Magnetic dipole excitations in nuclei: Elementary modes of nucleonic motion

The nucleus is one of the most multifaceted many-body systems in the Universe. It exhibits a multitude of responses depending on the way one “probes” it. With increasing technical advancements of beams at the various accelerators and of detection systems the nucleus has, over and over again, surprised us by expressing always new ways of “organized” structures and layers of complexity. Nuclear magnetism is one of those fascinating faces of the atomic nucleus discussed in the present review. We shall not just limit ourselves to presenting the by now large data set that has been obtained in the past two decades using various probes, electromagnetic and hadronic alike and that presents ample evidence for a low-lying orbital scissors mode around $3\mathrm{MeV}$, albeit fragmented over an energy interval of the order of $1.5\mathrm{MeV}$, and higher-lying spin-flip strength in the energy region $5–9\mathrm{MeV}$ in deformed nuclei nor to the presently discovered evidence for low-lying proton-neutron isovector quadrupole excitations in spherical nuclei. To the contrary, the experimental evidence is put in the perspectives of understanding the atomic nucleus and its various structures of well-organized modes of motion and thus enlarges the discussion to more general fermion and bosonic many-body systems.

Figure 1

Schematic representation of the magnetic dipole strength distribution in even-even heavy deformed nuclei and its model character. From 301.

Figure 2

Schematic representation of the magnetic dipole strength distribution in even-even light nuclei and its interpretation in terms of the shell model. The subscripts $\pi$ and $\nu$ denotes the proton and neutron contributions, respectively. From 301.

Figure 3

Schematic representation of the $M1$ and associated $E2$ transitions of states with symmetric (lhs) and mixed-symmetric (rhs) wave functions in vibrational nuclei near closed shells.

Figure 4

Backward angle ${}^{156}\mathrm{Gd}(e,e^{\prime })$ spectra indicating the almost uniform excitation of many low-spin states in the measured energy region except for a strongly excited $J^{\pi }=1^{+}$ state seen in the $30\mathrm{MeV}$ spectrum at $E_x=3.075\mathrm{MeV}$ and a known $J^{\pi }=3^{-}$ state at $E_x=1.825\mathrm{MeV}$ in the $50\mathrm{MeV}$ spectrum. The excitation of the state at $E_x=3.075\mathrm{MeV}$ is due to the scissors mode (43).

Figure 5

Comparison of different scattering probes for ${}^{156}\mathrm{Gd}$. High-resolution nuclear fluorescence (upper part) and inelastic electron scattering (middle part) spectra showing a strongly excited $J^{\pi }=1^{+}$ state (hatched) and several weaker $1^{+}$ states, all marked by arrows. These states are conspicuously absent in the inelastic proton-scattering (lower part) spectrum (301).

Figure 6

Comparison of the experimental form factor for the transition into one of the $J^{\pi }=1^{+}$ scissors mode states in ${}^{164}\mathrm{Dy}$ with form factors predicted in QRPA (324). The dashed curve denotes the orbital part only (301).

Figure 7

Experimental asymmetries $ϵ=(N_{\perp }-N_{\parallel })∕(N_{\perp }+N_{\parallel })$ of Compton-scattered photons determined by a five-detector polarimeter. The experimental data are compared with the calculated polarization sensitivity of the setup given by the solid lines. Three strongly excited $M1$ transitions into states around $3\mathrm{MeV}$ excitation energy in the deformed nucleus ${}^{150}\mathrm{Nd}$ have been identified. For comparison an $E1$ transition into a state at $3.4\mathrm{MeV}$ in the spherical nucleus ${}^{142}\mathrm{Nd}$ is also shown. From 188.

Figure 8

Two parts of a NRF spectrum of ${}^{154}\mathrm{Sm}$. The ratio of the areas of peaks linked by brackets, corresponding to the ground state and the $2^{+}$ transitions, allows the assignment of the $K$ quantum number to the excited spin-one state. In the upper part two examples for $J^{\pi };K=1^{-};0$, and in the lower part two examples for $J^{\pi };K=1^{+,-};1$ assignments are shown (302).

Figure 9

Frequency distribution of experimental branching ratios for about 320 transitions observed in NRF in deformed rare-earth and actinide nuclei. One notes two distinct maxima at 0.5 for $K=1$ and at 2.0 for $\mathrm{K}=0$ spin-one levels. Adapted from 393.

