Low-temperature properties of BaWO4 based on experimental heat capacity in the range 5.7–304u202fK

Article abstract of DOI:10.1016/j.jallcom.2019.06.197

The heat capacity of barium tungstate single crystal was measured by the adiabatic method in the range of 5.7–304 K. No anomalies in the heat capacity associated with the phase transition were found. The analysis of the functional behavior of heat capacity near zero revealed a low-frequency peak in the phonon density of states of BaWO₄. An approach was proposed to allow quantitative description of the acoustic wing of the phonon density of states. The approximation of experimental data by an analytical expression was done, including the temperature region near 0 K. The Debye temperature at absolute zero has been obtained. The thermodynamic functions were calculated: entropy, enthalpy increment and Gibbs free energy.

Full text of DOI:10.1016/j.jallcom.2019.06.197

Journal of Alloys and Compounds 802 (2019) 235e243
Contents lists available at ScienceDirect
Journal of Alloys and Compounds
journal homepage: http://www.elsevier.com/locate/jalcom
Low-temperature properties of BaWO4 based on experimental heat
capacity in the range 5.7e304 K
,
A.E. Musikhin a *, M.A. Bespyatov a, V.N. Shlegel a, O.E. Safonova b
a Nikolaev Institute of Inorganic Chemistry, Siberian Branch of the Russian Academy of Sciences, 3,Acad Lavrentiev Ave, Novosibirsk, 630090, Russia
b Sobolev Institute of Geology and Mineralogy, Siberian Branch of the Russian Academy of Sciences, 3,Akad Koptyug Pr., Novosibirsk, 630090, Russia
a r t i c l e i n f o
a b s t r a c t
Article history:
The heat capacity of barium tungstate single crystal was measured by the adiabatic method in the range
of 5.7e304 K. No anomalies in the heat capacity associated with the phase transition were found. The
analysis of the functional behavior of heat capacity near zero revealed a low-frequency peak in the
phonon density of states of BaWO4. An approach was proposed to allow quantitative description of the
acoustic wing of the phonon density of states. The approximation of experimental data by an analytical
expression was done, including the temperature region near 0 K. The Debye temperature at absolute zero
has been obtained. The thermodynamic functions were calculated: entropy, enthalpy increment and
Gibbs free energy.
Received 19 February 2019
Received in revised form
12 June 2019
Accepted 16 June 2019
Available online 18 June 2019
Keywords:
Heat capacity
Adiabatic calorimetry
Thermodynamic functions
Laser materials
© 2019 Elsevier B.V. All rights reserved.
Barium tungstate
1. Introduction
been studied. Its structure [12e15], luminescent and optical prop-
erties [16e19], elastic properties [20], thermal expansion, and
Currently, materials based on molybdate and tungstate single
crystals are the subject of extensive research due to their unique
optical properties that account for a variety of their practical ap-
plications. Among them, barium tungstate, BaWO4, attracts a strong
interest both in science and practice. The double beta decay in-
vestigations are giving the rise of a multitude of dedicated exper-
iments in nuclear and particle physics. Barium contains two
conductivity [14,15,21] were examined. Spectral and other prop-
erties of BaWO4 were examined by computational methods
[19,22e24]. The literature reports experimental heat capacity ob-
tained using the DSC method in the range of 122e322 K [22]; there
are no experimental data available for temperatures below 122 K.
The heat capacity data obtained by computational methods are also
known [22,23]. However, the data on heat capacity presented in
these papers differ considerably. The aforesaid necessitates further
research on heat capacity and thermodynamic functions of BaWO4
at low temperatures.
In this paper, we report experimental data on heat capacity of
barium tungstate single crystal obtained by the adiabatic method in
the temperature range of 5.7e304 K. The results were used to
calculate thermodynamic functions and to determine a number of
other characteristics of this compound.
potentially 2
b
active isotopes, 130Ba and 132Ba [1,2]. The 130Ba
isotope is of particular interest because its double beta decay is not
yet detected unambiguously [3e5]. BaWO4 crystals are used to
create solid-state lasers [6e9]. An important advantage of such
crystals as barium tungstate is that their structure allows alloying
with laser-active rare earth metal ions [10]. This enables creation of
new laser media that allow generating coherent radiation for a
discrete set of wavelengths and implementing various combina-
tions of radiated frequencies. Paper [11] reports that, among all
crystals with scheelite structure, BaWO4 is the most effective
crystal for creating Raman lasers with tunable emission frequency.
