# Nb-substitution suppresses the superconducting critical temperature
of pressurized MoB2

J. Lim  Department of Physics, University of Florida, Gainesville, Florida
32611, USA  S. Sinha  Department of Physics, University of Florida,
Gainesville, Florida 32611, USA  A. C. Hire  Department of Materials Science
and Engineering, University of Florida, Gainesville, Florida 32611, USA
Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA
J. S. Kim  Department of Physics, University of Florida, Gainesville, Florida
32611, USA  P. M. Dee  Department of Physics, University of Florida,
Gainesville, Florida 32611, USA  R. S. Kumar  Department of Physics,
University of Illinois Chicago, Chicago, Illinois 60607, USA  D. Popov  HPCAT,
X-ray Science Division, Argonne National Laboratory, Argonne, Illinois 60439,
USA  R. J. Hemley  Departments of Physics, Chemistry, and Earth and
Environmental Sciences, University of Illinois Chicago, Chicago, Illinois
60607, USA  R. G. Hennig  Department of Materials Science and Engineering,
University of Florida, Gainesville, Florida 32611, USA  Quantum Theory
Project, University of Florida, Gainesville, Florida 32611, USA  P. J.
Hirschfeld  Department of Physics, University of Florida, Gainesville, Florida
32611, USA  G. R. Stewart  Department of Physics, University of Florida,
Gainesville, Florida 32611, USA  J. J. Hamlin  Department of Physics,
University of Florida, Gainesville, Florida 32611, USA

###### Abstract

A recent work has demonstrated that MoB2, transforming to the same structure
as MgB2 ($P6/mmm$), superconducts at temperatures above 30 K near 100 GPa [C.
Pei et al. Natl. Sci. Rev., nwad034 (2023)], and Nb-substitution in MoB2
stabilizes the $P6/mmm$ structure down to ambient pressure [A. C. Hire et al.
Phys. Rev. B 106, 174515 (2022)]. The current work explores the high pressure
superconducting behavior of Nb-substituted MoB2 (Nb0.25Mo0.75B2). High
pressure x-ray diffraction measurements show that the sample remains in the
ambient pressure $P6/mmm$ structure to at least 160 GPa. Electrical
resistivity measurements demonstrate that from an ambient pressure
$T_{\mathrm{c}}$ of 8 K (confirmed by specific heat to be a bulk effect), the
critical temperature is suppressed to 4 K at 50 GPa, before gradually rising
to 5.5 K at 170 GPa. The critical temperature at high pressure is thus
significantly lower than that found in MoB2 under pressure (30 K), revealing
that Nb-substitution results in a strong suppression of the superconducting
critical temperature. Our calculations indeed find a reduced electron-phonon
coupling in Nb0.25Mo0.75B2, but do not account fully for the observed
suppression, which may also arise from inhomogeneity and enhanced spin
fluctuations.

##  I Introduction

The discovery of superconductivity at a critical temperature
$T_{\mathrm{c}}=$39\text{\,}\mathrm{K}$$ in MgB2 [1] two decades ago sparked
great interest in diborides amongst the scientific community. The
superconductivity in this material is widely believed to be conventional in
nature, i.e., deriving from the electron-phonon interaction. The high critical
temperature has been attributed at least partly to high phonon energy scales
related to the presence of low mass (light) elements and to the significant
covalent character of the states near the Fermi surface [2, 3].

A great deal of effort was focused on increasing the $T_{\mathrm{c}}$ to
higher values by chemical substitution or pressure. These attempts were
unsuccessful. Pressure causes a monotonic decrease in the $T_{\mathrm{c}}$ of
MgB2 [4, 5]. Similarly, partial substitutions on the Mg or B sites invariably
cause a reduction of $T_{\mathrm{c}}$ [6, 7]. A number of structurally similar
borides or borocarbides were also investigated, but none of these exhibited
$T_{\mathrm{c}}$ values comparable to those found in MgB2. A gradual decrease
in further exploration of diboride superconductors followed. On the other
hand, the search for high superconducting critical temperatures in light
element compounds has been recommenced following the discovery of remarkably
high $T_{\mathrm{c}}$ values in pressurized hydrides [8, 9, 10].

The recent discovery of superconductivity in MoB2 with a $T_{\mathrm{c}}$
reaching as high as $32\text{\,}\mathrm{K}$ at
$110\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ has renewed the interest in
diborides [11]. However, it has been suggested that the mechanism of high
$T_{\mathrm{c}}$ in MoB2 is significantly different than that in MgB2 [12]. At
ambient pressure MoB2 exists in an $R\overline{3}m$ structure, which is non-
superconducting at low pressure. Above
$25\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, however, superconductivity
appears, with the highest $T_{\mathrm{c}}$ achieved in the $P6/mmm$ phase (the
same structure as MgB2) at $110\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. These
results led us to examine whether other diborides might also exhibit
remarkably high critical temperatures at elevated pressures. In a recent paper
[13], we reported that WB2 reaches a maximum $T_{\mathrm{c}}$ of
$\sim$$17\text{\,}\mathrm{K}$ at pressures near
$90\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. Unlike MoB2, bulk WB2 adopts a
$P6_{3}/mmc$ structure over the entire measured pressure range to at least
$145\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. Our findings suggested that the
superconducting nature of WB2 derives from stacking faults in a MgB2-like
structure.

An interesting question is whether the superconducting critical temperature of
pressurized MoB2 can be enhanced through chemical substitution. Our initial
work in this direction has focused on examining the effects of partial Nb
substitution on the Mo sites because NbB2 occurs with $P6/mmm$ structure in
which MoB2 superconducts above 30 K near 100 GPa. Recently, we showed, via
density functional theory calculations, that phonon free energy stabilizes the
$P6/mmm$ structure relative to the $R\overline{3}m$ structure at high
temperatures across the NbxMo1-xB2 series [14]. We were able to successfully
synthesize Nb-substituted MoB2 in the $P6/mmm$ structure at ambient pressure
via arc-melting. The resulting compounds, Nb1-xMoxB2, where
$x=0.1,0.25,0.5,0.75,0.9$, were superconducting with Nb0.25Mo0.75B2 having the
highest $T_{\mathrm{c}}$ of $8\text{\,}\mathrm{K}$ in the series. Specific
heat measurements on the $x=0.25$ sample demonstrate bulk superconductivity
and also showed a high upper critical field close to $7\text{\,}\mathrm{T}$
[14]. In the present study, we further investigate the superconductivity in
Nb-substituted MoB2 ($x=0.25$) through a combination of high-pressure
electrical resistivity and x-ray diffraction measurements to pressures as high
as $170\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$.

##  II Methods

At lower pressures ($<$ $2\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$), we used a
piston cylinder cell for resistivity measurements [15], with the
Nb_0.25Mo_0.75B2 sample
($\sim$1.0$\times$1.0$\times$$0.4\text{\,}\mathrm{m}\mathrm{m}^{3}$) mounted
in the van der Pauw configuration. A solution of n-pentane:isoamyl alcohol
(1:1 ratio) was used as the pressure medium. Details on the use of the piston
cylinder cell can be found in Ref. [16].

For higher pressure resistivity measurements, a micron-sized Nb_0.25Mo_0.75B2
sample
($\sim$40$\times$40$\times$$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{3}$)
was placed in a gas membrane-driven diamond anvil cell (OmniDAC from Almax-
EasyLab). A ruby crystal ($20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in
diameter) was used for pressure calibration [17] below
$80\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. At higher pressures, the pressure
was determined using the Raman spectrum of the diamond anvil [18]. Pressure
was measured at $10\text{\,}$ and $292\text{\,}\mathrm{K}$ during each cooling
cycle within an error estimation of 5%. Two opposing diamond anvils (type Ia,
1/6-carat, $0.15\text{\,}\mathrm{mm}$ central flats) and a cBN-epoxy,
soapstone insulated Re metal gasket were used for the four-probe method (see
inset in Fig. 1). The diamond anvil cell was then placed inside a customized
continuous-flow cryostat (Oxford Instruments). For each temperature-dependent
resistivity measurement, pressure was applied at room temperature. The sample
was then cooled to $1.8\text{\,}\mathrm{K}$ before being warmed back to room
temperature at a rate of
$\sim$$0.25\text{\,}\mathrm{K}\mathrm{/}\mathrm{m}\mathrm{i}\mathrm{n}$. The
measurements were performed with an excitation current of
$0.3\text{\,}\mathrm{mA}$. Further details of the non-hydrostatic high-
pressure resistivity techniques are given in Refs. [19, 13].

High pressure x-ray diffraction measurements were performed on a powdered
piece of Nb_0.25Mo_0.75B2 sample at beamline 16-BM-D at the Advanced Photon
Source, Argonne National Laboratory. The x-ray beam had a wavelength of
$0.41\text{\,}\mathrm{\text{Å}}$ ($30\text{\,}\mathrm{keV}$) in Runs 1 and 2,
which was focused to a
3$\times$$4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$ (FWHM) spot at
the sample. A MAR345 image plate detector calibrated with a CeO2 standard was
used to record the diffracted intensity with the typical exposure time of 60
to 120 seconds per image. Neon was used as the pressure medium, and pressure
was determined both using an online ruby fluorescence measurement [17] up to
$40\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ as well as the equation of state
of Au grains [20] loaded into the sample chamber up to
$162\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ within an error estimation of 2%.
DIOPTAS [21] software was used to convert the 2D diffraction images to 1D
diffraction patterns which were further analyzed by Rietveld [22] and Le Bail
[23] methods using GSAS-II software [24].

