Heat capacity and thermodynamic functions of CaMoO4 at low temperatures

Article abstract of DOI:10.1016/j.jallcom.2015.09.171

The heat capacity of a single crystal of calcium molybdate CaMoO₄ was measured by the adiabatic method in the range 6-310 K. No anomalies in the heat capacity associated with the phase transition were found. Based on the obtained data, thermodynamic functions (entropy, enthalpy, and Gibbs free energy) were calculated. Special features of the heat capacity behavior near zero temperatures as well as special features of the phonon density of states at low frequencies were detected. The low-frequency peak in the phonon density of states of CaMoO₄ was identified, which is associated with the phonon scattering on the regular fragments of the structure. The Debye temperature at absolute zero was calculated.

Full text of DOI:10.1016/j.jallcom.2015.09.171

Accepted Manuscript
Heat capacity and thermodynamic functions of CaMoO at low temperatures
4
A.E. Musikhin, V.N. Naumov, M.A. Bespyatov, V.N. Shlegel
PII:
S0925-8388(15)31147-6
10.1016/j.jallcom.2015.09.171
JALCOM 35446
DOI:
Reference:
To appear in: Journal of Alloys and Compounds
Received Date: 20 July 2015
Revised Date: 17 September 2015
Accepted Date: 18 September 2015
Please cite this article as: A.E. Musikhin, V.N. Naumov, M.A. Bespyatov, V.N. Shlegel, Heat capacity
and thermodynamic functions of CaMoO at low temperatures, Journal of Alloys and Compounds
4
(2015), doi: 10.1016/j.jallcom.2015.09.171.
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ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT
Heat capacity and thermodynamic functions of
CaMoO at low temperatures
4
*
A.E. Musikhin , V.N. Naumov, M.A. Bespyatov, V.N. Shlegel
Nikolaev Institute of Inorganic Chemistry, Siberian Branch of Russian Academy of Sciences,
3
, Acad. Lavrentiev Ave., Novosibirsk 630090, Russia
E-mail: musikhin@niic.nsc.ru
Abstract
The heat capacity of a single crystal of calcium molybdate CaMoO was measured by the adiabatic method
4
in the range 6–310 K. No anomalies in the heat capacity associated with the phase transition were found. Based
on the obtained data, thermodynamic functions (entropy, enthalpy, and Gibbs free energy) were calculated.
Special features of the heat capacity behavior near zero temperatures as well as special features of the phonon
density of states at low frequencies were detected. The low-frequency peak in the phonon density of states of
CaMoO was identified, which is associated with the phonon scattering on the regular fragments of the
4
structure. The Debye temperature at absolute zero was calculated.
Keywords: heat capacity, adiabatic calorimetry, thermodynamic functions, cryogenic scintillators, calcium
molybdate.
1

ACCEPTED MANUSCRIPT
1
. Introduction
The molybdate compounds are the subject of extensive research due to their unique optical properties,
which determine a number of practical applications. In this work, we study a single crystal of calcium
molybdate CaMoO . It may be used, for example, as elements of Raman lasers [1,2], as acousto-optic filters
4
[
3,4] or as scintillators [5–7]. In addition, the CaMoO single crystal has important scientific applications related
4
to the possibility of its use as a sensitive cryogenic phonon scintillation detector for both the study of
1
00
neutrinoless double-beta decay of Mo isotope [8,9] and use in experiments devoted to direct search for dark
matter [10].
The above-mentioned description determines the study of different properties of calcium molybdate,
including its thermodynamic characteristics. Optical and luminescent properties of this material are now well
investigated (see, for example, [11–14]); the electronic structure was studied, for example, in [15,16]; the elastic
constants were calculated in [17,18] by measuring the velocity of ultrasonic waves; in [19] the assumption was
made about the existence of a second-order phase transition at temperature 52 K. By simulation using the lattice
dynamics method, the authors of [20] calculated a number of the CaMoO characteristics, including the phonon
4
density of states, the heat capacity, the entropy and the Debye temperature at absolute zero.
Note that the low-temperature adiabatic calorimetry, developed by Nernst and his colleagues in 1910,
remains beyond competition when it comes to obtaining high-precision data on heat capacity in a wide range of
low temperatures. Also note that the experimentally determined heat capacity is the main tool and the final
judgment when testing phase transitions. In this work, we present the experimental data on the CaMoO single
4
crystal heat capacity obtained by adiabatic method in temperature range 6–310 K, the analysis of the results, and
the calculation of a number of important characteristics of the object in interest.
2
. Experimental
2
.1. Sample
The single-crystal sample of CaMoO was obtained by the Czochralski method in conditions of low
4
temperature gradients. Deeply purified (the purification procedure is described in [21]) molybdenum oxide
2

