# Design of the PMT underwater cascade implosion protection system for JUNO

Miao He<EMAIL_ADDRESS>Institute of High Energy Physics, Beijing 100049, China
Zhonghua Qin  Institute of High Energy Physics, Beijing 100049, China
Shaojing Hou  Institute of High Energy Physics, Beijing 100049, China
Xiaoping Jing  Institute of High Energy Physics, Beijing 100049, China
Hongbang Liu  Guangxi University, Nanning 530004, China  Zunjian Ke  Institute
of High Energy Physics, Beijing 100049, China  Diru Wu  Institute of High
Energy Physics, Beijing 100049, China  University of Chinese Academy of
Sciences, Beijing 100049, China.  Wan Xie  Institute of High Energy Physics,
Beijing 100049, China  Mehang Xu  Institute of High Energy Physics, Beijing
100049, China  Fang Chen  Institute of High Energy Physics, Beijing 100049,
China  Junguang Lu  Institute of High Energy Physics, Beijing 100049, China
Yuekun Heng  Institute of High Energy Physics, Beijing 100049, China  Jiawen
Zhang  Institute of High Energy Physics, Beijing 100049, China  Xiaoyan Ma
Institute of High Energy Physics, Beijing 100049, China  Zhipeng Du  Naval
Research Institute, Beijing 100161, China

###### Abstract

Photomultiplier tubes (PMTs) are widely used underwater in large-scale
neutrino experiments. As a hollow glass spherelike structure, implosion is
unavoidable during long-term operation under large water pressure. There is a
possibility of cascade implosion to neighbor PMTs due to shockwave. Jiangmen
Underground Neutrino Observatory designed a protection structure for each
20-inch PMT, consisting of a top cover, a bottom cover, and their connection.
This paper introduces the requirement and design of the PMT protection system,
including the material selection, investigation of manufacture technology, and
prototyping. Optimization and validation by simulation and underwater
experiments are also presented.

##  1 Introduction

Jiangmen Underground Neutrino Observatory (JUNO) is designed for the
determination of the neutrino mass ordering, precision measurement of neutrino
oscillation parameters, and other physics motivations [1]. The JUNO detector
consists of 20 kiloton liquid scintillator and more than 40 thousand
photomultiplier tubes (PMTs), immersed in a 44 m deep water pool [2, 3]. The
expected energy resolution is 3% @ 1 MeV to precisely measure the fast
oscillation structure in the reactor antineutrino spectrum. The lifetime of
JUNO is expected to be larger than 20 years.

All PMTs used in JUNO were made of quartz glass evacuated to $\sim 10^{-5}$
Pa. Therefore, pressure coming from water and the atmosphere on the PMT is up
to 0.54 MPa. The large hydrostatic pressure may break a PMT glass bulb during
long-term operation. In this case, shockwave will be generated because of
water rushing into the vacuum and then reversely ejecting in a short time. It
propagates in water and creates cascade implosion to a large number of PMTs as
a chain reaction, which happened in Super-Kamiokande in 2001 [4]. Super-
Kamiokande rebuilt the full detector including all $\sim$11,000 20-inch PMTs
in 2006 with a protection structure assembled to each PMT, made of acrylic on
the top and fiberglass-reinforced plastic on the bottom [5]. This structure
will largely reduce the water flow in case of implosion of a single PMT, and
thus prevent the generation or significantly reduce the strength of the
shockwave to avoid the chain reaction. This design has been inherited by
Hyper-Kamiokande [6], while fiberglass is going to be replaced with stainless
steel for less light emission and radioisotopes.

Simulation and experimental studies for the 12-inch PMT underwater implosion
and the impact on the neighbor PMTs were carried out by LBNE [7, 8, 9]. The
energy of the shockwave is in proportion to the product of water pressure and
the PMT volume, while the peak pressure at a certain distance $L$ from the
implosion center drops as $\sim L^{-1}$. Similar studies done by JUNO in 2015
[10] using 20-inch PMTs in 0.5 MPa water showed the peak pressure to the
neighbor PMT surface was about 20 MPa, and the chain reaction always happened
if there is no protection to the imploded PMT. It took about 10 ms for the
shockwave to generate since the PMT started crushing. The propagation speed of
shockwave in water was measured as 1481 m/s like the sound and the magnitude
dropped around $L^{-1.2}$, the latter depending on the shape and crushing time
of the glass bulb, as well as neighbor PMTs and other mechanical structure.

The water depth in JUNO is similar to Super-Kamiokande, while the number of
20-inch PMTs in JUNO is almost twice of Super-Kamiokande and they are more
closely packed. Therefore it is more challenging to deploy the protection
structure. Both design and manufacture need to be reviewed. In addition,
transparency of the protection structure in front of the PMT photocathode is
more critical for JUNO since it affects the detected number of photons and
then the energy resolution. In this paper, we will present the R&D of the
protection system in JUNO. The requirement will be introduced in Sec. 2.
Design and optimization are presented in Sec. 3, including the material
selection, manufacturing technology investigation, prototyping, underwater
tests, and simulation. The final design and validation are discussed in Sec.
4. Sec. 5 is the summary and perspective.

##  2 JUNO requirements on implosion protection

The design of the JUNO detector is shown in Fig. 1 (a). Liquid scintillator is
filled in a 12-cm thick acrylic vessel with an inner diameter of 35.4 m, which
is supported by a stainless steel structure through 590 connecting nodes. The
majority of scintillation light is between 380 nm and 550 nm, with the most
probable value around 420 nm [11]. 17,612 20-inch PMTs and 25,600 3-inch PMTs
are installed surrounding the liquid scintillator, providing 77.9% optical
coverage. All the detector is in a water pool, which also serves as a water
Cherenkov detector after equipping with 2,400 20-inch PMTs to tag cosmic
muons. There is another plastic scintillator detector on top of the water pool
providing precision tracking of cosmic muons.

The total number of $\sim$20,000 20-inch PMTs consists of two types:
$\sim$15,000 Micro-Channel Plate (MCP) PMTs from North Night Vision Technology
Co., Ltd (NNVT) and $\sim$5,000 dynode PMTs from Hamamatsu Photonics K.K.
(HPK). Taking the NNVT PMT as an example (Fig. 1 (b)), the glass bulb is made
of an $\sim$3 mm thick ellipsoid with a neck. The total volume in the glass
bulb is $\sim$50 L. The HPK PMT has a similar shape and dimensions but the
neck is larger, resulting in $\sim$10 L more volume.

(a)

Figure 1: (a) Schematic drawing of the JUNO detector. 20 kiloton liquid
scintillator is contained in an acrylic spherical vessel which is supported by
a stainless steel truss through 590 supporting nodes. 20,012 20-inch PMTs and
25,600 3-inch PMTs are installed on the stainless steel structure and are
surrounded by earth magnetic field compensation coils. All of the detector
components are immersed in a 44-m high water pool. A plastic-scintillator
based detector is on top of the water pool to track cosmic muons. (b)
Schematic drawing of the 20-inch PMT from North Night Vision Technology Co.
Ltd.

The minimum clearance between two 20-inch PMTs is only 25 mm in JUNO to
achieve the required optical coverage, which is almost an order of magnitude
smaller compared to Super-Kamiokande (40% coverage [4] and $\sim$20 cm
clearance). This is the major constraint for JUNO that the thickness of the
protection structure needs to be optimized and precisely controlled. This
small clearance also results in a factor of two larger peak pressure on the
neighbor PMT surface between JUNO and Super-Kamiokande. The JUNO requirements
on cascade implosion protection are as follows:

  * •

50 m pedestal water pressure resistant. Unlike the hydrostatic pressure on the
PMT, there will be pedestal pressure on the protection structure when the
inside PMT implodes.

  * •

No chain reaction in case of a single PMT implosion. Hydrostatic water
pressure tolerance of the JUNO 20-inch PMT is about 1 MPa [12], and we assume
the implosion threshold from shockwave is larger than 1 MPa because its impact
time is only tens of microseconds.

  * •

Accommodate 25 mm clearance.

  * •

Light absorption $<$ 1% at 420 nm and $<$7% at 380 nm. Taking acrylic as an
example, these requirements correspond to $>$91% and $>$85% transparency in
air at 420 nm and 380 nm, respectively, where there is an approximate 8% loss
of light due to reflection between acrylic and air. In water, reflected light
will be smaller than 1% because of the closer refractive index to air.

  * •

Compatible with pure water.

  * •

Low background. At least an order of magnitude lower radioactivity compared
with the PMT itself.

##  3 Preliminary design and manufacture investigation

The conceptual design of the JUNO PMT protection is similar to Super-
Kamiokande. Each PMT will be equipped with a pair of covers, consisting of a
top cover and a bottom cover, as shown in Fig. 2. The top cover is a shell
combined with a semi-ellipsoid and a short cylinder. It is transparent, with
the material investigated and selected from several candidates. The bottom
cover is a truncated ellipsoid connected to a cylinder, made of stainless
steel. There are a few holes on each cover (not shown in the figure) to let
water in and air out. Six hooks or screws connect the two covers. A stainless
steel shell filled with sealants provides waterproofing for the electrical
components on the bottom of the PMT, and this shell is fixed by a pair of
clamps. There are several plastic gaskets between the PMT and two covers for
support. There are two connecting bars on the stainless steel cover serving as
the interface to the detector’s main structure. In this paper, we only
introduce the design of the top cover, the bottom cover, and the connection
structure in between, while other waterproofing or mechanical structure will
be reported in the future. In addition, the design was done based on the NNVT
PMT and was assumed to be applicable to the HPK PMT.

Figure 2: Conceptual design of the PMT protection system. The top cover is
transparent and the bottom cover is made of stainless steel. The two covers
are connected by six hooks. Several plastic gaskets are used as buffer between
the PMT glass and the protection cover. Another stainless steel shell filled
with sealants is on the bottom of the PMT for the waterproofing and is fixed
by a pair of clamps. The full PMT unit is installed on the detector main
structure through two connecting bars.

###  3.1 Material selection for the top cover

Polymethylmethacrylate (PMMA), also known as acrylic, is a transparent
thermoplastic widely used in neutrino experiments as a container of the
detector target, either water such as SNO [13], or organic liquid such Daya
Bay [14], Double Chooz [15], and RENO [16]. In particular, Super-Kamiokande
used acrylic to produce the PMT protection cover after the accident. The water
compatibility was found to be excellent and the radioactivity can be easily
controlled to be better than 10 ppt [13, 17]. On the other hand, acrylic is
fragile and the elongation at break is only a few percent, which means a
little deformation could end up with a fracture. Therefore, two other
materials with much larger elongation at break, Polycarbonate (PC) and
Polyethylene terephthalate (PET) were also suggested. The major mechanical
parameters of these three materials were investigated and compared in Table 1.

Material | PMMA | PC | PET
---|---|---|---
Refractive index | 1.49 | 1.58 | 1.58-1.64
Elongation at break (%) | 2.5-4 | 100-150 | 60-165
Tensile strength (MPa) | 80 | 55-75 | 80
Tensile modulus (MPa) | 2400-3300 | 2300-2400 | 2000-4000
Izod impact strength (J$\cdot$m-1) | 16-32 | 600-850 | 13-35
Table 1: Comparison of transparent material for the top cover. All data are
taken from Ref. [18] for consistency. It is worth to note that mechanical
performances also depend on the manufacture technology. In particular, tensile
strength of PMMA shown here corresponds to casting, while injection molding
typically gives lower result similar to PC, which will be shown in Table 2.

A few PC and PET sheets were made with different thicknesses, and the
transparency in air was measured with a commercial spectrophotometer, with the
results at 420 nm shown in Fig. 3. The uncertainty was estimated as 0.5%, by
measuring the same sample several times and by comparing it with another
transparency measurement system [19]. The light absorption was assumed to be
linear to such a small thickness and fitted by $y=p0+p1\cdot x$, where $y$
represents the transparency and $x$ represents the sample thickness. The PET
samples were also sent to National Institute of Metrology (NIM) and the
measurements gave very consistent results. From the fitting, the light
absorption was found to be 1.9% per mm for PET and 0.8% per mm for PC. Taking
into account both the light absorption and the optical coverage, to reach the
same level of detected light with 10 mm acrylic cover, the thickness of the
PET cover and PC cover can only be 1.5 mm and 3 mm, respectively. Samples of
PET covers with 1.5-5 mm thickness and PC covers with 3 mm thickness were
produced while all of them failed in the underwater tests, which will be
introduced in Sec. 3.5. In the end, acrylic was chosen to be the material for
the top cover.

