# Magnetic dipole excitations of 50Cr

H. Pai  hari.vecc<EMAIL_ADDRESS>Institut für Kernphysik,
Technische Universität Darmstadt, D-64289 Darmstadt, Germany  T. Beck
Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt,
Germany  J. Beller  Institut für Kernphysik, Technische Universität Darmstadt,
D-64289 Darmstadt, Germany  R. Beyer  Institut für Strahlenphysik, Helmholtz-
Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany  M. Bhike  Department of
Physics, Duke University, Durham, North Carolina 27708, USA  Triangle
Universities Nuclear Laboratory, Durham, North Carolina 27708, USA  V. Derya
Institut für Kernphysik, Universität zu Köln, D-50937 Köln, Germany  U. Gayer
Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt,
Germany  J. Isaak  ExtreMe Matter Institute EMMI and Research Division, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
Frankfurt Institute for Advanced Studies FIAS, D-60438 Frankfurt am Main,
Germany  Krishichayan  Department of Physics, Duke University, Durham, North
Carolina 27708, USA  Triangle Universities Nuclear Laboratory, Durham, North
Carolina 27708, USA  J. Kvasil  Institute of Particle and Nuclear Physics,
Charles University, CZ-18000, Prague 8, Czech Republic  B. L$\rm\ddot{o}$her
Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt,
Germany  ExtreMe Matter Institute EMMI and Research Division, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  V.
O. Nesterenko  Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Moscow region, 141980, Russia  N. Pietralla  Institut für
Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany  G.
Martínez-Pinedo  Institut für Kernphysik, Technische Universität Darmstadt,
D-64289 Darmstadt, Germany  GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, D-64291 Darmstadt, Germany  L. Mertes  Institut für
Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany  V.
Yu. Ponomarev  Institut für Kernphysik, Technische Universität Darmstadt,
D-64289 Darmstadt, Germany  P.-G. Reinhard  Institut für Theoretische Physik
II, Universität Erlangen, D-91058, Erlangen, Germany  A. Repko  Institute of
Particle and Nuclear Physics, Charles University, CZ-18000, Prague 8, Czech
Republic  P. C. Ries  Institut für Kernphysik, Technische Universität
Darmstadt, D-64289 Darmstadt, Germany  C. Romig  Institut für Kernphysik,
Technische Universität Darmstadt, D-64289 Darmstadt, Germany  D. Savran
ExtreMe Matter Institute EMMI and Research Division, GSI Helmholtzzentrum für
Schwerionenforschung GmbH, D-64291 Darmstadt, Germany  Frankfurt Institute for
Advanced Studies FIAS, D-60438 Frankfurt am Main, Germany  R. Schwengner
Institut für Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, D-01328
Dresden, Germany  W. Tornow  Department of Physics, Duke University, Durham,
North Carolina 27708, USA  Triangle Universities Nuclear Laboratory, Durham,
North Carolina 27708, USA  V. Werner  Institut für Kernphysik, Technische
Universität Darmstadt, D-64289 Darmstadt, Germany  J. Wilhelmy  Institut für
Kernphysik, Universität zu Köln, D-50937 Köln, Germany  A. Zilges  Institut
für Kernphysik, Universität zu Köln, D-50937 Köln, Germany  M. Zweidinger
Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt,
Germany

###### Abstract

The low-lying $M1$-strength of the open-shell nucleus 50Cr has been studied
with the method of nuclear resonance fluorescence up to 9.7 MeV, using
bremsstrahlung at the superconducting Darmstadt linear electron accelerator
S-DALINAC and Compton backscattered photons at the High Intensity $\gamma$-ray
Source (HI$\gamma$S) facility between 6 and 9.7 MeV of the initial photon
energy. Fifteen $1^{+}$ states have been observed between 3.6 and 9.7 MeV.
Following our analysis, the lowest $1^{+}$ state at 3.6 MeV can be considered
as an isovector orbital mode with some spin admixture. The obtained results
generally match the estimations and trends typical for the scissors-like mode.
Detailed calculations within the Skyrme Quasiparticle Random-Phase-
Approximation method and the Large-Scale Shell Model justify our conclusions.
The calculated distributions of the orbital current for the lowest
$1^{+}$-state suggest the schematic view of Lipparini and Stringari (isovector
rotation-like oscillations inside the rigid surface) rather than the scissors-
like picture of Lo Iudice and Palumbo. The spin M1 resonance is shown to be
mainly generated by spin-flip transitions between the orbitals of the
$fp$-shell.

###### pacs:

21.10.Re; 23.20.Lv; 25.20.Dc; 21.60.Jz; 27.50.+e

##  I Introduction

During the last decades, investigation of strong magnetic dipole ($M1$)
transitions in medium-mass nuclei was of a keen interest ar ; Har01 ; dz ; npr
; rev ; Hei10 . First of all, this was caused by a diversity of various
physical effects related to these excitations: quenching of spin strength and
impact of non-nucleonic degrees of freedom, relation to Gamow-Teller
transitions and relevant astrophysical problems, isospin degrees of freedom of
valence-shell excitations, peculiarities of the orbital scissors-like mode,
cross-shell and l-forbidden transitions, impact of complex configurations,
etc. Furthermore, these nuclei are accessible to modern large-scale shell
model calculations lssm_Talmi ; lssm_Gaurier  that have the capability to
describe the effects mentioned above. Theoretical efforts and advanced
experiments have finally led to a significant progress in understanding the
features of $M1$ excitations in this mass region Hei10 .

Open-shell $fp$-shell nuclei are particularly important in this activity fe ;
cu ; mn . Description of these nuclei is rather complicated because here
deformation and pairing effects come to play. At the same time, these effects
make the physics of $fp$-shell nuclei much richer. In particular, deformation
results in appearance of the low-energy orbital scissors-like $M1$ mode
($1^{+}_{sc}$) Iudice ; ic ; db ; NLI97 .

