|  Magnetic structure and properties of the honeycomb antiferromagnet
[Na(OH2)3]Mn(NCS)3
---|---
|  Madeleine Geers,a,b Thomas B. Gill,c Andrew D. Burnett,d Euan N. Bassey,e‡
Oscar Fabelo,b Laura Cañadillas-Delgado,b Matthew J. Cliffe,∗a †
|

##

††footnotetext: a School of Chemistry, University Park, Nottingham, NG7 2RD,
United Kingdom Email<EMAIL_ADDRESS>b
Institut Laue Langevin, 71 avenue des Martyrs CS 20156, 38042 Grenoble Cedex
9, France††footnotetext: c School of Electronic and Electrical Engineering,
University of Leeds, Leeds, LS2 9JT, UK††footnotetext: d School of Chemistry,
University of Leeds, Leeds, LS2 9JT, UK††footnotetext: e Yusuf Hamied
Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge
CB2 1EW††footnotetext: ‡ Present Address: Materials Department and Materials
Research Laboratory, University of California, Santa Barbara, California
93106, United States of America††footnotetext: † Electronic Supplementary
Information (ESI) available: Additional experimental details for synthesis,
single crystal neutron diffraction measurements and analysis, second harmonic
generation, mCIF. Research data and analysis notebooks are available at the
Nottingham Research Data Management Repository DOI: 10.17639/nott.7396.. See
DOI: 10.1039/b000000x/

We report the magnetic structure and properties of a thiocyanate-based
honeycomb magnet [Na(OH2)3]Mn(NCS)3 which crystallises in the unusual low-
symmetry trigonal space group $P\overline{3}$. Magnetic measurements on powder
samples show this material is an antiferromagnet (ordering temperature
$T_{\mathrm{N,mag}}=18.1(6)\,$K) and can be described by nearest neighbour
antiferromagnetic interactions $J=-11.07(4)\,$K. A method for growing neutron-
diffraction sized single crystals (>10 mm3) is demonstrated. Low temperature
neutron single crystal diffraction shows that the compound adopts the
collinear antiferromagnetic structure with $T_{\mathrm{N,neut}}=18.94(7)\,$K,
magnetic space group $P\bar{3}^{\prime}$. Low temperature second-harmonic
generation (SHG) measurements provide no evidence of breaking of the centre of
symmetry.

Honeycomb magnets, due to their high symmetry and two-dimensionality, can host
a wide range of magnetic phases, including monolayer ferromagnetism in CrI31,
proximate Kitaev spin liquid states in RuCl3 and Na2Co2TeO6 2, 3, complex spin
textures in FeCl34, and magnetoelectricity in Co2Mo3O8.5

Molecular magnets are of particular interest for low-dimensional magnetism as
it is often more straightforward to produce well-isolated low-dimensional
connectivity, and the molecular ligands offer the potential for a high degree
of tunability.6 Honeycomb molecular magnets are no exception to this,
particularly those based on chelating ligands such as (ox^2-=C2O4^2-),7, 8, 9
tetraoxolene (C6O4R2^$n$-)10 amongst others.11, 12 These structures often have
multiple different paramagnetic species producing ferrimagnetic order, whether
this is alternating metal sites A[M^IIM^III](ox)3,7 A[M^IIM^III](C6O4R2)313,
14 or the presence of radical ligands in A_2[M2(C6O4R2)3],10 where A is charge
balancing cation (typically alkylammonium). Equally, distorted honeycomb
structures, with multiple different exchange pathways are common.12, 15, 16
Examples of ideal honeycomb lattice magnets are thus comparatively rare.

Fig. 1: Neutron single crystal structure of [Na(OH2)3]Mn(NCS)3 (D19 at the
ILL, 25 K).17 (a) View along the $c$ axis. Space group diagram for $P\bar{3}$
overlaid, $\overline{1}$: white circle, $3$: black triangle. (b) View along
$b$ axis. Hydrogen bonds shown as red lines.

