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Computing collective excitations of nuclei
Computer Physics Communications 22 (1981) 177—185 North-Holland Publishing Company
177
COMPUTING COLLECTIVE EXCITATIONS OF NUCLEI Z. SZYMANSKI
*
Institute of Theoretical Physics, University of Warsaw, ul. Hoza 69, P1-00-681 Warsaw, Poland
The atomic nucleus is a quantal many-body system of nucleons interacting with strong forces. It differs considerably both from such quantal systems as the atom which does not exhibit collective excitations, as well as from macroscopic media which do not show shell structure. The finite and not too large number of particles in the nucleus manifests itself by the important role played by the individual nucleonic orbits. The whole rich variety of nuclear motions can be understood in terms of these orbits. In this review article the nuclear collective excitations of various types and symmetries are studied as a coherent motion of nucleons characterized by their individual orbits. The interaction of nuclear elementary excitations results in the existence of collective vibrations in nuclei. If the interaction is especially strong, the corresponding mode may become unstable thus leading to spontaneously broken symmetry in the system. Examples of typical nuclear collective modes are given. In particular, the modes related to nuclear shape and those connected with building up high angular momentum are discussed in a more detailed way.
1. Introduction The atomic nucleus is a quantal many-body system of neutrons and protons interacting mainly with strong interactions. It differs considerably both from such quantal systems as for example the atom, which contains a central body (atomic nucleus) and thus does not exhibit collective excitations, as well as from macroscopic media that contain practically an infinite number of constituents and thus do not show a shell structure. The atomic nucleus is a fmite system of a not too large number of particles and, consequently, is characterized by pronounced surface effects. Moreover, the finite character of the nucleus implies the important role played by the individual nucleonic orbits moving in the nuclear avearge field. It turns out that the whole rich variety of nuclear motions and their characteristic features in the nucleus can be successfully described by the elementary excitations of the individual-nucleon type. In particular, the nuclear collective motions such as vibrations or rotations can be understood in terms of the coherent motions of nucleons exploring the set of available nucleonic *
Supported in part by the Polish—US Maria SktodowskaCurie Fund, No. P-F7F037P.
orbits. On the other hand, the stability (or instability) of various vibrational modes determines the symmetry of the nuclear field which, in turn, forms a base for the individual nucleonic orbits. In this review article we shall make an attempt to present examples of the most typical nuclear collective motions and to elucidate their intimate relations with the nucleonic orbits. Although the physical picture underlying the analysis seems to be relatively simple, we shall see that the actual computations may be quite involved in real nuclei as contrasted with the simplified models.
2. Individual nucleonic orbits We shall start our considerations by discussing the characteristics of typical nucleonic orbits in the nucleus. Elementary excitations in infinite media are often described as plain waves. Since the nuclear field in real nuclei turns out to be finite in space, it seems more appropriate to characterize the nudeonic orbits by states of sharp angular momentum rather than linear momentum. The potential of a three-dimensional h.o. (harmonic oscillator) may be discussed as the simplest example. Eigenstates of the
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Z. Szymaiiski / Collective excitations of nuclei
h.o. Hamiltonian (1) H= T + 21Mw2r2 may be represented as vectors IN!Aft) where the quantum numbers N, 1, A and ~2denote the principal h.o. quantum number, orbital angular momentum, its projection on the quantization axis, and total angular momentum projection (&~2= A 1 ~), respectively, It is well known that for even N, the quantum number I is also even and vice versa. As an example let us take the orbits: IN, 1 = N, A = N~&T~= A ±~> and
(in case of set (III) we have assumed that N is even, otherwise we could have = 1,! = 3, etc., instead ofl” 0,2,...). It chosen is clear 1that ordering in occupation in the orbits indicated by set (I) favours an oblate distortion of the matter distribution, while set (II) corresponds to a prolate distribution and, finally, set (III) is rather neutral with respect to either of these two trends. The above example shows how the order of filling-in the individual nucleonic orbits may influence the bulk properties of the nucleus and its resulting symmetry. Sets (I), (II) and (III) indicated above are obviously not the only possibilities. One could in principle combine nudeonic orbits according to some other principle. If for instance a large angular momentum is to be accommodated in the nucleus the following set of orbitals
IiV~1 =N, A =N, fl = A ± which are concentrated near the equatorial plane of the nucleus (perpendicular to the quantization axis), corresponding to classical orbits circling along the equator in two opposite senses. On the other hand the orbit tN,l=N,A0,~2’~±~) corresponds to a distribution that produces a rather prolate shape of the nucleus with respect to the quailtization axis, Configurations involving several nucleons may occupy neighbouring nucleonic orbits according to the exclusidn principle. In particular, three types of sets are possible IN,
1
1= N, A = ±N,~2= A ±
IN, l=N— 2, A= ±(N—2), ~
=
A ±~
=
N, A = 0
=
(I)
I
I
(II) IN, l=N
—
2, A = 0, ~
=
—
2, ~2= A ±
IN,l=N— 4,A=N— 4~fl
=
A±
is more likely to settle. It favours a strong alignment of angular momentum along the positive direction in the quantization axis. Similarly, one may imagine other orderings that would tend to favour the breaking of other nuclear symmetries. If a definite symmetry is already broken right from the beginning then the spherical Hamiltonian (and its characteristic orbits) have to be replaced by those
example theanisotropic case ofa nuclear that is h.o. similar to that ofinthe axially field symmetric
N,n 3,A,~Z) with two independent h.o. oscillators: in the z-direction with quantum number n3 and in the x, y direction with quantum number
±~>
or else 1)l IN,l0,A0,f~2±2 IN, I = 2, A = 0, ~ ±~>,
2, A = N
H=T+~’M{wi(x2+y2)+w~z2~, (2) the single nucleonic orbits may be described by vectors
+1’
IN,1N,Al,~A±~>
—
corresponding to broken symmetry in the field. For
~ IN, 1
IJV~1 = N
(3) The multiplet of states corresponding to the sphericalN 6 (i = 13/2) shell is shown in fig. 1 for the elongated nuclear shape (w 3 > w1). Here, the nN—n3.
