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Prog. Part. NucL Phys. Vol. 34, pp.291-307, 1995

Copyright0 1995 ElsevierScience Ltd

0146-6410(95)00025-9

Printed

n &eat Britain.All rightsreserved

0146-6410/95 $29.00

Magnetic Excitations in Deformed Nuclei

R. NOJAROV

Insiitufjlir Theoretische Physik Vniversitllt Tiibingen Auf der Morgenstelle 14

D-72076 Tiibingen Germany

ABSTRACT

Cross sections for inelastic electron scattering and energy distributions of Ml and E2 strengths of

K = l+ excitations in titanium, rare-earth, and actinide nuclei are studied microscopically within

QRPA. The spin Ml strength has two peaks, isoscalar and isovector, residing between the low- and

high-energy orbital Ml strength.

The latter is strongly fragmented and lies in the region of the

IVGQR, where the (e, e) cross sections are almost one order of magnitude larger for E2 than for Ml

excitations. Comparison with the quantized isovector rotor allows the interpretation of all the orbital

Ml excitations at both low and high energies as manifestation of the collective scissors mode.

KEYWORDS

Ml excitations in deformed nuclei, cross sections for inelastic electron scattering, energy distribution

of Ml (spin and orbital) and E2 strengths, quantized isovector rotor, scissors mode, isovector giant

quadrupole resonance, quasiparticle random-phase approximation.

GENERAL FEATURES AND INTERPRETATION PROBLEMS

Magnetic monopole excitations are strictly forbidden because of the nonexistence of magnetic charge.

Thus, the strongest magnetic excitations, i. e.

those with lowest multipolarity, are magnetic dipole

(Ml) excitations. They have been extensively studied in the past mainly in spherical nuclei through

inelastic electron scattering at backward angles, see e. g. the review article of Raman et al. (1991).

Recent experiments use also polarized tagged photons (Laszewski et al. 1988) and polarized protons

(Lisantti et al. 1991).

Some general features of the Ml excitations can be easily understood by examining the Ml operator,

which consists of orbital and spin parts:

MI = & +s;s,+& I =

&[J,

+ Cs; l)S, + s Ll,

gf = 0.7gf(free),

gi(free) = 5.5855,

gi(free) = -3.8263,

(1)

where gi, g:

are the spin gyromagnetic factors for protons and neutrons. A particle-hole Ml excitation

involves the transition matrix element of the Ml operator (1)

b

e ween two different single-particle

* Supported by the Deutsche Forschungsgemeinschaft.

** Permanent address: Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of

Sciences, BG-1784, Sofia, Bulgaria. E-mail: [email protected]

297

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R. Nojarov

states. In a spherical basis they are eigenstates of the total angular momentum J whose transition

matrix elements vanish due to the spherical symmetry. Thus, the orbital and spin matrix elements are

always equal in magnitude and out-of-phase. The Ml operator obeys in a spherical basis the selection

rules AN = 0, An = 0, Ae = 0, Aj = 0, fl, where N = 2n + L. It has, therefore, non-vanishing

matrix elements only between spin-orbit partners with j = e * l/2 and Aj = 1. These are spin-flip

transitions, but one should note that the matrix elements of the orbital angular momentum

L

are

not vanishing. Such orbital

contributions are present in spherical nuclei in the very weak isoscalar

Ml peak, while the much stronger isovector Ml strength is dominated by spin contributions to the

Ml transition matrix elements. This is due to the large value of the isovector spin gyromagnetic ratio

g; - g: = 9.4 relative to gi - gz = I, as noticed by Morpurgo (1958).

In addition to these predominantly spin transitions between two different j-shells, two new kinds of

mainly orbital Ml excitations with 1K = l+l become possible in deformed nuclei at low and high

energy:

i) The deformation splitting of the spherical j-shells gives rise to low-energy particle-hole exci-

tations within the same j-shell, characterized by large orbital and small spin matrix elements.

ii) The orbital angular momentum L has matrix elements between different major shells with

AN = 2, giving rise to high-energy orbital Ml excitations, which do not exist in spherical nuclei.

The spin contributions are negligible in the latter case, because of the AN = 0 spin selection rule.

In contrast to spherical nuclei, the experimental information on l+

excitations in deformed nuclei

was very scarce in the past, see e. g.

the introduction of Nojarov et al. (1988). Their extensive

experimental study started with the identification of low-lying orbital Ml excitations through a high-

resolution inelastic electron scattering on the linear accelerator in Darmstadt (Bohle et al., 1984a),

see the review articles of Richter (1991, 1993, 1994). Energies and B(M1) values of these states were

determined with high precision v i nuclear resonance fluorescence, reviewed by Kneissl (1992, 1994).

