THERMAL DIFFUSIVITIES Of BBO AND KTP HIGHTEMPERATURE FLUXES MEASURED BY THE ÅNGSTRÖM'S
METHOD
E. Mojaeva,#, G. Orra, E. Dul’kina, M. Tseitlinb and M. Rotha
a
Department of Applied Physics, The Hebrew University, Jerusalem 91904, Israel
b
Department of Applied Physics, Ariel University Center of Samaria, Ariel 44837,
Israel
ABSTRACT
A modification of the Ångström’s temperature wave method for measuring the
thermal properties of materials has been employed for in situ experimental evaluation
of thermal diffusivities of high-temperature solutions. High-temperature solutions
used for the growth nonlinear optical BBO (BaB2O4) and KTiOPO4 (KTP) crystals
have been studied, namely BBO-Na2O, BBO-NaF and KTP-K6P4O13 (KTP-K6). The
results show that the thermal diffusivity of the BBO-Na2O solution increases by two
orders of magnitude, from  = 410-3 to 410-1 cm2/s, along the liquidus line when the
Na2O content is changed from 0 (pure BBO melt) to 28%. In the case of the BBONaF solution, the thermal diffusivity increases to about 3.510-2 cm2/s when the NaF
content reaches 20% and remains practically constant upon further increase of the
solvent concentration, namely within the entire range of interest for the BBO crystal
growth. The thermal diffusivity of the KTP- K6 solutions is also relatively low, ~
410-2 cm2/s, and does not change when the K6 concentration is varied from 50 to
70% (which occurs during the crystal growth run from a self-flux). The values of the
thermal diffusivities obtained are discussed in terms of the heat transport in the melt
and the thermal stability of the crystal/melt interface against temperature fluctuations.
INTRODUCTION
Single crystals of barium metaborate -BaB2O4 (BBO) and potassium titanyl
phosphate KTiOPO4 (KTP) are among the most widely used materials in nonlinear
optical applications, such as second harmonic generation (SHG) and optical
parametric oscillation (OPO) of high pulse power lasers (1,2), and electro-optic
devices (3). In spite of the recent advances in improving their properties, larger,
inclusion-free and optically more uniform crystals are required.
Both BBO and KTP crystals are grown from high-temperature fluxes by the
widely used top-seeded solution growth (TSSG) technique (4,5). Successful crystal
growth from high-temperature solutions or melts is subject to solving a wide set of
physical problems. Full comprehension of the processes occurring in melts during
growth requires an integrated knowledge of a wide range of thermodynamic, kinetic,
and transport phenomena, and in particular of the momentum, heat and mass transfer
in the liquid phase. Thus, in liquids with free surfaces, natural convection can be
caused by temperature gradients resulting in buoyancy and nonuniform surface
tension (6). The unbalanced buoyancy and surface tension forces result in excessive
potential energy. A ratio of this energy to the dissipation factors (viscous drag and
heat conductance) is given by the Rayleigh number (in the case of buoyancy) and by
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the Marangoni number (in the case of nonuniform surface tension) as:
 gh3
Ra 
Ta

d
D
Mi  ( )
Tr
dT 2
(1)
(2)
where  is the volume expansion coefficient, g is acceleration due to gravity ,  is
the thermal diffusivity,  is the kinematic viscosity, h is a characteristic length
(depth of the flux),  is the surface tension, D is the crucible diameter,  is the flux
density, a and r are the axial and radial temperature differences respectively.
Eqs.(1) and (2) show that the melt thermal diffusivity ( ) is one of the
important parameters influencing the fluid flow, since it determines the rate of heat
transfer during temporal temperature variation, or simply the rate of temperatures
change. The thermal diffusivity is related to the thermal conductivity , specific heat
(at constant pressure) Cp and density ρ as follows:


C p
.
(3)
If the values of , Cp and  are known, it is possible to calculate  of any substance
from eq. (3), but one should bear in mind that all these parameters are temperature
dependent (7).