Figure 10

Members of a $K^{\pi }=1^{+}$ rotational band observed in the single-particle transfer reaction ${}^{165}\mathrm{Ho}({}^3\mathrm{H},{}^4\mathrm{He}){}^{164}\mathrm{Dy}$. The (not populated) $1^{+}$ bandhead lies at an excitation energy of $2.543\left(13\right)\mathrm{MeV}$ which can be identified with the $1^{+}$ state energy of $2.539\mathrm{MeV}$, observed in nuclear resonance fluorescence and inelastic electron scattering experiments (301).

Figure 11

The summed orbital $M1$ strengths observed in even-even Sm isotopes and plotted versus the square of the deformation parameter $\delta$ (391). See also Fig. 24 for the systematics of rare-earth nuclei including the present data.

Figure 12

Pictorial representation of the scissors mode. We view the mode in a proton-neutron two-fluid model (left part) and in a presentation where an inert core exists and only a small part of the proton and neutron fluids take part in the scissors motion (right part). The proton-neutron rotational oscillation is the basis of the two-rotor model which is the rotational analog of the semiclassical model of the giant electric dipole resonance.

Figure 13

Schematic representation of a coupled proton-neutron system with boson numbers $N_{\pi }=1$, $N_{\nu }=1$. The symmetric states (in the box) and the antisymmetric states are drawn at the right-hand side of the figure. Adapted from 150.

Figure 14

Orbital $M1$ strength versus mass number of the Sm isotopes. The results from full IBM-2 calculations (dash-dotted line) of 323 are shown, together with results in the SU(3) limit using a $Z=50$ subshell closure (solid line) and a $Z=64$ subshell closure (dashed line) (302).

Figure 15

Summed orbital $B\left(M1\right)$ strength from the ground state calculated in a Monte Carlo shell-model approach versus the corresponding $B(E2;0_1^{+}\to 2_1^{+})$ values in the Ba isotopes with mass numbers $A=138–150$. From 333.

Figure 16

Calculated QRPA magnetic dipole strength distribution compared to the experimental data point of the strongest transition observed in ${}^{156}\mathrm{Gd}$ (middle part) and orbit-to-spin ratios of the $M1$ transition matrix elements (bottom part). The magnitude of these ratios is shown and negative ratios are indicated by a dot on the top of the bar. The average overlap of the calculated $1^{+}$ states with the scissors mode state (denoted $R$) is displayed in the upper part. From 260.

Figure 17

The complete magnetic dipole response of ${}^{156}\mathrm{Gd}$ up to $10\mathrm{MeV}$ excitation energy, calculated using the QRPA approach. The individual levels have been folded with Gaussian functions. From 387 and 388.

Figure 18

$M1$ transition strengths in ${}^{160}\mathrm{Gd}$. (a) Experimental data, (b) pure pseudo-SU(3) scheme, and (c) complete pseudo-SU(3) calculation. The twist mode results when considering triaxial proton and neutron distributions which allow additional rotation around their principal axes. It can be combined with the scissors mode into a scissors plus twist mode. From 33.

Figure 19

Doorway picture of the scissors mode. Upper part: Schematic picture of the scissors mode embedded into the background of dense qp states (no coupling). Lower part: Level density for 2qp and 4qp $1^{+}$ states for Gd isotopes calculated using the Nilsson model. Adapted from 150.

Figure 20

Damping and fragmentation of the scissors mode. Upper part: Scissors $M1$ strength damped into the background of 2qp and 4qp states shown in lower part of Fig. 19, averaged over intervals of $100\mathrm{keV}$. The black part indicates transitions with a relative strength larger than 1%. Adapted from 150. Lower part: Experimentally measured orbital $M1$ strength distribution (390).

Figure 21

Level spacing distribution of scissors mode states in heavy deformed nuclei (152 states from 13 nuclei). (a) Nearest-neighbor spacing distribution, (b) cumulative nearest-neighbor distribution, (c) number variance ${\Sigma }^2\left(L\right)$, and (d) spectral rigidity ${\Delta }_3\left(L\right)$, where $L$ denotes the length of the sequence. The histograms and open circles display the data. Poissonian behavior and expectations from the Gaussian orthogonal ensemble (Wigner) are shown as short and long-dashed lines, respectively (84).

