# Calibration of fisheye camera using entrance pupil

###### Abstract

Most conventional camera calibration algorithms assume that the imaging device
has a Single Viewpoint (SVP). This is not necessarily true for special imaging
device such as fisheye lenses. As a consequence, the intrinsic camera
calibration result is not always reliable. In this paper, we propose a new
formation model that tends to relax this assumption so that a Non-Single
Viewpoint (NSVP) system is corrected to always maintain a SVP, by taking into
account the variation of the Entrance Pupil (EP) using thin lens modeling. In
addition, we present a calibration procedure for the image formation to
estimate these EP parameters using non linear optimization procedure with
bundle adjustment. From experiments, we are able to obtain slightly better re-
projection error than traditional methods, and the camera parameters are
better estimated. The proposed calibration procedure is simple and can easily
be integrated to any other thin lens image formation model.

Index Terms—  camera calibration, entrance pupil, bundle adjustment, feature
extraction

##  1 Introduction

Camera calibration is the estimation of a camera’s mapping function between a
set of known world points and their measured image coordinates. The parameters
that define this mapping are usually divided into two categories: intrinsic
and extrinsic parameters. The intrinsic parameters represent the internal
characteristics of the image lens and sensors, while the extrinsic parameters
model the relative position and orientation of the camera to the 3D world [1].

Fig. 1: Illustration of the varying Entrance Pupil (EP). The entrance $c$
coincide with the optical axis $O_{c}$. The intersection of the incidence rays
originated from world points $P^{(1)},P^{(2)},P^{(3)}$ intersect the optical
axis at off-axis shift $c^{(1)},c^{(2)},c^{(3)}$ from the EP.

Fig. 2:  Image formation model for fisheye cameras under Non-Single Viewpoint
(NSVP). $(P^{(1)},P^{(2)})$ represent the world point projected onto the image
plane at $(\tilde{p}^{(1)},\tilde{p}^{(2)})$ in the presence of lens
distortion. Assuming no lens distortion, the projected image points are
$(p^{(1)},p^{(2)})$. These point rays both form incidence angles
$(\theta^{(1)},\theta^{(2)})$ with the optical axis.

Most of the popular conventional imaging system assume that all incidence ray
from a world point must pass through the optical axis in order to conform with
a Single Viewpoint (SVP) cameras, under thin lens model [2]. However, the
single viewpoint assumption is not always true since imaging system are made
of various optical elements and/or mirrors. In figure 1, we illustrate a Non-
Single Viewpoint (NSVP) imaging using a fisheye lens. The locus of all the
viewpoints in such system setup form a caustic depicted with thick curve line.
The dotted lines represent the incidence ray originated from world points
$(P^{(1)},P^{(2)},P^{(3)})$ with varying incidence angles from the optical
axis. The incidence rays meet at off-axis shift $(c^{(1)},c^{(2)},c^{(3)})$
from the Entrance Pupil (EP) denoted as $c$ which is equivalent to the optical
center $O_{c}$. Only the incidence angle that coincides with the optical axis
meets at the EP in the figure illustration. As the incidence angle increase
away from the optical axis, the off-axis shift value increase as well.

For the majority of optical lenses with narrow field of view, the varying EP
is small and thus can be ignored during the intrinsic camera calibration.
However, this is not the case for special imaging device such as catadioptric
and fisheye lenses. A fisheye lens consists of several refractions and
reflections in the imaging formation process as a result of large number of
optical elements. Therefore, a single viewpoint assumption for fisheye camera
calibration in [3, 4] implies an unrealistic constraint on the entrance pupil
location, which impacts the parameter’s sensitivity and correctness. There are
literature that incorporate EP variation, such as [5, 6, 7, 8] using thin lens
and [9] using thick lens for perspective cameras. Also, [10] using thin lens
for fisheye cameras. Most of the thin lens model either have extreme
complexity or more suitable for catadioptric imaging system only.

A new camera model based on varying entrance pupil is proposed for fisheye
lens calibration. We introduce EP correction strategy to bring NSVP to a SVP
imaging model. In the proposed model, the extrinsic part of the model
incorporates the EP parameters that are used to correct the incidence ray from
world points, under a non-linear optimization through bundle adjustment
procedure. The experimental validation on calibration data shows slightly
better reprojection error over two representative state of the art [3, 10]
that are under SVP and NSVP categories respectively.

In the next section, we present state of the art image formation model and
necessary equations needed to understand the proposed method. The section
starts with the equidistant projection formulation [3], followed by the
optical lens modeling for fisheye lenses. The proposed image formation model
and the suitable calibration procedure is detailed in section 3. In section 4,
we perform experiments on calibration image data, and comparison with two
representative state of art methods. In the last section, we summarize the
paper and explain the future direction of this work.

