††thanks: Corresponding author<EMAIL_ADDRESS>Corresponding
author<EMAIL_ADDRESS>
# Improving the resolving power of Isochronous Mass Spectrometry by employing
an in-ring mechanical slit

J .H. Liu  Key Laboratory of High Precision Nuclear Spectroscopy and Center
for Nuclear Matter Science, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  X. Xu  Key
Laboratory of High Precision Nuclear Spectroscopy and Center for Nuclear
Matter Science, Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  P. Zhang  Key Laboratory of High Precision Nuclear
Spectroscopy and Center for Nuclear Matter Science, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, China  P. Shuai  Key Laboratory of High Precision Nuclear Spectroscopy
and Center for Nuclear Matter Science, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  X. L. Yan  Key Laboratory of High
Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  Y. H. Zhang  Key Laboratory of High Precision Nuclear Spectroscopy and
Center for Nuclear Matter Science, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
M. Wang  Key Laboratory of High Precision Nuclear Spectroscopy and Center for
Nuclear Matter Science, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  Yu. A.
Litvinov  Key Laboratory of High Precision Nuclear Spectroscopy and Center for
Nuclear Matter Science, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, Darmstadt, 64291 Germany  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  H. S.
Xu  Key Laboratory of High Precision Nuclear Spectroscopy and Center for
Nuclear Matter Science, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  K. Blaum
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany  T. Bao  Key Laboratory of High Precision Nuclear Spectroscopy and
Center for Nuclear Matter Science, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  H. Chen  Key Laboratory of High
Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, China  X. C. Chen  Key Laboratory of High
Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  R. J. Chen  Key Laboratory of High Precision Nuclear Spectroscopy and
Center for Nuclear Matter Science, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  C. Y. Fu  Key Laboratory of High
Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  D. W. Liu  Key Laboratory of High Precision Nuclear Spectroscopy and
Center for Nuclear Matter Science, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
W. W. Ge  Key Laboratory of High Precision Nuclear Spectroscopy and Center for
Nuclear Matter Science, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  R. S. Mao  Key Laboratory of High Precision
Nuclear Spectroscopy and Center for Nuclear Matter Science, Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  X. W. Ma
Key Laboratory of High Precision Nuclear Spectroscopy and Center for Nuclear
Matter Science, Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  M. Z. Sun  Key Laboratory of High Precision Nuclear
Spectroscopy and Center for Nuclear Matter Science, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, China  X. L. Tu  Key Laboratory of High Precision Nuclear Spectroscopy
and Center for Nuclear Matter Science, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  Y. M. Xing  Key Laboratory of High
Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  J. C. Yang  Key Laboratory of High Precision Nuclear Spectroscopy and
Center for Nuclear Matter Science, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  Y. J. Yuan  Key Laboratory of High
Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  Q. Zeng  East China University of Technology, Nanchang 100049, People’s
Republic of China  X. Zhou  Key Laboratory of High Precision Nuclear
Spectroscopy and Center for Nuclear Matter Science, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, China  X. H. Zhou  Key Laboratory of High Precision Nuclear
Spectroscopy and Center for Nuclear Matter Science, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  W. L. Zhan  Key
Laboratory of High Precision Nuclear Spectroscopy and Center for Nuclear
Matter Science, Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  S. Litvinov  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, Darmstadt, 64291 Germany  T. Uesaka
RIKEN Nishina Center, RIKEN, Saitama 351-0198, Japan  Y. Yamaguchi  RIKEN
Nishina Center, RIKEN, Saitama 351-0198, Japan  T. Yamaguchi  Department of
Physics, Saitama University, Saitama 338-8570, Japan  A. Ozawa  Insititute of
Physics, University of Tsukuba, Ibaraki 305-8571, Japan  B. H. Sun  School of
Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191,
China

###### Abstract

Isochronous Mass Spectrometry (IMS) in heavy-ion storage rings is an excellent
experimental method for precision mass measurements of exotic nuclei. In the
IMS, the storage ring is tuned in a special isochronous ion-optical mode.
Thus, the mass-over-charge ratios of the stored ions are directly reflected by
their respective revolution times in first order. However, the inevitable
momentum spread of secondary ions increases the peak widths in the measured
spectra and consequently limits the achieved mass precision. In order to
achieve a higher mass resolving power, the ring aperture was reduced to 60 mm
by applying a mechanical slit system at the dispersive straight section. The
momentum acceptance was reduced as well as better isochronous conditions were
achieved. The results showed a significant improvement of the mass resolving
power reaching $5.2\times 10^{5}$, though at the cost of about 40% ion loss.

