INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS
J. Phys. G: Nucl. Part. Phys. 30 (2004) 1089–1098
PII: S0954-3899(04)78217-1
Nuclear charge radii of neutron deficient titanium
isotopes 44 Ti and 45 Ti
Yu P Gangrsky1, K P Marinova1, S G Zemlyanoi1, I D Moore2,
J Billowes2, P Campbell2, K T Flanagan2, D H Forest3, J A R Griffith3,
J Huikari4, R Moore2, A Nieminen4, H Thayer3, G Tungate3 and J Äystö4
1
2
3
4
FLNR Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
Schuster Laboratory, University of Manchester, Manchester M13 9PL, UK
School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
Accelerator Laboratory, University of Jyväskylä, SF-405 51, Finland
Received 23 March 2004
Published 2 August 2004
Online at stacks.iop.org/JPhysG/30/1089
doi:10.1088/0954-3899/30/9/009
Abstract
Optical isotope shifts of the unstable 44,45 Ti isotopes, as well as those of stable
46−50
Ti, have been investigated by collinear laser spectroscopy on fast ion
beams using an ion guide isotope separator with a cooler-buncher. Changes
in mean square charge radii across the neutron 1f 7/2 shell are deduced. The
evolution of the even-N Ti nuclear radii shows a generally increasing tendency
with decreasing neutron number. This behaviour is significantly different to
that of the neighbouring Ca isotopes which exhibit a symmetric parabolic
behaviour across the shell. The trend of the Ti nuclear radii is consistent with
the predictions of the relativistic mean-field theory. The charge radius of 44 Ti
is also compared to predictions of a 40 Ca + α cluster model.
1. Introduction
The titanium isotopes studied in the present work are from the very interesting region near Z =
20 and between the neutron shell closures at N = 20 and 28. The unusual mass dependence of
the charge radii of the calcium (Z = 20) isotopes is well known [1, 2]. The parabolic-shaped
dependence in the Ca chain is characterized by a maximum in the middle of the shell and by
the surprising fact that the two doubly magic nuclei 40 Ca and 48 Ca have almost identical mean
square (ms) charge radii. The behaviour of the charge radii across the 1f7/2 shell has been
ascribed to changes of nuclear deformation [2–4], rather than any change in the volume of the
nuclear charge. The observed correlation between nuclear charge radii and binding energies
has been discussed by Zamick [5].
Although the symmetric parabolic behaviour of the calcium radii is qualitatively echoed
by the neighbouring odd-Z potassium chain [6], distinct differences include the amplitude
0954-3899/04/091089+10$30.00 © 2004 IOP Publishing Ltd Printed in the UK
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Yu P Gangrsky et al
of the mid-shell maximum and the odd–even staggering, which are both much smaller for
the potassium chain. Such gross structural differences are unusual for neighbouring isotope
chains [7, 8]. Further information on the nuclear radii trends in the region is provided by
the stable isotopes of titanium, 46−50 Ti (Z = 22) [9, 10], which cover only the upper half of
the neutron 1f7/2 shell. An earlier result on 44 Ti [11] suggested a continuing increase of Ti
charge radii with decreasing neutron number. It is important in the context mentioned above
not only to confirm this unusual result but also to extend the measurements to 42−45 Ti in the
lower half of the 1f7/2 shell.
The finite size and shape of the nuclear charge distribution perturbs the atomic structure.
Changes in the nuclear radius produce isotope shifts on optical transitions of different isotopes
that can be resolved by well-established techniques of laser spectroscopy. We have used the
method of collinear-beam laser spectroscopy to measure the nuclear ground state charge radii
in a more extended isotopic chain of titanium with the eventual aim of covering the whole
νf7/2 shell. This paper is devoted to a precise study of the isotope shift of titanium isotopes
ranging from 44 Ti to 50 Ti. The nucleus 44 Ti is of particular interest since, in contrast to other
Ti isotopes, it is predicted to have a dominant 40 Ca + α cluster structure [12–15] in its ground
state.