Figure 22

$E2$ and summed $M1$ strength in the even-even rare-earth nuclei. Upper part: $E2$ transition strengths of the $2_1^{+}$ states indicated in the figure versus $P$. Lower part: The same for the summed $M1$ strength in the energy range $E_x=2.5–4\mathrm{MeV}$. The solid lines correspond to fits explained in 293.

Figure 23

Orbital $M1$ strength versus the square of deformation in the Sm isotopes. The experimental points (black circles with error bars) are compared to theoretical predictions from 135 (circles), 114 (crosses), 164 (diamonds), 151 (squares), 231 (triangles), and 316 (stars) (305). Note that the experimental points do overlap with the triangles in a number of cases, making the distinction difficult.

Figure 24

Experimental $M1$ scissors mode strengths as a function of the deformation parameter ${\beta }_2^2$. (a) The straight line is a least-squares fit assuming intercept zero. (b) Prediction of an IBM-2 sum rule for the scissors mode strength of all even-even stable nuclei from Nd to W (363).

Figure 25

Relation between the experimental summed $M1$ strength of all $1^{+}$ states below $4\mathrm{MeV}$ for ${}^{\mathrm{142,146}–150}\mathrm{Nd}$ (squares), ${}^{\mathrm{144,148}–154}\mathrm{Sm}$ (triangles), ${}^{160–162}\mathrm{Dy}$ (circles), and the variation in the quantity $\delta \left(\frac{P}{N-1}⟨r^2⟩\right)$ related to the isotope shift (154).

Figure 26

Schematic representation of the effect of deformation on the orbital $\left(lj\right)\to \left(lj\right)$ transition strength. The Fermi level is denoted by the hatched lines (301).

Figure 27

Sum-rule analysis of the scissors mode in rare-earth nuclei. Excitation energy (upper part) and transition strength systematics (lower part) are presented. Experimental values (solid circles) and parameter-free predictions (open triangles) are shown for the mean excitation energy (upper part) and the summed $M1$ strength (lower part). The deformation dependence of the moment of inertia leads to the proportionality of the excitation energies of the scissors mode and the IVGDR as indicated by the dashed line (88).

Figure 28

The nuclear orbital and spin magnetic dipole response in a medium-heavy, a heavy, and a very heavy nucleus, derived from experiments with electromagnetic and hadronic probes, respectively (304).

Figure 29

Forward angle inelastic proton-scattering spectrum taken at $200\mathrm{MeV}$ incident energy on ${}^{154}\mathrm{Sm}$. The shaded area constitutes the giant spin magnetic dipole resonance. In the inset, the extracted $B\left(M1\right)$ strength (in units ${\mu }_N^2∕80\mathrm{keV}$) distribution is shown. Adapted from 106.

Figure 30

Angular distribution of the summed double-humped structures observed in ${}^{154}\mathrm{Sm}$, ${}^{158}\mathrm{Gd}$, and ${}^{168}\mathrm{Er}$. The dashed and full curves result from DWBA calculations based on the indicated particle-hole excitations (302).

Figure 31

Differential cross section and transverse spin-flip probability for inelastic polarized proton scattering on ${}^{154}\mathrm{Sm}$. The hatched areas show the double-humped GDR. Visible on the low-energy side of the GDR is the spin-flip $M1$ resonance between 5 and $12\mathrm{MeV}$ excitation energy and at $E_x=23.4\mathrm{MeV}$ for the IVGQR. The arrows visualize the connection between the electric resonances and dips in the spin-flip probability (305).

Figure 32

Magnetic dipole response of several deformed rare-earth nuclei determined by inelastic electron, photon, and proton scattering (305).

Figure 33

Spin magnetic dipole strength distributions in ${}^{238}\mathrm{U}$, ${}^{208}\mathrm{Pb}$, and ${}^{154}\mathrm{Sm}$. The center of gravity of the excitation energy of the two peaks representing the main strength follows a simple $A^{-1∕3}$ law, characteristic for spin-flip excitations between spin-orbit partners. The experimental strength distribution for ${}^{208}\mathrm{Pb}$ has been combined from inelastic electron and photon scattering experiments (305).