ACCEPTED MANUSCRIPT
−
3
(
(
MoO , with the impurity content not more than 6 · 10 wt%) and commercial high purity calcium carbonate
3
CaCO , 99.99 wt%) were used as the initial reagents. A mixture of molybdenum oxide and calcium carbonate
3
was mixed thoroughly to ensure a homogeneous composition. Synthesis and melting were carried out in a
platinum crucible of 70 mm in diameter. The crucible filled with a mixture was placed in the growing oven of
the crystal growing plant and heated at a rate of 25 degrees per hour to a temperature exceeding the melting
point by 20 degrees. The obtained sample was subjected to quantitative chemical analysis; the results are shown
in Table 1.
Table 1. Laser mass spectrometric analysis results.
Element
wt%
Element
wt%
Element
wt%
Element
wt%
−
5
−5
−5
−5
Na
Mg
Al
Si
Cl
Be
Ba
B
≤ 4·10
n/a (2·10 )
K
V
≤ 4·10
n/a (2·10 )
n/a (2·10 )
Ni
Cu
Zn
As
Cd
Rb
Nb
Ag
La
n/a (2·10 )
n/a (2·10 )
n/a (2·10 )
n/a (2·10 )
n/a (3·10 )
n/a (2·10 )
n/a (3·10 )
n/a (3·10 )
Sn
Sb
Te
W
Pb
Ra
Th
U
n/a (3·10 )
n/a (3·10 )
n/a (3·10 )
≤ 4·10
n/a (4·10 )
n/a (4·10 )
n/a (4·10 )
n/a (4·10 )
−
5
−5
−5
−5
−5
−5
−5
−5
≤ 4·10−
Cr
Mn
Fe
Co
Zn
Ga
Sr
4
−5
−5
−4
≤ 3·10
n/a (2·10 )
−
5
−4
−5
−5
3·10
1·10
−4
−5
−5
−5
≤ 1·10
n/a (2·10 )
n/a (2·10 )
−
5
−5
−5
−5
n/a (3·10 )
−
5
−5
−5
−5
1·10
n/a (2·10 )
−4
−5
−5
P
≤ 2·10
5·10
n/a (3·10 )
n/a – no impurity was detected, the impurity detection limit is shown in brackets; relative standard uncertainty of the
results of laser mass spectrometry is 0.15−0.25.
The results of the X-ray structural analysis of the sample in interest showed that it belongs to the scheelite
(
CaWO ) structural type and has a I4 /a space group, with cell parameters a = 5.226 Å, c = 11.43 Å. The unit
4
1
2
−
cell of the crystal contains MoO4 anions, which form tetrahedrons slightly compressed along the c-axis,
2
+
between which the Ca cations surrounded by eight oxygen ions are positioned (Fig. 1). The results of our
structural analysis agree with the data shown in [22] very well.
3

ACCEPTED MANUSCRIPT
Fig. 1. Structure of CaMoO crystal.
4
2
.2. Measurement of heat capacity
Fig. 2. The single crystal sample of CaMoO in the form of a hollow cylinder (right) prepared for the
4
experiment to measure the heat capacity and calorimetric ampoule (left). The scale is given in cm.
4

ACCEPTED MANUSCRIPT
The obtained single crystal of CaMoO was used to fabricate a sample in the shape of a hollow cylinder
4
(
Fig. 2) with the size corresponding to the internal volume of the calorimetric ampoule. The sample weight was
4
0.624 g. The heat capacity C (T) of this sample was measured with the use of demountable calorimetric
P,m
ampoule [23] made of nickel by the adiabatic method on the setup described in [24].
The methodology of filling an ampoule with a substance, as well as the filling of the ampoule with heat
exchange gas (helium), are all described in the work [23]. The results of heat capacity measurements for a
standard substance (benzoic acid) deviate from data [25] by less than 1% in the range 5–20 K, by less than 0.3%
in the range 20–80 K, and by less than 0.15% in the range 80–273 K. The molar mass used in the calculation of
the molar heat capacity and thermodynamic functions was determined from the formula CaMoO as
4
−
1
2
00.02 g mol . The heat capacity has been measured at 85 points in the range 6–310 K. The obtained data are
presented in Table 2. Analysis of the obtained experimental results gives no indications for the existence of any
special features in C (T) behavior associated with phase transitions.
P,m
a
−1
Table 2. Experimental values of heat capacity for a single crystal of CaMoO (molar mass M = 200.02 g mol ) at
4
pressure p = 0.1 MPa.
T,
K
C
,
T,
K
C
,
T,
K
C
,
T,
K
C
,
P,m
P,m
−1
P,m
−1
P,m
−1
−
1
−1
−1
−1
−1
J · mol · K
J · mol · K
J · mol · K
67.82
69.97
72.01
73.84
75.61
77.27
79.00
80.66
82.21
83.77
85.38
86.97
88.44
89.84
91.20
92.51
93.81
94.99
96.12
J · mol · K
100.06
101.03
102.01
103.05
103.98
104.92
105.92
106.90
107.80
108.60
109.40
110.27
111.16
111.85
112.59
113.42
114.15
114.97
115.72
Series 1
65.62
69.12
72.35
75.44
78.81
82.34
85.67
89.21
93.30
28.71
130.02
134.75
139.35
143.58
147.70
151.99
156.45
160.82
165.10
169.63
174.39
179.07
183.67
188.21
192.68
197.09
201.45
205.76
210.01
226.07
230.17
234.40
238.83
243.20
247.54
251.86
256.35
260.81
265.03
269.21
273.56
278.09
282.58
286.99
291.34
295.65
300.11
304.87
6
.21
.54
0.0308
0.0858
0.1443
0.235
0.354
0.486
0.790
1.402
2.216
3.207
4.366
5.648
7.200
8.926
10.79
12.91
15.04
17.39
31.25
8
33.52
1
1
1
1
1
1
2
2
2
3
3
3
4
4
4
4
0.06
1.51
2.96
4.20
6.34
9.39
2.40
5.29
8.17
0.93
3.91
6.98
0.08
3.37
6.56
9.91
35.72
38.05
40.48
42.72
45.03
47.67
Series 2
78.74
81.81
85.73
89.99
94.39
98.94
103.26
107.71
112.31
37.99
40.13
42.75
45.54
48.31
51.06
53.62
56.31
58.88
5