(III)
Z. Szymahski / Collective excitations of nuclei
179
3. Dispersion relations, response function
~
~,S4-
-+~ -
I
def~
_________________________________________ 3
2
i
2
3
Fig. 1. Multiplet of states corresponding to the spherical
TheWe ground-state asexcitations well asthe the elementary excitations they for mentary vidual nuclear would shall nucleonic nucleonic collective now tend describe orbits. to occupy vibrations Since the mechanism nucleons derived lowest in terms from available are of offermions, the formation the eleindiorbits. This is valid in the absence of the interaction. of the system are schematically illustrated in fig. 2a,b. In the ground-state 10) all lowest orbits are occupied up to a certain energy X (Fermi surface) while the elementary excitations I ph) are of the particle—hole type and may be characterised by their energy (e~ eh) as well as the basic matrix element ~ (4) Here, the operator C)IZ ~ corresponds to a given sym—
N = 6 (i = 13/2) shell for the elongated nuclear shape
(~ > wj).
metry and a defmite type of nuclear distortion. The matrix element (4) determines the strength of a rele-
standard notation: s, p, d, f, g, h, i,j, corresponds to orbital angular momentum 1 = 0, 1,2,3,4, 5,6, 7, ..., respectively. Obviously, the harmonic oscillator as a model for the nuclear field presents itself as a crude simplification which can only be employed to gain a preliminucleonic motion in the individual orbits. In a more nary qualitative orientation in the structure of realistic description of the nuclear field, several effects such as the existence of a strong spin—orbit coupling, finite range of the nuclear-force potential characterized by the depth V. radius R, boundary diffuseness, etc., have to be taken into account. Several detailed formulae for nuclear potentials, e.g. Nilsson, Woods—Saxon, folded Yukawa, etc., have been extensively employed by many investigators. In most cases, however, these potentialshave been diagonalized in the basis of the h.o. eigenvectors. In practical computations in heavy nuclei which may be strongly distorted either by large spatial deformation and/or considerably high angular momentum, a basis is usually needed that contains all the h.o. shells up to typically N 15. The order of matrices to be diagonalized may then be roughly of the order of 500 X 500. If neither parity nor time-reversal invariance holds this number may have to be doubled, ...
___________
___________
___________
___________
____________
____________
_________
_________
___________
__________
Ii
I I
— —
—
—
11111 I
___________
II
III
I
I
I ii
— — _________
_________ ~—
__________
__________
lj
-~ ___________ __________
_________
___________
___________
—
—
—
_________
I
(c)
Fig. 2. Nuclear collective vibrations; (a) ground state, (b) elementary excitations, (c) strong transitions (p1 ~i,r NIh>.
Z. Szymaiiski / Collective excitations of nuclei
180
vant nuclear single-particle transition (for example electromagnetic transition) and in the same time its characteristic selection rules take care of the sym-
with arrows connecting the (p. h) pairs. It is obvious that chemical potential X~corresponds to the soft nucleus while X1 and X3 do not.
metry in the distortion of the particle—hole state defmed with respect to the ground state. If we deal for example with the nuclear elongation mode (which corresponds to an axially symmetric distortion2in Y the nuclear field), an operator of the type ~)fl~ r 20 would be an 2Yappropriate choice and the matrix elements (p1 r 20 Ih) would correspond to the elementary electric quadrupole transitions.2 Y In the same time, selection rules following from the r 20 operator take care that the only Iph) vectors involved into the wave function are those favouring an axially symmetric quadrupole deformation. Similarly, the operator r2(Y 22 + Y2~)produces a nonaxial deformation while h/~ is the alignment of angular momentum in the direction of the x-axis, etc. Now, if there exist many low-energy particle—hole states Iph)with large matrix elements (pl~11fNIh) the system turns out to be soft (if ever stable) with respect to the corresponding collective mode. This property may depend very sensitively on the number of particles (or in other words on the location of the Fermi surface X) as illustrated in fig. 2c, where the strong transitions (p 1C)1~~I h) are marked by lines
In the mathematical treatment of the vibration each (p. h) pair is treated as an elementary oscillator. The oscillators are then coupled by the specific residual force leading to the Hamiltonian for coupled oscillators
—
—
_\~
K
H=
+
—~--~
+
~ \/
?(1’1A~1~,~1, (5)
where ir• and ~, are the momentum and coordinate of the ith oscillator of frequency w1, a~denotes the interaction strength ~17(~ is the pair. matrix element 1~X Ih) of the ith and particle—hole Diagonalisa(p1 ~) tion of H leads to the dispersion equation 1
=
—icxF(w)
(6)
for the normal modes. Function 2 I~~t .12 ~, F(~~)) = 2 I
—
(7)
W
is called the response function of the system. Graphical solutions of eq. (6) are illustrated in fig. 3. It is seen that the poles w1 of the response function corre-
~N
Fig. 3. Graphical solutions of eq. (6).
181
Z. Szymahski / Collective excitations of nuclei
spond to the energies of the undisturbed (ph) excitations while the strength c1111 at each pole is related to the corresponding residue in F(w) at w = w1. Fig. 3 illustrates the graphical solutions of eq. (6) as the intersection points of the response function curve F(w) with the horizontal straight line at ordinate l/~ These solutions determine the excitations of the interacting system: the vibrational frequency as the collective root of eq. (6) and other noncollective roots. The collective root is characterised as lying far enough from all the poles w,. In the case of a very soft system (IK~Ilarge) the collective root lies very low. In the limiting case, the corresponding root may become imaginary which corresponds to instability of the given mode and, consequently, to spontaneously broken symmetry. Numerical solution of eq. (6) does not usually pose serious problems. Vibrations of neutrons and protons may be treated separately as two coupled motions. In this case the single equation (6) has to be replaced by a 2 X 2 determinant. If several different vibrational modes are coupled together the order of the determinant may be further increased. Nevertheless, it is not very difficult to solveofthe corresponding dispersion equation as a function a smgle parameter wand to determine the roots of eq. (6). —
~.
