Magnetic moment of Ag-104(m) and the hyperfine magnetic field of Ag in Fe using nuclear magnetic resonance on oriented nuclei
V.V. Golovko, I.S. Kraev, T. Phalet, B. Delaure, M. Beck, V. Yu. Kozlov, S. Coeck, F. Wauters, P. Herzog, Ch. Tramm, D. Zakoucky, D. Venos, D. Srnka, M. Honusek, U. Koester, N. Severijns
MMagnetic moment of Ag m and hyperfine magnetic field of Ag in Fe using nuclearmagnetic resonance on oriented nuclei V.V. Golovko, ∗ I.S. Kraev, T. Phalet, B. Delaur´e, M. Beck, V.Yu. Kozlov, S. Coeck, F. Wauters, P. Herzog, Ch. Tramm, D. Z´akouck´y, D. V´enos, D. Srnka, M. Honusek, U. K¨oster,
4, 5 and N. Severijns Instituut voor Kern- en Stralingsfysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Helmholtz-Institut f¨ur Strahlen- und Kernphysik, Universit¨at Bonn, 53115 Bonn, Germany Nuclear Physics Institute, ASCR, 250 68 ˇReˇz, Czech Republic Institut Laue Langevin, 6 rue Jules Horowitz, F-38042 Grenoble Cedex 9, France ISOLDE, CERN, 1211 Gen`eve 23, Switzerland (Dated: October 19, 2018)Nuclear magnetic resonance (NMR/ON) measurements with β and γ ray detection have beenperformed on oriented Ag g,m nuclei with the NICOLE He- He dilution refrigerator setup atISOLDE/CERN. For Ag g ( I π = 5 + ) the γ -NMR/ON resonance signal was found at ν =266 . < | B hf (Ag Fe ) | = 44.709(35) T. A detailed analysis of other relevant data available in the literatureyields three more values for this hyperfine field. Averaging all four values yields a new and precisevalue for the hyperfine field of Ag in Fe i.e. | B hf (Ag Fe ) | = 44.692(30) T. For Ag m ( I π = 2 + ), theanisotropy of the β particles provided the NMR/ON resonance signal at ν = 627.7(4) MHz. Usingthe new value for the hyperfine field of Ag in Fe this frequency corresponds to the magnetic moment µ ( m Ag) = + 3 . µ N , which is significantly more precise than previous results. The mag-netic moments of the even-A − Ag isotopes are discussed in view of the competition betweenthe ( π g / ) − / + ( ν d / ν g / ) / + and ( π g / ) − / + ( ν d / ν g / ) / + configurations. The magneticmoments of the ground and isomeric states of Ag can be explained by an almost complete mixingof these two configurations.
PACS numbers: 21.10.Ky, 21.60.-n, 27.60.-j, 76.60.Jx
I. INTRODUCTION
Nuclear magnetic moments are an important tool forthe study of nuclear structure as they provide a sensi-tive test of the nuclear coupling scheme. The light even-A − Ag isotopes provide a very interesting case.In the heavier − Ag isotopes the ( π g / ) − / + pro-ton group and the ( ν d / ) − / + neutron state couple toproduce I π = 1 + and I π = 6 + ground and isomericstates, respectively. In the lighter − Ag isotopes theground and isomeric states have, respectively, I π = 5 + and I π = 2 + and are resulting from a mixture of the( π g / ) − / + and ( π g / ) − / + proton groups coupling toa ( νd / νg / ) n / + neutron configuration, where n = 5or 7. (Henceforth, for brevity, the neutron occupation n will be omitted). Thus, whereas in most cases isomericstates have excitation energies of several hundred keVand a higher spin than the corresponding ground state,the silver isotopes , Ag provide a different picture:the excitation energies of the isomeric states do not ex-ceed a few keV, and their spins are lower than those ofthe ground states. The exact nature of the mixing of ∗ Electronic address: [email protected]; Present ad-dress: Department of Physics, Queen’s University, Stirling Hall,Kingston, ON, Canada, K7L3N6 different configurations to produce the wave functions ofthese isomeric and ground states turns out to be an in-triguing problem, to which precise values of the nuclearmagnetic moments of these states can provide importantinformation.The nucleus
Ag has a I π = 5 + ground state withhalf-life t / = 69 min, and a I π = 2 + isomeric state with t / = 33 . I π = 5 + ground state the magnetic moment wasdetermined with high accuracy in a γ -NMR/ON mea-surement as µ = 3.914(8) µ N [1] and later also in anatomic beam magnetic resonance experiment, i.e. µ =3.919(3) µ N [2]. For the isomeric state the only valuefor the magnetic moment that is available in the litera-ture, i.e. µ = +3.7(2) µ N [3], is not very precise though.Note that by careful comparison of the works of Ames etal [3], Greenebaum and Phillips [4], and van Walle [5] wefound that the value µ = 4 . µ N listed for Ag m in Table 3.8 of Ref. [5], and which was copied in Ref. [6]and later also in Ref. [7], is in fact the value that wasobtained by Greenebaum and Phillips [4] for Ag m .Previously already, the proton ground state configura-tion of Ag was found to be a mixture of the ( π g / ) − / + and ( π g / ) − / + proton groups [3]. A clear indication fora transition from ( π g / ) − / + being the dominant com-ponent in the wavefunction, to ( π g / ) − / + was foundwhen going from Ag (with I π = 7 / + ) to Ag (with I π = 9 / + ), and also when going from Ag to