So far the experimental data on the $M1$ $1^{+}_{sc}$ strength in the
$fp$-region were limited to a few axially deformed nuclei: 46,48Ti (Z=22)ar
and 56Fe (Z=26) fe . It was found that the $1^{+}_{sc}$ in these nuclei is
mainly represented by one low-energy (in the region 3.4-4.4 MeV) state with
$K^{\pi}=1^{+}$. As expected Iudice ; ic ; db ; NLI97 , this state is
characterized by a strong $M1$ transition to the ground state. The theoretical
analysis ar ; li ; fe  indicates that the $1^{+}_{sc}$ in this mass region is
of a mixed (orbital + spin) character, although with a dominant orbital
contribution. Shell-model calculations fe  demonstrate a considerable
dependence of the description on a subtle balance of different effects such as
equilibrium deformation, pairing, interaction parameters, etc.

For a better understanding of the $1^{+}_{sc}$ features in $fp$-shell nuclei,
we certainly need experimental data for deformed Cr (Z=24) isotopes placed
between the Ti and Fe chains. $J^{\pi}=1^{+}$ states of 50Cr are known from
previous ($p$,$p^{\prime}$) experiments ppr  and for three of them
$\gamma$-ray transitions have been measured in nuclear resonance fluorescence
(NRF) en ; rm . $B(M1)$ values are known only for two $1^{+}$ states of 50Cr.
A comprehensive $M1$ strength distribution is missing. It is the purpose of
the present study to provide data for a more comprehensive $M1$ strength
distribution for the deformed nucleus 50Cr.

This nucleus was studied in photon scattering experiments using NRF uk ; np ;
fr . The experiment was performed at the superconducting Darmstadt electron
linear accelerator S-DALINAC sdl  using unpolarized bremsstrahlung with a
continuous spectral distribution from the Darmstadt High Intensity Photon
Setup (DHIPS) ks . Parity quantum numbers of spin-1 states were determined
using the High Intensity $\gamma$-Ray Source higs1  operated by the Triangle
Universities Nuclear Laboratory and the Duke Free Electron Laser Laboratory
(DFELL) at Duke University in Durham, NC, USA.

In the present experiment, the spectrum of 50Cr up to 9.7 MeV and the
corresponding M1 strength were determined. The M1 strength for the lowest
$K^{\pi}=1^{+}$-state with excitation energy of 3628.2 keV was obtained and
the $B(M1)$ values for 12 strongly excited $1^{+}$ states were measured for
the first time.

Two theoretical models were used for further interpretation: Skyrme
Quasiparticle Random-Phase-Approximation (QRPA) Ring ; Repko  and Large-Scale
Shell Model (LSSM) lssm_Gaurier . The self-consistent QRPA and LSSM rather
well reproduce the experimental axial quadrupole deformation
$\beta_{\rm{exp}}$=0.2897(44) bnl  of 50Cr and provided qualitatively close
results. The calculations agree upon the feature that the 3628.2-keV state has
predominantely orbital character with a minor spin admixture. This means that
the present experiment provides observation of the $1^{+}_{sc}$ in Cr
isotopes. The new data essentially supplement the $1^{+}_{sc}$ systematics in
$fp$-shell nuclei and, as shown below, lead to a better understanding of
$1^{+}_{sc}$ features in this mass region. Besides our analysis of the orbital
flow implies a closer correspondence to the schematic view of Lipparini and
Stringari lip_str_83  corresponding to isovector rotation-like oscillations of
the nucleons inside the rigid nuclear surface than to the scissors-like
picture of Lo Iudice and Palumbo Iudice .

The paper is organized as follows. In Sec. II, the experimental method and
data analysis are outlined. In Sec. III, the theoretical analysis of the
results is given. Sec. IV provides a summary.

##  II Experimental Method and Data Analysis

In NRF measurements, the excitation mechanism is purely electromagnetic.
Therefore, intrinsic properties like spin, parity, and transition
probabilities can be determined from the measured quantities (angular
distribution, polarization asymmetries, $\gamma$-ray energy, and intensity uk
; np ; fr ) in a model-independent way. For details of the NRF method, basic
relations between the detected number of events and energy-integrated cross-
sections, transition widths as well as $M1$ transition strengths, we refer the
reader to the reviews by Metzger fr  and Kneissl and coworkers uk ; np . Spin
quantum numbers, cross-sections $I_{i,0}$ (energy-integrated scattering cross
sections for exciting a state $i$ and deexciting this state to the ground
state 0), ground-state transition widths ${\Gamma_{0}}$, and transition
strengths $B(M1)$ were measured at the DHIPS setup.

A series of NRF experiments was performed at TU Darmstadt and at Duke
University. The photon-scattering experiment with unpolarized bremsstrahlung
from the Darmstadt linear electron accelerator S-DALINAC sdl  was performed at
the Darmstadt High Intensity Photon Setup (DHIPS) ks . Two measurements on
50Cr were performed with bremsstrahlung at endpoint energies of 7.5(1) MeV and
9.7(1) MeV, respectively. The measurement with 7.5(1) MeV bremsstrahlung has
been carried out to identify transitions via intermediate states.
Bremsstrahlung has been produced by completely stopping the intense electron
beam from the S-DALINAC’s injector in a thick copper radiator target. The
generated photon beam passes a massive copper collimator resulting in a beam
spot with a size of about 2.5 cm diameter at the NRF target position. Target
nuclei are excited by the resonant absorption of photons and subsequently
decay either directly or via intermediate states to the ground state.