###### pacs:

23.20.En, 23.20.Lv, 27.60.+j

##  I Introduction

The nuclear mass is one of fundamental properties of a nucleus. It provides
valuable information on the nuclear binding energy that embodies the summed
effect of all interactions among ingredient nucleons. Accurate mass values
play an essential role in various research subjects ranging from chemistry to
stringent tests of weak interaction and the Standard Model. In general, the
required accuracy depends on the research subject being explored Blaum . For
example, investigations of the evolution of shell closures requires relative
mass precision better than $\delta m/m~{}=~{}10^{-6}$, while the test of the
conserved vector current (CVC) hypothesis requires a relative precision better
than $10^{-8}$. Most of the nuclei with unknown masses are far from the valley
of $\beta$-stability. Hence, their precision mass measurements are constrained
by their low production cross-sections and short half-lives. Isochronous mass
spectrometry (IMS) established in storage rings has been proven to be a
powerful tool for mass measurements of exotic nuclei with short half-lives
even down to several tens of microseconds zq ; bhsun1 . Furthermore, IMS
allows for sensitive detection of a single ion by using a time-of-flight (ToF)
detector trot ; detector . In recent years, many important questions
concerning nuclear structure and nuclear astrophysics have been addressed by
applying IMS at the Experimental Storage Ring (ESR) at GSI, Darmstadt, Germany
JSt ; PRep ; GSI ; YuriReview ; RKn2  and the Experimental Cooler-Storage Ring
(CSRe) at IMP, Lanzhou, China tuxiaolin ; tuprl ; zyh ; IMP ; Yan13 ; peng14 ;
xx ; ZhangReview ; zpplb ; xing18 ; fu18 ; zpprc ; smz . Moreover, several new
storage ring facilities aiming at the IMS mass measurements are being
constructed or planned ZhangReview ; Walker .

Figure 1: (Colour online) The layout of the CSRe. The positions of the ToF
detector and the introduced mechanical slit are indicated.

For particles stored in a ring, their revolution times ($T$) depend on their
mass-to-charge ratios ($m/q$) and momenta ($p$). In a first-order
approximation one can write Wollnik ; Hausmann ; Hausmann2 ; Franzke :

$\displaystyle\frac{\Delta
T}{T}=\frac{1}{\gamma_{t}^{2}}\frac{\Delta(m/q)}{m/q}+(\frac{1}{\gamma_{t}^{2}}-\frac{1}{\gamma^{2}})\frac{\Delta
p}{p},$ (1)

where ${\gamma}$ is the relativistic Lorentz factor of the stored particles
and ${\gamma_{t}}$ denotes the transition energy of the storage ring, an ion-
optical parameter determined by the ring lattice. Various efforts are made to
minimize the second term on the right hand side of the above equation Franzke
, which directly affects the achievable mass resolving power. In IMS, the
storage ring is tuned into the isochronous ion-optical mode, characterized by
a certain ${\gamma_{t}}$ value. The ions are injected with energies
corresponding to ${\gamma}\approx\gamma_{t}$. As a consequence, the revolution
times for these species are directly determined by their $m/q$ and are (in
first order) independent of their momentum spreads. It is clear that due to a
limited magnetic rigidity ($B\rho=mv\gamma/q$) acceptance of the ring, this
condition is fulfilled for a limited range of $m/q$ values. The narrow $m/q$
region in which the isochronous condition ${\gamma_{t}}\approx\ {\gamma}$ is
roughly fulfilled is called the isochronous window window . Typically, the
absolute value of phase-slip factor ${\eta}$, defined as
$1/{\gamma_{t}^{2}}-1/{\gamma^{2}}$, is as small as $10^{-3}$ in the
isochronous window and increases rapidly depending on the proximity of
${\gamma}$ to ${\gamma_{t}}$. The above considerations assume that
${\gamma_{t}}$ is constant over the entire acceptance of the ring. In
practice, due to the field imperfections and the chromatic aberrations of
magnetic elements, the parameter ${\gamma_{t}}$ has a dependence on the closed
orbit (magnetic rigidity). For more details, the reader is referred to Fig.
4.19 in Ref. thesis  and Fig. 3 in Ref. Dolin  where ${\gamma_{t}}$ as a
function of $B\rho$ is illustrated for the case of the ESR. There are
investigations on how to minimize such nonlinearities by introducing higher
multipole magnetic fields like octupoles or even decupoles Sergey . Thus, the
large momentum spread due to the nuclear reaction process and the non-constant
${\gamma_{t}}$ contribute to the spread of the measured revolution times and
limit the mass resolving power. To achieve a higher mass resolution, a pioneer
technique, called _B_ $\rho$-tagging, was realized at GSI 10 ; 11 ; 12 .
There, a narrow slit system was utilized at the second dispersive focal plane
of the in-flight fragment separator FRS to define (restrict) the _B_ $\rho$
spread of transmitted and injected fragments to
$\Delta(B\rho)/(B\rho)=1.5\times 10^{-4}$, while the injection acceptance of
the ESR is $\Delta(B\rho)/(B\rho)\approx 10^{-3}$. As a result, mass resolving
power of about $5\times 10^{5}$ was achieved though at a cost of dramatically
reduced statistics.

Figure 2: (Colour online) Standard deviations of the measured revolution time
peaks, $\sigma(T)$. Data from three experimental settings after correction for
the effect of magnetic field instabilities are shown. Open circles and open
triangles represent results from the settings without using the slit with
${\gamma}_{t}$ = 1.396, 1.400, respectively. Solid squares show results from
${\gamma}_{t}$ = 1.400 setting using the silt system.

To restrict simultaneously the momentum spread and the parameter ${\eta}$, a
metallic slit was installed in the storage ring CSRe. Fig. 1 illustrates the
schematic view of the CSRe, in which the positions of the slit and the ToF
detector are also shown. This technique has been utilized in the experiment
aiming at mass measurements of 58Ni projectile fragments. By application of
the in-ring slit, the mass resolving power about $5.2\times 10^{5}$ (sigma
value) has been achieved, to be compared to $1.8\times 10^{5}$ tuxiaolin ; zyh
in previous experiments without using the slit. As a highlighted result, low-
lying isomers in 44V and 52Co are well resolved from the corresponding ground
states. The mass values and their interpretation have been discussed in Refs.
xx ; zpplb ; zpprc . In this contribution, unpublished details of the
experiment and data analysis are presented.