##  2 Image Formation Model

In the figure 2, $i=1,\ldots,n$ represents the number of world points
projected onto the lens. $O(X,Y,Z),O_{c}(X_{c},Y_{c},Z_{c})$ represent the
world and camera origins. $o(x,y),o_{p}(u,v)$ represent the image and pixel
coordinates origin, $f$ is the camera’s focal length and $r$ is the distance
between a projected image point $p$ and origin $o$. In addition, $O_{c}$
correspond to the optical center and entrance pupil $c$ of the camera/lens,
and $Z_{c}-$axis describe the camera’s optical axis. In the figure, the
incidence ray for every object point should pass through the optical center
under SVP assumption for thin lens modeling.

Extrinsic : The camera point $P_{c}^{(i)}(X_{c},Y_{c},Z_{c})$ is derived from
world point $P^{(i)}(X,Y,Z)$ using equation (1). The $3\times 3$ rotation
matrix $\mathcal{R}$ and $3\times 1$ translational vector $\mathcal{T}$ are
the extrinsic parameters that determine the external pose of a camera to the
world coordinates [2, 1]. Thanks to equation (1), the incidence angle
$\theta^{(i)}$ under SVP with the optical axis is defined as
$\theta^{(i)}=cos^{-1}\frac{Z_{c}^{(i)}}{\|\overrightarrow{O_{c}P_{c}^{(i)}}\|}$
.

$\begin{split}\begin{bmatrix}X_{c}\\\ Y_{c}\\\
Z_{c}\end{bmatrix}^{(i)}=[\mathcal{R}|\mathcal{T}]\begin{bmatrix}X\\\ Y\\\
Z\\\ 1\end{bmatrix}^{(i)}\end{split}$ (1)

Intrinsic $\&$ Lens distortion : In the presence of optical distortions and
using equidistant projection [10, 3, 2], distorted image point
$\tilde{p}^{(i)}(\tilde{x},\tilde{y})$ is expressed in equation (2), where
angle $\varphi=tan^{-1}\frac{y}{x}$, radial distance
$r(\theta)=\theta+k_{1}\theta^{3}+k_{2}\theta^{5}+k_{3}\theta^{7}+k_{4}\theta^{9}$
[3], and $k_{1},k_{2},k_{3},k_{4}$ are the lens distortion parameters.

$\begin{split}\tilde{x}=r(\theta).cos(\varphi),\\\
\tilde{y}=r(\theta).sin(\varphi)\\\ \end{split}$ (2)

As a final step, image point $\tilde{p}^{(i)}(\tilde{x},\tilde{y})$ is
transformed to its pixel coordinate location as :
$\tilde{u}=f_{x}.\tilde{x}+sk.\tilde{y}+u_{0}$ and
$\tilde{v}=f_{y}.\tilde{x}+v_{0}$ respectively. The parameter $sk$ is the skew
factor, $(f_{x},f_{y})$ are the focal lengths in pixels, $(u_{0}$, $v_{0})$
are the principal point in pixels [1].

##  3 Proposed Model using Entrance Pupil

Some fisheye calibration methods [3, 11, 4] put a lot of unnecessary pressure
on the optical lens model to simplify the distortions introduced as a result
of the varying EP. This tend to affect the accuracy of the calibration and
proper estimation of the distortion parameters. Most especially, closer object
to the camera tends to be the most impacted by the entrance pupil variation.
This problem is investigated using a 3D reconstruction of some measured object
in an industrial application. As a result, we are motivated in this work to
separate the EP formation from the intrinsic modeling of the system. With this
approach, EP is then modeled as part of the camera extrinsic parameters.

Extrinsic : For the non-single viewpoint in Fig. 2, the incidence angle is
calculated as
$\theta^{(i)}=cos^{-1}\frac{Z_{c}^{(i)}}{\|\overrightarrow{c^{(i)}P_{c}^{(i)}}\|}$.
The EP variation is then modeled as a function of $\theta$ by using higher
degree of odd polynomial. This is a reasonable assumption when one assume
there is no decentering in the lens system [10, 12]. Our approach ensure that
the EP model is separated from image coordinate system during future
optimizations, and thus is modeled as part of the extrinsic parameters. As a
result, we do not need to worry about decentering effect.

The proposed solution is better illustrated in Figure 3. We move the off-axis
$c^{(i)}$ to optical center $O_{c}$ in a series of non-linear optimization.
This is equivalent to moving point $P^{(i)}$ along the $Z_{c}-$axis using
parameter $E(\theta^{(i)})$ which is
$E(\theta)=e_{1}*\theta^{3}+e_{2}*\theta^{5}+e_{3}*\theta^{7}+e_{4}*\theta^{9}$,
where $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$ are the entrance pupil parameters.
Using the parameter $E(\theta)$, our extrinsic model is rewritten as in
equation (3).