ACCEPTED MANUSCRIPT
5
5
6
4.39
8.36
1.76
20.57
23.45
25.93
116.72
121.00
125.43
61.26
63.42
65.62
214.45
219.31
222.51
97.35
98.49
99.26
310.10
116.66
a
Standard uncertainty for temperature u(T) = 0.01 K; relative standard uncertainty for the heat capacities
ur(CP,m) = 0.01 (6 < T < 20 K); 0.003 (20 K < T < 80 K); 0.0015 (80 < T < 310 K).
3
. Results and discussion
3
.1. Calculation of thermodynamic functions
To calculate integral thermodynamic functions, the obtained heat capacity data were extrapolated to zero
and experimental values of C (T) was smoothed. Smoothing is done to eliminate random experimental
P,m
uncertainty and leads to a more accurate calculation of the integral thermodynamic functions. In so doing, we
assumed that heat capacity of CaMoO had no anomalous contributions below 6 K.
4
To extrapolate the heat capacity to zero temperature we used the technique described in [26]. The C (T)
P
dependence near to zero is represented as the sum of the heat capacity as per the Debye model, C (T), and heat
D
capacity as per the Einstein model, C (T):
E
C (T) = (1− η) ⋅C (T) + η ⋅C (T)
,
(1)
P
D
E
where η is the proportion of the vibrational modes of the heat capacity as per the Einstein model. Eq. (1) is
transformed as follows:
CP
 C

E
−
1= η ⋅
−1
(2)


CD

CD

and after the substitution of variables:
Y(θ ) =
CP
CD
CE
CD
−1
;
X (θ ,θ ) =
−1
,
(3)
(4)
D
D
E
is converted into the homogeneous linear equation:
Y(θ ) = η X (θ ,θ )
.
E
D
D
By varying the unknown parameters, θ and θ , we can obtain physically reasonable description of the
D
E
experimental heat capacity below a certain temperature T . The θ = 446.2 K and θ = 97.1 K parameters give
0
D
E
the best description of the CaMoO heat capacity in X, Y (3) coordinates below T ≈ 17 K (see Fig. 3). The slope
4
0
of the straight line (4) determines the η = 0.0159.
6

ACCEPTED MANUSCRIPT
Fig. 3. Heat capacity in coordinates Y(θ ) versus X(θ ,θ ), where θ and θ are variable parameters, which
D
D
E
D
E
related to description of heat capacity within the framework of the Debye and Einstein models respectively
see equations (2–4)). The points are experimental values with indication of combined standard uncertainty;
(
the straight line is the best description of experimental values using Eq. (4) in range 6–17 K, which was
obtained with the parameters θ = 446.2 K и θ = 97.1 K.
D
E
2
The obtained description of the heat capacity in coordinates C/T versus T is shown in Fig. 4. It shows that the
experimental points are well described by Eq. (1). We consider this description as a smoothed heat capacity
(S)
CP,m (T) of CaMoO below T . Apparently, at T → 0 all vibrational modes change to acoustic oscillations
4
0
because Einstein contribution becomes vanishingly small. Accordingly, Eq. (1) can be written as follows:
4
3
4
π
T
C (T ) = (1 −
η
)
⋅
⋅
.
3
D
(5)
P
5
θ
On the other hand, the heat capacity at T → 0 obeys the Debye’s law:
4
3
4
π
T
C (T ) =
⋅
.
(6)
(7)
P
3
5
Θ (0)
D
After solving this equation system (5) and (6) we obtain the expression for the Debye temperature at zero:
1
/ 3
Θ (0) = θ /(1− η)
,
D
D
that links Θ (0) with a θ variable parameter. As a result, for the Debye temperature at zero, the value of
D
D
Θ (0) = 448.6 K was obtained, which characterizes the slope of the dotted line shown in Fig. 4.
D
7

