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A modified generator coordinate model to study nuclear vibrations and rotations
V o l u m e 160B, n u m b e r 1,2,3
PHYSICS LETTERS
3 October 1985
A M O D I F I E D GENERATOR C O O R D I N A T E M O D E L T O S T U D Y NUCLEAR VIBRATIONS AND R O T A T I O N S A.B. V O L K O V Department of Physics, McMaster Umverszty, Hamdton, Ontario, Canada, L8S 4M1 Received 27 March 1985 A simple mlcroscopac determmantal type basis is used to generate oscfllator-hke wave functions and assooated rotational b a n d s These wave functions can be used to mvestagate other p h e n o m e n a
The original Bohr-Mottelson model [1-3] describing nuclear vibrations and rotations has had tremendous success in explaining the basic features of an incredibly large amount of data. There have been a number of attempts to reformulate the model on a more microscopic basis. One of the most promising approaches would appear to be the generator coordinate method ( G C M ) developed by a number of investigators based on the pioneer work of Hill, Wheeler and Griffin [4,5]. The method and the difficulties associated with the method are well described in the book of Ring and Schuck [6] which also includes references to previous work. A modified form of the G C M will be used in this work to demonstrate the nature of quadrupole shape vibrations and provide a set of microscopic wave functions for different angular momenta which can be used to investigate electromagnetic cascades, (t, p) and (p, t) reaction strengths, etc. The ideal calculation would use a basis of constrained H a r t r e e - F o c k wave functions [7] evaluated for different deformations or determinantal wave functions generated by a sophisticated form of the Woods-Saxon potential [8]. However, the difficulties and time associated with calculating the necessary quantities for medium or heavy nuclei require fairly drastic simplifications. Therefore a simple three-dimensional cartesian harmonic oscillator model with different possible oscillator constants is used.
In order to avoid discontinuities associated with sharp level crossings due to changing deformations, simple BCS pairing wave functions [9] are used for neutrons and protons separately. A = 1 2 / A 1/2 MeV where A is the total number of particles used for each deformation. The product wave function for N neutrons and P protons is given by = I-I ll,'
+ v.(a,fl,)a+~(a,fl,)a+(.,fl,)][O).
(1)
v represents the quantum numbers appropriate to the three cartesian wave functions and the spin and isospin value describing a given single particle state. ~ is the time reversed state, a, is an oscillator constant for the x and y coordinates and fl, is associated with the z coordinate which leads to a state which is essentially cylindrically symmetric. The a,, fl, are chosen to give a set of 40 states which span oblate to prolate quadrupole expectation values. Up to 240 single-particle states are used to define the BCS wave functions (b,. The wave function at each deformation can be characterized by its quadrupole expectation value. The a , , / 3 are normalized at each deformation so as to give a constant volume for the system. The set of wave functions (1) form a nonorthogonal basis. The solution of the energy eigenvalue problem in such a non-orthogonal basis represents a discretized form of the GCM. Because of the non-orthogonality of the basis, the expan-
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 160B, number 1,2,3
PHYSICS LETTERS
generates an orthonormal representation with eigenvalues qp fairly close to the actual quadrupole values since everything is tied to the diagonal matrix elements and the calculated overlap matrix
sion coefficients of the resultant energy wave function cannot be directly interpreted as probability amplitudes. Therefore, it is convenient to first use the non-orthogonal basis to solve the problem.
a~qp = qp~qp,
3 October 1985
A,j. (2)
The non-orthogonal normalized wave functions in the laboratory system are defined for even I as
where Q is the quadrupole moment operator and
~qp= E Cpt~P,(.,,~,).