Ag [5]. a r X i v : . [ nu c l - e x ] J un For the isomeric state Ag m with I π = 2 + a mixing ofthese two proton groups was assumed on the basis of thenot very precise magnetic moment value from an atomicbeam experiment [3]. We have therefore performed a newmeasurement on Ag m in order to get a more precisevalue for the magnetic moment of this isomeric state, thuspermitting to shed more light on its exact configurationas well as on the evolution of the proton configurationin the − Ag isotopes. The method of nuclear mag-netic resonance on oriented nuclei was used, observingthe destruction of the β particle emission asymmetry byradio frequency radiation ( β NMR/ON). The nuclei wereoriented with the method of low temperature nuclear ori-entation [8]. Ag m was obtained from the decay of Cd parentnuclei ( t / = 57 . Ag ground state was found to bepresent in the sample as well, we measured in the sameexperimental run also the resonance signal for g Ag.The fact that the magnetic moment of this state hadpreviously been determined already with good accuracyfrom hyperfine structure measurements, yielding µ =3.913(3) µ N [2], allowed us to obtain a new and precisevalue for the hyperfine field of Ag impurities in Fe.Note, finally, that the magnetic moment of m Ag de-termined here also served as calibration for a measure-ment of the β emission asymmetry in the decay of thisisotope, for the study of isospin mixing [9]. II. EXPERIMENTAL DETAILSA. Sample preparation and detection set-up
The Ag isotopes used for the NMR/ON studies re-ported here were obtained from the decay of the
Cdprecursor produced with the ISOLDE facility [10, 11].The radioactive
Cd ( t / = 57.7 min) was producedwith a 1.4-GeV proton beam (8 · protons per pulse,staggered mode) from the CERN Proton SynchrotronBooster accelerator (PS Booster), bombarding a tin liq-uid metal target [12]. Reaction products diffused out ofthe target and effused via a hot transfer line to the ionsource. The atoms were ionized to 1 + ions, extracted,accelerated to 60 keV and then mass separated by theISOLDE General Purpose Separator (GPS). Finally, the Cd beam was transported through the beam distribu-tion system and implanted into a 125 µ m thick, 99.99 %pure iron foil soldered onto the cold finger of the He- He refrigerator of the NICOLE low-temperature nuclearorientation set-up [13, 14]. The iron foil supplied byGoodfellows TM was first polished and then annealed un-der a hydrogen atmosphere at ≈ ◦ C for about 6 hours.Prior to the implantation of the
Cd beam a polariz-ing magnetic field of B ext = 0.5 T was applied with thesuperconducting split coil magnet in the refrigerator, inorder to fully magnetize the iron foil. This field was thenlowered to B ext =0.1008(3) T to reduce its influence on source rod10 mKsource rf-coil (cid:2)(cid:3) detector (cid:2)(cid:3) detectorB (cid:4)(cid:3) detectorB coil FIG. 1: (Color online) Schematic lay-out of the experimentalset-up. The radioactive sample is prepared by implanting theISOLDE beam, that is not shown here and is arriving perpen-dicular to the plane of the drawing, in the Fe foil on the Cusample holder. A polarizing magnetic field B ext is created bythe split-coil superconducting magnet. The RF coil providesa field perpendicular to B ext for the NMR/ON measurements.The β particles are observed with planar HPGe detectors in-stalled inside the 4K-shield of the refrigerator at an angle ofabout 15 ◦ with respect to the magnetic field B ext . Large vol-ume HPGe γ detectors are installed outside the refrigerator. the trajectories of the β particles. The fact that part (i.e.about 10% [15]) of the saturation magnetization of theFe foil was lost when the field was reduced to this lowervalue slightly reduced the sensitivity but did not furtheraffect the measurements.The geometry of the experimental set-up was very sim-ilar to the one we used for our previous β -NMR/ON ex-periments [9, 17, 18] and is shown schematically in Fig-ure 1. The angular distribution of beta particles was ob-served with two planar HPGe particle detectors [19, 20]with a sensitive area of about 110 mm , mounted insidethe 4 K radiation shield of the refrigerator at a distance ofabout 32 mm from the sample. Operating these detectorsinside the 4K radiation shield, i.e. without any materialbetween source and detector, avoids energy losses and re-duces scattering of the β particles. They were mountedat an angle of 15 ◦ with respect to the magnetization axis(viz. the horizontal external magnetic field axis) in or-der to minimize the influence of scattering effects in theFe host foil. Thin isolated copper wires (about 13 cmlong) connected the detectors to the preamplifiers thatwere placed outside the refrigerator, resulting in an en-ergy resolution of about 3 keV for 1 MeV β particles.Angular distributions of γ rays were observed withthree large volume HPGe detectors with an energy res-olution of about 3 keV at 1332 keV. These were placedoutside the refrigerator, two along the magnetization axis(see Fig. 1) and one perpendicular to it. They servedto observe the γ rays of the Ag isotopes, as well as tomonitor the temperature of the sample by observing theanisotropy of the 136 keV γ ray from a calibrated CoFenuclear orientation thermometer [21, 22] that was sol-dered on the back side of the sample holder.