$\begin{split}\begin{bmatrix}X_{c}\\\ Y_{c}\\\ Z_{c}\\\
\end{bmatrix}^{(i)}=\begin{bmatrix}\mathcal{R}&\mathcal{T}\end{bmatrix}\begin{bmatrix}X\\\
Y\\\ Z+E(\theta)\\\ 1\\\ \end{bmatrix}^{(i)}\end{split}$ (3)

Fig. 3: Our correction for the entrance pupil variation. The world points
$(P^{(1)},P^{(2)})$ are moved in a series of optimizations to an optimal
target plane using parameters $E(\theta^{(1)})$ and $E(\theta^{(2)})$.

These complete the proposed extrinsic model to correct the varying EP. The
entrance pupil variation most impact the Z-axis, so the proposed correction is
significant, while the X and Y-axis corrections are left to be corrected as
part of radial distortion in section 2. For lenses with large distortions
other than radial ones, various models that incorporates tilt, tangential
distortions can be easily integrated to this model [12, 4]

###  3.1 Calibration Procedure

For the calibration procedure, we follow the popular planar based calibration
method in [1]. A planar calibration target with known feature points that is
moved at different poses from the camera to be calibrated is used. Let us
denote $i=1,\ldots,n$ number of feature points, and $j=1,\ldots,m$ number of
poses. A world point and its corresponding distorted image pixel can then be
represented as $P^{(i,j)}$ and $\tilde{p}^{(i,j)}$ respectively.

The calibration procedure consist of three main steps: (i) feature extraction
[13], (ii) initial camera parameters estimation [3], (iii) Non-linear
refinement of initial camera parameters with the proposed model.

Feature Extraction : We exert the correspondence between points $P^{(i,j)}$ on
the calibration target and the projected image pixels $\tilde{p}^{(i,j)}(u,v)$
using the method proposed in [13].

Initial camera parameters estimation : We retrieve some initial camera
parameters: intrinsic $(f_{x},f_{y},u_{0},v_{0},k_{1},k_{2})$ and extrinsic
$(\mathcal{R}_{(j)},\mathcal{T}_{(j)})$ for each pose $j$ using method
proposed in traditional fisheye calibration [3].

Non-linear optimization with EP: The remaining parameters that have not been
initialized from earlier step are set to zero i.e ($sk=0$, $k_{3},k_{4}=0$,
$e_{1},\ldots,e_{4}=0$). Finally, all these parameters are refined in equation
(4) by minimizing the reprojection error between the true image pixels
$\tilde{p}^{(i,j)}$ and the reprojected one $\hat{p}^{(i,j)}$.

$\begin{split}min\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{n}\|\tilde{p}^{(i,j)}-\hat{p}^{(i,j)}(\hat{\mathcal{K}},\hat{\mathcal{R}}_{(j)},\hat{\mathcal{T}}_{(j)},\hat{e}_{1},\ldots,\hat{e}_{4},\\\
\hat{k}_{1},\ldots,\hat{k}_{4})\|^{2}\\\ \end{split}$ (4)

where
$\hat{\mathcal{K}}:\\{\hat{f}_{x},\hat{f}_{y},\hat{sk},\hat{c}_{x},\hat{c}_{y}\\}$
are the estimated intrinsic parameters, $\\{\hat{k}_{1},\ldots,\hat{k}_{4}\\}$
are the estimated radial distortion parameters, and
$\\{\hat{\mathcal{R}}_{(j)},\hat{\mathcal{T}}_{(j)},\hat{e}_{1},\ldots,\hat{e}_{4}\\}$
are the estimated extrinsic parameters. The optimization is done with bundle
adjustment using Levenberg-Marquardt’s method [14]. Please note that the world
point is corrected at each iteration to give the estimated $\hat{p}^{(i,j)}$.

##  4 Experiments

The proposed calibration method has been evaluated on both real and synthetic
dataset. The experiments have been done with a calibration target in a working
distance of between $100~{}to~{}150~{}mm$ from the camera to be calibrated.
This working distance is required to be able to accurately measure the
capability of the system in an extreme setup. The calibration target is made
up of (11$\times$7) 77 features in total (see figure 4). The acquired image
exhibit large optical distortion as a result of the fisheye lens. The feature
pixels are the Center of Gravity (CoG) of the projected circular patterns, and
have been extracted using the method proposed in [13].

Fig. 4: An example image for pose $j$ of the calibration target. The extracted
feature pixels are the center of gravity of the projected circular pattern.