There are also numerous methods of determining thermal diffusivity from the
experiment independently (8-15). One of them is the general dynamic method of
measuring thermal parameters, used by Ångström in 1861 (and published in English
in 1863 (16)). In his experiments on the thermal conductivity Ångström used a long
metallic bar of a small cross-section, one end of which was subjected to periodic
changes in temperature (16). The method is based on direct experimental simulation
of the finite difference approximation to the one-dimensional heat conduction
equation. By analyzing at two points the amplitudes and phases of a sinusoidal heat
wave propagated in a semi-infinite metal rod one can determine the thermal
diffusivity directly and independently of the external conditions. It can be shown (17)
that the thermal diffusivity is then given by
l2
(4)

T ( 1   1 ) ln  B1 C1 
where l is a distance between the points x1 and x2 , T is the period of the temperature
changes, 1,  and B1, C1 are the phases and amplitudes of the heat wave
respectively.
Various modifications of the Ångström's method have been applied
to different solid and liquid materials (18-23).
In the present work, we have applied the Ångström's temperature wave
method to determine in situ the thermal diffusivities of several high-temperature oxide
fluxes, namely BBO-Na2O, BBO-NaF and KTP-K6P4O13, typically used for the
BBO and KTP single crystal growth. The  values relevant for the real crystal growth
temperature intervals have been determined in order to make the results useful for a
phenomenological hydrodynamic flow analysis and further mathematical modeling.
184
EXPERIMENTAL
The experiment is based on applying a sinusoidally varying thermal flux
incident on one face of a one-dimensional specimen and convectively cooling its
opposite face. This results in a sinusoidally varying temperature on the cooled face
with a measurable phase lag between the incident and transmitted waves that depends
on the material properties and the heat transfer coefficient. The block diagram of the
experimental setup is presented on Fig.1.
It is based on a resistively heated furnace equipped with a programmable Eurotherm
temperature controller with the temperature stable to  0.1°C. The temperature
distribution field in the furnace was controlled by Pt-Pt/Rh thermocouples. The hightemperature fluxes under study were synthesized in a Pt crucible of 10 mm diameter
and 100 mm height which was ultimately filled to just above 3/4 of its height. The
crucible was placed on a pedestal in the central part of the furnace as shown in Fig.1.
In order to prevent vertical heat convection, the temperature of the flux surface was
kept at about 2-3oC higher temperature relatively to crucible bottom. The radial
temperature gradient in the flux did not exceed 0.5oC.
Fig.1. Experimental setup for measuring thermal diffusivities of the molten
solutions.
Periodic cooling of the lower crucible end was achieved by using an air
compressor with rotating shutter (stroboscope). An Alumina tube of 6 mm diameter
(less than the crucible diameter) was used to transport the pulses of cold air to the
crucible bottom through the axial hole drilled in the pedestal. There was no direct
contact between the crucible vertical surface and the air blown in. The compressed air
was at room temperature, and when the hot crucible bottom was exposed to such air
pulse a sinusoidal heat wave propagated through the molten flux along the crucible
185
axis direction. Thermocouples placed along the heat wave path allowed monitoring
the time dependence of temperature changes. The shutter was driven by a custom
designed controller which permitted to vary the duration of air blow pulses (heat
waves) and change their amplitudes at a constant air flow. The amplitude of the heat
waves could also be varied by the air blow pressure control. The combined control
over the shutter rotation frequency and the air pressure has allowed obtaining heat
waves in a sinusoidal form within a 0.4 - 15oC temperature range with time periods
from 0.5 to 30 s. An example of typical traces of the temperature waveforms (in
voltage units) obtained from the lower thermocouple Tc1 and the upper thermocouple
Tc2 immersed in a BBO/Na2O flux is shown in Fig.2, for a measurement with a time
period T = 6.3 s. These traces can be very emulated by the sine functions.
Fig.2. Typical temperature waveforms recorded by the thermocouples Tc1 and
Tc2; parameters indicated in the figure are described in the text and by
eq. (4).
The distance between the two Pt-Pt/Rh thermocouples (Tc1 and Tc2)
measuring the amplitude and phase difference of the thermal waves, l = x1 – x2, could
be varied from 2 to 50 mm, and the accuracy in determining their vertical locations
was  0.5 mm. In practice, distances (l) smaller than 10 mm were used for two main
reasons. It has been shown for the BBO melt that the amplitude of transient
temperature fluctuation decreases exponentially while propagating through the melt
(24). We have verified this observation by measuring the amplitude of the thermal
waveform as a function of the distance from the crucible bottom (for the constant
compressed air pressure and shutter rotation frequency, and the results are shown in
Fig.3 for a 80%BBO/20%Na2O flux. Obviously, l = 5-6 mm would be suitable for
obtaining well measurable temperature waveforms by both Tc1 and Tc2
thermocouples. Attempts to increase the Tc1 and Tc2 temperature differences by
overcooling the crucible bottom showed that when the Tc1 temperature waveform
amplitude reached the value of 25°C, the melt became quasi-frozen due to local
186
supercooling, and it was impossible to reveal any differences in the phase shifts in
temperature oscillations.