ACCEPTED MANUSCRIPT
2
Fig. 4. Heat capacity of CaMoO in coordinates C /T versus T . The circles are experimental values; the
4
P,m
solid line presents the results of approximation by Eq. (1) with parameters θ = 446.2 K, θ = 97.1 K and
D
E
3
η = 0.0159; the dashed line corresponds to the Debye law (С ~ T ) with Θ (0) = 448.6 K.
D
The smoothing of experimental values of heat capacity above the temperature T was carried out by the
0
Rumshitsky method [26,27]. Root-mean-square deviation of the experimental values from the smoothed
(S)
CP,m (T) curve for CaMoO equals to: 0.01 in the range 6–15 K; 0.002 in the range 15–35 K and 0.0003 in the
4
range 35–310 K. Relative deviation of experimental values from the smoothed curve of heat capacity is shown
in Fig. 5.
(
S)
Fig. 5. Relative deviation of experimental values of heat capacity C (T) from the smoothed curve C
(T)
P,m
P,m
for calcium molybdate.
(S)
Using the smoothed heat capacity CP,m (T), we calculated integral thermodynamic functions at constant
pressure: entropy ꢀS (T), enthalpy ꢀH (T), and Gibbs free energy ꢀG (T) in temperature range 0–310 K.
P,m
m
m
These thermodynamic functions are presented in Table 3.
8

ACCEPTED MANUSCRIPT
a
Table 3. Thermodynamic functions at pressure p = 0.1 MPa for a single crystal of CaMoO (molar mass
4
−1
M = 200.02 g mol ) in the temperature range 0–310 K.
(
S)
T,
C
,
ꢀS
,
ꢀHm,
−ꢀGm,
P,m
−
P,m
−1
1
−1
−1
−1
−1
K
J·mol ·K J·mol ·K
J·mol
J·mol
2
4
6
8
1
1
1
1
1
2
2
2
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
0.001033
0.00827
0.0280
0.0680
0.1429
0.271
0.466
0.738
1.098
1.551
2.099
2.741
3.476
4.299
5.204
7.789
10.75
13.99
17.44
21.02
24.64
28.27
31.86
35.40
38.87
42.27
45.55
48.68
51.72
54.68
57.62
60.33
62.94
65.42
67.80
70.08
72.31
74.45
76.49
78.46
80.34
82.16
83.91
85.61
87.25
88.84
90.39
91.90
93.36
96.15
98.65
100.99
103.28
105.49
107.60
0.000345
0.00276
0.0093
0.0222
0.0446
0.0811
0.137
0.216
0.323
0.461
0.634
0.843
1.091
1.378
1.705
2.696
3.926
5.376
7.027
8.856
10.84
12.96
15.18
17.50
19.90
22.36
24.87
27.41
29.99
32.58
35.19
37.82
40.44
43.06
45.67
48.27
50.86
53.44
56.00
58.54
61.06
63.56
66.04
68.49
70.93
73.34
75.73
78.10
80.44
85.07
89.60
94.04
98.38
102.64
106.82
0.000519 0.000171
0.00828
0.0419
0.134
0.337
0.741
1.47
2.66
4.48
7.11
0.00275
0.0139
0.0442
0.109
0.232
0.446
0.794
1.327
2.105
3.193
4.664
6.591
9.053
12.13
23.03
39.49
62.65
93.58
133.22
182.40
241.83
312.13
393.80
487.26
592.9
710.9
841.6
985.1
1141.5
1310.9
1493.5
1689.1
1898
0
2
4
6
8
0
2
4
6
8
0
5
0
5
0
5
0
5
0
5
0
5
0
5
00
05
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
00
10
20
30
40
50
60
10.75
15.57
21.77
29.53
39.02
71.33
117.5
179.3
257.8
353.9
468.0
600.3
750.6
918.8
1104
1307
1527
1763
2014
2280
2560
2855
3164
3485
3818
4162
4518
4885
5263
5650
6047
6453
6869
7292
7725
8165
8613
9069
9532
10480
11454
12452
13473
14517
15583
2120
2355
2602
2863
3137
3423
3722
4034
4358
4694
5043
5403
5776
6160
6557
7384
8258
9176
10138
11143
12191
9

