Low-temperature thermal properties and features of the phonon spectrum of lutetium tetraboride

Article abstract of DOI:10.1016/j.jallcom.2014.06.030

The coefficients of thermal expansion to the c axis (α _∥, α_⊥) were measured for lutetium tetraboride over the temperature range 4.2-300 K. The heat capacity data for lutetium tetraboride were used for the calculation of tetraboride phonon spectrum moments and also for the development of a simplified tetraboride spectrum model. The use of the heat capacity and thermal expansion data allowed the temperature changes of the Gru?neisen parameters Γ, Γ_∥, Γ_⊥ for tetraboride to be calculated. As a result of the approximation of Γ_⊥(T), Γ _∥(T) temperature dependencies in accordance with the chosen phonon spectrum model have been found: the anomalies of Γ_⊥(T), Γ_∥(T) are at about 25 K and then drop at lower temperatures due to the Einstein vibrations of boron sublattices.

Full text of DOI:10.1016/j.jallcom.2014.06.030

Journal of Alloys and Compounds 613 (2014) 170–174
Contents lists available at ScienceDirect
Journal of Alloys and Compounds
journal homepage: www.elsevier.com/locate/jalcom
Low-temperature thermal properties and features of the phonon
spectrum of lutetium tetraboride
a
a,
⇑
a
a
b
V.V. Novikov , N.V. Mitroshenkov , A.V. Matovnikov , D.V. Avdashchenko , A.V. Morozov ,
c
c
L.M. Pavlova , V.B. Koltsov
a
Bryansk Petrovsky State University, 14 Bezhitskaya St., Bryansk 241037, Russia,
Russian Timiryazev State Agrarian University, 49 Timiryazevskaya St., Moscow 127550, Russia
National Research University of Electronic Technology ‘‘MIET’’, Moscow 124498, Russia
b
c
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 7 May 2014
Received in revised form 5 June 2014
Accepted 5 June 2014
Available online 14 June 2014
k
The coefficients of thermal expansion to the c axis (a , a\) were measured for lutetium tetraboride over
the temperature range 4.2–300 K. The heat capacity data for lutetium tetraboride were used for the cal-
culation of tetraboride phonon spectrum moments and also for the development of a simplified tetrab-
oride spectrum model. The use of the heat capacity and thermal expansion data allowed the temperature
changes of the Grüneisen parameters
As a result of the approximation of
sen phonon spectrum model have been found: the anomalies of
drop at lower temperatures due to the Einstein vibrations of boron sublattices.
Ó 2014 Published by Elsevier B.V.
C
,
C
k
,
C
\
for tetraboride to be calculated.
(T) temperature dependencies in accordance with the cho-
(T), (T) are at about 25 K and then
\ k
C (T), C
Keywords:
Rare earth alloys and compounds
Heat capacity
Thermal expansion
Thermal analysis
C
\
C
k
1
. Introduction
6
Boron sub-lattice is made up of B octahedra (the structural unit
of the boron sub-lattice of rare-earth hexaborides), connected by
chains of boron atoms, which, together with the octahedron ribs,
form the distorted heptangular rings. This ring-shaped arrange-
ment of boron atoms is characteristic of rare-earth diborides. A
tetraboride lattice is represented as alternate layers of metal and
boron atoms (Fig. 1) [13].
4
Tetraborides of rare-earth elements (RB compounds, in which
R stands for the rare-earth element), in addition to the high hard-
ness and melting temperature typical of all rare-earth borides, also
possess a number of properties making them rather interesting
from the general scientific and practical perspectives. The applica-
tion-oriented interest in rare-earth tetraborides is primarily condi-
tioned by the magnetic and structural transformations occurring in
most of them [1–10]. As a result of these transformations, a
number of physical and thermodynamic characteristics change in
discrete steps. Such changes in their properties can be used for pro-
ducing various switching devices, magnetic cooling devices and so
forth [11]. Of great theoretical interest is the research of the order-
ing processes in the tetraborides’ magnetic sub-systems, which are,
Above the rings of boron atoms there are metal atoms which
form squares and triangles in the plane perpendicular to the crystal
3+
c axis. This mutual arrangement of R paramagnetic ions corre-
sponds to the so-called Shastry–Sutherland lattice (SSL), the most
characteristic feature of which is a geometric frustration, that is
the impossibility of complete ordering in the system of atomic
magnetic moments up to the absolute zero [14] and, as a conse-
quence, the presence of zero-point entropy [15–17].