##  II Experiment and Results

The experiment was conducted at the Heavy Ion Research Facility in Lanzhou
(HIRFL) and Cooler Storage Ring (CSR) accelerator complex. The high-energy
part of the facility consists of the heavy ion synchrotron CSRm, the
experimental ring CSRe coupled to CSRm by an in-flight fragment separator
RIBLL2 Xia jiawen . The short-lived nuclei of interest were produced in
projectile fragmentation reaction of 58Ni19+ primary beams at a relativistic
energy on a beryllium-9 target with a thickness of 15 mm. At these energies,
the produced fragments are fully ionized. The fragments were selected by
RIBLL2 within a certain _B_ ${\rho}$ acceptance. A cocktail beam was injected
into the CSRe. In our context, the CSRe has a relatively large _B_ ${\rho}$
injection acceptance. The transition energy of CSRe was set to ${\gamma}_{t}$
= 1.400 in the isochronous ion-optical mode. In order to set the best
isochronous condition for 52Co27+, which is the primary goal in this
experiment, the ring was set to a fixed magnetic rigidity of _B_
${\rho}(^{52}$Co${}^{27+})$ = 5.8474 Tm, calculated for $\gamma$ =
${\gamma}_{t}$ = 1.400. Also the magnetic rigidity of the beam-line RIBLL2 was
set to this value to allow for an optimal transmission. The energy of the
primary beam 58Ni19+ was selected to be 467.91 MeV/u according to the
calculation via LISE++ program lise  so that the 52Co27+ ions had the most
probable velocity with $\gamma$ = 1.400 after the exit from the beryllium
target.

Figure 3: (Colour online) The ${\beta}$-functions and the dispersion function
of the CSRe as a function of the orbit length. The thick black line, red
dotted line and green dashed line represent the ${\beta}$x, ${\beta}$y and _D_
x functions, respectively. The positions where the slit and the TOF detector
are installed are indicated on the top with a red rectangle and blue
rectangle, respectively.

A ToF detector was used to measure the revolution times of the stored ions.
The detector is based on the detection of secondary electrons released from
the surface of a carbon foil installed inside the ring aperture detector . The
stored ions penetrate the foil at each revolution. Ideally, the electrons are
released at each passage of an ion through the detector. The electrons were
guided to a set of micro-channel plates (MCPs) by perpendicularly arranged
electrostatic and magnetic fields. The timing signals were recorded directly
by a high-performance digital oscilloscope Tektronix DPO 71254. For each
injection the recording (sampling) time was set to 300 $\mu$s, to be compared
to 200 $\mu$s in previous experiments tuprl ; zyh ; window . The revolution
times of each ion was extracted from the timing signals. After correction for
the time drifts due to instabilities of magnetic fields, masses of nuclides of
interest were obtained from the final revolution time spectrum. Our
conventional procedure for the data analysis has been described in detail in
Refs. tuxiaolin ; zpprc .

After 12-hours data accumulation (see blue open circles in the Fig. 2), it has
been found that nuclides with revolution times around 616 ns had the minimum
standard deviation $\sigma(T)$, while the revolution time of 52Co27+ ions was
about 614 ns. Since this is a good indicator for the isochronous condition,
the ring was obviously not optimized for 52Co27+. According to the measured
experimental data, the transition energy ${\gamma}_{t}$ of the CSRe was about
1.396. This slight deviation of ${\gamma}_{t}$ from the aimed value was mainly
caused by the imperfections of the ring magnetic fields. A first order
optimization of the ion-optical isochronous setting was made via modifications
of the quadrupole magnetic field strengths gx . The current of a family of
quadrupoles, MG43Q1 and MG44Q1, was increased by 0.4$\%$. In this way, the
${\gamma}_{t}$ of the CSRe was corrected to 1.400. The success of this
optimization was confirmed after a 8-hours data accumulation (see red
triangles in the Fig. 2), where the nuclides with revolution times around 613
ns showed the minimum standard deviation $\sigma(T)$.

Figure 4: (Colour online) The revolution time spectra of 52Co zoomed in a time
window of 613.892 ns $\leq$ _t_ $\leq$ 613.904 ns. The top and bottom panels
show the spectra before and after the application of the slit, respectively.
The displayed results in the latter panel are from double-Gaussian chi-squares
fitting.

The excitation energy of the low-lying isomeric state in 52Co is about 390
keV, which is inferred from its mirror nucleus 52Mn 52Mn  regardless of
isospin-symmetry breaking. According to Eq. 1, the corresponding difference of
revolution times between the isomeric and ground states in 52Co is about 3 ps.
However, from the results of the described two settings, the minimum standard
deviation is about 1 ps for the nuclides with the best isochronous condition.
To achieve better resolution of the isomer from the ground state in the
revolution time spectrum, a higher mass resolving power is needed.

For this purpose, a mechanical slit limiting the ring aperture has been
installed. In principle, it should be installed at the place where the
dispersion is large. Its actual position was determined according to the
simulation for $\beta$-functions and dispersion function of the ion-optical
setting of the CSRe as shown in Fig. 3. The dispersion at the position of the
slit was estimated to be 20 cm/%. The width of the slit set to be was 60 mm,
corresponding to the momentum acceptance of the CSRe of $\Delta p/p$
$\sim~{}\pm$ 0.15$\%$, while this value was $\sim~{}\pm$ 0.2$\%$ in the
previous experiments under the same optical settings but without the
application of the slit zyh .