ACCEPTED MANUSCRIPT
2
2
2
2
2
3
70
109.58
110.92
112.20
114.94
118.88
122.04
122.75
126.54
16669
17015
17774
18897
19825
20037
21195
13280
13631
14409
15578
16560
16786
18033
73.15 110.15
80
90
98.15 114.61
00
10
111.40
113.16
114.93
116.62
3
a
Standard uncertainty for temperature u(T) = 0.01 K; relative combined standard uncertainty for the smoothed values
of heat capacity u (CP,m) = 0.013 (T < 20 K); 0.003 (20 K < T < 80 K); 0.0015 (T > 80 K).
r
3
.2. A comparison of the heat capacity of CaMoO with the known data
4
The authors of [20], based on calculations using lattice dynamics method, calculated the phonon density of
(LD)
(LD)
states g(ω) and the heat capacity at constant volume CV,m (T) and constant pressure CP,m (T) of CaMoO4.
(S)
(LD)
The comparison of our data for the heat capacity CP,m (T) and the calculated data CP,m (T) is shown in Fig. 6.
(LD)
The heat capacity CP,m (T) below 175 K was calculated from the phonon density of states of [20] in Fig. 4a.
We estimate that, below this temperature, the C (T) − C (T) difference does not exceed the experimental
P
V
(
LD)
uncertainty; it agrees with the data shown in [20]. Above 175 K, the CP,m (T) values are obtained directly from
Fig. 5 shown in [20].
(
LD)
Fig. 6. Absolute (a) and relative (b) deviation of the calculated heat capacity CP,m (T) [20] from smoothed
(S)
experimental heat capacity CP,m (T) (this work) for calcium molybdate. Here difference of heat capacities is
(LD)
(S)
ꢀ
C (T) = C
(T) − CP,m (T).
P,m
P,m
Generally, we can note that the heat capacity obtained by calculations in [20] fairly well agrees with the
experimental results. As can be seen in Fig. 6, the relative deviation above 65 K is less than 3%; however,
below 65 K, when the temperature is decreased, it can reach 50%. We believe that the observed discrepancy
between the heat capacities is determined by the physical model on which simulations are based.
10

ACCEPTED MANUSCRIPT
In the study of the pulse electron paramagnetic resonance and dielectric characteristics, the authors of [19]
have suggested that ferroelastic second-order phase transition at temperature 52 K occurs in CaMoO . However,
4
the analysis of our heat capacity data does not reveal any signs of phase transition in the wide range around
5
2 K (see Fig. 7). In addition, the results of analysis of a number of CaMoO characteristics in [20] do not
4
confirm the presence of this phase transition.
Fig. 7. Experimental heat capacity of CaMoO in coordinates C /T versus T in the temperature range 6–
4
P,m
3
10 K.
3
.3. Heat capacity at liquid-helium temperatures and some features of phonon density of
states at low frequencies
When analyzing the behavior of the CaMoO heat capacity at low temperatures it is convenient to use an
4
α(T) function discussed in [26,28]. This function characterizes the degree of the heat capacity increase at low
α(T)
temperatures, when it can be represented as C(T) ~ T ; it is calculated by the following formula:
ln
[
C(T )/C(T )
]
2
1
α
(T) =
(8)
ln
(
T2 /T1
)
where T = (T + T )/2, T and T are temperatures, for which the heat capacity is known. The smaller is the
1
2
1
2
difference between T and T , the better the α(T) function corresponds to the degree of the heat capacity
1
2
increase. Here, the index reflecting the isobaric nature of the heat capacity is omitted.
Function α(T) is commonly used to analyze the heat capacity curve at low temperatures for revealing the
degree of anisotropy of the crystalline lattice for compounds under consideration. For three-dimensional crystal
structures α(T) = 3, for two-dimensional and one-dimensional structures α(T) is equal to 2 and 1,
1
1

ACCEPTED MANUSCRIPT
correspondingly; quasi-two-dimensional and quasi-one-dimensional structures may have intermediate values of
the α(T) index at low temperatures. The index α(T) can be used to define the range of validity of the Debye law
(
where α(T) = 3). Besides, behavior of the index α(T) allows to detecting low-frequency peaks (“bosonic
peaks”) in the phonon density of states, if they exist. This is evidenced by the values of function α(T) > 3, which
is considered as a characteristic feature of the boson peak presence. This is caused by the fact that Einstein heat
capacity, which can be considered as a good approximation in description of the low-frequency peak in the
density of states, increases exponentially.
(S)
To calculate function α(T), we use smoothed heat capacity CP,m (T). Fig. 8 shows behavior of α(T) for
calcium molybdate and for the Debye model at Θ = 448.6 K. It is seen that the range of validity of the Debye
D
3
law (C ~ T ) should have extended to ~25 K. However, α(T) for this compound demonstrates a peak in the
vicinity of 12 K, which points to the presence of the low-frequency peak in the phonon density of states of
2
CaMoO single crystal. This peak appears against the background of acoustic wing (g ~ ω ) in the phonon
4
density of states and has its maximum near the frequency ω = 97.1 K.
The presence of low-frequency peak in the phonon density of states suggests that there is a characteristic
energy of phonon scattering at regular fragments of the structure of this compound. Such behavior is typical for
crystals having well-ordered structures in the long-range order. The presence of the low-frequency peak can be
related to the fact that some structural fragments in a well-ordered crystalline structure vibrate as an entity. Such
structural fragments can be linked with a unit cell as well as its separate parts, in those cases where the crystal
has a rather complicated structure (see, for example, [20,29]). We believe that the structural fragment of the
considered compound may be [CaMoO ] . This fragment is a half of the unit cell (see Fig. 1), while the second
4
2
half is mirror symmetrical relative to the first. This repeating fragment is the heaviest in the structure of calcium
molybdate. With the increase in mass of the fragment M, other things being equal (same rigidity of the crystal),
–
0.5
its frequency ω as known decreases inversely proportional to the square root of its mass (ω ~ M ).
12