4. Nuclear elongation Out of all the nuclear collective modes related to the change in nuclear shape the ellipsoidal axiallysymmetric distortion is probably the most pronounced and best investigated. For small deviations from the spherical distribution of matter this mode overlaps with the quadrupole (X = 2, ~u= 0) distortion that is connected with operator CJ7~= r2 Y 20. The softness of the nucleus related to this mode turns out to depend very sensitively on the degree to which nuclear shells are filled. The more nucleons outside closed shells, the softer the system becomes. Nuclei with unfilled shells both for protons and neutrons may even become unstable in this mode thus leading to nonspherical equilibrium shapes. Let us discuss 2the Y features of nuclear vibrations in the quadrupole (r 20) mode and the possible onset of the stable nonspherical shape in the simplified case of a single harmonic oscillator shell with N denoting
its main quantum number. There are ~7 ~(N + 1) X + 2) degenerate pair-states in the shell. In the case of many particles occupying degenerate (or close lying) levels the situation is slightly more complicated as compared to that represented in fig. I. The complication is caused by the onset of the superfluid phase in the nucleus following from the strong correlations between pairs of nucleons. Without going into a detailed description of the superfluid correlations let us only mention that the elementary excitations of the system are no longer of the particle—hole type. They are instead pairs of dressed excitations (quasiparticles). For a single degenerate shell the two quasiparticle energy which should replace the energies hw1 in eq. (7) may be estimated as constant quantitiesof the order of G~2where G denotes the strength of a pairing force. Moreover, the matrix element ~ entering the response function is also modified by a certain factor determined by the existence of the pair correlations. In the case of a degenerate h.o. shell the resulting response function turns out to have only one pole at energy G~Zand the strength depends sensitively on the degree shell is filled 2flc)z ~ tojwhich the~ h.o. [(G~\2 21 2 G F(w) = —s--— ~l ~c-) I ~ (I x ~ I9~~I2, (8)
(N
~
,
—
—
where 9Z denotes the number of valence particles (0 ~ ~ (N + l)(N + 2)). The presence of a term 9Z(l 9Z/2~l)in the response function is very essential. It shows that the strength of the transition at the pole increases considerably as the shell is being filled and reaches its maximum when the shell is half occupied. Stability in the elongation mode depends on the strength ~ of the relevant two-body force. It turns —
out that for estimates corresponding to a realistic situation in nuclei the vibration already becomes unstable at 9~/2f~ 0.2. At this point, eq. (6) no longer has a real collective solution for w with the response function F(w) given by eq. (8) and the spherical symmetry is spontaneously broken. Once this point is reached, the representation of spherical h.o. orbits is no longer useful and one can rather characterize the individual nucleonic orbits by a deformed li~rmonicoscillator (2) with stationary states IN, n~,A, ~L)
Z. Szymahski / Collective excitations of nuclei
182
(low frequency mode). The other possibility would be connected with the coupling between the major h.o. shells. The major h.o. shell with the main quantum number N may couple with the two neighbouring shells N ±2. In this way the giant quadrupole resonance could be described (high frequency mode). We shall not follow this direction in the present article.
Nt i
A ii =
N
_______________________
~-
S. Other components in the shape distortion
~‘
N
-
p7~NH
— -
N-f
/1
0
deFo i.,. —
Fig. 4. Competition between the spherical and deformed shape.
discussed in section 2. The competition between the spherical and deformed shape is illustrated in fig. 4. When the h.o. shell is partly filled the onset of deformation becomes a result of a balance between the steeply down-going orbits with n~= N + 1, A = 0 against the up-going orbits with n~= 0 and A = N tending to restore spherical symmetry. In the calculations corresponding to a more realistic case the full response function F(w) with the inclusion of all mechanisms leaving the’h.o. degeneracy has to be considered. In addition, the superfluid correlations have to be included by solving a nonlinear set of equations following from the treatment of a pairing force. Alternatively, one may solve the problem of orbits in the deformed potential and consider the total nuclear energy as a function of the deformation computed by means of some method that takes into account the shell structure in the nucleus (Strutinsky averaging procedure). Such methods will not be discussed in the present article. We have described the quadrupole vibration and the possible onset of nuclear deformation as following mostly from the existence of the valence nucleons
5.1. Hexadecapole mode (X,,u = 4, 0) This is the mode determined by the r4 Y40 operator that modifies 2 Ythe elongation degree of freedom charactensed by r 20. A slight modification of the prolate ellipsoidal shape provided by the presence of the hexadecapole component in the nuclear field tends to make the two tips of the ellipsoid less pointed and in the same time a neck along the “waistline” (i.e. the equatorial plane) of the nucleus is formed, see fig. 5. We shall not analyse this degree of freedom in its detailed relation with the properties of nuclear orbits, shell fifing, etc. Instead, we shall only mention that the inclusion of the hexadecapole mode may be very important in the discussion of the very heavy nuclei that may undergo fission. It turns out that the delicate
-- --
______
/
-
.
. .
.
-
.
.
Fig. 5. Modification of the prolate ellipsoidal shape provided by the presence of the hexadecapole component in the nuclear field.
Z. Szymauiski / Collective
excitations of nuclei
183
distortion. It turns out that around neutron number larger than 130 there occur many pairs of nucleonic orbits such as [404 fl —[514 [402 ~]—[512
fl, fl,
[402 fl—[512 ~], a Ii I-
a a a
[400 4]—[510
5:
that lie close to the Fermi surface and interact across it via rather strong octupole strength (~(~ r3Y3o) with l,~n5=l,~A=0and~S~=0,TheN=4
a 0
0
Fig. 6. The “lowest energy cut” through the potential energy surface for some actinide nuclei.