(a)

(b)

Figure 3: Transparency at 420 nm in air of (a) PET and (b) PC as a function of
their thicknesses.

###  3.2 Manufacture technology investigation and prototyping for the top
cover

Three technologies were tried to produce the acrylic top cover.

  * •

Casting: Almost all of the acrylic sheets in the market are made by casting,
where methyl methacrylate (MMA) forms a longer molecular chain by the
polymerization reaction and turns into PMMA. A dedicated mold was made by a
very thick acrylic plate with aluminum film pasted on the surface. It took
four days to produce one cover because of the slow polymerization reaction
rate. In addition, a large number of bubbles were generated during this
process and were hard to be excluded in such kind of mold.

  * •

Thermoforming: An acrylic sheet pressurized by a stainless steel mold at a
temperature between 100 and 150 ∘C. The process took a few hours. More than 10
samples were produced. The thickness was measured by an ultrasonic device at
different positions and the deviation was found to be larger than 50%.

  * •

Injection molding: Dry acrylic pellets were fed into an injection machine. At
a temperature of about 240 ∘C, they became semifluid and were injected into a
steel mold at a temperature of 70 ∘C and a successively decreased pressure
from 140 MPa to 60 MPa. After several minutes, it became solid and can be
removed from the mold.

The molds and products for all of these three technologies are shown in Fig.
4. Injection molding was found to be the best for the acrylic cover because of
the high production rate and precise control of the dimensions with a
deviation smaller than 1 mm.

Figure 4: Manufacture of acrylic covers by three different technologies.

###  3.3 Study of injection molding

Many companies are providing PMMA raw materials for injection molding. Most of
them carry anti-UV components and thus can not be used for the protection
cover. Therefore, material screening was done firstly by only selecting types
that are explicitly labeled as UV-transmission in their specifications. 13
types of materials were selected with their major mechanical parameters
summarized in Table 2. The transparencies for all of them are labeled as
larger than 92% in air for 3.2 mm thick standard specimens.

Company | Brand | Type | Tensile | Tensile | Elongation
---|---|---|---|---|---
|  |  | strength (MPa) | modulus (MPa) | at break (%)
Evonik [20] | PLEXIGLAS® | IM20 | 77 | 3300 | 5.5%
7N | 73 | 3200 | 3.5%
| PLEXIGLAS® | V920-UVT | 69 | 3100 | 5%
Altuglas | VS-UVT | 65 | 2900 | 5%
International [21, 22] | V825-UVA5A | 70 | 3100 | 5%
| ALTUGLAS® | VSUVT | 66 | 2900 | 5%
MITSUBISHI [23] | ACRYPET® | VH5 | 61 | 3300 | 3%
TF8 | 59 | 3300 | 3%
TF9 | 57 | 3200 | 2%
Chimei [24] | ACRYREX® | CM211 | 65 | 2700 | 5.5%
Sumitomo [25] | SUMIPEX® | MG5 | 75 | 3100 | 3%
MGSS | 73 | 3100 | 2%
MGSV | 70 | 3100 | 2%
Table 2: Mechanical specifications of different UV-transmission PMMA material.
All data come from companies official websites.

Since UV transmission is critical for JUNO, materials from Table 2 were
purchased from the market as many as possible. 3-mm thick acrylic plates were
made with them and the transparency as a function of wavelength was measured
by the spectrophotometer. One of the results is shown in Fig. 5 where the
sample was made by Evonik IM20 and the transparency is 91.9% at 420 nm and
89.8% at 380 nm. To demonstrate the UV-absorption, another sample’s result is
also shown which was made by MITSUBISHI VH001 and contains a UV absorber
therefore the transparency decreases quickly from 91.7% at 420 nm to 17.5% at
380 nm. Taking into account the mechanical performances and price, two
materials, IM20 from Evonik and MGSV from Sumitomo were selected as
candidates. Both of them have transparency around 92.0% at 420 nm, and have
been used to produce acrylic cover prototypes and tested successfully, which
will be introduced in Sec. 4.4.

Figure 5: Examples of transparency scanning results in air. IM20 from Evonik
is UV-transparent while VH001 from MITSUBISHI contains UV absorber.

Hygroscopicity is one of the properties of acrylic. Water absorption can be
considered as diffusion with the coefficient changes with the temperature
[26]. However, the mass fraction of absorbed water at the equilibrium state
does not depend on the volume or shape of acrylic, nor the temperature either.
It was measured around 2% [27, 28]. Some studies suggested that water as a
polar molecule has a plasticization effect and may introduce some stress on
the acrylic thus reducing the mechanical performances [29].

A batch of standard specimens with the shape following ISO 527 were injection
molded with an arbitrary PMMA material. Their weight was measured as 11.54 g
with a standard deviation of 0.02 g, then some of them were put into ultra-
pure water at normal pressure and the rest into tap water at 0.5 MPa pressure,
both at room temperature. In the following 200 days, several of them were
taken out and measured again. The increase in the weight represented the water
absorption as shown in the top panel of Fig. 6. The absorption rates were very
similar for different water and different pressures. The water mass fraction
was 1.60% after 200 days and was close to saturation. The weight of two
acrylic covers was also measured before and after 207 days in tap water at
normal pressure and room temperature. The absorption rate was smaller than the
standard specimen because of its larger thickness.

The tensile stress and tensile modulus were also measured for these specimens
with the standard GB/T 1040.2-2006, as shown in the middle and the bottom
panels of Fig. 6. No decrease for either of the two parameters was found
during these 200 days. Since the water absorption in our experiment was
already 80% compared to the saturation, a significant reduction in the
mechanical performance is not expected. On the other hand, during the long-
term operation of JUNO, the protection cover will work in the environment
almost without any UV light at a stable temperature 21$\pm$1 ∘C, and the
protection structure will not suffer any external stress except for its weight
in water, the aging should not be a problem either.

Figure 6: The top panel is the weight increase due to water absorption as a
function of time for different acrylic samples and water pressures. The middle
panel and the bottom panel show the change of tensile strength and tensile
modulus of acrylic specimens as a function of time in water under normal
pressure.

###  3.4 Investigation and prototyping of the bottom cover

The bottom cover is used to support the top cover and to provide an interface
to the detector’s main structure. Stainless steel type 304 was chosen because
of many advantages: 1) commonly used material in water with low background
easily to be $\sim$ppb level; 2) good mechanical performance with a large
ultimate strength of more than 500 MPa; 3) good processability with many
techniques; 4) compatible with pure water; 5) low cost compared to stainless
steel type 316 with similar mechanical performance.

Different technologies are investigated for producing the bottom cover,
including rolling, stamping, rotary extrusion, etc. A prototype with a semi-
ellipsoid shape was made by stamping as shown in Fig. 7 (a) and another one
with a cone shape by rolling in Fig. 7 (b). Rolling is cheaper for prototyping
since it does not require a mold. However, it is very difficult to produce an
ellipsoid by rolling even splitting the bottom cover into two halves. On the
other hand, the cone-shaped bottom cover shows significantly lower mechanical
performance according to the simulation study in Sec. 4.1. Therefore, stamping
was used for the main body and the rest connecting structure was welded.
Demagnetization after stamping is needed since there will be a magnetic field
because of stainless steel stretching, which has a significant impact on the
photoelectrons collection in the PMT.

(a)

(b)

Figure 7: Bottom cover prototypes made by two different technologies including
(a) stamping and (b) rolling.

###  3.5 Underwater tests

The underwater experiments were designed to test the full PMT protection
system. A steel-made water tank was produced to provide up to 1 MPa water
pressure. The tank looks like a horizontal located cylinder, as shown in Fig.
8, with an inner diameter of 3 m and a length of 5 m. During the experiments,
water was injected to a level of 4/5 and the rest 1/5 was kept in air. After
that, the pressure was increased with the help of an air pump, typically
stopped at 0.5 MPa with respect to the atmospheric pressure. Since the air
volume is two orders of magnitude larger than a PMT (typically 0.05 m3), the
decrease in the water pressure is negligible even if several PMTs break at the
same time. There are four acrylic windows on the tank allowing recording of
the experiment by the high-speed camera with the help of high-intensity light
out of the tank.

Figure 8: Pressure tank made of steel used for the underwater test. The inner
volume is 35 m3. Four acrylic windows on the wall of the tank are used for the
high-speed camera or the light.

In the traditional way like Super-Kamiokande, a PMT with protection covers was
imploded underwater and the neighbor PMTs were used to justify where there was
a chain reaction. In this scenario, only one pair of top and bottom covers can
be tested at a time. To increase the test efficiency, a new setup was
implemented as shown in Fig. 9. A 3$\times$3 PMT module was produced of
stainless steel. The central PMT was installed without any protection cover or
only with the bottom cover. Two 14$\times$14 cm2 stainless steel plates with
four screws at the corner were installed around the equator of the PMT face to
face, driven by a hydraulic device to implode the PMT underwater. The
shockwave was therefore generated and used as a trigger to implode neighbor
PMTs, and thus their protection covers will stand under a pedestal water
pressure. Once they fail, there will be secondary shockwave which can be
measured by pressure sensors. With this setup, up to six sets of protection
covers can be tested simultaneously.

The shockwave propagates in the water like sound. When it comes to the
boundary between water and acrylic or stainless steel, part of the energy is
reflected which can be calculated as

$r=(\frac{\rho_{2}c_{2}-\rho_{1}c_{1}}{\rho_{2}c_{2}+\rho_{1}c_{1}})^{2},$ (1)
$t=1-r=\frac{4\rho_{2}c_{2}\rho_{1}c_{1}}{(\rho_{2}c_{2}+\rho_{1}c_{1})^{2}},$
(2)

where $\rho_{1}$, $\rho_{2}$ represent material densities and $c_{1}$, $c_{2}$
represent speed of sound in two materials. $r$ and $t$ are the reflected and
transmitted shockwave energy fractions, respectively. Considering water
density 1 g/cm3, speed of sound in water 1483 m/s, acrylic density 1.19 g/cm3,
speed of sound in acrylic 2730 m/s, and there are two boundaries when the
shockwave goes through the acrylic cover, the transmitted energy fraction is
calculated as 74%. It means $\sim$1/4 of the shockwave energy will be
reflected by the acrylic cover and possibly bring some damages, thus this
experiment is at a very extreme case. The rest of shockwave energy will
completely impact on the PMT because there is vacuum inside the PMT thus the
shockwave can not propagate further.

Figure 9: Setup of the PMT underwater protection experiment at the
optimization stage. The central PMT without any protection cover plays as a
trigger to implode neighbor PMTs all equipped with the protection structure.

(a)

(b)

Figure 10: PMT underwater experiments. The central PMT was a trigger which
created shockwave and imploded all neighbor PMTs in protection covers. (a) Two
acrylic covers made of injection molding. (b) Six top covers made of different
material, manufacture technology and thickness. The bottom panels show the
shockwave measurements by four pressure sensors.

Six underwater experiments were done in 2016 and 2017 to find an optimized
thickness for the top cover, while the bottom covers were kept the same. Two
typical experiments were introduced here. In Fig. 10 (a), two PMTs both with
11.5 mm thick injection molding acrylic covers were installed close to the
central trigger PMT. Four pressure sensors were deployed 0.53 m-0.75 m to
their nearest PMT centers. After the implosion, the two neighbor PMTs were
broken as expected, while their protection covers survived. In particular,
there are several cracks on the acrylic covers all starting from the
connection screws, which will be discussed in Sec. 4.3. Pressures of the
shockwave as a function of time are shown in the bottom panel of Fig. 10 (a).
The peak at 1.083 s corresponds to the implosion of the central PMT. There are
small oscillations in the next 10 ms presumably due to multiple reflections of
the primary shockwave on the inner surface of the pressure tank. After that,
no peak pressure was observed therefore there was no secondary shockwave
generated by the neighbor PMTs, and thus the protection requirement was
satisfied. More details of this experiment including the high-speed camera
data can be found in Ref. [30].

This result gives two conclusions. First, it approves that once the protection
cover does not collapse in case of an implosion of a PMT, there will be almost
no shockwave generated thus no chain reaction happens. Second, in the most
extreme case that one protection cover collapses together with its inside PMT
because of any production defection, there will be a chain reaction but only
limited to a small range unless a second protection cover collapses as well.
Therefore qualities of the products have to be controlled to make sure the
possibility of the second case is negligible.