Fig.3. Amplitude attenuation upon thermal wave propagation from the
crucible bottom into the melt (at an oscillation period T = 3.2  0.1 s).
The problem of extracting weak periodic temperature oscillation signals in
high-temperature melts was solved by using an appropriate electronic compensation
scheme for monitoring the thermocouple voltage signals. The scheme included a high
stability compensation block with a 6 V dry battery used as a source of constant
voltage. The compensated signal was amplified and sent to the terminal of the data
acquisition unit Model DT322. All experimental data were processed using computer
software.
RESULTS AND DISCUSSION
Thermal diffusivities of high-temperature solutions with compositions relevant
for growing single crystals of BBO and KTP have been determined experimentally.
BBO crystals are usually grown from fluxes because they crystallize in the
crystallographically noncentrosymmetric -phase, suitable for nonlinear optical
applications, far below its 1095°C melting point, namely below 925°C. According to
the phase diagrams (25,26), this temperature corresponds to the 0.79BBO/0.21Na2O
and 0.69BBO/0.31NaF compositions for the conventional Na2O and more advanced
(less viscous) NaF fluxes respectively. In fact, a series of three compositions along the
liquidus have been studied for both cases. The compositions were chosen as
following: xBBO-(1x)Na2O (x = 0.72 , 0.79, 0.88 ) and xBBO-(1x)NaF (x = 0.58,
0.69, 0.78).
Sinusoidal temperature waveforms corresponding to the periodic cooling, of
the type shown in Fig.2, have been used to calculate the thermal diffusivities for each
flux composition using eq. (4). The results for the two BBO fluxes using Na2O and
NaF as solvents, namely the thermal diffusivities as a function of the solvent
concentration, are shown in Fig.4. The thermal diffusivity value of the pure BBO melt
187
is taken from the literature (24), while the measurement have been performed for
specific fluxes with composition/temperature combinations corresponding to the
respective liquidus curves. The results show that starting from a relatively low value
of  = 0.004 cm2/s for the BBO melt, the thermal diffusivity increases almost
logarithmically along the liquidus line in the case of the Na2O solvent. At 800°C, or a
temperature at which the growth BBO crystals from the Na2O flux is typically
terminated, the thermal diffusivity is two orders of magnitude higher than that of pure
BBO melt. This fact, in addition to the parallel (although not as dramatic) increase in
the kinematic viscosity (26), indicates that the Rayleigh becomes smaller at the end of
the growth run and the buoyancy driven convection reduces. The flow stability may
be thus disposed, also due to the reduction in the axial temperature gradient, and the
solid/liquid interface breakdown can be observed (27).
Fig. 4. Thermal diffusivity as a function of the solvent concentration in the
BBO-Na2O (full squares) and BBO-NaF (open squares) solutions.
The liquidus temperatures corresponding to the specific compositions
are indicated as well.
It is noteworthy that the lack of knowledge about the thermal diffusivity
variation as a function of composition in the xBBO-(1x)Na2O melts (as well as their
viscosity) has lead some workers to erroneous conclusions about the stability of these
melts against small transient temperature fluctuations. Thus Wang et al. (24) implied
that at a typical growth temperature of 880°C the melt thermal diffusivity was of the
order of 410-3 cm2/s and the dynamic viscosity of about 2 P yielding a monstrous
Prandtl number (Pr =  /) of over 120. According to our new data on the thermal
diffusivity (Fig. 4) and also measured viscosities of the BBO-Na2O melt as a function
of the solvent concentration (26), the Pr can be calculated for the entire temperature
interval employed for growth of BBO crystals. The results of the calculations are
given in Fig. 5, and they show that the Pr changes linearly and not dramatically within
this interval and is as small as ~ 2.15 at 880°C. The results of Fig. 5 may be important
for future computer modeling of the BBO crystal growth.