(3)
l
This requires the solution of the matrix equation
T'. [ { qJ,lQle?j) - qA,j] g = O,
(4)
J where
0/'M= q~,(a,fl,)[(2I + 1)/8~r2]t/2D~o(f~),
(9)
where D~t 0 is the usual Wigner rotation matrix function of the Euler angles f~. The rotationally invariant hamiltonian is defined in the intrinsic system as n = ncoulomb + Hsurfa~ + Hparude + Hrotauonal,
A,, = (~,1~,)
(5)
is the overlap matrix. In order to simplify the calculation two approximations, which do not destroy the essential microscopic nature of the calculation, are employed. In the first, the overlap (norm) matrix is defined as the simple product function overlap
The diagonal matrix elements of this hamiltonian are given by
1-1,,] _ -_ (0," MIH[O]" M) = e, + (1/2l,)1(1 + 1), (10) where ~, = (~,('~,/~,)1 (~Co~or~b +/-/surface
A,, - l-I [ u fi a,~,)u fi %g )
4- Hparucle) kb, ( at/~,)).
+ o~( a,fl,)o~( ctgflj )a,j ] , a,g = I(Ola~(a,fl,)a+(ajflj)lO)l
2,
(6)
which would be correct for orthogonal sets of single-particle states and is nearly correct for adjacent deformations. This approximation neglects other contributing permutations for larger deformation differences but in any event these overlaps are small. The second simplification defines the off-diagonal matrix element of Q as ('/',[QI~:> = 1(@, + @j),A,j,
(7)
where ~, = <~,10[(/,,>
(8)
is the easily calculated quadrupole expectation value for the ith deformation. Since the @, are a monotonically increasing set of quadrupole expectation values, the error in (7) is probably small, but in any event (7) can be thought of as defining a modified quadrupole operator Q which then 6
(11)
The Coulomb and surface energies are calculated with a density and surface consistent with each ~,(cqfl,) and normalized to liquid drop values at zero deformation. The single-particle contribution is given by a simple summation of harmonic oscillator single-particle energies weighted by the pairing occupation factors v2 appropriate for each deformation. The rotational hamiltonian is of the Bohr form [10] with the moment of inertia l, determined by the quadrupole expectation value for the deformation. The one constant of proportionality is determined by a fit, for the final solutions, to the I = 2 energy in the ground state rotational band. The set of diagonal energies e, are an approximation to the energies that would be found by a constrained Hartree-Fock calculation and behave very much like published energy curves. However, the shell effects are found to be larger than more realistic calculations which is a well known consequence of the harmomc oscillator approximation.
Volume 160B, number 1,2,3
PHYSICS LETTERS
In order to solve the energy eigenvalue problem, it is necessary to obtain appropriate off-diagonal matrix elements for H in the non-orthogonal basis (9). The H matrix, to avoid the extreme difficulty of explicit calculations, is taken to be
3 October 1985 12
2.5
.......
~
I0
2.0
~
(o,,.
I0
........
2
. . . . . . .
0
8
I0
~
[}c0(e , + e,) +
=
½(1/2l, + 1/21j)I(I +
A
1)
> .......
(12)
+ f ( q , , O,)] A,,,
4
where
8
8
- - 6 .......
2 4
~ 0
n.LtJ
Z
- - 4
hi
f(tff,, qj)
=
Cl(q,-
~j)2
(13)
is often used in G C M calculations. A slightly more complicated form is used in this calculation, but reduces to (13) for relatively small shape differences. The constant c o is used to obtain more realistic shell effect energies. It is important to note that H~ has the calculated diagonal energy values and is proportional to the calculated (and microscopic) overlap factors A,j. The final results for any given nucleus tend to remain qualitatively the same for a fairly large variation of the various parameters in the hamiltonian matrix as long as they are chosen to give comparable excitation energies for the system. Finally, the standard eigenvalue problem
Ht/,~,, = E,,ttk~,,
(14)
is solved, with
¢~. Ed~p¢%[(2/ + 1)/8~2]1/2DIo(~~), =
(15)
P where the ~pq are the orthonormal "quadrupole" states, and the dn/p are now true probability amplitudes. The final results cannot be used to fit specific real nuclei because of the use of a cartesian basis and the neglect of spin-orbit interactions. However, the qualitative results do resemble the behavior of real nuclei to a remarkable extent. Thus the cartesian closed shell nucleus H0~, 5 0 ~ " 7 0 is 204 spherical while ~34~ and prolate 50 ~ x84 80A124 are deformed, etc. Some of the results for 134A 50 z . 84 a r e shown in fig. 1 and fig. 2. States have been calculated for up to I = 24. Some of the spectra for the first three bands are shown in fig. 1 for two calculations in which
1.0
.......