Another experiment with six neighbor PMTs is shown in the top panel of Fig. 10
(b), where the numbers represent PMTs and their protection covers. Acrylic
covers 2 and 5 were made of injection molding, the same as in the previous
experiment. 3 and 4 were made of thermoforming, with thicknesses of 8.5 mm and
6.5 mm, respectively. 1 and 6 were made of PC by blow molding [31] with a
thickness of 3 mm. After the implosion of the central PMT 0, all six neighbor
PMTs broke completely. Acrylic covers 2 and 3 stood at their original places
with only some cracks. Acrylic cover 5 also survived but down to the ground
together with the stainless steel cover thus they can not be seen in this
photo, because the connection between the stainless steel cover and the
experiment module was not tight enough. Acrylic cover 4 and two PC covers 1
and 6 were all collapsed and broke into pieces.

There are three peaks clearly seen in the pressure sensors data shown in the
bottom panel of Fig. 10 (b). The first peak around 0.847 s corresponds to the
trigger PMT. About 15 ms later, the second peak appears in particular for
pressure sensors P1 and P4, deployed near the two PC covers 1 and 6, which
means they collapsed almost simultaneously with their inner PMTs. Then after
100 ms, the third peak takes place, corresponding to acrylic cover 4 and
recorded by the nearby pressure sensor P2. Pressure sensor P3 was on top of
the central PMT and was found dead after the experiment, presumably due to
water leakage, thus the measured pressure magnitude is very small while the
time structure is consistent with other sensors. Apparently, the thickness of
the cover plays an important role against water pressure and the threshold of
the thickness should be between 6.5 mm and 8.5 mm. More important, the 6.5 mm
thick acrylic cover resisted water pressure for about 100 ms before
collapsing. This period depends on the water pressure and the thickness of the
cover. If we can enlarge the inlet area on the stainless steel to let the
water go inside faster, the risk of protection cover failure could be reduced,
while the primary shockwave due to water flushing in will be increased.
Therefore it needs optimization based on the simulation and experiments.

All the underwater experiment results at this stage are summarized in Table.
3, including all of the three materials and four production technologies we
investigated. They are sorted in thickness on the top of the cover. Because
the thickness is not uniform in particular for the covers made of
thermoforming and the simulation suggested that the collapse always starts on
the top which will be discussed in Sec. 4.1. The threshold of the implosion of
the top cover was thus determined as 7 mm as all 9 samples larger than this
thickness succeeded while all but one sample smaller than this thickness
failed.

Material | Technology | Thickness on top (mm) | Number of samples | Number of failure
---|---|---|---|---
PMMA | Injection molding | 11.5 | 5 | 0
PMMA | Bending | 9.9 | 1 | 0
PMMA | Casting | 8.5 | 1 | 0
PMMA | Bending | 8.5 | 1 | 0
PMMA | Bending | 7.5 | 1 | 0
PMMA | Bending | 6.8 | 2 | 1
PMMA | Bending | 6.7 | 1 | 1
PMMA | Casing | 6.5 | 1 | 1
PC | Blow molding | 3.0 | 3 | 3
PETG | Blow molding | 5.0 | 1 | 1
PETG | Blow molding | 3.0 | 1 | 1
PETG | Blow molding | 1.5 | 1 | 1
Table 3: Summary of underwater PMT protection experiments

##  4 Final design and validation

###  4.1 Simulation study

For the thin spherical structure such as the protection cover, buckling is the
most common failure mode in case of external pressure. The classical critical
buckling pressure $p_{b}$ of a spherical shell can be expressed as [32]

$p_{b}=\frac{2E}{\sqrt{3(1-\nu^{2})}}\frac{t^{2}}{r^{2}},$ (3)

where E is the tensile modulus, $\nu$ is Poisson’s ratio, $t$ and $r$ are the
thickness and middle radius of the shell, respectively. The buckling
performance for a hollow structure with other shapes is not as good as a
sphere and thus needs sophisticated simulation study. In particular, since $r$
is limited by the radius of PMT, with a given material, $t$ is the most
sensitive parameter for the buckling pressure and thus needs to be optimized.

Finite Element Analysis (FEA) was done taking into account the geometry for
both top and bottom covers as two semi-ellipsoids. The inner diameter at the
equator was fixed at 512 mm, leaving 2 mm clearance to the PMT. The inner
height of the top cover was 186 mm as a baseline, and 14 holes with a diameter
between 5 mm and 10 mm were implemented in the simulation. The connection
between the top and bottom covers was set as tight. Two independent simulation
analyses using different software were done and compared. Deformation and
tensile stress of the protection cover were simulated in the condition of a
0.5 MPa uniform pedestal load, with an example shown in Fig. 11, where the
thickness was set as 12 mm for the top cover and 3 mm for the bottom cover.

(a)

(b)

(c)

Figure 11: Finite Element Analysis for a 12 mm thick top cover and 3 mm thick
bottom cover with a 0.5 MPa uniform pedestal load. (a) Colors represent
tensile stress in the unit of MPa. (b) Tensile stress around the holes on the
top cover. (c) Colors represent displacement in the unit of mm.

A buckling failure happens when the deformation is too large that the
structure suddenly collapses. A typical buckling failure for an acrylic cover
is shown in Fig. 12 (a), where the buckling starts on the top with many cracks
generated. The critical buckling pressure as a function of the acrylic cover
thickness is shown in Fig. 12 (b). The error bars were set as 0.1 MPa,
corresponding to the scanning step of the load and the difference between the
two analyses. Extrapolating the simulated results to 0.5 MPa buckling pressure
with a proportion to the square of the thickness, as suggested in Eq. 3, gives
6.0 mm, which looks very close to the buckling failure threshold obtained in
Table. 3.

Since the buckling always starts on the top where there is almost no
limitation on the thickness because of detector installation, a non-uniform
protection cover was designed with a gradually reduced thickness from top to
bottom. Such a design can be easily realized by injection molding. The FEA
showed that when the thickness was set as 11 mm on the top and 9 mm at the
equator, the buckling pressure was 1.7 MPa, which is three times larger than
the largest water pressure in JUNO and the same as a uniform acrylic cover
with a thickness of 11 mm. This reduction of 2 mm at the equator gives 1%
increase in the optical coverage. The minimum clearance between two acrylic
covers will be 3 mm.

A higher top cover with an inner height 200 mm was also simulated based on the
9-11 mm non-uniform thick design. The buckling pressure was found to be
further improved to 2.0 MPa, because it is closer to a sphere. Considering the
difficulty of manufacture, larger heights were not considered.

The tensile stress was also studied and found to be much lower than the
fracture threshold of acrylic. However, the connection between the top and
bottom covers was simplified in the FEA. The stress situation is complicated
in the case of a PMT implosion and is hard to be simulated. The study and
optimization of the connection will rely on the experiment and will be
discussed in Sec. 4.3.

The critical buckling pressure as a function of stainless steel cover
thickness is shown in Fig. 12 (c), where the top cover was set as 9-11 mm non-
uniform thick and 200 mm high. 2 mm thickness was found to be good enough as
it resists 1.8 MPa water pressure. Increasing the thickness to 3 mm does not
help too much since the top cover starts buckling at 2.0 MPa. At 0.5 MPa
pressure, the water flow rate through inlets is about 20 m/s according to
Bernoulli’s Principle Formula. Taking into account the resistance of the
collapsing PMT glass, if there are four holes 100 mm in diameter on the bottom
cover, it will take 10-20 ms for water to fill the PMT volume in case of an
implosion. There is negligible impact on the buckling failure with these holes
according to the FEA. A cone-shaped bottom cover was also simulated by an
independent analysis and the critical buckling pressure was found to be 40%
lower compared with the semi-ellipsoid one, therefore it was not used for
further simulation or prototyping.

(a)

(b)

(c)

Figure 12: Finite Element Analysis for buckling failure of protection covers.
(a) An example of buckling failure for an 11 mm thick acrylic cover under 2.4
MPa pedestal pressure. Colors represent stress in the protection cover. The
deformation was enlarged by a factor of three for better visualization. (b)
Critical buckling pressure as a function of acrylic cover thickness. (c)
Critical buckling pressure as a function of stainless steel cover thickness.
The last data points at 3 mm thickness corresponds to buckling failure of the
top cover.

###  4.2 New prototyping for acrylic covers

A new injection mold was made according to the final 9-11 mm nonuniform
design, with an inner height of 200 mm. Tens of prototypes were produced and
found to be solid and smooth without any bubbles inside. However, the steel of
the mold was selected to be soft to reduce the cost, and thus the surface can
not be polished as good as SPI-A3 [33] or better. As a result, the acrylic
cover samples look a little fuzzy but it does not have any impact on the
mechanical test. The high-transparent acrylic cover has been made with a new
mold for mass production, which will not be discussed in this paper.

###  4.3 Design of connection

In the past underwater experiments, all of the acrylic covers were found to
have some cracks around the connection points, as shown in Fig. 13. There were
two processes. When the PMT was imploded, the 0.5 MPa water pressure brought
110 kN force on the acrylic cover, which was supported by six screws with a
diameter of several millimeters if the connection was too tight. The contact
area between these screws and the acrylic was too small to avoid damage to the
acrylic, and there were always cracks starting from the top of the connection
holes. After that, water went inside through the big holes on the stainless
steel cover, rolled, and pushed the acrylic cover outwards. The force turned
into the bottom of the connection holes and sometimes made a notch.

The connection holes on the acrylic cover were optimized as 10 mm in diameter.
Six hooks with 6 mm diameter going through the holes pull the acrylic cover
down to the flange of the stainless steel cover, one of them shown in the top
panel of Fig. 14 (a). With a 1 mm thick gasket surrounding the hole, there is
still 2 mm space between the hole and the hook on the top, to avoid direct
contact in case of a pedestal water pressure. The distance between the bottom
of the hole to the edge of the acrylic cover is 30 mm, three times the
diameter of the hole, to reduce the possibility of fracture of the acrylic
when the water goes inside.

Another connection structure was designed as a stainless steel ring with an
inner diameter a little smaller than the PMT, and fixed with six screws to the
stainless steel cover, as shown in Fig. 15 (a). In this case, there are no
connection holes on the acrylic cover but this ring will shadow some light
thus it was treated as a backup plan. Both of these connections were
prototyped and tested in the underwater experiments in Sec. 4.4.

Figure 13: Cracks generated on the acrylic cover in the underwater
experiments, all starting from the connection points between top and bottom
covers.

###  4.4 Experimental validation

Experimental validation of the final design was done with a different setup.
The central PMT was protected by both the acrylic and the stainless steel
cover. One or two PMTs also with protection covers were installed nearby at a
minimum clearance of 5 mm, and were used to directly check if there was a
chain reaction when the central PMT was imploded. The two implosion plates
were moved from the equator to the lower hemisphere and went upwards through
the holes on the stainless steel cover to implode the PMT. Another static
plate was installed between the acrylic cover and the PMT on the top. The top
hole on the acrylic cover was enlarged to 30 mm to let the fixing screw in. In
some of the experiments, pressure sensors were also installed to get a
quantitative measurement of the shockwave.

One of the experiments is shown in Fig. 14 (a). The acrylic cover was made by
MGSV and fixed with six hooks. After the implosion, the acrylic cover survived
and both neighbor PMTs were safe, proving that there was no chain reaction.
Only small cracks were found on the bottom of three of the connection holes,
which was completely understood and showed that our optimization of the
connection worked well. This experiment was recorded by a high-speed camera at
3,000 frames per second, with four representative frames shown in Fig. 14 (b).
At time 0, cracks start on the glass bulb near the implosion plate. 4 ms
later, cracks are all over the glass bulb. 14 ms is the middle of PMT
collapse. At 28 ms, the volume inside the protection cover is filled with
water. The water filling time is larger than the calculation in Sec. 4.1
because two injects were partially blocked by the implosion plates.

(a)

(b)

Figure 14: (a) Underwater experiment to validate the final design, with the
acrylic cover fixed by six hooks. Two plastic tapes in blue were pasted on the
acrylic cover to help the high-speed camera trace possible cracks. The picture
in the bottom panel was taken after the implosion of the central PMT. (b)
Pictures recorded by the high-speed camera during the implosion of the central
PMT.