This method effectively improves the precision of the revolution time
measurement in comparison with the other settings as demonstrated in Fig. 2.
The obtained standard deviations of the revolution times are shown as black
squares in this figure. There were two changes that should be addressed. The
first one was that the revolution time, of which nuclide have the minimum
$\sigma(T)$, shift from 613 ns to 614 ns even though the current of any magnet
was not change at all. As discussed in the introduction, this shift is mainly
due to the none constant value of $\gamma_{t}$ in the full momentum
acceptance. The $\gamma_{t}$ value discussed here was an average result. By
using the slit, the $\gamma_{t}$ parameter was restricted in a smaller
momentum acceptance, and thus the average value of $\gamma_{t}$ changed. The
second point is that the resolving power was improved by about a factor of two
for 52Co. Fig. 4 clearly illustrates that the low-lying isomer in 52Co with
$E_{x}$ of about 390 keV was now well resolved from the ground state.
Furthermore, both the nuclides of interest and nuclides used as references in
the mass calibration procedure benefited from this method. As a result, the
statistical error and the fitting error in the mass calibration could be
reduced leading to an unprecedented mass precision of 5 $\sim$ 10 keV reached
so far in the IMS for short-lived nuclei.

Figure 5: (Colour online) Beam loss as a function of the storage time ranging
from 100 $\mu$s to 300 $\mu$s. All ions with revolution time in the window
from 608 ns to 620 ns were counted. $N_{(t>T_{s})}$ is the average number (per
injection) of the ions whose storage time $t$ longer than a given $T_{s}$. The
blue solid-line and red dash-line represent the results from the setting of
${\gamma}_{t}$ = 1.400 before and after employing the slit, respectively.

Similar to the _B_ $\rho$-tagging method applied at the FRS-ESR, the negative
consequence of the application of the slit is that the available orbitals in
the CSRe were reduced leading to the loss of valuable particles due to the
smaller acceptance. We compared the average number of survived ions before and
after the application of the slit. As shown in Fig. 5, a great reduction of
about 40% was clearly seen after using the slit. Furthermore, the continuous
decease of the average number in both cases reveals that the beam gradually
lose when circling in the ring. In the previous experiments, only those ions
which circulated for more than about 300 revolutions (186 $\mu$s) in CSRe were
considered in the data analysis tuxiaolin . Obviously, some ions do not
survive for so long time. Before using the slit, the average number decreased
by 8% (from 14.25 to 13.13) when compared ions with storage time longer than
100 $\mu$s and 186 $\mu$s, while decreased by 18% (from 8.83 to 7.20) after
using the slit. The loss of ions corresponding to the storage time was
amplified by using the slit.

The uncertainty of the revolution time of each stored ion, that was extracted
from periodic timing signals, contains two contributions, the finite emittance
(defined as a deviation from the reference particle) of the ion and the time
resolution of the ToF detector. The influence of the former can be eliminated
by averaging the data from a large number of revolutions. The uncertainty
contribution from the latter has been estimated in Ref. zq  to be

$\displaystyle{\delta T}{\approx}\frac{3.64\sigma}{\sqrt{\varepsilon M^{3}}},$
(2)

where $\varepsilon$ is the detection efficiency, $\sigma$ (about 50 ps) is the
time resolution of the ToF detector, and $M$ is the number of turns that ions
were stored in the ring. $\varepsilon$ varies from 20% to 90% depending on the
total number of timing signals in the detector in one injection and on the
proton number of the ion detector ; zhangwei . In this experiment, it is about
50% for ions with proton number around 20. For an ion that was stored for more
than 100 $\mu$s, namely for more than 150 turns, the uncertainty of the
revolution time was better than 0.1 ps. This value shall be compared to the
standard deviations of the revolution time peaks in the final spectrum of
larger than 0.5 ps.

Finally, all ions that circulated for more than 100 $\mu$s were used, leading
to an increase of statistics by about 20%. Our present results show that the
limitation set on the storage times in the data analysis of the previous
experiments was too conservative.

Figure 6: (Colour online) Ratio of the numbers of stored ions in two settings
for different nuclides. _N_ and _N_ 0 are the average numbers of ions (per
injection) for a given nuclide in the setting of ${\gamma}_{t}$ = 1.400 but
with and without the slit, respectively. The red line in the figure has no
meaning and just guide the eyes.

The loss of ions for different species of nuclide, caused by the application
of the slit, was further investigated as shown in Fig. 6. The normalized
number for each nuclide is all smaller after using the slit. About 30%-45% of
ions were lost in the revolution-time window from 612 ns to 616 ns, where the
nuclei of interest were mainly located (indicated by the grey-shadowed region
in Fig. 6).