The name of the fisheye lens camera device used for the calibration test is
witheld due to intellectual property issue. However, it is a virtual reality
device that is made up of 8 cameras with spatial resolution of 2048 $\times$
2048\. Each fisheye lens has a Field of View (FOV) of 195 ∘ and focal length
of 3.5mm. We have made several test on all of the 8 cameras but only the
result of one principal camera is documented in this paper due to page
limitation. The proposed calibration is evaluated with Root Mean Square (RMS)
error between the true image pixels and reprojected ones using the estimated
camera parameters, over 25 calibration images.

In the remaining part of this section, we present the results of the proposed
calibration method and compare the re-projection error with the representative
state of art algorithms: (i) Kannala et al. [3] that is based on SVP approach
and (ii) Gennery [10] that is based on NSVP approach. We have chosen Kannala
et al. [3] as the compared fisheye calibration method over Scaramuzza et al.
[11] because it provides better result under our current experimental setup
[13]. All the evaluated methods are based on thin lens model which is the main
focus in this work.

Analysis of real image test: The result and comparison between the
experimented state of art methods and our approach are shown in Table 1.
Different imaging conditions have been abbreviated in the caption of the
table. The estimate of the principal point for our method is not very far from
the physical principal point ($\frac{(u_{0},v_{0})}{2}$), even with the
incorporation of the entrance pupil variation.

By analysing the whole table, one can see that our calibration method provides
the lowest mean reprojection error. The reprojection error ( 0.1956 ) is only
slightly lower than Gennery method [10] but still validates the proposed
method.

Kannala [3] Gennery [10] Our Method $(f_{x})$ 587.9521 597.3014 591.7301
$(f_{y})$ 593.1520 594.0581 592.0340 $sk$ 0.2314 0.1526 0.1978 $(u_{0})$
1020.3841 1013.7982 1013.2001 $(v_{0})$ 1028.0564 1024.2808 1025.0300 $e_{1}$
- 0.0712 0.0851 $e_{2}$ - -0.1092 -0.2577 $e_{3}$ - 0.2917 0.3016 $e_{4}$ -
0.4861 0.5368 $k_{1}$ 0.0152 -0.0096 0.0109 $k_{2}$ 0.0161 0.0028 -0.0013
$k_{3}$ -0.0232 -0.0132 0.0008 $k_{4}$ 0.0065 0.0027 -0.0004 $error$ 0.2913
0.2115 0.1956 $std$ 0.0891 0.0792 0.0735

Table 1: The experimental results and comparism between the proposed
calibration method (ours) and state of art methods under SVP and NSVP.
($error$\- Mean error, $std$ \- standard deviation error in pixels).

Analysis of synthetic test: To demonstrate further the validity of the
proposed model, we made synthetic experiment validation using exact setup
under the real image test. We use synthetic generated feature points in order
to eliminate any bias from circular feature detection under camera
calibration. Table 2 shows the standard deviation of the calibration
parameters. In the table, we summarize the distortion parameters (radial and
EP), so we only display their mean values. The smallest mean values between
the compared methods for each camera parameter is illustrated in bold font.
From the outcome of the experiments, our method provides the lowest standard
deviation of all the compared methods for the focal length and principal point
estimation.

Kannala [3] Gennery [10] Our Method $(f_{x},f_{y})$ 0.1145 0.9275 0.9081 $sk$
0.1982 0.2140 0.2047 $(u_{0},v_{0})$ 0.3023 0.2312 0.2147 $e_{1},\ldots,e_{4}$
- 0.0313 0.0305 $k_{1},\ldots,k_{4}$ 0.0298 0.0311 0.0354

Table 2: Mean standard deviation of the camera parameters estimated using
synthetic data feature points. The smallest mean values is illustrated in bold
font.

##  5 Conclusion

We propose a new camera model that integrates entrance pupil variation for
fisheye camera calibration. We model the entrance pupil variation as part of
extrinsic and thus separate it from the intrinsic that includes the
conventional lens distortion models. The proposed method gives accurate
entrance pupil and other camera parameters estimate when applied on both real
and synthetic images.

By taking into account the Entrance Pupil variation, one ensure that the
pixels located at the highly distorted extremities of the fisheye lens, and
especially image objects that are very close to the lens are corrected. This
improves the calibration result and it is expected to greatly improve future
3D reconstruction.