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Fig. 8. The special features of the heat capacity behavior at low temperatures and the density of states at low
frequencies. (a)—temperature dependence α(T) calculated for CaMoO (solid line) and for the Debye model
4
2
(
dashed line); (b)—the phonon density of states g(ω) [20] in coordinates g(ω)/ω versus ω.
2
Using the data on calculation of g(ω) published in [20], we have built a g(ω)/ω relationship as a function of
2
ω, as shown in Fig. 8b. A plateau observed at low frequencies (Fig. 8b) suggests that g(ω) ~ ω ; a plateau of
3
α(T) at low temperatures (Fig. 8a) corresponds to C(T) ~ T . That is, for small x values, where x = ћω/(k T), both
B
functions obey the Debye’s law. As can be seen in Fig. 8b, at ~100 K frequency, a peak is observed in g(ω)
[
20]; it agrees well with our results for the determination of the “boson peak” frequency, which is 97.1 K.
2
Interestingly, the α(T) function obtained from the low-temperature heat capacity and g(ω)/ω as a function
of ω have a similar shape. Based on [30], at small x the phonon density of states is proportional to the derivative
of the heat capacity g(ω) ~ dC(T)/dT. This explains the observed similarity of the function shapes shown in
Fig. 8a and Fig. 8b, taking into account that α(T) is also a (logarithmic) derivative of the heat capacity.
3
.4. Debye temperature
An important characteristic of the solid body is the Debye temperature at absolute zero Θ (0), which allows
D
describing the low-temperature thermodynamic properties associated with the acoustic branches of the lattice
oscillations. If Θ (0) is known, we can precisely calculate the velocity of sound, averaged over all acoustic
D
branches, as well as to evaluate the elastic properties of the crystal. The Θ (0) value allows to quantitatively
D
describe the low-frequency wing of the phonon density of states. The Debye temperature at zero can be
calculated based on the experiments on measuring the velocity of ultrasound waves. It can also be determined
from the phonon density of states obtained by experimental or computational methods. A common method for
1
3

ACCEPTED MANUSCRIPT
calculating Θ (0) which yields high accuracy uses precisely measured heat capacity data in the low-temperature
D
region.
The publication [20] contains data on calculating the Debye temperature at absolute zero. Therewith, Θ (0)
D
was calculated using different approaches based on experiments on measuring the velocity of ultrasonic waves
[
17,18]. The authors of [20] also estimated, using the lattice dynamics method, the heat capacity of CaMoO4,
which was also used to calculate Θ (0). These results are presented in Table 4.
D
Using our data on heat capacity, we calculated the Debye temperature at zero, Θ (0) = 448.6 ± 2.0 K (see
D
Table 4). The above-mentioned uncertainty was estimated based on experimental measurement uncertainty of
the heat capacity, taking into account possible variations when described by Eq. (1), related to the choice of the
approximation interval. As can be seen in Table 4, our data on the Debye temperature are the closest to the data
published in [20] using the calculated dependence of the shear modulus and elasticity modulus on the
temperature (at T → 0).
Table 4. The calculated Debye temperature at zero compared with the published data.
Θ (0)
D
Source data
Ref.
[17]
[17]
The values for directly calculating of Θ (0)
Ref.
[18],[20]
[20]
D
(
K)
38±2
46
95±9
The velocities of ultrasonic waves
experimental)
4
Average bulk and shear modules
(
The velocities of ultrasonic waves
experimental)
Calculated bulk modulus and shear modulus
versus temperature
4
(
A quasi-harmonic lattice dynamics
approach (calculation)
3
Calculated lattice heat capacity
[20]
The low-temperature heat capacity
Experimental heat capacity extrapolated to
zero temperature by Eq. (1)
Our
calculation
4
48.6±2.0
This work
(
experimental)
4
. Conclusions
For the first time, the precise experimental data on the heat capacity of calcium molybdate in the range 6–
10 K have been obtained. Based on these data, the integral thermodynamic functions (entropy, enthalpy, and
3
Gibbs free energy) of CaMoO have been calculated. The description of the heat capacity in the low-
4
temperature region based on physically justified equation allowed us to reveal the specific features regarding the
low-frequency region of the phonon density of states of the single crystal of interest. In our opinion, the specific
features associated with the low-frequency peak in the g(ω) spectrum are caused by a characteristic energy of
14

