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Instability of the quasi-particle vacuum and band-crossing in high-spin rotational states
Volume 49B, number 2
PHYSICS LETTERS
INSTABILITY
OF THE QUASI-PARTICLE
1 April 1974
VACUUM AND BAND-CROSSING
IN HIGH-SPIN ROTATIONAL
STATES
L. LIN, R.M. DREIZLER Institute for Theoretical Physics, University of Frankfurt/M, W.Germany
and
B.E. CHI Physics Dept., State University of New York at Albany, Albany, New York, USA
Received 4 March 1974 It is shown that the energy of the quasi-particle excitation in a rotational system may become negative when the angular frequency to reaches a certain value. As a result, the energy of the excited state may be lower than the energy of the quasi-particle vacuum. It is suggested that this may serve as a natural explanation of the recently observed band-crossing phenomena where the levels of the "excited-band" become lower then the levels of the "ground-band" after a certain value of angular momentum. Preliminary calculation agrees well with the experimental result obtained in Juelich for 1 5 6 D y ' The collective rotational properties of a nuclear system in a rotating frame can be described, at least qualitatively, by the semi-classical cranking Hamiltonian:
H= E%-
)qq < I~xl~> c,~c~
(1)
Here a and a are the Nilsson single-particle state and its time reversal conjugate, X is the Fermi energy, G is the pairing constant, and w is the angular frequency. When w is zero, we can use the standard BogoliubovBCS treatment for the system. When co is not zero, the dynamics of the system is modified through the influence o f the Coriolis interaction and deviates from that o f the simple BCS case such that all the physical quantities become functions o f w. The effective pairing strength and the energy gap begin to decrease [1]. The quasiparticle vacuum is modified as well as the quasi-particle excitations. Among all the orbitals, those near the Fermi level are most sensitive to the Coriolis force. In some cases, the quasi-particle energy near the Fermi level may become close to zero, which theoretically corresponds to gapless superconductivity [2]. It may also become negative as the Coriolis force becomes stronger. For these values o f
the angular frequency the quasi-particle vacuum will then no longer represent the lowest state o f the eveneven system. It is suggested that this consequence may serve as a natural interpretation of the recently observed band-crossing phenomena: The energies of the excited band are lower than the energies in the groundstate-band after a certain value o f angular m o m e n t u m [3]. In the BCS quasi-particle representation, the Hamiltonian can be written as [e.g. 4]
H(w)= Uo(~O)+ ~ ~>0
E(a+a+a+a~,) (2)
a,13>0 Here U0(w ) is the unperturbed vacuum energy, E~ = x / ( e , , - X)2 + A2 is the BCS quasi-particle energy, a s = u~C~ - v^.C + is the quasiparticle transformation, A = GE
Volume 49B, number 2
PHYSICS LETTERS
Next, the. Heisenberg equations of motion for the quasi-particle operators are given as:
d th dt
(3)
= P(w) a
where
,,
co/<~A,~
- E ao¢--og<~fi~<~ i
'
and c~,/3 > 0. We notice that S ~ is symmetrical in ~, 17and Aa~ is anti-symmetrical, so P(co) is a symmetric matrix. Therefore, it can be diagonalized by an orthogonal transformation. Let the new normal modes be represented as:
ba
= ~ (foLjfl+g~ea~), ~>0
1 April 1974
eq. (3), it is seen that only 2N states in the system are coupled together through the matrix P(co) which has the dimension 2N, When we diagonalize the matrix P(co), we get 2N independent modes in the new representation (eq. (4a), (4b)) with eigenvalues W~'s and W~'s, so that there is no spurious state. The eigenvalues of P(co) are related to the "rotational quasi-particle" energies with respect to the new vacuum These new quasi-particle energies (in the rotating frame) are represented by eq. (5a) and (5c). Furthermore, it can be shown that W~ = W~ [5]. In the new representation, the Hamiltonian H(co) becomes:
H ( ~ ) = WO(CO) + ~ (b+~b~+b~b~) (6) a>0 there W0(~ ) can be shown to be equal to U0(¢o ) + £ a > 0 ( E a - Wa). The energy operator for the system is given as:
(4a) -- v0( ) +
. (4b) + = ~ ] (fc,~a~-t-g<~oao) ~>0 The equations of motion for these new normal modes are:
a>0
e&:a
+a;as)
b +
ihb<~ = [b ,H] = W b
(5a)
iti {)+ ~ = [b+,H] = - W b +
(5b)
ill Da = [b~,H] = W~b6<
(5c)
i~ b + = [b ,H] = - W - b a
(5d)
If we substitute eq. (4a) and eq. (4b) into eq. (5a) and eq. (5d) respectively, we obtain
gcl~ and
Now, let the dimension of our basis be 2N, that is, we have N a's and N a's. There are 2N independent single-particle modes the system can take. From 118
=
Wo(W)+a~O W04(b+b Ot Ot +b+b-)+ OL Og
"
('O]X "
(7)
The new vacuum state of the system [~0(w)), which satisfies b~i'I'0(w )) = 0, is the state with no "rotational quasi-particle", For an even-even system, the state with 2-"rotational quasi-particle" can be represented The energy of the new aslqs2<~(co))= bab~l@0(co)). + + vacuum is (qs0(o))iH0[ @0(u~)) = W0(~ ) + W(@O(aJ)l]xl qs0(w)), and the energy of the 2-"rotational quasi-particles" state is Wo(~O) + 2Wa + w(qs2al/x[ @2<~). This excited state, provided (~0lJxlg'0) ~ (~2,,[/xl~2a), may become lower in energy than the vacuum if Wa is negative for a certain value of co. If taken at face value this means that the original ansatz and philosophy breaks down. We obtain, however, a very strong hint for the point at which band-crossing occurs. A preliminary numerical calculation shows that, for 156Dy, the lowest qausi-particle energy Wa(co) for neutrons becomes negative at ~w = 0.3 MeV (for details see ref. [5] ). The rotational quasi-particle vacuum is therefore not stable according to our description. The recent experimental result of Juelich for 156Dy shows that the levels of the excited band fall below the corresponding levels of the groundstate-
Volume 49B, number 2
PHYSICS LETTERS
band [3]. From the experimental data one extracts the frequency h w ~ 0.32 for J = 16. In this calculation the deformation is taken to be = 0.3 in accordance with the B ( E 2 : 2 + ~ 0 +) value [6]. The pairing constants Gn, Gp are chosen to produce An, Ap ~ 0.92 at w = 0. The spherical basis used for generating the Nilsson orbitals contains for the protons all levels in the N = 2, 3,4, 5 oscillator shells plus 0j ls/2 and 0i 13/2 and for the neutrons all levels in the N = 3,4, 5, 6 plus 0j 1s/2 . The parameters of the Nilsson model l ' s and l 2 term are those given b y ref. [7] (~ = 0.0637, 12n -- 0.6,/2p = 0.42) for the rare earth nuclei. The values o f An(W ) and Ap(6O) are chosen to minimize ( 4 0lH(w)l 4 0) for the protons and the neutrons respectively. One o f the authors (L.L.) would like to thank Professor A. Faessler and Dr. K. Goeke for valuable discussions. We also thank the Hochschulrechenzentrum of the University Frankfurt for the use o f their facilities.
1 April 1974
References [1] B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett. 5 (1960) 511. [2] L. Lin, M.S. thesis (1967), Case-Western Univ., Cleveland, Ohio, USA; A. Goswami, L. Lin and G. Struble, Phys. Lett. 25B (1967) 451. [3] R.M. Lieder et al., to be published; T.L. Khoo et al~ Phys. Rev. Lett. 31 (1973) 1146; H. Beuscher et al., Z. Physik 263 (1973) 201; [4] G.E. Brown, Unified theory of nuclear model (NorthHolland Publishing Co., Amsterdam, 1964). [5] L. Lin, R.M. Dreizler and B. Chi, to be published. [6] P.H. Stelson and L. Grodzins, Nucl. Data, Sect. A, 2 (1965) 21. [7] I. Ragnarson, Intern. Conf. on The properties on nuclei far from the region of beta-stability, Leysin 1970; CERN report 70-30.