Meaningfully, we found that the _N_ /_N_ 0 has a positive correlation with the
revolution time. A possible explanation for this dependence is that the
momentum distribution for each nuclide is different, as discussed in Ref.
momentum ; mom2 . Without the slit, nuclei within the entire acceptance of
CSRe can be stored in the ring. After applying the slit, the momentum
distributions are restricted in a smaller range. In this experiment, the
nuclei with longer revolution times are in general heavier and closer to the
projectile, thus have narrower momentum distribution compared with the nuclei
with shorter revolution times. Thus, the _N_ /_N_ 0 is larger for nuclei with
longer revolution time. To testify this hypothesis, we plan to measure the
actual momentum distributions of nuclei in the ring by using a double-TOF
detector system window ; sp_nimb  in the forthcoming experiments.

##  III Summary and perspective

We presented some details of the experiment and data analysis of isochronous
mass measurements at the CSRe with the application of an in-ring slit system.
Owing to the slit, the momentum distribution was reduced and a better
isochronicity has been achieved. The results have shown that the IMS with an
applied slit can lead to a significantly improved mass resolving power. In
this experiment we achieved $m/\Delta m=5.2\times 10^{5}$ (sigma value).
However, the application of the slit leads to the loss of about 40% of ions.
Our method was further successfully applied in experiments addressing 112Sn
projectile fragments xing18 , where a slit with a narrower, 50 mm opening, has
been introduced resulting in even higher mass resolving power.

In the past few years, the IMS at the CSRe has been extended by the
installation of a double-TOF detector system. Several pilot experiments have
been done xymtof . With the new set-up, the velocity of each stored ion can be
measured in addition to its revolution time. Thus the $\gamma_{t}$ as a
function of the orbit length could accurately be measured crj ; wwge . With
the latter developments, the mass resolving power of the IMS will likely be
further improved without losing statistics.

##  IV ACKNOWLEDGMENTS

The authors thank the staffs in the accelerator division of IMP for providing
stable beam. This work is supported in part by the National Key R&D Program of
China (Grant No. 2018YF A0404401 and No. 2016YFA0400504), the National Nature
Science Foundation of China (Grants No. 11605249, 11605248, 11605252,
11505267, 11575112, and 11575007), the CAS External Cooperation Program (Grant
No. GJHZ1305), the CAS through the Key Research Program of Frontier Sciences
(Grant No. QYZDJ-SSW-SLH005), the Key Research Program of the Chinese Academy
of Sciences (No. XDPB09), the Helmholtz-CAS Joint Research Group (HCJRG-108),
and by the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (Grant No 682841 “ASTRUm”). Y.A.L.
acknowledges the support by the CAS President’s International Fellowship
Initiative Grant (2016VMA 043). K.B. acknowledges support by the Nuclear
Astrophysics Virtual Institute (NAVI) of the Helmholtz Association. X.X.
thanks the support from CAS "Light of West China" Program.

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# CaloGraph: Graph-based diffusion model for fast shower generation in
calorimeters with irregular geometry

Dmitrii Kobylianskii<EMAIL_ADDRESS>Nathalie Soybelman
Etienne Dreyer  Eilam Gross  Weizmann Institute of Science, Israel

###### Abstract

Denoising diffusion models have gained prominence in various generative tasks,
prompting their exploration for the generation of calorimeter responses. Given
the computational challenges posed by detector simulations in high-energy
physics experiments, the necessity to explore new machine-learning-based
approaches is evident. This study introduces a novel graph-based diffusion
model designed specifically for rapid calorimeter simulations. The methodology
is particularly well-suited for low-granularity detectors featuring irregular
geometries. We apply this model to the ATLAS dataset published in the context
of the Fast Calorimeter Simulation Challenge 2022, marking the first
application of a graph diffusion model in the field of particle physics.

††preprint: APS/123-QED

##  I Introduction

Simulation plays an essential role in interpreting collision data from the
Large Hadron Collider (LHC) experiments and testing alignment with theoretical
predictions. The unique set of challenges entailed in simulating collision
data, including high-dimensional feature space and lack of tractable
likelihood models, have inspired a range of deep learning solutions [1, 2]. In
particular, for simulating particle interactions in the detector, the core
challenge is limited computational resources, dominated by the extreme detail
needed to model particle showers in the calorimeter. Here, the traditional
approach of Monte Carlo simulation based on Geant4 [3] is robust but highly
resource intensive – occupying the largest fraction of time in the ATLAS
simulation chain [4]. In future high-luminosity LHC runs, calorimeter
simulations will need to cope with an order of magnitude higher data rate,
potentially becoming the limiting factor for physics analysis without
significant progress in the field [5].

Many efforts have been employed in order to speed up calorimeter simulations
significantly. While fast parameterized shower models have been successfully
deployed at LHC experiments [6, 7], they are limited in accuracy. More
recently, the emergence of deep generative models has led to their great
popularity and potential in tackling this task. The first generative model
applied to calorimeter simulations was CaloGAN [8]. It represented the
calorimeter as a 3D image and used a Generative Adversarial Network (GAN) to
generate them. Building on the success of this work, GANs were already
implemented in the ATLAS fast simulation AtlFast3 [9].