3
+
on the one hand, conditioned by the characteristic properties of R
For the successful study of the features of electronic and mag-
netic sub-systems of the magnetic materials, it is often necessary
to single out the influence of the phonon sub-system on the values
of the properties under study. To evaluate the lattice contribution
to the characteristics of magnetic materials, the method of com-
parison with the non-magnetic isostructural analogue is usually
employed. For the family of rare-earth tetraborides these diamag-
paramagnetic ions and, on the other hand, by the peculiarity of
their spatial location caused by the unique crystalline structure
of RB
Tetraborides of rare-earth elements are characterized by the
tetragonal crystalline structure of the UB
type, D 54 h – P4/mbm.
The unit cell contains four formula units [12].
4
compounds.
4
netic analogues are lanthanum and lutetium tetraborides (LaB
4
and
LuB respectively).
As follows from the phase diagrams and our experiments, it is
much more difficult to synthesize the single-phased sample of
4
⇑
Corresponding author.
E-mail addresses: vvnovikov@mail.ru (V.V. Novikov), weerm@yandex.ru
N.V. Mitroshenkov).
(
http://dx.doi.org/10.1016/j.jallcom.2014.06.030
925-8388/Ó 2014 Published by Elsevier B.V.
0

V.V. Novikov et al. / Journal of Alloys and Compounds 613 (2014) 170–174
171
cuvette, reached the temperature regulator which kept the cuvette temperature in
the range of 5–300 K with an accuracy of not less than 0.05 K. To find the values for
the a and c lattice parameters of lutetium tetraboride we experimentally worked
out the angle positions for h Bragg reflections (214) and (271) at room tempera-
ture: 2h214 = 141.6° and h271 = 144.6°. These reflections belong to the high-angle
region of scattering and are well-represented. At room temperature, the Bragg
2
2
angles of scattering for 28 reflexes were evaluated. Then the ahkl(cos hhkl), chkl(cos -
hkl) dependencies were calculated, which were extrapolated to zero. The ordinate
h
intersection of atr(300 K) = 0.70336 nm, ctr(300 K) = 0.39732 nm were considered
the true values of the a and c parameters for lutetium tetraboride at room temper-
ature. Differences
considered as constant values within the whole temperature range and were used
as corrections to the values of ah1,h2(T), ch1,h2(T). Here h , h are the reflex scattering
angles (214) and (271), respectively, within the range of 4.2–300 K.
The heat capacity of the LuB sample in the range of 2–300 K was measured in
D
a = atr(300 K)ꢀah1,h2(300 K),
D
c = ctr(300 K)ꢀch1,h2(300 K) were
1
2
4
the adiabatic vacuum calorimeter produced by Termax, the construction of which is
similar to the one described above [18]. The sample temperature in the process of
the calorimetric experiment was measured using a ferrous-rhodium thermometer
with an accuracy of ±0.05 K. The experimental adiabatic conditions were automat-
ically maintained. In each heating cycle, the sample temperature increased by
0
2
.2–0.5 K. The inaccuracy of the heat capacity evaluation amounted to 3% at
–20 K. At 60 K it decreased to 1% and remained within this range up to the room
4
Fig. 1. The unit cell of RB .
temperatures. The difference in the calibration measurement results of the sample
of electrolitic copper, annealed and melted in the vacuum, from the recommended
values [19] did not exceed the inaccuracy indicated above.
lanthanum tetraboride than lutetium tetraboride. Therefore, to
study the phonon properties of rare-earth tetraborides at low tem-
4
peratures, the lutetium tetraboride was chosen (LuB ).