To better understand the superconducting properties of Nb_0.25Mo_0.75B2 under
pressure we calculate the Allen-Dynes $T_{\mathrm{c}}$ at
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. The electron-phonon coupling
constant, $\lambda$, was calculated from Eliashberg spectral function,
$\alpha^{2}F(\omega)$, obtained using the tetrahedron method as implemented in
the density functional theory (DFT) code Quantum Espresso [25, 26, 27]. We use
the Perdew–Burke-Ernzerhof functional for the exchange-correlation energy in
the DFT calculations [28]. The virtual crystal approximation was used with the
optimized norm-conserving pseudopotentials [29, 30]. A $k$-point mesh of
$20\times 20\times 20$ and a $q$-point mesh of $4\times 4\times 4$ was used in
the calculations.

##  III Results

Figure 1: Resistivity of Nb_0.25Mo_0.75B2 versus pressure to
$171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ at 10, 150, and
$292\text{\,}\mathrm{K}$. The resistivity curves show no noticeable change
with pressure indicating the absence of any structural phase transition.
Pressures at $150\text{\,}\mathrm{K}$ were estimated, reflecting the small
changes between pressures measured at 10 and $292\text{\,}\mathrm{K}$. Inset
shows the microphotograph of the sample, a ruby pressure calibrant, and the
four-probe method looking through the upper diamond central flat (or culet).
The white scale bar indicates $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The pressure-dependent resistivity curves of Nb_0.25Mo_0.75B2 are shown in
Fig. 1 at 10, 150, and $292\text{\,}\mathrm{K}$. While increasing pressure at
$292\text{\,}\mathrm{K}$, the resistivity was measured simultaneously at that
temperature. However, the resistivity curves at 10 and
$150\text{\,}\mathrm{K}$ were extracted from the temperature-dependent
resistivity at different pressures (see inset in Fig. 2). There is no
significant change in resistivity with respect to pressure indicating the
absence of any structural phase transition. We also plot the resistivity in a
base 10 logarithmic scale showing that the resistivity smoothly decreases with
pressure (see Fig. S1 in the Supplemental Material [31]). The inset in Fig. 1
illustrates the four-probe electrical resistivity configuration in the diamond
anvil cell looking through the upper diamond used in these measurements.

Figure 2: Representative temperature-dependent resistivity curves of
Nb_0.25Mo_0.75B2 under pressure to
$171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ (measured at
$10\text{\,}\mathrm{K}$) clearly showing the zero resistivity of
superconducting transition between 1.8-$10\text{\,}\mathrm{K}$ during each
warming cycle. Inset shows the full 1.8-$292\text{\,}\mathrm{K}$ temperature
range studied.

Figure 2 shows selected temperature-dependent resistivity curves under
pressures up to $171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ (measured at
$10\text{\,}\mathrm{K}$) focusing on the superconducting transition.
Nb_0.25Mo_0.75B2 superconducts at ambient pressure with a $T_{\mathrm{c}}$ of
$8\text{\,}\mathrm{K}$ as reported by our recent study [14]. Zero resistivity
below the superconducting transition is observed in Nb_0.25Mo_0.75B2
throughout the whole pressure range studied. The superconducting transition
broadens significantly above $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. We
denote the transition width ($\Delta T_{\mathrm{c}}$) by vertical bars in Fig.
3. The resistivity curve at $171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ in
the inset of Fig. 2 ends at $200\text{\,}\mathrm{K}$, where the diamonds
failed during the warming cycle. Nevertheless, we managed to measure the
highest pressure at $171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ using diamond
anvil Raman at $10\text{\,}\mathrm{K}$ during the cooling cycle (see Fig. S2
in the Supplemental Material [31]).

Figure 3: Superconducting phase diagram of Nb_0.25Mo_0.75B2 to
$171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ (measured at
$10\text{\,}\mathrm{K}$). The superconducting transition temperature
($T_{\mathrm{c}}$) initially goes down until
$\sim$$50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ above which it monotonically
increases up to $171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. The upper and
lower vertical bars refer to $T_{\mathrm{c}}$(90%) and
$T_{\mathrm{c}}$(offset) respectively. The dashed line shows
$T_{\mathrm{c}}$(P) of elemental Nb for comparison [32]. Inset refers
$T_{\mathrm{c}}$(P) of pure MoB2 from Ref. [11]

The superconducting transition temperature ($T_{\mathrm{c}}$) of
Nb_0.25Mo_0.75B2 versus pressure to
$171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ from Run 1 (below
$2\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ including ambient pressure using a
piston-cylinder cell) and Run 2 (above
$2\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ using a diamond anvil cell) is
shown in Fig. 3. The $T_{\mathrm{c}}$(50%) is defined by the temperature
corresponding to the 50% of normal state resistivity value just above the
superconducting transition ($\sim$$10\text{\,}\mathrm{K}$), whereas the upper
and lower vertical bars refer to the 90% and 0%(offset) criteria,
respectively. The pressure-dependent superconducting transition temperature
($T_{\mathrm{c}}$(P)) initially decreases with pressure with a slope of
$-0.067(6)\text{\,}\mathrm{K}\mathrm{/}\mathrm{G}\mathrm{P}\mathrm{a}$ and
above $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ monotonically increases with
a slope of
$0.0097(6)\text{\,}\mathrm{K}\mathrm{/}\mathrm{G}\mathrm{P}\mathrm{a}$.
Interestingly, the slope change in $T_{\mathrm{c}}$(P) above
$50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ is accompanied by the significant
broadening of superconducting transition width ($\Delta T_{\mathrm{c}}$),
defined as the difference between $T_{\mathrm{c}}$(90%) and
$T_{\mathrm{c}}$(offset) (see the corresponding vertical bars). The
nonhydrostatic condition in the measurement partially contributes to the
broadening due to the presence of the pressure gradient. However, the sudden
increase above $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ suggests the effect
originates mainly from the sample itself. A comparison of $T_{\mathrm{c}}$(P)
between Nb_0.25Mo_0.75B2 and elemental Nb metal [32] is shown in Fig 3, which
clearly demonstrates that the superconductivity in Nb_0.25Mo_0.75B2 is not
associated with Nb inclusions. Previous work has demonstrated that this
material is a bulk superconductor [14].

Figure 4: Contour plot of XRD patterns of Nb_0.25Mo_0.75B2 to
$161\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ at room temperature from Runs 1
and 2. The ambient structure ($P6/mmm$) persists up to the highest pressure
without any structural phase transitions.

In order to determine the presence of any structural transitions, we have
performed synchrotron X-ray diffraction (XRD) measurements on powdered
Nb_0.25Mo_0.75B2 samples under high pressure and room temperature using Ne as
a pressure transmitting medium in diamond anvil cells (DACs). Figure 4 shows a
contour plot of XRD patterns whose intensities are normalized with the (101)
peak in Runs 1 and 2. The $P6/mmm$ structure at ambient pressure persists to
pressures as high as $161\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ as seen by
the continued presence of the three dominant peaks with (001), (100), and
(101) Miller indices. Vertically offset plots of the XRD patterns with respect
to pressure from Runs 1 and 2 are shown in Fig. S3 in the supplemental
material [31]. The peaks from the highly compressible Ne can be easily
distinguished from those from the sample. The reflections from both Ne
pressure medium and Re metal gasket are confirmed by their equation of state
[33, 34]. There is a small amount of unidentified second phase between 6-7
degrees marked by a white asterisk (*).

Figure 5: P-V-isotherm of Nb_0.25Mo_0.75B2 to
$161\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ at room temperature. Inset shows
the $c/a$ ratio versus pressure. There is a slope change above
$\sim$$50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ marked by a light blue
shaded area referring to the potential correlation with the slope change of
$T_{\mathrm{c}}$(P) in Fig. 3.

The resulting pressure-volume (P-V) curve of Nb_0.25Mo_0.75B2 in $P6/mmm$
structure at room temperature from Runs 1 and 2 is shown in Fig. 5 with the
$c/a$ ratio versus pressure in the inset. There is a slope change in the $c/a$
ratio above $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ marked by a light blue
shaded area, which seems to potentially correlate with the slope change in the
$T_{\mathrm{c}}$(P) in Fig. 3. Interestingly, the value of the $c/a$ ratio
plateaus above $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, meaning that $c$
lattice parameter begins to be less compressible. This may indicate that the
interaction between interlayers begins to play a significant role in the
$P6/mmm$ structure. The calculated $a$ and $c$ lattice parameters with respect
to pressure are shown in Fig. S4 [31]. The Vinet Equation of state [35] is
used to fit the P-V curve, which gives rise to an ambient volume
$25.8\text{\,}\mathrm{\text{Å}}^{3}$ ($V_{0}$), bulk modulus
$272\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ ($B_{0}$), and a derivative of
the bulk modulus of 4.1 ($B_{0}^{\prime}$). The bulk modulus of
Nb_0.25Mo_0.75B2 is comparable to that of MoB2
($296\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$) [36].

