Garvey-Kelson Relations for Nuclear Charge Radii
J. Piekarewicz,
1
M. Centelles,
2
X. Roca-Maza,
2
and X. Vi˜nas
2
1
Department of Physics, Florida State University, Tallahassee, FL 32306
2
Departament d’Estructura i Constituents de la Mat`eria and Institut de Ci`encies del Cosmos,
Facultat de F´ısica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
(Dated: December 2, 2009)
The Garvey-Kelson relations (GKRs) are algebraic expressions originally developed to predict
nuclear masses. In this letter we show that the GKRs provide a fruitful framework for the prediction
of other physical observables that also display a slowly-varying dynamics. Based on this concept,
we extend the GKRs to the study of nuclear charge radii. The GKRs are tested on 455 out of the
approximately 800 nuclei whose charge radius is experimentally known. We find a rms deviation
between the GK predictions and the experimental values of only 0.01 fm. This should be contrasted
against some of the most successful microscopic models that yield rms deviations almost three times
as large. Predictions—with reliable uncertainties—are provided for 116 nuclei whose charge radius
is presently unknown.
PACS numbers: 21.10.Ft,21.10.Dr,21.60.−n
Theoretical calculations of nuclear ground-state prop-
erties may be classified according to two primary trends.
One of them is the so-called macroscopic-microscopic ap-
proach that is naturally rooted in the Strutinsky energy
theorem [1]. According to this theorem [2], the nuclear
binding energy may be separated into two components—
one large and smooth and the other one small and fluc-
tuating. The largest contribution varies smoothly with
both mass (A) and atomic (Z) numbers and describes the
average trend of the nuclear masses, as in the liquid drop
model and its various refinements [3, 4]. In contrast, the
small contribution fluctuates due to quantal effects (e.g.,
shell corrections) that are often incorporated through an
independent particle model with a realistic potential that
also varies smoothly with A and Z. The macroscopic-
microscopic approach has enjoyed its greatest success in
the work of M¨oller et al. [5] and Duflo and Zuker [6].
The “competing” approach, falling under the general
rubric of mean-field models, consists of a microscopic
description in which the average nuclear potential and
the single-particle orbits are determined self-consistently.
Although mean-field models vary widely in sophistica-
tion, their tenet is an effective interaction or energy den-
sity functional that incorporates as much as possible of
the known nuclear dynamics. The effective interaction
is parametrized in terms of several empirical constants
(e.g., coupling constants and range parameters) that are
then fitted to a variety of ground-state observables.
A lesser known approach—in spite of its 40 year
existence—is the one by Garvey and Kelson [7, 8]. Rather
than attempting a global description of nuclear masses,
the formalism is based on local mass relations. The
Garvey-Kelson relations (GKRs) have been recently revi-
talized both because of an interest in understanding any
inherent limitation in the nuclear-mass models as well
as due to their possible applications in stellar nucleosyn-
thesis [9, 10, 11]. In doing so, it was discovered that the
GKRs—which are essentially parameter free—rival in ac-
curacy the most successful mass formulae available in the
literature [10, 11].
The Garvey-Kelson relations are derived from a few
simple physical principles (such as isospin symmetry) and
a central assumption of a nuclear mean field and residual
interaction that vary slowly with atomic number [7, 8].
Within this picture, Garvey and Kelson introduced the
following two local relations—each among the masses of
six neighboring nuclei—that allowed them to estimate an
unknown nuclear mass from those of its neighbors:
∆M
(1)
6
(N, Z) ≡ M(N + 2, Z − 2) − M (N, Z)
+ M(N, Z − 1) − M(N + 1, Z − 2)
+ M(N + 1, Z) − M (N + 2, Z − 1) = 0 , (1a)
∆M
(2)
6
(N, Z) ≡ M(N + 2, Z) − M(N, Z − 2)
+ M(N + 1, Z − 2) − M(N + 2, Z − 1)
+ M(N, Z − 1) − M(N + 1, Z) = 0 . (1b)
It is the aim of this letter to show that this method may
be successfully extended to nuclear charge radii—a fun-
damental property of atomic nuclei that is both slowly
varying and has a large experimental database [12, 13].