6
........
6 ........
05
.......
4 4
2
2 0
Theory 134 aoAa4 A=ZXexp
........
0.0
2
.......
z a : "" t a e x p
Q
Fzg 1 The partzal rotational spectra for a system with 50 protons and 84 neutrons based on three vlbraUonal band heads for two different values of A, A = Ae~p and A ~ 2Aex p The larger A leads to more coUectzve wave functmns w~th subsequent energy shafts
different values of A were used to calculate the occupation probabilities v~(a,fl,). The use of a larger A leads to a smoother variation in the overlap matrix A,~ which leads to greater collectivity, smaller deformations, and smaller effective moments of inertia. This is seen as an upward displacement of all states including the band heads. The parameters in (12) were chosen to fit the I = 2 ground state energy in zSaGd as well as the first two excited I = 0 states which leads to a spectrum resembling that of 154Gd. The distribution of the probability amplitudes d~tp defined by (14) and (15) are shown in fig. 2 for the case of the "microscopic" overlap A,j calculated using (6) and the case of a smooth exponential overlap chosen to reproduce the main decrease in A,j. N o other changes were made in the calculation. Each circle represents the dip appropriate to a quadrupole eigenvalue qp. ~bg and ~ are the first and second I = 0 band heads. The exponential overlap case displays probability amplitude distributions which are excellent
Volume 160B, number 1,2,3
PHYSICS LETTERS
WAVE FUNCTIONS
0,6
~04 ,oo -02 -04 I
I
I
I
I
I
3 October 1985
level crossings which occur as the deformation changes. The I - 14 wave function for the ground state band shows the result of centrifugal stretching which leads to an increase of the effective moment of inertia which leads to compression of the spectrum for high I values. The microscopic wave functions generated by these calculations have been used to calculate B(E2) values between eight rotational bands for up to 1 = 24. The calculated "r-ray intensity distribution due to 1104 possible transitions is found to closely resemble experimentally observed intensity patterns.
w04
This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. ~oo -02
References
-04
z6o
abe
oo,o~,o,,o~ ~,o~,,T 66o
z6o
Fig 2 The theorelacal probablhty amphtudes for the ground states, the ground state band I = 14 state, and beta band heads for the 50 proton, 84 neutron system w~th respect to elgenstates of the quadrupole operator for two different overlap matnces A,j. The simple exponenual case leads to harmomc oscdlator type wave functtons
representations of the first two harmonic oscillator states of a vibration. The quadrupole distribution is clustered around the prelate minimum of the e, given by (11). The distributions for the microscopic overlap case are similar but show effects due to irregularities in A u due to single-particle
[1] A Bohr, Mat Fys Medd Dan Vxd Selsk 26, no 14 (1952) [2] A Bohr and B.R Mottelson, Mat Fys Medd Dan Vld Selsk 27, no 16 (1953). [3] M.A Preston and R.K Bhadun, Structure of the nucleus (Ad&son-Wesley, Reading, M_A, 1975) Ch 9. [4] D L. Hall and J A Wheeler, Phys Rev 89 (1953) 1109 [5] J J. Gnttin and J A Wheeler, Plays Rev 108 (1957) 311. [6] P Rang and P Schuck, The nuclear many-body problem (Spnnger, Berhn, 1980) Ch 10 [7] P Rang and P Schuck, The nuclear many-body problem (Spnnger, Berhn, 1980) p 92 [8] P Rang and P Schuck, The nuclear many-body problem (Spnnger, Berlin, 1980) p 67 [9] L S. Kasshnger and R A Sorensen, Mat Fys Medd Dan Vld Selsk 32, no 9 (1960) [10] M.A Preston and R K Bhadun, Structure of the nucleus (Addison-Wesley, Reading, MA, 1975) p. 359
PHYSICS LETTERS
3 October 1985