As of today, quite a number of barium tungstate properties have
2. Experimental
2.1. Sample
Single crystal sample of BaWO4 was grown using the low ther-
mal gradient Czochralski technique [25,26]. Deeply purified (the
purification procedure is described in Ref. [27]) tungsten oxide WO3
* Corresponding author.
E-mail address: musikhin@niic.nsc.ru (A.E. Musikhin).
https://doi.org/10.1016/j.jallcom.2019.06.197
0925-8388/© 2019 Elsevier B.V. All rights reserved.

236
A.E. Musikhin et al. / Journal of Alloys and Compounds 802 (2019) 235e243
and commercial high-purity (content of the main component is
99.5 wt%) barium carbonate BaCO3 were used as the initial re-
agents. A mixture of tungsten oxide and barium carbonate was
mixed thoroughly to ensure a homogeneous composition. Syn-
thesis and melting were carried out in a platinum crucible of
70 mm in diameter. As a result, a colorless BaWO4 single crystal of
optical quality without any visual defects, inclusions or cracks was
obtained. For more detailed description of the growing process and
the quality of obtained BaWO4 crystals, see Ref. [28]. A quantitative
chemical analysis of the sample using a Thermo Fisher Scientific
iCAP Qc ICP-MS, coupled with an ESI NWR-213 laser ablation sys-
tem, was done. The results are shown in Table 1.
Barium tungstate belongs to the structural type of scheelite
(CaWO4) and has an I41/a space group. The unit cell of the crystal
contains WO24ꢀ anions, which form tetrahedrons slightly com-
pressed along the c-axis, between which the Ba2þ cations sur-
rounded by eight oxygen ions are positioned (Fig. 1). The sample
was studied by powder X-ray diffraction by Shimadzu XRD-7000
diffractometer (with Ni-filtered CuKa radiation in the 2q range of
Fig. 1. Crystal structure of BaWO4.
10ꢁe80ꢁ). The obtained powder pattern corresponds to literature
data for BaWO4 [29] (Fig. 2).
2.2. Experimental heat capacity
To measure heat capacity, a sample was made from a BaWO4
single crystal in the form of a hollow cylinder whose dimensions
corresponded to the internal volume of the calorimetric ampoule
(Fig. 3). The sample mass in a vacuum was 51.490 g. Molar mass of
385.18 g molꢀ1 was determined from the chemical formula of
BaWO4. A disassemblable calorimetric ampoule with the sample
was placed in a vacuum manipulator, then it was vacuumized, filled
with helium as a heat exchange gas (to 860 Pa), and sealed [30]. The
molar heat capacity Cp,m(T) of the sample was measured by the
adiabatic method using a plant which is described in detail in paper
[31,32]. The results of heat capacity measurements of a standard
substance (benzoic acid) deviate from reference data [33] by 1.0% in
the range of 5e20 K; 0.3% in the range of 20e80 K; and 0.15% in the
range of 80e273 K. The heat capacity of barium tungstate has been
measured at 82 points in the range of 5.7e304 K. The obtained data
are presented in Table 2. Analysis of the functional behavior of
BaWO4 heat capacity gives no indications for the existence of any
special features in Cp,m(T) associated with phase transitions.
Fig. 2. Experimental (black) and theoretical (red) X-ray powder diffraction pattern of
BaWO4. (For interpretation of the references to color in this figure legend, the reader is
referred to the Web version of this article.)
Table 1
The mass spectrometric analysis results of BaWO4 sample.
Element
wt%
Ag, La, Ni, Cd, Pb
Au, Sn, Y, Zr
Al
n/a (5$10ꢀ4
3$10ꢀ4
)
9$10ꢀ5
As, Sm, Mn, Ga, Ge, Sb
n/a (1$10ꢀ4
<1$10ꢀ3
)
Ba, Fea, Mga
Be, V, Cs, Co
Bi, Cu
Ce, Tb
Cr
n/a (2$10ꢀ5
6$10ꢀ4
)
n/a (7$10ꢀ6
n/a (1$10ꢀ3
)
)
Eu, Gd, Dy, Ho, Er, Tm, Yb, Lu,
Tl, Pr, U, Th, Ta, In
Li
Mo, P, Sc, Sr
Nb, Nd, Rb, Se
Rh
Ti*
Zn
n/a (1$10ꢀ5
n/a (9$10ꢀ5
4$10ꢀ3
)
)
6$10ꢀ5
8$10ꢀ4
<2$10ꢀ3
5$10ꢀ3
n/a e no impurity was detected, the impurity detection limit is shown in
brackets.