188
Fig. 5. Variation of the Pr along the liquidus of the BBO-Na2O binary
phasediagram.
Returning now to the problem of temperature fluctuations, we will refer to
starting composition for the BBO crystal growth, namely 80%BBO-20%Na2O, with
the  = 0.13 and  = 0.33 (26). In order for the temperature fluctuations not to affect
the crystal growth, the critical width of the temperature wave, l = (8)1/2 (where  is
the duration of temperature instability), should not exceed the width of the momentum
boundary layer, m = 1.613(/)1/2 (where  = 2n and n is the crystal rotation rate in
the r/s units) (24). With the typical crystal rotation rate of 2 rpm (27), m ≈ 2 cm.
Therefore, from the condition
2
 m
(4)
8
we calculate that if the duration of transient temperature fluctuation exceeds 1.22 s,
the temperature wave can penetrate to the growth interface and affect the quality of
the growing BBO crystal.
Fortunately, the thermal diffusivity of the BBO-NaF flux is substantially
smaller (the average value is ~ 3.510-2 cm2/s), and it is almost constant along the
entire liquidus part that is of interest for the BBO crystal growth. Apparently, the
duration of the temperature fluctuation must be four times longer, or about 5 s, in this
case in order to cause an instability at the BBO crystal/melt interface. This explains
the better quality and the higher yield of BBO crystals that can be grown from the
BBO-NaF flux. Table 1 summarizes the exact measured values of thermal
diffusivities of the BBO melt and various fluxes together with three KTP-K6Ti4O13
(KTP-K6) self-fluxes and other melts of nonlinear optical crystals, such as LiNbO3
and KNbO3. The latter exhibit clearly smaller values of thermal diffusivities and,
therefore, larger Pr, which is the origin of a number of hydrodynamic flow
instabilities typical for oxide crystal melts.
189
Table 1. Summary of measured thermal diffusivity () values of the BBO and KTP
solutions (with indication of the specific solution temperatures) together
with the k values of some pure oxide melts (from literature).
Melt Composition
Melt Temperature,
°C
Thermal Diffusivity,
cm2/s
Source
BBO
1,095
0.00042
(24)
88%BBO-12%Na2O
1,020
0.037  0.007
This work
80%BBO-20%Na2O
925
0.130  0.02
This work
72%BBO-28%Na2O
800
0.410  0.08
This work
78%BBO-22%NaF
982
0.036  0.012
This work
69%BBO-31%NaF
925
0.043  0.005
This work
58%BBO-42%NaF
856
0.029  0.005
This work
50%KTP-50%K6
920.7
0.040  0.012
This work
60%KTP-40%K6
968.4
0.036  0.006
This work
70%KTP-30%K6
1,008
0.040  0.003
This work
LiNbO3
1,257
0.0084
(24)
KNbO3
1,050
0.014
(24)
KTP crystals are mostly grown from self-fluxes, such as the conventional K6
flux, in an almost gradient-free environment with the fluid flow driven entirely by
forced convection due to crystal rotation. The fluid dynamics is not governed by
thermal diffusion, and the thermal diffusivity is not for the flow modeling. However,
the  values are still helpful in understanding the influence of possible temperature
fluctuations on the crystal/melt interface stability, like discussed above for the case of
BBO. The kinematic viscosity of a typical KTP-K6 flux is quite similar to that of
BBO,  ≈ 0.33 cm2/s (based on the data of (28): dynamic viscosity η = 78 cP and  =
2.35 g/cm3). Since the crystal (seed) rotation rate is usually much higher than in the
case of BBO, namely: n = 30 rpm (29), the width of the momentum boundary layer is
as low as 0.5 cm for the KTP-K6 flux. From Table 1, the flux thermal diffusivity is
practically constant, at least for the compositions studied, and we can adopt hereby a
 = 0.04 cm2/s value. In this case, the calculation based on eq. (4) shows that the
growth interface instability may be caused by a very short temperature fluctuation of
about 0.15 s. This is important, since periodic inversion of the crystal rotation is
routinely used in the KTP crystal growth (29,30), and it may be the main source of
compositional growth striations observed in KTP crystals (31). It is still important to
note that even if the defects exist, they are evenly distributed along the KTP crystal
growing out from a compositionally changing solution (5), since the thermal
diffusivity (see Table 1) is practically independent on the solvent concentration.