A M O D I F I E D GENERATOR C O O R D I N A T E M O D E L T O S T U D Y NUCLEAR VIBRATIONS AND R O T A T I O N S A.B. V O L K O V Department of Physics, McMaster Umverszty, Hamdton, Ontario, Canada, L8S 4M1 Received 27 March 1985 A simple mlcroscopac determmantal type basis is used to generate oscfllator-hke wave functions and assooated rotational b a n d s These wave functions can be used to mvestagate other p h e n o m e n a
The original Bohr-Mottelson model [1-3] describing nuclear vibrations and rotations has had tremendous success in explaining the basic features of an incredibly large amount of data. There have been a number of attempts to reformulate the model on a more microscopic basis. One of the most promising approaches would appear to be the generator coordinate method ( G C M ) developed by a number of investigators based on the pioneer work of Hill, Wheeler and Griffin [4,5]. The method and the difficulties associated with the method are well described in the book of Ring and Schuck [6] which also includes references to previous work. A modified form of the G C M will be used in this work to demonstrate the nature of quadrupole shape vibrations and provide a set of microscopic wave functions for different angular momenta which can be used to investigate electromagnetic cascades, (t, p) and (p, t) reaction strengths, etc. The ideal calculation would use a basis of constrained H a r t r e e - F o c k wave functions [7] evaluated for different deformations or determinantal wave functions generated by a sophisticated form of the Woods-Saxon potential [8]. However, the difficulties and time associated with calculating the necessary quantities for medium or heavy nuclei require fairly drastic simplifications. Therefore a simple three-dimensional cartesian harmonic oscillator model with different possible oscillator constants is used.
In order to avoid discontinuities associated with sharp level crossings due to changing deformations, simple BCS pairing wave functions [9] are used for neutrons and protons separately. A = 1 2 / A 1/2 MeV where A is the total number of particles used for each deformation. The product wave function for N neutrons and P protons is given by = I-I ll,'
+ v.(a,fl,)a+~(a,fl,)a+(.,fl,)][O).
(1)
v represents the quantum numbers appropriate to the three cartesian wave functions and the spin and isospin value describing a given single particle state. ~ is the time reversed state, a, is an oscillator constant for the x and y coordinates and fl, is associated with the z coordinate which leads to a state which is essentially cylindrically symmetric. The a,, fl, are chosen to give a set of 40 states which span oblate to prolate quadrupole expectation values. Up to 240 single-particle states are used to define the BCS wave functions (b,. The wave function at each deformation can be characterized by its quadrupole expectation value. The a , , / 3 are normalized at each deformation so as to give a constant volume for the system. The set of wave functions (1) form a nonorthogonal basis. The solution of the energy eigenvalue problem in such a non-orthogonal basis represents a discretized form of the GCM. Because of the non-orthogonality of the basis, the expan-
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 160B, number 1,2,3
PHYSICS LETTERS
generates an orthonormal representation with eigenvalues qp fairly close to the actual quadrupole values since everything is tied to the diagonal matrix elements and the calculated overlap matrix
sion coefficients of the resultant energy wave function cannot be directly interpreted as probability amplitudes. Therefore, it is convenient to first use the non-orthogonal basis to solve the problem.
a~qp = qp~qp,
3 October 1985
A,j. (2)
The non-orthogonal normalized wave functions in the laboratory system are defined for even I as
where Q is the quadrupole moment operator and
~qp= E Cpt~P,(.,,~,).