balance between nuclear surface and Coulomb energy may be seriously affected 4 Y in these nuclei (in the actinide region) if the r 4o mode is superimposed 2Y on the usual r 20 distortion. For example the potential energy surface may be a quite complicated function of the deformation parameters 132 and j34 determining the quadrupole and hexadecapole contnibution to the nuclear field. Fig. 6 illustrates the resulting “lowest energy cut” through the potential energy surface along a certain path in the plane (132, j34) for some actinide nuclei. We can see that certain nuclei may exhibit a two-humped fission barrier with a well (“second minimum”) inside. This peculiar behaviour in the fission barrier explains successfully the existence of shape isomers that deexcite through spontaneous fission. 5.2. Reflection asymmetry (A, p
fl,
=
3, 0)
We shall only discuss here the octupole (A, p
=
3, 0)
Obviously, such distortion the nuclear violates mode superimposed on the in existing stablefield elongation. the symmetry with respect to the equatorial plane of the nucleus. In the elongated nucleus the ellipsoid is then transferred into an asymmetric shape (“pear shape”). It turns out that a favourable situation for this mode may occur in heavy fissioning nuclei at the second hump in their fission barrier (fig. 6). This corresponds to the very elongated shape in nuclear field with an appreciable hexadecapole component in the
configurations appearing in this set are the “waistline” orbits while N = 5 partners are asymmetric with respect to the waistline. Consequently the strong interaction in these pairs tends to shift nuclear matter from the upper half of the nucleus down (or vice versa). This situation is illustrated in fig. 7. As an effect of the strong particle—hole interaction the second hump in the fission barrier (cf. fig. 6) may be sliced in some nuclei. This means that the nucleus starts its way to scission from a pear-shape saddle rather than from a shape that is symmetric with respect to reflection. This possibility has been attributed to the mass asymmetry in the fission products of some nuclei that exhibit the octupole instability (e.g. some isotopes of Ra, Th and U). The relevant numerical calculations of the above mode are rather involved since the presence of the (X, p) = (3, 0) component in the nuclear field means that the parity of the nucleonic states is mixed and, consequently, the order of matrices to be diagonalized is doubled.
9~ C51k %1
Fig. 7. half Shifting ofnucleus nuclearbecause matter from lowerinteraction half to thein upper of the of thethe strong N = 5 partners.
Z. Szymaiiski / Collective excitations of nuclei
184
6. High angular momentum states in nuclei (rotation and alignment of angular momentum) The mechanism of nuclear rotation and alignment of angular momentum is different from that in the case of vibration at least in two respects. First of all, there is no restoring force in the nuclear rotation that attempts to orient the nuclear symmetry axis in a direction that is fixed in space. Second, the nuclear rotating field breaks the invariance of the system with respect to time reversal. The Coriolis force that acts on the nucleons moving along their orbits in the rotating nuclear field seems to be the most important feature in the study of the rotational mode in the nucleus. The Coriolis interaction originating from a coupling of type I’ J of the nucleonic angular momentum .J with the total nuclear spin I tends to align nucleonic orbits in such a way that J becomes parallel to I. The nucleonic relevant matrix elements of the Coriolis force are (for a single f-shell) _________________
(Ill + 1 Ii~ I/f1) =
~v’(/ cZ)(/ + &2 + 1),
(/~2 li/i If&2) =
~ + 1)
—
—
(9)
One can immediately see that the strongly interacting orbits are those of largef and small ~l. These orbits are most easily aligned. In every h.o. shell N there
5
4
-
—
~v.
•
~)
-
-
•
the nucleus picks up more angular momentum. This effect has been called a back-bending as the curves representing some physical quantities “bend back” as functions of w (cf. fig. 8). The detailed treatment of rotating nuclei and the computation of the backbending effect in particular require extensive computation. As one of the possibilities, the Hartree—Fock—Bogolyubov (HFB) formalism (or its simplification known under the name of the independent quasiparticle formalism in a rotating nucleus) are employed. The HFB equations form a nonlinear set of equations for the energy eigenvalues E1 and eigenvectors that define the energy and structure of a quasiparticle. In a matrix notation they may be written as
(~8)
~~__—°°
—
exists one multiplet of orbits of highest angular momentum, for example the h = 11/2 orbits for the N= 5 shell (with 15), the i’~13/2 forN 6(1 = 6), / = 15/2 for N = 7 (1 7), etc. These orbits are in addi. tion shifted down by the strong spin—orbit coupling, so that the h = 11/2 orbit can be found in the nuclear spectrum among the N = 4 orbits, i 13/2 among N 5 orbits, etc. This property exhibits even more the peculiar character of these orbits. Let us study as an example the deformed orbits belonging to the i = 13/2 multplet (cf. fig. 1). These orbits are very important for the formation of aligned states in the rare earth nuclei. Each time the Fermi surface A passes such an orbit there is a considerable jump in the alignment of the nuclear angular momentum. This actually means that the existing collective angular momentum in the rotating nucleus is suddenly shifted from the collective mode into the intrinsic degrees of freedom. As a consequence, the collective angular velocity of rotation w may even decrease locally at such point. This is the reason for the apparent paradox discovered in some nuclear spectra. At the point of rapid alignment, w decreases although
\
1 A’ 1))(BW) =EI(())~ (10) where matrices(vv and ~ again depend on the solutions: _~*,
p 07
o
.
0
‘
1
I
.2
~
~
(MeV)
Fig. 8. “Back-bending” effect.
•
I
_hw(/1)07,
e~+4
~
(11) (12)
Z. Szymahski / Collective excitations of nuclei
with
185
(12)). Then, eq. (10) is to be solved for the eigenvalues E, and eigenvectors ~ Then eqs. (11) to (16) have
p0p
=
X0p
=
~ ~
(13)
~ B~WAW).
(14)
The set (10) is then completed by the additional (constraint) equations p00
=
(15)
0
and ~
all
(fi)0~p~o,
(16)
for the number of particles 9Z and angular momentum I. These two constraints enable one to fix two Lagrange multipliers A and w entering eq. (10). Now, the solution procedure can be described in the following way. First of all, a guess has to be made for the self-consistent nuclear field (second term in eq. (11)) and for the self-consistent pairing field ~ (eq.
to be verified. If the final values for the fields do not coincide with the original ones, a correction must be made and the hole procedure has to be reiterated. When the iteration is terminated one can use the fmal values of the quasiparticle energies E1 and veàtors to compute any physical quantity. In particular, eq. (16) provides us with the value of total angular
(2)
momentum. The system of HFB equations is rather difficult to solve in a general way. This follows mostly from its nonlinearity. In fact, only approximate solutions have been found numerically up to now for the determination of nuclear properties at high angular momentum. References [1] A. Bohr and B.R. Mottelson, Nuclear Structure, vols. 1 and 2 (W.A. Benjamin, 1969 and 1975). [21S.G. Nilsson and I. Ragnarsson, Elementary theory of nuclear structure, in preparation.
