Development of bakelite based Resistive Plate Chambers

S. Biswasa, S. Bhattacharyab, S. Boseb, S. Chattopadhyaya, S. Sahab, M.K.
Sharanb, Y.P. Viyogic

aVariable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata-700 064, India

bSaha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata-700 064, India

cInstitute of Physics, Sachivalaya Marg, Bhubaneswar, Orissa-751 005, India

Abstract

A Comparative study has been performed on Resistive Plate Chambers made of
different grades of bakelite paper laminates, produced and commercially
available in India. The chambers, operated in the streamer mode using argon :
tetrafluroethane : isobutane in 34:59:7 mixing ratio, are tested with cosmic
rays for the efficiency and the stability with cosmic rays. A particular grade
of bakelite (P-120, NEMA LI-1989 Grade XXX), used for high voltage insulation
in humid conditions, was found to give satisfactory performance with stable
efficiency of $>$ 96% continuously for more than 110 days. A silicone
treatment of the inner surfaces of the bakelite RPC is found to be necessary
for operation of the detector.

Key words: RPC; Streamer mode; Bakelite; Cosmic rays

PACS: 29.40.Cs

1\. Introduction

The Resistive Plate Chambers (RPCs), first developed by Santonico et al. [1]
using bakelite and by Yu. N. Pestov et al. [2, 3] and subsequently by others
[4], using silicate glass are used extensively in high energy physics
experiments. The RPCs are being considered for the following reasons a)
relatively low cost of materials used in making RPCs, b) robust fabrication
procedure and handling and c) excellent time and position resolution.
Primarily used for generating faster trigger for muon detection [5], time of
flight (TOF) [6, 7] measurement, and tracking capabilities in multi layer
configurations, they are successfully used in BELLE [8], BaBar [9], BESIII
[10], and several upcoming LHC experiments (ATLAS,CMS etc.) [11, 12]. RPCs are
used in neutrino experiments like OPERA where its excellent time resolution
and tracking capabilities are exploited [13]. The RPCs are also being explored
for use in PET imaging with TOF-PET [14], detection of $\gamma$-rays [15] and
neutrons [16, 17] over a large area.

The RPCs are made up of high resistive plates (e.g. glass, bakelite, ceramics
etc.) as electrodes, which help to contain the discharge created by the
passage of a charged particle or an ionizing radiation in a gas volume, and
pick-up strips are used to collect the resulting signals. Typical time
resolution for a single gap RPC is $\sim$ 1-2 ns. By reducing the gaps between
the electrodes or by using multi-gap configuration, time resolution in such a
detector can be reduced to $<$ 100 ps [18, 19].

The RPCs are operated in two modes, viz., the proportional mode and the
streamer mode [20]. Over the years, one of the main concerns with the use of
RPCs is their long term stability. In the proportional mode, a small amount of
charge is produced in the gas, which allows the RPC to recover in a relatively
shorter time to handle high counting rates ($\sim$ 1 KHz/cm2). Ageing effects
caused by the accumulated charge is also relatively less in this mode. In the
streamer mode, the amount of charge produced is considerably larger creating
induced signals of larger magnitude. But, the recovery time is larger and the
irreversible damage caused by the accumulated charge reduces the life of the
RPC. However, several remedial measures can be taken to prolong its life under
streamer mode of operation. Careful choice of materials, smoothness of the
surfaces to avoid localization of excess charges, surface treatment to reduce
the surface resistivity or providing alternate leakage path for post-streamer
recovery are adopted in the major high energy physics experiments. Prolonged
stable operation in streamer mode of the BELLE RPCs, though made of glass, is
a testimony to many serious efforts taken for the above cause [8].

The glass-based RPCs are found to be more stable mainly for low rate
applications, even though some erosion effects are found for such cases,
particularly when these are operated in the streamer mode [20]. This has been
attributed to the corroding of the glass surface due to the large charge build
up in the streamer mode of operation. However, in the proportional mode of
operation, the detectors can be operated for longer period. At the end of
nineties, it was found that the RPCs based on bakelites show serious ageing
effects reducing the efficiency drastically [21]. Detailed investigations
revealed that the use of linseed oil for the surface treatment in such cases
was the main reason for this ageing effect [22, 23]. Efforts were subsequently
made to look for alternatives to linseed oil treatment, or to develop bakelite
sheets which can be used without the application of linseed oil [24]. It has,
however, been found that for several ongoing and future applications (e.g. CMS
detectors), bakelite based RPCs are chosen as preferred options mainly due to
cheaper cost of fabrication.

In the proposed India-based Neutrino Observatory (INO), the RPCs have been
chosen as the prime active detector for muon detection in an Iron Calorimeter
(ICAL) [25]. As proposed presently, ICAL is a sampling calorimeter consisting
of 140 layers of magnetized iron, each of 60 mm thickness, using RPCs of 2m
$\times$ 2m area as active media sandwiched between them. A 50 Kton ICAL is
expected to consist of about 27000 RPC modules. For ICAL RPCs, main design
criteria are (a) good position resolution, (b) good timing resolution (c) ease
of fabrication in large scale with modular structure and most importantly (d)
low cost. Detailed R & D are being performed on glass RPCs for this
application. In this article, we report a parallel effort on building and
testing of the RPC modules using the bakelite obtained from the local
industries in India. The aim of the study is to achieve stable performance of
the RPC detector for prolonged operation.

The paper is organized as follows. In the next section, we describe the method
of assembly of the RPC modules. The section 3 contains measurements of the
bulk resistivity of the bakelite sheets used in this application while the
cosmic ray set-up used in our experiment are discussed in section 4. The
results of the study are reported in section 5.

2\. Construction of the RPC modules

Figure 1: Schematic diagram of a resistive plate chamber.

A schematic view of the assembled RPC modules is shown in the Fig. 1\. Two 300
mm $\times$ 300 mm $\times$ 2 mm bakelite sheets are used as electrodes. The
inner surfaces of the two sheets are separated by a 2 mm gap. Uniform
separation of the electrodes are ensured by using five button spacers of 10 mm
diameter and 2 mm thickness, and edge spacers of 300 mm $\times$ 8 mm $\times$
2 mm dimension, both being made of polycarbonate. Two nozzles for gas inlet
and outlet, also made of polycarbonate, are placed as part of the edge
spacers. All the spacers and nozzles are glued to the bakelite sheets using
Araldite® epoxy adhesive (grade: AY 103∗∗MP, made by Huntsman A.M., (Europe)).
The 2 mm thick active gas gap of the RPC modules are leak-checked using argon
and helium sniffer probes. The edges of the bakelite sheets are sealed by
applying a layer of the epoxy adhesive to prevent permeation of moisture.

After proper cleaning, a graphite coating is made on the outer surfaces of
bakelite sheets to distribute the applied voltage uniformly over the entire
RPC. A gap of 10 mm from the edges to the graphite layer is maintained to
avoid external sparking. The surface resistivity varies from 500
K$\Omega$/$\Box$ to 1 M$\Omega$/$\Box$ for different samples. The graphite
coating, applied by using a spray gun, however, results in a non-uniformity
(less than 20%) for a particular coated surface. Two small copper foils $\sim$
20 $\mu$m thick are pasted by kapton tape on both the outer surfaces for the
application of high voltage. The HV connectors are soldered on these copper
strips. Equal HVs are applied on both the surfaces.

In order to collect the accumulated induced charges, pick-up strips are placed
above the graphite coated surfaces with minimum air-gap. The pick-up strips
are made of copper (20 $\mu$m thick), pasted on one side of 10 mm thick foam.
The area of each strip is 300 mm $\times$ 30 mm with a separation of 2 mm
between two adjacent strips. The pick-up strips are covered with 100 $\mu$m
thick kapton foils to insulate them from the graphite layers. The ground
plane, made of aluminium, is pasted on the other side of the foam. The signals
from different strips are sent through a ribbon cable, followed by RG-174/U
coaxial cables using proper impedance matching.

The gases used in the RPC are mixtures of Argon, Isobutane and
Tetrafluroethane (R-134a) in varying proportion. The gases are pre-mixed,
stored in a stainless steel container and sent to the detector using stainless
steel tubes. A typical flow rate of 0.4 ml per minute resulting in $\sim$ 3
changes of gap volume per day is maintained by the gas delivery system. A
systematic analysis was made for R-134a and Isobutane before use in the system
by a Prisma Quadstar 422 Residual Gas Analyzer. The composition was found to
be 98.83% for R-134a with 0.75% O2 and 0.41% N2 and 98.93% for Isobutane with
1.07% H2.

3\. Measurement of bulk resistivity of bakelite

The bakelite sheets are phenolic resin bonded paper laminates. In the present
work, three types of bakelite sheets have been used to build as many modules.
They are (a) mechanical grade bakelite (P-1001), (b) Superhylam grade and (c)
electrical grade (P-120).

The P-1001 and P-120 grade bakelites are manufactured by Bakelite Hylam, India
and the Superhylam grade is obtained from the other manufacturer Super Hylam,
India. The sheets of P-1001 and P-120 are matt finished whereas superhylam is
glossy finished. The P-1001 has good mechanical properties whereas the P-120
has good mechanical and electrical properties under humid conditions prevalent
in India.

The bulk resistivity of the electrode plates of the RPC is an important
parameter [26]. The high resistivity helps in controlling the time resolution,
singles counting rate and also prevents the discharge from spreading through
the whole gas. We have measured the bulk resistivities of the bakelite sheets
via the measurement of the leakage current.