B. Angular distribution formalism
The angular distribution of radiation emitted from anaxially symmetric ensemble of oriented nuclei is givenby [23] W ( θ ) = 1 + f (cid:88) λ B λ ( µB tot /k B T, I ) U λ A λ Q λ P λ (cos θ ) , (1)where f represents the fraction of the nuclei that expe-rience the full orienting hyperfine interaction, while therest (1 − f ) is supposed to feel no orienting interactionat all; the B λ describe the nuclear orientation; the U λ are the deorientation coefficients which account for theeffect of unobserved intermediate radiations; the A λ arethe directional distribution coefficients which depend onthe properties of the observed radiation and the nuclearlevels involved; the Q λ are solid angle correction factorsand P λ (cos θ ) are the Legendre polynomials. The detec-tion angle θ is measured relative to the direction of thesaturation magnetization axis of the Fe host foil whichis defined by the applied horizontal magnetic field. Theorientation coefficients B λ depend on the temperature ofthe sample T , the spin I and the magnetic moment µ ofthe oriented state, and on the total magnetic field B tot the nuclei experience (with k B the Boltzmann constant).This field is given by B tot = B hf + B ext (1 + K ) − B dem (2)where B hf is the hyperfine magnetic field of Ag in Fe (seeSec.IV.A), B ext is the externally applied field, K is theKnight shift parameter and B dem is the demagnetizationfield [24].Three measurements of the Knight shift for silver iniron are reported in the literature. All three determinedthe Knight shift in γ -NMR/ON experiments, yielding K ( Ag F e ) = − . K ( g Ag F e ) = − . K ( m Ag F e ) = − . K (Ag F e ) = − . γ rays only λ -even terms occur in Eq. (1). Forpositrons from allowed β decays only the λ = 1 term ispresent and Eq.(1) transforms to [23] W ( θ ) = 1 + f vc B A Q cos θ, (3)where v/c is the positron velocity relative to the speed oflight.Experimentally the angular distribution is obtained by W ( θ ) = N cold ( θ ) N warm ( θ ) , (4) with N cold , warm ( θ ) the count rates when the sample is’cold’ (about 10 mK; oriented nuclei) or ’warm’ (about1K; unoriented nuclei). In on-line experiments, wherethe count rates vary with beam intensity, it is customaryto construct a double ratio, combining count rates in twodifferent detectors in order to eliminate effects of beamintensity fluctuations and avoid the need to correct forthe lifetime of the isotope. In the present work the doubleratio R = (cid:20) N (15 ◦ ) N (165 ◦ ) (cid:21) cold / (cid:20) N (15 ◦ ) N (165 ◦ ) (cid:21) warm (5)was used for the β particles and R = (cid:20) N (0 ◦ ) N (90 ◦ ) (cid:21) cold / (cid:20) N (0 ◦ ) N (90 ◦ ) (cid:21) warm (6)for γ rays. All data were corrected for the dead timeof the data acquisition system using a precision pulsegenerator. C. NMR/ON resonance set-up and formalism
For the NMR/ON measurements an RF oscillatingfield was applied perpendicular to the external magneticfield (see Fig. 1). The NMR coil producing this oscillat-ing field consisted of a pair of two-turn coils mounted ona teflon frame that was fixed inside the 300 mK radiationshield surrounding the sample. The coil was fed from thetop of the refrigerator by coaxial cables connected to adigital Marconi frequency generator with a range from10 kHz to 3.3 GHz. In addition to the NMR coil a pick-up coil for monitoring the RF signal was present too. Alinear RF amplifier with a constant gain of 46 dBm wasinstalled between the frequency generator and the RFcoil. The intensity of the RF signal was kept as small aspossible in order to avoid too strong RF heating and sub-sequent reduction in anisotropy for the β particles and γ rays.The resonance measurements were performed at sam-ple temperatures in the range from 8 to 12 mK by scan-ning the radio frequency, ν , and observing the variationof the anisotropy of the β particles and γ rays emittedby the Ag nuclei. At resonance, transitions between theZeeman split nuclear sublevels are induced which par-tially equalize the originally unequal populations of thesublevels, thus reducing the degree of nuclear orientationand therefore the magnitude of the anisotropy R − ν res is related to the nuclearmagnetic moment through the relation ν res [MHz] = (cid:12)(cid:12)(cid:12)(cid:12) . µ [ µ N ] B tot [T] I [ (cid:126) ] (cid:12)(cid:12)(cid:12)(cid:12) (7)When pro mille precision is required and the nucleus issituated in a medium other than vacuum and being sub-ject to an external applied magnetic field, the nuclearmagnetic moment value has to be corrected for diamag-netism. This is a reduction of the field experienced by thenuclei due to the polarization of the medium, which leadsto an apparent reduction in the nuclear magnetic dipolemoment if no correction to the applied field strength ismade. The corrected magnetic moment value is thengiven by µ corr = µ uncorr (cid:20) − σ B ext B ext + B hf (cid:21) − (8)with σ the diamagnetic correction. For silver nuclei σ =0.005555 [6]. III. DATA COLLECTION AND ANALYSIS