(3)
l
This requires the solution of the matrix equation
T'. [ { qJ,lQle?j) - qA,j] g = O,
(4)
J where
0/'M= q~,(a,fl,)[(2I + 1)/8~r2]t/2D~o(f~),
(9)
where D~t 0 is the usual Wigner rotation matrix function of the Euler angles f~. The rotationally invariant hamiltonian is defined in the intrinsic system as n = ncoulomb + Hsurfa~ + Hparude + Hrotauonal,
A,, = (~,1~,)
(5)
is the overlap matrix. In order to simplify the calculation two approximations, which do not destroy the essential microscopic nature of the calculation, are employed. In the first, the overlap (norm) matrix is defined as the simple product function overlap
The diagonal matrix elements of this hamiltonian are given by
1-1,,] _ -_ (0," MIH[O]" M) = e, + (1/2l,)1(1 + 1), (10) where ~, = (~,('~,/~,)1 (~Co~or~b +/-/surface
A,, - l-I [ u fi a,~,)u fi %g )
4- Hparucle) kb, ( at/~,)).
+ o~( a,fl,)o~( ctgflj )a,j ] , a,g = I(Ola~(a,fl,)a+(ajflj)lO)l
2,
(6)
which would be correct for orthogonal sets of single-particle states and is nearly correct for adjacent deformations. This approximation neglects other contributing permutations for larger deformation differences but in any event these overlaps are small. The second simplification defines the off-diagonal matrix element of Q as ('/',[QI~:> = 1(@, + @j),A,j,
(7)
where ~, = <~,10[(/,,>
(8)
is the easily calculated quadrupole expectation value for the ith deformation. Since the @, are a monotonically increasing set of quadrupole expectation values, the error in (7) is probably small, but in any event (7) can be thought of as defining a modified quadrupole operator Q which then 6
(11)
The Coulomb and surface energies are calculated with a density and surface consistent with each ~,(cqfl,) and normalized to liquid drop values at zero deformation. The single-particle contribution is given by a simple summation of harmonic oscillator single-particle energies weighted by the pairing occupation factors v2 appropriate for each deformation. The rotational hamiltonian is of the Bohr form [10] with the moment of inertia l, determined by the quadrupole expectation value for the deformation. The one constant of proportionality is determined by a fit, for the final solutions, to the I = 2 energy in the ground state rotational band. The set of diagonal energies e, are an approximation to the energies that would be found by a constrained Hartree-Fock calculation and behave very much like published energy curves. However, the shell effects are found to be larger than more realistic calculations which is a well known consequence of the harmomc oscillator approximation.
Volume 160B, number 1,2,3
PHYSICS LETTERS
In order to solve the energy eigenvalue problem, it is necessary to obtain appropriate off-diagonal matrix elements for H in the non-orthogonal basis (9). The H matrix, to avoid the extreme difficulty of explicit calculations, is taken to be
3 October 1985 12
2.5
.......
~
I0
2.0
~
(o,,.
I0
........
2
. . . . . . .
0
8
I0
~
[}c0(e , + e,) +
=
½(1/2l, + 1/21j)I(I +
A
1)
> .......
(12)
+ f ( q , , O,)] A,,,
4
where
8
8
- - 6 .......
2 4
~ 0
n.LtJ
Z
- - 4
hi
f(tff,, qj)
=
Cl(q,-
~j)2
(13)
is often used in G C M calculations. A slightly more complicated form is used in this calculation, but reduces to (13) for relatively small shape differences. The constant c o is used to obtain more realistic shell effect energies. It is important to note that H~ has the calculated diagonal energy values and is proportional to the calculated (and microscopic) overlap factors A,j. The final results for any given nucleus tend to remain qualitatively the same for a fairly large variation of the various parameters in the hamiltonian matrix as long as they are chosen to give comparable excitation energies for the system. Finally, the standard eigenvalue problem
Ht/,~,, = E,,ttk~,,
(14)
is solved, with
¢~. Ed~p¢%[(2/ + 1)/8~2]1/2DIo(~~), =
(15)
P where the ~pq are the orthonormal "quadrupole" states, and the dn/p are now true probability amplitudes. The final results cannot be used to fit specific real nuclei because of the use of a cartesian basis and the neglect of spin-orbit interactions. However, the qualitative results do resemble the behavior of real nuclei to a remarkable extent. Thus the cartesian closed shell nucleus H0~, 5 0 ~ " 7 0 is 204 spherical while ~34~ and prolate 50 ~ x84 80A124 are deformed, etc. Some of the results for 134A 50 z . 84 a r e shown in fig. 1 and fig. 2. States have been calculated for up to I = 24. Some of the spectra for the first three bands are shown in fig. 1 for two calculations in which
1.0
.......