Titanium is a low-Z element for which the field shift component of the isotope shift,
which provides the information on changes in ms charge radii, is a small effect. The isotope
shift is dominated by the mass shift arising from the change in the recoil energy of the
nucleus with neutron number. Optical measurements on radioactive titanium isotopes thus
require high precision and extreme sensitivity because of the radioisotope production rates.
The experimental requirements are met by the use of a recently developed high-sensitivity
technique [16] involving the cooling and bunching of radioactive ions in preparation for the
laser measurement.
2. Experimental details
Experiments were performed using the online ion guide isotope separator (IGISOL) facility
at the Cyclotron Laboratory, University of Jyväskylä. The ion-guide ion source is particularly
well suited for providing beams of refractory elements such as titanium. A full description
of the experimental apparatus—the IGISOL, the laser spectroscopy set-up and the coolerbuncher—may be found elsewhere [16–18].
The neutron deficient isotopes 44 Ti (T1/2 = 49 y) and 45 Ti (T1/2 = 184.8 m) were produced
in the 2n and 1n channels of the 45 Sc(p, xn)46−x Ti reaction. The beam flux was of the order of
1500 ions s−1 for 45 Ti but was only 100–200 ions s−1 for 44 Ti. The experiments were performed
using the method of collinear laser spectroscopy on bunched ion beams. A gas-filled linear
Paul trap on a 40 kV platform was used to cool and trap the ions for 500 ms with high
efficiency. The ion collection could be released in a bunch of ≈20 µs duration and delivered
as a low-emittance (3π mm mrad) beam to the laser spectroscopy station (figure 1) [19].
The stability of the cooler platform voltage is crucial for the energy spread (and therefore
the resolution). The observed spectra were broadened by an energy spread arising from a
voltage ripple on the cooler platform. The broadening corresponds to an energy spread of
roughly 7 eV.
Ionic ensembles, reaccelerated from the trap, were overlapped with a counter-propagating
laser beam by inserting a 1 mm aperture at the centre of the beam line weakly focusing both
beams through the aperture. The 1 mm aperture was then withdrawn during the measurements.
A Hamamatsu R5900P-03-L16 photomultiplier viewed the overlap region and was used
to monitor resonance fluorescence. A significant reduction in the photon background was
Nuclear charge radii of neutron deficient titanium isotopes 44 Ti and 45 Ti
1091
Ion
beam
Laser
beam
Lenses
Segmented
PMT
Microchannel
Plates
Figure 1. Schematic view of the laser spectroscopic station.
achieved by electronically gating the photomultiplier signal such that the photon events were
only accepted if they arrived while the ion bunch was in front of the detector [20]. This
allowed a suppression of the random background by a factor of 2 × 104 .
Laser light was provided by a frequency-doubled Spectra-Physics 380D dye laser
locked to a chosen molecular iodine absorption line. The isotope shift and hyperfine structures
of the titanium isotopes were measured in the transition d2 s 4 F3/2 (0 cm−1 ) → d2 p 4 F3/2
(30 837 cm−1 ) of the Ti II spectrum at λ ∼ 324.2 nm. The laser power was 2 mW. An 18 mm
long overlap region was imaged, via a quartet of fused-silica lenses, onto the photomultiplier.
Online calibrations were performed on the reference isotope 48 Ti, which was released from a
coated skimmer electrode by electric discharge during the experiment.
The online measurements were complemented by off-line work on the stable isotopes.
During these measurements, the ripple on the cooler platform voltage was successfully
suppressed (using a 40 kV, 100 nF capacitor) and the hyperfine structures of the odd isotopes
were well resolved. The effective linewidths were reduced from 120 MHz, observed during
the earlier 44,45 Ti measurements, to 50 MHz for the example of 47 Ti (approaching the
power-broadened natural linewidth of ∼30 MHz). All ripple-broadened structures could
be satisfactorily fitted using a common Gaussian profile for each hyperfine component.
Pure Lorentzian profiles were found to best describe the observed lineshapes following the
suppression of the voltage ripple.
