5
I,O
o
o,5
1,o
r
(EcEz) in
2fl z
(BI3~ J
Fig. 1. Cuts of the contour plot of the estimated two-dimensional gamma-energy correlation spectra across the diagonal in the energy regions indicated. The correlation obtained by using only stretched E2 gamma-rays is shown by the shaded area. See text for the details.
values around the yrast line are shown in fig. 2b. Though the minimum of the potential energy lies at 3' = 0 °, it is seen that at moderately higher spin the energy spectrum shows a clear tendency toward that of a
"/-unstable shape. For reference to the energy scale used, we note that the calculated transition energy from the lowest 2 + state to the ground state is 0.60 in units of 2h2/B[32. In fig. 1 the correlation obtained by
Volume 150B, number 1,2,3
PHYSICS LETTERS
3 January 1985
a 2h---L-z)i V(v) in(~BI3~
0
1'5°
- 5~ -I0
30°
45*
t
b ENERGY in ~,BOo2] J
110 100 90 80 - -
w
- -
70 - -
m
6O
50
- -
" ~
_ _
~
Z
B
- -
w
D
- -
- -
- -
p
m
- m
/,0 - -
D
30
-
_ _
- -
- -
m
n
m
- -
B
m
20
E m
m
w
10
o
_
-o
--
2
t.
6
8
10
12
1~
16
18
20
22
2~
26
>I Fig. 2. (a) The potential Y(~,),which is used in the present numericalexamples. (b) Estimated energyeigenvaluesaround the yrast line. using only stretched E2 gamma-rays is shown by the shaded area. It is seen that the filling in the central valley comes totally from the correlation between the stretched ( A I = 2) and the non-stretched ( A I :/= 2) transitions, the major part of which comes from the A1 = 0 transitions. (In the present model calculation the probability in which two non-stretched transitions occur in a given cascade is pretty small.) When we take a different entry-line, for example 11 = 20,12 = 24
and/or n 1 = 3, n 2 = 8, a very similar pattern of filling in the central valley is obtained as a function o f E 1 + E2, and the only difference which appears is the height of the ridges (namely, the number of the correlations between the stretched transitions). A choice of a different potential V(7), for example, the set ( V 1 = ---6.0, V2 = +3.0) in units of 2h2/B~2 produces a pattern of filling in the central valley which is qualitatively very similar to fig. 1.
Volume 150B, number 1,2,3
PHYSICS LETTERS
The values of angular momentum I in the present calculation should not be directly compared with experimental Z-values, since a number of important degrees of freedom are missing in our model calculation and, especially, we have not included aligned angular momentum of particles. A given deexcitation process in the present calculation such as the one shown in fig. 1 is accompanied with a large number of particle configurations. If we assume that the aligned particles behave as spectators, then, around yrast states, we may just add the total amount of the aligned angular momentum of particles in each configuration to our present Z-values, in order to obtain experimental Z-values which can be compared. Though there has been no clear-cut evidence for the deviation of nuclear shape from axial-symmetry around the ground-states of medium or heavy nuclei, the deviation is generally expected at high spin. There have been publications [7] in which the result of the deviation in discrete gamma-ray yrast-spectroscopy is discussed in connection with available experimental data. The present calculation is clearly too simple to simulate the nuclear system at high spin well above the yrast line, but it shows one of the consequences of the deviation of nuclear shape from axial-symmetry, which are expected to appear in the analysis of the continuum gamma-rays. In order to obtain the filling in the central valley, we need: (1) the nuclear shape is very r-soft but not completely y-unstable, (2) the transition energies have to be larger than certain values, and (3) the deexcitation process goes
10
3 January 1985
through states which he appreciably higher than the yrast line. We note that the contribution of the nonstretched gamma-rays to the filling in the central valley in the correlation spectra can be checked, in principle, by measuring the angular correlation. One of the authors (I.H.) is grateful to Ben R. Mottelson for discussions. References [l] See, e.g., R.M. Diamond and F.S. Stephens, Annu. Rev. Nucl. Part. Sci. 30 (1980) 85 ; B. Herskind, Proc. Intern. Conf. on Nuclear physics, vol. 2, eds. P. Blasi and R.A. Ricci (Tipografia Compositori, Bologna, 1983) p. 117. [ 21 See, e.g., 0. Andersen, J.D. Garrett, G.B. Hagemann, B. Herskind, D.L. Hillis and L.L. Riedinger, Phys. Rev. 43 (1979) 687; M.A. Deleplanque, Phys. Ser. 24 (1981) 158. t31 R.A. Sorensen, Proc. 1982 INS Intern. Symp. on Dynamics of nuclear collective motion, ed. K. Ogawa and K. Tanabe (Mt. Fuji, 1982). [41 A. Bohr and B.R. Mottelson, Nuclear structure, vol. II (Benjamin, New York, 1975). [51 A. Bohr, Dan. Mat. Fys. Medd. 26 (1952) no. 14. [61 N. Onishi, Genshikaku Kenkyu 17 (1972) 151 [in Japanese ] ; N. Onishi, I. Hamamoto, S. Aberg and A. Ikeda, to be published. [71 E.g., I. Hamamoto and B.R. Mottelson, Phys. Lett. 132B (1983) 7; R. Bengtsson, H. Frisk, F.R. May and J.A. Pinston, Nucl. Phys. A415 (1984) 189.