Scattered $\gamma$-rays were detected by three HPGe detectors with 100%
efficiency relative to a standard $3\verb+"+\times 3\verb+"+$ NaI detector at
a $\gamma$-ray energy of 1.3 MeV. They are placed at polar angles of
$90^{\circ}$ and $130^{\circ}$ with respect to the incident beam. The
detectors were surrounded by lead and BGO Compton suppression shields. The
first measurement was performed with 2.0 g of an isotopically enriched 50Cr
target (96.416% enriched) using an end-point energy of $E_{0}$ = 9.7(1) MeV
and average electron beam currents of about 20 $\mu$A. In this measurement
data were taken for about 131 hours. For energy and photon-flux calibrations
399.3 mg of enriched 11B (99.52% enriched) were used that were irradiated
simultaneously. The efficiency of the two HPGe detectors was determined with a
56Co source up to about 3500 keV energy. For higher photon energies the gamma-
ray detection efficiencies were extracted from a simulation using the Geant4
toolkit g4 . The second measurement was performed with the same target mass
using an end-point energy at $E_{0}$ = 7.5(1) MeV and average electron beam
currents of about 33 $\mu$A. In this measurement, data were taken for about
105 hours.

Figure 1: Photon scattering spectra of the 50Cr($\gamma,\gamma^{\prime}$)
reaction from DHIPS between 3000 keV and 10000 keV for the $130^{\circ}$
detector, measured at $E_{0}$ = 9.7 MeV. Ground-state $M1$ transitions of 50Cr
are indicated by arrows.

Photon scattering spectra of the 50Cr($\gamma,\gamma^{\prime}$) reaction from
DHIPS between 3000 and 10000 keV for the detector at $130^{\circ}$, measured
at $E_{0}$ = 9.7(1) MeV are shown in Fig. 1. We have observed 33 excited
states of 50Cr with spin quantum numbers $J=1$. In this work, we focus on
$J^{\pi}$=$1^{+}$ states. $M1$ transitions to the ground state of 50Cr are
indicated in Fig. 1 by arrows. In the following, transitions corresponding to
direct decays to the ground state are called elastic transitions and those
decaying via intermediate states will be referred to as inelastic transitions.

Figure 2: Experimental asymmetry for the $M1$ transitions observed at
HI$\gamma$S.

Parity quantum numbers of spin-1 states were determined at the High Intensity
$\gamma$-Ray Source higs1  operated by the Triangle Universities Nuclear
Laboratory and the Duke Free Electron Laser Laboratory (DFELL) at Duke
University in Durham, NC, USA. This facility provides a quasi-monoenergetic
beam of nearly completely linearly polarized photons from Laser Compton
Backscattering (LCB) in the entrance channel. For details of the analysis
procedure, we refer the reader to Refs. nph ; npm . The measurement was
performed with 2.0 g of an isotopically enriched 50Cr target (96.416%
enriched). The target was centered between four HPGe detectors which were
placed in a cross-like geometry at polar angles of $90^{\circ}$ with respect
to the incident beam, two of them have 100% relative efficiency and the other
two have 60% relative efficiency, respectively. In this setup, two detectors
were placed in the (horizontal) polarization plane of the incident
$\gamma$-ray beam at azimuthal angles $\varphi=0^{\circ}$ and
$\varphi=180^{\circ}$ and the other two were placed perpendicular to it at
$\varphi=90^{\circ}$ and $\varphi=270^{\circ}$, respectively.

The experimental asymmetries $\varepsilon$ for the $M1$ transitions observed
in the measurements at HI$\gamma$S are shown in Fig. 2. The asymmetry
$\varepsilon$ is defined by the ratio of the measured and efficiency-corrected
peak intensities $A_{\parallel}$ and $A_{\perp}$, within and perpendicular to
the polarization plane, respectively:

$\varepsilon=\frac{A_{\parallel}-A_{\perp}}{A_{\parallel}+A_{\perp}}=q\cdot\Sigma.$
(1)

Here, $\Sigma$ is the analyzing power and $q$ is the experimental sensitivity
which accounts for the finite opening angles of the detectors and the finite
size of the target. The analyzing power $\Sigma$ of NRF amounts to $+1$ for a
$J^{\pi}=1^{+}$ state and $-1$ for $J^{\pi}=1^{-}$ state, respectively. For
this setup, the experimental sensitivity $q$ amounts to 0.86(1). In total 13
$M1$ excitations of 50Cr have been observed in this experiment.