Figure 2: The volume resistivity ($\rho$) as a function of the applied voltage for three grades of bakelites. Table 1: Mechanical and electrical properties of different grades of bakelite. Trade | NEMA | BS-2572 | Density | Electrical | Surface | Bulk
---|---|---|---|---|---|---
Name | LI-1989 | Grade | (g/cc) | strength | finish | resistivity
| Grade |  |  | (kV/mm) |  | ($\Omega$-cm)
P-1001 | X | P1 | 1.38 | 3.5 | Matt | 6.13 $\times$ 1010
Superhylam | - | P2 | 1.72 | 9.5 | Glossy | 1.25 $\times$ 1011
P-120 | XXX | P3 | 1.22 | 9.5 | Matt | 3.67 $\times$ 1012

This measurement is performed at the same place and the same environment where
the RPCs have been tested. The test set up is kept in a temperature and
humidity controlled room. These two parameters have been monitored during the
experiment and are nearly the same around that time. The bulk resistivities of
different grade materials at 4 kV are tabulated in Table 1. The volume
resistivity($\rho$) vs. voltage(V) characteristics of different grade
materials are shown in Fig. 2. It is clear from the figure that the bulk
resistivity is considerably higher for the P-120 grade bakelite. For the
P-1001 grade, resistivity is much lower and it cannot sustain high voltage
above 4 kV. Therefore, the P-1001 grade is not considered further for building
up the RPC. The superhylam grade, though having lower bulk resistivity than
P-120, would stand high voltages up to 6 kV. This is also considered for the
fabrication of RPC detector.

4\. Cosmic ray test setup

Figure 3: Schematic representation of the cosmic ray test setup.

Fig. 3 shows the schematic of the setup for testing the RPC modules using
cosmic rays. Three scintillators, two placed above the RPC plane and one
placed below are used for obtaining the trigger from the incidence of the
cosmic rays. The coincidence between scintillator I (350 mm $\times$ 250 mm
size), scintillator II (350 mm $\times$ 250 mm size) and the finger
scintillator(III) (200 mm $\times$ 40 mm size) is taken as the Master trigger.
Finally, the signal obtained from the pick-up strip of the chamber is put in
coincidence with the master trigger obtained above. This is referred to as the
coincidence trigger of the RPC. The width of the finger scintillator is made
smaller than the total width of the two adjacent readout strips, thereby
needing a correction for dead zones in between two readout strips.

The high voltage to the RPC, are applied at the ramping rate of 5 V/s on both
the electrodes. The streamer pulses are obtained starting from the high
voltage of 5 kV across the RPC. The high voltage is applied to some of the
RPCs by using the CAEN Mod.N470 unit and to the others using the CAEN
Mod.N471A unit. The leakage current as recorded by the high voltage system is
studied. The signals from two consecutive strips covered by the scintillator
III are ORed to form the final signal, for the next part of the pulse
processing.

The Philips Scientific leading edge discriminators (Model 708) are used for
the scintillators and the RPC pulses. Various thresholds are used on the
discriminators to reduce the noise. For our final results, a threshold of 40
mV is used on the RPC signal. We have used a CAMAC-based data acquisition
system LAMPS, developed by Electronics Division, Bhabha Atomic Research
Centre, Trombay, India. Counts accumulated in a CAEN (Model 257) scalar over a
fixed time period are recorded at regular intervals, and saved in a periodic
log database. The temperature and the humidity are monitored at the time of
measurement.

5\. Results

An important and obvious goal of any RPC detector development is to study the
long term stability with high efficiency. In that spirit, the following
studies are performed in the cosmic ray test bench of the RPC detectors.

The efficiency of the RPC detector, taken as the ratio between the coincidence
trigger rates of the RPC and the master trigger rates of the 3-element plastic
scintillator telescope as mentioned in sec.4, is first studied by varying the
applied HV for each detector. The rates are calculated from data taken over 30
minutes duration for each HV setting. The temperature and humidity during
these measurement are recorded to be about 22-25∘C and 63-65% respectively.
The average master trigger rate is $\simeq$ 0.005 Hz/cm2. The variation of
efficiency with applied HV is shown in the Fig. 4 and that of the singles
counting rates with the HV is shown in the Fig. 5. It is seen that for both
the bakelite grades, the efficiency has increased from 20% to 75% as HV is
ramped up from 6.5 kV to 6.8 kV. The efficiency for the superhylam grade
gradually increases and reaches the plateau at $\sim$ 96% from 7.5 kV, while
that of the P-120 grade reaches a maximum of $\sim$ 79% at 7.2 kV and then
decreases steadily up to $\sim$ 35% as the HV is increased to 9 kV. The
singles counting rates in both the cases, however, have increased more or less
exponentially with sudden jumps around 6.5-7.0 kV (see Fig. 5), i.e. near the
points where the efficiency becomes uniform (in case of superhylam) or starts
to decrease (in case of P-120). This possibly indicates the onset of a
breakdown regime that recovers in a reasonable time for the superhylam grade
RPC but works the other way for the P-120 grade bakelite RPC. It should,
however, be noted that the singles counting rate and the leakage current of
the superhylam RPC are both larger than those of the P-120 RPC, which are
expected on the basis of smaller bulk resistivity of the superhylam grade
bakelite.

Figure 4: The efficiency as a function of high voltage for two RPCs (silicone
coated & uncoated P-120 and Superhylam grade bakelite) obtained with a gas
mixture of Argon (34%) + Isobutane (7%) + R-134a (59%). The threshold for the
RPCs are set at 40 mV for P-120 and 50mV for superhylam.
Figure 5: Singles counting rate as a function of high voltage.

In order to investigate the reason for the above phenomena, and taking cue
from the fact that superhylam surfaces are glossy finished while the P-120
surfaces are matt finished, we have dismantled the detectors and made surface
profile scans over a 5 mm span of the surfaces using DekTak 117 Profilometer.
These scans, done also for the P-1001 grade, are shown in Fig. 6. It is
clearly seen that the three surfaces have a short range variation (typically
$\sim$ 0.1 $\mu$m length scale) and a long range variation (typically $\sim$ 1
$\mu$m length scale). The long range surface fluctuation, which is a measure
of non - uniformity, averaged over several scans are: 0.84 $\pm$ 0.12 $\mu$m
(P-120), 0.49 $\pm$ 0.17 $\mu$m (superhylam) and 0.88 $\pm$ 0.09 $\mu$m
(P-1001). Thus the long range fluctuations,within the limits of experimental
uncertainties, are nearly the same. On the other hand the short range
fluctuations, a measure of surface roughness, are: 0.64 $\pm$ 0.06 $\mu$m
(P-120), 0.17 $\pm$ 0.02 $\mu$m (superhylam), and 0.63 $\pm$ 0.13 $\mu$m
(P-1001), and thus indicate a superior surface quality of the superhylam
grade.

Figure 6: Linear surface profile scans of the three grades of bakelite sheets.

To explore a remedial measure, we have applied a thin layer of viscous
silicone fluid (chemical formula : [R2SiO]n, where R = organic groups such as
methyl, ethyl, or phenyl) [coefficient of viscosity = 5500 cP, manufactured by
Metroark Limited, Kolkata, India] on the inner surfaces of the P-120 bakelite
sheets. About 1 gm of the fluid is applied over 300 mm $\times$ 300 mm area.
Based on the specific gravity (1.02 at 23∘C) of the fluid, the estimated
coating thickness would be $\sim$ 10 $\mu$m. This material is chosen for the
following reasons: a) very low chemical reactivity with the gases used; b)
good thermal stability over a wide temperature range (from -100 to 250 ∘C); c)
very good electrical insulator; d) excellent adhesion to most of the solid
materials, and e) low vapour pressure, which is essential for stable operation
over a reasonable time period. The silicone treated surfaces are kept under
infrared lamp for 24 hours to allow the viscous fluid to fill all the micro-
crevices on the surface. The reassembled detector is tested at the same set-
up. The results of efficiency and singles count rate measurements, shown in
the Figs. 4 and 5, indicate a remarkable improvement in the performance of the
P-120 detector. The efficiency reaches from 20% to 75% as the HV is increased
from 5.7 kV to 6.2 kV, while the singles count rate, as a whole has decreased
by a factor of 5\. This indicates quenching of micro-discharge after silicone
treatment, which is very much desirable for functioning of the detector. The
efficiency in this case reaches $>$ 95% plateau at 7 kV.

It is worth noting that surface treatment with insulating / non-polar liquid
as a remedial measure was first demonstrated for the BaBar RPCs. However,
formation of stalagmites by polymerisation of uncured linseed oil droplets had
created conducting paths through the gap, thereby causing irreversible damage
to the bakelite plates [22]. The process of linseed oil treatment was later
changed by increasing the proportion of eptane as a thinner to produce a
thinner coating (10-30 $\mu$m) on the inner surface [27]. Our observation that
silicone coating of the inner surfaces aides the proper functioning of our
P-120 bakelite RPC detector once again confirms the importance of smooth
surface finish of the inner surfaces.

To judge the improvement in the overall performance of the RPC detector, we
have measured the leakage current through the RPC detector with and without
silicone coating and the plot of these as a function of the applied HV as
shown in the Fig. 7. Both the plots show a common feature that the current-
voltage curves have two distinctly different slopes. While the gas gap behaves
as an insulator in the lower range of applied voltage and hence the slope over
this span scales as the conductance of the polycarbonate spacers, at higher
range of voltage, the gas behaves as a conducting medium due to the formation
of the streamers. Therefore, the slope over this range scales as the
conductance of the gas gap. It is seen that the slope in the higher range of
voltage is much steeper for the RPC without silicone coating and hence it
points to the fact that some sort of uncontrolled streamers are being formed
in the gas gap causing a degradation of the efficiency. This possibly does not
happen in the RPC detector with silicone coating.