3. Results
Examples of the resonance spectra, converted to frequency relative to the centroid of 48 Ti
(section 4), are shown in figure 2. The centroid of the 44 Ti resonance peak was established
by fitting the data with a Gaussian profile, which was found to describe most adequately the
lineshape. The hyperfine structure of 45 Ti (with a total splitting of about 180 MHz) was
not resolved. From measurements of the stable Ti isotopes and those for the atomic 45 Ti
(I = 7/2) ground state, reported in [21], it is possible to estimate hyperfine parameters for
the states involved in the studied ionic transition. In [21], the magnitude and the ratio of
the magnetic dipole moment and the electric quadrupole moment of 45 Ti were determined
to be |µ| = 0.095(2) nm and |Qs | = 0.015(15) b with µ/Qs > 0. Hyperfine parameters
in the ion, scaled from the measured moments, were found to be Al = ±5.82(12) MHz
and Bl = ∓1.0(10) MHz for the lower level of the investigated transition and Au =
∓9.18(19) MHz and Bu = ±1.0(10) MHz for the upper level, under the assumption of zero
hyperfine anomaly. Different fits to the frequency converted 45 Ti spectrum were attempted
to measure the centroid of the transition: (i) a simple Gaussian profile covering the whole
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Yu P Gangrsky et al
60
50
2500
44
45
47
50
40
2000
40
1500
Counts
30
30
20
1000
20
10
0
-2000
500
10
-1600
-1200
0
-1600
-1200
-800
0
-2000
0
2000
Frequency (MHz)
Figure 2. Optical resonances in the unstable 44 Ti and 45 Ti and in stable 47 Ti isotopes as a function
of frequency relative to the centroid of the 48 Ti resonance. The total accumulation time was 16.8 h
for the 44 Ti spectrum and 9.9 h for the 45 Ti. Fitted lineshapes, using Gaussian profiles, are shown
for 44 Ti and 45 Ti resonances. A fitted spectrum, using Lorentzian profiles, displaced for clarity, is
shown for the 47 Ti structure.
unresolved spectrum and (ii) assuming Bl = Bu = 0, a fixed Al /Au ratio, and calculated
relative intensities of the hyperfine components fitted with Gaussian profiles. Both methods
lead to consistent results. Under the assumptions in the latter method two, comparable and
symmetric, minima in the multi-dimensional χ 2 space were found corresponding to solutions
with Al equal to ±4.6(6) MHz. The discrepancy with the scaled value of Al suggests, subject
to the validity of the assumptions in the fitting, a substantial hyperfine anomaly in this system
(highly likely due to the very small magnetic moment of 45 Ti [21]).
To confirm the absence of any systematic inaccuracies in the system, stable isotope
shift measurements were also performed. In the King-plot analysis, the isotope shifts
measured in the optical transition of Ti ions showed perfect consistency with those measured
in optical transitions of titanium atoms obtained earlier by a high-accuracy off-line laser
technique [9, 10].
4. Analysis of data
During collinear spectroscopy, the laser frequency was locked to a fixed reference frequency,
νl . Fine-tuning of the optical frequency in the ions’ frame was achieved by changing the
ion-beam velocity with a Doppler tuning voltage, applied to the laser–ion interaction region.
As the ion beams and the laser beams were counter-propagating, the Doppler-shifted transition
frequency may be expressed as
1−β
ν(β) = νl = ν0 1 − β2
Nuclear charge radii of neutron deficient titanium isotopes 44 Ti and 45 Ti
where
1093
eU (eU + 2mA c2 )
(eU + mA c2 )
and ν0 is the resonance frequency of the optical transition in the rest frame of the ions. The
relative ion-beam velocity β is described in the relativistically correct formula: eU is the beam
kinetic energy, corresponding to the peak position of a resonance line of the isotope A, and
mA is the mass of the isotope. In all calculations, mA has been taken from the data compiled
in [22].
In order to extract ms charge radii changes in the titanium system, the small nuclear
field shift must be accurately determined. The two main experimental limitations, affecting
the absolute accuracy, were found to be the laser frequency stabilization and the absolute
determination of the total acceleration voltage. Special care was taken to keep the laser
frequency stabilized, to prevent mode-hops and to eliminate long-term instabilities [23]. The
drift in the total acceleration voltage could be monitored to a relative accuracy of ±0.1 V.