a
Impurity detection is limited by spectral interference. Relative standard
Fig. 3. The sample of BaWO4 single crystal (right) and calorimetric ampoule (left). The
uncertainty of the results is 0.15.
ruler scale is in cm.

A.E. Musikhin et al. / Journal of Alloys and Compounds 802 (2019) 235e243
237
Table 2
Experimental values of heat capacitya of BaWO4 single crystal (molar mass M ¼ 385.18 g molꢀ1) at pressure p ¼ 0.1 MPa in the 5.7e304 K temperature range.
T/ K
Cp,m/ J molꢀ1 Kꢀ1
T/ K
Cp,m/ J molꢀ1 Kꢀ1
T/ K
Cp,m/ J molꢀ1 Kꢀ1
T/ K
Cp,m/ J molꢀ1 Kꢀ1
5.69
7.07
8.22
9.42
0.1088
0.2116
0.359
0.591
0.850
1.181
1.793
2.600
3.455
4.643
6.493
8.389
10.14
12.28
14.72
18.08
21.23
23.63
26.29
29.16
32.47
51.53
55.80
59.84
63.82
68.02
72.43
76.99
78.85
81.75
82.87
86.55
87.65
91.14
92.18
95.50
96.50
99.75
101.04
105.80
110.38
114.80
36.08
39.49
42.59
45.50
48.41
51.33
54.23
55.30
57.12
57.76
59.92
60.51
62.45
62.94
64.78
65.21
66.93
67.41
69.82
72.04
74.05
119.10
123.62
128.37
132.99
137.50
141.92
146.40
151.27
156.36
161.36
166.28
171.12
175.88
180.69
185.53
190.31
195.03
199.70
204.32
208.88
213.40
75.86
77.69
79.50
81.28
83.00
84.47
86.06
87.56
89.14
90.63
92.03
93.40
94.66
95.95
97.16
98.37
99.46
100.51
101.62
102.51
103.34
217.88
222.31
226.99
231.83
236.66
241.49
246.29
251.04
256.01
260.94
265.57
270.16
274.71
279.24
284.03
289.04
294.00
298.93
304.16
e
104.37
105.50
106.50
107.25
108.33
109.38
110.20
110.94
112.01
112.81
113.57
114.30
115.05
115.86
116.54
117.30
118.31
118.98
119.53
e
10.43
11.44
12.86
14.41
15.81
17.50
19.82
22.03
23.89
26.12
28.62
31.97
35.17
37.63
40.46
43.56
47.28
e
e
a
Standard uncertainty for temperature u(T) ¼ 0.01 K; relative standard uncertainty for the heat capacity ur(Cp,m) ¼ 0.01 (5 < T < 20 K); 0.003 (20 K < T < 80 K); 0.0015
(80 < T < 310 K).
Fig. 4. Molar heat capacity of BaWO4 as a function of temperature in the range of
5.7e304 K. Open circles are our experimental data; filled circles are experimental data
[22]; dash line is calculated heat capacity [22]; dotted line is calculated heat capacity
[23].
Fig. 5. Experimental molar heat capacity of BaWO4: circles are our data (adiabatic
method); unfilled rhombuses are data [22] (DSC method); filled rhombuses are cor-
rected data [22].
The comparison of the heat capacity data obtained with the data
of [22,23] is shown in Fig. 4. The experimental results of paper [22]
are lower than our data and deviate by a value varying from 8% to
5% in the range of 122e304 K. The computational data obtained in
Ref. [22] are slightly lower than the experimental data reported in
the same paper, and, as seen from the figure, are much lower than
our data obtained when examining a wide temperature range. The
data of [23] for temperatures above 200 K are in accord with our
data, but below 200 K an increasing deviation is observed as the
temperature decreases, this deviation being 10% at ~25 K (heat
capacity data for comparison were taken from Fig. 2 of paper [23]).