3
. Results and discussion
2
. Experiment
p 4
In Table 1, one can see the smoothed values of C (T) LuB molar
The polycrystalline sample of lutetium tetraboride was synthesized by the
method of boron-thermal reduction of metal from its oxide in a vacuum [17]:
heat capacity in the temperature range under study. Fig. 2a and b
3
2
p p p
illustrate the experimental dependencies of C (T), C /T (T ). C (T)
Lu þ 11B ¼ 2LuB þ 3BO "
2
O
3
4
dependency (Fig. 2a) has the features typical of diamagnetic rare-
earth borides within the range of low temperatures [20]. One can
clearly see the smooth anomaly of the C (T) curve within the range
p
For the synthesis we used lutetium oxide of 99.9% purity produced by Mosreak-
tiv, as well as the elemental boron of 99% purity produced by ErmakKhim. The
synthesis was carried out in the vacuum electric furnace manufactured by
Termotekhnik ML. At the first stage of synthesis, we annealed the stoichiometric
of 25–75 K and the dependency close to the linear at the higher
temperatures (100–300 K). The bell-shaped maximum at the
1
mixture of oxide and boron at a temperature of T = 1473 K for 3 h at a pressure
ꢀ
1
3
2
of 10 Pa. The X-ray diffraction pattern of the synthesized sample obtained by
the X-ray diffractometer DRON-7 (Burevestnik Research and Manufacturing
p
dependency C /T (T ) (Fig. 2b) signals the presence of Einstein
3
2
components for LuB4 heat capacity. The increase in the C /T (T )
p
Association) in Co Ka radiation was compared to the ASTM data and contained
curve is conditioned by the contribution of the electron gas into
the reflections of tetraboride, oxide and metal phases (Fig. 2a). To eliminate the
excess phases, the sample, previously comminuted to a powdery condition and then
compacted into a tablet, underwent additional annealing at a temperature of
the boride heat capacity: Cel = 0.00017 J/g at. K.
We approximated the phonon spectrum of lutetium tetraboride
by the following expression:
T
1
= 1973 K for 1 h. After the second annealing, the X-ray diffraction pattern con-
tained the reflexes of the only phase – the LuB phase (Fig. 2b). According to the
chemical analysis, the synthesized sample contains 19.74% lutetium, and 80.27%
boron. The crystalline lattice parameters for the LuB sample were as follows:
4
3
3
Gð
x
x
Þ ¼ a
1
x
þ a
2
x
<
þ a
3
dð
x
ꢀ
x1EÞ þ a
4
dð
x
ꢀ
x
2EÞ;
4
a = 0.70336 nm, c = 0.39732 nm (according to [12] a = 0.7036 nm, c = 0.3974 nm).
The a and c lattice parameters for lutetium tetraboride in the temperature range
of 5–300 K were determined by the Debye–Scherrer method using the X-ray diffrac-
<
x
1 max
x
x
2 max
ð1Þ
tometer DRON-7.0 in Co K
a
radiation with Bragg–Bretanno focusing. From the syn-
1 2
where a , a are determined from the values of the characteristic
thesized LuB powder, a tablet 13 mm in diameter and 1 mm thick was formed by
4
Debye temperatures of h1D, h2D; and
which Debye parabolic phonon spectra are cropped. The values of
these spectra as well as the a , a coefficients, were chosen for the
best correspondence to the experimental data.
1m
x , x2m are frequencies at
means of pressing with a small amount of an organic binding agent. The tablet was
placed in the copper cuvette of the X-ray helium cryostat, which was equipped with
a
constantan wire heater 0.05 mm thick. The signal from the ‘Constantan–
3
4
Copper +0.1% Iron’ thermocouple, whose junction point was adjusted to the copper
3
2
p p
Fig. 2. Lutetium tetraboride heat capacity (a) C (T) dependency and (b) C /T (T ) dependency.