Table 1:  Transitions observed in the 50Cr($\gamma$,$\gamma^{\prime}$) reaction with an endpoint energy of 9.7 MeV. Experimental uncertainties of the excitation energies are less then 0.5 keV.  $E_{x}$111The difference between the the excitation energy ($E_{x}$) and transition energy (gamma-ray energy ($E_{\gamma}$)) is due to the nuclear recoil. |  $E_{\gamma}$111The difference between the the excitation energy ($E_{x}$) and transition energy (gamma-ray energy ($E_{\gamma}$)) is due to the nuclear recoil. |  $\frac{W(90^{\circ})}{W(130^{\circ})}$ | $J^{\pi}$ |  $I_{i,0}$ | $\Gamma_{0}$ |  $\frac{\Gamma_{i}}{\Gamma_{0}}$222Branching ratio. |  $B(M1)\uparrow$ |  $B(M1)\uparrow$ |  ${\Gamma_{0}}$red
---|---|---|---|---|---|---|---|---|---
(keV) | (keV) |  |  | (eVb) | (eV) |  | ($\mu^{2}_{N}$) | ($\mu^{2}_{N}$) | (meV/$\rm MeV^{3}$)
3628.2 | 3628.0 | 0.74(2) |  $1^{+}$333From Refs. ppr ; hs ; ar . | 120(5) | 0.205(9) |  | 1.113(49) | 0.99(12)444From Ref. en . | 4.29(19)
| 2845.0 |  |  |  |  | 0.49(1) |  |  |
4997.0 | 4996.7 | 0.78(6) |  $1^{(+)}$555From Ref. nds . | 16.2(12) | 0.070(7) |  | 0.145(15) | - | 0.56(6)
| 4213.8 |  |  |  |  | 1.0(1) |  |  |
5931.2 | 5930.8 | 0.92(9) | $1^{+}$ | 23.9(20) | 0.073(6) | - | 0.091(7) | - | 0.35(3)
7600.8 | 7600.2 | 0.80(10) | $1^{+}$ | 66.4(73) | 0.334(37) | - | 0.197(22) | - | 0.76(8)
7645.7 | 7645.1 | 0.69(17) | $1^{+}$ | 23.2(28) | 0.118(14) | - | 0.068(8) | - | 0.26(3)
7948.1 | 7947.4 | 0.78(3) | $1^{+}$ | 197.7(96) | 1.382(79) |  | 0.714(41) | - | 2.75(16)
| 7164.5 |  |  |  |  | 0.27(2) |  |  |
8045.8 | 8045.1 | 0.77(10) | $1^{+}$ | 42.2(47) | 0.238(26) | - | 0.118(13) | - | 0.46(5)
8121.5 | 8120.8 | 0.76(20) | $1^{+}$ | 16.4(20) | 0.094(11) | - | 0.045(5) | - | 0.18(2)
8528.1 | 8527.4 | 0.70(8) | $1^{+}$ | 96(11) | 0.85(11) |  | 0.353(48) | - | 1.36(18)
| 7743.1 |  |  |  |  | 0.39(6) |  |  |
8885.6 | 8884.8 | 0.82(8) | $1^{+}$ | 77.7(68) | 0.534(47) | - | 0.197(17) | 0.277(74)555From Ref. nds . | 0.76(7)
9007.9 | 9007.0 | 0.88(13) | $1^{+}$ | 40.5(48) | 0.286(34) | - | 0.101(12) | - | 0.39(5)
9208.3 | 9207.4 | 0.77(20) | $1^{+}$ | 50(12) | 0.369(89) | - | 0.123(30) | - | 0.47(11)
9409.5 | 9408.5 | 0.78(14) | $1^{+}$ | 105(17) | 0.81(13) | - | 0.252(41) | - | 0.97(16)
9579.1 | 9578.1 | 0.97(25) | $1^{+}$ | 37.7(78) | 0.301(62) | - | 0.089(18) | - | 0.34(7)
9719.1 | 9718.1 | 0.85(11) | $1^{+}$ | 173(21) | 1.42(17) | - | 0.402(49) | - | 1.55(19)
Figure 3: The experimental $B(M1)$ values for the ground state transitions.

##  III Results and discussion

###  III.1 Experimental results

The experimental results are summarized in Table 1 and Figure 3. In the
present work, we have deduced $M1$ strengths in 50Cr for the first time
through the ($\gamma$,$\gamma^{\prime}$) reaction in addition to the
3628.0-keV transition, whose $M1$ strength has previously been measured via
($\gamma$,$\gamma^{\prime}$) en  and ($e$,$e^{\prime}$) ppr ; hs ; ar
experiments albeit with larger uncertainties. In the present experiment, the
uncertainty of the cross section has been reduced for the 3628.0-keV
transition. Nine of the observed excited states of 50Cr may be already known
from a ($p$,$p^{\prime}$) experiment ppr  and three states (3628.2, 7645.7 and
8885.6 keV) were known from ($\gamma$,$\gamma^{\prime}$) en ; rm  experiments.
For twelve $1^{+}$ states of 50Cr, our data provide first information on their
$\gamma$-decay transitions.

Figure 4: B(M1)$\uparrow$ values for $1^{+}_{sc}$ states of 46,48Ti ar  and
56Fe fe  along with the 3.628 MeV state of 50Cr. The experimental deformation
parameters $\beta_{\rm{exp}}$ bnl  are given for each nucleus.

Using linearly polarized quasi-monoenergetic photons at HI$\gamma$S, positive
parity quantum numbers were assigned to 13 states apart from the 3628.2-keV
state. The definite positive parity quantum number assignment for the
3628.2-keV state has been adopted from earlier ($p$,$p^{\prime}$ ) and
($e$,$e^{\prime}$) ppr ; hs ; ar  experiments. Three new $M1$ transitions
(5930.8, 8045.1 and 8120.8 keV) have been identified in the present work and
for thirteen $1^{+}$ states $M1$ excitation strengths have been obtained for
the first time.

Figure 5: Experimental $1^{+}_{sc}$ energies (top) and B(M1)-values (bottom)
as compared to estimations (2)-(3) and (4)-(5) in 46,48Ti ar , 50Cr (present
work) and 56Fe fe . The experimental deformation parameters $\beta_{\rm{exp}}$
are taken from bnl .

The experimental $B(M1)$ values into the ground state are shown in Figure 3.
The $M1$ strength distribution exhibits an intriguing pattern: an isolated,
rather strong $M1$ excitation at low energy at 3.6 MeV is separated from an
accumulation of fragmented $M1$ strength above 7 MeV. A strongly excited
$J^{\pi}$ = $1^{+}$ state at an excitation energy of 3628.2 keV with a
strength of $B(M1)\uparrow$ = 1.113(49) $\mu^{2}_{N}$ is observed. The rest of
the observed $M1$ excitations in 50Cr is expected ppr  to be generated by spin
excitations, mainly from $f_{7/2}\rightarrow f_{5/2}$ orbitals like in 52Cr
pai . We will first concentrate on the $1^{+}$ state at 3628.2 keV. This state
could correspond to a moderately collective scissors-like mode. This mode has
been predicted Iudice ; ic  and then discovered in ($e$,$e^{\prime}$)
reactions by A. Richter et al. db  in heavy deformed nuclei. As mentioned
above, the $1^{+}_{sc}$ state was also reported in the open $fp$ shell nuclei
46Ti, 48Ti ar  and 56Fe (Z=26) fe .

