PHYSICAL REVIEW C VOLUME 44, NUMBER 1 JULY 1991
Magnetic moments around the Z=40shell closure: 'Y, Zr, and Nb
I.Berkes, M. DeJesus, B.Hlimi, *M.Massaq, and E.H.Sayouty Institut dePhysique Nucleaire deLyon, IN2P3/CNRS etUniUersite Claude Bernard,
43,Bddu 11Nouernbre 1918,69622 Villeurbanne CEDEX,France
K.Heyde
Institute forTheoretical Physics and Institute forNuclear Physics, Proeftuinstraat 86,9000Gent, Belgium (Received 25 July 1990)
Magnetic moments around the Z=40shell closure have been established using nuclear magnetic reso- nance on oriented nuclei in iron. From the resonance frequencies we established
Il ("Y;2+)I=&.01(+-is)p~, Ip("Zr; -',')1=1.»3(»)p~, Ip("»;2')I=6.1S3(5)p~. The results ob- tained are discussed in the framework ofparticle-core coupling and, inmore detail, using the Nilsson de- formed single-particle model. Itis shown that for certain deformation regions the measurement ofthe magnetic moment can give information on the nuclear quadrupole deformation.
I. INTRODUCTION II.EXPERIMENTS
The odd-mass proton nuclei Y,Nb, and Tclie close to the Z=40and X=50closed shells. They are interesting candidates for the study of the dependence of the —',+ ground-state magnetic moments on the quadrupole defor- mation determined by the number of valence nucleons outside the closed-shell configuration. Adding nucleons to the Z=40, N=50 configuration implies modification of the spherical shell-model magnetic moment through configuration mixing. One can use either spherical particle-core coupling calculations or start from the be- ginning from a deformed single-particle field in, e.g., the Nilsson model. We study the variation ofthe magnetic dipole moment p versus the equilibrium ground-state de- formation for both odd mass proton and neutron nuclei.
One of the measures of the quadrupole deformation comes from the knowledge of the quadrupole moment.
In nuclear orientation this quantity can be measured in two ways: alignment ofthe nuclei in a noncubic single crystal (integral method), or sublevel resonance in a mag- netic single crystal (e.g., Co). Both methods require the knowledge ofthe electric-field gradient ofthe impurity in the host, in addition, the integral method often suffers from the presence ofnonsubstitutional sites ofthe impur- ity in the host.
As mentioned above, the magnetic rrioment is an in- direct measure ofthe nuclear deformation. In low tem- perature nuclear-orientation (NO) precise magnetic mo- ments can be established only with nuclear magnetic res- onance on oriented nuclei (NMR/ON). Several attempts have been made to resonate yttrium nuclei, either on yt- trium produced by direct recoil implantation into iron with the s5Rb(a,2n) Y reaction [1]or by performing NMR/ON on recoil-implanted. Zr or on its decay prod- uct Y [2]. NMR failed in all these experiments. In fact, Zr and Yoxidize easily, especially just after implant- ed ions stop in a still hot microenvironment. Therefore, in this experiment all nuclei are decay products of neutron-rich Rb isotopes.
A. Source preparation
The on-line separator ISOLDE-2 of CERN yields very strong Rb beams over a large range ofmasses [3].Mass- 91,-95, and -97 rubidium isotopes have been implanted with an energy of60keV into pure iron foils. No stable isotopes are collected at these masses, so the implantation densities are low. The iron foils were coated on their back sides with Ga-In eutectic solder, pressed against the cold finger of the NICOLE dilution refrigerator with a thin plastic plate and loaded together with a CoCO or a CoI'e nuclear thermometer into the refrigerator. A de- tailed description ofthis refrigerator is published in Ref.
I:4].
B.Data accumulation and results
Pulse-height spectra ofy rays were taken with 2 or 3 intrinsic Ge detectors in the direction ofthe polarizing field (0'and 180)and perpendicular toit (90 ). The data were registered on the magnetic mass storage unit ofthe data acquisition system which controlled the experiment.
The data ofthe 0' and 180' detectors were summed in the evaluations.
The variation ofthe y-ray intensity with temperature has been evaluated for several y rays. The temperature ofthe sample has been obtained from the orientation of the Conuclear thermometer.
The NMR/ON technique is described in detail in Ref.