In another experiment, IM20 was used to make the acrylic cover and it was
fixed with a stainless steel ring as an alternative connection structure, as
shown in Fig. 15 (a). It was also successful since the neighbor PMT was
undamaged after the implosion of the central PMT. A long going-through crack
was found on the acrylic cover below the stainless steel ring, which was also
understood, that when the water went inside and pushed the acrylic cover
upwards, because of the constraint by the ring, the stress on the acrylic was
so large and beyond the fracture strength. A pressure sensor was deployed in
this experiment on the top 60 cm far from the center of the imploded PMT. A
peak was found with an amplitude of around 0.15 MPa after the baseline
subtraction shown in Fig. 15 (b). That was created by the influx of water
through the holes on the stainless steel cover. Extrapolating this number to
the neighbor PMT glass shell as $L^{-1.2}$ results in 0.37 MPa, a factor of 50
reductions of shockwave compared to the peak pressure in case of no
protection.

(a)

(b)

Figure 15: (a) Underwater experiment to validate the final design, with the
acrylic cover of the imploded PMT fixed by the stainless steel ring . (b)
Shockwave measured by the pressure sensor.

##  5 Summary and perspective

A protection system consisting of 20,000 shells with a connection structure
was designed for JUNO 20-inch PMTs. Each shell is made of a top acrylic cover
and a bottom stainless steel cover, with the shape accommodating both NNVT
PMTs and HPK PMTs. High transparency and high mechanical performances are the
major requirements for the protection cover. In addition, due to 77.9% optical
coverage in JUNO, almost a factor of two compared to other experiments with a
similar scale in the world, dimensions of the protection cover need to be
optimized and precisely controlled. PMMA (acrylic), PC, and PET were compared
as candidates for the top cover, and acrylic was chosen mainly because of its
better transparency. Prototypes of acrylic covers were made by casting,
thermoforming, and injection molding. Injection molding was finally selected
because of the high production rate and precise control of the dimensions.
Thirteen UV-transmission PMMA materials were investigated. IM20 from Evonik
and MGSV from Sumitomo were used for prototyping and both of them were
validated by underwater experiments. Hygroscopicity was also studied for
acrylic and the decrease in mechanical performances due to water absorption
was found to be negligible. Prototypes of stainless steel covers were made by
rolling and stamping.

More than 10 underwater experiments at 0.5 MPa proceeded which can be divided
into two stages. In the first stage, a novel experiment scheme was carried out
by imploding multiple protected PMTs using shockwave simultaneously and thus
tested a number of protection covers made by different materials or
technologies and with different thicknesses at the same time. It largely
improved the testing efficiency and reduced the experiment cost. As a result,
a thickness of around 7 mm was determined to be the minimum requirement.
Simulation studies with FEA were done concerning buckling failure and tensile
failure. For the acrylic cover, a nonuniform thickness design was finalized,
with 11 mm on the top and 9 mm on the bottom. Two different shapes of the
stainless steel cover were compared and a semi-ellipsoid shape was chosen
because of better mechanical performance and the thickness was optimized as 2
mm. The critical buckling pressure for the full structure is three times
larger than the 50 m water pressure. Four inlet holes with 100 mm diameter
were designed on the stainless steel cover to let the water fill in within 10
ms. Connections between the top and bottom covers were also optimized to
minimize the risk of acrylic fracture in case of an implosion.

Based on the final design, new prototypes of acrylic covers were produced and
validated in the second stage of underwater experiments, where only one
protected PMTs was imploded. The neighbor PMTs were safe and the cascade
implosion was avoided. Measurement of pressure sensors showed 0.37 MPa peak
pressure extrapolated to the neighbor PMTs, which was a factor of 50
reductions of shockwave and far below the PMT implosion threshold. The final
design and prototyping of the protection system were proved.

Both the acrylic covers and stainless steel covers are under production. A
sampling test shows the transparency of the acrylic cover is larger than 91%
in air, and the manufacturing precision is better than 1 mm. Radioactivity of
a few samples was measured and the background contribution from acrylic covers
and stainless steel covers is O(10-4) and O($10^{-2}$) respectively compared
to the PMT itself. We plan to publish another paper on the engineering design,
mass production, quality control, and performances once the mass production
finishes.

##  6 Acknowledgments

This work was supported by the Strategic Priority Research Program of the
Chinese Academy of Sciences, Grant No. XDA10011100. We thank North Night
Vision Technology Co., Ltd for providing all PMTs in the underwater
experiments. We thank Naval Research Institute, Naval University of
Engineering, and Zhejiang University for the simulation studies. Naval
Research Institute also provided the high-pressure water tank, high-speed
camera, and pressure sensors.

## References

  * [1] JUNO, F. An et al., Neutrino Physics with JUNO,  J. Phys. G 43, 030401 (2016), arXiv:1507.05613.
  * [2] JUNO, Z. Djurcic et al., JUNO Conceptual Design Report,  (2015), arXiv:1508.07166.
  * [3] JUNO, A. Abusleme et al., JUNO physics and detector,  Prog. Part. Nucl. Phys. 123, 103927 (2022).
  * [4] Super-Kamiokande, Y. Fukuda et al., The Super-Kamiokande detector,  Nucl. Instrum. Meth. A 501, 418 (2003).
  * [5] Super-Kamiokande, K. Abe et al., Solar neutrino results in Super-Kamiokande-III,  Phys. Rev. D 83, 052010 (2011), arXiv:1010.0118.
  * [6] Hyper-Kamiokande, K. Abe et al., Hyper-Kamiokande Design Report,  (2018), arXiv:1805.04163.
  * [7] LBNE, J. Goon et al., The Long Baseline Neutrino Experiment (LBNE) Water Cherenkov Detector (WCD) Conceptual Design Report (CDR),  (2012), arXiv:1204.2295.
  * [8] M. Diwan et al., Underwater implosions of large format photo-multiplier tubes,  Nucl. Instrum. Meth. A 670, 61 (2012).
  * [9] J. Ling et al., Implosion chain reaction mitigation in underwater assemblies of photomultiplier tubes,  Nucl. Instrum. Meth. A 729, 491 (2013).
  * [10] Z. Qin,  Status of PMT protection,  https://juno.ihep.ac.cn/cgi-bin/Dev_DocDB/ShowDocument?docid=1296, internal, 2016.
  * [11] JUNO, Daya Bay, A. Abusleme et al., Optimization of the JUNO liquid scintillator composition using a Daya Bay antineutrino detector,  Nucl. Instrum. Meth. A 988, 164823 (2021), arXiv:2007.00314.
  * [12] JUNO, A. Abusleme et al., Mass Testing and Characterization of 20-inch PMTs for JUNO,  (2022), arXiv:2205.08629.
  * [13] SNO, J. Boger et al., The Sudbury neutrino observatory,  Nucl. Instrum. Meth. A 449, 172 (2000), arXiv:nucl-ex/9910016.
  * [14] Daya Bay, F. P. An et al., The Detector System of The Daya Bay Reactor Neutrino Experiment,  Nucl. Instrum. Meth. A 811, 133 (2016), arXiv:1508.03943.
  * [15] Double Chooz, Y. Abe et al., Indication of Reactor $\bar{\nu}_{e}$ Disappearance in the Double Chooz Experiment,  Phys. Rev. Lett. 108, 131801 (2012), arXiv:1112.6353.
  * [16] RENO, J. K. Ahn et al., Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment,  Phys. Rev. Lett. 108, 191802 (2012), arXiv:1204.0626.
  * [17] JUNO, A. Abusleme et al., Radioactivity control strategy for the JUNO detector,  JHEP 11, 102 (2021), arXiv:2107.03669.
  * [18] http://www.goodfellow.com.
  * [19] X. Yang et al., The measurement system of acrylic transmittance for the JUNO central detector,  Rad. Det. Tech. Meth. 4, 284 (2020).
  * [20] http://www.plexiglas-polymers.com.
  * [21] http://www.plexiglas.com.
  * [22] http://www.altuglas.com.
  * [23] http://www.acrypet.com.
  * [24] http://www.chimeicorp.com.
  * [25] http://www.sumitomo-chem.com.sg.
  * [26] M. Braden, The absorption of water by acrylic resins and other materials,  Journal of Prosthetic Dentistry 14 (1964).
  * [27] A. Tsuboi et al., Water absorption characteristics of two types of acrylic resin obturators,  Journal of Prosthetic Dentistry 94, 382 (2005).
  * [28] M. N’Diaye et al., Water absorption of poly(methyl methacrylate) measured by vertical interference microscopy,  Langmuir 7, 31 (2012).
  * [29] Y. Liu et al., Experimental Study on Hydroscopicity of Organic Glass (in Chinese),  Glass 06-0032-03, 1003 (2016).
  * [30] Y. Jiang et al., Experimental and numerical study on implosion protection of large format photo-multiplier tubes,  International Journal of Mechanical Sciences 163, 105099 (2019).
  * [31] S. L. Belcher,  https://www.sciencedirect.com/topics/engineering/blow-molding.
  * [32] S. P. Timoshenko and J. M. Gere,  Theory of Elastic Stability, 2 ed. (McGraw Hill, New York, 1961).
  * [33] https://www.spifinish.com/technology.


# Topology of higher-dimensional black holes’ thermodynamics in massive
gravity

Deyou Chen<EMAIL_ADDRESS>School of Science, Xihua University, Chengdu
610039, China  Key Laboratory of High Performance Scientific Computation,
Xihua University, Chengdu 610039, China  Yucheng He<EMAIL_ADDRESS>School of Science, Xihua University, Chengdu 610039, China  Key Laboratory of
High Performance Scientific Computation, Xihua University, Chengdu 610039,
China  Jun Tao<EMAIL_ADDRESS>Center for Theoretical Physics, College of
Physics, Sichuan University, Chengdu, 610065, China

###### Abstract

In the recent work [Phys. Rev. Lett. 129, 191101 (2022)], the topological
number was found to be a universal number independent of the black hole’s
parameters. In this paper, we study topological numbers for five-, six- and
seven-dimensional anti-de Sitter black holes in the ghost-free massive
gravity. We find that when the black holes are charged, they have the same
topological number, and this number is independent of their parameters’
values. For the uncharged black holes, their topological numbers are 0 or 1,
and the specific values are determined by the values of the black holes’
parameters. Since $k$ and $c_{0}^{2}c_{2}m^{2}$ appear together in the
generalized free energy in the form of $k+c_{0}^{2}c_{2}m^{2}$, where $k$
characterizes the horizon curvature and $c_{2}m^{2}$ is the coefficient of the
second term of massive potential associated with the graviton mass, this
result is applicable to the black holes with the spherical, Ricci flat or
hyperbolic horizons. This work shows that the parameters of the ghost-free
massive gravity play an important role in topological classes of black holes.

##  I Introduction

Topology is an important mathematical tool. When using this tool, the details
of the research objects are ignored, and only their generic properties are
focused on. The generic properties are revealed by topological quantities.
Cunha, Berti and Herdeiro first used the topological approach to study light
rings of ultracompact objects (UCOs) CBH1 . They found that an UCO must have
at least two light rings, one of them is stable and the other is unstable.
This approach was extended to the four-dimensional, axisymmetric
asymptotically flat black hole and a significant result was obtained in CBH2 .
The result shows that there is at least one standard light ring outside the
black hole for each rotation sense. The radial stability of light rings is
very important for black holes’ shadows, and non-extremal and extremal black
holes exist in our universe. Given these, some important improvement was
implemented and universal properties of light rings(spheres) were gotten in
GG1 ; SWW1 ; GG2 ; GG3 . In GG4 , the authors also put forward an topological
approach to study the black hole shadow caused by the existence of the light
rings or light spheres.