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# Magnetic properties and magnetocaloric effect in a Laves phase compound
$\mbox{Er}_{2}\mbox{Rh}_{3}\mbox{Si}$

Baidyanath Sahu  André M. Strydom  Highly Correlated Matter Research Group,
Department of Physics, University of Johannesburg, PO Box 524, Auckland Park
2006, South Africa Corresponding author<EMAIL_ADDRESS>
###### Abstract

The magnetic properties and magnetocaloric effect of an intermetallic compound
$\mathrm{Er_{2}Rh_{3}Si}$ was investigated by performing temperature
dependence of dc$\textendash$ and ac$\textendash$magnetic susceptibility and
isothermal magnetizations. Polycrystalline $\mathrm{Er_{2}Rh_{3}Si}$ compound
with $\mathrm{Mg_{2}Ni_{3}Si}$-type of rhombohedral Laves phases (space group
$R\overline{3}m$, hR18) structure was synthesized by arc-melting method. The
ac$\textendash$ and dc$\textendash$magnetic susceptibility results revealed
that the compound exhibited ferromagnetic order with $T_{C}$ $\approx$ 18 K.
The obtained isothermal magnetic entropy changes ($\Delta S_{m}$), and
relative power cooling for a change of magnetic field 0$\textendash$9 T are
10.1 $\mathrm{J.~{}kg^{-1}.~{}K^{-1}}$, and 255 J. kg-1 respectively. Arrott
plots and also the universal scaling plot by rescaled temperature axis and
normalized $\Delta S_{m}$ collapse onto a one curve for independent of
magnetic fields, confirming a second order ferromagnetic to paramagnetic phase
transition in $\mathrm{Er_{2}Rh_{3}Si}$.

###### keywords:

Ferromagnet; Magnetic susceptibility ; Universal scaling plot; Magnetocaloric
effect

, and

##  1 INTRODUCTION

Study of magnetocaloric effect (MCE) of a magnetic material is an important
for the application in solid state magnetic refrigeration technology. Solid
state magnetic refrigeration has advantages over the commercial gas
compression/expansion as it is environmental friendly.

MCE is an intrinsic magneto-thermodynamic phenomena in a magnetic material,
which describes the changing of temperature with applicant of external
magnetic field in an adiabatic environment. MCE materials can be classified
from the characteristics value of the isothermal magnetic entropy change
($\Delta S_{m}$), adiabatic temperature change ($\Delta T_{ad}$), and relative
power cooling (RCP) by changing of magnetic field [1, 2, 3, 4, 5]. Research on
MCE of rare earth based ternary intermetallics magnetic compounds are devoted
low temperature application in magnetic refrigerator due to the relatively low
transition temperatures [4, 6].

Rhombohedral Laves phases for binary ($\mathrm{RT_{2}}$; R = rare-earths
elements and T = transition metals) have attracted interest due to their
fantastic magnetic properties and magnetocaloric effect [7, 8, 9, 10]. Our
previous studies reveals that ternary Laves phases ($\mathrm{R_{2}T_{3}X}$; X
= S and P block elements) intermetallic compounds can posses good MCE [11,
12]. It is also reported that Eribium (Er) based ternary intermetallic
compounds are attractive for the large/giant MCE properties [13, 14, 15]. In
this paper, a systematic magnetic study is presented in details for the
understanding of magnetic properties in the ternary Laves phases compound,
$\mathrm{Er_{2}Rh_{3}Si}$. The MCE of $\mathrm{Er_{2}Rh_{3}Si}$ has been
investigated using magnetic isotherms. A second order ferromagnetic to
paramagnetic phase transition has been observed in $\mathrm{Er_{2}Rh_{3}Si}$.

##  2 EXPERIMENTAL DETAILS

Polycrystalline sample of $\mathrm{Er_{2}Rh_{3}Si}$ was synthesized by
arc$\textendash$melting technique on a water cooled copper hearth under high
purity argon atmosphere. High purities constituent elements (purities are
99.99 wt %) were used for this sample. The ingot was flipped and remelted
several times to ensure homogeneous mixing of the constituents. The weight
loss after final melting was less than 0.5 wt.%. The ingot was wrapped with
tantalum foil, encapsulated in an evacuated quartz tube and annealed at 1273 K
for one week and then quenched in cold water. The room temperature powder
X-ray diffraction (XRD) patterns was collected using Cu$\textendash$Kα
radiation of Rigaku X-ray diffractometer. The Powder XRD pattern was analyzed
by Rietveld refinement method using the “FULLPROF” software [16, 17]. Both
dc$\textendash$ and ac$\textendash$ magnetic measurements were performed on a
commercial Dynacool physical property measurement system (PPMS), attached with
a vibrating sample magnetometer (VSM) option (made by Quantum Design, USA).
Temperature dependence of dc$\textendash$magnetization ($M(T)$) was measured
under zero field cooled (ZFC) and field cooled (FC) protocol [11].

##  3 RESULTS AND DISCUSSIONS

###  3.1 X-ray diffraction

Figure 1: The experimentally observed powder XRD pattern of
$\mathrm{Er_{2}Rh_{3}Si}$ along with the Rietveld refinement fit using space
group $R\overline{3}m$, hR18.