[5]. The frequency of the NMR signal generator was
modulated with a continuous up-down ramp at 30 Hz.
Supposing that the hyperfine field distribution around the resonance field isGaussian, the measured curve takes the convoluted form [5]
2
v+Av V Vo
f(v)= U—tv gv'~—exp cr
This curve has been fitted to the measured points.
1991 The American Physical Society
MACxNETIC MOMENTS AROUND THE Z=40SHELL CLOSURE: 105 TABLE I. Increase of FWHM versus relative modulation
width 2hv/FWHM in the NbFe experiment. The precision of the experimental FWHM ratio isabout 10%.
FWHM„„„
FWHM
2.0
O U
M 0)
2hv/FWHM 0.9 1.85
3.6
Calc.
1.20 1.75 3.6
Exp.
1.15 1.75
3.6 o~ 1.0
100%
The internal full width at half maximum, FWHM=2cr&ln2, is extracted directly from the fit. In general, it is assumed that while the total modulation width 2hv is small with respect tothe FWHM, the width ofthe convoluted f(v)curve FWHM„„„isapproximate- ly given by FWHM+2b, v [5,6].
In Fig. 1 we present the convoluted FWHM„„,versus 2hv/FWHM. It isclear that the increase ofFWHM„„,
with modulation width is much less important than the prediction ofthe linear approach. We also estimated the ideal destruction of the anisotropy, supposing that the radio-frequency power is always sufhcient. Figure 1 shows that for a single line resonance a relative modula- tion width of 2hv/FWHM=0. 6 increases only by 9%
the measured FWHM, while the calculated destruction of anisotropy (intensity ofthe NMR signal) becomes higher than 50%. Moreover, the full width at one-tenth max- imum increases only by 2.3%%uo. The variation ofthe mea- sured FWHM ofthe NbFe resonance with modulation width ispresented in Table I.
Ifthe spin-lattice relaxation time ofthe nucleus in the lattice isnot negligible as compared to the time needed to sweep the frequency over the frequency range ofthe mea- surements, the NMR resonance signal shows a broaden- ing in the direction ofthe frequency sweep. In all NMR resonances an equal number of sweeps with increasing and decreasing frequency were summed. This method prevents a center-of-gravity displacement due to this thermal relaxation. If the spin-lattice relaxation time is small relative to the time ofsweep over the resonance, the folded line shape will not differ from the convoluted f(v)
function, but the FWHM may be increased.
The external polarizing field for the NMR/ON mea- surements was always 0.096(2) T. The temperature ofthe Zr and Nb samples under rf power was between 6.5 and 9 mK, and about 25mK for 'Y
Resonance in high-frequency coupling between the radio-frequency system and the sample may increase the temperature of the sample in certain frequency ranges
0.5
0.5 1.0 1.5 2hz/FWHM
FIG. 1. Increase ofthe convoluted full width at half max- imum (FWHM„„„)with respect tothe intrinsic one versus rela- tive modulation width (continuous line) and the linear approach (broken line). In the lower part we present the calculated reso- nant destruction ratio.
and thus destroy the nuclear orientation. The absence of such thermal resonances has been checked in each mea- surement. We also ensured that the radio-frequency power applied was sumcient to saturate the resonance sig- nal.
The resonance frequency v is related to the nuclear g factor by the relation
v= gPx [B„f+(I+K)B,„,],
where B&f is the hyperfine field acting on the host nu- cleus, B„,the external polarizing field, and Ka possible small Knight shift.
The results ofNMR/ON measurements are resumed in Table II. The error on the resonance frequency extrapo- lated to zero external polarizing field is increased to take into account an eventual Knight shift. Itshould be noted that the destruction ratio ofthe anisotropies is very low, probably due to the source quality. Hereafter we give some details for each measurement.
'Y .—On account of its half-life (T,&2=48 m) the orientation of 'Y,adecay product ofthe 9.5hr half-life TABLEII. Results ofNMR/ON measurements. v: resonance frequency. v,„„:resonance frequency extrapolated to zero exter- nalpolarizing Geld. Precisions ofFWHM and the destruction ratios Dare about 10%.