Table 1 shows the computed moments of phonon frequencies, the electron-phonon
coupling parameter, and the Allen-Dynes $T_{\mathrm{c}}$
($T_{\mathrm{c}}^{\mathrm{AD}}$) for NbB2 (at 0 and
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$), Nb_0.25Mo_0.75B2 (at 50 and
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$), and MoB2 (at
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$). According to these
calculations, 25% Nb-substitution results in a moderate (roughly 30%)
suppression of $T_{\mathrm{c}}$ compared to pure MoB2 at
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. This occurs primarily due to a
suppression of the electron-phonon coupling. Interestingly, the calculated
$T_{\mathrm{c}}^{\mathrm{AD}}$ for Nb_0.25Mo_0.75B2 at both 50 and
$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ appear to be overestimations when
compared to the experimental $T_{\mathrm{c}}$. Contrary to the observed
experimental trend, we found that $T_{\mathrm{c}}^{\mathrm{AD}}$ decreases as
the pressure increases. Note that our x-ray diffraction results indicate that
at $100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, Nb_0.25Mo_0.75B2 and MoB2
adopt the same $P6/mmm$ structure.

Table 1:  Calculated superconducting parameters. The critical temperatures, $T_{\mathrm{c}}$ were calculated using the Allen-Dynes equation with $\mu^{*}=0.16$. All the calculations utilized the $P6/mmm$ structure. The DOS at the Fermi level is in units of states/eV/unit cell volume. (* indicates calculation was performed using the experimental lattice parameters.) Material | P | $N(E_{\mathrm{F}})$ | $\omega_{\text{log}}$ | $\langle\omega^{2}\rangle$ | $\lambda$ | $T_{\mathrm{c}}^{\mathrm{AD}}$
---|---|---|---|---|---|---
| (GPa) |  | (K) | (K) |  | (K)
NbB2 | 0 | - | 354 | 502.6 | 0.75 | 8.86
NbB2 | 100 | 0.795 | 577.1 | 767.4 | 0.48 | 1.65
Nb_0.25Mo_0.75B2 | 50 | 1.16 | 268.8 | 426.5 | 1.41 | 23.33
Nb_0.25Mo_0.75B2 | 100 | 0.99 | 362.2 | 542.7 | 1.02 | 20.14
Nb_0.25Mo_0.75B2* | 100 | 0.90 | 419.8 | 608.3 | 0.94 | 19.58
MoB2 | 100 | 1.14 | 283.3 | 452.5 | 1.48 | 29.17

##  IV Discussion

One question that still follows from our experiment is why Nb-doped MoB2 has a
significantly lower transition temperatures than MoB2 over the same pressure
range studied in Ref. [11]. Much of the answer to this question can be gleaned
from the literature on NbB2, MoB2, and alloyed transition metal diborides. We
will focus on those findings which are most relevant for superconductivity,
starting with the density of states (DOS) near the Fermi level. When compared
with NbB2, MoB2 has a higher DOS near the Fermi level (Table 1) and a higher
fraction of electrons occupying antibonding states [37, 38]. This difference
helps to explain why, at ambient/low pressure, MoB2 is a less stable diboride,
preferring the trigonal $R\bar{3}m$ space group symmetry with alternating
puckered boron planes instead of the hexagonal $P6/mmm$ structure realized by
NbB2 [37]. In addition, MoB2 has a higher isotropic electron-phonon coupling
constant than NbB2 [39, 40, 41, 42, 43]. Here, we would like to point out that
the calculated electron-phonon coupling for NbB2 at ambient pressure of
$\lambda\sim 0.43$ in Singh [44] is a result of poorly converged calculations
[45, 41], and our calculated value agrees with Heid _et al._ [41].

Another interesting aspect of the present study is that the experimentally
realized suppression of $T_{\mathrm{c}}$ is at odds with the
$T_{\mathrm{c}}^{\mathrm{AD}}$ obtained using the Allen-Dynes formula. The
theory and experiment both qualitatively agree that Nb-substitution reduces
the $T_{\mathrm{c}}$ in MoB2 at high pressure (Table 1) compared with MoB2.
However, there is significant quantitative disagreement in the magnitude of
$T_{\mathrm{c}}$ between the two results. Experimentally, we found that
Nb_0.25Mo_0.75B2 at 100 GPa exhibits only about 30% of the $T_{\mathrm{c}}$ of
pure MoB2 at the same pressure (Table 1). In contrast, the Allen-Dynes
equation predicts that the Nb-substituted sample should exhibit about 70% of
the $T_{\mathrm{c}}$ of pure MoB2 (i.e., for Nb_0.25Mo_0.75B2,
$T_{\mathrm{c}}^{\mathrm{AD}}=19.58$ \- 20.14 K; for MoB2
$T_{\mathrm{c}}^{\mathrm{AD}}=29.17$ K ). In other words, the Allen-Dynes
$T_{\mathrm{c}}^{\mathrm{AD}}$ prediction works reasonably well for pure MoB2,
but it fails to capture the strong reduction in $T_{\mathrm{c}}$ for Nb-doped
MoB2.

Performing the same calculation for the $T_{\mathrm{c}}^{\mathrm{AD}}$ of
stoichiometric NbB2 at ambient pressure reveals a similar overestimation.
However, in that case, the degree of overestimation is difficult to gauge
since the experimental literature for stoichiometric NbB2 is rife with
inconsistencies. Some papers report $T_{\mathrm{c}}$’s between 0.62 K and 9 K
[46, 47, 48, 49], and many others report an absence of superconductivity down
to the lowest temperatures measured [50, 51, 52, 53, 54, 55]. There is
considerably more evidence for finite $T_{\mathrm{c}}$’s up to 8-11 K in
_nonstoichiometric_ NbB2, characterized by increasing the ratio of B to Nb
(enabled by Nb-vacancies) [50, 56, 51, 49, 57, 58, 59, 52, 54, 55] or
decreasing this ratio via B-vacancies [60, 61]. Assuming that stoichiometric
NbB2 does not superconduct experimentally, except possibly at minimal
temperatures, the Allen-Dynes prediction of $T_{\mathrm{c}}^{\mathrm{AD}}=$
8.86 K becomes a rather severe overestimation.

In light of the sensitivity to inhomogeneity and vacancy formation in NbB2, we
point out that MoB2 is also susceptible to metal vacancy formation, which
generally lowers the electronic density of states [43]. Taken together, we
cannot rule out the role of inhomogeneities due to vacancies in the alloyed
sample. Our calculations show that the tendency for metal vacancy formation in
Nb0.25M0.75B2 ($E_{\mathrm{vf}}=0.214$ eV) is even more likely than in NbB2
($E_{\mathrm{vf}}=1.794$ eV). The presence of vacancies on the $4d$-atom site
could lower the DOS at the Fermi level, reducing $T_{\mathrm{c}}$. While we do
not include these effects in our calculations of the Eliashberg function, we
suspect they play a role in the discrepancy between theory and experiment.

Another potential pathology leading to $T_{\mathrm{c}}$ predictions larger
than experiment could stem from spin fluctuations absent from the present
formalism. Several 3$d$ transition metals like V and Cr are better known to
have significant spin fluctuations [62, 63, 64, 65, 66]. While Nb is generally
considered a conventional electron-phonon superconductor, some claim that spin
fluctuations effects are essential for estimating $T_{\mathrm{c}}$ [67, 64].
We have used a modified McMillan formula defined in Eqn. (2) of Ref. [68] to
estimate the electron-paramagnon coupling constant required to match the
experimental $T_{\mathrm{c}}$ of Nb_0.25Mo_0.75B2 (100 GPa), obtaining
$\lambda_{\mathrm{sf}}\sim 0.15$. By comparison, to match a
$T_{\mathrm{c}}<0.1$ K in NbB2 (0 GPa) would require
$\lambda_{\mathrm{sf}}>0.26$. These values are comparable to results for Nb in
Ref. [67] and provide at least a partial explanation for the $T_{\mathrm{c}}$
mismatch. Recent theoretical work on the itinerant antiferromagnet CrB2
suggests that spin fluctuations are suppressed under pressure, giving rise to
electron-phonon-mediated superconductivity at higher pressures [66]. It is
unclear if Nb_0.25Mo_0.75B2 exhibits analogous behavior in the pressure
dependence of $T_{\mathrm{c}}$ in part due to the unknown role of other
effects like disorder. Further theoretical investigations are necessary to pin
down the sources of the overestimation of $T_{\mathrm{c}}$, which is outside
the scope of this study.

Our measured $T_{\mathrm{c}}$ values are comparable to those reported in many
other stoichiometric and nonstoichiometric ternary diboride compounds (at
ambient/low pressure), such as Mo0.95Sc0.05B2 ($T_{\mathrm{c}}\approx$ 4.8 K)
[69], Mo0.96Zr0.04B2 ($T_{\mathrm{c}}\approx$ 5.9 K) [38], Zr0.96V0.04B2
($T_{\mathrm{c}}\approx$ 8.7 K) [70], Zr0.96Nb0.04B2 ($T_{\mathrm{c}}\approx$
8.1 K) [71], relevant doped binaries such as Nb1-xB2 ($T_{\mathrm{c}}\approx$
9.2 K) [51], NbBx ($T_{\mathrm{c}}\approx$ 9.4 K) [47], and many other borides
of Mo and Nb in the range $T_{\mathrm{c}}\approx 0$ to 11.2 K [50]. There is
considerably less literature studying diborides under pressures near 100 GPa,
so it isn’t easy to draw complete comparisons with the references above.