Fig. 1 shows the experimental and refined powder XRD patterns of
$\mathrm{Er_{2}Rh_{3}Si}$ compound at room temperature. The Rietveld analysis
was done using a rhombohedral phase with space group $R\overline{3}m$. The
Rietveld refinement analysis revealed that $\mathrm{Er_{2}Rh_{3}Si}$
crystallizes in $\mathrm{Mg_{2}Ni_{3}Si}$$\textendash$type of rhombohedral
Laves phases. The obtained lattice parameters of a unit cell are a = b =
5.556(2) Åand c = 11.829(3) Å. These obtained lattice parameters are in good
agreement with previously report values [18].

###  3.2 Magnetic properties

Fig. 2 shows the temperature dependence of ZFC and FC dc$\textendash$magnetic
susceptibility ($\chi(T)$ = $M(T)/H$) for different applied magnetic fields.
The ZFC and FC of $\chi(T)$ for all fields show a typical paramagnetic to
ferromagnetic (PM-FM) transition. The Curie temperatures ($T_{C}$) $\approx$
18 K was derived from the peak value of d$\chi(T)$/d$T$ $vs.$ $T$ of FC curves
(shown in inset(a) of Fig. 2). Below $T_{C}$, the ZFC and FC show a kink at 4
K, which is probably due to the spin reorientation at low temperature in
$\mathrm{Er_{2}Rh_{3}Si}$. The external field dependent thermomagnetic
irreversibility below $T_{C}$ is observed in ZFC and FC
dc$\textendash$magnetic susceptibility curves. This thermomagnetic
irreversibility may be as a result of the magnetic anisotropy and the domain
wall pinning effect in $\mathrm{Er_{2}Rh_{3}Si}$ [19, 20].

Figure 2: Temperature dependence of ZFC and FC dc$\textendash$magnetic
susceptibility ($\chi(T)$) under different external magnetic field. Inset (a):
d$\chi(T)$/d$T$ of the FC curve for magnetic field of 0.5 T. Inset (b):
Temperature variation of inverse $\chi(T)$ of the FC curve for external field
of 0.2 T along with the Curie$\textendash$Weiss fitting line. Inset (c):
Temperature variation real part of ac$\textendash$susceptibility for two
different frequencies 500 and 1000 Hz, while applied ac$\textendash$driven
field 0.3 mT and zero external dc$\textendash$magnetic field.

Inset(b) of Fig. 2 shows the inverse dc$\textendash$magnetic susceptibility
($\chi^{-1}(T)$) under an applied field of 0.2 T. The linear behavior of
$\chi^{-1}(T)$ deviates below 80 K. This deviation may be due to the
influenced of crystalline field of $\mathrm{Er^{3+}}$. The $\chi^{-1}(T)$ of T
$\textgreater$ 80 K, was expressed by Curie-Weiss law $\chi^{-1}(T)$ =
(${T-\theta_{P}}$)/C, where $\mathrm{C=N\mu^{2}_{eff}/3k_{B}}$ is known as
Curie constant and $\theta_{P}$ is the Weiss paramagnetic temperature. The
fitting line of Curie$\textendash$Weiss law to the experimental data is shown
in inset(b) of Fig. 2. The least$\textendash$squares (LSQ) fit on experimental
data yielded a parameter $\theta_{P}$ = -24.16 K. The observed negative value
reveals that antiferromagnetic interaction are associated with ferromagnetic
ordering in $\mathrm{Er_{2}Rh_{3}Si}$. The calculated $\mathrm{\mu_{eff}}$ =
9.85 $\mathrm{\mu_{B}}$/$\mathrm{Er}$. This obtained $\mathrm{\mu_{eff}}$ is
slightly higher than the the theoretical value for a free trivalent
$\mathrm{Er^{3+}}$, $\mathrm{g_{J}[J(J+1)]^{1/2}}$ = 9.58 $\mathrm{\mu_{B}}$
for J = 15/2, may be the result of crystal field effect (CEF). The obtained
$\mathrm{\mu_{eff}}$ value indicate that 4f$\textendash$ shell electrons of
$\mathrm{Er^{3+}}$ ions are the predominant magnetic species for
$\mathrm{Er_{2}Rh_{3}Si}$.

To further confirm the magnetic transition, ac$\textendash$susceptibility was
measured. Inset(c) of Fig. 2 shows the real part of the
$\chi^{\prime}_{ac}(T)$ data recorded at two different
ac$\textendash$frequencies; 500 and 1000 Hz. The $\chi^{\prime}_{ac}(T)$
exhibits a single peak at 18 K due to ferromagnetic transition. There is no
significant difference is observed in $\chi^{\prime}_{ac}(T)$ for two
different frequencies, indicating that there no signature of spin or cluster
glass behavior in $\mathrm{Er_{2}Rh_{3}Si}$.