Figure 7: Current as a function of the applied voltage for RPC made by P-120
grade bakelite.

We have also examined the effect of discriminator threshold setting on the
efficiency curves of the RPC with silicone treated surfaces. These are plotted
in the Fig. 8. It is clear that the efficiency curves do not depend much on
the threshold setting from 20 mV to 80 mV, except that the efficiency plateau
is marginally higher at the lowest threshold setting of 20 mV.

Figure 8: Efficiency versus high voltage for different thresholds for silicone
coated P-120 grade bakelite RPC with silicone coating.

The effect of humidity on the efficiency curves has also been studied. This
measurement has been done at relative humidities of 58% and 67% of the
laboratory environment and at the same room temperature of $\sim$ 23∘C. These
curves, plotted in the Fig. 9, indicate no effect of humidity on the
efficiency. However, the leakage currents, measured simultaneously and plotted
in the Fig. 10, are a bit larger at higher humidity. This observation
indicates that charge leakage through the exterior surfaces may be
contributing more at higher humidity.

Figure 9: Efficiency versus high voltage for different humidities for silicone
coated P-120 grade bakelite RPC.
Figure 10: Current versus high voltage for different humidities for silicone
coated P-120 grade bakelite RPC.

The long term stability of the bakelite RPCs has been studied using the same
cosmic ray test set-up. The coincidence trigger counts of the RPCs and the
master trigger counts, accumulated over every 2 hours, have been recorded
continuously for more than 6 months. The room temperature and the relative
humidity have been controlled to keep them less than 24∘C and 80%,
respectively. When the humidity was larger, particularly during the monsoon
season, the test set-up was shut down till it came down to 80% and below. The
singles count rates of the RPCs have also been recorded simultaneously. The
Figs. 11 and 12 depict the variation of efficiency and singles count rates
over the above mentioned period for both the grades of RPCs. The superhylam
grade RPC has worked with an efficiency of $>$ 95% which remained steady for
25 days, but beyond that, it deteriorated gradually to $\sim$ 86% efficiency
within next 13 days. The singles count rate, however, has increased from day
one from 1 Hz/cm2 to 10 Hz/cm2 within 10 days, and then it increased slowly
over the next 28 days. After that period, the singles count rate shot up to
$>$ 30 Hz/cm2. The leakage current gradually increased from 3-4 $\mu$A to $>$
10 $\mu$A within that period. This RPC was discontinued after 38 days and the
silicone coated P-120 grade RPC was then mounted. The efficiency measured was
$\sim$ 96% and above and has remained steady for more than 110 days. The
singles count rate also has remained steady around 0.1 Hz/cm2. The leakage
current was found to be marginally dependent on temperature and humidity,
though it has remained steady at $\sim$ 400 nA during the operation.

Figure 11: Efficiency as a function of time for two RPC prototypes.
Figure 12: The single counting rate as a function of time for the two RPC
prototypes.

The superhylam grade RPC has also been tested again after a gap of a few
months. It has shown the same higher leakage current ($>$ 10 $\mu$A) and lower
efficiency ($\sim$ 86%) indicating that some intrinsic breakdown of the bulk
material may have taken place.

6\. Conclusion

We have made a comparative study of bakelite RPCs made from two different
grades of bakelite. The RPC, made of superhylam grade bakelite with melamine
coated glossy finished surface is found to have a shorter life. On the other
hand, the RPC made from P-120 grade bakelite with matt finished surfaces,
which are coated with a thin layer of viscous silicone fluid, are found to
work steadily for more than 110 days showing a constant efficiency of $>$ 96%
without any degradation. The detector is found to be less immune to variation
in humidity which makes it a viable alternative to semiconductive glass based
RPC.

7\. Acknowledgement

We are thankful to Prof. Naba Kumar Mandal of TIFR, India and Prof. Kazuo Abe
of KEK, Japan for their encouragement and many useful suggestions in course of
this work. We are also grateful to Dr. C. Bhattacharya, Mr. G.S.N. Murthy, Mr.
M.R. Dutta Majumdar, Mr. S.K. Thakur and Mr. S.K. Bose of VECC for their help
in the work. We acknowledge the service rendered by Mr. Avijit Das of SINP for
surface profile scans of the bakelite sheets used by us. We would like to
thank the SINP workshop for making the components of the detectors, and Mr.
Ganesh Das of VECC for meticulously fabricating the detectors. Finally we
acknowledge the help received from the scientific staff of Electronics
Workshop Facility of SINP for building the gas flow control and delivery
system of the gas mixing unit used in this study.

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[orcid=0000-0002-9312-8569] [1]

[orcid=0000-0001-7750-1522]

[1]Corresponding author

1]organization=Department of Physics, Florida International University,
city=Miami, citysep=, postcode=33199, state=FL, country=USA

2]organization=Department of Physics and Astronomy, Ghent University,
city=Gent, postcode=B9000, country=Belgium

# Phase-space distributions of nuclear short-range correlations

W. Cosyn<EMAIL_ADDRESS>J. Ryckebusch<EMAIL_ADDRESS>[  [

###### Abstract

Nuclear short-range correlations (SRCs) induce high-momentum/high-energy
fluctuations in the nuclear medium. In order to assess their impact on nuclear
bulk properties, like nuclear radii and kinetic energies, it is instrumental
to determine how SRCs are distributed in phase space as this sheds light on
the connection between their appearance in coordinate and momentum space.
Using the lowest-order correlation operator approximation (LCA) to include
SRC, we compute two-dimensional nuclear Wigner quasiprobability distributions
$w(r,k)$ to locate those $({r},{k})$ phase-space regions that are most heavily
impacted by SRCs. The SRC-induced high-momentum components find their origin
in a radial range that is confined to the nuclear interior. Significant SRCs
strength is generated in the full momentum range $0\leq k\lesssim
5~{}\text{fm}^{-1}$ covered in this work, but below the Fermi momentum those
are dwarfed by the mean-field contributions. As an application of $w(r,k)$, we
focus on the radial dependence of the kinetic energy $T$ and the momentum
dependence of the radius $r_{\text{rms}}$ for the symmetric nuclei
${}^{12}\text{C}$, ${}^{40}\text{Ca}$ and the asymmetric nucleus
${}^{48}\text{Ca}$. The kinetic energy almost doubles after including SRCs,
with the largest increase occurring in the nuclear interior $r\lesssim 2$ fm.
The momentum dependence of the $r_{\text{rms}}$ teaches that the largest
contributions stem from $k\lesssim 2$ fm-1, where the SRCs induce a slight
reduction of the order of a few percent. The SRCs systematically reduce the
${}^{48}\text{Ca}$ neutron skin by an amount that can be 10%.

###### keywords:

nuclear short-range correlationsWigner distributions nuclear radii

##  1 Introduction

The size of an atomic nucleus [1, 2] and how protons and neutrons are
spatially arranged for various proton-to-neutron ratios [3, 4, 5, 6] are
topics of continued great interest in the precision era of nuclear physics.
Detailed nuclear-structure studies have shown that long-range correlations
connected with core-breaking effects have a substantial impact on computed
proton and neutron nuclear radii and are sources of uncertainties in advanced
nuclear-structure calculations. For example, a systematic study for
${}^{48}\text{Ca}$ [1] indicated that in ab initio theory with a family of
modern chiral effective forces, the variations in the computed proton and
neutron radii can be of the order of 10%. The impact of short-range
correlations (SRCs) on bulk nuclear properties like radii is not that well
known and has recently been addressed in [7]. In that paper, qualitative
arguments are developed as to why the omission of SRCs may have a non-
negligible impact on the computed radii of neutron-rich nuclei and it was
suggested that more quantitative calculations are in order.

Wigner distributions [8, 9] provide a distinct view on the spatial and
momentum structure of quantum systems and are widely applicable including in
subatomic physics [10]. It is a subject of great interest in non-perturbative
quantum chromodynamics (QCD) [11, 12, 13, 14, 15, 16, 17, 18]. The QCD Wigner
distribution provides information about the joint position-momentum
distributions of partons in the nucleon and as such can be considered as the
mother partonic distribution from which all others can be derived. Selected
results of Wigner distributions for finite nuclei have been presented in Refs.
[19, 20, 21, 22]. A recent calculation using realistic potentials highlighted
the influence of SRCs on the deuteron’s Wigner distribution [23].

In this work we aim at providing a study of the position-momentum structure of
SRCs by presenting calculations of Wigner distributions for finite nuclei. The
SRCs have been connected with fat momentum tails in the nuclear momentum
distributions [24, 25, 26, 27]. The spatial structure of the SRCs in finite
nuclei has received less attention and will be one of the topics of discussion
here. To quantify the impact of SRCs, we use the lowest-order correlation
operator approximation (LCA) [28, 29, 30] that is based on a number of
assumptions: (i) the scale separation between the long-range and short-range
nuclear correlations; (ii) the universal local character that make SRCs a
property that can be imposed on the mean-field behavior through the operation
of universal operators [31, 32, 33]. The LCA shares these assumptions with
alternate theoretical approaches to quantify the impact of SRCs, including the
generalized contact formalism (GCF) [34, 35]. As LCA shifts the complexity
from the wave functions to the operators, it can be used for SRCs estimation
in nuclear structure and nuclear reaction applications. Since the LCA
formalism does not account for long-range correlations, our focus is on the
relative contribution of SRCs to nuclear bulk properties like point-nucleon
radii and kinetic energies. We deem that our calculations serve as an
important comparative benchmark for highlighting the impact of SRCs on nuclear
bulk properties.

