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On the effect of quadrupole pairing interaction on the backbending behaviour in rotating nuclei
Volume 53B, number 4
PtlYSICS LETTERS
23 December 1974
ON THE EFFECT OF QUADRUPOLE PAIRING INTERACTION ON THE BACKBENDING BEHAVIOUR IN ROTATING NUCLEI J. KRUMLINDE Department o f Mathematical Physics, Lund lnstttute o f Technology, Lund, Sweden
Z. SZYMAI~ISKI Institute for Nuclear Research, Warsaw, Poland
Received 25 October 1974 The influence of the quadrupole component in the nucleon-nucleon pairing force on the rotational nuclear motion is investigated with a simple model in connection with the backbending phenomenon. The inclusion of this component although necessary in the realistic calculation does not affect considerably the qualitative picture of the phenomenon.
Experiments initiated by Johnson, Ryde and Sztarkier [1 ] and developed by many other investigators (for reference to experiments and theoretical approaches see [2]) show that in many nuclei there is a singular behaviour in the nuclear moment of inertia as function of the angular velocity of the rotational motion (the backbending effect). This effect can be understood as a sudden alignment of nucleonic angular momenta in the direction of the total angular momentum of the rotating nucleus due to the Coriolis force. The short range pairing force producing superfluid correlations in the nucleus tends to counteract the Coriolis alignment [3]. Employing a simple particles plus rotor model with the valence nucleons filling the two degenerate single particle levels and interacting via the ordinary (monopole) pairing force we have attempted [ 4 - 6 ] to investigate the structure of the backbending phenomenon. On the other hand, it has been observed that the employment of the ordinary (monopole) pairing force seems to be unable to explain the well-known attenuation of the effective Coriolis matrix elements that couple the nuclear rotational bands [ 7 - 1 0 ] . Moreover, the nuclear moments of inertia as calculated with the inclusion of the monopole pairing only, underestimate the actual values in nuclei by several percent. The gauge noninvariance of the monopole pairing force was indicated [7] as a possible reason for the 322
above difficulties. The inclusion of the higher order multipole terms in the short range pairing interaction restoring the gauge invariance may thus be important. In particular the X = 2 (quadrupole) pairing component has been shown [7-10] to remove to a large extent the discrepancies mentioned above. Thus, it appears that the quadrupole pairing force is an important component that may influence considerably the structure of nuclear rotation. Having this in mind we attempt in this paper to investigate the Influence of the quadrupole pairmg component on the nuclear backbending effect within our solvable two-level model. The total expression for the quadrupole pairing involves various (X = 2,/~) terms characterized by the strength constants G20 6u = 0), G21 (/a = +1) and G22 (/a = +2). In the two level model all terms can be expressed by the generators of R(8), the rotation group in eight dimensions [5]. In order to simplify the discussion we shall limit ourselves to the restricted problem involwng R(5) symmetry [4, 5]. As shown below, the/a ---0 and /a = +1 components of the pairing quadrupole tensor can be expressed in terms of R(5) operators. The /a = + 1 components seem to be most relevant to nuclear rotation. The/a = 0 component, although not directly related to nuclear rotation may influence significantly the static pairing correlations in the nucleus [11, 12]. Our Hamiltonian is
PHYSICS LETTERS
Volume 53B, number 4 H = Hsp t Hp t z1$;’
t Hbzpo’ + &(Z-ix)2
(I)
H6”p” = -G,l
Introducing the creation and annihilation operators (a:, CZJand (b:, by) for the upper and lower shells, respectively, we obtain in the two-level model [4] (see also the tensor classification in [5], table III)
HSP=e c(apySa2a,-b:b,-bZB,-)= ”
”
(+z:_+b;b;)
Hb’p”’= -G,,
= -GoO(K++A+)(K_
~(baay+byae) ”
(4)
~(~;a:_-b;b;) V
c (Uta,-bFb,) ”
= - G&C+-A+)(K_
2f(Ko-A,)
(9
-A_)
and j,=:
C(aynV+b,bV) ”
c (a;b;+a;b;) v
=-G,,W+W_
(2) HP =-GO, c
23 December 1974
I_.
C(~:b,-a:_b,-tb:a,-b:a,)=3(f,tf_) ”
(6) where 2e denotes the distance between the two levels,
+A_)
t
0
26
2
4
26
6
6
26
0
2
L
6
6
30
26
26
26
Fig. 1. The curves of moment of inertia versus square, w2 of the angular velocity for various values of the single particle splitting, E = 0.0.1, 0.2 and 0.3 (in arbitrary units). The parameters of the pairing force strength are arranged in a matrix Goo,Gao,Gal =
(10,00,00) (10,00,05I (10,00,10) (lO,OS,OO) (10,05,05) (lO,OS,lO) X 1O-2 ( (10,10,00) (10,10,051(10,10,10) I
.
referring to the corresponding sections of the figure.