Duan’s topological current $\phi$-mapping theory is famous for its wide
application in various physical systems DL1 ; DL2 . Using his theory, Wei, Liu
and Mann proposed another important topological approach to research local and
global properties of black holes WLM . In this approach, the black hole
solutions are seen as defects in the thermodynamic parameter space. The
defects are related to zero points of a field at $\vec{x}=\vec{z}$ in a space
and researched in terms of winding numbers. The winding numbers reveals local
thermodynamic characteristics of the black holes. The positive/negative
winding numbers characterize the local stability/instability of the black
holes. The global properties are determined by the topological charges. Each
black hole is endowed with a topological charge, and then these black holes
can be divided into different classes according to the values of the
topological charges. A key point in WLM is that a vector field was constructed
with a generalized free energy. A similar generalized free energy was defined
in LW1 ; JWY1 ; JWY2 and used to research on the black hole phase transitions.
Since this approach was proposed, it has immediately attracted people’s
attention. Based on this approach, some new viewpoints on the topological
approach have been proposed, the topological arguments on the black holes in
the complex spacetimes or in the different ensembles were performed SWW2 ;
SWW3 ; YBM ; BLT ; CWL1 ; ZYF ; FJW ; DWW1 ; DWW2 ; DWW3 ; DWW4 ; LLZW ; DZ1 ;
DZ2 ; DZ3 ; LW2 ; GP1 ; GP2 ; GP3 ; ZCHLY ; AAGS ; XWWY ; HMM . The results
obtained in these work strongly support the viewpoint proposed in WLM that the
topological number for the black hole branches at an arbitrary given
temperature is a universal number independent of the black hole’s parameters.
These researches are shining with new light on black holes physics.

Our concern is whether the topological numbers are independent of all black
holes’ parameters. In this paper, we use the topology approach to study the
topological numbers for high-dimensional, charged and uncharged black holes in
the ghost-free massive gravity, specifically five-, six-, and seven-
dimensional black holes. In the calculation, we change the values of the black
holes’ parameters and calculate their topological numbers. The purpose is to
verify our concern. We find that the topological numbers for the charged black
holes are independent of their parameters’ values, while the topological
numbers for the uncharged black holes depend on the values of their
parameters. Einstein’s general relativity (GR) is a low energy effective
theory. The UV completeness requires that GR be modified to meet physical
descriptions in the high energy region. Massive gravity is one of the modified
gravity theories. In this gravity, the graviton is endowed with mass. A
spherically symmetric black hole solution with a negative cosmological
constant was obtained in DV . Its thermodynamical properties in the extended
phase space were researched in CHPZ ; XCH1 ; XCH2 ; DYC .

The rest of this paper is organized as follows. In the next section, we
briefly review the solution of the higher-dimensional black holes in the
massive gravity and discuss its thermodynamical properties. In Sec. III, we
calculate the topological numbers for the charged and uncharged black holes by
changing the values of the black holes’ parameters. Sec. IV is devoted to our
conclusion and discussion.

##  II Higher-dimensional black hole solution in massive gravity

The action for an $(n+2)$-dimensional ghost-free massive gravity is given by
DV

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int{dx^{n+2}\sqrt{-g}\left[R+\frac{n(n+1)}{l^{2}}-\frac{F^{2}}{4}+m^{2}\sum_{i=1}^{4}{c_{i}u_{i}(g,f)}\right]},$
(1)

where $F=F^{\mu\nu}F_{\mu\nu}$ and $F_{\mu\nu}$ is the electromagnetic tensor,
the terms including $m^{2}$ are the massive potential associate with graviton
mass, $f$ is a fixed symmetric tensor called as the reference metric. $c_{i}$
are constants, and their values are discussed in CHPZ . $u_{i}$ are symmetric
polynomials of the eigenvalues of the $(n+2)\times(n+2)$ matrix
$\mathcal{K}_{\nu}^{\mu}=\sqrt{f^{\mu\alpha}g_{\alpha\nu}}$:
$u_{1}=[\mathcal{K}]$, $u_{2}=[\mathcal{K}]^{2}-[\mathcal{K}^{2}]$,
$u_{3}=[\mathcal{K}]^{3}-3[\mathcal{K}][\mathcal{K}^{2}]+2[\mathcal{K}^{3}]$ ,
$u_{4}=[\mathcal{K}]^{4}-6[\mathcal{K}^{2}][\mathcal{K}]^{2}+8[\mathcal{K}^{3}][\mathcal{K}]+3[\mathcal{K}^{2}]^{2}-6[\mathcal{K}^{4}]$.
The square root in $\mathcal{K}$ denotes
$(\sqrt{A})_{\nu}^{\mu}(\sqrt{A})_{\lambda}^{\nu}=A_{\lambda}^{\mu}$ and
$[\mathcal{K}]=\mathcal{K}_{\mu}^{\mu}$.

This action admits a solution of static black hole with the spacetime metric
CHPZ

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}h_{ij}dx^{i}dx^{j},$ (2)

and reference metric $f_{\mu\nu}=diag(0,0,c_{0}^{2}h_{ij})$, where $c_{0}$ is
a positive constant, and $h_{ij}dx^{i}dx^{j}$ is the line element for an
Einstein space with the constant curvature $n(n-1)k$. $k=1$, $0$ or $-1$
corresponds to a spherical, Ricci flat or hyperbolic topology horizons of the
black hole, respectively. According to the reference metric, we get the
symmetric polynomials

$\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle\frac{nc_{0}}{r},\quad\quad
u_{2}=\frac{n(n-1)c_{0}^{2}}{r^{2}},$ $\displaystyle u_{3}$ $\displaystyle=$
$\displaystyle\frac{n(n-1)(n-2)c_{0}^{3}}{r^{3}},\quad
u_{4}=\frac{n(n-1)(n-2)(n-3)c_{0}^{4}}{r^{4}}.$ (3)

The expression of the metric function $f(r)$ is

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
k+\frac{r^{2}}{l^{2}}-\frac{16\pi M}{nV_{n}r^{n-1}}+\frac{(16\pi
Q)^{2}}{2n(n-1)V_{n}^{2}r^{2(n-1)}}+\frac{c_{0}c_{1}m^{2}r}{n}+c_{0}^{2}c_{2}m^{2}$
(4)
$\displaystyle+\frac{(n-1)c_{0}^{3}c_{3}m^{2}}{r}+\frac{(n-1)(n-2)c_{0}^{4}c_{4}m^{2}}{r^{2}},$

where $l^{2}$ is related to the cosmological constant $\Lambda$ as
$l^{2}=-\frac{n(n+1)}{2\Lambda}$. $M$ and $Q$ are the black hole’s mass and
charge, respectively. $V_{n}$ is the volume spanned by coordinates $x^{i}$.
This black hole describes a static charged spacetime. The event horizon
$r_{h}$ is determined by $f(r)=0$. We rewrite the mass with the horizon and
charge as follows

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{nV_{n}r_{h}^{n-1}}{16\pi}\left[k+\frac{r_{h}^{2}}{l^{2}}+\frac{(16\pi
Q)^{2}}{2n(n-1)V_{n}^{2}r_{h}^{2(n-1)}}+\frac{c_{0}c_{1}m^{2}r_{h}}{n}+c_{0}^{2}c_{2}m^{2}\right.$
(5)
$\displaystyle\left.+\frac{(n-1)c_{0}^{3}c_{3}m^{2}}{r_{h}}+\frac{(n-1)(n-2)c_{0}^{4}c_{4}m^{2}}{r_{h}^{2}}\right].$

The Hawking temperature, entropy and electric potential at the event horizons
are given by

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{V_{n}r_{h}^{n}}{4},\quad\quad\quad\Phi_{e}=\frac{16\pi
Q}{(n-1)V_{n}r_{h}^{n-1}},$ $\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi
r_{+}}\left[\frac{(n+1)r_{h}^{2}}{l^{2}}+\frac{(16\pi
Q)^{2}}{2nV_{n}^{2}r_{h}^{2(n-1)}}+c_{0}c_{1}m^{2}r_{h}+(n-1)c_{0}^{2}c_{2}m^{2}+(n-1)k\right.$
(6)
$\displaystyle\left.+\frac{(n-1)(n-2)c_{0}^{3}c_{3}m^{2}}{r_{h}}+\frac{(n-1)(n-2)(n-3)c_{0}^{4}c_{4}m^{2}}{r_{h}^{2}}\right],$

respectively. The thermodynamics in the extended phase space of the massive
gravity were studied in CHPZ ; XCH1 ; XCH2 ; DYC , where the cosmological
constant was seen as a variable related to pressure,
$P=-\frac{\Lambda}{8\pi}=\frac{n(n+1)}{16\pi l^{2}}$, and its conjugate
quantity is a thermodynamic volume $V$. Then the mass is regarded as the
enthalpy KRT1 ; KRT2 ; KRT3 ; KRT4 ; KRT5 . For this black hole, its
thermodynamic volume is $V=\frac{V_{n}r_{h}^{n+1}}{n+1}$. It was found that
the van der Waals-like phase transition exists the black hole not only with
the spherical black holes but also with the Ricci flat and hyperbolic
horizons. When $c_{1}$, $c_{2}$, $c_{3}$ and $c_{4}$ are seen as extensive
parameters for the mass, and their conjugate quantities are

$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\frac{V_{n}c_{0}m^{2}r_{h}^{n}}{16\pi},\quad\quad\quad\quad\quad\quad\Phi_{2}=\frac{nV_{n}c_{0}^{2}m^{2}r_{h}^{n-1}}{16\pi},$
$\displaystyle\Phi_{3}$ $\displaystyle=$
$\displaystyle\frac{n(n-1)V_{n}c_{0}^{3}m^{2}r_{h}^{n-2}}{16\pi},\quad\quad\Phi_{4}=\frac{n(n-1)(n-2)V_{n}c_{0}^{4}m^{2}r_{h}^{n-3}}{16\pi}.$
(7)

Obviously, these quantities obey the first law of thermodynamics

$\displaystyle dM=TdS+VdP+\Phi_{e}dQ+\sum_{i=1}^{4}\Phi_{i}dc_{i}.$ (8)

When the cosmological constant and parameters $c_{i}$ are fixed, the terms
$VdP$ and $\Phi_{i}dc_{i}$ disappear and the mass is the internal energy.

##  III Thermodynamic topology of higher-dimensional black holes

In this section, we first review the topological method proposed in WLM , and
then change the values of the black holes’ parameters to study the
thermodynamic topology for the different higher-dimensional black holes in the
ghost-free massive gravity. The topological numbers for these black holes are
obtained.

###  III.1 Topological current

According to Ref. WLM , the generalized free energy is defined by

$\displaystyle\mathcal{F}=E-\frac{S}{\tau},$ (9)

where $E$ and $S$ are the energy and entropy of a black hole system,
respectively. $\tau$ is a variable and can be seen as the inverse temperature
of the cavity enclosing the black hole. This free energy is off-shell except
at $\tau=\frac{1}{T}$. To calculate the topological numbers, a vector is
constructed as

$\displaystyle\phi=\left(\frac{\partial\mathcal{F}}{\partial
r_{h}},-\cot\Theta\csc\Theta\right).$ (10)

In this vector, $0<r_{h}<+\infty$ and $0\leq\Theta\leq\pi$. Zero points of the
vector obtained at $\tau=1/T$ and $\Theta=\pi/2$ correspond to the on-shell
black hole solution. Other points are not the solutions of Einstein field
equations, are the off-shell states. $\phi^{\Theta}$ diverges at $\Theta=0$
and $\Theta=\pi$, which leads to that the direction of the vector is outward.

Using Duan’s $\phi$-mapping topological current theory DL1 ; DL2 , we define a
topological current

$\displaystyle
j^{\mu}=\frac{1}{2\pi}\varepsilon^{\mu\nu\rho}\varepsilon_{ab}\partial_{\nu}n^{a}\partial_{\rho}n^{b},$
(11)

where $\mu,\nu,\rho=0,1,2$, $a,b=1,2$,
$\partial_{\nu}=\frac{\partial}{x^{\nu}}$ and $x^{\nu}=(\tau,r_{h},\Theta)$.
$\tau$ is seen as a time parameter of the topological defect. $n^{a}$ is a
unit vector defined by
$\left(\frac{n^{r}}{||n||},\frac{n^{\Theta}}{||n||}\right)$. It is easily to
prove the current is conserved. Using the Jacobi tensor
$\varepsilon^{ab}J^{\mu}(\frac{\phi}{x})=\varepsilon^{\mu\nu\rho}\partial_{\nu}{\phi}^{a}\partial_{\rho}{\phi}^{b}$
and two-dimensional Laplacian Green function
$\Delta_{\phi^{a}}ln||\phi||=2\pi\delta^{2}(\phi)$, the current is rewritten
as

$\displaystyle j^{\mu}=\delta^{2}(\phi)j^{\mu}\left(\frac{\phi}{x}\right),$
(12)

which is nonzero only when $\phi^{a}(x^{i})=0$. Then a topological number in a
parameter region $\sum$ is calculated as

$\displaystyle
W=\int_{\sum}j^{0}d^{2}x=\sum_{i=1}^{N}\beta_{i}\eta_{i}=\sum_{i=1}^{N}w_{i},$
(13)

where $j^{0}=\sum_{i=1}^{N}\beta_{i}\eta_{i}\delta^{2}(\vec{x}-\vec{z}_{i})$
is the density of the current. $\beta_{i}$ is Hopf index which counts the
number of the loops that $\phi^{a}$ makes in the vector $\phi$ space when
$x^{\mu}$ goes around the zero point $z_{i}$. Clearly, this index is always
positive. $\eta_{i}$=sign$J^{0}(\phi/x)_{z_{i}}=\pm 1$ is the Brouwer degree.
$w_{i}$ is the winding number for the $i$-th zero point of the vector in the
region and its values is independent on the shape of the region. In the
researches, people found the this value is determined by the (un-)stable black
holes. In the following, we will study the topological numbers for the higher-
dimensional black holes in the massive gravity by using the different values
of the black holes’ parameters.