The spin Ml strength was studied recently with (p,p)

reactions (Frekers et al., 1990; Richter 1991,

1993, 1994), providing also information on the high-energy orbital Ml strength (Richter, 1993, 1994).

The interpretation of the orbital Ml excitations in deformed nuclei is still controversial. They were

identified initially (Bohle et al., 1984a, b) with the scissors mode predicted by the two-rotor model of

Lo Iudice and Palumbo (1978, 1979). Neutrons and protons are assumed in this collective model to

perform isovector (out-of-phase) rotational oscillations around an axis perpendicular to the nuclear

symmetry axis. The collective model of Bohr and Mottelson (1975) was not able to predict such a

mode, because it was restricted to isoscalar degrees of freedom where the I+ state is purely spurious.

The quantum collective model for spherical nuclei was extended to isovector vibrations by Faessler

(1966). The isovector 1+ mode in deformed nuclei, described classically by the two-rotor model

(Lo Iudice and Palumbo, 1978, 1979), was quantized recently (Nojarov, 1994) in canonical relative

variables.

Microscopic calculations within the quasiparticle random-phase approximation (QRPA) using sepa-

rable interactions support the interpretation of the orbital 1+ excitations in deformed nuclei in terms

of isovector rotational vibrations or a weakly collective scissors mode (Nojarov and Faessler, 1988,

1990, 1993; Faessler, 1992). The residual interaction is constructed from the quadrupole operators

Qz,*r, generating infinitesimal rotations around an axis perpendicular to the nuclear symmetry axis.

It should be stressed that the orbital QRPA 1+ excitations acquire a well-pronounced scissors char-

acter only after restoration of the rotational invariance (Nojarov and Faessler, 1988) violated by the

deformation of the mean field.

The interpretation in terms of scissors mode is supported by some works using microscopic approaches

(e. g. De Coster and Heyde, 1991), but rejected by others (e. g. Hamamoto and Aberg, 1984, 1986;

Speth and Zawischa, 1988, 1989; Zawischa and Speth, 1990). In the above works of Zawischa and

Speth the scissors mode is found within QRPA at high energy, Etheor. 22-24 MeV. It represents the

Ml response of the collective motion related with the isovector giant quadrupole resonance (IVGQR).

We are going to present here some of our results related with the electromagnetic excitations of

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l+ states in deformed nuclei and supporting the interpretation of all the orbital Ml excitations as

manifestation of the scissors mode.

LOW-ENERGY ORBITAL EXCITATIONS AND

SPIN STRENGTH DISTRIBUTION

Our QRPA results are obtained with a deformed, axially-symmetric Woods-Saxon potential, BCS

pairing, and separable quadrupole and spin residual interactions (Faessler and Nojarov, 1990). The

rotational invariance, violated by the deformation of the mean field,

is restored in RPA using a

symmetry-restoring procedure (Nojarov and Faessler, 1988).

1.5

1.0

0.5

0.0

1.0

- 0.5

N

3

- 0.0

-

g1.0

a

0.5

0.0

1.0

0.5

5Gd

lsaGd

1

6 6

E(MeV)

Fig. 1.

Energy distribution of Ml strength in rare-earth nuclei (Sarriguren et al.,

1994). Single, mainly orbital, QRPA excitations below 4 MeV with B(Ml)>

0.1 p$ are displayed as bars and compared to (7,-y) experimental data

(circles) from Pitz et al. (1989, 1990), Friedrichs et al. (1992), Margraf et

al. (1993), and Ziegler et al. (1993). The predominatly spin Ml strength

above 4 MeV (histograms) is compared to

(p,p)

data (dots with error bars)

from Frekers et al. (1990) and Richter (1991, 1993).

Results for several rare-earth nuclei are dispayed in Fig. 1. The theoretical B(M1) values of single l+

excitations are plotted as bars. They are in a good agreement with data from high-precision (7,~)

experiments (circles with error bars).

These are predominantly low-energy orbital Ml excitations,

lying between 2 and 4 MeV.

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R. Nojarov

The energy distribution of the calculated spin Ml strength is given by histograms in bins of 80 keV

and compared to data from inelastic proton scattering. Most of the spin strength resides between

5 and 10 MeV. Its typical double-bumped structure is reproduced qualitatively. The analysis of

neutron and proton spin matrix elements of the involved QRPA excitations has shown (Sarriguren et

al., 1993, 1994; Nojarov et aI., 1993) that the weaker lower peak is mainly isoscalar, while the higher-

lying strong peak is isovector. The spin strength is similar in this respect to the case of spherical

nuclei (Lipparini and Richter, 1984; Laszewski et al., 1988). Only the isovector spin strength shows

up in both peaks (Fig. l), because the isoscalar one is strongly damped by the small value of the

isoscalar spin gyromagnetic factor, as noted by Morpurgo (1958)

and mentioned in the introduction.