Nucleus v (MHz) v,„„(MHz) FWHM (MHz) D (%) Bqf(T) Ref. (B)
"Zr
97Nb
9+
25+
29+
2
308.57(5) 96.55(4) 275.92(4)
309.54(6) 96.88(5)
276.92(5) 0.55
16 38 16
—30.4(+, )
—28.1(5)
—26.57(2)
1.33(+3) 0.452(8) 1.367(1)
6.01(+is) 1.131(20) 6.153(5) 'Seethe text.
106 I.BERKESetaI.
QA4
g 0.42..
C)
0.40-.
0.38
306 308
FREQ {MHz)
3io
FIG.2. Anisotropy I(0')/I(90 )ofthe 556-keV y radiation of 'Y Feversus rffrequency. Modulation: +0.25 MHz. Fit- ted values: see Table II.
'Sr, is fully relaxed. Supposing that the
isomeric transition has a pure M4 multipolarity, the sub- stitutional fraction (fraction ofnuclei subject to the total hyperfine field, while 1 f propor—tion does not experi- ence any hyperfine field) isf=0.86(4).
No appreciable anisotropies could be seen in the decay ofthe oriented 'Srnucleus to 'Y.
Contrary to A =95and 2=97samples, the resonance width is very large in this sample (Fig. 2).
Marest, Haroutunian, and Berkes measured the mag- netic moment of Y:6.10(+,6)pz [7]. After the nuclear orientation group in Munich succeeded in producing a resonance of yttrium in iron, the Bonn group also resonated Y implanted at low temperature into iron and found a resonance frequency 314(1) MHz [8]. This is
somewhat higher than could have been expected from the old magnetic field value of 28.6(6) T [9]. Therefore we prefer to evaluate the magnetic field from the resonance in iron and the magnetic moment of Y:BIf(YFe)
= —30.4(+i6) T. The negative sign is from Ref.
[10]. The ratio of the magnetic moments of Y to 'Y is not subject to the higher error on p: From our resonance frequency and that from Bonn we can deduce
ip( Y )/p( 'Y )i=1.014(3).
Zr.—The substitutional fraction has been evaluated from the known orientation pattern ofthe daughter nu- cleus [6] Nb asf=0.75(2). The resonance width ofthe NbFe resonance was about 0.7 MHz, and the destruc- tion ratio about 20%. The increase of both of these quantities in Zris due torelaxation (Fig. 3).
With the NMR frequencies (Table II) we can deter- mine from the integral orientation pattern the (E2/M 1)
mixing parameters oftwo y transitions in Nb:
5(742 keV;—,'~—,')= —0.14(2),
5(757 keV;—,'~—,')= +0.10(3) .
The hyperfine field ofZr in ZrFe2 can be reevaluated from the integral orientation measurement of Krane et al. [11]~BIf (Zr, ZrFe2)~ =17(5)T.
The hyperfine field ofZr in iron has been determined by perturbed angular distribution at room temperature
[12].This room temperature value has been extrapolated to low temperature using the precise ratio for Bhf(FeFe )
from the Mossbauer effect [13] Bhf(300 K)/Bi,f(4
K)=0.9746(1). As the temperature dependence of Bhf(ZrFe) may be somewhat different from that ofiron, we increase the error on the correction factor 0.975(10) and use for the evaluation of the magnetic moment BI, (Zr, Fe,0K)=—28. 1(5) T.
Nb.—Nb has been oriented in ZrFez by Krane et al. [11].As they prepared their radioactive sample by
x105
4.95 7.x101
490
DCo
o+4.85-
?.0..
C)
CQ
+6.9
6.8 4.80
96 98
FREQ (MHz)
100 6.7. .
274 275 276
FREQ (MHz)
277 278
FIG.3. Intensity ofthe 757-keV y radiation of ZrFe in the direction of polarizing field, versus rffrequency. Modulation:
+0.5MHz. Fitted values: see Table II.
FIG.4. Intensity ofthe 658-keV y radiation of NbFe in the direction ofpolarizing field, versus rf frequency. Modulation:
+0.25MHz. Fitted values: seeTable II.
MAGNETIC MOMENTS AROUND THEZ=40SHELL CLOSURE: 107 in situ neutron irradiation, itcan be assumed that in their
source all Nb nuclei occupy substitutional sites. Com- paring their integral orientation pattern with ours, we can deduce f=0.72(7) for our sample. The resonance is presented on Fig. 4.
Our more precise magnetic moment value lies within the error bars quoted by Krane et al.:7.5(14)pz.