New developments in the field of generative models bringing new models to the
market triggered a multitude of new developments for calorimeter simulations
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. The Fast Calorimeter Simulation
Challenge 2022 [20, 21, 22] was developed to provide a framework for a
comparison of all ongoing efforts. Normalizing flows (NFs) and invertible
neural networks (INNs), as well as various diffusion-based models, showed
promising results, while often exhibiting a compromise between accuracy and
speed. Beyond the choice of network architecture, the data representation
poses an important aspect as well. While, for example, CaloScore [15, 16] and
CaloDiffusion [12] rely on image representation, CaloClouds [10, 11] and
CaloFlow [19] utilize point clouds. Point clouds offer a natural
representation of calorimeters and particle physics data in general.
Generative models of point clouds proved their effectiveness in the task of
jet simulations [23, 24, 25, 26, 27], where each point represents a jet
constituent. For calorimeter simulations, points represent energy depositions.
Such a representation is independent of the underlying geometry and,
therefore, computationally more efficient than a sparse 3d image. However,
voxelization needs to be applied to convert the point cloud into detector
cells, which can introduce a bias and affect the performance. This method is,
therefore, mostly suitable for high-granularity calorimeters, for example, at
the proposed CLIC detector [28].

Figure 1: Event display from the pion dataset in graph form. The nodes
represent voxels, and their size relates monotonically to the deposited
energy. Each colour represents a different calorimeter layer.

In this work, we propose CaloGraph: a denoising diffusion model that
represents calorimeter data as a graph, as shown in Fig. 1. Graph neural
networks (GNNs) showed wide success in HEP applications [29], including
clustering tracking detector hits into tracks [30, 31, 32] or classifying jets
based on constituents [33, 34, 35]. Particularly, graph representations of
calorimeters are used for the reconstruction of particle flow candidates [36,
37, 38]. Until now, their use for generative tasks remained unexplored. The
motivation for a graph representation of the calorimeter is similar to the one
for point clouds, with the additional advantage of edges incorporating
relations between neighbouring objects and allowing information transfer.
Furthermore, unlike the point cloud, the graph structure is fixed, and nodes
directly represent detector cells, avoiding the need for voxelization and
associated performance losses. However, a large number of edges will lead to a
significant memory need, posing a limitation to this method for high-
granularity detectors. In most inference tasks with GNNs, the number of nodes
is predefined by the input graph, whereas in our case, nodes with nonzero
energy are predicted by the generation task and thus not known in advance. It
is therefore necessary to define a fixed initial graph that sets an upper
limit on the number of nodes with nonzero energy. We therefore present results
for the ATLAS-like dataset 1 of the calorimeter challenge [20].

We note that applications of graph-based diffusion have thus far been
restricted to generation of molecules, proteins, and materials [39]. Typically
in these cases, the network is used to predict the graph adjacency matrix and
also node and edge features in some cases [40]. In our application, we take
the graph structure as given and only predict node features with the diffusion
model.

##  II Dataset

CaloGraph was trained and evaluated using Dataset 1 from the Fast Calorimeter
Simulation Challenge 2022 [20]. This dataset comprises Geant4-simulated
particle showers in a model of the ATLAS calorimeter and was used to train
AtlFast3 [9]. The dataset comprises two sets of calorimeter showers—one for
photons and one for pions.

The calorimeter consists of 533 voxels and 7 layers in the case of the pion
dataset and 368 voxels and 5 layers for photons. The voxelization leads to an
irregular distribution of radial and angular bins $N_{r}\times N_{\alpha}$
among the layers given as follows:

pions $\displaystyle\qquad 8\times 1,\;10\times 10,\;10\times 10,\;5\times
1,\;5\times 1,\;15\times 10,\;16\times 10,\;10\times 1$ photons
$\displaystyle\qquad 8\times 1,\;16\times 10,\;19\times 10,\;5\times
1,\;5\times 1$ (1)

Photon and pion energies range from 256 MeV to 4.2 TeV, increasing in powers
of two. For each energy level, 10,000 showers are provided, with a reduced
amount for very high energies. In total 242,000 (241,600) showers are provided
for the photon (pion) dataset.

###  II.1 Graph creation

Each calorimeter shower is represented by a graph in the following way. An
example from the pion sample is shown in Fig. 1. The graph nodes depict the
calorimeter voxels with the fixed coordinates
$\\{\eta_{i},\phi_{i},\text{layer}_{i}\\}$ as input features and the deposited
energy $E_{i}$ as ground truth. As shown, the voxels comprise rings of equal
$R_{i}=\sqrt{\phi_{i}^{2}+\eta_{i}^{2}}$ which are used to determine the graph
edges. Each voxel is connected to its two neighbouring voxels on the same ring
and up to two nearest neighbors on concentric rings. Connections between
layers occur for voxels located in the outermost and innermost rings in terms
of $R$. In layers without angular binning, the rings each correspond to a
single voxel and are connected concentrically, as depicted by straight lines
in Fig. 1. The innermost voxels in these layers are connected to the innermost
ring of voxels in the layers below and above, where available. In total, we
have 1478 (2210) edges for the photon (pion) graphs.

###  II.2 Data preprocessing

Before the network operates on the graph, a normalization process is applied
to the node features. The $\eta$ and $\phi$ coordinates are normalized to
achieve a zero mean and a standard deviation of 1. The energy normalization
process is illustrated in the pseudo-code of Algorithm 1. The deposited
energies in the voxels $E_{i}$ are normalized relative to the incoming energy
$E_{inc}$ with an additional factor $f=3.1$ for photons and $f=6.4$ for pions.
This factor is essential to account for scenarios where $E_{inc}$ is less than
$E_{i}$. Following this, a logit transform is executed with $\alpha=10^{-6}$.
Finally, the resulting values undergo further normalization to obtain a zero
mean and a standard deviation of 1. The incoming energy $E_{inc}$ undergoes
min-max normalization to be in the range $[0,1]$.