The isovector peak consists of almost purely spin excitations, while the isoscalar one contains also

some orbital admixtures. The latter are not included in the histograms, since only the spin strength

is deduced from the (p,p) experiments whose data are plotted for comparison.

Fig. 2.

Ml spectrum of 23sU. Theoretical QRPA excitations with B(Ml)> 0.1 &

(bars in the lower two plots) are compared in the bottom plot with (e, e)

data from Heil et al. (1988). The energy region between 2 and 3 MeV from

the middle plot is displayed versus an enlarged energy scale in the bottom

plot in order to see better the agreement with experiment. Bars in the top

plot: squared overlaps with the scissors state.

The same general features are manifested also in heavier deformed nuclei, as seen from the Ml

spectrum of 238U in Fig. 2. Instead of histograms for the spin strength alone, the total (spin +

orbital) B(M1) values of single excitations up to 10 MeV are plotted. As seen from the top plot

of Fig. 2, the five strongest low-lying orbital excitations, shown in more detail in the bottom plot,

overlap altogether 40 with the collective scissors state.

FORM FACTORS FOR INELASTIC ELECTRON SCATTERING

Transverse Ml transition densities are obtained microscopically from the QRPA wave functions of

different l+ excitations. They are used to calculate the corresponding DWBA form factors for back-

ward inelastic electron scattering (Heisenberg and Blok, 1983). The theoretical form factors of the

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301

00

0.5

1.0

1.5

2.0

q

WI

- total

--- orbdxsl : / I1

10-e ,1,,..1,,,.

i

1

0 05 4,,: 15

Wl

Fig. 3. Transverse Ml form factors of 48Ti in DWBA for (e, e) at scattering angle

0 = 165 versus the effective momentum transfer (Nojarov ct al., 1991 .

The QRPA form factors (full curves) of strong Ml excitations at 3.78 MeV

(1.h.s. plot) and 7.2 MeV (r.h.s. plot) are compared with experimental data

(dots with error bars or arrows) from Guhr et al. (1990) and Richter (1990),

respectively. The latter form factor was predicted before the subsequently

published experimental data. Dot-dashed curves: the collective scissors form

factor in the 1.h.s. plot and the orbital QRPA contribution in the r.h.s. plot,.

8

Fig. 4.

Ml transition densities of 48Ti for the scissors state (1.h.s. plot) and the

QRPA excitation at 3.78 MeV (r.h.s. plot) from Nojarov et al. (1991). The

total transition density (continuous curves) is the sum of the proton con-

vection (dot-dashed) and magnetization (neutrons: short-dashed, protons:

long-dashed) current contributions.

8

two strong Ml excitations in 48Ti, shown in Fig.

3, agree well with the experimental data even at

high momentum transfer. The state at 7.2 MeV is almost purely spin-flip transition, because the

orbital contribution (dot-dashed curve in the r.h.s. plot) is very small. Its QR.PA form factor was

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302

R. Nojarov

predicted before the corresponding experimental data became available.

The transition density of the Ml excitation at 3.78 MeV from Fig. 3 is displayed in the r.h.s. plot

of Fig. 4 and compared to the scissors transition density in the 1.h.s. plot. It is seen that the two

transition densities and their three components are very similar to each other, apart from some minor

differences in the proton currents deep inside the nucleus for r < 2 fm. Such differences will be relevant

for the (e, e) form factor only at high transferred momentum. Although the scissors transition density

has larger amplitudes (note the different scales of the two plots), its shape is very similar to that

of the QRPA excitation. This should be expected from the large overlap of the low-energy orbital

excitations with the scissors mode, seen in the top plot of Fig. 2 on the example of 238U. Larger spin

contributions are present in lighter nuclei even at low energy, but the similarity with the scissors form

factor and transition density is still well manifested, as seen from Figs. 3 and 4. This is due to the

fact that the scissors mode is less collective in lighter nuclei and, therefore, more similar to the weakly

collective QRPA excitations (Nojarov and Faessler, 1993).

12 3

8 9 1

Fig. 5.

Ml transition densities of the strongest low-lying orbital Ml excitation ob-

tained in QRPA: at 2 MeV in 232Th (dot-dashed curve in the upper plot,

Nojarov et al., 1993), at 3.3 MeV in

54Sm (continuous curve in the lower

plot, Faessler et al. 1990). Scissors Ml transition densities: continuous

curve in the upper plot and dot-dashed curve in the lower plot.