III. DISCUSSION
In nuclei near to double- or single-closed shell situa- tions, the ground-state configuration will be dominated in most cases by specific and rather pure spherical shell- model configurations. Near Z =40and N =50for odd- proton configurations the 1g9/2 orbital determines the ground or isomeric states for many odd-mass Y,Nb, and Tc nuclei. On the other hand, the odd-neutron configuration isnot as pure a single-particle configuration as is the proton configuration, although near X=51,53,
and even 55 the 2d~/2 orbital is still an important com- ponent inthe I =—,'+ground state.
Even though the spherical shell-model configuration dominates the nuclear structure in most cases, trends in a series ofisotopes or isotones point toward the importance ofquadrupole admixtures even very close to the Z =40,
N=50 region. It has been shown in the Tl region [14]
N=58
and in the In region [15]that particle-core coupling can give a satisfactory explanation of the variation in the magnetic dipole moment p. Concentrating on the major configuration (3s,&zin odd-mass Tl nuclei, Ig9/p in odd- mass In nuclei) it was shown that second-order perturba- tion theory explains the qualitative variation of p
through a relation ofthe form [16]
p(j ')=(1 —~')~(j ')+~'&2+~j'
j~=jlp.l2+ej' ', Jm =J&,
where j '—:3s,&z, lg9/2 (Tl and In, respectively) and a
describes the core-coupled admixture l2+j ', j& into
the main lj '& single hole configuration. This amplitude
a is inversely proportional to E„(2,), so a specific varia- tion ofP(j ')is implied. Applying this procedure to the Z=40, %=50region, in particular for the odd-proton nuclei where the lg9/2 (Y) or lg9/z (Nb, Tc) configurations are dominant, the correction to the single particle magnetic dipole moment p indicates a decrease
with increasing neutron number for a given isotope, even considering the depletion ofthe main single particle com- ponent. This variation is qualitatively illustrated in Fig.
5.
A difFerent approach takes the quadrupole proton- neutron corrections into account by considering aslightly deformed single-particle Nilsson potential. Here the quadrupole deformation e2 implies a particular variation of p for a given Q Nilsson-model configuration. For odd-proton nuclei, where the 1g9/p spherical orbital is dominant, the ground-state configuration and the mag- netic moment will depend very much on the sign ofthe deformation. On the oblate side (see Figs. 6 and 7) the Fermi level for Y and Nb nuclei stays very close to the
0 =—',+ orbital and a very pure state remains over a
l~
1.
+
LaJ
I I
I I I I I I I I I
\
'I J
\ ~
I
V
N=54
N=50
4 6
4J 1
5.4 5.2
CL
I
5.0)
LJ
F)~ 48~
O{f) V)
1/2+ 9/2+
1/2
-F4.1$+
,7/2
3Q -1Q (3I2
54 56 58 40 42 44 46 48
FIG. 5. The variation of 1/E„(2&+) (MeV ') for the even- even nuclei inthe Z=40,N=50—58 mass region. This quanti- ty 1/A~~) is proprotional to the particle-core admixture coefficient a, modifying the 1g9/2 magnetic moment through second-order perturbation theory.
I ' ' I ' ' ' I
-0.4 —0.2 0.0 0.2 O.4 QUADRUPOLE DEFORMATION
(~2.«)
FIG. 6. Part of the Nilsson model single-particle level scheme for the nucleus 40Zr as function ofthe quadrupole defor- mation e2. On the horizontal axis we indicate, however, (G2,64), since we have calculated along a line minimizing the total- energy surface in the hexadecapole deformation e4. The single- particle levels are identified via the remaining good quantum numbers 0 . The Fermi level A,for the corresponding Zand X
values ofY,Nb, Tc,and Rh isotopes isalso drawn (Ref.[16]l.
108 I.BERKESetal.
46
Z
0.08 0.12 0.14 0.15 0, 18 0.20
0.25
0.50 0.32
40
36 44 46 48 50 52 54 56 58 60 62
N
FIG. 7. Contour plot for the ground-state equilibrium (e&) quadrupole deformation, constructed from Ref. [23]. The re- gion ofoblate shapes isindicated by dashed lines, the density in- creasing with increasing oblate deformation e~.
rather large deformation interval. One can inspect in Fig. 7 (where the equilibrium deformation has been plot- ted on a contour diagram [15]),that for Y nuclei with
46~%~52 very small oblate shapes occur and thus an almost constant magnetic moment results. This is also consistent with the experimental ratio between the mo- ments of Y and 9'Y~as discussed in Sec.IIB.