The systematic error on the isotope shifts, due to these uncertainties, is less than 1 MHz for
two neighbouring Ti isotopes. This is of the order of, or even smaller than, the experimental
statistical errors. The more substantial systematic error related to the absolute calibration
of the total acceleration has, as shown in [24], a similar mass dependence on the nuclear
recoil correction (the mass shift) and the effect of this uncertainty can be eliminated by the
non-optical calibration of the ms charge radii.
48,A
, proportional to the ms nuclear radii
The isotope shift consists of the field shift δνFS
change, and a term accounting for the finite mass of the nucleus, the so-called mass shift,
48,A
[25]:
δνMS
β=
48,A
48,A
δν 48,A = δνFS
+ δνMS
= F δr 2 48,A +
mA − m48
MMS
mA m48
where the usual convention for the sign δν 48,A = ν A − ν 48 is adopted. The relativistic field
shift between two isotopes or isomers in an optical transition for light elements is given by
F δr 2 48,A = π a03
|(0)|2
f (Z)δr 2 48,A
Z
where
|(0)|2 = β|(0)|2ns
is the change of the electron charge density at the nucleus between lower and upper states
of the optical transition, |(0)|2ns is the electron density in a ns state and β is the screening
factor (see, e.g., [25]). The function f (Z) is sensitive to the course of the Dirac wavefunction
at the extended nucleus [26]. F is the so-called electronic factor. For the investigated
d2 s 4 F3/2 → d2 p 4 F3/2 optical transition in Ti II two terms, |(0)|2ns and β, in the electronic
factor were calculated using the general relativistic atomic structure program (GRASP)
Dirac–Fock package [23]. GRASP predicts high purity (99.5%) for the atomic ground state
function d2 s 4 F3/2 when optimized in a d2 s + ds2 + d3 basis alone. For the purpose of the
estimate a pure d2 s → d2 p transition was assumed and far-configuration mixing was taken to
be negligible. The procedure is outlined in detail in [23]. Although the screening parameter β
was calculated under these restricted conditions the value obtained, 1.08, is perfectly consistent
with the normally applied estimates for s → p and analogous d2 s → d2 p transitions. The
absolute accuracy of the electron density |(0)|2ns in the ground state can only be determined
by comparison with experimental results. In zinc (Z = 30), the GRASP package overestimates
the density by ≈10%. The value may therefore be considered accurate to approximately this
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Yu P Gangrsky et al
Table 1. Isotope shifts and charge radii in the Ti isotope chain. Errors in parentheses are statistical;
errors in brackets, shown separately, are systematic uncertainties (for explanation, see the text).
The earlier data of [10] with their statistical errors only are shown for comparison. The absolute
values of charge radii, calculated using the method of [7], are also presented (with associated
statistical errors).
A
δν 48,A (MHz)
λ = 324.2 nm
δr 2 48,A (fm2 )
present work
44
45
46
47
48
44
44
−1558(17)
−1100(8)
−765(3)
−363(2)
–
+400(4)
+731(3)
+0.143(37) [25]
+0.013(17) [18]
+0.110(7) [12]
+0.030(4) [6]
–
−0.139(9) [6]
−0.160(7) [11]
δr 2 48,A (fm2 )
[10]
r 2 1/2 (fm)
+0.108(6)
+0.018(8)
–
−0.138(12)
−0.165(9)
3.6185(38)
3.6005(27)
3.6139(24)
3.6029(24)
3.5987(23)
3.5793(25)
3.5764(24)
level which is consistent with other often applied uncertainties. When combined with the
Zimmerman [27] estimate of f (Z), a value F = −460(46) MHz fm2 is derived with an
adopted error of 10%.
The interpretation of isotope shifts (IS) in light elements is complicated by the large mass
shifts and the difficulty in reliably calculating the correlated electron contribution to these
terms (the specific mass shifts). In such cases, the field and mass shifts are often separated
using empirical methods [24] based on independent measures of the field shifts. Two values
for the changes in nuclear ms charge radii δr 2 48,46 and δr 2 48,50 can be obtained from the
model-independent analysis of muonic and electron scattering data by Wohlfahrt et al [28]
(see also [9]):
δr 2 48,46 = 0.108(6)[7] fm2
and
δr 2 48,50 = −0.165(9)[7] fm2 .