Valuable insights into the success of the GKRs [10, 11]
may be gained by recalling that according to Strutinsky’s
energy theorem [1, 2], the nuclear mass function M(N, Z)
may be written as M =
f
M +δM, where
f
M and δM denote
the smooth and fluctuating parts of the mass function,
respectively. Viewing Eqs. (1) in this light, they can be
cast as ∆M
6
= ∆
f
M
6
+ ∆(δM )
6
. Now, a Taylor series
expansion of the smoothly-varying (and largest) contri-
bution
f
M may be performed around the reference point
arXiv:0912.0503v1 [nucl-th] 2 Dec 2009
2
(N, Z) [14] that yields
∆
f
M
(1)
6
(N, Z) =
∂
3
f
M
∂Z
2
∂N
−
∂
3
f
M
∂Z∂N
2
+ O(∂
4
f
M) , (2a)
∆
f
M
(2)
6
(N, Z) =
∂
3
f
M
∂Z
2
∂N
+
∂
3
f
M
∂Z∂N
2
+ O(∂
4
f
M) . (2b)
The above result indicates that the particular linear com-
binations of masses involved in the GKRs induce strong
cancellations in the Taylor expansions—making the rela-
tions insensitive to the underlying (liquid drop) function
as well as to its first and second derivatives. Hence, as
long as successive derivatives of the underlying function
become progressively smaller (indeed, as in the liquid-
drop formula [14]), the GKRs should be satisfied to a
very good approximation. Ultimately then, the level of
accuracy of the GKRs will depend on the extent to which
the fluctuating contributions δM get cancelled out in
Eqs. (1). This we expect to be specific to the system un-
der consideration. In the case of atomic nuclei, the rather
smooth local behavior with N and Z of the mean-field
potentials used to calculate the quantum contributions—
plus the fact that in Eqs. (1) the interactions between
nucleons cancel to first order in an independent-particle
picture [7, 8]—ensures a strong cancellation of the quan-
tum fluctuations among neighboring nuclei.
The above discussion suggests that the success of the
GKRs for nuclear masses may be extended to other ob-
servables that are driven by a similar underlying physics.
In this letter we focus on the nuclear charge radius. The
charge radius is a nuclear structure observable that is
known with exquisite accuracy for a few nuclei in the pe-
riodic table [12, 13]. The systematic measurement of the
charge distribution of nuclei started with the pioneering
work of Hofstadter in the late 1950’s [15] and continues to
this day with the advent of powerful continuous electron
beam facilities [16]. Although the experimental situation
for charge radii lags behind that of nuclear masses, an
extensive database of almost 800 charge radii already ex-
ists [13]. Moreover, the advent of novel technologies and
facilities to perform electron scattering off short-lived iso-
topes, such as ELISe [17] and SCRIT [18], may extend
the data well beyond current limits in the coming years.
From the theoretical side, the most sophisticated ap-
proaches to charge radii are based on either macroscopic-
microscopic models [19, 20, 21, 22, 23] or microscopic
mean-field formulations using effective interactions [24,
25, 26, 27, 28, 29, 30]. When some of these models are
used to compare against experimental data, the rms de-
viations lie in the 0.03 to 0.06 fm range [23]. In this letter
we will show that an approach based on local relations of
the Garvey-Kelson type represents a very attractive and
robust alternative.
Nuclear charge radii display, as in the case of masses,
small fluctuations on top of a fairly smooth average be-
havior [13, 14]. This may be illustrated by employing the
108 112
116
120 124
A
4.56
4.6
4.64
4.68
R
ch
(fm)
Experiment
HFBCS-MSk7
Garvey-Kelson
Sn isotopes
FIG. 1: (Color online) Comparison between theoretical pre-
dictions and experimental values [13] for the charge radius of
the Sn-isotopes.
liquid-drop inspired formula proposed in Ref. [20]. When
such a formula is fitted to the charge radii of the close
to 800 nuclei included in the recent 2004 compilation by
Angeli [13] we obtain,
R
ch
(N, Z) = 0.4980 + 0.8754 A
1/3
− 0.9845 α + 0.2703 A
1/3
α
2
fm . (3)
Here α ≡ (N − Z)/A is the neutron-proton asymmetry
of the nucleus and the fit produces the moderate rms de-
viation of 0.041 fm. This liquid-drop inspired formula
is particularly useful to estimate the derivatives of the
charge radius, as in Eqs. (2). Using a representative
set of nuclei—ranging from
16
O to
208
Pb—we found the
third-order derivatives of (3) to be suppressed by 4 to 6
orders of magnitude relative to R
ch
(N, Z) itself. More-
over, given that mean-field formulations provide quanti-
tatively accurate predictions of ground-state properties,
we expect—as in the case of masses—strong cancellations
of the fluctuating contributions to the GKRs for nuclear
charge radii. These facts open the possibility of applying
the GKRs to the study of nuclear charge radii.