6
........
6 ........
05
.......
4 4
2
2 0
Theory 134 aoAa4 A=ZXexp
........
0.0
2
.......
z a : "" t a e x p
Q
Fzg 1 The partzal rotational spectra for a system with 50 protons and 84 neutrons based on three vlbraUonal band heads for two different values of A, A = Ae~p and A ~ 2Aex p The larger A leads to more coUectzve wave functmns w~th subsequent energy shafts
different values of A were used to calculate the occupation probabilities v~(a,fl,). The use of a larger A leads to a smoother variation in the overlap matrix A,~ which leads to greater collectivity, smaller deformations, and smaller effective moments of inertia. This is seen as an upward displacement of all states including the band heads. The parameters in (12) were chosen to fit the I = 2 ground state energy in zSaGd as well as the first two excited I = 0 states which leads to a spectrum resembling that of 154Gd. The distribution of the probability amplitudes d~tp defined by (14) and (15) are shown in fig. 2 for the case of the "microscopic" overlap A,j calculated using (6) and the case of a smooth exponential overlap chosen to reproduce the main decrease in A,j. N o other changes were made in the calculation. Each circle represents the dip appropriate to a quadrupole eigenvalue qp. ~bg and ~ are the first and second I = 0 band heads. The exponential overlap case displays probability amplitude distributions which are excellent
Volume 160B, number 1,2,3
PHYSICS LETTERS
WAVE FUNCTIONS
0,6
~04 ,oo -02 -04 I
I
I
I
I
I
3 October 1985
level crossings which occur as the deformation changes. The I - 14 wave function for the ground state band shows the result of centrifugal stretching which leads to an increase of the effective moment of inertia which leads to compression of the spectrum for high I values. The microscopic wave functions generated by these calculations have been used to calculate B(E2) values between eight rotational bands for up to 1 = 24. The calculated "r-ray intensity distribution due to 1104 possible transitions is found to closely resemble experimentally observed intensity patterns.
w04
This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. ~oo -02
References
-04
z6o
abe
oo,o~,o,,o~ ~,o~,,T 66o
z6o
Fig 2 The theorelacal probablhty amphtudes for the ground states, the ground state band I = 14 state, and beta band heads for the 50 proton, 84 neutron system w~th respect to elgenstates of the quadrupole operator for two different overlap matnces A,j. The simple exponenual case leads to harmomc oscdlator type wave functtons
representations of the first two harmonic oscillator states of a vibration. The quadrupole distribution is clustered around the prelate minimum of the e, given by (11). The distributions for the microscopic overlap case are similar but show effects due to irregularities in A u due to single-particle
[1] A Bohr, Mat Fys Medd Dan Vxd Selsk 26, no 14 (1952) [2] A Bohr and B.R Mottelson, Mat Fys Medd Dan Vld Selsk 27, no 16 (1953). [3] M.A Preston and R.K Bhadun, Structure of the nucleus (Ad&son-Wesley, Reading, M_A, 1975) Ch 9. [4] D L. Hall and J A Wheeler, Phys Rev 89 (1953) 1109 [5] J J. Gnttin and J A Wheeler, Plays Rev 108 (1957) 311. [6] P Rang and P Schuck, The nuclear many-body problem (Spnnger, Berhn, 1980) Ch 10 [7] P Rang and P Schuck, The nuclear many-body problem (Spnnger, Berhn, 1980) p 92 [8] P Rang and P Schuck, The nuclear many-body problem (Spnnger, Berlin, 1980) p 67 [9] L S. Kasshnger and R A Sorensen, Mat Fys Medd Dan Vld Selsk 32, no 9 (1960) [10] M.A Preston and R K Bhadun, Structure of the nucleus (Addison-Wesley, Reading, MA, 1975) p. 359