190
CONCLUSIONS
Ångström’s temperature wave method has been applied to the in situ
mesurements of thermal diffusivities of high-temperature solutions routinely used for
growing the nonlinear optical BBO and KTP crystals. The difference from the original
Ångström’s method was that a vertical molten liquid column was the subject of
studies instead of a semi-infinite metal rod. A detailed analysis of the propagation of a
sinusoidal heat wave through high-temperature solutions was carried out. The
resulting temperature waveforms obtained at two points, namely the differences in
their amplitudes and phases, were studied. It was concluded that the distance between
the two measurement points should not exceed 5-6 mm due to the fast attenuation of
the heat wave in the molten liquid. Surprisingly, the finite difference approximation to
solving the one-dimensional heat conduction equation, as suggested by Ångström,
could still be applied, and the thermal diffusivities of the various high-temperature
solutions could be evaluated successfully. The results show that the thermal
diffusivity of the BBO-Na2O solution increases by two orders of magnitude, from  =
410-3 cm2/s for the pure BBO melt to 410-1 cm2/s for the 72%BBO-28%Na2O
solution. A similar, but slower initial increase of the thermal diffusivity is observed
upon adding the solvent in the case of the BBO-NaF solution, but once the solvent
reaches the concentration of 20%, the thermal diffusivity stabilizes at a constant level
of about 3.510-2 cm2/s. The thermal diffusivity of the KTP-K6 solution is practically
constant in the range of the K6 solvent concentrations varying fro 50 to 79%, and its
average value,  ~ 410-2 cm2/s, is almost as low as that of BBO-NaF. In the case of
BBO-NaF, the lower thermal diffusivity (in comparison with the BBO-Na2O solution)
is the main reason for the better stability of the crystal/melt interface against
temperature fluctuations. In the case of KTP-K6, due to the high rotation rate needed
for the compositional homogenization of the solution, the width of the momentum
boundary layer is small, and periodic temperature fluctuations lead to the formation of
growth striations even though the thermal diffusivity is relatively low.
ACKNOWLEDGMENTS
This research was supported by the Israel Science Foundation under the grant
#156/05.
REFERENCES
1. Armstrong D, Alford W, Raymond T, Smith A, Bowers M:
‘Parametric amplification and oscillation with walkoff-compensating crystals’.
J. Opt. Soc. Am. B: Opt. Phys.1997 14 (2) 460-74 .
2. Zhang T, Yao J, Zhu X, Zhang B, Li E, Zhao P, Li H, Wang P :‘Widely tunable,
high-repetition-rate, dual signal-wave optical parametric oscillator by using
two periodically poled crystals’. Opt. Commun.2007 272 (1) 111-15.
3. Roth M, Tseitlin M, Angert N:’ Oxide Crystals for Electro-Optic Q-Switching of
Lasers’. Glass Phys. Chem.2005 31 (1) 86-95.
4. Perlov D,Roth M:’ Low-temperature synthesis of starting materials for β-barium
metaborate (β-BBO) crystal growth’. J. Cryst. Growth 1993 130 (3-4) 686-89.
5. Angert N, Tseitlin M, Yashchin E, Roth M: ‘Ferroelectric phase transition
temperatures of KTiOPO4 crystals grown from self-fluxes’. Appl. Phys. Lett.
1995 67 (13) 1941-43.
191
6. Normand C, Pomeau Y, Velarde M: ‘Convective instability: A physicist’s
approach’. Rev. Modern Phys. 1977 49 (3) 581-624.
7. Soltanolkotabi M, Bennis G, Gupta R:’ Temperature dependence of the thermal
diffusivity of GaAs in the 100–305 K range measured by the pulsed photothermal
displacement technique’. J. Appl. Phys. 1999 85 (2) 794-99.
8. Broussely M, Levick A, Edwards G:’ A Novel Comparative Photothermal
Method for Measuring Thermal Diffusivity’. Int. J. Thermophys 2005 26 (1)
221-32.
9. Sparavigna A, Giachello G, Omini M, Strigazzi A.: ‘High-sensitivity capacitance
method for measuring thermal diffusivity and thermal expansion: Results on
aluminum and copper’. Int. J. Thermophys. 1990 11 (6) 1111-26.