The experimental data of [22] on heat capacity in the 122e322 K
range were obtained by the DSC method. Considering the specifics
of the DSC method, the data obtained may have systematic de-
viations which, as a first approximation, are usually associated with
the shifting of the experimental curve by a certain constant value
3. Calculation
3.1. Description of heat capacity near zero temperature
A general expression for the heat capacity of solid in harmonic
approximation is written as
∞
ð
CðTÞ ¼ 3N kB gð
u
Þ
J
ð
u
; TÞd
u;
(1)
0
where
J(u,T) is the Einstein function:
ꢀ
ꢁ2
ꢀ
ꢁ
u
u
Jð
u
; TÞ ¼
sh2
;
(2)
2T
2T
d
С. Assuming that the data of [22] may have a systematic deviation,
N is the number of atoms in the system under consideration, kB
we calculated this correction which was
d
С ¼ 6.6 J molꢀ1 Kꢀ1. The
is the Boltzmann constant, ħ is the Planck's constant, g(u) is the
phonon states density normalized to unity.
In the region of low temperatures (below the temperature cor-
responding to the maximum entropy increment), irrespective of
the shape of g(u), heat capacity C(T) can be represented as a power
result of [22] obtained with this correction taken into account is
shown in Fig. 5 together with our data. As can be seen, the two
curves agree with each other very well in the 122e322 K range,
with the relative root-mean-square deviation of the data being
0.13%.
function:

238
A.E. Musikhin et al. / Journal of Alloys and Compounds 802 (2019) 235e243
the dependence a(T) allows us to obtain important information that
is necessary to select a physically reasonable approach when
describing the low-temperature heat capacity.
CðTÞ ¼ ATaðTÞ
;
(3)
where A is a certain factor,
a
(T) is a dimensionless value that
To find out the behavior of heat capacity of barium tungstate at
characterizes the degree of increase in heat capacity versus tem-
perature. In the interval under consideration, (T) changes from
about 3 to 1. When the relationship of C(T) is known, the exponent
low temperatures, we used the function of a(T), which was calcu-
lated based on experimental points by formula (5) (Fig. 6, circles).
As can be seen from the figure, in the 5e15 K range, there are values
a
a
(T) can be calculated by the formula:
of a(T) > 3, which, as was noted above, testifies to presence of a low-
frequency peak in the phonon density of states of BaWO4 single
crystal. Such behavior is fairly typical and is usually observed in
examining perfect single-crystal samples, see, for example, [38,39].
ꢂ
ꢃ
ꢄ
CðTÞ
T
T*
a
ðTÞ ¼ lim ln
ln
:
(4)
CðT*Þ
T*/ T
Taking account of the observable values of a(T) > 3, we used a
If the function of C(T) is defined by a known set of points (Ti, Ci),
the following formula can be used to calculate a(T):
physically reasonable approach to describe heat capacity Cp(T) near
zero of temperatures [39]; within the framework of this approach
the relationship of Cp(T) is represented as a sum of heat capacity of
the Debye model CD(T) and of the Einstein model CE(T):
ꢃ
C2
C1
T2
T1
aðTÞ ¼ ln
ln
;
(5)
where T ¼ (T1 þ T2)/2 and T1, T2 are temperatures of the closest
experimental points to which C1 and C2 heat capacities correspond.
CpðTÞ ¼ ð1 ꢀ
h
Þ,CDðTÞ þ
h
,CEðTÞ;
(8)
It is evident that the accuracy of calculation of the function of
a(T)
decreases significantly as the difference between T1 and T2 in-
creases; however, the estimated values of a(T) can be derived from
the experimental heat capacity of Cp(T).