Due to the rather long half-life of both isotopesand their rather large magnetic hyperfine interactionstrengths T int = | µB/k B I | , i.e. T int (cid:39)
13 mK for Ag g and T int (cid:39)
30 mK for Ag g , relaxation of the nuclearspins to thermal equilibrium was sufficiently fast and noproblems with incomplete spin-lattice relaxation [28–30]were expected.The decay of the 5 + 104 Ag g ground state (Figure 2)is distributed over many EC/β + branches resulting inthree rather intense γ rays with energies of 555.8 keV,767.6 keV, and 941.6 keV and intensities of 93%, 66%and 25% , respectively. Whereas the 555.8 keV transi-tion is also clearly present in the decay of the isomericstate, the β decay of the isomer contributes very little(i.e. only (cid:39) γ raysare therefore especially well suited for the determinationof the magnetic moment of Ag g in a γ -NMR/ON ex-periment.For Ag m the main EC/β + branch represents (cid:39) + level at555.8 keV in Pd [16]. All other branches representless than 2% each of the total decay strength. As the555.8 keV γ transition depopulating the 2 + first excitedstate is also one of the stronger γ lines in the decay of Ag g (see Fig. 2) and the next most intense γ transitionin the decay of Ag m has an intensity of only about 4%,it was decided to use the β transition to the first excitedstate to search for the resonance signal of m Ag.Typical γ ray and β particle spectra are shown inFigs. 3 and 4, respectively. A. γ -NMR/ON on g Ag F e
The nuclear magnetic resonance signal for the groundstate Ag g was obtained by observing the change ofthe anisotropy of the 555.8, 767.6, 941.6 keV γ rays as afunction of the frequency of the applied RF field. Using atriangular modulation frequency of 100 Hz, a modulation Pd Ag =4279(4) keV β + Q =4279(4) keV β + Q ∈ , + β + β I ∈ I log(ft) E . . E . . . . . . . . . . . . . . . . . . . . . . . . FIG. 2: Partial decay scheme of Ag g showing only those γ transitions with intensities larger than 5% and the relatedlevels and EC/β decay branches. The most intense γ lines,with energies of 555.8 keV, 767.6 keV and 941.6 keV, wereused for the γ -NMR/ON measurements. C oun t s P u l s e r k e V k e V k e V . k e V . k e V . k e V FIG. 3: Typical γ spectrum for Ag g,m , obtained in 300 s ofcounting. The 511 keV positron annihilation line, the 122 keVand 136 keV γ lines of the Co F e nuclear thermometer, andthe 555.8 keV, 767.6 keV, 941.6 keV γ ray lines that were usedfor the γ -NMR/ON measurements on g Ag, are indicated.Most γ lines belong to the decay of g,m Ag (see [16]). Thestrong line between the 555.8 keV and the 767.6 keV lines isthe 709 keV internal transition in
Cd. C oun t s P u l s e r
122 keV136 keV511 keV 2708 keV
FIG. 4: Typical β spectrum for Ag m , obtained in 300 sof counting. The endpoint energy for the decay of Ag m (i.e. 2708 keV) is indicated. At low energy the 122 keV and136 keV γ rays from the Co F e nuclear thermometer, the511 keV annihilation peak and the 555.8 keV γ ray from the Ag g,m decays are also visible. bandwith of 0.5 MHz and an RF signal level of -32 dBm,the center RF frequency was varied in steps of 0.5 MHzover the resonance search region from 263.5 MHz to270 MHz. The search region could be chosen so nar-row since two precise and mutually consistent values areavailable in the literature for both the hyperfine field ofAg in Fe, i.e. | B hf (Ag F e ) | = 44.7 T [25, 34], and themagnetic moment of Ag g , i.e. µ = 3 . µ N [1, 2]. Twoscans were performed, stepping the frequency region inboth upward and downward directions, in order to avoidpossible shifts of the (effective) resonance centers due toa finite spin-lattice relaxation time. At each frequencydata were accumulated for 300 s.An NMR effect was observed for all three γ rays (seee.g. Fig. 5). Table I summarizes the results of the fits ofa Gaussian line and a linear background to the data forthe three different lines with the MINUIT minimizationpackage [32]. The weighted average of the three cen-tral frequencies, obtained in a magnetic field of B ext =0.1008(3) T, is | ν res ( Ag m F e ) | = 266 . . (9)This value is in agreement with, but an order of mag-nitude more precise than the previous result obtainedin a similar experiment by Vandeplassche et al. [1], i.e. ν res = 266 . B. β -NMR/ON on Ag m Fe To determine the magnetic moment of the isomericstate the change of the β particle anisotropy as a func- TABLE I: γ -NMR/ON resonance centre frequencies ( ν res ) ob-tained for the three most intense γ rays in the decay of Ag g .Energy (keV) ν res (MHz)555.8 266 . . . . tion of the RF frequency was observed. The only valuefor the magnetic moment of Ag m available in the liter-ature is not very precise, i.e. µ ( Ag m ) = +3.7(2) µ N [3].The magnetic moment was therefore first determined,prior to our measurement but during the same beamtime, by scanning the frequency of the first of the twolasers used to selectively ionize Ag atoms in the RILISion source [33]. On-line analysis of this laser scan yielded µ ( Ag m ) = 3 . µ N , (10)corresponding to a frequency ν res = 630 ±