The experimental value of the quadrupole deformation parameter
$\beta_{\rm{exp}}$ for 50Cr is 0.2897(44) bnl , which indicates that this
nucleus is well deformed and hence can support the formation of a $1^{+}_{sc}$
state. The ratio of the $M1$ transition rates of the $1^{+}$ level at 3628.2
keV to the ground state and the first-excited $2^{+}$ state (at 783.32 keV
excitation energy) is found to be $\frac{B(M1;1^{+}_{1}\to
2^{+}_{1})}{B(M1;1^{+}_{1}\to 0^{+}_{1})}$ =
$\frac{\Gamma_{1}}{\Gamma_{0}}\cdot$$(\frac{E_{\gamma 0}}{E_{\gamma 1}})^{3}$
= 1.02(2) with $E_{\gamma 0,(1)}$ being the $\gamma$-ray transition energy to
the ground (first-excited) state. This value is larger by a factor of 2
compared to the value of 0.5 obtained by employing the Alaga rule al . This
discrepancy hints at some degree of K mixing, perhaps 50Cr exhibits some
degree of gamma-softness sh1 .

Since in 50Cr neutron and proton Fermi levels are in the same open subshell,
spin magnetism is also expected in the lowest $1^{+}$ state. Therefore, the
state at 3628.2 keV should be of a mixed orbital/spin nature. It is also
excited in ($p$,$p^{\prime}$) reactions ppr , which points to a sufficient
contribution of spin-$M1$ strength.

The available experimental results for the $1^{+}_{sc}$ in $fp$-shell nuclei,
including the present data, are exhibited in Fig. 4. It is instructive to see
how much these data match early Har01 ; Hei10 ; NLI97 ; Iud_Rich

$\displaystyle E$ $\displaystyle=$ $\displaystyle 66\delta A^{-1/3}\rm{MeV},$
(2) $\displaystyle B(M1)$ $\displaystyle=$ $\displaystyle
0.0042EA^{5/3}\delta^{2}g_{r}^{2}\;\mu^{2}_{N},$ (3)

and modified Pie98

$\displaystyle E$ $\displaystyle=$ $\displaystyle
13.4\sqrt{1+(3\delta)^{2}}A^{-1/3}\rm{MeV},$ (4) $\displaystyle B(M1)$
$\displaystyle=$ $\displaystyle
0.66\frac{\delta^{3}}{1+(3\delta)^{2}}A^{4/3}g_{\rm{eff}}^{2}\;\mu^{2}_{N},$
(5)

empirical estimations for $1^{+}_{sc}$ energies and $B(M1)$ values. Here,
$\delta=0.946\beta$ is the deformation parameter; $g_{r}=2Z/A$ is the orbital
factor, $g_{\rm{eff}}=c_{g}g_{r}$ with $c_{g}$=0.8 Pie98 . The deformation
parameter $\beta$ in Figs. 4-7 is related to the value
$B(E2)\uparrow=B(E2,0^{+}_{1}\to 2^{+}_{1})$ as Raman
$\beta_{\rm{exp}}=\frac{4\pi}{3ZR^{2}_{0}}\sqrt{B(E2)\uparrow/e^{2}}$ with
$R_{0}=1.2\;A^{1/3}\rm{fm}$. As seen below from Fig. 5, estimations (2)-(3)
and (4)-(5) give for $fp$-shell nuclei rather close results.

The data are compared to the estimates according to Eqs. (2)-(3) and (4)-(5)
in Fig. 5. We see in panel (a) that the experimental values lie somewhat below
the estimation but closely follow the linear dependence on the deformation
parameter, which is a strong fingerprint of a scissors-like excitation. Our
data on 50Cr well fit this trend. Following the panel (b), the correspondence
of the experimental and estimated $B(M1)$-values is also quite satisfactory.
Some deviations from the phenomenological estimates can be explained by that
in $fp$-nuclei the states are determined by a few two quasiparticle (2qp)
components and the orbital motion is supplemented by a significant spin
admixture (see discussion below). Altogether one may state that our results
for the state at 3.628 MeV rather well fit the $1^{+}_{sc}$ systematics, which
justifies the assignment of predominantly scissors-like nature to this state.
In the next section, our theoretical analysis presents additional arguments in
favor of this conclusion.

###  III.2 Theoretical analysis

The $M1$ strength in 50Cr was analyzed within the Skyrme Quasiparticle Random-
Phase-Approximation (QRPA) Ring ; Repko  and a Large-Scale Shell Model (LSSM)
lssm_Gaurier .

####  III.2.1 Skyrme QRPA

Results of the self-consistent Skyrme-QRPA calculations are presented in Figs.
6-9 and Tables II and III. The calculations are performed with a QRPA
restricted to axial symmetry Repko  for axially deformed nuclei. The Skyrme
QRPA method is fully self-consistent since i) both the mean field and the
residual interaction are derived from the same Skyrme functional and ii) the
residual interaction includes all the functional contributions as well as the
Coulomb direct and exchange terms. The Skyrme parameterization SGII SGII  is
used. The volume $\delta$-force pairing is treated at the BCS level Ben00 .
The parameter of the equilibrium axial quadrupole deformation $\beta$=0.30 is
determined by minimization of the total energy of the system. The calculated
$\beta$ is in a satisfactory agreement with the experimental value
$\beta_{\rm{exp}}$=0.2897(44) bnl . The spurious strength is concentrated
below 2 MeV (not shown in Figs. 6, 7 and 9) and does not noticeably affect the
results. Note that applications of the self-consistent Skyrme QRPA to magnetic
orbital and spin excitations in deformed nuclei are still very limited and
mainly devoted to heavy deformed nuclei, see e.g. Nest_M1 ; Nest_M1_Nd . To
our knowledge, the present Skyrme QRPA exploration is the first one for medium
deformed nuclei.