1
72
V.V. Novikov et al. / Journal of Alloys and Compounds 613 (2014) 170–174
Table 1
Lattice parameters a (nm), c (nm) and heat capacity C
p
4
(J/mol K) of LuB .
T, K
a
c
C
p
T, K
a
c
C
p
2
4
6
8
0
0
0
0
0
0
0
0
0
–
–
0.0023
100 0.70282 0.39687 26.7758
110 0.70284 0.39689 29.1590
120 0.70286 0.39691 31.3624
140 0.70290 0.39694 37.0540
160 0.70294 0.39699 42.6100
180 0.70299 0.39703 47.7670
200 0.70304 0.39708 53.1099
220 0.70310 0.39712 57.8709
0.70275 0.39680 0.0078
0.70275 0.39680 0.0190
0.70275 0.39680 0.0390
0.70274 0.39680 0.0767
0.70274 0.39679 1.1740
0.70274 0.39680 4.6068
0.70275 0.39680 8.8911
0.70276 0.39681 12.6719 240 0.70316 0.39717 62.7623
0.70277 0.39682 16.4428 260 0.70322 0.39722 67.0953
0.70278 0.39683 19.2591 280 0.70329 0.39727 71.5559
0.70279 0.39684 21.9727 300 0.70336 0.39732 76.1250
0.70281 0.39686 24.2005
1
2
3
4
5
6
7
8
9
Fig. 3.
D 4
v (n) dependency on moment number for LuB phonon spectrum.
4
The resulting heat capacity of LuB in the given approximation
is represented as the sum of Debye and Einstein components and
the electron gas contribution:
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
T
T
T
T
h
E2
C ¼ a
þ a
The satisfactory approximation of the
1
C
D
þ a
2
C
D
þ a
3
C
E
þ a
4 E
C
h
D1
h
D2
h
E1
5
T
ð2Þ
C
p
(T) experimental
dependency for lutetium tetraboride was achieved with the
following parameter values: hD1 = 1140 K, hD2 = 211 K, hE1 = 130 K,
h
0
E2 = 190 K,
.00017 J g at.
a
1
= 0.7,
a
2
= 0.06,
a
3
= 0.085,
a
4
= 0.07,
5
a =
ꢀ
1
ꢀ2
K
.
The highly informative characteristics of the crystal phonon
sub-system are the moments of its phonon spectrum [21]:
4
Fig. 4. LuB heat capacity approximations at high temperatures by the moments of
phonon spectrum (points – experimental data, 1 – one moment, 2 – two moments,
3 – three moments).
Z
1
1
n
ln
v
Gð
mÞdv
6N
0
ꢀ
ꢁ
n
Fig. 4 shows LuB4 dependency curves C (T) within the high
temperature range calculated by the Thirring method [23] using
the obtained moments as compared to the experimental low-
temperature data. As can be seen from the figure, the use of
moments of higher order makes it possible to better approximate
to the experimental data.
p
h
k
ꢁ
l
n
¼
ln
;
where N is the Avogadro constant,
quency distribution function (phonon spectrum), h is the Planck
constant and k is the Boltzmann constant. To compare the moments
m is frequency, G(m) is the fre-
D
of different order it is customary to use the moment function m (n):
ꢂ
ꢃ
1
n
4. Thermal expansion
1
3
v
D
ðnÞ ¼
ðn þ 3Þln
4
The experimental values of a and c lattice parameters for LuB ,
In the Debye approximation
The value for the lutetium tetraboride was similarly calcu-
lated [22].
m
D
(n) = const.
obtained in the present research, are represented in Fig. 5. The
smoothed values of a(T) and c(T), smoothed by least squares
method, are represented in Table 1.
l
i
The resulting calculations of the phonon spectrum moments of
lutetium tetraboride are represented in Table 2.