One promising material family to search for honeycomb magnets are the metal
thiocyanates, which show a range of low dimensional magnetic properties due to
the layered structural connectivity and have interactions as strong as
comparable halides.18, 19 In this paper we investigate the magnetic properties
of [Na(OH2)3]Mn(NCS)3 (NaMn(NCS)3.3H2O),20 formed of ideal two-dimensional
honeycomb [Mn(NCS)3]- layers with the hexagonal void filled by 1D [Na(OH2)3]+
rods [Fig. 1]. It crystallises in the low-symmetry but still trigonal
$P\overline{3}$ space-group, comparatively rare in inorganic materials (fewer
than 0.25% of ICSD structures21). As a result, the Mn(II) sites do not lie on
inversion centres, which raises the possibility that magnetic ordering may
break the centre of symmetry and hence induce simultaneous magnetic and
electric order (Type II multiferroicity).22 Although the structure of this
compound has been reported, because it is very sensitive to humidity very
little is known about its physical properties.20 We report a method for the
synthesis of large [Na(OH2)3]Mn(NCS)3 single crystals (12 mm3) of suitable
quality for neutron diffraction measurements, which mitigates the strong
humidity dependence of this compound. Using a combination of bulk magnetic
property measurements, low-temperature single crystal neutron diffraction and
SHG measurements, we have determined its magnetic properties. We find that
[Na(OH2)3]Mn(NCS)3 orders with the classical honeycomb antiferromagnetic
ground state, ordering into the centrosymmetric collinear antiferromagnetic
$P\bar{3}^{\prime}$ space group.

We synthesised [Na(OH2)3]Mn(NCS)3 using a method adapted from that of
Biedermann _et al._ 20, using the salt metathesis of manganese(II) sulfate and
barium thiocyanate with additional sodium thiocyanate in aqueous solution in a
$1:1:1$ ratio. After filtering the precipitated BaSO4 and removing the water,
a pale green-yellow powder was obtained, deliquescent in ambient conditions,
which can be recovered from its own solution by heating at no more than
$60$°C. The powder sample is also sensitive to dry environments, and will
decompose to Mn(NCS)2 and NaNCS on heating.

On concentration _in vacuo_ , aqueous solutions of [Na(OH2)3]Mn(NCS)3 change
colour sequentially from colourless to pale pink, pale green and eventually to
aqua blue [ESI Fig. 1]. We found that the optimal concentration for crystal
growth is when the solution is an intense green colour, just before it begins
turning blue. More concentrated, blue, solutions very rapidly crystallised,
which led to many smaller crystals. Less concentrated solutions were not
sufficiently supersaturated for crystal growth [ESI Fig. 2]. On leaving this
green solution to stand for 24 h at $7$°C we obtained large regular-hexagonal
crystals. As the supersaturated solution absorbs water from the air, the
concentration falls below the level required for optimal crystal growth after
24 h. The harvested crystals from this initial crystallisation were then used
as seeds for future crystallisation with the growth solution being
reconcentrated to the optimal concentrations. After nine iterations of seeded
crystallisation we obtained pale green, hexagonal crystals of
[Na(OH2)3]Mn(NCS)3 suitable for neutron diffraction experiments: $4\times
3\times 1$ mm3 [Fig. 3 ($\mathrm{a}$)]. Due to the humidity sensitivity of
this compound, we explored a variety of conditions for storage using saturated
aqueous salt solutions to control the atmosphere: $54$% (Mg(NO3)2(aq.)), $39$%
(NaI(aq.)), $32$% (CaCl2(aq.)) and $23$% (KCH3COO(aq.)).23, 24 We found that
the crystals were indefinitely stable at $32$% humidity (>23 months). We
confirmed the phase purity of powder samples with powder X-ray diffraction and
the quality of smaller single crystals was checked using ambient temperature
X-ray single crystal diffraction.

Fig. 2: (a) Magnetic susceptibility, $\chi$ measured from 2-300 K (black),
with the high temperature fits using the Curie-Weiss (teal) and Curély
equation (pink) for a classical honeycomb magnet. Inset: temperature
derivative of the susceptibility in the vicinity of the transition. (b)
Isothermal magnetisation data, measured between $+5.00$ and $-5.00$ T at $2$
K. Inset: close-up of data between $+0.1$ and $-0.1$ T

Previous work has established that manganese thiocyanate compounds show
moderately strong magnetic interactions 18, 19, and so to investigate the
magnetic properties of [Na(OH2)3]Mn(NCS)3 and uncover any magnetic phase
transitions, we carried out magnetic susceptibility ($\mu_{0}H=0.01$ T) and
isothermal magnetisation measurements on a powder sample ($2$ K, $-5$ to $+5$
T). We found that the magnetic susceptibility increases on cooling until a
broad maximum is reached around $25$ K. Below this temperature, the
susceptibility decreases until a sharp drop at the ordering temperature,
$T_{\mathrm{N}}=18.1(6)$ K [Fig. 2$(\mathrm{a})$]. The temperature dependence
of the moment was fitted to the Curély model of a nearest neighbour classical
honeycomb magnet12, using the following Hamiltonian:

$\mathcal{H}=-\frac{1}{2}\sum_{i,j}JS_{i}\cdot S_{j},$ (1)

where $|S|=\sqrt{S(S+1)}$, and the sum is only over nearest neighbours. In
this convention, antiferromagnetic superexchange will give negative $J$. We
fitted the data over the full temperature range, with an additional $1/T$ term
to account for paramagnetic impurities due to sample hydration, giving
$J=-10.0(2)\,$K and $g=1.8$ with 7.0(2)% paramagnetic impurities. We also
fitted just the high temperature regime ($T>60\,$K), using both the Curély
model, giving $J=-11.07(4)\,$K, $g=2.02$, and using the Curie-Weiss model
which gives a Curie-Weiss temperature, $\theta_{\mathrm{CW}}=-51.4(1.4)\,$K,
equivalent to $J=-17.4(5)\,$K and $g=2.09$, the overestimated Curie-Weiss
temperature being typical of analysis of low-dimensional magnets. The presence
of a small fraction of hydrated impurity means the extracted values of $g$ are
unreliable.

The isothermal magnetisation does not saturate up to $5.00(1)$ T, consistent
with the moderately strong antiferromagnetism, and there is no observable
hysteresis within the error limits of the measurements (<1mT) [Fig. 2(b)]. A
change in the gradient at $2.0(3)$ T is suggestive of a spin reorientation
transition [ESI Fig. 8].

The combination of the negative $\theta_{\mathrm{CW}}$ and lack of hysteresis
in the isothermal magnetisation data, suggests that [Na(OH2)3]Mn(NCS)3 has
ground-state antiferromagnetic order. If there is a ferromagnetic component
due to spin canting, which could explain the up-turn in the low temperature
susceptibility data, it must be small in magnitude.

Fig. 3: Summary of the refined magnetic structure in $P\bar{3}^{\prime}$ at
$2$ K (D19, ILL).17 (a) $F_{\mathrm{obs}}$ against $F_{\mathrm{calc}}$ plot.
Inset: the measured single crystal. (b) Magnetic structure viewed along the
[110] direction showing the antiferromagnetic arrangement of moments (pink
arrows). [Mn(NCS)3]- framework is shown as a wireframe. (c) Temperature
dependence of the $100$ Bragg reflection, which has a significant magnetic
contribution, fitted to a power law (black line), $\beta=0.31(3)$. (d)
Alternate view of the magnetic structure, with the magnetic lattice
(connecting Mn atoms) shown.

To determine the low temperature nuclear and magnetic structure of
[Na(OH2)3]Mn(NCS)3, single crystal neutron diffraction measurements were
performed (D19, ILL).17 Full datasets were collected above ($25$ K) and below
($2$ K) the ordering temperature of $T_{\mathrm{N}}=18.1(6)$ K. The data at
$25$ K ($\lambda=1.45$ Å, $631$ independent reflections) could be integrated
with the same space group as the reported structure at ambient temperature,
$P\bar{3}$, indicating the absence of any structural transitions. The atom
positions and anisotropic displacement parameters were refined freely, with a
final $\chi^{2}=23.20$, obtained by refinement using the FullProf programme.25
From each H2O molecule one hydrogen is directed towards a nitrogen, whilst the
other hydrgoen is positioned towards a sulfur atom with $\angle$ O-H$\cdots$N
$=153.291(2)^{\circ}$ and $\angle$ O-H$\cdots$S $=164.258(2)^{\circ}$. The
distances of the hydrogens to the acceptor atoms are _d_ ${}_{\mathrm{H1\cdots
N}}=2.355(5)$ Å and _d_ ${}_{\mathrm{H2\cdots S}}=2.326(7)$ Å, consistent with
previous reports.26

In the low temperature dataset ($2$ K) all the Bragg reflections ($651$ unique
reflections) could be indexed by the nuclear space group $P\bar{3}$, implying
the propagation vector is k = $(0,0,0)$. The potential magnetic space groups
with this propagation vector were identified using the Bilbao Crystallographic
Server,27, 28, 29 $P\bar{3}^{\prime}$, $P\bar{3}$, $P3$, $P\bar{1}^{\prime}$,
$P\bar{1}$ and $P1$ in BNS notation.28 We initially focussed on the three
maximal symmetry magnetic space groups: $P\bar{3}^{\prime}$, $P\bar{3}$ and
$P3$. Of these, the $P\bar{3}$ magnetic space group only allows for a
collinear ferromagnetic order and so was neglected as it is inconsistent with
our magnetic property measurements; $P\bar{3}^{\prime}$ permits only collinear
antiferromagnetic order and $P3$ allows antiferromagnetic, ferromagnetic or
ferrimagnetic arrangement of the moments as it contains two distinct Mn(II)
sites.