1:procedure Normalize($E_{i},E_{inc}$)

2: $E_{i}^{\prime}\leftarrow\frac{E_{i}}{f\cdot E_{inc}}$

3: $E_{i}^{\prime}\leftarrow\alpha+(1-2\alpha)E_{i}^{\prime}$

4: $E_{i}^{\prime}\leftarrow\ln{\frac{E_{i}^{\prime}}{1-E_{i}^{\prime}}}$

5: $E_{i}^{\prime}\leftarrow\frac{E_{i}^{\prime}-\mu}{\sigma}$

6: return $E_{i}^{\prime}$ $\triangleright$ Return normalized energy

7:end procedure

Algorithm 1 Deposited energy normalization procedure

##  III Model description

CaloGraph is a score-based graph diffusion model, where a diffusion process is
designed to perturb the original node features $x$ slowly with Gaussian noise
addition. The neural network learns to undo the noising process by estimating
the amount of noise added $\epsilon_{t}$ and solving the corresponding
differential equation. In this way, we can start from pure noise
$x_{T}\sim\mathcal{N}(0,\mathbb{I})$ and sequentially denoise it to sample
from the original data distribution $x_{0}\sim p_{data}$.

The diffusion formulation of CaloGraph follows the DDIM approach [41] closely.
We take as input a noised graph with disturbed energy
$E_{t}=\sqrt{\bar{\alpha}_{t}}E_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon$, where
$t$ is the diffusion time step, $\epsilon\sim\mathcal{N}(0,\mathbb{I})$ and
$\bar{\alpha}_{t}$ are given by a cosine noise schedule adopted from [42].
During training, the GNN learns to predict the added noise $\epsilon$
conditioned on the voxel coordinates $C$ and the incoming particle energy
$E_{inc}$. The loss is defined as:

$L=\mathbb{E}_{t,E_{0},\epsilon}\big{[}\left\|\epsilon_{\theta}(E_{t};t,C,E_{inc})-\epsilon\right\|^{2}\big{]}$
(2)

During the inference process, we start with the graph, where the deposited
energies are given by pure noise $E_{T}\sim\mathcal{N}(0,\mathbb{I})$. Then,
we use our network and the PNDM sampler from [43] to estimate the noise needed
to be removed to iteratively denoise the deposited energy.

Figure 2: Architecture of CaloGraph. The input noised graph is updated with
the embedded conditional input comprised of the sampled time step and the
incoming particle energy. Message-passing is applied on the updated graph, and
the output is combined with the conditional input and the pre-message-passing
node features to predict the noise.

The CaloGraph architecture is depicted in Fig. 2. The network takes as input
the noised target graph with the node features, including the cell positions
($\eta_{i},\phi_{i}$, layeri) and the noised cell energy $E_{t}$ as described
above. Initially, the cell layer undergoes an update through a learnable
embedding. Following this, both position and noised energy are processed
through a multi-layer perceptron (MLP). The updated node features are then
combined with the conditional input, comprising the embedded, uniformly
sampled time step and the incoming particle energy updated via another MLP.
The resulting combined output is passed through another MLP, giving the
updated graph. Subsequently, four rounds of message-passing neural networks
(MPNN) [44] are applied. The output of the message passing is combined through
a skip connection with the input to the MPNN and the conditional input. The
resulting node representation vectors are passed through the final MLP to
predict the noise. We trained separate models for the photon and pion samples
using the same architecture and hyperparameters shown in Tab. 1, without
dedicated optimization. The models were implemented using PyTorch and DGL
(Deep Graph Library) [45].

Hyperparameters
---
batch size | 200
optimizer | AdamW
lr | $10^{-3}$
# of epochs | 700
# of time steps | 50
Network sub-parts | Parameters
Embedding part | 155 710
MPNN (4 rounds) | 519 750
Noise predictor | 148 157
Total | 823 617
Table 1: Network hyperparameters of the CaloGraph model

##  IV Results

To evaluate the performance of CaloGraph, we consider different high-level
features that reflect how well the shower shapes are learned. We can define
the centre of energy and its width for a layer $n$ in the angular direction
$\eta$:

$\langle\eta\rangle_{n}=\frac{\sum_{i}\eta_{n,i}E_{n,i}}{\sum_{i}E_{n,i}},\qquad\qquad\sigma_{\langle\eta\rangle,n}=\sqrt{\frac{\sum_{i}\eta_{n,i}^{2}E_{n,i}}{\sum_{i}E_{n,i}}-\langle\eta\rangle^{2}_{n}}$
(3)

where the sum over $i$ goes over all layer voxels and $E_{i}$ is the generated
energy deposition of voxel $i$. We show these quantities for layer 2 in the
upper panels of Fig. 3 for the photon test dataset and respectively Fig. 4 for
the pion test dataset. The distributions are modelled well for both datasets,
with a percent-level discrepancy with respect to the Geant4 baseline. The peak
at 0 width stems from events that have at most one active cell in the layer.

Figure 3: Distribution high-level features in the photons sample.

Figure 4: Distribution high-level features in the pions sample.