Within the biggest part of the temperature range under study
on a(T) and c(T) dependencies, there are no explicit anomalies.
Close to the absolute zero – at about 20 K – both a(T) and c(T)
dependencies have their minimum. In the area lower than 10 K
the temperature decrease is accompanied by lattice expansion.
This anomalous behaviour of the crystalline structure can be con-
ditioned by different reasons. In particular, the negative expansion
at the lowest temperatures can be caused, for instance, by the
influence of the conduction electrons, as well as by the peculiari-
ties of the crystalline lattice sub-system [24]. The revealed anom-
alies are more distinct on the temperature dependencies of the
The value of
l
2
determines the limit value of the characteristic
ꢄ
ꢅ
1
3
1
h
2
l
2
temperature at infinity: h ¼ k
¼ 1081 K.
3
The temperature changes of LuB
4
S(T) entropy make it possible
to determine the value of
m
D
(0), and consequently the mean fre-
1
quency of the phonon spectrum: e
3
h
v
q
¼ hv
D
ð0Þ [21]. For LuB
4
1
3 ꢀ1
v
q
= 1.22 ꢂ 10
c .
The energy of the lattice zero-point oscillations (E
Z
) is deter-
1
3 ꢀ1
mined from the moment
l
1
: v
q
= 1.22 ꢂ 10
c .
m
D
(n) dependency is represented in Fig. 3.
a c
coefficients of linear a , a and volume b thermal expansion (Fig. 6).
Table 2
The phonon spectrum moments of lutetium tetraboride.
(ꢀ2, 5), sꢀ
2.5
l
(ꢀ2), sꢀ2
l
(ꢀ1, 5), s
ꢀ1.5
l
(ꢀ1), sꢀ2.5
l
(ꢀ0,5), s
ꢀ2.5
l
(1), s
1.58 ꢂ 10
l
(2), s2
l
(4), s4
1.37 ꢂ 10
l
(6), s6
6.88 ꢂ 10
l
.76 ꢂ 10ꢀ
32
2.87 ꢂ 10ꢀ26
5.4 ꢂ 10
ꢀ20
1.18 ꢂ 10ꢀ13
3.15 ꢂ 10
ꢀ7
13
3.04 ꢂ 10ꢀ26
53
79
1

V.V. Novikov et al. / Journal of Alloys and Compounds 613 (2014) 170–174
173
Fig. 7. Grüneisen parameters for lutetium tetraboride.
The peculiarities of
\ k
C (T), C (T) dependencies are: (1) the
Fig. 5. a(T), c(T) lattice parameters for lutetium tetraboride at 5–300 K. Open circle
the experimental data, the solid curves – the smoothed dependencies.
–
distinct maximum at about 25 K; (2) the steep downfall with the
transition to the negative area at the further temperature decrease.
To understand which factors influence this behaviour of the
C
\
(T),
k
C (T) dependencies, we calculated these values using
Eq. (3), which, according to approximation (1), is as follows:
k
1
C
D1
C
k
k1 þ k
2
C
D2
C
k2 þ k
3
C
C
E1
C
E1 þ k
k3 þ k
4
C
E2
C
k4
C
C
k
¼
¼
:
ð6Þ
ð7Þ
1
C
D1 þ k
2
C
D2 þ k
3
4 E2
C
k
1
C
D1
C
?1 þ k
2
C
D2
C
?2 þ k
D2 þ k
3
3
C
C
E1
C
?3 þ k
4
C
E2
C
?4
?
:
k
1
C
D1 þ k
2
C
E1 þ k
4 E2
C
Here k
ity (Eq. (2)); and CDi, CEi are Debye and Einstein functions of heat
capacity. Since (T) within the whole temperature range,
we used one of the equations, Eq. (7) for the selection of the values
calc. The best correspondence to the experimental (T), (T)
dependencies was achieved at the following values:
= ꢀ10;
= 1.2; = 14.9;
= ꢀ5.5.