These values include, shown separately in the square brackets, a systematic error
arising from the nuclear polarization [28]. The two independent values for δr 2 48,A and
the two corresponding experimental isotope shifts δν 48,A can, in principle, be combined to
determine the constants MMS and F uniquely. Such a procedure however results in substantial
uncertainties (about 30% for F). The theoretically calculated value of F, −460(46) MHz fm2 ,
is found to be consistent with that suggested by the non-optical values and if adopted permits
an accurate determination of the one unknown mass-shift constant, MMS = 788(6) GHz amu.
The error in MMS reflects the 10% uncertainty adopted for the electronic factor and the
uncertainties on the muonic radii.
The experimental values of the isotope shifts and the corresponding changes in the ms
nuclear charge radii derived with the F and MMS values above are compiled in table 1. The
accuracy with which changes in the mean square radii were extracted from the measured
isotope shifts is determined by: (i) the errors of the measurements which are shown in
parentheses and arise from the statistical errors of the isotope shifts; (ii) the error on the
estimation of the mass-shift constant. The absolute values of the rms charge radii are given
in the last column of table 1. These values were obtained using a combination of optical
and non-optical experimental data in a manner outlined in [7]. Figure 3 shows the ms charge
radii changes in the Ti isotope chain as a function of mass number. The systematic errors can
result in a shift in the general slope of the plot, but relative effects between the isotopes are
insensitive to the calibration uncertainties. The evolution of the ms charge radii in the neutron
deficient part of the isotope chain is in qualitative agreement with preliminary measurements
Nuclear charge radii of neutron deficient titanium isotopes 44 Ti and 45 Ti
1095
Figure 3. Mean square nuclear charge radii versus neutron number for the Ti isotopes. Left
panel: open circles denote the experimental data with two enveloping dashed lines indicating the
systematic uncertainties, a Talmi–Zamick fit [5, 29] to the data is indicated by the crosses. Right
panel: the ms radius predictions of the RMF theory [30, 31] (filled triangles).
Figure 4. Mean square nuclear charge radii versus neutron number for K and Ca (with data taken
from the review paper of Otten [32]) and for Ti isotopes (from this work). The model-independent
values of absolute charge radii, r 2 1/2 , are taken from the data of [28] and [33]. The error bars
represent statistical uncertainties alone.
reported in [11] and the nuclear radius of 44 Ti is observed to be comparable or greater than
that of 46 Ti.
5. Discussion
Figure 4 compares the charge radii trends of the Ti, Ca [1, 2] and K [6] isotopes across the 1f7/2
shell. Although some underlying parabolic dependence of charge radii on neutron number is
evident for all three chains, the differences between the isotonic neighbours are much greater
than one might expect for such a restricted mass region.
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Yu P Gangrsky et al
The Talmi–Zamick model [5, 29] provides a quantitative, shell-based, description of the
charge radii trends in the Ca isotopes. Although strictly only applicable at magic-proton shell
closures, it has often been empirically extended, for example, by Wohlfahrt et al [28], and
used to fit radial trends in other isotope chains. Figure 3 shows a Talmi–Zamick form of fit for
the measured Ti isotopes. Although the agreement over the measured isotopes is reasonable,
the three-parameter model fails to reproduce the charge radii of 47 Ti and only fits the 44 Ti
radius at the limit of statistical and systematic errors. Furthermore, it is clear that the true
test of the fit will only be possible when data for the last two members of the 1f7/2 shell
are available. These data points are equally vital for comparisons with the predictions of the
self-consistent relativistic mean field (RMF) theory. The predictions of the RMF theory have
been particularly successful in describing both charge radii in the near-spherical Pb nuclei
(where its ability to reproduce the kink at the N = 126 shell was a notable achievement) and
radii in regions of the strong nuclear deformation, such as the Kr, Sr and Zr isotope chains.
The (even) Ti isotopes were selected as a particular case for study by Lalazissis et al [30, 31].