The kinetic energy is connected with the momentum structure whereas the radii
are connected with the spatial structure. With the Wigner distributions one
gains access to the momentum structure of radii and the spatial structure of
the kinetic energy, and how those are impacted by SRCs. Alternate approaches
[35] have addressed the SRCs in both coordinate and momentum space. Wigner
distributions provide a unique window of insight into the phase-space
distributions as they provide information in both variables simultaneously.

In what follows, we first develop a formalism to compute Wigner distributions
that include the effect of SRCs. We then proceed with the presentation of the
results of numerical calculations for the separated proton and neutron Wigner
distribution for the nuclei ${}^{12}\text{C}$, ${}^{40}\text{Ca}$ and
${}^{48}\text{Ca}$. Those distributions form the basis to elucidate the phase-
space dependence of SRCs in the proton and neutron radii and kinetic energies.

##  2 Formalism

The Wigner distribution is the central quantity of interest in this work. The
quantum Wigner operator has the following spectral decomposition in three-
dimensional coordinate ($\equiv\bm{r}$) or momentum ($\equiv\bm{k}$) space:

$\widehat{w}(\bm{r},\bm{k})=\frac{1}{(2\pi)^{3}}\int
d\bm{x}\;e^{i\bm{k}\cdot\bm{x}}\;\Bigl{|}\bm{r}-\frac{\bm{x}}{2}\Bigr{\rangle}\Bigl{\langle}\bm{r}+\frac{\bm{x}}{2}\Bigr{|}\\\
=\frac{1}{(2\pi)^{3}}\int
d\bm{q}\;e^{-i\bm{q}\cdot\bm{r}}\;\Bigl{|}\bm{k}+\frac{\bm{q}}{2}\Bigr{\rangle}\Bigl{\langle}\bm{k}-\frac{\bm{q}}{2}\Bigr{|}\,.$
(1)

In the coordinate space representation, $\bm{k}$ is Fourier conjugate to the
relative coordinate $\bm{x}$ of the off-diagonal spatial density, while
$\bm{r}$ is Fourier conjugate to the relative momentum $\bm{q}$ in the off-
diagonal momentum density. The Wigner quasiprobability distribution for a pure
state $\Psi$ is the expectation value of the $\widehat{w}(\bm{r},\bm{k})$:

$w(\bm{r},\bm{k})=\langle\Psi|\,\widehat{w}(\bm{r},\bm{k})\,|\Psi\rangle.$ (2)

The integral of the Wigner operator over $\bm{r}$ $(\bm{k})$ yields the
momentum (spatial) density operator

$\displaystyle\hat{n}(\bm{k})=\int
d\bm{r}\;\widehat{w}(\bm{r},\bm{k})=|\bm{k}\rangle\langle\bm{k}|\,,$ (3)
$\displaystyle\hat{\rho}(\bm{r})=\int
d\bm{k}\;\widehat{w}(\bm{r},\bm{k})=|\bm{r}\rangle\langle\bm{r}|\,.$ (4)

Expectation values of operators $\widehat{F}$ can be written as the phase-
space average of the Wigner distribution

$\langle\widehat{F}\rangle=\iint
d\bm{r}d\bm{k}\;w(\bm{r},\bm{k})f(\bm{r},\bm{k})\,,$ (5)

where the Wigner transform $f(\bm{r},\bm{k})$ of $\widehat{F}$ is defined as

$\displaystyle f(\bm{r},\bm{k})$ $\displaystyle\equiv\int
d\bm{x}\;e^{-i\bm{k}\cdot\bm{x}}\;\Bigl{\langle}\bm{r}-\frac{\bm{x}}{2}\Bigr{|}\,\widehat{F}\,\Bigl{|}\bm{r}+\frac{\bm{x}}{2}\Bigr{\rangle}\,,$
$\displaystyle=\int
d\bm{q}\;e^{i\bm{q}\cdot\bm{r}}\;\Bigl{\langle}\bm{k}+\frac{\bm{q}}{2}\Bigr{|}\,\widehat{F}\,\Bigl{|}\bm{k}-\frac{\bm{q}}{2}\Bigr{\rangle}\,.$
(6)

In this work, we assume spherical symmetry and present results for the
$w(r,k)$ which are obtained after integrating $w(\bm{r},\bm{k})$ of Eq. (2)
over the solid angles of the spatial and momentum coordinates

$w(r,k)\equiv\iint d\Omega_{\bm{r}}\,d\Omega_{\bm{k}}\;w(\bm{r},\bm{k})\,.$
(7)

The Wigner distribution $w(r,k)$ is the quasiprobability distribution in the
nucleon’s radial coordinate and momentum. It is normalized to the number of
nucleons $A$

$\int r^{2}dr\int k^{2}dk\;w(r,k)=A\,.$ (8)

An outline of the derivation of the Wigner distribution in the LCA is given in
App. A. Through the introduction of SRCs operators, many-body operators
between Slater-determinant states are generated. In LCA, the many-body
operators generated from one-body operators are truncated at the level of two-
body operators and the $w(r,k)$ can be separated in a proton and a neutron
part by considering the four isospin pair combinations (see Eq. (16))

$w(r,k)=\left[w_{pp}(r,k)+w_{pn}(r,k)\right]+\left[w_{nn}(r,k)+w_{np}(r,k)\right]\,.$
(9)

Hereby, the $w_{pn}$ has two categories of contributions. The first category
stems from an uncorrelated proton and neutron that are both described as
quasi-particles in the mean field. The second category is the SRCs one whereby
the tagged proton and neutron are correlated through one or a product of two
correlation operators. Similar discussions hold for the other three pair
combinations.

To obtain the nuclear rms radius and kinetic energy, we consider scalar
operators $\hat{r}^{2},\hat{k}^{2}$ which have Wigner transforms $r^{2},k^{2}$
respectively. We can use Eq. (5) to extract the radial dependence of the
kinetic energy operator in coordinate space, and that of the rms radius in
momentum space. The first method permits to calculate the quasi-expectation
value of $\hat{T}=\hat{k}^{2}/2m$ ($\hat{r}^{2}$) at a given position
(momentum)

$\displaystyle T(r)=\Bigl{<}\widehat{T}(r)\Bigr{>}=\frac{\int
k^{2}dk\;\tfrac{k^{2}}{2m}\;w(r,k)}{\int k^{2}dk\;w(r,k)}\,,$ (10)
$\displaystyle
r_{\text{rms}}(k)\equiv\sqrt{\langle\hat{r}^{2}(k)\rangle}=\sqrt{\frac{\int
r^{2}dr\;r^{2}w(r,k)}{\int r^{2}dr\;w(r,k)}}\,.$ (11)

Due to quantum effects in the Wigner distribution, these quasi-expectation
values of positive definite operators can be negative. These variables allow
to infer the magnitude of the nucleon kinetic energy at a certain $r$, or the
size of the nuclear rms radius with a certain $k$. Because of the $r$-($k$-)
dependence in both numerator and denominator, the $T(r)$
($r^{2}_{\text{rms}}(k)$) do not integrate to the full $T$
($r^{2}_{\text{rms}}$).

A distribution that is both reminiscent of the spatial structure of the
kinetic energy (momentum structure of the nuclear radius) and provides the
proper scalar quantity after integration, can be obtained in the second method
that is based on the computation of the densities $\rho_{T}(r)$ and
$\rho_{r^{2}}(k)$:

$\displaystyle\rho_{T}(r)\equiv\frac{r^{2}\int\,k^{2}dk\;\tfrac{k^{2}}{2m}\;w(r,k)}{\int
r^{2}dr\int k^{2}dk\;w(r,k)}\,,$ $\displaystyle\int
dr\rho_{T}(r)=\bigl{<}\widehat{T}\bigr{>}=T\,;$ (12)
$\displaystyle\rho_{r^{2}}(k)\equiv\frac{k^{2}\int r^{2}dr\;r^{2}w(r,k)}{\int
k^{2}dk\int r^{2}dr\;w(r,k)}\,,$ $\displaystyle\int
dk\rho_{r^{2}}(k)=\langle\hat{r}^{2}\rangle=r_{\text{rms}}^{2}\,.$ (13)

The $\rho_{T}(r)$ encodes the contribution to the kinetic energy at given
nucleon radial coordinate $r$ and was also considered in Ref. [20].
Analogously, $\rho_{r^{2}}(k)$ encodes the contribution to the nuclear radius
squared at given momentum $k$. In this work we assess the impact of SRCs on
the $T(r)\,(r_{\text{rms}}(k))$ and $\rho_{T}(r)\,(\rho_{r^{2}}(k))$, each
offering complementary insight in what actually happens to bulk nuclear
properties after including high-momentum/high-energy fluctuations in a model.