17 MHz, whichwas then used as the search region for the β -NMR/ONexperiment.In order to maximize statistics, the destruction of the β anisotropy effect R = 1 − W (15 ◦ ) W (165 ◦ ) (11)was monitored in the energy region from 600 keV to theendpoint at 2708 keV. In spite of the fact that the mag-nitude of a β anisotropy decreases towards lower energiesbecause of its dependence on v/c (see Eq. (3)), the sizeof the anisotropy was still rather large, i.e. R (cid:39) . γ rays. A triangularmodulation frequency of 100 Hz was again used. Sev-eral frequency scans were performed. For most of thesethe frequency step width was 2 MHz and the modula-tion amplitude ± −
42 dBm. For each scan the frequency region wasalways stepped both in upward and in downward direc-tions. No difference between the center frequencies wasfound for passes in opposite directions, indicating thatrelaxation effects were indeed negligible.The first scans were performed with continuous modu-lation of the RF signal. These data were again fittedusing a simple Gaussian with a constant background.Thereafter four scans were performed by recording thespectrum for each frequency first without and immedi-ately thereafter with frequency modulation. This FM-off/FM-on mode was used to obtain a better definitionof the line shape. This was considered to be helpful sinceresonance lines in iron hosts at high frequencies are ratherbroad due to the inhomogeneous broadening of about 1%.Fig. 6 shows the destruction, S , of the β asymmetry as
263 264 265 266 267 268 269 2700.60.620.640.660.680.7263 264 265 266 267 268 269 2700.60.620.640.660.680.7
Frequency MHz( ) W ( ) / W ( ) (cid:2)(cid:2) (a)
263 264 265 266 267 268 269 2700.480.50.520.540.560.580.60.620.64263 264 265 266 267 268 269 2700.480.50.520.540.560.580.60.620.64
Frequency MHz( ) W ( ) / W ( ) (cid:2)(cid:2) (b) FIG. 5: (Color online) On-line γ -NMR/ON curves for the 767.6 keV (a) and the 941.6 keV (b) γ lines of Ag g . The ratio ofthe pulser-normalized γ anisotropies W (0 ◦ ) /W (90 ◦ ) is plotted as a function of RF frequency. Data points for two scans (onein upward and one in downward direction) are superposed. The integrated destruction of γ anisotropy is about 25% in bothcases. The corresponding resonance frequencies are given in Table I. a function of frequency. This destruction S is defined asthe difference between the ratio W (15 ◦ ) /W (165 ◦ ) withFM on and FM off normalized to the ratio with FM off: S = (cid:16)(cid:104) W (15 ◦ ) W (165 ◦ ) (cid:105) FM on − (cid:104) W (15 ◦ ) W (165 ◦ ) (cid:105) FM off (cid:17)(cid:104) W (15 ◦ ) W (165 ◦ ) (cid:105) FM off (12)The total destruction observed was about 4 %. Thisvalue is smaller than in our previous experiments with Cu ( S ≈ As ( S ≈ Ag g resonances (i.e. S ≈ β parti-cles of the non-resonant g Ag F e in the energy regionused for analysis. Nevertheless, the resonance frequencyvalue could still be determined with good accuracy. Ta-ble II lists the results for the various scans that wereperformed. The weighted average resonance frequency,again obtained in a magnetic field of B ext = 0.1008(3) Tis ν res ( Ag m F e ) = 627 . . (13) IV. RESULTSA. Hyperfine field of Ag impurities in Fe
To extract a precise magnetic moment value for Ag m from the NMR/ON frequency obtained, a precise valuefor the hyperfine magnetic field of Ag impurities in Fehost, B hf (Ag F e ), is required. Two values are quotedin the literature. The first, | B hf (Ag F e ) | = 44.72(2) T,
610 620 630 640 650615 625 635 645
Frequency (MHz) -0.0100.0000.0100.020-0.0050.0050.015 D es t r u c t i on ( a r b i t r a r y un i t s ) FIG. 6: β -NMR/ON resonance curve for Ag m F e . Datafrom three frequency scans using the FM-off/FM-on sequencehave been summed. The destruction of anisotropy is definedin Eq. (12). The total destruction observed was about 4 %. was obtained in a NMR/ON experiment with Ag m in Fe [34]. However, from the reported resonance fre-quency of 203.75(10) MHz in a field of 0.220 T, and withthe magnetic moment of 3.604(4) µ N used by the au-thors, one obtains | B hf (Ag F e ) | = 44.72(5) T, i.e. thesame value but with a significantly larger error bar. Itseems that the error on the magnetic moment was ne-glected in Ref. [34]. Later, Eder et al. [25] quoted thevalue | B hf (Ag F e ) | = 44.69(5) T which they deduced fromthe NMR/ON frequency ν ( Ag m F e ) = 204.78(2) MHzreported in Ref. [35] and the magnetic moment value | µ ( Ag m ) | = 3.607(4) µ N from Refs. [36] and [37]. Inthe mean time several new hyperfine interaction mea- TABLE II: β -NMR/ON results for various scans of m Ag.Every measurement consists of two scans over the frequencyregion from 613 MHz to 647 MHz, one in upward and theother in downward direction. The measurements were