Figure 6: Total (a), orbital (b), and spin (c) M1 strengths K=1 calculated
within Skyrme QRPA with the force SGII.

Figure 6 presents calculated $B(M1)$ values obtained for the total (orbital +
spin), orbital, and spin $M1$ transitions $I^{\pi}_{K}=0^{+}_{0,{gs}}\to
1^{+}_{1,\nu}$ from the ground state to the $\nu$-th QRPA state with
$K^{\pi}=1^{+}$. In the spin part, the familiar quenching factor of 0.7 is
used. The calculations result in a $1^{+}$ state at 2.71 MeV with a large $M1$
excitation strength of $B(M1)$$\uparrow$=1.52 $\mu_{N}^{2}$. By comparing
$B(M1)$ values in the plots a), b) and c), it is easy to see that the magnetic
strength of this state is produced by a constructive interference of the
orbital and spin parts. The orbital strength dominates which indicates a
predominantly scissors-like character of the state. The mainly scissors-like
nature of this state is also supported by a satisfactory agreement of its
excitation energy and $B(M1)$ strength with the systematics for the
$1^{+}_{sc}$ as a function of nuclear deformation and mass number. Note that
the calculated 2.71-MeV state is not purely orbital but has some spin
admixture, which is typical li  for light and medium deformed nuclei. Despite
a noticeable energy difference, the 2.71-MeV state corresponds to the
experimentally observed state at 3.628 MeV because both of them deliver
experimentally and theoretically a maximal $B(M1)$ strength in the energy
range 2-4 MeV.

Figure 7: Proton (a), isoscalar (b), and isovector (c) E2 strengths K=1
calculated within the Skyrme QRPA with the force SGII. In panel (b), high
peaks at 3.9 and 5.5 MeV (with B(E2)=116 and 101 $\rm{e}^{2}\rm{fm}^{4}$,
respectively) are marked by crosses.

The scissors-like character of the calculated 2.71-MeV, $1^{+}$ state can be
additionally confirmed by comparison of the proton, isoscalar ($\Delta$T=0)
and isovector ($\Delta$T=1), $\Delta$K=1 B(E2;$I^{\pi}_{K}=0^{+}_{0,gs}\to
2^{+}_{1,\nu}$) values of the associated 2${}^{+}_{1,\nu}$ state belonging to
the same band. These values were obtained with the effective charges
$e_{\rm{eff}}^{p}=1,e_{\rm{eff}}^{n}=0$;
$e_{\rm{eff}}^{p}=e_{\rm{eff}}^{n}=1$; and
$e_{\rm{eff}}^{p}=-e_{\rm{eff}}^{n}=1$, respectively. The B(E2) strengths are
exhibited in Fig. 7. For the convenience of comparison with Fig. 6, the
strengths are plotted as a function of the excitation energies
$E_{1^{+}_{1,\nu}}$ for the QRPA states with K=1 (The energies of
($0^{+}_{0,gs}\to 2^{+}_{1,\nu}$) transitions can be roughly estimated in the
approximation that the moment of inertia of K=1 bands is the same as in the
ground-state band. Then the rotational increment
($E_{2^{+}_{1,\nu}}$-$E_{1^{+}_{1,\nu}}$) to $E_{1^{+}_{1,\nu}}$ is $\sim$522
keV.). Fig. 7 shows that, for the band based on the calculated 2.71-MeV state,
we get a negligible B(E2, $\Delta$T=0) but sizeable B(E2, $\Delta$T=1) value.
This clearly emphasizes the isovector excitation character of the state.
Moreover, the B(E2, $\Delta$T=1) value calculated for this state is the
largest for all states below 6 MeV. This observation points to the close
relation of this state to the underlying quadrupole deformation. Both aspects
match the isovector nature of the scissors-like mode which is usually viewed
as rotation-like out-of-phase oscillations of the deformed proton and neutron
subsystems.

The nature of the 2.71-MeV state is further inspected in Fig. 8, where
distributions (proton, neutron, isoscalar and isovector) of the orbital
transition current of this state are exhibited. The distributions represent
the proton (p) and neutron (n) transition densities of the convection current,
as well as their sums ($\Delta$T=0) and differences ($\Delta$T=1). For the
convenience of the view, the currents in all the panels are equally scaled so
as to exhibit mainly the strong flows. The figure shows that the orbital
motion is basically represented by rotation-like out-of-phase oscillations of
protons and neutrons. Indeed in the regions of most intense flow (at the left
and right nuclear surface), the protons and neutrons move in the opposite
directions (compare plots a) and b)). In these regions, the $\Delta$T=1 motion
dominates over the $\Delta$T=0 one. It is remarkable that the $\Delta$T=1
motion is most pronounced in the region of the nuclear equator and weak at the
nuclear poles. This is surprising since the scissors-like picture suggests the
opposite. One may suggest that the orbital motion in 2.71-MeV state actually
corresponds not to the scissors-like flow proposed by N. Lo Iudice and F.
Palumbo Iudice  but to an alternative picture from E. Lipparini and S.
Stringari lip_str_83 , where the isovector motion takes place inside the rigid
surface and is maximal in the equator region. These two pictures are analogous
to the Goldhaber-Teller GT  and Steinwedel-Jensen SJ  treatments of the
electric isovector giant dipole resonance, respectively. Our self-consistent
Skyrme-QRPA results more closely correspond to the Lipparini-Stringari picture
for the low-energy orbital $M1$ mode.

Now let’s briefly discuss the states at $E>$ 4 MeV, i.e. above the scissors-
like mode. As seen from Fig. 6, these states are mainly of a spin nature with
only a few exceptions at $\sim$4 and 6.5-7.5 MeV. The spin states form two
groups at $\sim$5 and $\sim$8 MeV, that constitute altogether the K=1 branch
(with both K=$\pm$1 contributions) of the $M1$ spin giant resonance.