##  3 Results

The LCA has the following inputs: (i) a set of universal strength correlation
functions $f_{p}(r_{12})$ entering the matrix elements of Eq. (17); (ii) the
HO frequency for which we adopt a global parameterization of the form

$\hbar\omega=c_{1}A^{-\frac{1}{3}}-c_{2}A^{-\frac{2}{3}}\,.$ (14)

As in previous publications [28, 29, 30], we use the Argonne VMC correlation
functions [36] for the tensor and spin-isospin correlation functions, and use
two options for the central correlation function $f_{c}(r_{12})$: a hard one
that is computed with the aid of the Reid potential (denoted $f_{c}[R]$) [37],
and the softer Argonne correlation function ($f_{c}[V]$) [36]. We stress that
with the $f_{c}[R]$ we obtain momentum distributions that are very similar to
those obtained in ab initio calculations [29]. Furthermore, the choice for
$f_{c}[R]$ was a data driven one, as 12C$(e,e^{\prime}pp)$ data could be
described with this choice [38]. For the HO frequency of Eq. (14), we have in
previous LCA works systematically used the “default” values
$c_{1}=45~{}\text{MeV},c_{2}=25~{}\text{MeV}$, a choice that we refer to as
$\hbar\omega[d]$. As we quantify the effect of SRCs on nuclear radii in this
work and wish to quantify uncertainties stemming from the model parameters, we
also explore other values of $c_{1},c_{2}$ by fitting them to the measured
nuclear rms charge radii of ${}^{4}\text{He}$, ${}^{9}\text{Be}$,
${}^{12}\text{C}$, ${}^{16}\text{O}$, ${}^{27}\text{Al}$, ${}^{40}\text{Ca}$,
${}^{48}\text{Ca}$, ${}^{56}\text{Fe}$, ${}^{108}\text{Ag}$,
${}^{197}\text{Au}$ and ${}^{208}\text{Pb}$ [39]. This was done using a
standard minimum $\chi^{2}$ fit. These inputs carry the label
$\hbar\omega[f]$. In this work, the IPM corresponds with the HO model that can
be formally reached after setting all correlation operators equal to zero in
LCA. We consider this HO model as the benchmark against which to measure the
impact of SRCs.

Figure 1: Top: Ratio of model charge radii $r_{c}$ (including the point rms radius correction of [1]) and data [39]. Bottom: One-body momentum distribution $n(k)$ for 48Ca as computed in LCA with the four model variants detailed in Table 1.  Table 1: Summary of the model inputs used in this work. Left column labels are explained in the text. Label | $f_{c}$ |  $c_{1}$ [MeV] |  $c_{2}$ [MeV]
---|---|---|---
IPM $\hbar\omega[d]$ | none | 45 | 25
IPM $\hbar\omega[f]$ | none | $40.0\pm 1.2$ | $2.8\pm 5.1$
LCA $\hbar\omega[d]/f_{c}[R]$ | Reid | 45 | 25
LCA $\hbar\omega[f]/f_{c}[R]$ | Reid | $36.7\pm 1.1$ | $4.1\pm 4.8$
LCA $\hbar\omega[d]/f_{c}[V]$ | VMC | 45 | 25
LCA $\hbar\omega[f]/f_{c}[V]$ | VMC | $36.5\pm 0.4$ | $0.01\pm 1.0$

As a robustness check and sensitivity analysis, we consider two choices for
the two inputs to LCA. Together with the two HO frequency variants of the IPM
we are left with six models that are listed in Table 1. In Fig. 1 we compare
the computed charge radii to data for these models. As we compute point-
nucleon radii, corrections (see Ref. [1]) were applied before comparing to
data. Note that the fits produce $c_{1}$ that are somewhat smaller than the
default value and a small $c_{2}$. We stress that the variation of
$\hbar\omega$ across the various model variants does not exceed 13%. In Fig.
1, we see that the IPM $\hbar\omega[d]$ model provides reasonable agreement
with the measured rms radii. The LCA calculations with those HO parameters
produce rms radii that are a few percent smaller whereby the strongest impact
of the SRCs is observed for the LCA model that uses a hard central correlation
function. For medium and heavy nuclei, the fitted IPM and LCA $\hbar\omega[f]$
parameterizations yield almost identical charge radii, in overall good
agreement with the data. For the light nuclei there are larger deviations
between the predicted radii obtained with the $\hbar\omega[d]$ and
$\hbar\omega[f]$ parameters. With the $\hbar\omega[f]$ HO frequency the
measured radius for ${}^{12}\text{C}$ can be reproduced. The larger deviations
between computed and measured radii for 4He and 9Be can be attributed to the
absence of long-range effects in the LCA. For these light nuclei, also the
center-of-mass (c.o.m.) corrections can be sizable [40].

The numerical cost of LCA scales polynomially with mass number $A$ which makes
it applicable throughout the mass table. In this work, our focus is on
${}^{12}\text{C}$, ${}^{40}\text{Ca}$, and ${}^{48}\text{Ca}$. The latter
nucleus allows us to assess the SRC effects in asymmetric nuclei. For
${}^{48}\text{Ca}$, the bottom panel in Fig. 1 shows the LCA one-body momentum
distribution for the four model variants. Contrary to the radii, the choice of
HO frequency has an almost negligible effect on the momentum distribution. The
choice for the central correlation function, on the other hand, mainly affects
the momentum distribution for momenta larger than half the nucleon mass
$k\gtrsim 3~{}\text{fm}^{-1}$. As reported in Refs. [29, 30], as relatively
little strength is present at those momenta this difference results in a
variation of the order of a few percent in comparisons with quantities
extracted from electron scattering data. For the LCA momentum distributions
for ${}^{12}\text{C}$ and ${}^{40}\text{Ca}$ (shown in Ref. [29]), similar
remarks hold for the sensitivity to the two input sources as for
${}^{48}\text{Ca}$.

Over the last decade, constraints of the SRCs models have been improved
through the increased availability of two-nucleon knockout data from proton-
and electron-nucleus experiments in selected kinematics [41, 42, 43, 44, 45,
46, 47]. Thanks to its flexibility, the LCA model could be well tested against
results for the isospin [29] and mass [48, 30] dependence of SRCs. The LCA
provides a good basis for accurate SRCs modeling across the nuclear mass
table. As an illustration of this we mention the so-called $a_{2}$ scaling
factors that are extracted from inclusive electron-nucleus data and can be
connected to the aggregated effect of SRCs in nucleus $A$ relative to the
deuteron. A recent 48Ca$(e,e^{\prime})$ / 40Ca$(e,e^{\prime})$ measurement has
addressed the isospin structure of SRCs [49] and provided the result $0.971\pm
0.012$ for the $a_{2}(^{48}\text{Ca})/a_{2}(^{40}\text{Ca})$ ratio of measured
cross sections per nucleon. Loosely speaking, this implies that per nucleon
there is about 3% less impact from SRCs in 48Ca as compared to 40Ca. The LCA
predictions for this quantity (that were published before the data [30]) can
be extracted from the high-momentum tails of the LCA momentum distribution for
${}^{40}\text{Ca}$ and ${}^{48}\text{Ca}$. The computed numbers for the 48Ca /
40Ca ratio are 4.89/4.99=0.98 (see Table I of Ref. [30]) which is in close
agreement with the data.

Figure 2: Two-dimensional proton and neutron Wigner distributions $w(r,k)$ for
${}^{12}\text{C}$, ${}^{40}\text{Ca}$, and ${}^{48}\text{Ca}$ as computed in
the IPM and the LCA using the $\hbar\omega[d]/f_{c}[R]$ input. For the two
symmetric nuclei, the proton (p) and neutron (n) result are identical. Mean-
field results (top row); full LCA results (middle row); SRCs contribution to
LCA results (bottom row). For each of the 12 distributions, the white vertical
(horizontal) line denotes the corresponding $r_{\text{rms}}$
($k_{\text{rms}}$). For the IPM and full LCA results (top and middle rows) the
$w(r,k)$ is normalized to the total number of protons/neutrons (6/6 for
${}^{12}\text{C}$; 20/20 for ${}^{40}\text{Ca}$; 20/28 for
${}^{48}\text{Ca}$), while the normalization of the bottom panel is equal to
the LCA prediction for the number of SRCs protons/neutrons (1.8/1.8 for
${}^{12}\text{C}$; 7.6/7.6 for ${}^{40}\text{Ca}$; 8.0/9.9 for
${}^{48}\text{Ca}$).

Figure 2 shows the results for the two-dimensional Wigner distribution
$w(r,k)$. These LCA numerical results were subjected to several checks. One-
body momentum distributions $n(k)$ obtained using Eq. (3) were confronted with
those from direct computation [30]. Similarly, the rms radii obtained with Eq.
(13) that involves the $w(r,k)$ were compared with those obtained through
direct calculations with the rms radius operator. Finally, the normalization
of the four isospin pair combinations $(pp,nn,np,pn)$ of the LCA $w(r,k)$ of
Eq. (9) are constrained by $Z$ and $N$.111 Note that for these normalizations
this is not a trivial result, as we expand the denominator of the operator
matrix elements to the same lowest order in the LCA and no normalization is
artificially imposed. We find sub-percent deviations that can be attributed to
the truncation of the summation over the quantum numbers
$\left(U,M_{U}\right)$ in Eq. (19) and the introduction of finite grid sizes
in $(r,k)$. Note that in computing matrix elements with $w(r,k)$ one
multiplies it with $r^{2}k^{2}$, see for example Eqs. (12) and (13).