where heat capacities Cp(T), CD(T), and CE(T) are normalized to unity
(we will use index m to designate molar heat capacity, for example,
Cp,m(T)), value
h reflects the share of the vibrational modes of the
Another form of the expression for
formula (3). Taking the derivative of function (3) and assuming that
(T) practically does not change in the small neighborhood of
a(T) can be obtained from
Einstein heat capacity. As in known, the Debye heat capacity taking
account of the second term of the small-parameter expansion, T/
qD ≪ 1, is written as:
a
temperature T, we obtain:
vCðTÞ
¼ ATa,
a
Tꢀ1
:
(6)
ꢅ
ꢆ
CDðTÞ ¼
3 ꢀ 3 eꢀ
;
(9)
p4
T
qD
qD
T
vT
4
5
q
D=T
Replacing ATa with C(T) the left-hand side of (6) in accordance
with equation (3), we obtain:
where qD is the Debye characteristic temperature. The Einstein heat
capacity is described by the equation:
ꢃ
vCðTÞ CðTÞ
aðTÞ ¼
:
(7)
vT
T
Here, we omitted the heat capacity index, because formulas (3)e(7)
can be used for analysis of both isobaric heat capacity and isochoric
heat capacity. Formulas (4) and (7) are fully equivalent and provide
a more accurate description where a smoothed relationship of C(T)
is present.
Analysis of the heat capacity curve using the function a(T) near
zero temperatures allows us to conclude about some features of the
compound under consideration [34,35]. For example, the depen-
ꢀ
ꢁ2
ꢀ
ꢁ
qE
2T
qE
2T
CEðTÞ ¼
sh2
;
(10)
where qE is the Einstein characteristic frequency, expressed in K.
Equation (8) allows describing the experimental heat capacity of
Cp(T) in a wider temperature range, as compared to only using the
Debye law, and makes the heat capacity extrapolation to 0 K more
reliable. By dividing both members of equation (8) by CD(T) and
changing the variables:
dence
a(T) of isostructural compounds has the same functional
behavior [35]. Also value of the degree of increase in heat capacity
allows determining the degree of anisotropy of the crystal lattice of
the compound under consideration. In isotropic structures
a
¼ 3, in
two-dimensional and one-dimensional structures
a
¼ 2 and 1,
respectively [36], and in quasi-two-dimensional and quasi-one-
dimensional structures, intermediate values may be observed as
temperatures approach zero [37]. The value of
mining the scope of validity of the Debye law, in which the degree
of increase in heat capacity is three. The function (T) may be an
a(T) allows deter-
a
indicator of the presence of additional components in the heat
capacity of a solid or reflect some features of C(T) behavior. The
relationship of
peak”) in the density of states g(
narrow Einstein frequency band implemented against the back-
ground of the acoustic wing of g( ). This is evident from the values
of the function of (T) > 3. This is explained by the fact that the
Einstein heat capacity, which can be regarded as a good approxi-
mation in describing the low-frequency peak in g( ), grows
exponentially, which is much faster than the dependence T 3. Thus,
a
(T) enables revealing a low-frequency peak (“boson
u
) if it exists. Such a peak is a
u
a
Fig. 6. Temperature dependence of function a(T) for BaWO4: circles are the calculation
based on experimental heat capacity Cp(T) by formula (5); solid line is the calculation
for smoothed heat capacity C(ps)(T) by formula (4) or (7). Dash line represents the
calculation for Debye model at QD(0) ¼ 272.2 K.
u

A.E. Musikhin et al. / Journal of Alloys and Compounds 802 (2019) 235e243
239
CpðTÞ
CDðTÞ
YðqDÞ ¼
XðqD
ꢀ 1;
(11)
CEðTÞ
;
qEÞ ¼
ꢀ 1
CDðTÞ
expression (8) is transformed into a homogeneous linear equation
YðqDÞ ¼
h
XðqD
;
qEÞ:
(12)
The best description of the experimental points by equation (12)
in coordinates (11) was found by varying the unknown parameters
of qD, qE and monitoring the minimum of the sum of squared de-
Fig. 8. Molar heat capacity of BaWO4 in the C/T versus T2 coordinates up to 15 K: circles
are experimental values; straight line is the description by equation (8). Dash line
corresponds to the Debye law at QD(0) ¼ 272.2 K.
viations. This description of BaWO4 heat capacity was achieved
with the values of the parameters qD ¼ 267.5 K and qE ¼ 81.2 K
(Fig. 7). The slope of straight line (12) determines the value of
h
¼ 0.05158. Thus, the description of the experimental heat ca-
pacity of BaWO4 below the temperature T0 x 15 K was obtained.