177
COMPUTING COLLECTIVE EXCITATIONS OF NUCLEI Z. SZYMANSKI
*
Institute of Theoretical Physics, University of Warsaw, ul. Hoza 69, P1-00-681 Warsaw, Poland
The atomic nucleus is a quantal many-body system of nucleons interacting with strong forces. It differs considerably both from such quantal systems as the atom which does not exhibit collective excitations, as well as from macroscopic media which do not show shell structure. The finite and not too large number of particles in the nucleus manifests itself by the important role played by the individual nucleonic orbits. The whole rich variety of nuclear motions can be understood in terms of these orbits. In this review article the nuclear collective excitations of various types and symmetries are studied as a coherent motion of nucleons characterized by their individual orbits. The interaction of nuclear elementary excitations results in the existence of collective vibrations in nuclei. If the interaction is especially strong, the corresponding mode may become unstable thus leading to spontaneously broken symmetry in the system. Examples of typical nuclear collective modes are given. In particular, the modes related to nuclear shape and those connected with building up high angular momentum are discussed in a more detailed way.
1. Introduction The atomic nucleus is a quantal many-body system of neutrons and protons interacting mainly with strong interactions. It differs considerably both from such quantal systems as for example the atom, which contains a central body (atomic nucleus) and thus does not exhibit collective excitations, as well as from macroscopic media that contain practically an infinite number of constituents and thus do not show a shell structure. The atomic nucleus is a fmite system of a not too large number of particles and, consequently, is characterized by pronounced surface effects. Moreover, the finite character of the nucleus implies the important role played by the individual nucleonic orbits moving in the nuclear avearge field. It turns out that the whole rich variety of nuclear motions and their characteristic features in the nucleus can be successfully described by the elementary excitations of the individual-nucleon type. In particular, the nuclear collective motions such as vibrations or rotations can be understood in terms of the coherent motions of nucleons exploring the set of available nucleonic *
Supported in part by the Polish—US Maria SktodowskaCurie Fund, No. P-F7F037P.
orbits. On the other hand, the stability (or instability) of various vibrational modes determines the symmetry of the nuclear field which, in turn, forms a base for the individual nucleonic orbits. In this review article we shall make an attempt to present examples of the most typical nuclear collective motions and to elucidate their intimate relations with the nucleonic orbits. Although the physical picture underlying the analysis seems to be relatively simple, we shall see that the actual computations may be quite involved in real nuclei as contrasted with the simplified models.
2. Individual nucleonic orbits We shall start our considerations by discussing the characteristics of typical nucleonic orbits in the nucleus. Elementary excitations in infinite media are often described as plain waves. Since the nuclear field in real nuclei turns out to be finite in space, it seems more appropriate to characterize the nudeonic orbits by states of sharp angular momentum rather than linear momentum. The potential of a three-dimensional h.o. (harmonic oscillator) may be discussed as the simplest example. Eigenstates of the
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Z. Szymaiiski / Collective excitations of nuclei
h.o. Hamiltonian (1) H= T + 21Mw2r2 may be represented as vectors IN!Aft) where the quantum numbers N, 1, A and ~2denote the principal h.o. quantum number, orbital angular momentum, its projection on the quantization axis, and total angular momentum projection (&~2= A 1 ~), respectively, It is well known that for even N, the quantum number I is also even and vice versa. As an example let us take the orbits: IN, 1 = N, A = N~&T~= A ±~> and
(in case of set (III) we have assumed that N is even, otherwise we could have = 1,! = 3, etc., instead ofl” 0,2,...). It chosen is clear 1that ordering in occupation in the orbits indicated by set (I) favours an oblate distortion of the matter distribution, while set (II) corresponds to a prolate distribution and, finally, set (III) is rather neutral with respect to either of these two trends. The above example shows how the order of filling-in the individual nucleonic orbits may influence the bulk properties of the nucleus and its resulting symmetry. Sets (I), (II) and (III) indicated above are obviously not the only possibilities. One could in principle combine nudeonic orbits according to some other principle. If for instance a large angular momentum is to be accommodated in the nucleus the following set of orbitals
IiV~1 =N, A =N, fl = A ± which are concentrated near the equatorial plane of the nucleus (perpendicular to the quantization axis), corresponding to classical orbits circling along the equator in two opposite senses. On the other hand the orbit tN,l=N,A0,~2’~±~) corresponds to a distribution that produces a rather prolate shape of the nucleus with respect to the quailtization axis, Configurations involving several nucleons may occupy neighbouring nucleonic orbits according to the exclusidn principle. In particular, three types of sets are possible IN,
1
1= N, A = ±N,~2= A ±
IN, l=N— 2, A= ±(N—2), ~
=
A ±~
=
N, A = 0
=
(I)
I
I
(II) IN, l=N
—
2, A = 0, ~
=
—
2, ~2= A ±
IN,l=N— 4,A=N— 4~fl
=
A±
is more likely to settle. It favours a strong alignment of angular momentum along the positive direction in the quantization axis. Similarly, one may imagine other orderings that would tend to favour the breaking of other nuclear symmetries. If a definite symmetry is already broken right from the beginning then the spherical Hamiltonian (and its characteristic orbits) have to be replaced by those
example theanisotropic case ofa nuclear that is h.o. similar to that ofinthe axially field symmetric
N,n 3,A,~Z) with two independent h.o. oscillators: in the z-direction with quantum number n3 and in the x, y direction with quantum number
±~>
or else 1)l IN,l0,A0,f~2±2 IN, I = 2, A = 0, ~ ±~>,
2, A = N
H=T+~’M{wi(x2+y2)+w~z2~, (2) the single nucleonic orbits may be described by vectors
+1’
IN,1N,Al,~A±~>
—
corresponding to broken symmetry in the field. For
~ IN, 1
IJV~1 = N
(3) The multiplet of states corresponding to the sphericalN 6 (i = 13/2) shell is shown in fig. 1 for the elongated nuclear shape (w 3 > w1). Here, the nN—n3.