Inspecting the results of Fig. 2, one observes that in IPM the $w(r,k)$
extends over the entire radial range and over a well-constrained $k$-range
that is almost identical for all three nuclei considered. The SRCs generate a
fat momentum tail in $w(r,k)$ that is mainly confined to the interior of the
nucleus. Correspondingly, the high-momentum SRC contributions are distributed
in a narrower $r$ range than the mean-field contributions. Indeed, the weight
of the fat tails diminishes with increasing $r$, with the largest weight at
$r<r_{\text{rms}}$, and hardly any high-momentum components for $r\gtrsim
2r_{\text{rms}}$. This is a reflection of the fact that in LCA the SRCs are
chiefly generated from correlation operators acting on IPM nodeless relative
$S$-pairs [50]. Obviously, the wave functions for these $S$-pairs have a
finite density at relative $r_{12}=0$ and are very much confined to the
nuclear interior. The panels in the bottom row of Fig. 2 consistently show
that the strength of the SRCs contribution to the LCA $w(r,k)$ shifts to
smaller $r$ compared to the IPM. Note that the use of the term _SRC_ does not
imply a momentum cut here. There are significant contributions to the SRCs
part from $k<k_{F}$ as the bottom row in Fig. 2 shows. The normalization of
the SRC part (1.8/6 nucleons for ${}^{12}\text{C}$) cannot be directly
connected with experimental measures for the number of SRC pairs as those
exclusively refer to high-momentum SRCs. The maximum of the Wigner
distribution (and the rms radius) for the SRCs contribution sits at smaller
values of $r$ in comparison to the IPM and LCA ones. We verified that SRCs
make up almost 100% of the LCA result for $k>2~{}\text{fm}^{-1}$ at any $r$.
As mentioned earlier the absolute SRC contribution diminishes strongly for
large $r$ at these higher momenta. This might raise the question to what
extent the SRCs in the interior can be probed in scattering experiments, where
the ejected particles are subject to strong final-state interactions (FSI)
while traversing the nuclear medium. A previous study [51] has shown, however,
that reactions probing correlated nucleon pairs are still sensitive to the
nuclear interior, even after correcting for FSI effects. This is in contrast
to single-nucleon knockout reactions which become much more surface dominated
after FSI corrections.

For ${}^{12}\text{C}$, the two-dimensional Wigner distributions $w(r,k)$ in
Fig. 2 has a distinctive negative region for small $r$ and $k$, illustrative
of quantum effects. For the two calcium isotopes, the area with $w(r,k)<0$ is
also located at small $r$ and $k$. The proton $w(r,k)$ are almost identical in
${}^{40}\text{Ca}$ and ${}^{48}\text{Ca}$ (note the log scale), whereas for
neutrons in ${}^{48}\text{Ca}$ the emergence of a surface-located neutron skin
is clearly visible. In the forthcoming, it will be shown that the relative
impact of SRCs on the proton and neutron kinetic energies differs in an
asymmetric nucleus [52, 53, 45, 29]. Compared to the deuteron results for
$w(r,k)$ of Ref. [23], the fat momentum tails in LCA for finite nuclei extend
over a larger momentum range and do not display the oscillations. These
differences are likely attributed to the smoothing effect from pair c.o.m.
motion and the fact that many pairs are contributing.

Figure 3: Kinetic energy and rms radius of 12C. Top row: momentum dependence
of rms radius (all nucleons). Bottom row: radial dependence of kinetic energy
per nucleon (all nucleons). The left column shows the expectation value of the
operator at a specific $r$ or $k$ [Eqs. (10), (11)]. The right column plots
the density of the corresponding observable in $r$ or $k$ [Eqs. (12), (13)].
All curves use the $\hbar\omega[d]$ HO inputs. “LCA SRC” curves show the
contribution from SRCs to the “LCA total” result. The difference between the
two is not equal to the IPM curve because both the IPM and LCA total result
use distributions normalized to the total number of nucleons. The grey
vertical bands in the lower row cover the range of point rms radii for the
depicted model inputs, see Table 2.

We now discuss the numerical results for the Wigner-based functions that are
introduced to investigate the spatial dependence of the kinetic energy ($T(r)$
and $\rho_{T}(r)$ of Eqs. (10) and (12)) and the momentum dependence of the
nuclearradii ($r_{\text{rms}}(k)$ and $\rho_{r^{2}}(k)$ of Eqs. (11) and
(13)). In what follows, we systematically compare IPM with LCA results and we
stress that both are obtained for identical normalizations of the $w(r,k)$.
The ${}^{12}\text{C}$ results for the $\hbar\omega[d]$ model variants of Table
1 are shown in Figure 3. Similar results are obtained with the
$\hbar\omega[f]$ HO frequency. The SRCs substantially increase the
$T(r\lesssim 4~{}\text{fm})$ and $\rho_{T}(r\lesssim 4~{}\text{fm})$, whereby
the effect gradually decreases with growing radial distance $r$. For $r>4$ fm,
the IPM and LCA $T(r)$ and $\rho_{T}(r)$ coincide, showing again the minor
role played by SRCs beyond the high-density regions in the nuclear interior.
The uncertainty on $T(r)$ connected with the choice of the central correlation
function is largest for $r=0$ and is of the order of 20% on the SRC
contribution and 10% on the LCA result. For $r<r_{\text{rms}}$ the SRCs
account for roughly 60-70% of the kinetic energy. The $\rho_{T}(r)$
demonstrates that the kinetic energy receives significant contributions from
$r>r_{\text{rms}}$. For the rms radius, the overall impact of the SRCs is far
more modest than what is observed for the kinetic energies. The LCA
predictions for $\rho_{r^{2}}(k)$ are below the IPM one except for
$k>4~{}\text{fm}^{-1}$. Overall this implies smaller radii when including SRCs
using the same HO frequency, as was shown in Fig 1. From the $\rho_{r^{2}}(k)$
one infers that both the IPM and LCA contributions to $r_{\text{rms}}$ from
$k>2~{}\text{fm}^{-1}$ are very small despite the fact that the LCA
$\rho_{r^{2}}(k)$ is much larger than the IPM one for high momenta. We look at
these high-momentum contributions in more detail in Fig. 6.

Figure 4: ${}^{40}\text{Ca}$ and ${}^{48}\text{Ca}$ radial dependence of
kinetic energy expectation value $T(r)$ (top row) and density $\rho_{T}(r)$
(bottom row). All LCA results use the $\hbar\omega[d]/f_{c}[R]$ input, IPM
uses $\hbar\omega[d]$. Proton and neutron results are identical in
${}^{40}\text{Ca}$. Normalization of “LCA SRC” as in Fig. 3. Legends
(linestyle/color) apply to all panels. The vertical bands cover the range of
point rms radii for the depicted models, see Table 2. Figure 5: As in Fig. 4
but now for the momentum dependence of the $r_{\text{rms}}(k)$ expectation
value (top row) and $\rho_{r^{2}}(k)$ density (bottom row).

In Figs. 4 and 5, IPM and LCA results for the combinations
$\left[T(r),\rho_{T}(r)\right]$ and
$\left[r_{\text{rms}}(k),\rho_{r^{2}}(k)\right]$ for ${}^{40}\text{Ca}$ and
${}^{48}\text{Ca}$ are shown. Our major focus here is how SRCs impact the
proton and neutron features and the role of the neutron excess in
${}^{48}\text{Ca}$. To this end we discuss the numerical results of the IPM
model with the $\hbar\omega[d]$ HO frequency and compare them with results of
the LCA model with the $\hbar\omega[d]/f_{c}[R]$ input. In general, similar
trends as for the ${}^{12}\text{C}$ results of Fig. 3 are observed. Comparing
the radial dependence of the kinetic energy $T(r)$ for the protons and
neutrons in ${}^{48}\text{Ca}$ in Fig. 4, we can make the following
observations. In the IPM neutrons possess more kinetic energy than protons for
all $r$. For the SRC contribution to the LCA result, the situation is
reversed. As far as the SRCs contributions are concerned, protons are more
kinetically energetic for all $r$. The weighted sum of the IPM and SRCs
contributions results in the radial dependence of the LCA $T(r)$. In the deep
interior $r\lesssim 1.8$ fm of the nucleus, where the SRC effects are in full
swing, protons have more kinetic energy, while for $r\gtrsim 1.8$ fm neutrons
have more. The difference is very small, and in the deep surface region $r>6$
fm there is no effect from the SRC. The ${}^{48}\text{Ca}$ $\rho_{T}(r)$ shows
that counted per nucleon the IPM protons in the nuclear interior contribute
more to the bulk kinetic energy than the neutrons, while the situation is
reversed in the exterior. In the LCA, the first effect is enhanced while the
second is reduced. The combination of these effects results in more
kinetically energetic protons than neutrons after including SRCs (see Table
2).

In line with the carbon results of Fig. 3, the radius $r_{\text{rms}}(k)$ of
the calcium isotopes in Fig. 5 reaches a plateau at $k\gtrsim
2~{}\text{fm}^{-1}$. The ${}^{48}\text{Ca}$ neutron radii are larger than the
proton ones for momenta below the Fermi one, but in the momentum region
dominated by the tensor correlations ($1.5~{}\text{fm}^{-1}\lesssim k\lesssim
2.5~{}\text{fm}^{-1}$) the LCA proton and neutron radii almost coincide. As
with ${}^{12}\text{C}$, the $\rho_{r^{2}}(k)$ indicate that high-momentum
nucleons only contribute marginally to the rms radius, even in the LCA.

The resulting kinetic energies $T$ and rms radii $r_{\text{rms}}$ are
summarized in Table 2 for the different model variants. The mentioned values
are directly obtained through phase-space integration of the $w(r,k)$ (see
Eqs. (12) and (13)). The 40Ca results can be compared to those of recent ab
initio calculations (Tables V and VI of [27]). The calculations with the AV18
two-body force lead to $T=32.29~{}\text{MeV}$ and
$r_{\text{rms}}=3.41~{}\text{fm}$. In LCA we find values of $T$ and
$r_{\text{rms}}$ that are slightly smaller but the deviation is at most 10%.
We stress that in LCA we do not have long-range correlations. In that light it
is important to note that the results of [27] indicate that three-body forces
decrease the $T$ by about 1 MeV and increase the $r_{\text{rms}}$ by about 0.1
fm.