119
PHYSICS LETTERS
INSTABILITY
OF THE QUASI-PARTICLE
1 April 1974
VACUUM AND BAND-CROSSING
IN HIGH-SPIN ROTATIONAL
STATES
L. LIN, R.M. DREIZLER Institute for Theoretical Physics, University of Frankfurt/M, W.Germany
and
B.E. CHI Physics Dept., State University of New York at Albany, Albany, New York, USA
Received 4 March 1974 It is shown that the energy of the quasi-particle excitation in a rotational system may become negative when the angular frequency to reaches a certain value. As a result, the energy of the excited state may be lower than the energy of the quasi-particle vacuum. It is suggested that this may serve as a natural explanation of the recently observed band-crossing phenomena where the levels of the "excited-band" become lower then the levels of the "ground-band" after a certain value of angular momentum. Preliminary calculation agrees well with the experimental result obtained in Juelich for 1 5 6 D y ' The collective rotational properties of a nuclear system in a rotating frame can be described, at least qualitatively, by the semi-classical cranking Hamiltonian:
H= E%-
)qq < I~xl~> c,~c~
(1)
Here a and a are the Nilsson single-particle state and its time reversal conjugate, X is the Fermi energy, G is the pairing constant, and w is the angular frequency. When w is zero, we can use the standard BogoliubovBCS treatment for the system. When co is not zero, the dynamics of the system is modified through the influence o f the Coriolis interaction and deviates from that o f the simple BCS case such that all the physical quantities become functions o f w. The effective pairing strength and the energy gap begin to decrease [1]. The quasiparticle vacuum is modified as well as the quasi-particle excitations. Among all the orbitals, those near the Fermi level are most sensitive to the Coriolis force. In some cases, the quasi-particle energy near the Fermi level may become close to zero, which theoretically corresponds to gapless superconductivity [2]. It may also become negative as the Coriolis force becomes stronger. For these values o f
the angular frequency the quasi-particle vacuum will then no longer represent the lowest state o f the eveneven system. It is suggested that this consequence may serve as a natural interpretation of the recently observed band-crossing phenomena: The energies of the excited band are lower than the energies in the groundstate-band after a certain value o f angular m o m e n t u m [3]. In the BCS quasi-particle representation, the Hamiltonian can be written as [e.g. 4]
H(w)= Uo(~O)+ ~ ~>0
E(a+a+a+a~,) (2)
a,13>0 Here U0(w ) is the unperturbed vacuum energy, E~ = x / ( e , , - X)2 + A2 is the BCS quasi-particle energy, a s = u~C~ - v^.C + is the quasiparticle transformation, A = GE
Volume 49B, number 2
PHYSICS LETTERS
Next, the. Heisenberg equations of motion for the quasi-particle operators are given as:
d th dt
(3)
= P(w) a
where
,,
co/<~A,~
- E ao¢--og<~fi~<~ i
'
and c~,/3 > 0. We notice that S ~ is symmetrical in ~, 17and Aa~ is anti-symmetrical, so P(co) is a symmetric matrix. Therefore, it can be diagonalized by an orthogonal transformation. Let the new normal modes be represented as:
ba
= ~ (foLjfl+g~ea~), ~>0
1 April 1974
eq. (3), it is seen that only 2N states in the system are coupled together through the matrix P(co) which has the dimension 2N, When we diagonalize the matrix P(co), we get 2N independent modes in the new representation (eq. (4a), (4b)) with eigenvalues W~'s and W~'s, so that there is no spurious state. The eigenvalues of P(co) are related to the "rotational quasi-particle" energies with respect to the new vacuum These new quasi-particle energies (in the rotating frame) are represented by eq. (5a) and (5c). Furthermore, it can be shown that W~ = W~ [5]. In the new representation, the Hamiltonian H(co) becomes:
H ( ~ ) = WO(CO) + ~ (b+~b~+b~b~) (6) a>0 there W0(~ ) can be shown to be equal to U0(¢o ) + £ a > 0 ( E a - Wa). The energy operator for the system is given as:
(4a) -- v0( ) +
. (4b) + = ~ ] (fc,~a~-t-g<~oao) ~>0 The equations of motion for these new normal modes are:
a>0
e&:a
+a;as)
b +
ihb<~ = [b ,H] = W b
(5a)
iti {)+ ~ = [b+,H] = - W b +
(5b)
ill Da = [b~,H] = W~b6<
(5c)
i~ b + = [b ,H] = - W - b a
(5d)
If we substitute eq. (4a) and eq. (4b) into eq. (5a) and eq. (5d) respectively, we obtain
gcl~ and
Now, let the dimension of our basis be 2N, that is, we have N a's and N a's. There are 2N independent single-particle modes the system can take. From 118
=
Wo(W)+a~O W04(b+b Ot Ot +b+b-)+ OL Og
"
('O]X "
(7)
The new vacuum state of the system [~0(w)), which satisfies b~i'I'0(w )) = 0, is the state with no "rotational quasi-particle", For an even-even system, the state with 2-"rotational quasi-particle" can be represented The energy of the new aslqs2<~(co))= bab~l@0(co)). + + vacuum is (qs0(o))iH0[ @0(u~)) = W0(~ ) + W(@O(aJ)l]xl qs0(w)), and the energy of the 2-"rotational quasi-particles" state is Wo(~O) + 2Wa + w(qs2al/x[ @2<~). This excited state, provided (~0lJxlg'0) ~ (~2,,[/xl~2a), may become lower in energy than the vacuum if Wa is negative for a certain value of co. If taken at face value this means that the original ansatz and philosophy breaks down. We obtain, however, a very strong hint for the point at which band-crossing occurs. A preliminary numerical calculation shows that, for 156Dy, the lowest qausi-particle energy Wa(co) for neutrons becomes negative at ~w = 0.3 MeV (for details see ref. [5] ). The rotational quasi-particle vacuum is therefore not stable according to our description. The recent experimental result of Juelich for 156Dy shows that the levels of the excited band fall below the corresponding levels of the groundstate-
Volume 49B, number 2
PHYSICS LETTERS
band [3]. From the experimental data one extracts the frequency h w ~ 0.32 for J = 16. In this calculation the deformation is taken to be = 0.3 in accordance with the B ( E 2 : 2 + ~ 0 +) value [6]. The pairing constants Gn, Gp are chosen to produce An, Ap ~ 0.92 at w = 0. The spherical basis used for generating the Nilsson orbitals contains for the protons all levels in the N = 2, 3,4, 5 oscillator shells plus 0j ls/2 and 0i 13/2 and for the neutrons all levels in the N = 3,4, 5, 6 plus 0j 1s/2 . The parameters of the Nilsson model l ' s and l 2 term are those given b y ref. [7] (~ = 0.0637, 12n -- 0.6,/2p = 0.42) for the rare earth nuclei. The values o f An(W ) and Ap(6O) are chosen to minimize ( 4 0lH(w)l 4 0) for the protons and the neutrons respectively. One o f the authors (L.L.) would like to thank Professor A. Faessler and Dr. K. Goeke for valuable discussions. We also thank the Hochschulrechenzentrum of the University Frankfurt for the use o f their facilities.
1 April 1974
References [1] B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett. 5 (1960) 511. [2] L. Lin, M.S. thesis (1967), Case-Western Univ., Cleveland, Ohio, USA; A. Goswami, L. Lin and G. Struble, Phys. Lett. 25B (1967) 451. [3] R.M. Lieder et al., to be published; T.L. Khoo et al~ Phys. Rev. Lett. 31 (1973) 1146; H. Beuscher et al., Z. Physik 263 (1973) 201; [4] G.E. Brown, Unified theory of nuclear model (NorthHolland Publishing Co., Amsterdam, 1964). [5] L. Lin, R.M. Dreizler and B. Chi, to be published. [6] P.H. Stelson and L. Grodzins, Nucl. Data, Sect. A, 2 (1965) 21. [7] I. Ragnarson, Intern. Conf. on The properties on nuclei far from the region of beta-stability, Leysin 1970; CERN report 70-30.
119