per-formed with different stepsize and for different measurementtimes. The weighted average of the seven individual measure-ments (i.e. 14 scans) is indicated as well.Meas. Modulation ν res (MHz) Step (MHz) Coll. time (s)1 on 628.3(8) 2 1502 on 627.0(8) 2 3003 on 628.3(9) 5 3004 off/on 627.2(7) 2 3005 off/on 628.0(11) 2 1506 off/on 629.1(29) 2 1507 off/on 628.1(19) 2 150 ν res surements on Ag m , Ag m and Ag g have been re-ported, allowing one to deduce four precise values for B hf (Ag F e ), as will be discussed in the following para-graphs. An overview is given in Table III.Two values for B hf (Ag F e ) can be obtained from datafor Ag m . The two magnetic moment values for Ag m reported in Refs. [38] and [39] agree well and, after beingcorrected for diamagnetism [6], lead to the weighted av-erage value | µ ( Ag m ) | = 3.608(3) µ N . Combining thiswith the resonance frequency for this isotope in Fe hostreported in Ref. [35], i.e. ν ( Ag m F e ) = 204.78(2) MHz,Eq. (7) yields for the hyperfine field of Ag impurities inFe the value | B hf (Ag F e ) | = 44 . . (14)Note that no correction for the Knight shift is required asthe resonance frequency was quoted for B ext = 0, while acorrection for B dem can be neglected at the present levelof precision since a very thin foil was used.Further, the nuclear magnetic moment of Ag m citedabove, i.e. | µ ( Ag m ) | =3.608(3) µ N , and the resonancefrequency | ν ( m Ag F e ) | = 203.75(10) MHz reported inRef. [34], yield | B tot (Ag F e ) | = 44.451(43) T. The fre-quency was obtained with a 3 µ m thin foil, rendering thecorrection for B dem again negligible, and in B ext = 0.220T. Taking then also the Knight shift parameter K = -0.046(5) into account (which was not yet known at thetime of Ref. [34]) one derives from Eq. (2) | B hf (Ag F e ) | = 44 . . (15)A third value for | B hf (Ag F e ) | is obtained from datafor m Ag: NMR/ON resonance frequencies for this iso-tope were determined in both a Ag and an Fe host.Eder et al. [26] obtained | ν ( Ag m F e ) | = 210.57(3) MHzin B ext = 0 with a stack of foils of 2 µ m thickness(such that B dem = 0). Ohya et al. [40] reported | ν ( Ag m Ag ) | = 56.128(26) MHz in an external fieldof about 12 T. The exact value of this applied field could be deduced from the resonance frequency of Ag m in Ag for the same field setting, which was found at54.640(1) MHz [41].Using for the magnetic moment of m Ag theweighted average value from Refs. [38] and [39], i.e. | µ ( Ag m ) uncorr | =3.588(3) µ N (this time not correctedfor diamagnetism as the nuclear orientation in theNMR/ON experiments of Refs. [40] and [41] was in-duced solely by an external magnetic field), Eq. (7) yields | B tot | =11.987(10)T. The ratio r = 3.7516(18) of the twoabove mentioned frequencies for Ag m can be writtenas r = µ ( Ag m ) corr | B hf (Ag Fe ) | µ ( Ag m ) uncorr | B tot | = 1 . | B hf (Ag Fe ) | . , (16)with µ ( Ag m ) corr /µ ( Ag m ) uncorr = 1.005586 the fac-tor for the diamagnetic correction [6], so that | B hf (Ag F e ) | = 44 . . (17)Finally, a hyperfine field value can also be ob-tained from data for Ag g . Indeed, combining ourNMR/ON resonance frequency for Ag g in Fe, i.e. | ν ( Ag m F e ) | = 266.70(5) MHz, with the magneticmoment value | µ ( Ag m ) | = 3.919(3) µ N from opti-cal hyperfine spectroscopy measurements [2], Eq. (7)yields | B tot (Ag F e ) | = 44.639(35) T. Correcting this re-sult according to Eq. (2) for the external magneticfield B ext =0.1008(3) T, for the Knight shift parame-ter K = − µ m thick Fe foil used, i.e. B dem = 0.026(5) T (a 20%error was adopted in order to account for approximationsmade in the equations used to calculate B dem [9]), resultsin | B hf (Ag F e ) | = 44 . . (18)When combining the four above mentioned hyperfinefields to one weighted average, the correlations betweenthe first three values, which all rely on the same valuefor the magnetic moment of m Ag, were duly takeninto account by incorporating the full covariance matrix(e.g. [46]). Using then further standard error propagationtechniques yields for the magnetic hyperfine field of Agimpurities in Fe the value | B hf (Ag F e ) | = 44 . Ag m in thenext section. TABLE III: Input data leading to the hyperfine magnetic field for Ag impurities in Fe and the values obtained from these.line measured/deducedquantity value Ref. remark1 | µ ( Ag m ) | . µ N a [38] ABMR b | µ ( Ag m ) | . µ N a [39] BF-NMR/ON c | µ ( Ag m ) | . µ N weighted average of above two values4 | ν ( Ag m F e ) | . B ext = 0 T5 | B hf (Ag Fe ) | . | ν ( Ag m F e ) | . B ext = 0.220 T7 | B hf (Ag Fe ) | . | ν ( Ag m F e ) | . B ext = 0 T9 | ν ( Ag m Ag ) | . c in B tot = 11.987 T10 | B hf (Ag Fe ) | . | ν ( Ag g F e ) | . B ext = 0.1008 T12 | µ ( Ag g ) | . µ N [2] optical spectroscopy13 | B hf (Ag Fe ) | . | B hf (Ag Fe ) | . a Corrected for diamagnetism according to [6]. b Atomic beam magnetic resonance. c Brute-force nuclear magnetic resonance on oriented nuclei.