(III)
Z. Szymahski / Collective excitations of nuclei
179
3. Dispersion relations, response function
~
~,S4-
-+~ -
I
def~
_________________________________________ 3
2
i
2
3
Fig. 1. Multiplet of states corresponding to the spherical
TheWe ground-state asexcitations well asthe the elementary excitations they for mentary vidual nuclear would shall nucleonic nucleonic collective now tend describe orbits. to occupy vibrations Since the mechanism nucleons derived lowest in terms from available are of offermions, the formation the eleindiorbits. This is valid in the absence of the interaction. of the system are schematically illustrated in fig. 2a,b. In the ground-state 10) all lowest orbits are occupied up to a certain energy X (Fermi surface) while the elementary excitations I ph) are of the particle—hole type and may be characterised by their energy (e~ eh) as well as the basic matrix element ~ (4) Here, the operator C)IZ ~ corresponds to a given sym—
N = 6 (i = 13/2) shell for the elongated nuclear shape
(~ > wj).
metry and a defmite type of nuclear distortion. The matrix element (4) determines the strength of a rele-
standard notation: s, p, d, f, g, h, i,j, corresponds to orbital angular momentum 1 = 0, 1,2,3,4, 5,6, 7, ..., respectively. Obviously, the harmonic oscillator as a model for the nuclear field presents itself as a crude simplification which can only be employed to gain a preliminucleonic motion in the individual orbits. In a more nary qualitative orientation in the structure of realistic description of the nuclear field, several effects such as the existence of a strong spin—orbit coupling, finite range of the nuclear-force potential characterized by the depth V. radius R, boundary diffuseness, etc., have to be taken into account. Several detailed formulae for nuclear potentials, e.g. Nilsson, Woods—Saxon, folded Yukawa, etc., have been extensively employed by many investigators. In most cases, however, these potentialshave been diagonalized in the basis of the h.o. eigenvectors. In practical computations in heavy nuclei which may be strongly distorted either by large spatial deformation and/or considerably high angular momentum, a basis is usually needed that contains all the h.o. shells up to typically N 15. The order of matrices to be diagonalized may then be roughly of the order of 500 X 500. If neither parity nor time-reversal invariance holds this number may have to be doubled, ...
___________
___________
___________
___________
____________
____________
_________
_________
___________
__________
Ii
I I
— —
—
—
11111 I
___________
II
III
I
I
I ii
— — _________
_________ ~—
__________
__________
lj
-~ ___________ __________
_________
___________
___________
—
—
—
_________
I
(c)
Fig. 2. Nuclear collective vibrations; (a) ground state, (b) elementary excitations, (c) strong transitions (p1 ~i,r NIh>.
Z. Szymaiiski / Collective excitations of nuclei
180
vant nuclear single-particle transition (for example electromagnetic transition) and in the same time its characteristic selection rules take care of the sym-
with arrows connecting the (p. h) pairs. It is obvious that chemical potential X~corresponds to the soft nucleus while X1 and X3 do not.
metry in the distortion of the particle—hole state defmed with respect to the ground state. If we deal for example with the nuclear elongation mode (which corresponds to an axially symmetric distortion2in Y the nuclear field), an operator of the type ~)fl~ r 20 would be an 2Yappropriate choice and the matrix elements (p1 r 20 Ih) would correspond to the elementary electric quadrupole transitions.2 Y In the same time, selection rules following from the r 20 operator take care that the only Iph) vectors involved into the wave function are those favouring an axially symmetric quadrupole deformation. Similarly, the operator r2(Y 22 + Y2~)produces a nonaxial deformation while h/~ is the alignment of angular momentum in the direction of the x-axis, etc. Now, if there exist many low-energy particle—hole states Iph)with large matrix elements (pl~11fNIh) the system turns out to be soft (if ever stable) with respect to the corresponding collective mode. This property may depend very sensitively on the number of particles (or in other words on the location of the Fermi surface X) as illustrated in fig. 2c, where the strong transitions (p 1C)1~~I h) are marked by lines
In the mathematical treatment of the vibration each (p. h) pair is treated as an elementary oscillator. The oscillators are then coupled by the specific residual force leading to the Hamiltonian for coupled oscillators
—
—
_\~
K
H=
+
—~--~
+
~ \/
?(1’1A~1~,~1, (5)
where ir• and ~, are the momentum and coordinate of the ith oscillator of frequency w1, a~denotes the interaction strength ~17(~ is the pair. matrix element 1~X Ih) of the ith and particle—hole Diagonalisa(p1 ~) tion of H leads to the dispersion equation 1
=
—icxF(w)
(6)
for the normal modes. Function 2 I~~t .12 ~, F(~~)) = 2 I
—
(7)
W
is called the response function of the system. Graphical solutions of eq. (6) are illustrated in fig. 3. It is seen that the poles w1 of the response function corre-
~N
Fig. 3. Graphical solutions of eq. (6).
181
Z. Szymahski / Collective excitations of nuclei
spond to the energies of the undisturbed (ph) excitations while the strength c1111 at each pole is related to the corresponding residue in F(w) at w = w1. Fig. 3 illustrates the graphical solutions of eq. (6) as the intersection points of the response function curve F(w) with the horizontal straight line at ordinate l/~ These solutions determine the excitations of the interacting system: the vibrational frequency as the collective root of eq. (6) and other noncollective roots. The collective root is characterised as lying far enough from all the poles w,. In the case of a very soft system (IK~Ilarge) the collective root lies very low. In the limiting case, the corresponding root may become imaginary which corresponds to instability of the given mode and, consequently, to spontaneously broken symmetry. Numerical solution of eq. (6) does not usually pose serious problems. Vibrations of neutrons and protons may be treated separately as two coupled motions. In this case the single equation (6) has to be replaced by a 2 X 2 determinant. If several different vibrational modes are coupled together the order of the determinant may be further increased. Nevertheless, it is not very difficult to solveofthe corresponding dispersion equation as a function a smgle parameter wand to determine the roots of eq. (6). —
~.