In nonstoichiometric NbB2, increasing the B/Nb ratio tends to expand (shrink)
the $c$ ($a$) lattice parameter alongside a concomitant increase in
$T_{\mathrm{c}}$ [50, 56, 51, 49, 57, 58, 59, 52, 43, 54, 55]. This behavior
indicates that a smaller spacing along the $c$-axis is likely detrimental to
superconductivity in NbB2. Therefore, one can reasonably expect that the
$T_{\mathrm{c}}$ of NbB2 will decrease under pressure. Our $T_{\mathrm{c}}$
calculations further support this point, though the actual values are
overestimates. In contrast, experiments by C. Pei et al. show that the
$T_{\mathrm{c}}$ of MoB2 rises sharply with applied pressure beyond 25 GPa
until a structural transition near 70 GPa, where $T_{\mathrm{c}}$ continues to
increase with pressure (and the $c$ lattice parameter keeps decreasing) but at
a lower rate [11]. Hence to achieve a higher $T_{\mathrm{c}}$ value, both the
materials (NbB2 and MoB2) take advantage of different and opposing trends in
the lattice parameters. This difference possibly explains the relatively flat
$T_{\mathrm{c}}$ as a function of pressure observed in our experiments. Taken
together, we can see that the role of Nb in NbxMo1-xB2 is to increase the low-
pressure stability of the AlB2 structure ($P6/mmm$) without recreating other
conditions needed for the higher $T_{\mathrm{c}}$ observed in MoB2 under
pressure.

##  V Conclusions

In summary, we have studied the pressure-dependent superconducting transition
temperature of Nb_0.25Mo_0.75B2 in the same structure as MgB2 ($P6/mmm$).
Electrical resistivity measurements up to
$171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ reveal that $T_{\mathrm{c}}$
initially decreases with increasing pressure. Above
$50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, $T_{\mathrm{c}}$ increases
monotonically with a significant broadening of transition width $\Delta
T_{\mathrm{c}}$ up to the highest pressure. However, the ambient pressure
$T_{\mathrm{c}}$ of $8\text{\,}\mathrm{K}$ is the highest $T_{\mathrm{c}}$
observed up to at least $171\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$.
Synchrotron high-pressure XRD measurements up to
$161\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ show that the slope of the $c/a$
ratio changes above $50\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ within the
same $P6/mmm$ structure, indicating a potential correlation with the change in
slope of $T_{\mathrm{c}}$(P). Our theoretical findings show a reduction of
$T_{\mathrm{c}}$, due to the weakened electron-phonon coupling, in
Nb0.25Mo0.75B2 compared to pure MoB2 at high pressure, in qualitative
agreement with the experiment. However, these calculations underestimate the
observed suppression of $T_{\mathrm{c}}$, suggesting that additional factors,
such as inhomogeneity and spin fluctuations, may be present. High-pressure
studies of other substitutions into MoB2, which might enhance electron-phonon
coupling, would be interesting to explore, to determine whether
$T_{\mathrm{c}}$ values comparable to the $32\text{\,}\mathrm{K}$ observed in
MoB2 at $110\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ [11] can be realized at
low or ambient pressure.

## Acknowledgments

Work at the University of Florida was performed under the auspices of U.S.
Department of Energy Basic Energy Sciences under Contract No. DE-SC-0020385
and the U.S. National Science Foundation, Division of Materials Research under
Contract No. NSF-DMR-2118718. A.C.H. and R.G.H. acknowledge additional support
from the National Science Foundation under award PHY-1549132 (Center for
Bright Beams). We thank S. Tkachev (GSECARS, University of Chicago) for sample
gas loading for the x-ray diffraction measurements, and C. Kenney-Benson
(HPCAT) for technical assistance. R.S.K. and R.J.H. acknowledge support from
the U.S. National Science Foundation (DMR-2119308 and DMR-2104881). X-ray
diffraction measurements were performed at HPCAT (Sector 16), Advanced Photon
Source (APS), Argonne National Laboratory. HPCAT operations are supported by
the DOE-National Nuclear Security Administration (NNSA) Office of Experimental
Sciences. The beamtime was made possible by the Chicago/DOE Alliance Center
(CDAC), which is supported by DOE-NNSA (DE-NA0003975). Use of the gas loading
system was supported by COMPRES under NSF Cooperative Agreement No.
EAR-1606856 and by GSECARS through NSF grant EAR-1634415 and DOE grant DE-
FG02-94ER14466. The Advanced Photon Source is a DOE Office of Science User
Facility operated for the DOE Office of Science by Argonne National Laboratory
under Contract No. DE-AC02-06CH11357. High pressure equipment development at
the University of Florida was supported by National Science Foundation CAREER
award DMR-1453752.

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# ANALYSIS OF LONG-RANGE STUDIES IN THE LHC – COMPARISON WITH THE MODEL

D. Kaltchev  TRIUMF  Vancouver  Canada  W. Herr  CERN  Geneva  Switzerland

###### Abstract

We find that the observed dependencies (scaling) of long-range beam–beam
effects on the beam separation and intensity are consistent with the simple
assumption that, all other parameters being the same, the quantity preserved
during different set-ups is the first-order smear as a function of amplitude.

##  1 Introduction

###  1.1 The Proposed Method

In several Machine Development (MD) studies (see Ref. [1] and the references
therein), reduced crossing angles have been used to enhance long-range
beam–beam effects and thus facilitate their measurement. The basic assumption
made in this paper is that under such conditions, a single non-linearity, the
one caused by beam–beam, dominates the dynamics. Hence the method followed: we
choose some simple low-order dynamical quantity that characterizes phase space
distortion and assume that when this quantity is the same, the behaviour of
the system is the same. A most obvious candidate is the first-order smear –
the r.m.s. deviation of the phase-space ellipse from the perfect one. At a
fixed amplitude, smear is defined as the averaged generalized Courant–Snyder
invariant over the angle variable [2].

An analytical expression has previously been found [2] for the smear $S$ as a
function of amplitude $n_{\sigma}$. Suppose that the parametric dependence of
$S(n)$ on several beam–beam related parameters – the relativistic $\gamma$,
the number of particles per bunch $N_{b}$, the crossing angle $\alpha$, and
the normalized separations $n_{\rm l.r.}$ – is known. According to the above
assumption, for two machine configurations a and b one should have

$S(n_{\sigma};N_{b}^{a},n^{a}_{\rm
l.r.},\alpha^{a},\gamma^{a})=S(n_{\sigma};N_{b}^{b},n^{b}_{\rm
l.r.},\alpha^{b},\gamma^{b}).$ (1)

As a particular application of Eq. (1), we considered two experiments where
the intensities are $N_{b}^{a}$ and $N_{b}^{b}$. All other parameters being
the same, given $\alpha^{a}$, one can compute the expected $\alpha^{b}$. Our
task will be to show that the result agrees with observations.

###  1.2 Analytical Calculation of Invariant and Smear

Our derivation of $S(n_{\sigma})$ is based on the Lie algebraic method –
concatenation of Lie-factor maps – and is valid only to first order in the
beam–beam parameter and in one-dimension, in either the horizontal or the
vertical plane, but for an arbitrary distribution of beam–beam collisions,
head-on or long-range, around the ring.

For a ring with a single head-on collision point, Hamiltonian perturbation
analysis of the beam–beam interaction without or with a crossing angle has
been done by a number of authors, mostly in the resonant case. Non-linear
invariants of motion, both non-resonant and resonant, were analysed by Dragt
[3], with the one-turn map as observed immediately after the kick being

$R\;\mathrm{e}^{:{\mathit{F}}:}=\mathrm{e}^{:h:}\,.$ (2)

Here, $R=\mathrm{e}^{:f_{2}:}$ is the linear one-turn map and the kick factor
$F$ is the beam–beam potential (or Hamiltonian). For small perturbations and
far from resonances, particle coordinates in phase space are restricted on the
Poincaré surface of section

$h=\mathrm{const}.$ (3)

A detailed derivation of $h$ to first order in the beam–beam perturbation
strength can be found in A. Chao’s lectures:

$\displaystyle h(J,\phi)=-\mu
J+\sum_{n=-\infty}^{\infty}{c^{\rm(ho)}_{n}(J)\frac{n\,\mu}{2\sin\frac{n\mu}{2}}\mathrm{e}^{in(\phi+\mu/2)}}\,,$
(4)

where $\mu$ is the ring phase advance and $c^{\rm(ho)}_{n}(J)$ are
coefficients in the Fourier expansion of $F$, when the latter is rewritten in
action-angle coordinates $J,\phi$. The coefficients are shown to be related to
the modified Bessel functions. Analytical expressions for the invariant $h$,
the first-order smear, and the second-order detuning for the case of non-
linear multipole kicks distributed in an arbitrary way around the ring have
been derived by Irvin and Bengtsson [4]. Smear, the distortion of the ideal
phase-space ellipse, is formally defined in Ref. [5]. Finally, note that
extracting the smear is a natural step in the procedure that brings the map
into its normal form [6].

In Ref. [7], following the Lie algebraic procedure in Refs. [8] and [4], we
generalized Eq. (4) to describe multiple head-on kicks (IP1 and IP5) for the
case of the LHC. In Ref. [2], an expression was presented that was valid for
an arbitrary number of head-on (h.o.) and long-range (l.r.) collisions. This
expression, to be derived in detail next, has been used on several occasions
to interpret results from SixTrack simulations.