The implementation of the GK procedure for charge
radii follows closely the approach outlined by Barea and
collaborators for the case of nuclear masses [10]. For a
given nucleus, there are (depending on the availability of
experimental information) at most 12 possible estimates
of its charge radius [see Eqs. (1)]. All the available es-
timates are then averaged to produce a GK prediction
for the charge radius of the given nucleus. The result
is then compared (when available) to the experimental
value [13]. In the event that the experimental value is un-
available, a GK prediction is made for the charge radius
of such a nucleus that awaits experimental confirmation.
The isotopic chain in Tin with 18 experimentally mea-
sured charge radii provides an illustrative example of this
scheme. Fig. 1 shows the predictions for the Tin charge
radii—including the as yet unmeasured value for
107
Sn—
using the GK relations and the Hartree-Fock-BCS model
with the MSk7 interaction [25]. The HFBCS model of
Ref. [25] generates a rms deviation of only 0.0082 fm for
3
FIG. 2: (Color online) Absolute value of the difference be-
tween the GK estimate and the experimental value for the
charge radius of 455 nuclei as indicated by the color-coded
scale. Black squares denote GK predictions for 116 nuclei
whose charge radius has not yet been measured. These pre-
dictions are provided in tabular form in Ref. [31].
these isotopes. The authors of [25] have stressed that
such a good agreement is essentially parameter free, as
all their model parameters were fitted exclusively to nu-
clear masses. It is rewarding to see that the GKRs—with
a rms deviation of 0.0031 fm—work as good, if not better,
than the most sophisticated microscopic models available
to date. In what follows, we show that this success ex-
tends throughout the periodic table.
By proceeding as in the case of the Tin isotopes, GK
predictions were made for the charge radius of a total of
571 nuclei. From these, 455 can be compared against ex-
periment while 116 await experimental confirmation (our
GK predictions for the 116 nuclei whose charge radius is
yet unmeasured may be found in Ref. [31]). This in-
formation has been graphically encoded in Fig. 2. The
largest deviation between the GK prediction and experi-
ment is about 0.06 fm and this happens for only a hand-
ful of nuclei at the edges of two of the populated regions.
Most of the predictions are well below this largest value
and 331 of them fall within experimental error. Indeed,
the rms deviation obtained for the 455 charge radii is
of only 0.0097 fm. This may be compared against the
rms deviation of 0.0275 fm predicted by the microscopic
Hartree-Fock-Bogoliubov model HFB-8 [23] (albeit this
comparison includes the 782 experimentally measured
charge radii with Z ≥ 8 and N ≥ 8 [13]). Note that using
the same experimental data set, the new state-of-the-art
HFB mass formulas BSk17 [29] and D1M [30] yield sim-
ilar rms deviations (0.030 and 0.031 fm, respectively).
We next discuss our theoretical errors to better assess
the reliability and predictive power of the GKRs. In an-
swering the question of how the errors of the measured
charge radii affect the GK predictions, we avoid attach-
ing a theoretical error by simply adding (e.g., in quadra-
tures) the experimental errors associated to the 5 nuclei
required to make a single GK estimate. This method of
propagating errors is uncontrolled and misleading when
-0.02 -0.01 0 0.01 0.02
∆R
ch
(fm)
0
50
100
150
p (fm
−1
)
(a)
-0.02 -0.01 0 0.01 0.02
δR
ch
(fm)
0
25
50
75
(b)
Gaussian
Lorentzian
Dipole
FIG. 3: (Color online) (a) Distribution of the differences be-
tween theory and experiment (∆R
ch
) for the charge radii of
455 nuclei (see text). Gaussian, Lorentzian, and Dipole prob-
ability density functions have been fitted to the histogram.
(b) Statistical distribution of the experimental errors (δR
ch
)
for the 796 nuclei included in Angeli’s compilation [13].
applied locally, as the GKRs are satisfied with varying de-
grees of accuracy throughout the nuclear chart. Rather,
we adopt a global approach that provides statistically
reliable confidence levels. The histogram in Fig. 3(a)
displays the differences ∆R
ch
between the GK estimate
and the central experimental value for the charge radii
of the 455 nuclei for which the comparison was possible.
Clearly, the probability distribution is very narrow. To
extract faithful confidence levels we have fitted ∆R
ch
to
three different probability density functions (PDF): the
Gaussian, the Lorentzian, and the Dipole. The plot in-
dicates that the Gaussian PDF falls too fast. From the
remaining two, the Dipole gives a slightly better fit than
the Lorentzian so we adopt it henceforth. The Dipole
PDF is defined as
p(x; µ, σ) = (2σ
3
/π)
(x − µ)
2
+ σ
2
−2
, (4)
where in the present analysis the optimal values of the
mean and the standard deviation for ∆R
ch
are µ =
3.633 × 10
−3
fm and σ = 4.554 × 10
−3
fm, respectively.