Refinements were carried out with both the $P\bar{3}^{\prime}$ and $P3$ space
groups. The results of the refinements provided comparable models with similar
refinement statistics, $\chi^{2}=9.49$ ($P\bar{3}^{\prime}$) and
$\chi^{2}=9.8$ ($P3$) [ESI Figs. 6 & 7]. The significant difference between
these models is the loss of the inversion centres in the $P3$ space group,
which produces the two unique Mn sites. In both structures the Mn atoms are
positioned on a 3-fold rotation axis and thus the moments are constrained to
lie along _c_. The antiferromagnetic $P\bar{3}^{\prime}$ model has a moment
size, $gS$, of $4.9(2)\;\mu_{\mathrm{B}}$ per Mn, whereas in the $P3$ model,
the magnitude of the two moments are $-3.4(3)$ and $5.4(2)\;\mu_{\mathrm{B}}$.
These moments were closely correlated, and were only stable during the
refinements when the moments were constrained to refine between $-5.0$ and
$5.0\;\mu_{\mathrm{B}}$. This resulted in moments of $-4.0(2)$ and
$3.9(2)\;\mu_{\mathrm{B}}$. Within error these moments have the same
magnitude, although any uncompensated moment would result in a net
magnetisation along the _c_ axis. We also investigated the potential for lower
symmetry $\textbf{k}=0$ orderings, with canted moments or significant
magnetostructural distortions, which would require triclinic symmetry. We
integrated the data with triclinic symmetry, which produced a cell which was
trigonal within error, and then refined the magnetic and nuclear structures in
$P\bar{1}^{\prime}$ [ESI Fig. 4; ESI Table 1]. This model has a single Mn
site, $M=4.1(2)\;\mu_{\mathrm{B}}$, with the moment pointing primarily along
the _c_ axis with a significant canting, though the errors in this model
prevent reliable determination of moment direction. The atomic coordinates and
displacement parameters were refined freely, however there was no evidence
beyond error of any symmetry lowering beyond trigonal symmetry and the
refinement fit was significantly worse than the for the higher symmetry
models, with $\chi^{2}=51.3$. We therefore concluded that the triclinic models
were not required to describe this system.

To gain more detailed understanding of the transition we collected data
between $2$ and $25$ K with $0.25$ K steps to follow specific reflections. We
focussed in particular on the $100$ Bragg reflection, which has significant
magnetic contribution in the magnetically ordered stated. We fitted the
intensity of this reflection to a power law:

$I=M^{2}=A(T_{\mathrm{N}}-T)^{2\beta}+C,$ (2)

where _A_ is a proportionality constant, $T_{\mathrm{N}}$ is the ordering
temperature, $\beta$ is a critical exponent and $C$ the nuclear scattering
intensity. We found for the $100$ reflection that $\beta=0.31(3)$ and
$T_{\mathrm{N}}=18.94(7)$ K [Fig. 3(c)], which lies between the theoretical
results for a two-dimensional Ising antiferromagnet ($\beta=0.125$) and a
three-dimensional Heisenberg antiferromagnet ($\beta=0.367$).30 This suggests
that there is some degree of low dimensional character even in the vicinity of
the transition, as found for other two dimensional magnets: FeBr3
($\beta=0.324$),31 NiCl2 ($\beta=0.27$)32 and FeCl2 ($\beta=0.29$).33 Other
Bragg reflections, for example the $235$ reflection, show almost no change in
intensity with temperature as they have very limited contribution from
magnetic scattering [ESI Fig. 9]. The nuclear structure at $2$ K is very
similar to that at $25$ K, with no statistically significant deviations in
bond lengths.

Our refinements suggested that the magnetic structure is most likely the
centrosymmetric antiferromagnetic $P\bar{3}^{\prime}$, rather than the
noncentrosymmetric weak ferromagnetic $P3$. To confirm this model and to
determine if [Na(OH2)3]Mn(NCS)3 is centrosymmetric in its ground state, we
therefore carried out second harmonic generation measurements at 4 K and room
temperature, along with variable temperature measurements between 4 and 30 K.
Optical excitation was performed using Spectra Physics Spitfire Ace PA
amplified laser system (40 fs pulses at a 1 KHz repetition rate) at 800 nm (<
1 W average power) and 400 nm (< 500mW). We found no evidence of SHG signal
generation, suggesting the compound is centrosymmetric. The results suggest
that [Na(OH2)3]Mn(NCS)3 retains its inversion centres, therefore ordering in
the $P\bar{3}^{\prime}$ magnetic space group, and providing further evidence
that this compound is a good example of a classical Heisenberg honeycomb
antiferromagnet.