10. Beghi M, Luzzi L : ‘A thermoelastic method for measuring the thermal diffusivity
of Solids’. Measurem. Sci. Technol. 1999 10 1266-71.
11. Cowan R.: ‘Proposed Method of Measuring Thermal Diffusivity at High
Temperatures’. J. Appl. Phys. 1961 32 , 1363-70.
12. Vozár L, Hohenauer W :’ Flash method of measuring the thermal diffusivity. A
review’. High Temperatures - High Pressures 2003/2006 35/36 (3) 253-64.
13. Chirdon W, Aquino W, Hover K :’ A method for measuring transient thermal
diffusivity in hydrating Portland cement mortars using an oscillating boundary
temperature’. Cem. Concr. Res. 2007 37 , 680-90.
14. Hernández R, Tomás S, Cruz-Orea A, Sinencio F:’Method for measurement of
the thermal diffusivity in solids: Application to metals, semiconductors and thin
materials’. J. Appl. Phys. 1998 84, 6327-29.
15. Wada H, Watanabe M, Morimoto J, Miyakawa T:’Photopyroelectric (PPE)
determination of thermal diffusivity of Bi2Te2.85Se0.15 sintered thermoelectric
semiconductors’. J. Mat. Res. 1991 6 1711-14.
16. Ångström, Phil.Mag. 25, 130 (1863).
17. Carslaw H , Jaeger J : Conduction of Heat in Solids. Oxford:Clarendon Press,
1948 114-17.
18. Rudnev I.,Lyashenko V, Abramovich M :’Diffusivity of sodium and lithium’
Atomic Energy 1962 11(3) 877-80.
19. Bodzenta J, Burak B,Nowak M, Pyka M,Szaljako M, Tanasiewicz M
:’Measurement of the thermal diffusivity of dental filling materials using
modified Ǻgström’s method’. Dental Materials 2006 22 (7) 617-21.
20. Green A, Cowles L :’Measurements of thermal diffusivity of semiconductors by
Ǻgström’s method’. J. Sci. Instrum. 1960 37 349-51.
21. Lopez-Baeza E., Rubla J,Goldsmid H :’ Ǻgström’s thermal diffusivity method for
short Samples’. J. Phys. D: Appl. Phys. 1987 20 (9)1156-58.
22. Ebrahimi J: ‘Thermal diffusivity measurement of small silicon chips’.
Phys.D:Appl.Phys.1970 3 (2) 236-39.
23. Billington N :’ The Thermal Diffusivity of some Poor Conductors’. J. Sci. Instrum.
1949 26 20-3.
24 Wang B, Voigt A, Lu Z:’Effect of the Temperature Fluctuation of the Melt on βBaB2O4 (BBO) Crystal Growth’.Cryst.Res.Technol.2001 36 (11) 1239 -46.
25. Huang Q, Liang J :’Studies on flux systems for the single crystal growth of βBaB2O4’.J.Crystal Growth 1997 89 (3-4) 720-24.
26. Roth M.,Perlov D :’Growth of barium borate crystals from sodium fluoride
Solutions’. J. Cryst. Growth 1996 169 (4) 734-40.
27. Perlov D,Roth M: ‘Isothermal growth of β-barium metaborate single crystals by
192
28.
29.
30.
31.
continuous feeding in the top-seeded solution growth configuration’. J. Cryst
Growth 1994 137 (1-2) 123-27.
Vartak B, Kwon Y-I, Yeckel A, Derby J:’An analysis of flow and mass transfer
during the solution growth of potassium titanyl phosphate’. J. Cryst. Growth 2000
210 (4) 704-18.
Angert N, Kaplun L, Tseitlin M, Yashchin E, Roth M:’Growth and domain
structure of potassium titanyl phosphate crystals pulled from high-temperature
solutions’. J. Cryst. Growth 1994 137 (1-2) 116-22.
Vartak B, Yeckel A, Derby J:’Time-dependent, three-dimensional flow and mass
transfer during solution growth of potassium titanyl phosphate’. J. Cryst. Growth
2005 281(2-4) 391-406.
Sebastian M, Klapper H, Bolt R:’X-ray diffraction study of KTP (KTiOPO4)
crystals under a static electric field’. J. Appl. Cryst.1992 25 274-80.
193