Their calculations for the even Ti isotopes are plotted on the right side of figure 3 and show a
monotonic increase in ms radii from the N = 28 shell to below the N = 20 shell, unlike the
Talmi–Zamick fit. The calculated radii successfully reproduce the increasing ms radii at 44 Ti
and suggest that a proton skin forms in isotopes close to the drip line.
The RMF theory makes no predictions for odd nuclei and sheds no light on the
pronounced odd–even staggering of the charge radii seen in the Ca chain and for 45 Ti. The
predictions for Ca also fail to reproduce the parabolic radial trend in the even-N isotopes of
the chain. Despite this, and particularly as more experimental data become available, it may
be possible that only the behaviour of the Ca chain radii is poorly reproduced by the theory.
Lalazissis et al [30, 31] do not predict a noticeable kink in the charge radii systematics at
N = 20 for any isotope chain and this prediction is in agreement with all available experimental
data (which cover the Ar [24], K [6] and Ca [1, 2] chains). The observed behaviour of
the lightest Ti isotope measured, 44 Ti, also supports an increasing ms charge radius as the
N = 20 shell closure is approached. The formation of a proton skin and the applicability
of the RMF theory in this region thus remain an open, but experimentally answerable,
question.
For the particular case of the 44 Ti isotope, other quantitative calculations of the charge
radii must also be considered. A 40 Ca + α cluster structure has been predicted for the 44 Ti
ground state [14, 15]. The clustering of valence nucleons to form an α particle would act to
increase the rms radius only if the cluster rms distance from the 40 Ca core centre were greater
than the normal rms distance of the four valence nucleons (without the cluster correlations).
The increase is offset slightly because the rms radius is now defined with respect to the
centre of mass of the cluster system. Such cluster calculations have been carried out for
44
Ti [14, 15] and predict rms radii consistent with the present measurement. However, the
accuracy of the calculations is limited and the uncertainties are large compared to the apparent
discrepancy between the Ca and Ti (N = 22) isotone radii, which is at the level of 0.01 fm.
The calculations by Krasta et al [14] have an rms deviation with experimental values of at
least 0.07 fm (covering nine cluster nuclei below A = 44). For 44 Ti, they predict an rms radius
of 3.64 fm, compared to our experimental value of 3.618(4) fm (table 1). A parameter-free
calculation by Merchant [15] based on the cluster model of Buck et al [12] gives an rms value
of 3.71(4) fm, although he states this as an upper limit since the ground state is not a 100% pure
cluster configuration. It is evident that charge radii, even when measured with high precision,
do not constitute a meaningful test for the present cluster models. The clustering produces a
change in radius that is probably much less than the reliability of present model predictions.
Conversely, theoretical calculations of comparative variations between isotopes (such as the
Nuclear charge radii of neutron deficient titanium isotopes 44 Ti and 45 Ti
1097
RMF calculations discussed here) do provide a valuable framework for understanding charge
radii trends.
6. Concluding remarks
Isotope shifts and hyperfine structure measurements performed on seven titanium isotopes
have been used to evaluate the ms nuclear charge radii trend for the majority of the f7/2 shell at
Z = 22. The unstable 44 Ti and 45 Ti isotopes have been accessible as the result of the application
of a high-sensitivity laser spectroscopy technique [16] at the IGISOL facility, University of
Jyväskylä. The well-pronounced symmetric parabolic shape observed in the case of calcium is
not reproduced in the neighbouring titanium chain. The charge radii development in this chain
shows a generally increasing trend with decreasing neutron number (for the even-N systems).
Various theoretical models have been compared to the experimental results and the predictions
of the RMF theory [30, 31] appear to most closely reproduce the experimental trends.
The results give new information on the behaviour of the nuclear charge radius in a
scarcely investigated region immediately above the calcium (Z = 20) chain. The continuation
of the optical investigation of Ti isotopes to the lighter 43 Ti and to, especially, 42 Ti at the
N = 20 shell closure will provide a stringent test of available theoretical models.
Acknowledgments
This work has been supported by the Russian Foundation for Basic Research grant 01-0216455, by the INTAS grant 00-00463 and by a Joint Project grant from the Royal Society.
The authors are indebted to Yu Tz Oganessian, M G Itkis and Yu E Penionzhkevich for their
stimulating interest.
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