B. Nuclear magnetic moment of m Ag With the value for the hyperfine magnetic field of Agimpurities in Fe being established we can now deduce aprecise value for the magnetic moment of Ag m fromour resonance frequency | ν ( Ag m F e ) | = 627.7(4) MHz.Taking into account the external field B ext =0.1008(3) Tand the Knight shift parameter K = -0.046(5), and cor-recting for the demagnetization field opposite to the di-rection of the external field, i.e. B dem = 0.026(5) T (seeabove), the total magnetic field experienced by the nucleiis found to be | B tot | = 44.622(30) T. Eq. (7) then yieldsfor the magnetic moment | µ ( Ag m ) | = 3 . µ N . (20)Correcting for diamagnetism does not change this value.This is the first precision value for the magnetic momentof this isomer. V. MAGNETIC MOMENTS OF − AG The magnetic moments of the odd-odd even-A Ag iso-topes are listed in Table IV.Extended magnetic moment calculations for even-Aodd-odd Ag isotopes are not available in the literature.Table V presents an overview of the experimental val-ues for the magnetic moments of the ground and iso-meric states in the even-A − Ag isotopes as well asa comparison with values calculated with the additivity rule [49] µ I [ µ N ] = 12 I ( g n + g p )+ ( g n − g p ) [ I n ( I n + 1) − I p ( I p + 1)]2( I + 1) . (21)For this we used Schmidt single particle moments, modi-fied single particle moments, as well as the mean values ofexperimental magnetic moments of neighboring odd-evennuclei.The Schmidt single particle magnetic moments forthe π g / and ν d / orbitals that determine the spinsand magnetic moments of the light − Ag nuclei are+6.793 µ N and -1.913 µ N , respectively [50]. Modifiedsingle particle moments take into account the effects ofthe nuclear medium, i.e. configuration mixing, core po-larization and meson exchange [51] by using the effectivegyromagnetic ratios g mod l ( π ) = +1.1, g mod l ( ν ) = -0.05and g mod s = 0.7 g free s , i.e. g mod s ( π ) = 3.910 and g mod s ( ν )= -2.678, thus yielding µ mod ( π (g / )) = +6.355 µ N and µ mod ( ν (d / )) = -1.439 µ N . Finally, using experimental g factors from neighboring odd-even nuclei has the ad-vantage that configuration mixing and possible g factorquenching in the odd- A nuclei are automatically takeninto account. For the g factor of the g / protons, thevalue of the lower mass odd-A silver isotope was eachtime used. For the g factor of the d / neutrons the geo-metrical mean of the g factors of the neighboring isotopesof Ru, Pd and Cd with the neutron in the d / orbitalwere used.Note that nearly all neighboring odd-neutron Ru, Pd,Cd and Sn isotopes with N = 55, 57, 59, 61 and 63 have TABLE IV: Experimental magnetic moment values for theground and isomeric states of the even-A isotopes − Ag.Whenever a diamagnetic correction was applied this is indi-cated. In all other cases the correction was either negligiblysmall, or not necessary because of the method that was ap-plied or because of the large experimental error bar.Isotope
I µ ( µ N ) Ref. Ag g . a Ag g . b [1, 5] c . d +4 . e Ag g . e Ag g . f [43] c Ag g . f [43] c Ag m . e Ag m . c . g +3 . e Ag m . h . c . a . a Ag m . d Ag m . b , f [38] e . b , f [39] ha Low Temperature Nuclear Orientation (LTNO). b Recalculated using the hyperfine field value obtained in the pre-vious section. c Nuclear Magnetic Resonance on Oriented Nuclei (NMR/ON). d Optical hyperfine spectroscopy. e Atomic Beam Magnetic Resonance (ABMR). f Corrected for diamagnetism. g Resonant Laser Ionization Source technique (RILIS). h Brute-Force Nuclear Magnetic Resonance on Oriented Nuclei(BF-NMR/ON). a 5/2 + ground state. For the isotopes with N = 55 thisis due to the hole in the ν d / orbit. For the isotopeswith N = 57 to 63, filling the ν g / orbit, this indicatesthat with adding neutron pairs these predominantly cou-ple to spin 0, leaving the hole in the ν d / orbit, andthis all the way up to N = 64. This is why in the cal-culations of magnetic moments presented in Table V the( ν d / ) − configuration has always been used. The 7/2 + state related to a filled ν d / orbit and an odd numberof neutrons in the ν g / orbit is in almost all isotopesfound as an excited state at excitation energies rangingfrom 188 keV to 416 keV, except for Sn where it is theground state. It is not understood why this predominantcoupling of neutron pairs in the ν g / orbit occurs. How-ever, the configuration is complex as can be seen fromthe g factors of the odd-neutron states in this mass re-gion which are found to have values of about -0.30 (seecolumn 5 in Table V) which in absolute value is consid-erably smaller than the modified Schmidt value of -0.576for the ν d / orbit (the modified Schmidt g factor for the ν g / orbit is +0.242). Note that this issue is to some ex-tent also related to the current debate about the ground state of Sn being ν d / or ν g / (e.g. [52, 53]).As can be seen in Table V, for the heavier neutron defi-cient isotopes Ag,
Ag, and