4. Nuclear elongation Out of all the nuclear collective modes related to the change in nuclear shape the ellipsoidal axiallysymmetric distortion is probably the most pronounced and best investigated. For small deviations from the spherical distribution of matter this mode overlaps with the quadrupole (X = 2, ~u= 0) distortion that is connected with operator CJ7~= r2 Y 20. The softness of the nucleus related to this mode turns out to depend very sensitively on the degree to which nuclear shells are filled. The more nucleons outside closed shells, the softer the system becomes. Nuclei with unfilled shells both for protons and neutrons may even become unstable in this mode thus leading to nonspherical equilibrium shapes. Let us discuss 2the Y features of nuclear vibrations in the quadrupole (r 20) mode and the possible onset of the stable nonspherical shape in the simplified case of a single harmonic oscillator shell with N denoting
its main quantum number. There are ~7 ~(N + 1) X + 2) degenerate pair-states in the shell. In the case of many particles occupying degenerate (or close lying) levels the situation is slightly more complicated as compared to that represented in fig. I. The complication is caused by the onset of the superfluid phase in the nucleus following from the strong correlations between pairs of nucleons. Without going into a detailed description of the superfluid correlations let us only mention that the elementary excitations of the system are no longer of the particle—hole type. They are instead pairs of dressed excitations (quasiparticles). For a single degenerate shell the two quasiparticle energy which should replace the energies hw1 in eq. (7) may be estimated as constant quantitiesof the order of G~2where G denotes the strength of a pairing force. Moreover, the matrix element ~ entering the response function is also modified by a certain factor determined by the existence of the pair correlations. In the case of a degenerate h.o. shell the resulting response function turns out to have only one pole at energy G~Zand the strength depends sensitively on the degree shell is filled 2flc)z ~ tojwhich the~ h.o. [(G~\2 21 2 G F(w) = —s--— ~l ~c-) I ~ (I x ~ I9~~I2, (8)
(N
~
,
—
—
where 9Z denotes the number of valence particles (0 ~ ~ (N + l)(N + 2)). The presence of a term 9Z(l 9Z/2~l)in the response function is very essential. It shows that the strength of the transition at the pole increases considerably as the shell is being filled and reaches its maximum when the shell is half occupied. Stability in the elongation mode depends on the strength ~ of the relevant two-body force. It turns —
out that for estimates corresponding to a realistic situation in nuclei the vibration already becomes unstable at 9~/2f~ 0.2. At this point, eq. (6) no longer has a real collective solution for w with the response function F(w) given by eq. (8) and the spherical symmetry is spontaneously broken. Once this point is reached, the representation of spherical h.o. orbits is no longer useful and one can rather characterize the individual nucleonic orbits by a deformed li~rmonicoscillator (2) with stationary states IN, n~,A, ~L)
Z. Szymahski / Collective excitations of nuclei
182
(low frequency mode). The other possibility would be connected with the coupling between the major h.o. shells. The major h.o. shell with the main quantum number N may couple with the two neighbouring shells N ±2. In this way the giant quadrupole resonance could be described (high frequency mode). We shall not follow this direction in the present article.
Nt i
A ii =
N
_______________________
~-
S. Other components in the shape distortion
~‘
N
-
p7~NH
— -
N-f
/1
0
deFo i.,. —
Fig. 4. Competition between the spherical and deformed shape.
discussed in section 2. The competition between the spherical and deformed shape is illustrated in fig. 4. When the h.o. shell is partly filled the onset of deformation becomes a result of a balance between the steeply down-going orbits with n~= N + 1, A = 0 against the up-going orbits with n~= 0 and A = N tending to restore spherical symmetry. In the calculations corresponding to a more realistic case the full response function F(w) with the inclusion of all mechanisms leaving the’h.o. degeneracy has to be considered. In addition, the superfluid correlations have to be included by solving a nonlinear set of equations following from the treatment of a pairing force. Alternatively, one may solve the problem of orbits in the deformed potential and consider the total nuclear energy as a function of the deformation computed by means of some method that takes into account the shell structure in the nucleus (Strutinsky averaging procedure). Such methods will not be discussed in the present article. We have described the quadrupole vibration and the possible onset of nuclear deformation as following mostly from the existence of the valence nucleons
5.1. Hexadecapole mode (X,,u = 4, 0) This is the mode determined by the r4 Y40 operator that modifies 2 Ythe elongation degree of freedom charactensed by r 20. A slight modification of the prolate ellipsoidal shape provided by the presence of the hexadecapole component in the nuclear field tends to make the two tips of the ellipsoid less pointed and in the same time a neck along the “waistline” (i.e. the equatorial plane) of the nucleus is formed, see fig. 5. We shall not analyse this degree of freedom in its detailed relation with the properties of nuclear orbits, shell fifing, etc. Instead, we shall only mention that the inclusion of the hexadecapole mode may be very important in the discussion of the very heavy nuclei that may undergo fission. It turns out that the delicate
-- --
______
/
-
.
. .
.
-
.
.
Fig. 5. Modification of the prolate ellipsoidal shape provided by the presence of the hexadecapole component in the nuclear field.
Z. Szymauiski / Collective
excitations of nuclei
183
distortion. It turns out that around neutron number larger than 130 there occur many pairs of nucleonic orbits such as [404 fl —[514 [402 ~]—[512
fl, fl,
[402 fl—[512 ~], a Ii I-
a a a
[400 4]—[510
5:
that lie close to the Fermi surface and interact across it via rather strong octupole strength (~(~ r3Y3o) with l,~n5=l,~A=0and~S~=0,TheN=4
a 0
0
Fig. 6. The “lowest energy cut” through the potential energy surface for some actinide nuclei.