##  2 Derivation of the Invariant

###  2.1  Multiple Collision Points

The horizontal betatronic motion of a weak-beam test particle depends on its
initial amplitude $n_{\sigma}$ (in units of $\sigma$) and the collision set: a
set of all h. o. and l. r. collisions, also known as Interaction Points (IPs),
that this particle sees over a single revolution. Let us label the set with an
index $k$, limiting ourselves to only IPs located within the main interaction
regions IR5 (horizontal crossing) and IR1 (vertical crossing). In the case of
50 ns bunch spacing, $k$ ranges from 1 to 34, which includes 32 long-range IPs
($N_{\rm l.\,r.~{}}=32$).

The Lie map depends on the above-defined collision set through the normalized
separations $n^{(k)}_{x,y}=d^{(k)}_{x,y}/\sigma^{(k)}$ and the unperturbed
horizontal betatronic phases $\phi^{(k)}$ at the IPs. Here, $d_{x,y}$ is the
real-space offset of the strong-beam centroid in the $x$ or $y$ direction, and
it has been assumed that both the weak- and strong-beam transverse
distributions are round Gaussians of the same r.m.s. That is:

$\sigma^{(k)}=\sqrt{\beta^{(k)}\epsilon}\ \ \ \
(\beta^{(k)}_{x}=\beta^{(k)}_{y}\equiv\beta^{(k)}).$ (5)

In Eq. (5), $\beta^{(k)}$ are the beta functions and $\epsilon$ is the
emittance. It will be shown below that off-plane collisions contribute very
little to smear; thus after excluding these, the problem becomes one-
dimensional and may easily be illustrated (see Fig. 1). Here, $n^{(k)}_{x}$
are the strong-beam centroids in amplitude space: points
$(s^{(k)},n_{x}^{(k)})$, with $s$ being the distance to IP5 in metres.

Figure 1: A schematic view of weak- and strong-beam trajectories in real (top)
and amplitude (bottom) spaces. A reduced set is used: $N_{\rm
l.\,r.~{}}=8+8=16$ ($k=1,18$).

###  2.2 The Beam–Beam Hamiltonian

For a single collision (see Eq. (2)), by omitting the superscript $k$ in
$\sigma$ and $n_{x,y}$, the $x$-motion is described by a kick factor $F$ (or
Hamiltonian $H$) [2]:

$\displaystyle F=-H(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{P}(1-\mathrm{e}^{-\alpha})\frac{\mathrm{d}\alpha}{\alpha}\,=$
(6) $\displaystyle=$
$\displaystyle\overline{\gamma}+\Gamma_{0}(P)+\mathrm{ln}(P)\,,$ (7)
$P=P(x)=\frac{1}{2}\,\left[(n_{x}+\frac{x}{\sigma})^{2}+n_{y}^{2}\right]\,,$

where $F$ is in units of $\lambda\equiv\frac{N_{b}r_{0}}{\gamma}$, $r_{0}$ is
the classical particle radius, $\Gamma_{s}(P)\equiv\Gamma(s,P)$ denotes the
upper incomplete gamma function [9], and $\overline{\gamma}$ = 0.577216 is
Euler’s constant. The corresponding beam–beam kick is as follows:

$\displaystyle\Delta
x^{\prime}\equiv\frac{\mathrm{d}}{\mathrm{d}x}F(x)=\frac{\partial F}{\partial
P}\frac{\mathrm{d}P}{\mathrm{d}x}=$ $\displaystyle=$
$\displaystyle\frac{2(x+n_{x}\sigma)}{(x+n_{x}\sigma)^{2}+(n_{y}\sigma)^{2}}\left[1-\mathrm{e}^{-\frac{(x+n_{x}\sigma)^{2}+(n_{y}\sigma)^{2}}{2\sigma^{2}}}\right].$
(8)

The Fourier expansion of $H$ is as follows:

$H(n_{\sigma},\phi)=\sum_{m}{\cal C}_{m}\mathrm{e}^{im\phi}\,,$ (9)

where ${\cal
C}_{m}\equiv\frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{e}^{-im\phi}H\;\mathrm{d}\phi$.
These coefficients are easily computed numerically by using the implementation
of $\Gamma$ in Mathematica [2]. Further, analytical expressions in the form of
single integrals over Bessel functions have been derived in Ref. [10]. We
display these again in the simplified case $n_{y}=0$ (no off-plane
collisions):

$\displaystyle\left.{\cal
C}_{m}\right|_{n_{y}=0}=\displaystyle\int_{0}^{1}\frac{\mathrm{d}t}{t}\times$
$\displaystyle\begin{cases}\displaystyle[1-\mathrm{e}^{-\frac{t}{2}n_{x}^{2}}\mathrm{e}^{-\frac{t}{4}n_{\sigma}^{2}}\sum_{k=-\infty}^{\infty}I_{-2k}(t\,n_{\sigma}n_{x})I_{k}(-\frac{t}{4}n_{\sigma}^{2})]&\\\
\mbox{if }m=0\mbox{ \ \ and}\\\
\displaystyle-\mathrm{e}^{-\frac{t}{2}n_{x}^{2}}\mathrm{e}^{-\frac{t}{4}n_{\sigma}^{2}}\sum_{k=-\infty}^{\infty}i^{m}I_{m-2k}(t\,n_{\sigma}n_{x})I_{k}(-\frac{t}{4}n_{\sigma}^{2})&\\\
\mbox{if }m\neq 0.\end{cases}$

In the head-on case ($n_{x}^{(k)}=n_{y}^{(k)}=0$), the coefficients ${\cal
C}_{m}$ reduce to the $c^{\rm(ho)}_{m}$ from Ref. [8]. Note that in the most
interesting case, amplitudes near the dynamic aperture, both $n_{\sigma}$ and
$n_{x}$ and hence the Bessel function arguments are large ($\gg 1$).

Our first step is to remove the linear and quadratic parts
$F_{(1)}=\left.\frac{\partial F}{\partial x}\right|_{x=0}x$ and
$F_{(2)}=\frac{1}{2}\left.\frac{\partial^{2}F}{\partial
x^{2}}\right|_{x=0}\,x^{2}$. The non-linear kick factor and the corresponding
kick are as follows:

$\displaystyle F_{\rm nonl}=F-F_{(1)}-F_{(2)}\,\,,$ (10) $\displaystyle\Delta
x^{\prime}_{\rm nonl}\equiv\frac{d}{dx}F_{\rm nonl}(x)\,.$

As a next step, we rewrite Eq. (10) in action-angle coordinates $J,\phi$ by
substituting in it $x=\sqrt{2J\beta}\sin{\phi}=n_{\sigma}\sigma\sin{\phi}$,
where $n_{\sigma}=\sqrt{2I}=\sqrt{2J/\epsilon}$ is the test particle amplitude
(Eq. (A.1)). Next, we expand in Fourier series:

$\displaystyle\displaystyle F_{\rm
nonl}(n_{\sigma}\,\sigma\,\sin\phi)=c_{0}+\sum_{m\neq
0}c_{m}\mathrm{e}^{im\phi}\,.$ (11)

The coefficients $c_{m}$ are naturally the same as ${\cal C}_{m}$ above, with
the exception of $c_{1}$ and $c_{2}$, which contain additional $\mathrm{sin}$
and $\mathrm{sin}^{2}$ terms (see Eq. (A.1)).

###  2.3 Lie Map and Invariant

For an arbitrary set of collisions $n_{x}^{(k)}$, $\phi^{(k)}$ ($k=1,N$), we
represent the LHC lattice by a combination of linear elements and non-linear
kicks. It is shown in the Appendix that, to first order in $\lambda$, the Lie
map has the same form as the one for a single kick (2) – where, however, the
factor $F$ is given by the sum

$F\equiv\sum_{k=1}^{{N}}F_{\rm nonl}^{(k)}(n_{\sigma},\phi)$

and $F_{\rm nonl}^{(k)}$ are such that, compared to Eq. (11), the $k$th IP
participates with a phase shifted by $\phi^{(k)}$:

$\displaystyle F_{\rm nonl}^{(k)}(n_{\sigma},\phi)\equiv\left.F_{\rm
nonl}^{(k)}(x)\right|_{x\;\rightarrow
n_{\sigma}\,\sigma^{(k)}\,sin{(\phi+\phi^{(k)})}}=$ $\displaystyle=\sum_{m\neq
0}C_{m}^{(k)}\mathrm{e}^{im\phi}\,.$ (12)

The shift in phase means that the coefficients in Eq. (12) are simply related
to $c_{m}^{(k)}$: $C_{m}^{(k)}\equiv c_{m}^{(k)}\mathrm{e}^{im\phi^{(k)}}$ and
still satisfy $C_{-m}=C_{m}^{\star}$. Another important property of the
expansion is that only the oscillating part is taken (the $m=0$ term is
excluded). The invariant for multiple collision points is as follows (see the
Appendix):

$\displaystyle h(I,\phi)=-\mu
J-\lambda\displaystyle\sum_{k=1}^{N}\sum_{m=1}^{\infty}\frac{m\,\mu\,{c}^{(k)}_{m}(I)}{2\sin{(\frac{m\mu}{2})}}\mathrm{e}^{im(\phi+\mu/2+\phi^{(k)})}$
$\displaystyle+\;c.c.$

The surface of the section in phase space is given by
$h(I,\phi)=\mathrm{const}$. A natural initial condition is now imposed: that
the initial point in phase space for a particle starting at
$x_{0}=n_{\sigma}\sigma$ – that is, with an amplitude $I_{0}\equiv
J_{0}/\epsilon=n_{\sigma}^{2}/2$ – lies on the curve representing the
invariant:

$h(I,\phi)=h(I_{0},\pi/2),$ (13)

For a fixed $I_{0}$, this equation implicitly defines $I$ as a function of
$\phi$. It satisfies the initial condition $I(0)=I_{0}$:

$\displaystyle\displaystyle
I(\phi)=I_{0}+\sum_{k=1}^{{N}}\left(\mathrm{d}I^{(k)}(\phi)-\mathrm{d}I^{(k)}(0)\right)\,,$
$\displaystyle\mathrm{d}I^{(k)}(\phi)=$ (14)
$\displaystyle=\displaystyle\frac{\lambda}{\epsilon}\sum_{m=1}^{M}\left(\frac{m\,{c}^{(k)}_{m}(I_{0})}{2\sin{(m\mu/2)}}\mathrm{e}^{im(\mu/2+\phi-\phi^{(k)}+\pi/2)}+c.c.\right).$

Note that, to first order, the argument in ${c}^{(k)}_{m}$ has been replaced
with $I_{0}$. We have also separated the two sums so that
$\mathrm{d}I^{(k)}(\phi)-\mathrm{d}I^{(k)}(0)$ is the individual contribution
of the $k$th IP. In the same way, a different initial condition may be used
(more suitable for plots): $I(0)=I_{0}$, instead of $I(\pi/2)=I_{0}$.