###  III.2 Topological numbers for five-dimensional black holes

When $n=3$, the metric (2) describes a five-dimensional charged spacetime.
According to Eqs. (5) and (6), the mass and entropy of the black hole are

$\displaystyle
M=\frac{3V_{3}r_{h}^{2}}{16\pi}\left[k+c_{0}c_{1}m^{2}r_{h}+c_{2}c_{0}^{2}m^{2}+\frac{2c_{3}c_{0}^{3}m^{2}}{r_{h}}+\frac{8\pi
Pr_{h}^{2}}{3}+\frac{(16\pi Q)^{2}}{12V_{3}^{2}r_{h}^{4}}\right],$ (14)

$\displaystyle S=\frac{1}{4}V_{3}r_{h}^{3},$ (15)

respectively. Since the cosmological constant is a fixed constant, the black
hole’s energy is its mass. Inserting Eqs. (14) and (15) into Eq.(9), we get
the off-shell free energy

$\displaystyle\mathcal{F}=\frac{3V_{3}r_{h}^{2}}{16\pi}\left[c_{0}c_{1}m^{2}r_{h}+c_{2}c_{0}^{2}m^{2}+k+\frac{2c_{3}c_{0}^{3}m^{2}}{r_{h}}+\frac{8\pi
Pr_{h}^{2}}{3}+\frac{(16\pi
Q)^{2}}{12V_{3}^{2}r_{h}^{4}}\right]-\frac{V_{3}r_{h}^{3}}{4\tau}.$ (16)

Then the components of the vector $\phi$ are calculated as follows

$\displaystyle\phi^{r_{h}}$ $\displaystyle=$
$\displaystyle\frac{3V_{3}r_{h}\left(c_{2}c_{0}^{2}m^{2}+k\right)}{8\pi}+\frac{3c_{1}c_{0}m^{2}V_{3}r_{h}^{2}}{16\pi}+\frac{3c_{3}c_{0}^{3}m^{2}V_{3}}{8\pi}+2PV_{3}r_{h}^{3}-\frac{8\pi
Q^{2}}{V_{3}r_{h}^{3}}-\frac{3V_{3}r_{h}^{2}}{4\tau},$
$\displaystyle\phi^{\Theta}$ $\displaystyle=$
$\displaystyle-\cot\Theta\csc\Theta.$ (17)

Zero points are determined by $\phi^{r_{h}}=0$. We solve it and get

$\displaystyle\tau=\frac{12\pi V_{3}^{2}r_{h}^{5}}{32\pi
PV_{3}^{2}r_{h}^{6}+3c_{0}c_{1}m^{2}V_{3}^{2}r_{h}^{5}+6(c_{0}^{2}c_{2}m^{2}+k)V_{3}^{2}r_{h}^{4}+6c_{0}^{3}c_{3}m^{2}V_{3}^{2}r_{h}^{3}-128\pi^{2}Q^{2}}.$
(18)

Figure 1: Topological properties of the five-dimensional charged black hole,
where $a_{5}=1/(128\pi^{2})$, $b_{5}=-1$, $c_{5}=6$, $k=1$, $d_{5}=-15$ and
$Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=30$ is plotted in the right
picture. Zero points are at $(r_{h}/r_{0},\Theta)$= ($3.15,\pi/2$),
($10.36,\pi/2$) and ($28.99,\pi/2$), respectively.

Figure 2: Topological properties of the five-dimensional charged black hole,
where $a_{5}=\frac{1}{128\pi^{2}}$, $b_{5}=5$, $c_{5}=-10$, $k=1$, $d_{5}=1$
and $Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=5$ are plotted in the right
picture. The zero point is at $(r_{h}/r_{0},\Theta)$= ($6.55,\pi/2$).

For convenience, we order $a_{5}=Q^{2}/(V_{3}^{2}r_{0}^{4})$,
$b_{5}=c_{0}c_{1}m^{2}r_{0}$, $c_{5}=c_{0}^{2}c_{2}m^{2}$,
$d_{5}=c_{0}^{3}c_{3}m^{2}/r_{0}$. To show the zero points of $\phi^{r_{h}}=0$
in the plane $r_{h}-\tau$, we fix the black hole’s parameters at certain
values. In the left picture of Figure 1, there are three black hole branches
when $\tau_{1}<\tau<\tau_{2}$, a large black hole branch when $\tau<\tau_{1}$,
and a small black hole branch when $\tau>\tau_{2}$. A phase transition between
a small black hole and a large black hole (SBH/LBH) occurs in this region
$\tau_{1}<\tau<\tau_{2}$. For the large black hole region and the small black
hole region, the winding numbers for the zero points are $w=+1$. For the
intermediate black hole region, the winding number for the zero points is
$w=-1$. It has been proved that the positive (negative) winding numbers imply
the thermodynamic stability (instability) of the black holes. Thus the
topological number of this black hole is $W=1-1+1=1$. We order $\tau/r_{0}=30$
and plot the unit vector field $n$ in the right picture of Figure 1. There are
three zero points, zp1, zp2 and zp3. These points are at
$(r_{h}/r_{0},\Theta)$= ($3.15,\pi/2$), ($10.36,\pi/2$) and ($28.99,\pi/2$),
and their winding numbers are 1, -1 and 1, respectively. Then the topological
number is also $1$.

To further study the thermodynamic topology for this black hole with different
values of the parameters, we draw Figure 2. Obviously, $r_{h}$ monotonically
decreases with the increase of the inverse temperature $\tau$. This implies
that there is only a stable black hole and no phase transition occurs. There
is only one zero point in the right picture of Figure 2. Then the topological
number is $W=1$, which is completely consistent with the result obtained in
Figure 1. Although the black hole branches and zero points’ numbers in Figure
1 and Figure 2 are different, these data all indicate the same topological
number.

Figure 3: Topological properties of the five-dimensional uncharged black hole,
where $a_{5}=0$, $b_{5}=1$, $c_{5}=-10$, $k=1$, $d_{5}=-20$ and
$Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=4$ is plotted in the right
picture. Zero points are at $(r_{h}/r_{0},\Theta)$=($2.33,\pi/2$),
($9.94,\pi/2$) and ($51.64,\pi/2$), respectively.

Figure 4: Topological properties of the five-dimensional uncharged black hole,
where $a_{5}=0$, $b_{5}=5$, $c_{5}=-5$, $k=1$, $d_{5}=-10$ and
$Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=4$ is plotted in the right
picture. The zero points is at $(r_{h}/r_{0},\Theta)$=($5.64,\pi/2$).

Figure 5: Topological properties of the five-dimensional uncharged black hole,
where $a_{5}=0$, $b_{5}=1$, $c_{5}=5$, $k=1$, $d_{5}=10$ and
$Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=5$ is plotted in the right
picture. Zero points are at $(r_{h}/r_{0},\Theta)$=($12.40,\pi/2$) and
($34.17,\pi/2$), respectively.

Now we order $Q=0$ in the function 4 and discuss the thermodynamic topology of
the uncharged black hole in Figure 3 \- Figure 5. In the left picture of
Figure 3, there are three different black hole branches when
$\tau_{1}<\tau<\tau_{2}$, which represent the large, middle and small black
holes. The generation point and annihilation point are located at
$\tau_{1}/r_{0}$ and $\tau_{2}/r_{0}$, respectively. Therefore, there is a
SBH/LBH phase transition. For the large black hole branch and the small black
hole branch, the winding number for any zero point is $w=1$. For the the
intermediate black hole branch, the winding number is $w=-1$. Thus the
topological number is $W=1-1+1=1$. There is only one stable black hole and a
zero point in Figure 4, which implies there is no phase transition for this
black hole in this case. Its topological number is also obtained as $W=1$,
which is full in consistence with the results obtained for the charged black
hole.

In Figure 5, the annihilation point is at $\tau_{2}/r_{0}$. There are two
different black holes at the same temperature when $\tau<\tau_{2}$. The first
black hole with large radius is stable and has the winding number for any zero
point is $w=1$, the second black hole with small radius is unstable and has
the winding number for any zero point is $w=-1$. The topological number is
gotten as $W=1-1=0$, which is different from the result derived in Figure 3
and Figure 4. Comparing Figure 3, 4 and 5, we find that the topological number
for this uncharged black hole is $W=0$ or $1$, whose specific values are
determined by the values of the black hole’s parameters.

In this subsection, we found that there is a topological number for the five-
dimensional charged black hole with the different values of the parameters,
which is $W=1$. For the uncharged black hole, the black hole branches and zero
points’ numbers in Figure 3 \- Figure 5 are different, which lead to that its
topological numbers is $W=0$ or $1$.

###  III.3 Topological numbers for six-dimensional black holes

We now study the topological properties of the six-dimensional black holes in
the massive gravity. The purpose is to identify whether the black hole with
different parameters’ values has different topological numbers. Now $n=4$ in
the metric (2). The mass and entropy are

$\displaystyle
M=\frac{V_{4}r_{h}^{3}}{4\pi}\left[\frac{6c_{4}c_{0}^{4}m^{2}}{r_{h}^{2}}+\frac{3c_{3}c_{0}^{3}m^{2}}{r_{h}}+\frac{c_{0}c_{1}m^{2}r_{h}}{4}+c_{2}c_{0}^{2}m^{2}+\frac{4\pi
P}{5}r_{h}^{2}+\frac{(16\pi Q)^{2}}{12V_{4}^{2}r_{h}^{6}}+k\right],$ (19)

$\displaystyle S=\frac{1}{4}V_{4}r_{h}^{4},$ (20)

respectively. Inserting Eqs. (19) and (21) into Eq.(9), we get the generalized
free energy energy

$\displaystyle\mathcal{F}=\frac{V_{4}r_{h}^{3}}{4\pi}\left[\frac{6c_{4}c_{0}^{4}m^{2}}{r_{h}^{2}}+\frac{3c_{3}c_{0}^{3}m^{2}}{r_{h}}+\frac{c_{0}c_{1}m^{2}r_{h}}{4}+c_{2}c_{0}^{2}m^{2}+k+\frac{4\pi
Pr_{h}^{2}}{5}+\frac{(16\pi
Q)^{2}}{12V_{4}^{2}r_{h}^{6}}\right]-\frac{V_{4}r_{h}^{4}}{4\tau}.$ (21)

We use the definition of the vector and get

$\displaystyle\phi^{\text{rh}}=\frac{V_{4}\left[3r_{h}^{2}\left(c_{2}c_{0}^{2}m^{2}+k\right)+6c_{3}c_{0}^{3}m^{2}r_{h}+c_{1}c_{0}m^{2}r_{h}^{3}+6c_{4}c_{0}^{4}m^{2}+4\pi
Pr_{h}^{4}\right]}{4\pi}-\frac{8\pi
Q^{2}}{V_{4}r_{h}^{4}}-\frac{V_{4}r_{h}^{3}}{\tau}.$ (22)

Zero points are determined by the equation $\phi^{r_{h}}=0$. Then the relation
between $r_{h}$ and $\tau$ is gotten,

$\displaystyle\tau=\frac{4\pi
r_{h}^{7}}{-32\pi^{2}Q^{2}/V_{4}^{2}+6c_{4}c_{0}^{4}m^{2}r_{h}^{4}+6c_{3}c_{0}^{3}m^{2}r_{h}^{5}+3\left(c_{2}c_{0}^{2}m^{2}+k\right)r_{h}^{6}+c_{1}c_{0}m^{2}r_{h}^{7}+4\pi
Pr_{h}^{8}}.$ (23)

For convenience, we let $a_{6}=\frac{Q^{2}}{V_{3}^{2}r_{0}^{6}}$,
$b_{6}=c_{0}^{4}c_{4}m^{2}/r_{0}^{2}$, $c_{6}=c_{0}^{3}c_{3}m^{2}/r_{0}$,
$d_{6}=c_{0}^{2}c_{2}m^{2}$, $e_{6}=c_{0}c_{1}m^{2}r_{0}$, $k=1$ and
$Pr_{0}^{2}=0.001$ and draw Figure 6 \- Figure 9 in this subsection.