Figure 3: Magnetic field dependence of isothermal magnetization at different
temperature from 6 K to 40 K with a step of 2 K. Inset; field dependence
magnetization loop at 2 K.

To investigate the order of magnetic phase transition and $\Delta S_{m}$,
isothermal magnetization ($M(H)$) were measured at different temperature upto
a field value of 9 T. The recorded $M(H)$ is shown in Fig. 3. Inset of Fig. 3,
shows the magnetization loop at 2 K upto a field value of $\pm$7 T. There no
hysteresis is observed in $M(H)$ at 2 K, indicating the
$\mathrm{Er_{2}Rh_{3}Si}$ is a is a soft ferromagnetic material, which is
beneficial for the practical application in magnetic refrigerator [21]. One
can see that there is no tendency for saturation in the measured range. The
spontaneous magnetization was estimated by extrapolating higher field part to
zero external field yielding a value of 24.1 (emu/g)/Er = 2.9 $\mu_{B}$/Er.
The magnetization value at 7 T is 28.8 (emu/g)/Er = 3.5 $\mu_{B}$/Er for 2 K.
The obtained magnetization is smaller than the magnetic moment of a free
$\mathrm{Er^{3+}}$ ion (g$J$ = 9 $\mu_{B}$). The reduction of saturation value
may result from the magnetic anisotropy due to the CEF effects in
$\mathrm{Er_{2}Rh_{3}Si}$.

In order to get more information about nature of order of magnetic phase
transition, Arrott plots ($M^{2}~{}vs.~{}H/M$) were made from the isothermal
magnetization and is shown in Fig. 4. As per the Banerjee criterion [22], the
positive slops in $M^{2}~{}vs.~{}H/M$ indicate $\mathrm{Er_{2}Rh_{3}Si}$
compound undergoes second order ferromagnetic to paramagnetic phase
transition. The order of magnetic phase transitions is also confirmed by
employing universal scaling plots of normalized magnetic entropy change as
discussed later.

###  3.3 Magnetocaloric effect

The $\Delta S_{m}$ of $\mathrm{Er_{2}Rh_{3}Si}$ was calculated from the
isothermal magnetization curves by using the following Maxwell’s magnetic
thermodynamic relation [1, 2]

$\Delta S_{m}(T,H)=\int_{H_{i}}^{H_{f}}\left(\frac{\partial M}{\partial
T}\right)_{H^{\prime}}dH^{\prime},$ (1)

where $H_{i}$ is the initial magnetic field (zero in this present study) and
$H_{f}$ is the final field (9 T in this present experiment). Fig. 5 shows the
temperature variation $-\Delta S_{m}$ for different values of applied fields.
The obtained $-\Delta S_{m}$ values are positive for a wide range of
temperatures and a peak appears around the $T_{C}$, indicating paramagnetic to
ferromagnetic transition. The maximum value of isothermal magnetic entropy
changes $(-\Delta S^{max}_{m}$) gradually increases with increasing magnetic
field. It is found that the $-\Delta S^{max}_{m}$ values are 3.6 , 7.5, 9.1
and 10.1 $\mathrm{J.kg^{-1}.K^{-1}}$ for the change of magnetic field from
0$\textendash$2, 0$\textendash$5, 0$\textendash$7 and 0$\textendash$9 T
respectively.

Figure 4: Arrott plots of $M^{2}$ $vs.$ $H/M$ at different temperatures
ranging from 6 K to 40 K with a step of 2 K.

The relative cooling power (RCP) is also another important parameter to
quantify the heat transfer between the hot and cold reservoirs [23]. The RCP
is defined as the product of the maximum magnetic entropy change $|-\Delta
S^{max}_{m}|$ and the full width at half maximum ($\delta T_{FMHM})$ of
$\Delta S_{m}$ $vs.$ $T$ curves; RCP = $|-\Delta S^{max}_{m}|$ $\times$
$\delta T_{FMHM}$ [23]. The RCP was estimated from $\Delta S_{m}$ $vs.$ $T$
curves for different magnetic fields. The obtained RCP values gradually
increase with the increase of change in magnetic field. The RCP values are 40,
135, 200, and 255 J.kg-1 for a change of magnetic field 0$\textendash$2,
0$\textendash$5, 0$\textendash$7, and 0$\textendash$9 T respectively. The
magnetic transition temperature, $-\Delta S^{max}_{m}$ and RCP values at 5 T
of $\mathrm{Er_{2}Rh_{3}Si}$ and reported Er$\textendash$based ternary
intermetallic magnetic compounds are listed in Table 1 for the sake of
comparison. From these obtained $-\Delta S^{max}_{m}$ and RCP value, it can be
assumed that $\mathrm{Er_{2}Rh3Si}$ belongs to a class of considerable MCE
materials.