Figure 34

Experimental and theoretical spin magnetic dipole strength distributions in ${}^{154}\mathrm{Sm}$. Underneath the experimental data, theoretical predictions from various calculations are given (in descending order): QRPA from 387; 388, QTDA from 61; 63, QRPA from 315, and QRPA from 165.

Figure 35

Prediction of the full $M1$ strength distribution (summed in bins of $1\mathrm{MeV}$) including the high-energy part in ${}^{160}\mathrm{Gd}$. The dark-hatched zone represents spin strength, the light-hatched one orbital strength. In the lower part, only 2qp configurations up to $20\mathrm{MeV}$ are taken into account. In the upper part, a much higher energy cutoff for these 2qp configurations is imposed. From 262.

Figure 36

Spectra of the ${}^{165}\mathrm{Ho}(\gamma ,{\gamma }^{\prime })$ reaction in the energy region $E_{\gamma }=2.2–3.8\mathrm{MeV}$ observed with a Euroball Cluster detector placed under 130° with respect to the incident beam with an end point energy of $4.8\mathrm{MeV}$. Only the strongest transitions assigned to ${}^{165}\mathrm{Ho}$ are marked with lines; brackets connecting two peaks indicate decay branches to low-lying excited states. Other transitions are due to the calibration standard ${}^{27}\mathrm{Al}$ or result from background sources (178).

Figure 37

Comparison of the magnetic dipole strength in the odd-mass rare-earth nuclei (86). Upper part: Distribution of reduced ground-state decay widths. Lower part: Summed $B\left(M1\right)$ strengths assuming magnetic dipole character of all observed transitions in the energy range between 2.5 and $3.7\mathrm{MeV}$. Large differences in total strength, fragmentation, and the number of detected ground-state transitions is observed. Data are from 23, 245, 264, 320, and 178.

Figure 38

Ground-state decay width distributions for ${}^{154,155,156,157,158,160}\mathrm{Gd}$ extracted from photon scattering experiments. For the even-even nuclei all $\Delta K=1$ transitions are shown. In a number of cases, the $M1$ character has been determined by Compton polarimetry. Information from $(e,e^{\prime })$ form-factor measurements has also been considered. From 264.

Figure 39

Number of ground-state dipole transitions and their summed reduced width in deformed odd-mass rare-earth nuclei. Open bars refer to the data. Dotted bars stand for Monte Carlo generated strength distributions based on a statistical approach. Uncertainties of the random spectra have been estimated by 1000-fold repetition of the calculation. Error bars denote a $1\sigma$ deviation (178).

Figure 40

Ground-state decay width distributions for ${}^{163}\mathrm{Dy}$ from 23 and 265. The latter experiment had an order-of-magnitude improved sensitivity. While transition strength agree well for prominent excitations, many previously unresolved weak transitions are visible.

Figure 41

Theoretical and experimental $M1$ strength distribution in ${}^{167}\mathrm{Er}$. Bottom to top: Experimental $M1$ strength distribution, strength distribution summed up in the indicated energy bins, comparison with an IBFM calculation, and expected splitting of the energy spectrum through coupling of the unpaired particle (320).

Figure 42

Experimental and theoretical reduced width distributions. Experimental ground-state reduced width distribution in ${}^{157}\mathrm{Gd}$ (top), together with the QPNM predictions (339) for $M1$ (middle) and $E1$ (bottom) transitions to $K=1∕2,3∕2,5∕2$ final states displayed by full, dashed, and dotted lines, respectively (343).

Figure 43

High-resolution $(p,p^{\prime })$ and $(e,e^{\prime })$ spectra in ${}^{48}\mathrm{Ti}$. The $J^{\pi }=1^{+}$ states are marked by arrows (301).

Figure 44

High-resolution inelastic electron and proton-scattering spectra in ${}^{56}\mathrm{Fe}$ measured at TRIUMF. Lines above $6\mathrm{MeV}$ correspond to the excitation of $J^{\pi }=1^{+}$ states under the kinematic conditions of the two experiments (304).

Figure 45

Magnetic dipole strength distributions for ${}^{48}\mathrm{Ti}$ as calculated for the model spaces given on the lhs of the figure. From 220.