Figure 6: Topological properties of the six-dimensional charged black hole,
where $a_{6}=\frac{1}{32\pi^{2}}$, $b_{6}=25$, $c_{6}=-20$, $d_{6}=8$,
$e_{6}=-0.5$, $k=1$ and $Pr_{0}^{2}=0.001$. Zero points of the vector
$\phi^{r_{h}}$ in the plane $r_{h}-\tau$ are plotted in the left picture. The
unit vector field $n$ on a portion of the plane $r_{h}-\Theta$ at
$\tau/r_{0}=20$ is plotted in the middle and right pictures. Zero points are
at $(r_{h}/r_{0},\Theta)$=($1.25,\pi/2$), ($2.29,\pi/2$), ($2.79,\pi/2$),
($32.76,\pi/2$) and ($51.75,\pi/2$), respectively.

Figure 7: Topological properties of the six-dimensional charged black hole,
where $a_{6}=\frac{1}{32\pi^{2}}$, $b_{6}=-20$, $c_{6}=-30$, $d_{6}=10$,
$e_{6}=-1$, $k=1$ and $Pr_{0}^{2}=0.001$. Zero points of the vector
$\phi^{r_{h}}$ in the plane $r_{h}-\tau$ are plotted in the left picture. The
unit vector field $n$ on a portion of the plane $r_{h}-\Theta$ at
$\tau/r_{0}=20$ is plotted in the right picture. Zero points are at
$(r_{h}/r_{0},\Theta)$=($8.27,\pi/2$), ($32.32,\pi/2$) and ($59.58,\pi/2$),
respectively.

Figure 8: Topological properties of the six-dimensional uncharged black hole,
where $a_{6}=0$, $b_{6}=-1$, $c_{6}=-5$, $d_{6}=3$, $e_{6}=-0.5$, $k=1$ and
$Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=45$ is plotted in the right
picture. Zero points are at $(r_{h}/r_{0},\Theta)$=($3.33,\pi/2$),
($19.64,\pi/2$) and ($39.23,\pi/2$), respectively.

Figure 9: Topological properties of the six-dimensional uncharged black hole,
where $a_{6}=0$, $b_{6}=-1$, $c_{6}=-3$, $d_{6}=3.6$, $e_{6}=0.5$, $k=1$,
$Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=20$ is plotted in the
middle and right pictures. Zero points are at
$(r_{h}/r_{0},\Theta)$=($0.59,\pi/2$), ($0.81,\pi/2$), ($20.05,\pi/2$) and
($49.88,\pi/2$), respectively.

We first study the topological properties of the six-dimensional charged black
hole with different parameters’ values. In Figure 6, there are two generation
points and two annihilation points. The generation points are at
$\tau_{1}/r_{0}$ and $\tau_{2}/r_{0}$, respectively. The annihilation points
are at $\tau_{1}/r_{0}$ and $\tau_{2}/r_{0}$, respectively. These points
divide the black hole into five branches, namely, three stable black hole
branches and two unstable black hole branches, which leads to the topological
number $W=1-1+1-1+1=1$. Changing the parameters’ values, we find that there
are three black hole branches and three zero points in Figure 7. It is easily
to get that the topological number is $W=1$. Although their topological
numbers are the same in Figure 6 and 7, they have different phase transitions.
Clearly, this black hole has a topological number $W=1$ for the different
values of the parameters.

We then investigate the topological properties of the six-dimensional
uncharged black hole with different parameters’ values. There are three black
hole branches and three zero points in Figure 8, and then the topological
number for this black hole is $W=1$. This black hole has a LBH/SBH phase
transition. In Figure 9, there are one generation point and two annihilation
points, which divide the black hole into two stable black hole branches and
two unstable black hole branches. Four zero points are at
$(r_{h}/r_{0},\Theta)$=($0.59,\pi/2$), ($0.81,\pi/2$), ($20.05,\pi/2$) and
($49.88,\pi/2$), respectively. Thus its topological number is $W=1-1+1-1=0$.
Comparing Figure 8 and 9, we find that the topological number of this
uncharged black hole is $W=0$ or $1$, whose specific values are determined by
the values of the black hole’s parameters. This result is different from that
derived for the six-dimensional charged black hole.

Thus we conclude that the six-dimensional charged black hole has a topological
number $W=1$, while the topological number for the uncharged black hole in
this dimension has two possibilities, which is $W=0$ or $1$, and determined by
the values of the black hole’s parameters.

###  III.4 Topological numbers for seven-dimensional black holes

We continue to study the topological properties of the seven-dimensional black
holes in the massive gravity in this subsection. Then $n=5$ in the metric (2).
The mass and entropy are given by

$\displaystyle
M=\frac{5V_{5}r_{h}^{4}}{16\pi}\left[\frac{12c_{4}c_{0}^{4}m^{2}}{r_{h}^{2}}+\frac{4c_{3}c_{0}^{3}m^{2}}{r_{h}}+\frac{1}{5}c_{0}c_{1}m^{2}r_{h}+c_{2}c_{0}^{2}m^{2}+\frac{8\pi
Pr_{h}^{2}}{15}+\frac{(16\pi Q)^{2}}{40V_{5}^{2}r_{h}^{8}}+k\right],$ (24)

$\displaystyle S=\frac{1}{4}V_{5}r_{h}^{5}$ (25)

respectively. We use Eqs. (9), (24) and (25), and obtain the generalized off-
shell free energy

$\displaystyle\mathcal{F}=\frac{5V_{5}r_{h}^{4}}{16\pi}\left[\frac{12c_{4}c_{0}^{4}m^{2}}{r_{h}^{2}}+\frac{4c_{3}c_{0}^{3}m^{2}}{r_{h}}+\frac{c_{0}c_{1}m^{2}r_{h}}{5}+c_{2}c_{0}^{2}m^{2}+k+\frac{8\pi
Pr_{h}^{2}}{15}+\frac{(8\pi
Q)^{2}}{10V_{5}^{2}r_{h}^{8}}\right]-\frac{V_{5}r_{h}^{5}}{4\tau}.$ (26)

Adopting the definition $\phi^{r_{h}}=\frac{\partial\mathcal{F}}{\partial
r_{h}}$, we get the component of the vector

$\displaystyle\phi^{r_{h}}$ $\displaystyle=$
$\displaystyle\frac{5V_{5}r_{h}}{16\pi}\left[(c_{2}c_{0}^{2}m^{2}+k)r_{h}^{2}+c_{0}c_{1}m^{2}r_{h}^{3}+60c_{0}^{3}c_{3}m^{2}r_{h}+24c_{0}^{4}c_{4}m^{2}\right]+PV_{5}r_{h}^{5}$
(27) $\displaystyle-\frac{8\pi
Q^{2}}{V_{5}r_{h}^{5}}-\frac{5V_{5}r_{h}^{4}}{4\tau}$

Zero points is determined by $\phi^{r_{h}}=0$. We solve it and obtain the
relation between $r_{h}$ and $\tau$,

$\displaystyle\tau=\frac{20\pi V_{5}^{2}r_{h}^{9}}{16\pi
PV_{5}^{2}r_{h}^{10}+5c_{0}c_{1}m^{2}V_{5}^{2}r_{h}^{9}+20(c_{0}^{2}c_{2}m^{2}+k)V_{5}^{2}r_{h}^{8}+60c_{0}^{3}c_{3}m^{2}V_{5}^{2}r_{h}^{7}+120c_{0}^{4}c_{4}m^{2}V_{5}^{2}r_{h}^{6}-128\pi^{2}Q^{2}}=$
(28)

We order $a_{7}=\frac{Q^{2}}{V_{5}^{2}r_{0}^{8}}$,
$b_{7}=c_{0}^{4}c_{4}m^{2}/r_{0}^{2}$, $c_{7}=c_{0}^{3}c_{3}m^{2}/r_{0}$,
$d_{7}=c_{0}^{2}c_{2}m^{2}$, $e_{6}=c_{0}c_{1}m^{2}r_{0}$, $k=1$ and
$Pr_{0}^{2}=0.001$ and draw Figure 10 \- Figure 13 to discuss the topological
properties of the seven-dimensional black holes.

Figure 10: Topological properties of the seven-dimensional charged black hole,
where $a_{7}=\frac{1}{128\pi^{2}}$, $b_{7}=1$, $c_{7}=-2.1$, $d_{7}=1$,
$e_{7}=-0.1$, $k=1$ and $Pr_{0}^{2}=0.001$. Zero points of the vector
$\phi^{r_{h}}$ in the plane $r_{h}-\tau$ are plotted in the left picture. The
unit vector field $n$ on a portion of the plane $r_{h}-\Theta$ at
$\tau/r_{0}=20$ is plotted in the middle and right pictures. Zero points are
at $(r_{h}/r_{0},\Theta)$=($1.45,\pi/2$), ($1.68,\pi/2$), ($2.53,\pi/2$),
($8.15,\pi/2$) and ($59.84,\pi/2$), respectively.

Figure 11: Topological properties of the seven-dimensional charged black hole,
where $a_{7}=\frac{1}{128\pi^{2}}$, $b_{7}=-5$, $c_{7}=-3$, $d_{7}=3$,
$e_{7}=-0.5$, $k=1$ and $Pr_{0}^{2}=0.001$. Zero points of the vector
$\phi^{r_{h}}$ in the plane $r_{h}-\tau$ are plotted in the left picture. The
unit vector field $n$ on a portion of the plane $r_{h}-\Theta$ at
$\tau/r_{0}=40$ is plotted in the right picture. Zero points are at
$(r_{h}/r_{0},\Theta)$=($4.93,\pi/2$), ($26.30,\pi/2$) and ($51.54,\pi/2$),
respectively.

We first use different values of the parameters to discuss the topological
properties of the charged black hole. In Figure 10, there are two generation
points ($\tau_{1}/r_{0}$ and $\tau_{2}/r_{0}$) and two annihilation points
($\tau_{1}/r_{0}$ and $\tau_{2}/r_{0}$). They divide the black hole into three
stable black hole branches and two unstable black hole branches. Its
topological number is $W=1-1+1-1+1=1$. When the temperature is low, the black
hole is large. When the temperature is high, the black hole is small. There
are only one generation point and one annihilation point in Figure 11. These
points divide the black hole into three branches and three zero points appear.
Therefore, its topological number is $W=1-1+1=1$. Although this black hole has
different branches and zero points’ numbers for the different parameters’
values, it has the same topological number.

When the black hole is uncharged, $Q=0$ and $a_{7}=0$. Then the topological
properties of the uncharged black hole are discussed in Figure 12 and Figure
13. In the left picture of Figure 12, there are two annihilation points
($\tau_{2}/r_{0}$ and $\tau_{3}/r_{0}$) and one generation point
($\tau_{1}/r_{0}$), which divide the black hole into two stable black hole
branches and two unstable black hole branches. There are four zero points in
the middle and left pictures of Figure 12. Therefore, the topological number
is $W=1-1+1-1=0$. In Figure 13, there are three branches and zero points. Its
topological number is $W=1-1+1=1$. These two figures shows that this uncharged
black hole with different values of the parameters has a topological number
which is $W=0$ or $1$.

Therefore, the topological number for the seven-dimensional charged black hole
is $W=1$, while that for the uncharged black hole is $W=0$ or $1$.

Figure 12: Topological properties of the seven-dimensional uncharged black
hole, where $a_{7}=0$, $b_{7}=1$, $c_{7}=-1.9$, $d_{7}=0.5$, $e_{7}=-0.1$,
$k=1$ and $Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the
plane $r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$
on a portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=40$ is plotted in the
middle and right pictures. Zero points are at
$(r_{h}/r_{0},\Theta)$=($1.72,\pi/2$), ($4.58,\pi/2$), ($8.18,\pi/2$) and
($37.14,\pi/2$), respectively.