For ${}^{48}\text{Ca}$, one observes that all LCA model variants
systematically predict that the proton kinetic energy is about 1 MeV larger
than the neutron one, a phenomenon known as “kinetic energy inversion” [53].
For the rms radii, the LCA result shows a slight reduction of the different
radii for the $\hbar\omega[d]$ inputs, while SRC connected changes in the size
of the ${}^{48}\text{Ca}$ neutron skin are of the order 5-10%. Note that the
kinetic energy inversion for protons and neutrons in ${}^{48}\text{Ca}$ cannot
be directly inferred from the shapes of the $w(r,k)$ of Fig. 2. Indeed, the
difference between ${}^{48}\text{Ca}$ proton and neutron values for
$k_{\text{rms}}$ (which is related to $\langle T\rangle$ as $\langle
T\rangle=k_{\text{rms}}^{2}/2m$) is hardly visible in Fig. 2.

Overall, the impact of the SRCs relative to the IPM result for the radii and
kinetic energies is rather insensitive to the input choices of the
calculations. For the radii the largest uncertainties stem from the choice of
the HO parameter. As illustrated by the $\hbar\omega[d]$ results that use a
fixed HO frequency, the impact of SRCs is a reduction of the order 3-6%. The
SRCs systematically reduce the neutron skin. For the kinetic energies, the
impact of the SRCs is large and the uncertainty stemming from the choice for
the central correlation functions on the LCA result is of the order of 2-3
MeV. The choice for the oscillator parameter has a somewhat smaller (1 to 2
MeV) impact on the LCA kinetic energies.

Table 2: Computed kinetic energies per nucleon (in MeV) and point-nucleon rms
radii (in fm). The comparison between the computed and measured charge radii
for the different model variants is displayed in Fig. 1.

| ${}^{12}\text{C}$ | ${}^{40}\text{Ca}$ | ${}^{48}\text{Ca}$
---|---|---|---
Model | $T_{p,n}$ | $r_{p,n}$ | $T_{p,n}$ | $r_{p,n}$ | $T_{p}$ | $T_{n}$ |  $T_{n}$-$T_{p}$ | $r_{p}$ | $r_{n}$ |  $r_{n}$-$r_{p}$
IPM $\hbar\omega[d]$ | 16.1 | 2.46 | 16.5 | 3.36 | 15.7 | 18.0 | 2.2 | 3.44 | 3.68 | 0.237
LCA $\hbar\omega[d]$/$f_{c}[R]$ | 29.7 | 2.34 | 31.9 | 3.17 | 32.5 | 31.6 | -1.0 | 3.27 | 3.48 | 0.216
LCA $\hbar\omega[d]$/$f_{c}[V]$ | 26.8 | 2.40 | 28.9 | 3.22 | 29.5 | 28.7 | -0.8 | 3.31 | 3.53 | 0.221
IPM $\hbar\omega[f]$ | 18.3 | 2.30 | 17.2 | 3.29 | 16.2 | 18.5 | 2.3 | 3.39 | 3.63 | 0.234
LCA $\hbar\omega[f]$/$f_{c}[R]$ | 30.4 | 2.31 | 30.1 | 3.28 | 30.5 | 29.5 | -0.9 | 3.39 | 3.62 | 0.226
LCA $\hbar\omega[f]$/$f_{c}[V]$ | 28.7 | 2.32 | 27.8 | 3.28 | 28.2 | 27.5 | -0.8 | 3.39 | 3.62 | 0.227

Figure 6: Momentum cutoff dependence of rms radii. We show the ratio of point
rms radii calculated with a high-momentum cutoff $\Lambda$ to their
$\Lambda=+\infty$ values. ${}^{12}\text{C}$ and ${}^{40}\text{Ca}$ are shown
in the left panel; ${}^{48}\text{Ca}$ proton, neutron and neutron skin (n$-$p)
values in the right panel. IPM curves use the $\hbar\omega[d]$ input, LCA
curves use $\hbar\omega[f]/f_{c}[R]$. Other IPM/LCA inputs produce similar
curves.

So far, no momentum selection was imposed when investigating the influence of
SRCs on bulk nuclear properties. In order to keep the dimensionality of the
numerical _ab-initio_ calculations in check one often relies on softened
nucleon-nucleon interactions that involve a momentum cut-off. In light of
this, we wish to quantify the influence of SRCs on the nuclear radii as a
function of a momentum cut-off. To this end, we introduce a momentum cutoff
$\Lambda$ in the momentum integrals of Eq. (13), both in the numerator and
denominator. We show the relative change in the rms radii as a function of
$\Lambda$ in Fig. 6. The IPM (LCA) radii converge at $\Lambda\approx
2~{}\text{fm}^{-1}$ ($\Lambda\approx 4-5~{}\text{fm}^{-1}$). The momentum
range $2\lesssim k\lesssim 4~{}\text{fm}^{-1}$ shrinks the LCA rms radii by
about 1%. This might seem to be at odds with the obvious fact that at large
momentum the LCA $r_{\text{rms}}(k)$ of Fig. 5 overshoots the IPM one. These
$r_{\text{rms}}(k>2~{}\text{fm}^{-1})$ values, however, are smaller than those
for $r_{\text{rms}}(k<2~{}\text{fm}^{-1})$. This means that excluding the
$k>2~{}\text{fm}^{-1}$ (see $\rho_{r^{2}}(k)$ in Figs. 3 and 5), increases the
LCA rms radii. For ${}^{48}\text{Ca}$ the per nucleon influence of SRCs is
different for protons than for neutrons, resulting in a slight decrease of the
neutron skin on the order of 3% when including high-momentum SRCs with
$k>2~{}\text{fm}^{-1}$, independent of the chosen LCA model. This agrees with
the naive picture of the dominant tensor correlations pulling the high-
momentum protons close to the neutrons, see $r_{\text{rms}}(k)$ in Fig. 5.

##  4 Conclusion and Outlook

Most often nuclear short-range correlations in finite nuclei are addressed
from the perspective of distributions in momentum space and the operational
definition of SRCs is one whereby one refers to nucleons with a momentum well
above the Fermi momentum ($k\gtrsim k_{F}=1.2~{}\text{fm}^{-1}$). This implies
that the SRC terminology is exclusively used for nucleons that have a momentum
larger than $k_{F}$. In this work, we have added the spatial perspective by
presenting calculations of nuclear Wigner distributions $w(r,k)$ that include
the effect of SRCs. The calculations are conducted within the lowest-order
correlation operator approximation (LCA), a framework that has shown its
interpretative potential in dealing with observations.

Common features emerged for the $w(r,k)$ of the three nuclei
${}^{12}\text{C}$, ${}^{40}\text{Ca}$ and ${}^{48}\text{Ca}$ considered in
this work. The SRCs impact the $w(r,k)$ over the full momentum range
considered (from 0 up to the nucleon mass) and a radial range that is confined
to the deep nuclear interior. The SRC contribution to the Wigner distribution
has a $k_{\text{rms}}$ of about 1.6-1.7 fm-1, whereas the IPM $w(r,k)$ has
$k_{\text{rms}}\approx 0.8$ fm-1. The resulting LCA $w(r,k)$ that has both
mean-field and SRC contributions has $k_{\text{rms}}\approx 1.2$ fm-1. As the
SRCs are confined to the deep interior of the nucleus, the LCA $w(r,k)$ has an
$r_{\text{rms}}$ that is approximately 0.2 fm smaller than the IPM one when
using the same HO frequency.

From the $w(r,k)$ we computed a prototypical bulk nuclear momentum-space and
coordinate-space feature, namely the non-relativistic kinetic energy and the
point-nucleon radius. The SRCs almost double the proton and neutron kinetic
energies. Furthermore, the proton-neutron dominance of the SRCs gives rise to
peculiar effects in asymmetric nuclei with a neutron abundance whereby the
per-nucleon kinetic energy is larger for protons than for neutrons. The impact
of SRCs on proton and neutron radii is at the percent level. In asymmetric
nuclei the proton and neutron radii are impacted differently. In the
asymmetric nucleus ${}^{48}\text{Ca}$ studied here, the SRCs reduce the
neutron skin by an amount that is of the order of 5-10%, with high-momentum
SRC components accounting for 3%.

The Wigner distributions discussed here could provide valuable input in semi-
classical transport calculations for nuclear reaction cross sections [54, 55,
56] that wish to account for the effects of SRCs. It would be interesting to
see whether a model with a realistic description of both long-range and short-
range correlations leads to sizable changes in the nuclear radii.

The computational resources (Stevin Supercomputer Infrastructure) and services
used in this work were provided by the VSC (Flemish Supercomputer Center),
funded by Ghent University, FWO and the Flemish Government – department EWI.
We thank M. Sargsian for comments on an earlier draft.