The [Mn(NCS)3]- layers of [Na(OH2)3]Mn(NCS)3 stack uniformly along the _c_
axis, in contrast to the ABC stacking found in NaMnCl3.34 We found no evidence
of stacking faults, slip-stacking transitions or symmetry-lowering due to
stacking, in contrast to van der Waals materials such as MnPS3 and and RuCl3
which have monoclinic symmetry because of the layer sequence,35, 36 This is
likely due to the infinite rod [Na(OH2)3]^+_$n$ countercation, which forces
hydrogen bonds to the framework and force the layers to stack in alignment.

The importance of the trigonal symmetry of the cation, together with its
templating hydrogen bonds, can be seen by comparison with [1,3-Im]Mn(NCS)3
(1,3-Im = 1-ethyl-3-methyl imidazolium) which has the bulky 1,3-Im cation as
the charge balance to the [Mn(NCS)3]- honeycomb.37 The 1,3-Im cation has no
strong hydrogen bond donors and its size and bulk means the framework forms
irregular hexagons. This distortion away from perfect symmetry is often found
in molecular framework honeycomb magnets, for example Co(N3)2(bpg). DMSO (bpg
= 1,2-dipyridine-4-ethane-1,2-diol, DMSO = SO(CH3)2). 16, 15 The [Na(OH2)_3]+
countercation rod is not unique to [Na(OH2)3]Mn(NCS)3, occurring in a handful
of other compounds, _e.g._ [Na(OH_2)_3]_2[TeBr6].38 Neutron diffraction allows
a close look at the H-bond interactions between this cation and the framework.

The magnetic properties of [Na(OH2)3]Mn(NCS)3 are quite distinct from the
halide analogue NaMnCl3, which orders at $T_{\mathrm{N}}=6.5$ K with
ferromagnetic layers, coupled antiferromagnetically.39 [Na(OH2)3]Mn(NCS)3 has
an ordering temperature almost three times as high as NaMnCl3
($T_{\mathrm{N}}=18.1(6)$ K) and the net magnetic interactions are also
stronger, $|\theta_{\mathrm{CW}}|=-51(1)$ K, compared to NaMnCl3,
$|\theta_{\mathrm{CW}}|=-4.2$ K. The increase strength of interactions in the
thiocyanate compared to the halide despite the significantly longer
superexchange pathway (Mn-Cl-Mn compared to Mn-NCS-Mn) is consistent with
investigations of the binary compounds.18 This difference may be due to the
harder NCS- ligand having better orbital overlap with Mn^2+ and the single-
orbital nature of the superexchange pathway, rather than the two-orbital
pathway in manganous chlorides.

Our results establish that [Na(OH2)3]Mn(NCS)3 adopts the $P\bar{3}^{\prime}$
space group, but it is worth exploring implications of the $P3$ magnetic
symmetry with two uncompensated moments [ESI Fig. 10], particularly as
analogous structures with different metals may adopt different ground states.
$P3$ has no inversion centre and is polar, and so this structure (if it can be
engineered in analogues or through application of stimulus _e.g._ strain)
would have electrical polarisation produced directly by magnetic order (Type
II multiferroicity).22 Generally, type II multiferroics adopt complex magnetic
orderings, for example helical incommensurate states,40 or with structural
triangular lattice arrangements which introduce frustration, _e.g._ triangular
Sr3NiTa2O941 and Ba3MnNb2O9.42 It would be unusual for a collinear magnetic
ordering to induce this behaviour, and even more so for a molecular framework.
These results illustrate the importance of magnetic structure determination
for understanding the function of magnetic molecular framework compounds, as
also shown by other recent works on frameworks containing both Mn(II)43, 44,
45, 46 and other metals.47, 48, 49

In conclusion, we have determined the low temperature structure and magnetic
properties of [Na(OH2)3]Mn(NCS)3 by developing crystal growth methodology for
this humidity sensitive compound. Neutron diffraction data permitted the
hydrogen atoms to be located and hence the key role of the hydrogen bonding
network in framework structuration. Refinement of single crystal neutron
diffraction data shows that the compound magnetically orders in
$P\bar{3}^{\prime}$ magnetic space group, supported by SHG measurements for
which we saw no signal at the wavelengths we excited, consistent with this
centrosymmetric magnetic structure. The unusual low-symmetry trigonal space
group exhibited in this compound suggest that investigations into the
analogues incorporating metal ions which exhibit magnetic anisotropy (_e.g._
Co(II), Ni(II), Fe(II)) may lead to the complex spin structure and hence
multiferroic behaviour. Substitutions for metals which form metal thiocyanate
frameworks less sensitive to ambient humidity, _e.g._ Ni(II), may also lead to
more straightforward methods to grow and store these compounds.