balance between nuclear surface and Coulomb energy may be seriously affected 4 Y in these nuclei (in the actinide region) if the r 4o mode is superimposed 2Y on the usual r 20 distortion. For example the potential energy surface may be a quite complicated function of the deformation parameters 132 and j34 determining the quadrupole and hexadecapole contnibution to the nuclear field. Fig. 6 illustrates the resulting “lowest energy cut” through the potential energy surface along a certain path in the plane (132, j34) for some actinide nuclei. We can see that certain nuclei may exhibit a two-humped fission barrier with a well (“second minimum”) inside. This peculiar behaviour in the fission barrier explains successfully the existence of shape isomers that deexcite through spontaneous fission. 5.2. Reflection asymmetry (A, p
fl,
=
3, 0)
We shall only discuss here the octupole (A, p
=
3, 0)
Obviously, such distortion the nuclear violates mode superimposed on the in existing stablefield elongation. the symmetry with respect to the equatorial plane of the nucleus. In the elongated nucleus the ellipsoid is then transferred into an asymmetric shape (“pear shape”). It turns out that a favourable situation for this mode may occur in heavy fissioning nuclei at the second hump in their fission barrier (fig. 6). This corresponds to the very elongated shape in nuclear field with an appreciable hexadecapole component in the
configurations appearing in this set are the “waistline” orbits while N = 5 partners are asymmetric with respect to the waistline. Consequently the strong interaction in these pairs tends to shift nuclear matter from the upper half of the nucleus down (or vice versa). This situation is illustrated in fig. 7. As an effect of the strong particle—hole interaction the second hump in the fission barrier (cf. fig. 6) may be sliced in some nuclei. This means that the nucleus starts its way to scission from a pear-shape saddle rather than from a shape that is symmetric with respect to reflection. This possibility has been attributed to the mass asymmetry in the fission products of some nuclei that exhibit the octupole instability (e.g. some isotopes of Ra, Th and U). The relevant numerical calculations of the above mode are rather involved since the presence of the (X, p) = (3, 0) component in the nuclear field means that the parity of the nucleonic states is mixed and, consequently, the order of matrices to be diagonalized is doubled.
9~ C51k %1
Fig. 7. half Shifting ofnucleus nuclearbecause matter from lowerinteraction half to thein upper of the of thethe strong N = 5 partners.
Z. Szymaiiski / Collective excitations of nuclei
184
6. High angular momentum states in nuclei (rotation and alignment of angular momentum) The mechanism of nuclear rotation and alignment of angular momentum is different from that in the case of vibration at least in two respects. First of all, there is no restoring force in the nuclear rotation that attempts to orient the nuclear symmetry axis in a direction that is fixed in space. Second, the nuclear rotating field breaks the invariance of the system with respect to time reversal. The Coriolis force that acts on the nucleons moving along their orbits in the rotating nuclear field seems to be the most important feature in the study of the rotational mode in the nucleus. The Coriolis interaction originating from a coupling of type I’ J of the nucleonic angular momentum .J with the total nuclear spin I tends to align nucleonic orbits in such a way that J becomes parallel to I. The nucleonic relevant matrix elements of the Coriolis force are (for a single f-shell) _________________
(Ill + 1 Ii~ I/f1) =
~v’(/ cZ)(/ + &2 + 1),
(/~2 li/i If&2) =
~ + 1)
—
—
(9)
One can immediately see that the strongly interacting orbits are those of largef and small ~l. These orbits are most easily aligned. In every h.o. shell N there
5
4
-
—
~v.
•
~)
-
-
•
the nucleus picks up more angular momentum. This effect has been called a back-bending as the curves representing some physical quantities “bend back” as functions of w (cf. fig. 8). The detailed treatment of rotating nuclei and the computation of the backbending effect in particular require extensive computation. As one of the possibilities, the Hartree—Fock—Bogolyubov (HFB) formalism (or its simplification known under the name of the independent quasiparticle formalism in a rotating nucleus) are employed. The HFB equations form a nonlinear set of equations for the energy eigenvalues E1 and eigenvectors that define the energy and structure of a quasiparticle. In a matrix notation they may be written as
(~8)
~~__—°°
—
exists one multiplet of orbits of highest angular momentum, for example the h = 11/2 orbits for the N= 5 shell (with 15), the i’~13/2 forN 6(1 = 6), / = 15/2 for N = 7 (1 7), etc. These orbits are in addi. tion shifted down by the strong spin—orbit coupling, so that the h = 11/2 orbit can be found in the nuclear spectrum among the N = 4 orbits, i 13/2 among N 5 orbits, etc. This property exhibits even more the peculiar character of these orbits. Let us study as an example the deformed orbits belonging to the i = 13/2 multplet (cf. fig. 1). These orbits are very important for the formation of aligned states in the rare earth nuclei. Each time the Fermi surface A passes such an orbit there is a considerable jump in the alignment of the nuclear angular momentum. This actually means that the existing collective angular momentum in the rotating nucleus is suddenly shifted from the collective mode into the intrinsic degrees of freedom. As a consequence, the collective angular velocity of rotation w may even decrease locally at such point. This is the reason for the apparent paradox discovered in some nuclear spectra. At the point of rapid alignment, w decreases although
\
1 A’ 1))(BW) =EI(())~ (10) where matrices(vv and ~ again depend on the solutions: _~*,
p 07
o
.
0
‘
1
I
.2
~
~
(MeV)
Fig. 8. “Back-bending” effect.
•
I
_hw(/1)07,
e~+4
~
(11) (12)
Z. Szymahski / Collective excitations of nuclei
with
185
(12)). Then, eq. (10) is to be solved for the eigenvalues E, and eigenvectors ~ Then eqs. (11) to (16) have
p0p
=
X0p
=
~ ~
(13)
~ B~WAW).
(14)
The set (10) is then completed by the additional (constraint) equations p00
=
(15)
0
and ~
all
(fi)0~p~o,
(16)
for the number of particles 9Z and angular momentum I. These two constraints enable one to fix two Lagrange multipliers A and w entering eq. (10). Now, the solution procedure can be described in the following way. First of all, a guess has to be made for the self-consistent nuclear field (second term in eq. (11)) and for the self-consistent pairing field ~ (eq.
to be verified. If the final values for the fields do not coincide with the original ones, a correction must be made and the hole procedure has to be reiterated. When the iteration is terminated one can use the fmal values of the quasiparticle energies E1 and veàtors to compute any physical quantity. In particular, eq. (16) provides us with the value of total angular
(2)
momentum. The system of HFB equations is rather difficult to solve in a general way. This follows mostly from its nonlinearity. In fact, only approximate solutions have been found numerically up to now for the determination of nuclear properties at high angular momentum. References [1] A. Bohr and B.R. Mottelson, Nuclear Structure, vols. 1 and 2 (W.A. Benjamin, 1969 and 1975). [21S.G. Nilsson and I. Ragnarsson, Elementary theory of nuclear structure, in preparation.