To calculate the Debye characteristic temperature at zero QD(0),
we considered the obtained description of heat capacity Cp(T) in the
region where the Debye law is valid (at T / 0). Evidently, in this
case all vibrational modes change to acoustic vibrations, because
the Einstein contribution becomes vanishingly small, and only the
first summand remains in equation (8). Also, in accordance with the
Debye law, only the first summand remains in equation (9). Taking
the aforesaid into account, by substituting (9) to (8) we obtain an
equation which, in conjunction with the well-known expression
describing the Debye law, forms a set of two equations:
equation (8) was used to calculate the relationship of
BaWO4 (Fig. 6, solid line). So, Figs. 6e8 demonstrate a good
agreement of the experiment and its description by equation (8).
a(T) for
3.2. Acoustic component of phonon density of states
The heat capacity of a solid body contains 3NAn vibrational
modes (degrees of freedom). All vibrational modes are divided into
acoustic and optical modes associated with the respective com-
ponents of the vibrational spectrum. It is known that, in general, a
solid body has three acoustic branches with corresponding two
transverse modes and one longitudinal mode. Each such mode is
characterized by a certain acoustic speed with which a certain
Debye temperature may be associated, this temperature charac-
terizes the cutoff frequency of the respective acoustic branches
8
ꢅ
ꢆ
>
3
>
>
>
>
CpðTÞ
3Rn
4
5
p4
T
qD
¼ ð1 ꢀ
h
Þ
;
>
<
(13)
ꢅ
ꢆ
>
>
>
3
p4
T
C ðTÞ
4
5
p
>
>
¼
:
>
:
3Rn
QDð0Þ
(Fig. 9). So, the acoustic portion of the phonon density of states g(u)
in the continuum limit can be represented as:
The solution of set (13) in relation to QD(0) gives the expression
for the Debye temperature at zero:
gð
u
Þ ¼ a1u2 þ a2u2 þ a3u2 ¼ aeff u2
;
(15)
qD
pffiffiffiffiffiffiffiffiffiffiffiffi
QDð0Þ ¼
:
(14)
3
1 ꢀ
h
where aeff is a parameter that effectively take account of all the
three acoustic branches of the spectrum with the effective cutoff
frequency being determined from the normalization conditions.
Equation (15) accurately describes the spectrum in the region of
continuum description (see Fig. 9).
The value of QD(0) calculated for barium tungstate was
272.2 2.0 K.
The experimental heat capacity of BaWO4 (circles), its descrip-
tion by equation (8) (solid line), as well as the heat capacity of the
Debye model at QD(0) ¼ 272.2 K (dash line) are provided in Fig. 8 in
the C/T versus T2 coordinates. Also, the heat capacity description by
The right-hand side of equation (15) conforms to the Debye law,
) ~
g(u
u
2. Expressing frequency in Kelvins as
q
¼
u$(Z/kB), we can
write the normalization condition for g(
q):
Fig. 7. Heat capacity of BaWO4 in the Y versus X coordinates: circles are experimental
values; straight line is the description by equation (12) with parameters qD ¼ 267.5 K,
Fig. 9. Schematic representation of the three acoustic branches of the phonon density
of states.
qE ¼ 81.2 K and
h
¼ 0.05158.

240
A.E. Musikhin et al. / Journal of Alloys and Compounds 802 (2019) 235e243
QDð0Þ
∞
ð
ð
gðq
Þd
q
¼ aeff
q
2d
q
¼ 1:
(16)
0
0
Condition (16) allows to express aeff parameter via the Debye
temperature QD(0):
3
aeff
¼
:
(17)
(18)
Q3Dð0Þ
As a result, the expression for g(q) is written as:
3
gðq
Þ ¼ aeff q2
¼
q2
:
Q3Dð0Þ
Fig. 10. Acoustic wing of the phonon density of states of BaWO4. Upper curve is the
description within the framework of the Debye model; lower curve is the function
obtained by differentiation of heat capacity. The region of accurate continuum
description of the acoustic wing of g(q) is characterized by the coincidence of both
curves.
Taking account of the known value of QD(0) ¼ 272.2 K, we will
obtain the value of aeff ¼ 148.7 mKꢀ3 for BaWO4.
The paper [40] has established the relationship between heat
capacity and the density of states g(
fact that, at very low temperatures, the derivative of heat capacity
with respect to temperature is proportional to g( ). We can use
q) which is characterized by the
description of g(
continual description g(
q
) is valid. In doing so, the region of justice of the
) can extend even above 13 K, we can only
q
q
these results to find the said relationship in the region where the
establish that the boundary of the region of validity of such
Debye law is valid:
description lies in the frequency range of 13e81 K. As we believe,
the description of g(
does not characterize true behavior of the phonon density of states.