## Conflicts of interest

No conflicts of interest to declare.

## Acknowledgements

M.G. acknowledges the ILL Graduate School for provision of a studentship.
M.J.C. acknowledges funding from UKRI (EP/X042782/1) and the Hobday Bequest
School of Chemistry, University of Nottingham. We acknowledge the ILL for
beamtime under proposal numbers 5-41-1167. Raw data sets from ILL experiments
can be accessed via links provided in references. SHG measurements were
performed using equipment funded by EPSRC strategic equipment grant
EP/P001394/1.

## Author Contributions

M.G. and E.B. synthesised the samples; M.G. grew the crystals; M.G. and M.J.C.
carried out the magnetic measurements and the single crystal X-ray
experiments; M.G., O.F., L.C.D. and M.J.C. carried out the neutron diffraction
experiments and magnetic property analysis; T.B.G. and A.D.B. carried out the
second harmonic generation measurements; M.G., L.C.D. and M.J.C. wrote the
paper with contributions from all the authors.

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# Simple Power-law model for generating correlated particles

Tobiasz Czopowicz<EMAIL_ADDRESS>Jan Kochanowski University,
Kielce, Poland

###### Abstract

A search for the critical point of the strongly interacting matter by studying
power-law fluctuations within the framework of intermittency is ongoing. In
particular, experimental data on proton and pion production in heavy-ion
collisions are analyzed in transverse momentum space. In this regard, a simple
model with a power-law multi-particle correlations is introduced. The model
can be used to study sensitivity to detect power-law correlated particles in
the presence of various detector effects.

…

###### pacs:

….

##  I Motivation

One of the goals of the high-energy heavy-ion physics is to locate the
critical point (CP) in the phase diagram of the strongly interacting matter.
Theoretical studies suggest a smooth crossover transition at small
baryochemical potential $\mu_{B}$ and high temperature $T$ Aoki:2006we . At
lower $T$ and larger $\mu_{B}$, a first-order phase transition is expected
Asakawa:1989bq . The CP is a hypothetical end point of the first-order phase
transition that has properties of the second-order phase transition.

In the vicinity of CP, fluctuations of the order parameter become self-similar
Antoniou:2006zb , belonging to the 3D-Ising universality class. This can be
detected by studying particles’ fluctuations in the transverse momentum,
$p_{T}$, space within the framework of intermittency analysis by use of Scaled
Factorial Moments (SFM). A search for such power-law fluctuations was proposed
in Refs. Bialas:1985jb ; Bialas:1988wc ; Satz:1989vj ; Gupta:1990bi  and
experimental data on proton and pion multiplicity fluctuations have been
analyzed in transverse momentum space Anticic:2009pe ; Anticic:2012xb ;
Davis:2020fcy .

To study the sensitivity to detect power-law correlated particles in the
presence of various detector effects, a simple, fast model that can generate
particles with properties expected for the CP was developed and is presented
below.

##  II Power-Law Model

The Power-Law Model generates events with momenta of a given number of
particles with a given power-law correlation of their transverse momentum
difference and/or a given number of uncorrelated particles while maintaining a
given shape of single-particle inclusive $p_{T}$ distribution and a given
multiplicity distribution. Results, events with list of particles with their
momenta components, are stored in a text file and can be used for calculating
SFM or undergo further processing (e.g. momentum smearing to mimic detector’s
momentum resolution). The model is written in ANSI C with no external
dependencies. It uses SFC64 sfc64  random number generator.

###  II.1 Power-law correlation

The model allows for generating groups of correlated particles (pairs,
triplets, quadruplets, etc.). The correlations are introduced using the
average, over particles in a group, pair transverse momentum difference $S$.
For a given number of particles in a group $g$ that form
$g_{\text{p}}=g(g-1)/2$ pairs, $S$ is defined as

$S=\frac{1}{g_{\text{p}}}\sum\limits_{i=1}^{g}\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{g}|\Delta\overrightarrow{p_{T}}|_{i,j}\,.$ (1)

For a correlated pair, when $g=2$ and ${g_{\text{p}}=1}$, $S$ is equal to the
difference of the two particles transverse momenta,
$S=|\Delta\overrightarrow{p_{T}}|$.