323
Volume 53B, number 4
PHYSICS LETTERS
a is the inverse o f the core moment of inertia and K±, K o, A~, A o, f± and W± are the generators o f the group R(5) in the notation o f ref. [4]. The above Hamiltonian is then diagonalized within the lowest (symmetric) representation with the technique described in [4]. The results representing several sets of values of the parameters G00, G20 and G21 are shown in fig. 1, for different values of e. The actual reasonable estimates for the strength parameters o f the quadrupole pairing component, that seem to fit the experimental data best, correspond to [13]:
G20/Go0= 0.2, G21/Go0= 0.4. However, as can be seen from fig. 1 the qualitative picture does not change within the limits: from 0 to 0.5 G00 for both G20 and G21. We extend the range of variation for G20 and G21 up to G00, although it seems that the limiting values are unrealistically large. The main conclusion that can be drawn from this calculation is that the backbending behaviour in the curves remains with the quadrupole pairing included. This conclusion is valid within the reasonably large region o f variation for G20 and G21. We may also observe that the presence of the (X/~) = (21) component leads to an increase o f the moment of inertia at low angular momenta. This is in line with the previous results [8, 9, 14]. Furthermore, the increasing strength G21 of the (21) component tends to decrease somewhat the critical angular momentum I of the transition. The amount o f backbending depends on the level splitting e according to the pattern: smaller e - more backbending. With G21 increasing from 0 the amount of backbending increases up to a maximum and then decreases. This may be due to a certain balance between quadrupole pairing and the (particle-hole) quadrupole-quadrupole interaction :~nerating the single particle splitting. So, at certain values of the strength parameters they are almost cancelling each other. For realistic values o f e, G21 and G00 an increase in G21 increases backbending, thus working in the opposite direction as compared to an increase in e. Now, the increasing strength o f G20 causes a con-
324
23 December 1974
siderable decrease in the critical angular momentum /. In addition, its influence on the amount of backbending is quite similar to that o f G21. We are indebted to Professor B.R. Mottelson for fruitful discussions. One o f us (Z.S.) whishes to acknowledge Lund Institute of Technology for a travel grant.
References [1] A. Johnson, H. Ryde and J. Sztarkier, Phys. Lett. 34B (1971) 605; A. Johnson, H. Ryde and S. Hjorth, Nucl. Phys. A179 (1972) 753. [2] F.S. Stephens, Proc. 5th Summer school on nuclear physics 1972, Rudziska, Poland; R.A. Sorensen, Rev. Mod. Phys. 45 (1973) 353; Z Szymahskl, Proc. Intern. Conf. on Nuclear physics (Munich 1973), Vol. 2; J. Krumlinde, Proc. 6th Summer school, Masurian Lakes, September 1973, Poland, Nucleonika XIX (1974) 251; A. Johnson and Z. Szymahski, Phys. Rep. 7C (1973) 183, A. Faessler, Proc. 7th Summer school, Masunan Lakes, September 1974, Poland, to be published in Nucleonika. [3] B.R. Mottelson and J.G. Valatln, Phys. Rev. Lett. 5 (1960) 511. [4] J. Krumlinde and Z. Szymafiskx,Phys. Lett. 36B (1971) 157. [5] J. Krumlinde and Z. Szymahski, Ann. of Phys. (N.Y.) 79 (1973) 201. [6] J. Krumlinde and Z. Szymafiskl, Nuel. Phys. A221 (1974) 93. [7] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 2, to be pubhshed. [8] S.T. Belyaev, Selected topics in nuclear theory, ed. F. Janouch (IAEA, Vienna 1963). [9] I. Hamamoto, Nordita, preprint, Febr. 1974. [10] K. Hara, Proc. 7th Summer school, Masurian Lakes, September 1974, to be published in Nukleonika. [11] D.R. B~s and R.A. Broglia, Phys. Rev. C3 (1971) 2349. [12] D.R. B~s, R.A. Broglia and B. Nilsson, Phys. Lett. 40B (1972) 338. [13] R.A. Brogiia, D.R B~s and B. Nilsson, Phys. Lett. 50B (1974) 213. [14] A.B. Migdal, JETP 37 (1959); Nucl. Phys. 13 (1959) 655.