The lower panels give insights into the energy modelling. On the left, we see
the energy distribution in layer 2. For the photon sample, we see steps in the
histogram at high energies, which are an artefact of the discrete incoming
energies in the dataset. Apart from the discrepancy in some of those bins, the
network learns to reproduce the correct distribution. The same can be said for
the pion dataset, except for the mismodelling at energies below 500 MeV. A
shortcoming of the network can be observed in the total energy modelling. For
the photon dataset, the Geant4 distribution of the total deposited energy
normalized over the incoming energy is sharply peaked near 1. The network
prediction is similarly peaked but creates a wider distribution. This can be
seen in more detail in Fig. 5, where the ratio is shown for different incident
energies. The modelling is better for low incident energies where the
distributions are wider. The performance on the pion dataset is better. Due to
non-compensation and larger energy fluctuations associated with hadronic
showers, the total energy distribution is wider than for photons and is
captured better by the network. Overall, we have, at most, a discrepancy of
15% in the bulk of the data. The distributions for separated incident energies
in Fig. 6 are well-modelled for all energies above 500 MeV, the energy region
that appears to be problematic also for the per-layer distributions.

Figure 5: Ratio of total energy to the discrete values of $E_{inc}$ in the
photons sample.

An additional useful metric for the evaluation of a generative model is the
performance of a binary classifier trained to distinguish between Geant4 and
generated samples. The better the generative model, the harder it is to
distinguish them, and the worse the performance of the classifier. We train
two different classifiers for each dataset, one using low-level features and
the other using high-level features. For both cases, the network is a simple
MLP with two hidden layers with a size of 512. The low-level classifier takes
as input only the incoming energy and the voxel energies as a list. For the
high-level features, the network focuses more on the shower shape variables
than on the energy values only. We combine the centres of energy and their
widths in $\eta$ and $\phi$, which are defined in analogy to Eq. 3, together
with the deposited energies in each layer and the incoming energy and pass it
as input to the classifier. The classifiers are trained on a sample containing
equal proportions of showers from CaloGraph and Geant4. The area under the
receiver operating characteristic curve (AUC) for the different classifiers
for the pion and photon dataset are displayed in Tab. 2. A perfect generative
model would result in an AUC of 0.5. For both datasets, the high-level
classifier shows better results than the low-level one, reflecting the
previously discussed observation that the network correctly learns the shower
shape, whereas the total energy prediction shows room for improvement. We can
understand that, in this case, the AUC on the photon dataset is lower than for
the pions since the lower number of voxels makes it easier to learn the shape.
We find that the low-level classifiers’ performance is comparable for both
datasets.

Dataset | Low-level features | High-level features
---|---|---
photons | 0.81 | 0.63
pions | 0.80 | 0.72
Table 2: The AUC values for the DNN classifier trained to distinguish GEANT4
and generated showers for both datasets. Figure 6: Ratio of total energy to
the discrete values of $E_{inc}$ in the pions sample.

In addition to the accuracy of shower generation, it is critical to look at
the timing. In the tstacarch for fast surrogates, we want to minimize the
generation time per shower as much as possible. In general, while diffusion
models exceed competitive approaches in performance, their disadvantage is the
slow generation time caused by the iterative process of noise removal. This
can be mitigated with innovative sampling methods [46], for example,
progressive distillation as in [16]. The method involves training an
additional model that learns to halve the steps each time, which significantly
speeds up the generation but features a loss in performance. For CaloGraph,
however, further speedup does not seem necessary. The generation times per
shower for both datasets are depicted in Tab. 3 for different batch sizes on
CPU and GPU. We attribute our low inference time to the efficient graph
representation of the data and the small size of the network compared to other
diffusion models.

| photons | pions
---|---|---
Batch size | GPU | CPU | GPU | CPU
1 | 1.57 | 1.92 | 1.52 | 2.07
10 | 0.17 | 0.30 | 0.17 | 0.37
100 | 0.03 | 0.12 | 0.03 | 0.20
1000 | 0.01 | 0.17 | 0.02 | 0.35
Table 3: Generation time per shower in seconds for different batch sizes for
both datasets. The GPU studies were run on an NVIDIA TITAN RTX with 24GB VRAM,
and for the CPU on an Intel(R) Xeon(R) Gold 6234 at 3.30GHz

##  V Conclusion

We presented a novel diffusion model utilizing a graph neural network for the
task of generating calorimeter showers and demonstrated the performance on the
photon and pion dataset 1 of the Fast Calorimeter Simulation Challenge 2022.
In the case of irregular cell geometry, representing the calorimeter as a
graph simplifies the pre- and postprocessing of the input data significantly
and allows the generation of correct shower shapes due to the graph
connectivity and the passage of information between neighbouring cells.
Despite the use of diffusion, the inference time is fast due to the
compactness of the model. A shortcoming of the approach is the slight
mismodeling of the total deposited energy, especially in cases where it
follows a sharp distribution, as in the photon dataset. This can potentially
be mitigated in future work by adding a network that predicts the total energy
per layer before diffusing it among the voxels. We also note that with the
graph structure connecting all nearest neighbours, this model can primarily be
used for low-granularity detectors only. Considering different connectivity,
significantly reducing the number of edges and, therefore, also the memory
usage might allow a generalization of the method.

## Acknowledgement

We are grateful for the support provided by the collaborative Weizmann
Institute and MBZUAI research grant, as well as the Benoziyo Center for High
Energy Physics. Special thanks go to our colleagues at MBZUAI, especially
Shangsong Liang and Ruihong Zeng, for their insightful discussions. We also
extend our gratitude to Edward Shields for his valuable contribution.

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