A particularly useful concept is that of confidence in-
terval of size n defined as
CI(n) =
Z
µ+nσ
µ−nσ
p(x; µ, σ)dx . (5)
It represents the probability that a given ∆R
ch
will fall
within ±n standard deviations of the mean. For the
dipole PDF, CI(1) = 0.818 and CI(2) = 0.960. This
indicates that the difference between the GK predic-
tion for a nucleus (e.g., any of the 116 given in [31])
and the central experimental value falls in the range
∆R
ch
= (3.633 ± 4.554) × 10
−3
fm with an 82% confi-
dence level. We have repeated the statistical analysis
for the distribution of experimental errors. The various
PDFs are plotted in Fig. 3(b) using the same horizontal
scale as in 3(a). Evidently, the experimental distribution
of errors is significantly wider. Indeed, in fitting a dipole
4
form to it we obtain a mean of zero (the errors are sym-
metric) and a standard deviation that is more than twice
as large: σ
exp
= 9.427 ×10
−3
fm. These results suggest
that the GKRs provide a useful and reliable benchmark
for the calculation of nuclear charge radii. We trust that
our results may motivate the experimental community
to perform new measurements and to refine some of the
existing ones.
In summary, taking into account that the linear com-
binations of masses that enter into the Garvey-Kelson
relations are proportional to the third derivatives of the
slowly-varying part of the nuclear mass function plus a
remainder that comes from quantum fluctuations and is
locally small, we concluded that the GKRs could be suit-
ably extended to other observables obeying a similar un-
derlying physics. In this letter we showed that this is
indeed the case for the nuclear charge radius. Indeed, we
made a systematic implementation of the GKRs using the
existing experimental database of charge radii [13]. Of
the 455 GK predictions that could be compared against
experiment, an overall rms deviation of only 0.0097 fm
was obtained. Moreover, of these 455 predictions 331 fell
within experimental error. For comparison, one of the
best microscopic models available in the literature (the
HFB-8 model of [23]) yields a rms value of 0.0275 fm. In
addition, we were able to make predictions for 116 nu-
clei [31] whose charge radius is presently unknown. Fi-
nally, by performing a global statistical analysis, we at-
tached meaningful theoretical errors to our predictions.
A similar analysis of the experimental errors revealed a
standard deviation more than twice as large.
In the future, we want to examine to which extent
the GK relations may be of use in other finite quantum
systems, such as helium and metal clusters where the
energy systematics is amenable to be described by a semi-
classical mass formula plus quantum corrections on top
of it [1]. We also intend to extend the GK predictions
to uncharted areas of the table of nuclides with the help
of methods from the field of image reconstruction—note
that important first steps in this direction have already
been taken [32, 33]. This would be particularly attractive
for nuclear masses in regions of astrophysical interest and
for charge radii of short-lived radioactive nuclei.
J.P. is indebted to Profs. A. Frank and J. Hirsch
(UNAM) for their hospitality and for enlightening dis-
cussions on the Garvey-Kelson relations. J.P. also thanks
Prof. W. Mio from the Mathematics Department at
FSU for valuable discussions on image-reconstruction
techniques. Work supported in part by grants DE-
FD05-92ER40750 (U.S. DOE), FIS2008-01661 (Spain
and FEDER), and 2009SGR-1289 (Spain), and the Con-
solider Ingenio Programme CSD2007-00042. J.P. and
X.R. acknowledge grants 2008PIV00094 from AGAUR
and AP2005-4751 from MEC (Spain), respectively.
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![FIG. 3: (Color online) (a) Distribution of the differences between theory and experiment (∆Rch) for the charge radii of 455 nuclei (see text). Gaussian, Lorentzian, and Dipole probability density functions have been fitted to the histogram. (b) Statistical distribution of the experimental errors (δRch) for the 796 nuclei included in Angeli’s compilation [13].](/figures/fig-3-color-online-a-distribution-of-the-differences-between-14hrbvx2.webp)
![FIG. 2: (Color online) Absolute value of the difference between the GK estimate and the experimental value for the charge radius of 455 nuclei as indicated by the color-coded scale. Black squares denote GK predictions for 116 nuclei whose charge radius has not yet been measured. These predictions are provided in tabular form in Ref. [31].](/figures/fig-2-color-online-absolute-value-of-the-difference-between-3vvml2i2.webp)