##  Appendix A Wigner distribution in the LCA

For details of the formalism underlying the LCA, we refer to Refs. [50, 57,
28, 29, 30]. Here, we outline the major derivation steps of the nuclear Wigner
distribution in LCA. In the LCA, normalized and antisymmetrized two-particle
HO states with quantum numbers $\alpha_{i}\equiv
n_{i}l_{i}j_{i}m_{j_{i}}t_{i}(\bm{r}_{i})$ are expanded in coupled relative
and center-of-mass HO states using Moshinsky brackets

$\bigl{|}\alpha_{1}\alpha_{2}\bigr{\rangle}=C^{\mathcal{A}}_{\alpha_{1}\alpha_{2}}\Bigl{|}\mathcal{A}\equiv
n(lS)jm_{j}(\bm{r}_{12}),NLM_{L}(\bm{R}_{12}),TM_{T}\Bigr{\rangle}\,.$ (15)

See Ref. [50] for the full expressions of
$C^{\mathcal{A}}_{\alpha_{1}\alpha_{2}}$. Through the action of universal SRC
operators, one-body operators become effective two-body operators in the LCA.
The two-dimensional Wigner distribution of Eq. (7) can be computed with the
two-body operator

$\widehat{w}_{t_{1}t_{2}}(r_{1},k_{1})=\frac{1}{(2\pi)^{3}}\sum_{s_{1},s_{2}}\iint
d\Omega_{\bm{r}_{1}}d\Omega_{\bm{k}_{1}}\iint d\bm{q}\,d\bm{k}_{2}\;\\\ \times
e^{-i\bm{q}\cdot\bm{r}_{1}}\;\biggl{|}\bm{k}_{1}+\frac{\bm{q}}{2},s_{1}t_{1};\bm{k}_{2},s_{2}t_{2}\biggr{\rangle}\biggl{\langle}\bm{k}_{1}+\frac{\bm{q}}{2},s_{1}t_{1};\bm{k}_{2},s_{2}t_{2}\biggr{|}\,.$
(16)

Note that the separate nucleon-pair contributions
$t_{1}t_{2}\in\\{pp,nn,np,pn\\}$ to the matrix element are calculated. The
$(pp,pn)$ and $(nn,np)$ terms are combined to obtain the proton and neutron
Wigner distributions respectively. The LCA matrix elements between coupled
states adopt the form

$w^{\mathcal{A}\mathcal{A^{\prime}},\text{LCA}}_{t_{1}t_{2}}(r_{1},k_{1})=\Bigl{\langle}\mathcal{A}\,\Bigr{|}\left[f_{p}(r_{12})\widehat{O}^{p}(1,2)\right]^{\dagger}\\\
\times\widehat{w}_{t_{1}t_{2}}(r_{1},k_{1})\left[f_{q}(r_{12})\widehat{O}^{q}(1,2)\right]\Bigl{|}\,\mathcal{A}^{\prime}\Bigr{\rangle}\,,$
(17)

where $f_{p,q}(r_{12})\widehat{O}^{p,q}(1,2)$ are one of the three correlation
operators (central, tensor, spin-isospin) introduced in LCA. The strength
functions $f_{i}(r_{12})$ depend on the internucleon distance. Note that an
IPM calculation of the two-dimensional Wigner distribution of Eq. (17)
corresponds with replacing the correlation operator with the unity one. The
effect of the SRCs on the overall normalization is accounted for by dividing
LCA matrix elements by a norm calculated to the same order using Eq. (17) with
$\widehat{w}(r_{1},k_{1})\rightarrow 1$.

For the three included SRC operators, the action of the correlation operators
on the coupled states can be schematically written as

$\widehat{O}^{i}(1,2)\Bigl{|}\mathcal{A}\equiv
n(lS)jm_{j}(\bm{r}_{12}),NLM_{L}(\bm{R}_{12}),TM_{T}\Bigr{\rangle}\\\
=\sum_{l^{\prime}=|j-1|}^{j+1}O^{i}(S,T,j,l,l^{\prime})\\\
\times\Bigl{|}nl(l^{\prime}S)jm_{j}(\bm{r}_{12}),NLM_{L}(\bm{R}_{12}),TM_{T}\Bigr{\rangle}\,.$
(18)

The included correlation operators can only change the quantum number $l$ (to
$l^{\prime}$) in the angular part of the relative HO coordinate, while in the
radial part of the HO wave function it remains unchanged. No other quantum
numbers change. The coefficients $O^{i}$ in the decomposition of Eq. (18)
depend on $(S,T,j,l,l^{\prime})$, and only the tensor correlation operator
generates coefficients with $l\neq l^{\prime}$. The final result is obtained
using the following steps:

  1. 1.

Inserting two complete sets of coordinate space wave functions
$(\bm{r}_{1},\bm{r}_{2})$ in Eq. (17) and rewriting everything in relative
$(\bm{r}_{12})$ and c.o.m. $(\bm{R}_{12})$ coordinates.

  2. 2.

Inverse Fourier transforming the radial HO wave functions in the c.o.m.
coordinates.

  3. 3.

Expanding the plane waves in terms of Bessel functions and spherical
harmonics.

  4. 4.

Making use of spherical harmonic identities.

One obtains

$w^{\mathcal{A}\mathcal{A^{\prime}},\text{LCA}}_{t_{1}t_{2}}(r_{1},k_{1})=\sum_{l_{p}=|j-1|}^{j+1}\sum_{l^{\prime}_{q}=|j^{\prime}-1|}^{j^{\prime}+1}\text{O}^{p\dagger}(S,T,j,l,l_{p})\\\
\times\text{O}^{q}(S,T^{\prime},j^{\prime},l^{\prime},l_{q}^{\prime})\langle\tfrac{1}{2}t_{1}\tfrac{1}{2}t_{2}|TM_{T}\rangle\langle\tfrac{1}{2}t_{1}\tfrac{1}{2}t_{2}|T^{\prime}M^{\prime}_{T}\rangle\\\
\times\sum_{m_{S}}\langle l_{p}m_{l_{p}}Sm_{S}|jm_{j}\rangle\langle
l^{\prime}_{q}m_{l^{\prime}_{q}}Sm_{S}|j^{\prime}m^{\prime}_{j}\rangle\\\
\times\frac{2^{5}}{\pi^{2}}\sum_{um_{u}}\sum_{Km_{K}}\sum_{K^{\prime}m^{\prime}_{K}}i^{L-L^{\prime}-K+K^{\prime}}\begin{pmatrix}l_{p}&u&K\\\
m_{l_{p}}&m_{u}&m_{K}\end{pmatrix}\\\ \times\begin{pmatrix}l_{p}&u&K\\\
0&0&0\end{pmatrix}\begin{pmatrix}l_{q}^{\prime}&u&K^{\prime}\\\
m_{l_{q}^{\prime}}&m_{u}&m_{K^{\prime}}\end{pmatrix}\begin{pmatrix}l_{q}^{\prime}&u&K^{\prime}\\\
0&0&0\end{pmatrix}\\\
\times\sum_{Um_{U}}\hat{u}^{2}\hat{U}^{2}\hat{l}_{p}\hat{l}_{q}^{\prime}\hat{k}^{2}\hat{k}^{\prime
2}\hat{L}\hat{L}^{\prime}\begin{pmatrix}L&K^{\prime}&U\\\
M_{L}&m^{\prime}_{K}&m_{U}\end{pmatrix}\\\
\times\begin{pmatrix}L&K^{\prime}&U\\\
0&0&0\end{pmatrix}\begin{pmatrix}L^{\prime}&K&U\\\
M_{L^{\prime}}&m_{K}&m_{U}\end{pmatrix}\begin{pmatrix}L^{\prime}&K&U\\\
0&0&0\end{pmatrix}\\\
\times\xi_{p,nlN^{\prime}L^{\prime}}^{UuK}(r_{1},k_{1})\xi_{q,n^{\prime}l^{\prime}NL}^{UuK^{\prime}}(r_{1},k_{1})\,,$
(19)

where we used the notation $\hat{j}\equiv\sqrt{2j+1}$. In Eq. (19), the
correlation strength functions $f_{i}$ enter through

$\displaystyle\xi_{i,nlN^{\prime}L^{\prime}}^{UuK}(r_{1},k_{1})=(-1)^{N^{\prime}}\int
dPP^{2}\Pi_{N^{\prime}L^{\prime}}(P)$
$\displaystyle\qquad\qquad\qquad\qquad\times
j_{U}(\sqrt{2}Pr_{1})\chi_{i,nl}^{uK}(k_{1},P)\,,$ (20)
$\displaystyle\chi_{i,nl}^{uK}(k_{1},P)=\int
drr^{2}f_{i}(r)R_{nl}(r)j_{u}(\sqrt{2}k_{1}r)j_{K}(Pr)\,.$ (21)

The $R_{nl},\Pi_{NL}$ are defined through the radial parts of the normalized
HO wave function $\psi_{nlm_{l}}(\bm{r})$ and its Fourier transform
$\phi_{nlm_{l}}(\bm{k})$

$\displaystyle\psi_{nlm_{l}}(\bm{r})$
$\displaystyle=R_{nl}(r)Y_{lm_{l}}(\Omega_{r})\,,$
$\displaystyle\phi_{nlm_{l}}(\bm{k})$
$\displaystyle=\frac{1}{(2\pi)^{\frac{3}{2}}}\int
d^{3}re^{-i\bm{k}\cdot\bm{r}}\psi_{nlm_{l}}(\bm{r})$
$\displaystyle=i^{-l}(-1)^{n}\Pi_{nl}(k)Y_{lm_{l}}(\Omega_{p})\,.$ (22)

In the numerical evaluation of Eq. (19), the summation over $U$ is truncated
after $U>10$, while for the other variables all combinations with non-
vanishing $3j$-symbols are retained. This truncation in the variable $U$
results in normalization errors on the sub-percent level for heavy nuclei.

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