The correlations are introduced by generating $S$ according to the power-law
distribution:

$\rho_{S}(S)=S^{-\phi}\,,$ (2)

with a given exponent $0\leq\phi<1$. Due to scaling of power law, $S$ for sub-
groups of $g$ particles follow the same distribution (e.g. quadruplet of
particles correlated with $\phi$ consists of 4 triplets correlated with $\phi$
and 6 pairs also correlated with $\phi$).

###  II.2 Event multiplicity

Number of events, $N_{\text{events}}$, is one of the input (command-line)
parameters. Processing will stop after reaching the requested value. Number of
particles in each event is drawn from either a standard distribution (e.g
Poisson with a given expected value) or a custom distribution supplied in a
text file. Event multiplicity can also be set to a constant value.

###  II.3 Particles’ transverse momentum components

In order to generate transverse momentum components of each particle, the
following parameters are used:

  1. (i)

desired ratio of total number of correlated particles to all particles $r$
(default: 0.5),

  2. (ii)

power-law exponent $\phi$ (default: 0.8),

  3. (iii)

minimum and maximum value of the average pair momentum difference $S$ of
correlated particles (default: $S_{\text{min}}=0.0$, $S_{\text{max}}=1.2$
GeV/c),

  4. (iv)

number of correlated particle in group $g$ (default: 2),

  5. (v)

single-particle transverse momentum distribution $\rho_{p_{T}}(p_{T})$ (up to
$p_{T}{}_{\text{max}}=1.5$ GeV/c) in a text file (default:
$\rho_{p_{T}}(p_{T})=p_{T}exp(-6p_{T})$).

Uncorrelated particles are generated as long, as generating a correlated group
would still not exceed the ratio $r$. Then, a correlated group if generated.

####  II.3.1 Uncorrelated particles

Generating uncorrelated particle’s transverse momentum components takes
following steps:

  1. (i)

draw $p_{T}$ form the supplied transverse momentum distribution
$\rho_{p_{T}}$,

  2. (ii)

draw azimuthal angle $\varphi$ from a uniform distribution [$0,2\pi$),

  3. (iii)

calculate the components, as $p_{x}=p_{T}cos(\varphi)$ and
$p_{y}=p_{T}sin(\varphi)$.

####  II.3.2 Correlated particles

Before generating the first correlated group, the total number of correlated
particles is estimated, as

$n_{\text{c}}=N_{\text{events}}\cdot\langle N\rangle\cdot r\,,$

where $\langle N\rangle$ is the mean value of the requested multiplicity
distribution. Total number of correlated groups to be generated is
$n_{\text{g}}=n_{\text{c}}/g$.

Then, an array of $n_{\text{g}}$ values of $S$ following the power-law
distribution $\rho_{S}$ from Eq. 2 is generated and sorted in ascending order.
Each value is calculated using the inverse transform sampling method (with
additional constraints on minimum and maximum values of $S$) as

$S_{i}=\left[(S_{\text{min}})^{1-\phi}+\left((S_{\text{max}})^{1-\phi}-(S_{\text{min}})^{1-\phi}\right)R_{i}\right]^{\frac{1}{1-\phi}}\,,$

where $i=1\ldots n_{\text{g}}$ and $R_{i}$ are random numbers from a uniform
distribution $[0,1)$.

Also, a histogram $H$ of each correlated particle’s transverse momentum is
created by randomly drawing $n_{\text{c}}$ values of $p_{T}$ from
$\rho_{p_{T}}$.

In the $p_{x}-p_{y}$ plane, each correlated particle in a group, is evenly
positioned on a circle with diameter $S$, centered at their average $p_{T}$.
For each group a next value from the array of $S$ values is used. In
combination with the maximum value of $p_{T}$, 1.5 GeV/c, it determines
available range of values of single particle $p_{T}$. Next, a 1D probability
distribution of available $p_{T}$ values from histogram $H$ is constructed. It
is then used to draw average $p_{T}$ of particles in group.

Having the center (average $p_{T}$) and the diameter ($S$) of the circle in
$p_{x}-p_{y}$ plane, particles are placed evenly starting at a random
position. Then, components of their transverse momenta are calculated and
stored.

As a last step, histogram $H$ is updated by removing obtained values of
$p_{T}$.