PtlYSICS LETTERS
23 December 1974
ON THE EFFECT OF QUADRUPOLE PAIRING INTERACTION ON THE BACKBENDING BEHAVIOUR IN ROTATING NUCLEI J. KRUMLINDE Department o f Mathematical Physics, Lund lnstttute o f Technology, Lund, Sweden
Z. SZYMAI~ISKI Institute for Nuclear Research, Warsaw, Poland
Received 25 October 1974 The influence of the quadrupole component in the nucleon-nucleon pairing force on the rotational nuclear motion is investigated with a simple model in connection with the backbending phenomenon. The inclusion of this component although necessary in the realistic calculation does not affect considerably the qualitative picture of the phenomenon.
Experiments initiated by Johnson, Ryde and Sztarkier [1 ] and developed by many other investigators (for reference to experiments and theoretical approaches see [2]) show that in many nuclei there is a singular behaviour in the nuclear moment of inertia as function of the angular velocity of the rotational motion (the backbending effect). This effect can be understood as a sudden alignment of nucleonic angular momenta in the direction of the total angular momentum of the rotating nucleus due to the Coriolis force. The short range pairing force producing superfluid correlations in the nucleus tends to counteract the Coriolis alignment [3]. Employing a simple particles plus rotor model with the valence nucleons filling the two degenerate single particle levels and interacting via the ordinary (monopole) pairing force we have attempted [ 4 - 6 ] to investigate the structure of the backbending phenomenon. On the other hand, it has been observed that the employment of the ordinary (monopole) pairing force seems to be unable to explain the well-known attenuation of the effective Coriolis matrix elements that couple the nuclear rotational bands [ 7 - 1 0 ] . Moreover, the nuclear moments of inertia as calculated with the inclusion of the monopole pairing only, underestimate the actual values in nuclei by several percent. The gauge noninvariance of the monopole pairing force was indicated [7] as a possible reason for the 322
above difficulties. The inclusion of the higher order multipole terms in the short range pairing interaction restoring the gauge invariance may thus be important. In particular the X = 2 (quadrupole) pairing component has been shown [7-10] to remove to a large extent the discrepancies mentioned above. Thus, it appears that the quadrupole pairing force is an important component that may influence considerably the structure of nuclear rotation. Having this in mind we attempt in this paper to investigate the Influence of the quadrupole pairmg component on the nuclear backbending effect within our solvable two-level model. The total expression for the quadrupole pairing involves various (X = 2,/~) terms characterized by the strength constants G20 6u = 0), G21 (/a = +1) and G22 (/a = +2). In the two level model all terms can be expressed by the generators of R(8), the rotation group in eight dimensions [5]. In order to simplify the discussion we shall limit ourselves to the restricted problem involwng R(5) symmetry [4, 5]. As shown below, the/a ---0 and /a = +1 components of the pairing quadrupole tensor can be expressed in terms of R(5) operators. The /a = + 1 components seem to be most relevant to nuclear rotation. The/a = 0 component, although not directly related to nuclear rotation may influence significantly the static pairing correlations in the nucleus [11, 12]. Our Hamiltonian is
PHYSICS LETTERS
Volume 53B, number 4 H = Hsp t Hp t z1$;’
t Hbzpo’ + &(Z-ix)2
(I)
H6”p” = -G,l
Introducing the creation and annihilation operators (a:, CZJand (b:, by) for the upper and lower shells, respectively, we obtain in the two-level model [4] (see also the tensor classification in [5], table III)
HSP=e c(apySa2a,-b:b,-bZB,-)= ”
”
(+z:_+b;b;)
Hb’p”’= -G,,
= -GoO(K++A+)(K_
~(baay+byae) ”
(4)
~(~;a:_-b;b;) V
c (Uta,-bFb,) ”
= - G&C+-A+)(K_
2f(Ko-A,)
(9
-A_)
and j,=:
C(aynV+b,bV) ”
c (a;b;+a;b;) v
=-G,,W+W_
(2) HP =-GO, c
23 December 1974
I_.
C(~:b,-a:_b,-tb:a,-b:a,)=3(f,tf_) ”
(6) where 2e denotes the distance between the two levels,
+A_)
t
0
26
2
4
26
6
6
26
0
2
L
6
6
30
26
26
26
Fig. 1. The curves of moment of inertia versus square, w2 of the angular velocity for various values of the single particle splitting, E = 0.0.1, 0.2 and 0.3 (in arbitrary units). The parameters of the pairing force strength are arranged in a matrix Goo,Gao,Gal =
(10,00,00) (10,00,05I (10,00,10) (lO,OS,OO) (10,05,05) (lO,OS,lO) X 1O-2 ( (10,10,00) (10,10,051(10,10,10) I
.
referring to the corresponding sections of the figure.