Figure 5: Temperature variation of isothermal magnetic entropy changes ($\Delta S_{m}$) of $\mathrm{Er_{2}Rh_{3}Si}$ for different value of magnetic field. Table 1:  The transition temperature ($T_{M}$), the maximum value of magnetic entropy change ($\Delta S_{m}^{max}$), relative cooling power, RCP, under the change of magnetic field 0$\textendash$5 T for $\mathrm{Er_{2}Rh_{3}Si}$ together with ternary Er$\textendash$based intermetallic compounds.  Compound |  $T_{M}$ |  $-\Delta S_{m}$ |  RCP |  Ref
---|---|---|---|---
|  (K) |  ($\mathrm{J.kg^{-1}.K^{-1}}$) |  (J.kg-1) |
$\mathrm{Er_{2}Rh_{3}Si}$ |  18 |  7.5 |  135 |  PW
$\mathrm{Er_{2}Rh_{3}Ge}$ |  21 |  9.2 |  225 |  [11]
$\mathrm{Er_{2}Co_{2}Al}$ |  32/21 |  5.9 |  152 |  [24]
$\mathrm{Er_{2}Ni_{2}Ga}$ |  7.1 |  9.62 |  196 |  [25]
$\mathrm{ErRhSi}$ |  8.5 |  8.7 |  — |  [26]
$\mathrm{ErAgAl}$ |  14 |  10.5 |  261 |  [27]
$\mathrm{ErPtGa}$ |  5 |  8.0 |  159 |  [28]
$\mathrm{ErFeAl}$ |  55 |  6.1 |  311 |  [29]
PW: Present Work |  |  |  |
Figure 6: Universal scaling plot by normalized isothermal magnetic entropy
change ($\Delta S_{m}$/$\Delta S^{max}_{m}$) as a function of the rescaled
temperature $\theta$ around $T_{C}$ for different values of magnetic field of
$\mathrm{Er_{2}Rh_{3}Si}$.

Franco $et$ $al.,$ [30, 31, 32, 33, 34] have proposed an universal scaling
plot by normalizing temperature dependent isothermal magnetic entropy change
for a second order magnetic phase transition material. Universal scaling
curves are constructed by the normalized entropy change ($\Delta
S_{m}$/$\Delta S^{max}_{m}$) against rescaled temperature ($\theta$). The
$\theta$ is given by the following expression;

$\displaystyle\theta=(-(T-T_{C})/(T_{r1}-T_{C})),T<T_{C}$
$\displaystyle((T-T_{C})/(T_{r2}-T_{C})),T>T_{C},$ (2)

where $T_{r1}$ and $T_{r2}$ are the temperatures corresponding to $\Delta
S^{max}_{m}$/$\Delta S_{m}$ = 0.5 below and above $T_{C}$, respectively for
each applied field. Fig. 6 shows the Universal scaling plot for
$\mathrm{Er_{2}Rh_{3}Si}$. It is noted that the $\Delta S_{m}$/$\Delta
S^{max}_{m}$ $vs.$ $\theta$ curves nearly collapse onto a single master curve
curve independent of temperature and field, indicating a second order magnetic
phase transition in $\mathrm{Er_{2}Rh_{3}Si}$.

##  4 SUMMARY

The magnetic properties and magnetocaloric effect in the
$\mathrm{Er_{2}Rh_{3}Si}$ compound have been studied in the present work. The
ternary intermetallic compound $\mathrm{Er_{2}Rh_{3}Si}$ crystallizes in the
rhombohedral $\mathrm{Mg_{2}Ni_{3}Si}$$\textendash$type structure and shows a
paramagnetic to ferromagnetic phase transition with $T_{C}$ of 18 K. The
explored MCE from magnetic isotherms revealed that $-\Delta S_{m}$, and RCP
are 10.1 $\mathrm{J.~{}kg^{-1}.~{}K^{-1}}$, and 255 J. kg-1 respectively for a
change of magnetic field 0$\textendash$9 T. The existence of second order
ferro$\textendash$paramagnetic phase transition is confirmed from Arrott plots
and the universal scaling plots by normalized $\Delta S_{m}$ and rescaled
temperature axis.

## ACKNOWLEDGEMENT

This work is supported by Global Excellence and Stature (UJ-GES) fellowship,
University of Johannesburg, South Africa. AMS thanks the URC/FRC of UJ for
financial assistance.

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