Having in view the predominantly orbital nature of the scissors mode, one has to expect that the

above similarity should be well pronounced also in heavier nuclei, where the low-lying excitations have

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a stronger orbital character. This is confirmed on the example of rare-earth and actinide nuclei, shown

in Fig. 5. For each nucleus, the transition density of the strongest (low-lying orbital) Ml excitation

is very similar to the corresponding scissors transition density. In all the four cases in Fig. 5 the main

contribution arises from the (orbital) proton convection current. The scissors transition densities

have larger amplitudes, because of the strong collectivity. The scissors and the considered QRPA

excitations have the same leading two-quasiparticle components in their wave functions (Nojarov and

Faessler, 1993).

The (e, e) form factors of the strongest low-lying orbital Ml excitations in 238U and 154Sm are

displayed in Fig. 6. They agree with the experimental data, while the scissors form factors are much

larger, but exhibit again a similar shape, as in the case of transition densities compared in Fig. 5.

The main contribution to the QRPA form factor originates from the orbital part (dot-dashed curve

in the 1.11.~. plot), which is typical for heavy nuclei.

_ RlA

lo-@1 4 1 .\I

, .

0.0

0.5 1.0 1.5

Pm-Ierf

0

0.5

1.5 2

serf [Im-I

Fig. 6.

Theoretical Ml (e, e) form factors calculated in DWBA for a scattering

angle 0 = 165 and compared to experiment (dots with error bars) for 238U

(Heil et al., 1988) and

154Sm (Bohle et al., 1984b). Continuous curves:

QRPA form factor (Nojarov et al., 1993) of the two strongest low-lying Ml

excitations in 238U, shown in Fig. 2, and the strongest one in 154Sm (Faessler

et al., 1990). Dot-dashed curves: the convection current contribution to the

QRPA form factor of

23*U and the scissors form factor of 154Sm. Dashed

curve: the scissor form factor of

238U The form factors in the r.h.s. plot are

obtained from the transition densities in the lower plot of Fig. 5.

The comparison between scissors and QRPA wave functions, Ml transition densities, and (e, e) form

factors allows us to interpret the low-lying orbital Ml excitations in deformed nuclei as isovector

rotational vibrations or a weakly collective scissors mode (Nojarov and Faessler, 1988, 1990, 1993;

Faessler, 1992).

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R. Nojarov

Ml AND E2 e,e) CROSS SECTIONS OF HIGH-ENERGY

ORBITAL EXCITATIONS

The energy distributions of Ml and E2 strengths for QRPA excitations with K = 1+ in rsoGd

are displayed as histograms in Fig. 7. They are obtained from a symmetry-restoring interaction of

quadrupole type with a ratio r = -2 between its isovector and isoscalar coupling constants (Nojarov

et al., 1994b). The latter constant is calculated microscopically from the condition of rotational

invariance. It influences mainly the low-energy strength (E < 12 MeV) which is, therefore, relatively

well determined from first principles.

0

5 1 15 2 25 3

energy [MeV]

Fig. 7.

Energy distributions (contour histograms) of E2 (top plot) and Ml (lower

two plots) strengths of QRPA excitations with K = l+ in 16Gd (Nojarov

et al., 1994b). Shaded areas in the lower two plots: only orbital Ml strength.

The quasiparticle basis cut-off at 20 MeV in the bottom plot produces a sin-

gle schematic excitation (at 22 MeV), incorporating the whole high-energy

orbital Ml strength.

Only the K = fl contributions to the E2 strength are displayed in the top plot of Fig. 7, because

we are interested in E2 transitions accompanying the Ml excitations, whose strength is shown in

the lower two plots. The K = 1 component of the isoscalar giant quadrupole resonance (ISGQR),

seen in the top plot of Fig. 7, is peaked at 11 MeV with B(E2;0+0 -+ 2+1) = 0.12 e2b2 in the

region 9-11 MeV. The B(E2) value agrees with experimental data (Van der Woude, 1987, 1991) on

neighbouring nuclei. The isovector counterpart (IVGQR) is peaked between 22 and 23 MeV with

B(E2;0+0 + 2+1) = 0.15 e2b2 in the region 17-25 MeV. This result is in a qualitative agreement

with recent experimental data on

154Sm (Richter, 1994). The QRPA energy-weighted E2 strength

in the interval O-30 MeV exhausts 99.5 from the corresponding classical sum-rule for the

K =

1

component (Bohr and Mottelson, 1975), S(E2; class.) = 5.76 e* bMeV.