The smear $S(n_{\sigma})$ is now defined as the normalized r.m.s. of the
invariant – that is, $\sqrt{V}$, with $V$ being the variance:

$\displaystyle S(n_{\sigma})={\sqrt{V}}/{\langle\,I\,\rangle}\,,$
$\displaystyle V=\frac{1}{2\pi}\int(I-\langle I\rangle)^{2}\mathrm{d}\phi\,,\
\ \ \ \langle I\rangle=\frac{1}{2\pi}\int I\mathrm{d}\phi\,.$

##  3 Verification with tracking

As an example application, this section studies the very simple collision set
that still possesses all the symmetries with the l.r. set at 8 sigma, as
depicted in Fig. 2. Both IR5 and IR1 are included. The goal here is to test
the invariant $I(\phi)$ by tracking with a simple model built with kicks
$\Delta x^{\prime}_{\rm nonl}$ alternating with linear matrices and SixTrack.
The parameters are as follows: energy $3.5~{}$TeV, $N_{\rm b}=1.2\times
10^{11}$, and normalized emittance $\epsilon_{n}=2.5\times 10^{-6}$.

Figure 2: The sample set-up: three collisions in each IR5 and IR1. The l.r.
are set at 8 sigma.

Tracking single particles at various amplitudes with the simple model produces
the results shown in Fig. 3. A particle starts with $n_{\sigma}=3$, or $7$
($I_{0}=4.5$, or $24.5$). The $c_{m}$ are computed with an accuracy of
$10^{-7}$ – the value of $M$ in Eq. (2.3) is about $40$.

Figure 3: An invariant tested on a simple kick-matrix model. Black points:
turn-by-turn coordinates ($\phi,I$) for $10^{3}$ turns. Red: invariant
$I(\phi)$ (initials chosen so that $I(\pi/2)=I_{0}$).

Since the beam–beam potential changes the linear optics, we need to find the
linearly perturbed matched $\beta$-function value at the initial point for
tracking. For the plots in Fig. 3, this is done in a separate run, using a
linear kick $(\Delta x^{\prime})_{\rm lin}$ (only terms $\sim x^{2}$ in the
Hamiltonian). This is similar to what is done in SixTrack. The resultant
matched $\beta$ is used to define the initial coordinate $x_{0}$ (through
$n_{\sigma}$). The values of the smear are shown at the top of each plot.

Plotting the smear over a range of amplitudes with all three methods – model,
SixTrack, and analytical $S(n_{\sigma})$ – results in Fig. 4. Note that here
the images of the strong-beam centroids (see Fig. 1) are represented by
vertical grey lines drawn at $0$ and $8$ sigma.

Figure 4: Agreement with SixTrack.

Let us now look at the individual contributions to $I(\phi)$ of the six IPs at
three amplitudes chosen arbitrarily; say, $n_{\sigma}=1,3$, and $7$.

Figure 5: Individual contributions $dI^{(k)}(\phi)-dI^{(k)}(0)$ – color code
as in Fig. 2.

The excursions (w.r.t. $I_{0}$) of the individual invariant surfaces are shown
in Fig. 5. Here, $I(0)=I_{0}$. The colour code is as in Fig. 2, and in
addition for the head-ons we use solid black for IP5 and dashed for IP1. Near
the axis ($n_{\sigma}=1$), only the two head-ons contribute and, being of
opposite signs, almost compensate each other. At $n_{\sigma}=3$, one begins to
see long-range contributions that grow when $n_{\sigma}=7$. At such large
amplitudes, the compensation is no longer true. Magenta and green are barely
seen, meaning that the contribution of off-plane collisions is negligible.
Thus in the case of a test particle moving in the horizontal motion, the
contribution of all l.r. in IR1 can be neglected, and vice versa for vertical
motion and IP5.

##  4 The behaviour of the smear $S(n_{\sigma})$ near the dynamic aperture

Above some critical strength of beam–beam interaction – that is, quantities
$N_{\rm l.\,r.~{}}$ and/or $N_{\rm b}$ and/or an inverse crossing angle – the
first-order theory is no longer an adequate description of the smear. However,
as we will see, the behaviour of $S(n_{\sigma})$ may still be used as an
indication of the dynamic aperture, since it exhibits a local maximum near it.

Figure 6: Various combinations of numbers of long-range collisions and bunch
intensities to illustrate linear and non-linear behaviour:

| $N_{\rm l.\,r.~{}}$ | $N_{\rm b}$
---|---|---
(a) | $16$ | $1.2\times 10^{11}$
(b) | $32$ | $1.2\times 10^{11}$
(c) | $32$ | $1.6\times 10^{11}$
(d) | $32$ | $0.2\times 10^{11}$

What happens is that the linear behaviour – that is, the agreement between the
first-order $S$ and SixTrack at all amplitudes seen in Fig. 4 – is replaced by
what is shown in Figs. 6(a)–(c). The blue ($S(n_{\sigma})$) and the red
(SixTrack) curves depart from each other once $n_{\sigma}$ approaches
amplitudes near the strong-beam core, represented by the cluster of vertical
grey lines. At this point, the exact smear (red) exhibits a steep growth; thus
the dynamic aperture is likely to be close to this point, while $S$ goes
through a maximum and then through a minimum, thus forming a dip. Upon exiting
the core, past the last grey line, the red and blue curves almost re-merge. It
can be shown that the above property of $S(n_{\sigma})$ is a consequence of
the left–right symmetry of IR5 and IR1. Namely, the individual contributions
(such as the red and blue curves in Fig. 2) change sign or flip about the axis
each time $n_{\sigma}$ crosses a grey line. At this amplitude, $S(n_{\sigma})$
stops growing and goes through a maximum.

##  5 Analysis of long-range experiments

###  5.1 Dependence on Intensity and Crossing Angle

We set the parameters as at the MD: energy $3.5~{}$TeV,
$\epsilon_{n}=2.5\times 10^{-6}$ [11], and $\beta^{\star}=0.6$ m.

Of all the collision sets used at the MD, let us consider three: $N_{\rm
l.\,r.~{}}=32$, 24, and 16. For each of them, two parameters, the bunch
intensity $N_{\rm b}$ and the (half) crossing angle $\alpha$, uniquely define
the dependence of the first-order smear on amplitude $S(n_{\sigma};N_{\rm
b},\alpha)$ through the following procedure. First, being a first-order
quantity in $\lambda$, the smear is obviously proportional to the intensity:
$S\sim N_{\rm b}$. Second, the dependence of $n_{x,y}^{(k)}$ on the (half)
crossing angle $\alpha$ is given by the well-known scaling law:
$n_{x,y}^{(k)}\sim\alpha\sqrt{\beta^{\star}}$, where $n_{x,y}^{(k)}$ are taken
from some sample lattice built for $\beta^{\star}=0.55$ m and $\alpha=125$.
Finally, the phases $\phi^{(k)}$ are assumed to be independent of $\alpha$.

The dependence on the angle is presented in Fig. 7. Each blue branch
corresponds to $S(n_{\sigma};1.2\times 10^{11},\alpha)$ being taken over an
amplitude range where it is monotonically increasing; hence, as we already
know, it will remain in agreement with the tracking for any strength of the
beam–beam interaction.

Figure 7: The dependence of the smear(amplitude) graph on the parameter
$\alpha$ for $N_{\rm l.\,r.~{}}=32$ (top) and $N_{\rm l.\,r.~{}}=24$, $16$
(bottom). Each graph is restricted within a domain extending up to its first
maximum (red dot) (the entrance into the strong-beam core). Figure 8: The
collision sets for $N_{\rm l.\,r.~{}}$=24 and 16 are built by dropping the
first and last two or four elements from the full set ($N_{\rm
l.\,r.~{}}=32$).

Coming now to the MD, the observed losses during reduction of the crossing
angle in IP1 are shown in Figs. 10 and 10 [1].

Figure 9: An experiment with $N_{\rm b}=1.2\times 10^{11}$: losses start at
$\alpha_{1}\approx 87\;~{}\mu{\rm rad}$.