323
Volume 53B, number 4
PHYSICS LETTERS
a is the inverse o f the core moment of inertia and K±, K o, A~, A o, f± and W± are the generators o f the group R(5) in the notation o f ref. [4]. The above Hamiltonian is then diagonalized within the lowest (symmetric) representation with the technique described in [4]. The results representing several sets of values of the parameters G00, G20 and G21 are shown in fig. 1, for different values of e. The actual reasonable estimates for the strength parameters o f the quadrupole pairing component, that seem to fit the experimental data best, correspond to [13]:
G20/Go0= 0.2, G21/Go0= 0.4. However, as can be seen from fig. 1 the qualitative picture does not change within the limits: from 0 to 0.5 G00 for both G20 and G21. We extend the range of variation for G20 and G21 up to G00, although it seems that the limiting values are unrealistically large. The main conclusion that can be drawn from this calculation is that the backbending behaviour in the curves remains with the quadrupole pairing included. This conclusion is valid within the reasonably large region o f variation for G20 and G21. We may also observe that the presence of the (X/~) = (21) component leads to an increase o f the moment of inertia at low angular momenta. This is in line with the previous results [8, 9, 14]. Furthermore, the increasing strength G21 of the (21) component tends to decrease somewhat the critical angular momentum I of the transition. The amount o f backbending depends on the level splitting e according to the pattern: smaller e - more backbending. With G21 increasing from 0 the amount of backbending increases up to a maximum and then decreases. This may be due to a certain balance between quadrupole pairing and the (particle-hole) quadrupole-quadrupole interaction :~nerating the single particle splitting. So, at certain values of the strength parameters they are almost cancelling each other. For realistic values o f e, G21 and G00 an increase in G21 increases backbending, thus working in the opposite direction as compared to an increase in e. Now, the increasing strength o f G20 causes a con-
324
23 December 1974
siderable decrease in the critical angular momentum /. In addition, its influence on the amount of backbending is quite similar to that o f G21. We are indebted to Professor B.R. Mottelson for fruitful discussions. One o f us (Z.S.) whishes to acknowledge Lund Institute of Technology for a travel grant.
References [1] A. Johnson, H. Ryde and J. Sztarkier, Phys. Lett. 34B (1971) 605; A. Johnson, H. Ryde and S. Hjorth, Nucl. Phys. A179 (1972) 753. [2] F.S. Stephens, Proc. 5th Summer school on nuclear physics 1972, Rudziska, Poland; R.A. Sorensen, Rev. Mod. Phys. 45 (1973) 353; Z Szymahskl, Proc. Intern. Conf. on Nuclear physics (Munich 1973), Vol. 2; J. Krumlinde, Proc. 6th Summer school, Masurian Lakes, September 1973, Poland, Nucleonika XIX (1974) 251; A. Johnson and Z. Szymahski, Phys. Rep. 7C (1973) 183, A. Faessler, Proc. 7th Summer school, Masunan Lakes, September 1974, Poland, to be published in Nucleonika. [3] B.R. Mottelson and J.G. Valatln, Phys. Rev. Lett. 5 (1960) 511. [4] J. Krumlinde and Z. Szymafiskx,Phys. Lett. 36B (1971) 157. [5] J. Krumlinde and Z. Szymahski, Ann. of Phys. (N.Y.) 79 (1973) 201. [6] J. Krumlinde and Z. Szymafiskl, Nuel. Phys. A221 (1974) 93. [7] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 2, to be pubhshed. [8] S.T. Belyaev, Selected topics in nuclear theory, ed. F. Janouch (IAEA, Vienna 1963). [9] I. Hamamoto, Nordita, preprint, Febr. 1974. [10] K. Hara, Proc. 7th Summer school, Masurian Lakes, September 1974, to be published in Nukleonika. [11] D.R. B~s and R.A. Broglia, Phys. Rev. C3 (1971) 2349. [12] D.R. B~s, R.A. Broglia and B. Nilsson, Phys. Lett. 40B (1972) 338. [13] R.A. Brogiia, D.R B~s and B. Nilsson, Phys. Lett. 50B (1974) 213. [14] A.B. Migdal, JETP 37 (1959); Nucl. Phys. 13 (1959) 655.