###  II.4 Particles’ longitudinal momentum components

Both correlated and uncorrelated particles’ longitudinal momentum components
$p_{z}$ are calculated independently from $p_{T}$ from a center-of-mass
rapidity distribution. The following parameters are used:

  1. (i)

minimum and maximum value of the center-of-mass rapidity (default:
$y_{\text{min}}^{\text{CMS}}=-0.75$, $y_{\text{max}}^{\text{CMS}}=0.75$),

  2. (ii)

mass of particles $m$ (default: 0.938 GeV),

  3. (iii)

rapidity of the center-of-mass in laboratory frame (default:
$y_{\text{cms}}^{\text{LAB}}=2.88$).

The center-of-mass rapidity distribution is assumed to be uniform in a given
range and one value $y^{\text{CMS}}$ is chosen at random. Using a given
particle mass and generated $p_{T}$, transverse mass is calculated as

$m_{T}=\sqrt{(p_{T})^{2}+m^{2}}\,.$

Knowing rapidity of the center-of-mass in laboratory frame and transverse mass
allows to calculate $p_{z}$:

$p_{z}=m_{T}\cdot sinh(y^{\text{CMS}}+y_{\text{cms}}^{\text{LAB}})\,.$

###  II.5 Model performance

The key feature of the model is introducing a power-law correlation of
particles while preserving a given single-particle transverse momentum
distribution. To test it, 10000 events with different settings have been
generated and the relevant distributions are shown in Fig. 1. For $\phi=0.8$
and $\phi=0.5$ the model generates power-law distributions close to the ones
requested (_bottom_). Also, for the two requested transverse momentum
distributions, generated data follow them closely (_top_).

(a) $\rho_{p_{T}}(p_{T})=p_{T}exp(-6p_{T})$,
$\phi=0.8$, $r=0.5$, $g=2$, $N=2$

(b) $\rho_{p_{T}}(p_{T})=p_{T}exp(-6p_{T})$,
$\phi=0.5$, $r=0.5$, $g=2$, $N=2$

(c) $\rho_{p_{T}}(p_{T})=sin^{2}(2(p_{T})^{2})$,
$\phi=0.8$, $r=0.5$, $g=2$, $N=2$

Figure 1:  Transverse momentum (_top_) and $S$ (_bottom_) distributions for
10000 events generated with different model parameters. Transverse momentum
distributions are plotted separately for uncorrelated (_blue_) and correlated
(_red_) particles. The $S$ distribution is fitted with $f(S)=A\cdot S^{-\phi}$
and presented in log-log scale to reveal the power-law shape.

Generating these data sets took approximately 200ms each.

###  II.6 Scaled Factorial Moments for the model data

The main purpose of the model is to study SFM, which is used within
intermittency analysis as a tool for locating the CP. Therefore it must
generate particles with properties expected for the CP. One of the properties,
power-law correlation is explicitly built-in in the model. It would result in
a power-law dependence of SFM of the order $q$ with respect to the number of
$p_{x}$–$p_{y}$ cells $M^{2}$:

$F_{q}(M)\propto(M^{2})^{\phi_{q}}\,.$

Another feature of SFM in the vicinity of the CP is a linear relation between
exponents (intermittency indices) $\phi_{q}$. To test it with the model, 10000
events each containing $N=6$, all correlated particles ($r=1$, $g=6$,
$\phi=0.8$) have been generated. Then, SFM up to $q=6$ order were calculated
and fitted with a power law. Results are shown in the left panel of Fig. 2.
The obtained exponents $\phi_{2},\dots,\phi_{6}$ are presented in the right
panel. Clearly they do exhibit expected linearity.

Figure 2:  SFM of the orders from $q=2$ to $q=6$ (_left_) and power-law
exponents $\phi_{2},\dots,\phi_{6}$ obtained from their fits (_right_) for
10000 events with 6 correlated particles each, generated with the model.

##  III Summary and outlook

This work is motivated by experimental searches for the critical point of the
strongly interacting matter in heavy-ion collisions. A model introducing a
power-law correlation predicted in the vicinity of the CP was presented. The
expected scaling behavior of $F_{q}(M)$ in $M^{2}$ as well as a linear
relation of obtained intermittency indices is observed. Introducing
correlations between particles does not affect transverse momentum and
multiplicity distributions.

The model can be used to study the impact of detector effects (e.g.
acceptance, efficiency, resolution, etc.) on the behavior of the scaled
factorial moments.

###### Acknowledgements.

The author would like to express gratitude to Marek Gaździcki for the
motivation, help and critical comments. This work was supported by the Polish
National Science Centre grant 2018/30/A/ST2/00226.

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