ACCEPTED MANUSCRIPT
the phonon scattering on the regular fragments of the structure of the compound in interest. The Debye
temperature at absolute zero has been calculated. The obtained results have been compared with the published
data.
1
5

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References
1. T.T. Basiev, A.A. Sobol, Yu.K. Voronko, and P.G. Zverev, Opt. Mater. 15 (2000) 205–216,
DOI: 10.1016/S0925-3467(00)00037-9
2. V.I. Lukanin, and A.Y. Karasik, Opt. Commun. 336 (2015) 207–212, DOI: 10.1016/j.optcom.2014.10.012
3. V.Ya. Molchanov, Proc. SPIE 2643 (1995) 189–193, DOI: 10.1117/12.222742
4. V.I. Balakshy, K.R. Asratyan, and V.Y. Molchanov, J. Opt. A: Pure Appl. Opt. 3 (2001) S87–S92
1
0.1088/1464-4258/3/4/365
. V.B. Mikhailik, and H. Kraus, J. Phys. D: Appl. Phys. 39 (2006) 1181–1191, DOI: 10.1088/0022-
727/39/6/026
. V.B. Mikhailik, S. Henry, H. Kraus, and I. Solskii, Nucl. Instrum. Methods Phys. Res. A 583 (2007) 350–
55, DOI: 10.1016/j.nima.2007.09.020
5
6
7
3
3
. A.V. Veresnikova, B.K. Lubsandorzhiev, I.R. Barabanov, P. Grabmayr, D. Greiner, J. Jochum, M. Knapp,
C. Oßwald, R.V. Poleshuk, F. Ritter, B.A.M. Shaibonov, Y.E. Vyatchin, and G. Meierhofer, Nucl.
Instrum. Methods Phys. Res. A 603 (2009) 529–531, DOI: 10.1016/j.nima.2009.02.041
. S.J. Lee, J.H. Choi, F.A. Danevich, Y.S. Jang, W.G. Kang, N. Khanbekov, H.J. Kim, I.H. Kim, S.C. Kim,
S.K. Kim, Y.D. Kim, Y.H. Kim, V.V. Kobychev, V.N. Kornoukhov, J.I. Lee, J.S. Lee, K.B. Lee, M.K.
Lee, Y.H. Lee, S.S. Myung, J.H. So, V.I. Tretyak, and Y. Yuryev, Astropart. Phys. 34 (2011) 732–737,
DOI: 10.1016/j.astropartphys.2011.01.004
8
9
. J.H. So, H.J. Kim, V.V. Alenkov, A.N. Annenkov, H. Bhang, R.S. Boiko, O.A. Buzanov, D.M. Chernyak,
J.H. Choi, S. Choi, F.A. Danevich, K.V. Efendiev, A.M. Gangapshev, Yu.M. Gavryluk, A.M. Gezhaev,
Y.S. Hwang, H. Jiang, W.G. Kang, V.V. Kazalov, N.D. Khanbekov, G.B. Kim, S.K. Kim, S.C. Kim, Y.D.
Kim, Y.H. Kim, V.V. Kobychev, V.N. Kornoukhov, V.V. Kuzminov, H.S. Lee, J.I. Lee, J.M. Lee, K.B.
Lee, M.J. Lee, M.K. Lee, S.J. Lee, J. Li, X. Li, V.M. Mokina, S.S. Myung, A.S. Nikolaiko, S. Olsen, H.
Park, S.I. Panasenko, D.V. Poda, R.B. Podviyanuk, O.G. Polischuk, S.S. Ratkevich, Y. Satou, K. Tanida,
V.I. Tretyak, S.P. Yakimenko, and Q. Yue, IEEE Trans. Nucl. Sci. 59 (2012) 2214–2218,
DOI: 10.1109/TNS.2012.2200908
10. I. Bavykina, G. Angloher, D. Hauff, M. Kiefer, F. Petricca, and F. Pröbst, Opt. Mater. 31 (2009) 1382–
1387, DOI: 10.1016/j.optmat.2008.09.018
1
1. R. Grasser, E. Pitt, A. Scharmann, and G. Zimmerer, Phys. Stat. Sol. (B) 69 (1975) 359–368,
DOI: 10.1002/pssb.2220690206
1
2. C.E. Tyner, and H.G. Drickamer, J. Chem. Phys. 67 (1977) 4103–4115, DOI: 10.1063/1.435387
3. J.A. Groenink, C. Hakfoort, and G. Blasse, Phys. Stat. Sol. (A) 54 (1979) 329–336,
DOI: 10.1002/pssa.2210540141
1
16