The dependency calculated according to Eq. (6) is represented
in Fig. 7. While selecting ki values, it was revealed that the
experimental peculiarities of ji(T) dependencies, that is, the
distinct maximum around 20 K and the steep downfall at the
further temperature lowering, are shaped by varying the Ci3
i
are the coefficients of i-th mode contributions to heat capac-
C
\
ꢃ
C
k
C
C
\
C
k
C
i
C
1
Fig. 6. Temperature dependency of LuB
dash-and-dot line – (T); dotted line –
4
linear thermal expansion coefficient:
(T); solid line – b(T).
a
a
a
c
C
2
C
3
C
4
\i
C , C
4
The revealed peculiarities of LuB thermal expansion were ana-
lysed on the basis of calculating the temperature dependencies of
C
bV
the boride Grüneisen parameter,
expansion coefficient; C
of which, in the first approximation, we considered to be equal to
the isobaric heat capacity C . In the quasi-harmonic approximation,
the value represents the weight-average of the i-th mode of
C
¼ vTC
0
where b is the volume
V
, Ci4
v
is isochoric molar heat capacity, the value
values, that is, these peculiarities observed in the area of negative
expansion are conditioned by the Einstein oscillations of the boron
sub-lattice.
p
c
C
i
We made an attempt to eliminate the steep downfall of
dependencies at the lowest temperatures by means of considering
the electron contribution to LuB thermal expansion (Eq. (35) from
i
C (T)
the phonon spectrum, where the weight coefficients of each mode
are the contributions of those modes to the heat capacity [25]:
4
P
m
[27]). However, even at very large values of the electron Grüneisen
parameter (in which Ce amounts to 100 units and more) we failed
to find any significant influence of the electron contribution on
C
i
m
i
c
C
C
i
i
c
¼
P
ð3Þ
(T)
i
Two Grüneisen functions
C
\
,
C
k
correspond to the
a
a
(T),
a
c
i
C (T) dependency.
linear thermal expansion coefficients. For the tetragonal structure
they are determined as follows [25]:
5
. Conclusions
V
c
?ðTÞ ¼
ðC11 þ C11
Þ
a
a
þ C13
a
C
ð4Þ
The complex calorimetric and X-ray research of the properties
C
p
of lutetium tetraboride conducted for the first time enabled us to
define the type of the LuB phonon spectrum within the accepted
V
4
c
kðTÞ ¼
C
13
a
a
þ C33
a
C
ð5Þ
Debye–Einstein model, as well as to single out the anomalies of
the boride thermal characteristics and to ascertain the cause of
these anomalies.
C
p
Here, cxy are the adiabatic elastic constants and C
p
(T) is the heat
capacity at the constant volume.
The values of the phonon spectrum moments determined from
The values of
C
\
,
C
k
calculated according to the data on LuB
4
the C
p
(T) temperature dependency made it possible to single out
phonon sub-systems which
thermal expansion and elastic constants from [26] (Eqs. (4), (5))
the important characteristics of LuB
4
are represented in Fig. 7. As one can see from the figure, the values
are the characteristic temperatures at absolute zero and at infinity,
the mid-geometric frequency of the phonon spectrum, and the
lattice zero-oscillation energy.
of
\ k
C , C within the whole temperature range under study are
close to each other.

1
74
V.V. Novikov et al. / Journal of Alloys and Compounds 613 (2014) 170–174
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4
2
[
[
4
(
p 4
1) the temperature dependency C (T) of the LuB heat capacity
4
within the range of 2–300 K can be satisfactorily approxi-
mated by the sum of two Debye and two Einstein compo-
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the characteristic temperatures and the parts of the
described contributions to the boride complete heat capac-
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[
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2) the negative thermal expansion of lutetium tetraboride at
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The research was undertaken under the auspices of The
Ministry of Education and Science of the Russian Federation (State
Assignment No. 2014/426 for 2014-2016) and the Russian
Foundation for Basic Research (Grant No. 14-02-31692 mol_a).
[
[
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