Figure 10: An experiment with $N_{\rm b}=1.6\times 10^{11}$: losses start at
$\alpha_{1}\approx 96~{}\mu{\rm rad}$.

###  5.2 An Explanation of the Case $N_{\rm l.\,r.~{}}=32$ (Brown Curves)

For $N_{\rm l.\,r.~{}}=32$ (the full $50$-ns collision set shown in Fig. 8),
we need to explain the brown curves in Figs. 10 and 10. Here, losses are seen
to start at $\alpha_{1}\approx 87$ and $\alpha_{2}\approx 96~{}\mu{\rm rad}$,
respectively.

In view of our previous findings, the off-plane losses (in IR5) are neglected
and by using the postulate made in the Introduction (Eq. 1), we have:

$\displaystyle S(n_{\sigma};1.2\times
10^{11},\alpha_{1})=S(n_{\sigma};1.6\times 10^{11},\alpha_{2}),$ (15)

which is to be solved for the angles.

Figure 11: Graphs of  $S(n_{\sigma};1.2\times 10^{11},86)$ (red) and
$S(n_{\sigma};1.6\times 10^{11},96)$ (blue). The smear is seen to be $\approx
3\%$ at 1.5 $\sigma$.

Figure 11 shows that a good solution to Eq. (15) consists of the values
$\alpha_{1}=86$, $\alpha_{2}=96~{}\mu{\rm rad}$. Indeed, this figure shows
that Eq. (15) is fulfilled not in a single point, but for all amplitudes up to
1.5 $\sigma$, where the smear reaches $\approx 3\%$. What has happened, of
course, is that scaling by a factor $1.6/1.2$, but reducing the angle from
$\alpha_{2}$ to $\alpha_{1}$, has almost exactly preserved one particular blue
branch from Fig. 7. Conversely, small variations about this solution, say $\pm
5~{}~{}\mu{\rm rad}$, lead to deviations of red and blue curves, as shown in
Fig. 12.

Figure 12: Small variations about the solution $\pm 5~{}~{}\mu{\rm rad}$.

###  5.3 Explanation of Cases $N_{\rm l.\,r.~{}}$=24 and 16 (Green and Black)

For $N_{\rm l.\,r.~{}}$=24 and 16 (reduced collision sets in Fig. 8), one
needs to explain the green and black decay curves in Figs. 10 and 10. By
looking now at the bottom two plots in Fig. 7, we search for blue branches
that pass through the same maximum-smear point as found above: $3\%$ at 1.5
$\sigma$. The resultant branches are plotted in Figs. 13 and 14, with solution
angles as summarized in Table 1. Again, at least a qualitative agreement is
observed to the extent allowed by the resolution of Figs. 10 and 10.

Table 1: Angles of solutions for different intensities. $N_{\rm b}$ | Green | Black
---|---|---
$1.2\times 10^{11}$ | 65 | 53
$1.6\times 10^{11}$ | 83 | 72

Figure 13: $N_{\rm b}=1.2\times 10^{11}$.

Figure 14: $N_{\rm b}=1.6\times 10^{11}$.

Of the four plots in Figs. 13 and 14, on three occasions the 3%-smear line
intersects a monotonic part of $S(n_{\sigma})$ where, as we already know from
Section 4, there is an exact agreement with SixTrack. The rough indication for
the dynamic aperture, as the amplitude corresponding to a maximum of $S$, has
been used in only one case: $\alpha$=53.

## References

  * [1] R. Assmann et al., “Results of Long-Range Beam–Beam Studies – Scaling with Beam Separation and Intensity,” CERN-ATS-Note-2012-070 MD (2012).
  * [2] W. Herr and D. Kaltchev, “Analytical Calculation of the Smear for Long Range Beam–Beam Interactions,” Proc. PAC 2009.
  * [3] A.J. Dragt and O.G. Jakubowicz, “Analysis of the Beam–Beam Interaction Using Transfer Maps,” Proc. Beam–Beam Interaction Seminar, Stanford, CA, 22–23 May 1980, SLAC-R-541.
  * [4] J. Bengtsson and J. Irwin, “Analytical Calculation of Smear and Tune Shift,” SSC-232 (February 1990).
  * [5] M.A. Furman and S.G. Peggs, “A Standard for the Smear,” SSC-N-634 (1989).
  * [6] E. Forest, “Analytical Computation of the Smear,” SSC-95 (1986).
  * [7] D. Kaltchev, “On Beam–Beam Resonances Observed in LHC Tracking,” TRI-DN-07-9 (2007).
  * [8] A. Chao, “Lie Algebra Techniques for Nonlinear Dynamics,” http:www.slac.stanford.edu/ãchao
  * [9] M. Abramowitz and I.A. Stegun, _Handbook of Mathematical Functions_ (New York: Dover, 1972).
  * [10] D. Kaltchev, “Hamiltonian for Long-range Beam-Beam, Fourier coefficients,” TRIUMF Note (2012).
  * [11] The use of $\epsilon_{n}=2\times 10^{-6}$ as an alternative does not change any results.

##  6  Appendix

The non-linear kick factor in Eq. (10) is

$\displaystyle F_{\rm
nonl}(n_{\sigma},\phi)=\gamma+\Gamma_{0}(P)+\mathrm{ln}(P)-F_{(1)}-F_{(2)}\,,$
$\displaystyle
P=\frac{1}{2}\left((n_{x}+n_{\sigma}\sin{\phi})^{2}+n_{y}^{2}\right)\,,$
$\displaystyle
F_{(1)}=\frac{2n_{x}}{(n_{x}^{2}+n_{y}^{2})}\left(1-\mathrm{e}^{-\frac{n_{x}^{2}+n_{y}^{2}}{2}}\right)n_{\sigma}\sin{\phi}\,,$
$\displaystyle
F_{(2)}=\frac{-n_{x}^{2}+n_{y}^{2}+\mathrm{e}^{-\frac{n_{x}^{2}}{2}-\frac{n_{y}^{2}}{2}}\left(n_{x}^{2}+n_{x}^{4}-n_{y}^{2}+n_{x}^{2}n_{y}^{2}\right)}{\left(n_{x}^{2}+n_{y}^{2}\right)^{2}}\times$
$\displaystyle\times\,n_{\sigma}^{2}\sin^{2}\phi.\,\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm(A.1)}$

By following Ref. [4], the Lie map is given by an expression of the following
form:

$\displaystyle
M_{N+1}\mathrm{e}^{:f^{(N)}:}M_{N}\dots\mathrm{e}^{:f^{(2)}:}M_{2}\mathrm{e}^{:f^{(1)}:}M_{1}\,,$
$\displaystyle f^{(k)}(x)\equiv F_{\rm nonl}^{(k)}(x).$

Here, $M_{k}$ are linear operators and for brevity we have replaced $F_{\rm
nonl}^{(k)}(x)$ with $f^{(k)}(x)$ . We will show that since $F_{\rm nonl}$
depends only on the normalized coordinate $x/\sigma$, once we rewrite it in
terms of the eigen-coordinates at the $k$th kick, the local beta functions
$\beta^{(k)}$ disappear, while the phase $\phi^{(k)}$ is simply added to
$\phi$.

By reversing the order, the map transforming the test particle $(x,p_{x})$ for
one turn around the ring is

$\displaystyle{\cal
M}=M_{1}\mathrm{e}^{:f^{(1)}:}M_{2}\mathrm{e}^{:f^{(2)}:}\dots
M_{N}\mathrm{e}^{:f^{(N)}:}M_{N+1}=$
$\displaystyle=\mathrm{e}^{:\overline{M}_{1}f^{(1)}:}\mathrm{e}^{:\overline{M}_{2}f^{(2)}:}\dots\mathrm{e}^{:\overline{M}_{N}f^{(N)}:}\overline{M}_{N+1}.$

Reversal of the order means that in the first line all $f^{(k)}$ are now
functions of the same initial variables $(x,p_{x})$. In the second line,
accumulated linear maps $\overline{M}_{k}=M_{1}M_{2}...M_{k}$ have been
applied to transform the initial vector to the kick location. Thus, as a first
step, we have moved all kicks to the front of the lattice and
$\overline{M}_{N+1}$ is the total one-turn linear Lie operator.