Figure 46

Inelastic electron scattering form factors for a predominantly orbital (left-hand side) and predominantly spin-flip (right-hand side) $M1$ transition (301).

Figure 47

$M1$ strength distribution in ${}^{52}\mathrm{Cr}$ from (top to bottom) high-resolution $(e,e^{\prime })$ experiments (338) and large-scale shell-model calculations using the KB3G (287) and GXPF1 (176) interactions, respectively.

Figure 48

(Color online) Differential inelastic neutrino cross sections for ${}^{52}\mathrm{Cr}$ and initial neutrino energies $E_{\nu }=15$ and $25\mathrm{MeV}$. The solid histograms are obtained from the $M1$ data, the dashed ones from shell-model calculations. The bumps represent the ${\mathrm{GT}}_0$ strength shifted by the centroid energy of the resonance. The final neutrino energies are given by $E_f=E_{\nu }-E_x$ (200).

Figure 49

Measured $E2$ and $M1$ strengths in order to identify the $2_{1,ms}^{+}$ state in ${}^{94}\mathrm{Mo}$. (a) $B(M1;2^{+}\to 2_1^{+})$ values for the seven lowest-lying identified nonyrast $2^{+}$ states. (b) Corresponding $B(E2;0_1^{+}\to 2_i^{+})$ values. The error bars are displayed as boxes. From 103.

Figure 50

Inelastic electron and proton scattering off ${}^{94}\mathrm{Mo}$. Momentum-transfer dependence of the symmetric ($2_1^{+}$, upper part) and mixed-symmetric ($2_3^{+}$, lower part) one-phonon excitation cross sections in ${}^{94}\mathrm{Mo}$ in inelastic electron (left side) and proton (right side) scattering. The data (full squares) are compared to QPM (solid lines), shell-model (dashed lines), and IBM-2 (dotted lines) predictions (53).

Figure 51

Evolution of the total, orbital, and spin $B(M1;2_2^{+}\to 2_1^{+})$ values for the $N=52$ isotones. The results of recent shell-model calculations are compared with the data. From 171.

Figure 52

Excitation energies of the $2_1^{+}$ and $2_{ms}^{+}$ states in $N=80$ isotones. The splitting is a measure of the residual proton-neutron interaction (4).

Figure 53

Displacement field for the magnetic dipole low-lying rotational state. Results shown correspond to (a) a rigid rotation of the electrons with respect to the jellium background and (b) a rotation within a rigid surface. The direction of the unit vector $\widehat{\omega }$ ($z$ direction) points out of the plane. From 213.

Figure 54

Energy distribution of the $M1$ strength for Na clusters ranging from $N=15$ to 295. The deformation parameter $\delta$ characterizing the specific cluster is also given. From 256.

Figure 55

Results for the time evolution of an elliptic quantum dot with $N=20$ electrons, for a deformation value of $\eta =0.28$, and this using three different initial perturbations: pure rotation, orbital, and quadrupole distortions. In the left-side panels, the simulated $M1$ signal, expressed as $⟨{\widehat{L}}_z⟩$, is plotted as a function of time. The right-side panels show the corresponding $M1$ excitation strength. The middle right-side panel also shows the independent electron strength function (dashed line) and the small arrows indicate the position of the solutions given by Eqs. (49, 50). From 329.

Figure 56

Schematic comparison of the scissors motion in atomic nuclei with protons and neutrons generating the scissors motion, on a length scale of $10\mathrm{fm}$ (left side), and in Bose-Einstein condensates trapped in an external potential, the latter acting on a length scale of $100\mu \mathrm{m}$ (right side).

Figure 57

The classical frequency and the tilt angle for a trapped Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ atoms. From 242.

Figure 58

Spectrum of the ${}^{208}\mathrm{Pb}(\vec{p},{\vec{p}}^{\prime })$ reaction at $E=295\mathrm{MeV}$ and $\Theta =0°$ (368) measured with an energy resolution $\Delta E\simeq 25\mathrm{keV}$ (full width at half maximum).

Figure 59

(Color online) Schematic representation of the scissors mode in stable deformed nuclei (left) as compared to exotic nuclei with a weakly bound neutron skin (right). From 297.