Figure 13: Topological properties of the seven-dimensional uncharged black
hole, where $a_{7}=0$, $b_{7}=-5$, $c_{7}=-3$, $d_{7}=3$, $e_{7}=-0.5$, $k=1$
and $Pr_{0}^{2}=0.001$. Zero points of the vector $\phi^{r_{h}}$ in the plane
$r_{h}-\tau$ are plotted in the left picture. The unit vector field $n$ on a
portion of the plane $r_{h}-\Theta$ at $\tau/r_{0}=40$ is plotted in the right
picture. Zero points are at $(r_{h}/r_{0},\Theta)$=($4.93,\pi/2$),
($26.29,\pi/2$) and ($51.54,\pi/2$), respectively.

##  IV Conclusion and discussion

In this work, we studied the thermodynamic topology of the five-, six- and
seven-dimensional anti-de Sitter black holes in the ghost-free massive
gravity. The topological numbers for these black holes obtained in the work
are listed in Table 1. In the study, we first used the different values of the
five-dimensional black holes’ parameters and calculated their topological
numbers. Due to the different values, the numbers of generation points,
annihilation points and zero points are different. However, these data lead to
the same topological number $W=1$ for the five-dimensional charged black hole,
while the topological number for the uncharged black holes has two
possibilities, namely, $W=0$ or $1$. Its specific values are determined by the
values of the black holes’ parameters. This situation also occurs in the six-
and seven-dimensional black holes.

Black hole solutions | Topological number
---|---
5d charged BH | 1
5d uncharged BH |  $0$ or $1$
6d charged BH | 1
6d uncharged BH |  $0$ or $1$
7d charged BH | 1
7d uncharged BH |  $0$ or $1$
Table 1: 5d/6d/7d represent “a five-dimenional/six-dimenional/seven-
dimensional”, BH is the abbreviation for the black hole.

In the work, we only studied the thermodynamic topology of the spherically
symmetric (i.e. $k=1$) black holes. The reason is that $k$ and
$c_{0}^{2}c_{2}m^{2}$ appear together in the generalized free energy in the
form of $k+c_{0}^{2}c_{2}m^{2}$, where $k$ characterizes the horizon curvature
and $c_{2}m^{2}$ is the coefficient of the second term of massive potential
associated with the graviton mass. In the calculation, we considered the
combined effect of these two physical quantities. That is to say, as long as
the sum of the two quantities’ values is a constant, the topological numbers
calculated in this paper are the same. For example, $k=1$ and
$c_{0}^{2}c_{2}m^{2}=3$ in Figure 13, which yields $k+c_{0}^{2}c_{2}m^{2}=4$.
If we order $k=-1$ or $0$, thus $c_{0}^{2}c_{2}m^{2}=5$ or $4$, and the same
result can be gotten in Figure 13. In Figure 12, $k=1$ and
$c_{0}^{2}c_{2}m^{2}=0.5$, which leads to $k+c_{0}^{2}c_{2}m^{2}=1.5$. When
$k=-1$ or $0$, and then we get $c_{0}^{2}c_{2}m^{2}=2.5$ or $1.5$, and the
same result in Figure 12. Therefore, the topological numbers for the Ricci
flat and hyperbolic, charged and uncharged black holes in the five-, six- and
seven-dimensional massive gravity are also obtained. The charged black holes
have a same topological number $W=1$. For the uncharged black holes, their
topological numbers are $W=0$ or $1$. The topological number for the four-
dimensional black hole in the dRGT massive gravity was calculated in TS . He
found that the uncharged black hole has a topological number $W=0$.

In GP1 , Gogoi and Phukon have studied the thermodynamic topology of the four-
dimensional Dyonic AdS black hole in the different ensembles and found that
the topological class of this black hole is ensemble dependent. Therefore,
studying the thermodynamic topology of high-dimensional black holes in the
massive gravity in different ensembles is quite meaningful, which may lead to
an interesting result.

## References

  * (1)
  * (2) P.V.P. Cunha, E. Berti and C.A.R. Herdeiro, _Light-ring stability for ultracompact objects_ , _Phys. Rev. Lett._ 119 (2017) 251102.
  * (3) P.V.P. Cunha and C.A.R. Herdeiro, _Stationary black holes and light rings_ , _Phys. Rev. Lett._ 124 (2020) 181101.
  * (4) M. Guo and S. Gao, _Universal properties of light rings for stationary axisymmetric spacetimes_ , _Phys. Rev._ D 103 (2021) 104031.
  * (5) S.W. Wei, _Topological charge and black hole photon spheres_ , _Phys. Rev._ D 102 (2020) 064039.
  * (6) R. Ghosh and S. Sarkar, _Light rings of stationary spacetimes_ , _Phys. Rev._ D 104 (2021) 044019.
  * (7) H.C.D. L. Junior, J.Z. Yang, L.C.B. Crispino, P.V.P. Cunha and C.A.R. Herdeiro, _Einstein-Maxwell-dilaton neutral black holes in strong magnetic fields: topological charge, shadows and lensing_ , _Phys. Rev._ D 105 (2022) 064070.
  * (8) S.W. Wei, Y.X. Liu and R.B. Mann, _Intrinsic curvature and topology of shadows in Kerr spacetime_ , _Phys. Rev._ D 99 (2019) 041303(R).
  * (9) Y.S. Duan and M.L. Ge, _SU(2) gauge theory and electrodynamics of N moving magnetic monopoles_ , _Sci. Sin._ 9 (1979) 1072.
  * (10) Y.S. Duan, _The structure of the topological currenty_ , _SLAC-PUB-3301_ , (1984).
  * (11) S.W. Wei, Y.X. Liu and R.B. Mann, _Black hole solutions as topological thermodynamic defects_ , _Phys. Rev. Lett._ 129 (2022) 191101.
  * (12) R. Li and J. Wang, _Thermodynamics and kinetics of Hawking-Page phase transition_ , _Phys. Rev._ D 102 (2020) 024085.
  * (13) R. Li, K. Zhang and J. Wang, _Thermal dynamic phase transition of Reissner-Nordstrom Anti-de Sitter black holes on free energy landscape_ , _JHEP_ 2010 (2020) 090.
  * (14) S.W. Wei, Y.Q. Wang, Y.X. Liu and R. B. Mann, _Observing dynamic oscillatory behavior of triple points among black hole thermodynamic phase transitions_ , _Sci. China Phys. Mech. Astron._ 64 (2021) 270411.
  * (15) S.W. Wei and Y.X. Liu, _Topology of black hole thermodynamics_ , _Phys. Rev._ D 105 (2022) 104003.
  * (16) S.W. Wei and Y.X. Liu, _Topology of equatorial timelike circular orbits around stationary black holes_ , _Phys. Rev._ D 107 (2023) 064006.
  * (17) P.K. Yerra, C. Bhamidipati and S. Mukherji, _Topology of critical points and Hawking-Page transition_ , _Phys. Rev._ D 106 (2022) 064059.
  * (18) N.C. Bai, L. Li and J. Tao, _Topology of black hole thermodynamics in Lovelock gravity_ , _Phys. Rev._ D 107 (2023) 064015.
  * (19) C.H. Liu and J. Wang, _Topological natures of the Gauss-Bonnet black hole in AdS space_ , _Phys. Rev._ D 107 (2023) 064023
  * (20) Z.Y. Fan, _Topological interpretation for phase transitions of black holes_ , _Phys. Rev._ D 107 (2023) 044026.
  * (21) C.X. Fang, J. Jiang and M. Zhang, _Revisiting thermodynamic topologies of black holes_ , _JHEP_ 2301 (2023) 102.
  * (22) D. Wu, _Topological classes of rotating black holes_ , _Phys. Rev._ D 107 (2023) 024024.
  * (23) D. Wu, _Classifying topology of consistent thermodynamics of the four-dimensional neutral Lorentzian NUT-charged spacetimes_ , _Eur.Phys.J._ C 83 (2023) 365.
  * (24) D. Wu, and S.Q. Wu, _Topological classes of thermodynamics of rotating AdS black holesy_ , _Phys. Rev._ D 107 (2023) 084002.
  * (25) D. Wu, _Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes_ , arXiv: 2306.02324 [gr-qc].
  * (26) R. Li, C.H. Liu, K. Zhang and J. Wang, _Topology of the landscape and dominant kinetic path for the thermodynamic phase transition of the charged Gauss-Bonnet AdS black holes_ , arXiv:2302.06201 [gr-qc].
  * (27) Y.B. Du and X.D. Zhang, _Topological classes of BTZ black holes_ , arXiv:2302.11189 [gr-qc].
  * (28) Y.B. Du and X.D. Zhang, _Topological classes of black holes in de-Sitter spacetime_ , arXiv:2303.13105 [gr-qc].
  * (29) M. Zhang and J. Jiang, _Bulk-boundary thermodynamic equivalence: a topology viewpoint_ , arXiv:2303.17515 [hep-th]
  * (30) R. Li and J. Wang, _Generalized free energy landscapes of the charged Gauss-Bonnet AdS black holes in diverse dimensions_ , arXiv:2304.03425 [gr-qc].
  * (31) N.J. Gogoi and P. Phukon, _Thermodynamic topology of 4D Dyonic AdS black holes in different ensembles_ , arXiv:2304.05695.
  * (32) N.J. Gogoi and P. Phukon, _Topology of thermodynamics in R-charged black holes_ , _Phys. Rev._ D 107 (2023) 106009.
  * (33) P.K. Yerra, C. Bhamidipati and S. Mukherji, _Topology of critical points in boundary matrix duals_ , arXiv:2304.14988.
  * (34) M.Y. Zhang, H. Chen, H. Hassanabadi, Z.W. Long and H. Yang, _Topology of nonlinearly charged black hole chemistry via massive gravity_ , arXiv:2305.15674 [gr-qc].
  * (35) M.R. Alipour, M.A.S. Afshar, S. N. Gashti, J. Sadeghi, _Topological classification and black hole thermodynamics_ , arXiv:2305.05595 [gr-qc].
  * (36) Z.M. Xu, Y.S. Wang, B. Wu and W.L. Yang, _Riemann surface, winding number and black hole thermodynamics_ , arXiv:2305.05916 [gr-qc].
  * (37) T.N. Hung and C.H. Nam, _Topology in thermodynamics of regular black strings with Kaluza-Klein reduction_ , arXiv:2305.15910 [gr-qc].
  * (38) D. Vegh, _Holography without translational symmetry_ , arXiv:1301.0537 [hep-th].
  * (39) R.G. Cai, Y.P. Hu, Q.Y. Pan and Y.L. Zhang, _Thermodynamics of black holes in massive gravity_ , _Phys. Rev._ D 91 (2015) 024032.
  * (40) J.F. Xu, L.M. Cao and Y.P. Hu, _$P-V$ criticality in the extended phase space of black holes in massive gravity_, _Phys. Rev._ D 91 (2015) 124033.
  * (41) D.Y. Chen, _Thermodynamics and weak cosmic censorship conjecture in extended phase spaces of anti-de Sitter black holes with particles’ absorption_ , _Eur. Phys. J._ C 79 (2019) 353.
  * (42) D.C. Zou, Y.Q. Liu and R.H. Yue, _Behavior of quasinormal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity_ , _Eur. Phys. J._ C 77 (2017) 365.
  * (43) D. Kastor, S. Ray and J. Traschen, _Enthalpy and the mechanics of AdS black holes_ , _Class. Quant. Grav._ 26 (2009) 195011.
  * (44) B.P. Dolan, _Pressure and volume in the first law of black hole thermodynamics_ , _Class. Quant. Grav._ 28 (2011) 235017.
  * (45) D. Kubiznak and R.B. Mann, _P-V criticality of charged AdS black holes_ , _JHEP_ 1207 (2012) 033.
  * (46) S.H. Hendi, B. Eslam Panah and S. Panahiyan, _Einstein-Born-Infeld-Massive Gravity: adS-black hole solutions and their thermodynamical properties_ , _JHEP_ 1511 (2015) 157.
  * (47) S.H. Hendi, R.B. Mann, S. Panahiyan and B. Eslam Panah, _van der Waals like behaviour of topological AdS black holes in massive gravity_ , _Phys. Rev._ D 95 (2017) 021501(R).
  * (48) T. Sharqui, _Topological nature of black hole solutions in massive gravity_ , [arXiv: 2304.02889 [gr-qc].