We have calculated the magnetic dipole moments for the ground or isomeric states throughout the region around Z=40and X =50. Forodd-proton nuclei we use the Nilsson model with g factors g&=1.0, g,=5. 58,
g~ =0. At this point one has tobe careful, since the mag- netic dipole moment for a spherical Nilsson orbital does not lead to the corresponding Schmidt value when defor- mation becomes zero. Here, however, Coriolis mixing between all 0 orbitals, which is particularly strong for vanishing deformation, has been considered and this re- stores the magnetic moment to the spherical Schmidt value. The choice of the free gI and g, factors for odd- proton nuclei isjustified by the fact that near the Z =40,
N=50 shell closure the quenching of the g, factor in- duced by the perturbation theory through core polariza- tion is relatively small [17].The problem ofdetermining
the collective gz contribution is even more difticult. For strongly deformed systems, one uses quite often the hy- drodynamical value g~ =Z/A. Very recent studies on Kr point toward the need of severely reducing this value [18].Even in strongly deformed nuclei such as Dy, Er, and Yb, small values as low as g~ =0.2 were used
[19]. Calculations in the framework of the interacting boson model have pointed out the importance of taking into account only the valence nucleons and not the hy- drodynamical value, in order to describe moments in closed shell regions [20,21]. Here, the g factor for the low-lying collective states becomes g(2+)
=[g X +g N ]/zV where K and X are the numbers ofvalence neutron and proton pairs, g and g the associ- ated g factors, and %=X +X . This expression has a large range ofvalidity, as discussed in Ref. [20]. Making use ofthe microscopic estimates for the boson gyromag- netic factors (g =0, g =1), the above equation even simplifies into the expression g(2+)=N /(X +X ). If
we consider Z=40 as the subshell closure, then X„=O
forY,Zr, and Nb nuclei, and the g(2+)value, which can be identified to the gz factor in a collective geometric model, will be vanishingly small. The value ofgz =0has been used near Z=50 in particle-hole coupling model calculations too, with good results in the reproduction of magnetic dipole properties [15,21].
The need to take into account aquenching forthe neu- tron g, factor stems from the fact that in this case above
X=50 the neutron 2d5/~~2d
3/Q spin-Hip transition corrections occur quite low in energy and contribute in an important way to core polarization corrections for the spin g, factor. At the proton Z=40 shell the
1g9/p —+1g7/p spin-Aip excitation needs a much larger en- ergy (a spherical shell gap of LE=5 MeV) and subse- quently does not influence the proton g, factor in an im- portant way [22].
In Fig. 8 we present the magnetic moments for the
I =—',+ states as a function ofthe equilibrium deforma- tion ez taken from Mg01er and Nix [23].Here too, and in particular for the odd-mass Nb nuclei, we observe a steady decrease ofp(—,'+) with prolate quadrupole defor- mation as was the case in the particle core coupling ap- proach. In Fig. 8we also indicate the pure single-particle shell-model value of
p(lg9&z)=l +(g,/2) =6.79 for g,=5.58=g,&„,and p(lg9&z)=5.95g,=0.7g, &„,
as a reference value. These extreme spherical shell-model values give an indication that some quenching in the pro- ton g, factor is present. As already pointed out, the vari- ation of ihe magnetic moment with deformation is very small and thus, in this region magnetic moments are poor indicators for a determination of the quadrupole defor- mation. For prolate deformation, on the other hand (see Fig. 6for the Fermi level position), which becomes dom- inant near X=56 in most nuclei and already below
X=56forRb nuclei, more delicate situations occur. The
Fermi level is situated near the Q =—,', —,'+, —,'+ levels
(the small Q components originate from the lg9&z orbital) thus Coriolis mixing starts to play an important role and has been taken into account in all calculations. This isil- lustrated by the large variation in the moments for Nb,
Rh, and ' 'Rh. For ' 'Rh, in particular, a small vana- tion in ez and/or the core excitation energies gives a sub- stantial variation in pas illustrated by the hatched region in Fig. 8. The experimental moments, plotted in the same figure, conIIirrn the trend ofthe calculated moments,