Ag (with a 1 + groundstate and a 6 + isomeric state) the magnetic moment forthe isomeric state can be very well explained by a par-allel coupled ( π g / ) − / + ( ν d / ) − / + configuration. Thevalues calculated with the additivity relation (Eq. (21))using Schmidt single particle moments or modified singleparticle moments are systematically lower than the ex-perimental values, but the values calculated with experi-mental g factors are always very close to the experimentalresults.For the 1 + ground state of , , Ag, coming fromthe antiparallel coupling of the ( π g / ) − / + proton groupand the ( ν d / ) − / + neutron state, the additivity rulewith experimental g factors of neighboring isotopes yieldsvalues that are systematically about 0.3 to 0.5 µ N toolarge. Since the correction on this 1 + state due to coreexcitations is less than 0.1 µ N [50] this deviation is prob-ably due to the mixing of other states in the wave func-tion. The ( π g / ) − / + proton configuration is thus firmlyestablished for − Ag.For the lower mass , Ag isotopes Ames et al. [3]already pointed out that the wave function would con-sist of a ( ν d / ) − / + neutron configuration coupled toa mixed ( π g / ) − / + - ( π g / ) − / + proton configuration.The experimental magnetic moments of both the groundand the isomeric state in Ag are in between the val-ues calculated for the two pure proton-neutron configu-rations (see Table V) and support this suggestion. Themagnetic moment of Ag g ( I π = 5 + ) was previously al-ready determined with high precision in two independentexperiments [1, 2]. The measurement reported here nowprovides a precise value for Ag m ( I π = 2 + ) as well.This now permits a more detailed analysis of the con-figuration mixture in the 2 + , + doublet of states (only6.9 keV apart) in Ag. Writing the wave functions forboth states as (see [50]): ψ ( Ag g,m ) = α (cid:104) ( π g / ) − / + ( ν d / ) − / + (cid:105) + β (cid:104) ( π g / ) − / + ( ν d / ) − / + (cid:105) , (22)with α and β = √ − α the mixing amplitudes, theexpectation value of the magnetic moment operator be-comes < µ > = α (cid:68) µ (cid:104) ( π g / ) − / + ( ν d / ) − / + (cid:105)(cid:69) + β (cid:68) µ (cid:104) ( π g / ) − / + ν (d / ) − / + (cid:105)(cid:69) . (23)The mixing amplitudes, as obtained from the experimen-tal magnetic moments, are listed in the last column ofTable V and indicate a complete mixing of the two config-urations (i.e. α ≈ β ≈ / √ Ag g ( I π = 5 + )and almost complete mixing in Ag m ( I π = 2 + ). Per-forming the same analysis also for the doublet of 2 + and0 TABLE V: Magnetic moments of the even-A Ag isotopes. Experimental magnetic moments are compared with values obtainedwith the addition theorem (Eq. (21)) using either Schmidt values, µ Schadd , modified Schmidt values, µ modadd , or experimental momentvalues from neighboring isotopes, µ expadd . Note that in fact the ( ν d / ) − neutron configuration was used to calculate µ Schadd and µ modadd (see text). Since g n, exp was obtained from neighbouring isotopes with I π = 5/2 + , the values for µ expadd in column 8 takeinto account the fact that the effective neutron configuration is a mixture of ν d / and ν g / coupling to a spin 5/2 + . Further,since the Schmidt g-factors for ν d / and ν g / are -0.765 and 0.425, respectively, the fact that g n, exp varies between -0.252and -0.0331 (see column 5) indicates that a considerable amount of mixing between the ν d / and ν g / states to be present.(table adapted from Ref. [5]) A I π configuration g p, exp g n, exp µ Schadd µ N µ modadd µ N µ expadd µ N µ exp µ N Ref. µ exp mix.ampl. α or β (Eq.( 23))102 2 + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + . a − . b (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + . c − . d e this work 0.79(1) (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + f [2] 0.72(2) (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + . g − . h + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + i [40] –108 1 + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + . j − . k + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + . l − . m + (cid:104)(cid:0) π g / (cid:1) − / + (cid:0) ν d / ν g / (cid:1) / + (cid:105) + n [39] – a g factor of Ag (I π = 9/2 + ) from Ref. [2]. b Average of the g factors of Ru,
Pd and
Cd. c g factor of Ag (I π = 7/2 + ) from Ref. [2]. d Average of the g factors of Ru and
Cd. e Other determinations of this magnetic moment have yielded +3.7(2) µ N [3] and 3.7(1) µ N (this work, RILIS measurement). f Other determinations of this magnetic moment have yielded 3.916(8) µ N [1] (see also Table IV), and +4.0(2) µ N [3]. g g factor of Ag m (I π = 7/2 + ) from Ref. [2]. h Average of the g factors of Pd and
Cd. i Other determinations of this magnetic moment have yielded 3.709(4) µ N [25], 3.71(15) µ N [44], and 3.82(8) µ N [45]. j g factor of Ag m (I π = 7/2 + ) from Ref. [26]. k g factor of Cd from Ref. [47]. l g factor of Ag m (I π = 7/2 + ) from Ref. [26]. m g factor of Cd (level at 245 keV) from Ref. [48]. n Another determination of this magnetic moment has yielded 3.607(4) µ N [38]. + states of Ag m and Ag g , respectively (which dif-fer only 9.3 keV in energy), indicates also almost com-plete mixing of the two different proton-neutron config-urations in Ag m ( I π = 2 + ), although the error barsstill allow for a smaller contribution from the ( π g / ) − / + proton configuration. As for Ag g ( I π = 5 + ), the ex-perimental magnetic moment value suggests an almostpure ( π g / ) − / + configuration, although the large errorbar still allows for a sizeable mixing from the ( π g / ) − / + configuration (see Table V). A larger contribution of the( π g / ) − / + configuration in Ag g , as is apparently ob-served, would be in line with the fact that the groundstate in Ag has I π = 9/2 + and also with the magneticmoment of Ag. Indeed, the experimental magneticmoment of
Ag, i.e. µ = 5.627(11) µ N [2], is very closeto the value of 5.67 µ N that was calculated in Ref. [54] forthe lowest 9/2 + state in the odd Ag isotopes based on aconfiguration that is dominated by the ( π g / ) − / + pro-ton group. New and precise measurements of the mag- netic moments of the ground and isomeric state of Agas well as of the lower mass even-A Ag isotopes couldhelp to further clarify this. An interesting new methodin this respect, viz. in-gas-cell laser spectroscopy [55],was recently reported. This method is currently beingapplied to the isotopes − Ag [56].
ACKNOWLEDGEMENTS
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