ACCEPTED MANUSCRIPT
1
4. V.V. Mikhailin, A.N. Belsky, I.A. Kamenskikh, V.N. Kolobanov, P.A. Orekhanov, I.N. Shpinkov, D.A.
Spassky, and A.N. Vasil’ev, Nucl. Instrum. Methods Phys. Res. A 486 (2002) 367–373,
DOI: 10.1016/S0168-9002(02)00735-0
1
5. Y. Zhang, N.A.W. Holzwarth, and R.T. Williams, Phys. Rev. B 57 (1998) 12738–12750,
DOI: 10.1103/PhysRevB.57.12738
1
6. C. Pu, T. Liu, and Q. Zhang, Phys. Stat. Sol. (B) 245 (2008) 1586–1589, DOI: 10.1002/pssb.200743263
7. J.M. Farley, and G.A. Saunders, Solid State Commun. 9 (1971) 965–969, DOI: 10.1016/0038-
1
1098(71)90441-8
1
8. W.J. Alton, and A.J. Barlow, J. Appl. Phys. 38 (1967) 3817, DOI: 10.1063/1.1709023
9. J. Simon, J. Banys, J. Hoentsch, G. Völkel, R. Böttcher, A. Hofstaetter, and A. Scharmann, J. Phys.:
Condens. Matter 8 (1996) L359–L362, DOI: 10.1088/0953-8984/8/25/001
1
2
0. A. Senyshyn, H. Kraus, V.B. Mikhailik, L. Vasylechko, and M. Knapp, Phys. Rev. B 73 (2006) 014104,
DOI: 10.1103/PhysRevB.73.014104
2
1. L. Bergé, R.S. Boiko, M. Chapellier, D.M. Chernyak, N. Coron, F.A. Danevich, R. Decourt, V.Ya.
Degoda, L. Devoyon, A. Drillien, L. Dumoulin, C. Enss, A. Fleischmann, L. Gastaldo, A. Giuliani, M.
Gros, S. Herve, V. Humbert, I.M. Ivanov, V.V. Kobychev, Ya.P. Kogut, F. Koskas, M. Loidl, P. Magnier,
E.P. Makarov, M. Mancuso, P. de Marcillac, S. Marnieros, C. Marrache-Kikuchi, S.G. Nasonov, X.F.
Navick, C. Nones, E. Olivieri, B. Paul, Y. Penichot, G. Pessina, O. Plantevin, D.V. Poda, T. Redon, M.
Rodrigues, V.N. Shlegel, O. Strazzer, M. Tenconi, L. Torres, V.I. Tretyak, Ya.V. Vasiliev, M. Velazquez,
O. Viraphong, and V.N. Zhdankov, JINST 9 (2014) P06004, DOI: 10.1088/1748-0221/9/06/P06004
2. R.M. Hazen, and L.W. Finger, J. Phys. Chem. Solids 46 (1985) 253–263, DOI: 10.1016/0022-
2
3697(85)90039-3
23. V.N. Naumov, and V.V. Nogteva, Instrum. Exp. Tech. 28 (1985) 1194–1199
24. V.N. Naumov, G.I. Frolova, V.V. Nogteva, P.A. Stabnikov, and I.K. Igumenov, Russ. J. Phys. Chem. 74
(
2000) 1579–1584
5. N.P. Rybkin, M.P. Orlova, A.K. Baranyuk, N.G. Nurullaev, and L.N. Rozhnovskaya, Meas. Tech. 17
1974) 1021–1026
6. A.E. Musikhin, V.N. Naumov, M.A. Bespyatov, and N.V. Ivannikova, J. Alloys. Compd. 639 (2015)
45–148, DOI: 10.1016/j.jallcom.2015.03.159
2
2
2
2
2
(
1
7. L.Z. Rumshiskii, Matematicheskaya obrabotka rezul’tatov eksperimenta (Mathematical Processing of
Experiment Results), Nauka, Moscow, 1971
8. E.B. Amitin, Y.F. Minenkov, O.A. Nabutovskaya, V.N. Naumov, I.E. Paukov, and G. Krabbes, Russ. J.
Phys. Chem. 64 (1990) 942–944
9. A. Senyshyn, M. Hoelzel, T. Hansen, L. Vasylechko, V. Mikhailik, H. Kraus, and H. Ehrenberg, J. Appl.
Cryst. 44 (2011) 319–326, DOI: 10.1107/S0021889811004146
1
7

ACCEPTED MANUSCRIPT
30. V.N. Naumov, N.A. Nemov, and V.V. Nogteva, Phys. Lett. A 101 (1984) 414–418, DOI: 10.1016/0375-
9601(84)90617-0
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Highlights:
The low temperature heat capacity of CaMoO single crystal was measured.
4
Thermodynamic functions (entropy, enthalpy, and Gibbs free energy) were calculated.
The Debye temperature at absolute zero was calculated.
The characteristics of low-frequency part of the phonon spectrum were obtained.
The low-frequency “bosonic” peak in the phonon density of states was discovered.