Figure 8: Proton (a), neutron (b), isoscalar (c), and isovector (d) orbital
currents in x-z plane for the 2.71 MeV-state calculated in the Skyrme QRPA
with the force SGII. In all the panels, the currents are equally scaled to
demonstrate the most strong flows. The magnitude of the current is determined
by the arrow length.

A more general view of this resonance is given in Fig. 9 where the M1 strength
function

$S(M1;E)=\sum_{\nu}|\>\langle
1^{+}_{K,\nu}|\hat{M}(M1,K)|0\>\rangle|^{2}\>\xi_{\Delta}(E-E_{1^{+}_{K,\nu}})$
(6)

is presented. Here $|0\rangle$ is the ground state wave function,
$|1^{+}_{K,\nu}\rangle$ and $E_{1^{+}_{K,\nu}}$ are QRPA states and energies,
$\hat{M}(M1,K)$ is the M1 transition operator with the projection K,
$\xi_{\Delta}(E-E_{1^{+}_{K,\nu}})=\Delta/(2\pi[(E-E_{1^{+}_{K,\nu}})^{2}-\Delta^{2}/4]$
is the Lorentz smoothing with the averaging parameter $\Delta$=0.5 MeV.

Figure 9 shows that the spin resonance basically consists of two bumps at
$\sim$5 and $\sim$8 MeV from the K=1 branch and the bump at $\sim$11 MeV from
the branch K=0. Altogether spin excitations fill a broad region from 5-13 MeV,
which well matches a resonance region in LSSM calculations, see Fig. 10 below.
In our QRPA calculations, the spin resonance is not concentrated in one broad
bump but looks like a sequence of few well separated structures. This may be
expected from the fact that QRPA omits parts of the spin correlations. As seen
from Fig. 10, the inclusion of more correlations in the LSSM allows to mix the
separate structures and to form a broad regular resonance.

Figure 9a) shows that the interference of the orbital and spin excitations is
constructive at E$<$7 MeV and destructive at higher energies. As seen from
Fig. 9b), the excitations in the energy interval of our interest, E$<$10 MeV,
almost completely belong to the K=1 branch. In the energy interval 4.5-14 MeV
where the spin M1 resonance is localized, the calculated total M1 strength is
$\sim$5.9 $\mu_{N}^{2}$. This is in accordance with the experimental strengths
of 5-7 $\mu_{N}^{2}$ found in 24Mg and 28Si Har01 . For the interval 2-9.7 MeV
covered by the present experiment, the calculations give a total strength of
$\sim$6.2 $\mu_{N}^{2}$ which somewhat overestimates the total observed $M1$
strength $\sim$ 4.01 $\mu_{N}^{2}$.

Figure 9: (Color online) Strength functions in 50Cr smoothed by the Lorentz weight with $\Delta$=0.5 MeV. Panel (a): orbital (red dashed curve), spin (blue solid curve) and total orbital+spin (black bold curve) strengths. Contributions from all the projections (K=0 and 1) are included. Panel (b): K=0 (red dashed curve), K=1 (blue solid curve) and total (black bold curve) strengths. Table 2:  The energies $E$, $B(M1)$ values and two maximal 2qp components of particular $I^{\pi}_{K,\nu}$ QRPA states with K=1 and 0. The components are given in Nilsson quantum numbers $[Nn_{z}\Lambda]$. $\mathfrak{N}$ denotes the contribution of the component to the state norm. E | B(M1) | p/n | 2qp | $\mathfrak{N}$
---|---|---|---|---
MeV | $\mu_{N}^{2}$ |  |  |
K=1
2.71 | 1.52 | n | [303]7/2 [312]5/2 | 69$\%$
|  | p | [312]5/2 [321]3/2 | 27$\%$
5.01 | 1.20 | n | [321]3/2 [321]1/2 | 54$\%$
|  | p | [321]3/2 [321]1/2 | 33$\%$
7.70 | 1.30 | n | [312]5/2 [301]3/2 | 40$\%$
|  | n | [321]3/2 [310]1/2 | 29$\%$
K=0
11.0 | 1.56 | p | [312]5/2 [303]5/2 | 32$\%$
|  | n | [321]3/2 [301]3/2 | 25$\%$

It is instructive to see the structure of the QRPA states providing the
dominant contribution to the orbital and spin bumps in Fig. 9. This
information is given in Table 2. The table shows that the calculated
$1^{+}_{sc}$ at 2.71 MeV is formed by $\Delta$K=1 transitions between the
levels [303]7/2, [312]5/2 and [321]3/2. They are neighboring levels arising
due to the deformation splitting of the $1f_{7/2}$ subshell. Thus one
encounters here a typical scheme for orbital excitations. Note that the
collectivity of the 2.71-MeV state is low. Its two largest components exhaust
96$\%$ of the norm.

Table 2 also shows the structure of the calculated states at 5.01, 7.70 and
11.0 MeV that strongly contribute to the main parts of the spin M1 resonance
in Fig. 9. It is seen that these states are rather collective. Their two
largest components give altogether only 57-87$\%$ of the norm. Some of the
components, e.g. [312]5/2-[301]3/2 in the 7.70 MeV-state and
[321]3/2-[301]3/2, [312]5/2-[303]5/2 in the 11.0 MeV-state directly correspond
to the $1f_{7/2}-1f_{5/2}$ spin-flip transition. The strong deformation
splitting spreads the strength of $1f_{7/2}-1f_{5/2}$ transitions over
different QRPA states. Since the QRPA description does not include all
correlations, the spin $M1$ resonance is yet represented by a small number of
states and looks like a sequence of a few separated structures. The more
comprehensive picture of the spin $M1$ resonance is given below in terms of
the shell model which includes more correlations.