Let us denote matrices corresponding to Lie operators with hats; for example,
$\widehat{M}_{N+1}$. As a second step, with $\beta$, $\alpha$ being matched
Twiss parameters at the end of the lattice, one uses an $A_{0}$ transform that
transforms the ring matrix to a rotation (inserting identities ${\cal
A}_{0}{\cal A}^{-1}_{0}$ in between the exponents):

$\displaystyle\widehat{M}_{N+1}\stackrel{{\scriptstyle{\widehat{\cal
A}_{0}}}}{{\longrightarrow}}{\footnotesize\widehat{R}=\left(\begin{array}[]{cc}\cos\mu&\sin\mu\\\
-\sin\mu&\cos\mu\end{array}\right)},$ $\displaystyle{\widehat{\cal
A}_{0}}=\left(\begin{array}[]{cc}\sqrt{\beta}&0\\\
-\alpha/\sqrt{\beta}&1/\sqrt{\beta}\end{array}\right).$

The two steps above combined are equivalent to replacing the argument of $f$
by $\widetilde{x}_{k}$ – the eigen-coordinate at the $k$th location. To see
this, apply the ${\cal A}_{0}$ transform to both kick factor and coordinate:

${\cal A}_{0}\overline{M}_{k}f^{(k)}(x)=f^{(k)}({\cal
A}_{0}\overline{M}_{k}x)=f^{(k)}(\widetilde{x}_{k})\,,$
$\displaystyle{\widetilde{x}_{k}}\equiv{\cal
A}_{0}\overline{M}_{k}x=\sqrt{2\beta^{(k)}J}\sin{(\phi+\phi^{(k)})}\;.$

One can now drop the ${\cal A}_{0}$ on both sides of $\cal M$ and consider the
map:

$\displaystyle{\cal
M}=\mathrm{e}^{:\widetilde{f}^{(1)}:}\mathrm{e}^{:\widetilde{f}^{(2)}:}\dots\mathrm{e}^{:\widetilde{f}^{({N})}:}R,$
$\displaystyle\widetilde{f}^{(k)}(J,\phi)=f^{(k)}(\widetilde{x}_{k}),$
$\displaystyle R=\mathrm{e}^{:f_{2}:},\ \ \ \ \ \ \ {:f_{2}:}=-\mu J.$

To first order, one can just sum the Lie factors:

$\displaystyle{\cal M}\approx\mathrm{e}^{:F:}\;R=\mathrm{e}^{:h:},\ \ \ \ \ \
\ F\equiv\sum_{k=1}^{{N}}\widetilde{f}^{(k)}.$

By noting that above, as in Ref. [4], $R$ precedes the kick, while in Eq. (2)
and Ref. [8] the kick is assumed to be at the end of the lattice, our map is
identical to Eq. (2).

The first-order invariant $h$ is now found with the BCH theorem. Let us write
$F=\bar{F}+F^{\star}$, where $F^{\star}$ is the oscillating part. By taking
only $F^{\star}$:

$\displaystyle
h(J,\phi)=f_{2}+\frac{:f_{2}:}{1-\mathrm{e}^{-:f_{2}:}}F^{\star},\ \ \
{\rm(A.2)}$ $\displaystyle
F^{\star}\equiv\sum_{k=1}^{{N}}(\widetilde{f}^{(k)})^{\star},$

where according to Eq. (12),

$\displaystyle(\widetilde{f}^{(k)})^{\star}=\sum_{m\neq
0}C_{m}^{(k)}\mathrm{e}^{im\phi}=\sum_{m=1}^{\infty}\left(C_{m}^{(k)}\mathrm{e}^{im\phi}+cc\right)\,.$

A basic property of ${:f_{2}:}$ is to operate in a simple way on functions of
$J$, or eigenvectors $\mathrm{e}^{in\phi}$. Also, functions $G(f_{2})$ can
easily be applied to eigenvectors:

${:f_{2}:}\;\mathrm{e}^{in\phi}=i\,n\,\mu\,\mathrm{e}^{in\phi},$
$G({:f_{2}:})\mathrm{e}^{in\phi}=G(i\,n\,\mu\,)\mathrm{e}^{in\phi}.$

If we choose $G({\colon
f_{2}\colon})\equiv\frac{:f_{2}:}{1-\mathrm{e}^{:f_{2}:}}$, then we have:

$\displaystyle:G(f_{2}):\mathrm{e}^{im\phi}=$
$\displaystyle=G(im\mu)\mathrm{e}^{im\phi}=$
$\displaystyle=\frac{im\mu}{1-\mathrm{e}^{-im\mu}}\mathrm{e}^{im\phi}=$
$\displaystyle=\frac{i\,m\,\mu\,\mathrm{e}^{im\phi}}{\mathrm{e}^{i\,m\,\mu/2}-\mathrm{e}^{-i\,m\,\mu/2}}\mathrm{e}^{i\,m\,\mu/2}=$
$\displaystyle=\frac{m\,\mu\,\mathrm{e}^{im\phi}}{2\,\sin{(m\,\mu/2)}}\mathrm{e}^{i\,m\,\mu/2}.$

By substituting all these in Eq. (A.2) and using the property
$C_{m}^{(k)}=c_{m}^{(k)}\mathrm{e}^{im\phi^{(k)}}$, we obtain:

$\displaystyle h(J,\phi)=$ $\displaystyle=$ $\displaystyle-\mu
J-\displaystyle\lambda\;\sum_{k=1}^{N}\sum_{m=1}^{\infty}\left(\frac{m\;\mu\;c^{(k)}_{m}}{2\sin{(m\mu/2)}}\mathrm{e}^{im(\mu/2+\phi+\phi^{(k)})}+c.c.\right).$


119–126

# Galaxy and Mass Assembly (GAMA):
Selection of the Most Massive Clusters.

Héctor J. Ibarra-Medel1,2 Maritza Lara-López3 Omar López-Cruz1 the GAMA Team
1National Institute of Astrophysics, Optics and Electronic,
Luis Enrique Erro No 1, Tonantzintla, Puebla, México
email<EMAIL_ADDRESS>
2 email<EMAIL_ADDRESS>
3Institute of Astronomy, National Autonomous University of México,
Box 70-264, México City, México
email<EMAIL_ADDRESS>
(2014)

###### Abstract

We have developed a galaxy cluster finding technique based on the Delaunay
Tessellation Field Estimator (DTFE) combined with caustic analysis. Our method
allows us to recover clusters of galaxies within the mass range of $10^{12}$
to $10^{16}\ \mathcal{M}_{\odot}$. We have found a total of 113 galaxy
clusters in the Galaxy and Mass Assembly survey (GAMA). In the corresponding
mass range, the density of clusters found in this work is comparable to the
density traced by clusters selected by the thermal Sunyaev Zel’dovich Effect;
however, we are able to cover a wider mass range. We present the analysis of
the two-point correlation function for our cluster sample.

###### keywords:

methods: data analysis, surveys, galaxies: clusters: general, cosmology:
miscellaneous, cosmology: observations

††volume: 308††journal: The Zeldovich Universe Genesis and Growth of the
Cosmic Web††editors: Rien van de Weygaert, Sergei Shandarin, Enn Saar & Jaan
Einasto, eds.

##  1 Introduction

The GAMA survey (http://www.gama-survey.org/) is a multi-wavelength
spectroscopic survey that covers $\sim 360$ deg2, which includes $\sim
400,000$ galaxy redshifts down to a magnitude limit of $r_{AB}=19.8$ ([Driver
et al. (2011), Driver et al. 2011]). We chose three stripes within GAMA that
cover $\sim 144$ deg2 with $\sim 110,000$ galaxy spectra. These three
equatorial sky stripes are centred at 9h, 12h and 14.5h ([Driver et al.
(2011), Driver et al. 2011]).

We have implemented a new cluster finding technique to find overdensities and
estimate cluster masses, simultaneously. We find number galaxy overdensities
by using an adaptive method based on the Delaunay Tessellation Field Estimator
([Schaap & van de Weygaert(2000), platen, DTFE, Schaap & van de Weygaert 2000,
Platen 2009]), mass estimation is done using caustic analysis ([Serra et al.
(2011), Alpaslan et al. (2012), e.g., Serra et al. 2011, Alpaslan et al.
2012]). We use this method to detect clusters of galaxies within the mass
range of $10^{12}$ to $10^{16}\ \mathcal{M}_{\odot}$, up to $z=0.3$.

Figure 1:  The left panel present the comoving density traced by the cluster
found in this study (indicated by filled stars), we have compare our results
those of Menanteau et al, (2013) generated from a sample of massive clusters
selected by the Sunyaev-Zel’dovich effect (open triangles), the continuous
dashed line is a simple halo occupation model (HOD). In the right panel we
present a compilation of previous results on the characteristic scale for the
two-point cluster correlation as a function of cluster separation, the filled
big dots represents the results for our sample. We find agreement with
previous studies and models

##  2 Overview and Results

We have found 113 cluster within GAMA. For this sample we have estimated
positions, cluster redshifts, velocity dispersions, cluster sizes, and cluster
integrated luminosity. Our algorithm has been tested using the GAMA mock
catalogs ([Robotham et al. (2011), Robotham et al. 2011]). The calculation the
cluster luminosities have been generated by using the individual cluster-
galaxy luminosity functions (LF) corrected for completeness. We have evaluated
the cluster selection function by the application of a simple halo occupation
distribution (HOD) model. We want to stress that the density of clusters found
by mass selection methods (e.g., the Atacama Cosmology Telescope (ACT),
[Menanteau et al.$2013)$, Menanteau et al. 2013]) is comparable to one found
in this work; however, we have covered a larger mass range by more than three
orders of magnitude. In addition, we have generated the two-point correlation
for clusters of galaxies for our sample. We find broad agreement previous
observations and predictions ([Estrada et al. (2009), Estrada et al. 2009]).
We have generated the mass-to-light ratio (M/L) for the clusters and BCGs in
our sample, we find that a single power law
$\mathcal{L}\propto\mathcal{M}^{\eta}$ can describe . We found $\eta=0.6-1$
for clusters and $\eta_{BGC}=0.1-0.4$ for BCGs. These relations agree with the
results of [Lin et al. (2004), Lin et al.(2004)] and [Lin & Mohr(2004), Lin &
Mohr(2004)].

The sample found in this study can be used for further studies in galaxy
evolution and its relation with environment. We have shown that optical
surveys such as GAMA can be used to select cluster by mass. A sample of
cluster selected by our method can be used to traced baryon acoustic
oscillations using in a survey in which galaxies are selected in the same
fashion as GAMA but covering a larger volume.

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