The Ml strength distribution for K =

l+ excitations is shown in the lower two plots of Fig. 7, where

the orbital strength is plotted separately as shaded histograms. It is seen that the high-energy Ml

strength (above 17 MeV) is almost purely orbital. It follows roughly the energy distribution of the

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IVGQR. The two-quasiparticle basis was cut off at 20 MeV in the bottom plot to produce a single,

schematic QRPA excitation (at 22 MeV). It incorporates all the high-energy strength above 20 MeV

from the middle plot and has B(M1) = 3.6 pk,

in agreement with recent experimental estimates

(Richter, 1994). The same strength is distributed between 17 and 25 MeV in the more realistic case

from the middle plot. The largest B(M1) al ue of a single excitation from this energy region in the

middle plot is smaller than 0.25 p L, because of the considerable fragmentation at high energy.

The isovector rotor (Lo Iudice and Palumbo, 1978, 1979) was quantized recently (Nojarov, 1994)

iu relative canonical variables. The resulting scissors mode splits in the schematic two-level basis

of the deformed harmonic oscillator into low- and high-energy components, corresponding to trar-

sitions within t,he same shell with AN 0 and between shells with AN 2, respectively. It was

shown within the collective isovector rotor model (Nojarov, 1994) that the scissors mode exhausts

the whole non-spurious orbital Ml strength at low and high energy. This results strongly supports

the interpretation of all the orbital Ml excitations in deformed nuclei, at both low and high energies,

as manifestation of the scissors mode.

It, was shown (Zawischa and Speth, 1990) that the collective hydrodynamical model does not allow

the exist,ence of a low-energy scissors mode. However, after taking additionally the nuclear elasticity

within the same classical model into account, a low-energy mode was obtained as well (Zawischa and

Speth, 1993), carrying most of the orbital Ml strength. This result is already in agreement with

the predictions of the isovector rotor model, apart from the definite scissors nature of the low-energy

mode in t,he latter model.

I I I, I I,, , I I,, , I,,

-- Ml

\ _

\

\

\

I I

I, I, I I I I, I I I I I

\

50

75

100

incident energy [MeV]

Fig. 8.

DWBA (e, e) cross sections (0 = 165) for the single schematic QRPA

excitation at 22 MeV from the bottom plot of Fig. 7, plotted versus incident

electron energy (Nojarov et al., 199413). Dashed curve: Ml excitation with

Ih = l+l; dotted curve: E2 excitation with 1K = 2+1; continuous curve:

the sum of the Ml and E2 cross sections.

The Ml and E2 (e, e) cross sections of the schematic QRPA state at 22 Mev from the bottom plot

of Fig. 7 is plotted in Fig. 8. It is seen that the two cross sections have a very similar shape, but

even at backward scattering (0 = 165) the E2 cross section is about one order of magnitude larger

than the Ml cross section. Thus, the E2 electroexcitation is dominant in the high-energy region of

the IVGQR and the high-energy orbital Ml strength could hardly be seen through inelastic electron

scattering. We have found that substantial E2 contributions are present at intermediate momentum

transfer (0.4 < p < 0.6 fm-l) also in the (e, e) cross sections of low-energy orbital Ml excitations

(Nojarov et al.. 1994a; Dir&elder et al., 1994).

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R. Nojarov

CONCLUSIONS

We have studied theoretically the I 15 MeV) the cross sections are

one order of magnitude larger for E2 than for Ml electroexcitations, even at scattering angle of 165.

This is true at least for incident energies up to 150 MeV, for which calculations have been done.

The comparison between scissors and QRPA wave functions, Ml transition densities, and (e, e) form

factors allows us to interpret the low-lying orbital Ml excitations in deformed nuclei as isovector

rotational vibrations or a weakly collective scissors mode. The canonical quantization of the isovector

rotor in relative conjugate variables in the schematic basis of the deformed harmonic oscillator shows

that the scissors mode exhausts the whole non-spurious orbital Ml strength at both low and high

energy. These results strongly support the interpretation of all the orbital Ml excitations in deformed

nuclei, at both low and high energies, as fragmentation and manifestation of the collective scissors

mode.

The results presented here have been obtained in collaboration with many colleagues, to whom the

author is expressing his deep gratitude for their outstanding engagement: Amand Faessler, E. Moya

de Guerra, M. Dingfelder, P. Sarriguren, P. 0. Lipas, F. G. Scholtz, and M. Grigorescu. Discussions

with A. Richter, U. Kneissl, N. Lo Iudice, A. Raduta, H.-J. Wortche, and P. von Neumann-Cosel are

gratefully acknowledged. Thanks are due to J. Heisenberg for providing us with his DWBA code.

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