Figure 10: Total (a), orbital (b), and spin (c) M1 excitation strengths calculated within a large-scale shell-model with the KB3G interaction.  Table 3:  Calculated $B(E2,0^{+}_{1}\to 2^{+}_{1})$$\uparrow$ value and characteristics of the lowest 1${}^{+}_{1,\nu=1}$-state of 50Cr (energy E, total excitation strength B(M1)$\uparrow$, orbital strength B(M1)O, spin strength B(M1)S, and ratio R=B(M1)O/B(M1)S) as compared to the present experimental data. | Exp. | Skyrme QRPA | LSSM
---|---|---|---
B(E2)$\uparrow$$e^{2}b^{2}$ | 0.1052(32)111From Ref. bnl . | 0.11 | 0.108
E [MeV] | 3.628 | 2.71 | 3.54
B(M1) $[\mu^{2}_{N}]$ | 1.113(49) | 1.52 | 0.826
B(M1)O $[\mu^{2}_{N}]$ | - | 0.60 | 0.316
B(M1)S $[\mu^{2}_{N}]$ | - | 0.21 | 0.120
R | - | 2.9 | 2.6

####  III.2.2 Large-scale shell model

The results of the LSSM calculations are presented in Figure 10. We start with
a reminder of the computational procedures used in sh1 , which we follow here.
In the spherical shell model, 50Cr is described in a 0$\hbar$$\omega$ space,
i.e., ten particles are allowed to occupy all the states available in the $fp$
shell. The KB3G sh2  interaction is used. The single-particle energies are
taken from the experimental spectrum of 41Ca.

Figure 10 shows $B(M1)$ values obtained for the total (orbital \+ spin),
orbital, and spin $M1$ transitions. In the spin part, a quenching factor of
0.75 pvn  is used. The calculations give the state at 3.5 MeV with a sizeable
strength $B(M1)=0.826\ \mu_{N}^{2}$. The orbital strength dominates the 3.5
MeV state, which indicates the scissors-like character of the state. It should
be pointed out that in the LSSM the 3.5-MeV state is not purely orbital but
also has some spin admixture, which appears in the QRPA calculations. The
calculated 3.5-MeV state matches remarkably well with the experimentally
observed state at 3628.2 keV.

The LSSM calculations show that $M1$ transitions above 3.6 MeV are mainly of
spin character (see lower panel of Figure 10). The calculated summed $M1$
strength in the experimentally investigated energy range from 2 - 9.7 MeV
amounts to $\sim$5.6 $\mu_{N}^{2}$ as compared to the measured value of 4.1(1)
$\mu_{N}^{2}$. The total calculated strength in the resonance, energy interval
4.5 - 14 MeV, is $\sim$6.6 $\mu_{N}^{2}$ in good correspondence to the QRPA
result of 5.9 $\mu_{N}^{2}$.

In Table 3, the LSSM results are compared to QRPA ones and present
experimental data. The table shows that the LSSM better describes the level
energy than the Skyrme QRPA. Most probably this is because the LSSM takes into
account the coupling to complex configurations, omitted in the QRPA.

##  IV Summary and outlook

The low-lying $M1$-strength in the open-shell nucleus 50Cr has been determined
with the method of nuclear resonance fluorescence using bremsstrahlung at the
superconducting Darmstadt linear electron accelerator S-DALINAC and Compton
backscattered photons at the High Intensity $\gamma$-ray Source (HI$\gamma$S)
facility. Fifteen $1^{+}$ states have been observed between 3.6 and 9.7 MeV.
Further, 14 $\gamma$-ray transitions and 13 $B(M1)$ values have been measured
for the first time.

The experimental results were compared to calculations in the self-consistent
Skyrme Quasiparticle Random-Phase-Approximation (QRPA) Ring  and the Large-
Scale Shell Model (LSSM) lssm_Gaurier . The QRPA allowed to highlight the
basic features and origin of $1^{+}$ states while the LSSM demonstrated the
important role of complex configurations. Both models reproduce the similar
B(E2) values.

Following our theoretical analysis and comparison with other available results
for $fp$-shell nuclei the lowest $1^{+}$-state at 3.6 MeV was identified as an
isovector orbital mode with some spin admixture. In agreement with previous
studies for fp-shell nuclei fe , a constructive interference of the orbital
and spin contributions for this state was confirmed. This augments the
previous rare data (46,48Ti ar  and 56Fe fe ) on this orbital mode in the
$fp$-shell region. The obtained data match the estimations and trends for the
scissors-like mode in a satisfactory way.

It is also interesting that distributions of the orbital current, computed
within the Skyrme QRPA for the lowest $1^{+}$-state, favor the schematic
picture of Lipparini and Stringari lip_str_83  (isovector rotation-like
oscillations inside the rigid surface) rather than the scissors-like view of
Lo Iudice and Palumbo Iudice .

The spin excitations above the scissors-like mode were also inspected. The
QRPA calculation confirms that the spin $M1$ resonance in 50Cr is provided by
spin-flip transitions (mainly $1f_{7/2}-1f_{5/2}$) inside the $fp$-shell.

###### Acknowledgements.

The effort of the S-DALINAC crew at TU Darmstadt to provide an excellent
electron beam is gratefully acknowledged. We also thank the operators team of
Duke Free Electron Laser Laboratory for providing excellent beams. This work
was supported by the DFG under grant No. SFB 634 and by the Helmholtz
International Center for FAIR. V.O.N. thanks the DFG RE 322/14-1, Heisenberg-
Landau (Germany-BLTP JINR), and Votruba-Blokhintsev (Czech Republic-BLTP JINR)
grants. J.K. and A.R. are grateful for the support of the Czech Science
Foundation (P203-13-07117S).

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