So, quantitative description of the phonon density of states g( ) of
barium tungstate below ~13 K based on experimental heat capacity
and using a physically reasonable approach was obtained.
q) in accordance with equation (15) above 13 K
4
5
p4 T3
СV ðTÞ ¼
;
(19)
Q3Dð0Þ
q
where the isochoric heat capacity CV(T) equal to the isobaric heat
capacity Cp(T). The heat capacity (19) is normalized to unity. The
derivative of heat capacity with respect to temperature is written
as:
3.3. Approximation of experimental heat capacity
p4
3
To describe heat capacity Cp,m(T) in the range of 0e304 K, we
represented it as two components. One component of heat capac-
ity, CD(T), is described within the framework of the Debye model
which accurately corresponds to the found description near zero. It
is calculated in accordance with equation (1). In doing so, the
phonon density of states of the Debye model is given as:
vCV ðTÞ
vT
4
5
С0V ðTÞ ¼
¼
T2:
(20)
Q3Dð0Þ
C0V ðTÞ ¼
x
,gð Þ;
q
(21)
which establishes the relationship between the heat capacity and
density of states if the dimensionless factor is known. Using (18)
and (20) and taking account of the same dimension of temperature
and frequency, we have:
x
3
gð
qÞ ¼
q2; where 0 ꢂ
q
ꢂ
qD
(24)
qD3
The other component of heat capacity is described by a set of
4
5
p4
3
3
T2 ¼
x,
q2
;
(22)
Einstein heat capacities CE,i(T) corresponding to different charac-
teristic frequencies qE,i. Taking account of the above, the expression
for heat capacity is written as:
Q3Dð0Þ
Q3Dð0Þ
whence it follows that the factor
x
in equation (21) is equal to:
(23)
M
X
p4
CpðTÞ ¼ ð1 ꢀ
hÞ CDðTÞ þ
h
aiCE;iðTÞ;
(25)
4
x
¼
:
i¼1
5
Following the paper [40], we can write the equation.
Therefore, if we differentiate the normalized heat capacity with
respect to temperature, the obtained derivative will provide the
accurate description of the acoustic wing of the density of states
where i is the summation index running from 1 to M, M is the
selected number of Einstein modes, and ai is the factor that reflects
contribution of an individual Einstein mode. The heat capacities in
equation (25) are normalized to unity.
near zero frequencies when multiplied by the factor
x.
To be definite, let the share of the component of Debye heat
Within the framework of the described approach, we calculated
the acoustic wing of the phonon density of states of BaWO4 using
relationship (21). We took account in advance of the contribution to
the heat capacity of Cp(T) of the low-frequency Einstein mode
capacity CD(T) be equal to 3/18, i.e. (1 ꢀ
h
) ¼ 1/6, then
h
¼ 5/6. With
this in view, the effective Debye temperature calculated by formula
(14) will be equal to qD ¼ 149.8 K. The CD(T) is further subtracted
from the experimental heat capacity, and the result is approxi-
mated by a set of the Einstein heat capacities CE,i(T). There are many
approaches to choosing the characteristic frequencies of the Ein-
stein function to describe the heat capacity, including variation,
both in frequency and number of modes. We determined that the
first peak of the phonon density of states corresponds to the fre-
quency 81.2 K, and according to Refs. [22,23] the cutoff frequency is
(qE ¼ 81.2 K) found within the framework of the heat capacity
description by equation (12). The comparison of the obtained result
with the result of calculations within the framework of the con-
tinuum description of the density of states (18) is shown in Fig. 10.
As it follows from the figure, below 13 K both methods produce the
same result, which is in the region where the continuum

A.E. Musikhin et al. / Journal of Alloys and Compounds 802 (2019) 235e243
241
Table 3
The parameters qD, h, ai and qE,i, in which equation (25) describes the experimental
heat capacity of BaWO4.
Parameter
Value
Parameter
Value
qD
149.8 K
81.2 K
166 K
260 K
504 K
h
5/6
qE,1
qE,2
qE,3
qE,4
qE,5
a1
a2
a3
a4
a5
(6.15 0.03)ꢃ10ꢀ2
(14.63 0.08)ꢃ10ꢀ2
(21.59 0.11)ꢃ10ꢀ2
(33.97 0.16)ꢃ10ꢀ2
(32.19 0.24)ꢃ10ꢀ2
1306 K
in the range of 1300e1400 K. To describe heat capacity, 5 E func-
tions (I ¼ 5) were chosen, with the characteristic frequency of the
first function was set as qE,1 ¼81.2 K. The best description of the
heat capacity in the range of 0e304 K was found by varying the
remaining 9 parameters: a1 and ai; qE,i at i ¼ 2; 3; 4; 5. The corre-
spondence between the experimental data and the required heat
capacity description was determined by the relative sum of squared
deviations. The best description was obtained with the parameters
given in Table 3. The result of the BaWO4 experimental heat ca-
pacity description by equation (25) is shown in Figs. 11 and 12. The
Debye temperature QD(T) presented in Fig. 13 demonstrates very
good agreement between the experimental data and their
description by equation (25) at low temperatures.
Fig. 13. Debye temperature QD(T) of BaWO4 up to 25 K: circles e our experimental
data; solid line e description by equation (25).
mean-square deviation of the experimental points from the
smoothed curve Cp,m(T) for BaWO4 equals to: 0.008 in the range of
0e15 K and 0.0008 in the range of 15e304 K. Relative deviation of
the experimental values from the obtained heat capacity descrip-
tion is shown in Fig. 14. The nature of the deviations of the exper-
imental points from the approximating relationship allows us to
regard it as a smoothed curve that describes the experimental heat
capacity. Standard uncertainty for temperature u(T) is 0.01 K;
relative combined standard uncertainty for the smoothed values of
heat capacity uc,r(Cp) is 0.012 in the temperature range of 0e20 K,
0.003 in the range of 20e80 K, and 0.0017 in the range of 80e304 K.
It should be noted that, within the framework of a set of the Ein-
stein heat capacities, we cannot describe the heat capacity
adequately throughout the temperature region including the re-
gion near zero. This is mainly due to the fact that a continuous
spectrum of acoustic nature is poorly described by a finite set of
Einstein modes.
The resulting dependence Cp(T) is a smooth function, with any of
its i-th derivative is also a smooth function. Extrapolation to zero
was performed using a physically reasonable approach. Root-
3.4. Entropy, enthalpy and gibbs free energy
Using the results of the heat capacity Cp,m(T) description by
equation (25) with obtained parameters (Table 3), the thermody-
namic functions of BaWO4 (entropy, enthalpy increment and Gibbs
free energy) in the temperature range of 0e304 K were calculated
according to the following formulas:
Fig. 11. Molar heat capacity of BaWO4 as a function of temperature up to 304 K: circles
are our experimental data; solid line is description by equation (25); dash and dotted
lines correspond to the first and second terms of equation (25) associated with the
contribution of the Debye and Einstein heat capacities respectively.
T
ð
Cp;mðTÞ
Sp;mðTÞ ¼
d T;
(26)
T
0
Fig. 12. Molar heat capacity of BaWO4 up to 25 K: circles are our experimental data;
solid line is description by equation (25); dash and dotted lines correspond to the first
and second terms of (25) associated with the contribution of the Debye and Einstein
heat capacities respectively.
Fig. 14. Relative deviation of the experimental heat capacity values from its descrip-
tion by equation (25) (X-axis) for barium tungstate.

242
A.E. Musikhin et al. / Journal of Alloys and Compounds 802 (2019) 235e243
adnd.2001.0873.
T
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D
HmðTÞ ¼ HmðTÞ ꢀ Hmð0Þ ¼ Cp;mðTÞd T;
(27)
(28)
0
D
GmðTÞ ¼ HmðTÞ ꢀ T,Sp;mðTÞ:
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The numerical values of the thermodynamic functions are
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4. Conclusions
In this study, the heat capacity of a single crystal sample of
BaWO4 was measured by the adiabatic method in the range of
5.7e304 K for the first time. The obtained results have been
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Acknowledgement
This work was financially supported by the Ministry of Educa-
tion and Science of the Russian Federation and the Russian Foun-
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Appendix A. Supplementary data
Supplementary data to this article can be found online at
https://doi.org/10.1016/j.jallcom.2019.06.197.
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