FIRST-PRINCIPLE QUANTUM-CHEMICAL CALCULATIONS OF SEVERAL

THERMOMECHANICAL PARAMETERS OF BERYLLIUM CERAMICS

Yu. N. Makurin,1 I. R. Shein,1 M. A. Gorbunova,1

V. S. Kiiko,1 and A. L. Ivanovskii1

Translated from Novye Ogneupory, No. 10, pp. 38 – 42, October, 2006.

Original article submitted July 13, 2006.

The first-principle quantum-chemical calculations of elastic constants for beryllium oxide and their approxi-

mation (the Voigt – Reuss – Hill model) for a polycrystalline material are used to derive quantitative estimates

of several thermomechanical parameters of BeO ceramics: isothermal compression coefficient, sound veloc-

ity, the Debye temperature, and coefficients of linear and volume temperature expansion, as well as the tem-

perature dependence of molar heat capacity and thermal conductivity. The results are discussed in correlation

to available experimental data.

Ceramics based on beryllium oxide have found wide ap-

plication as functional and structural materials in contempo-

rary instrument engineering [1 – 4]. Calculating the service

parameters of BeO ceramics by methods of computational

quantum theory is of great practical interest, since in the case

of success it becomes possible to theoretically simulate a

wide class of functional and structural materials based on

BeO ceramics and alloyed with various impurities and to

predict their physicochemical properties responsible for their

service parameters.

The present study uses the results of first-principle quan-

tum-chemical calculations of elastic constants of single-crys-

tal BeO and their subsequent approximation for the polycrys-

talline state and determines several thermomechanical pa-

rameters of impurity-free BeO ceramics: isothermal com-

pression coefficient ÷, sound velocity v, Debye temperature

È, and coefficients of linear á and volume â temperature ex-

pansion, as well as the temperature dependence of molar heat

capacity cM(T) and thermal conductivity ë(T).

The initial data are the results of our calculations (using

the full-potential linear method of augmented plane waves

(FLAPW, code WIEN2K [5]) with generalized gradient ap-

proximation (GGA) of the exchange-correlation potential [6]

of the elastic constants Cij of wurtzite-type BeO. These

values correlated with available experimental data are given

in Table 1. The discrepancy between the calculated and mea-

sured Cij values ranges from 10 to 22%. It should be taken

Refractories and Industrial Ceramics Vol. 47, No. 5, 2006

310

1083-4877�06�4705-0310 © 2006 Springer Science+Business Media, Inc.

1Ural State Technical University (UPI), Yekaterinburg, Russia;

Institute of Solid-State Chemistry, Ural Branch of the Russian

Academy of Sciences, Yekaterinburg, Russia.

TABLE 1. Elastic Constants Cij, GPa, of Wurtzite-Type BeO Based on FLAPW-GGA Calculations Correlated with Experimental Data

Data

Constants

C11

C12

C13

C33

C44

C66

*1

Calculated by the authors of the present study 410.8 115.6 84.6 446.1 129.1 147.6

Experimental, published in the literature:

[7] 460.6 126.5 88.5 491.6 147.7 167.0

[8] 470 168 119 494 153 152

[9] 468 130 120 497 148 169

Mean value 466.2 141.5 109.2 494.2 149.6 162.7

Error ä*2, % 11.9 18.3 22.5 9.7 13.7 9.3

*1 C C C66 11 12

1

2

� �( ). *2 ä = (Cij

exp– C

ij

th)�C

ij

exp.

into account that all experimental data concern polycrystal-

line samples (i.e., they depend on the manufacture methods:

compaction and firing, as well as porosity, grain size, etc.)

and measured Cij values reported by different authors [7 – 9]

differ significantly: up to ~30 GPa (25 – 36%) for the con-

stant C13 (Table 1).

As polycrystalline BeO ceramics constitutes an aggre-

gated mixture of microcrystallites with a random mutual ori-

entation, a numerical estimate of its mechanical parameter

requires special averaging of values obtained in calculations

performed for a single crystal. The Voigt – Reuss – Hill pro-

cedure [10 – 12] is often used for this purpose. In the frame-

work of this approximation using elastic constants Cij of a

BeO single crystal, the shear moduli G and the compression

moduli B are determined as follows:

G =1

2(G

V+ G

R); B =

1

2(B

V+ B

R), (1)

where the asymptotically maximal (Voigt method) and mini-

mal (Reuss method) values of the shear modulus GV

and GR

and compression modulus BV

and BR

are found as follows:

G C C C C C CV

� � � � � �

1

302 4 12 12

11 12 33 13 55 66( ), (2)

GC C C C C C

B C C C CR

V

�

� �

� �

5

2

2

3

11 12 33 12

2 2

55 66

55 66 11 1

[( ) ]

[(2 33 12

2 2

55 662) ] ( )

,

C C C C� �

(3)

B C C C CV

� � � �

�

�

�

�

�

2

92

1

211 12 13 33

, (4)

BC C C C

C C C CR

�

� �

� � �

( ).

11 12 33 12

2

11 12 33 13

2

2 4(5)

The calculated values satisfactorily agree with experi-

mental data (Table 2).

The availability of G and B moduli opens up wide possi-

bilities for estimating a set of important thermomechanical

parameters of beryllium ceramics. Since many products

made of BeO ceramics are expected to serve at high pressure

(for instance, high-voltage insulators in chambers with mag-

netic reduction for obtaining magnetized hydrogen plasma,

in which a thermonuclear reaction takes place, and in other

technical devices), the isothermic compression coefficient ÷,

Pa–1, is very important. It is determined as follows:

� �

�

�

�

�

�

� � �

�

1 1504 10

12

V

dV

dP BT

. , (6)

where V is the cell volume.

According to available experimental data [16], the

temperature dependence ÷ is described by the following

equation:

÷ = a – 2bp + 3cp2,

for BeO, a = 4.13 � 10–12

Pa–1

, b = 4.3 � 10–22

Pa–1

, and

c = 0. Under low pressures p (<50 MPa), ÷ � a =

4.13 � 10–12

Pa–1

. It can be seen that the discrepancy between

the calculated and experimental [16] values is around 20%.

Chang and Cohen [15] report the experimental value

÷ = 4.76 � 10–12

Pa–1

, whose deviation from our calculation

is ~5%.

Another important characteristic of BeO ceramics used

in the metallurgical, nuclear, and radioelectronic sectors un-

der high or low temperatures (in the range of 77 – 1800 K)

as chemically and thermally stable efficient ultrasonic con-

centrators and sound ducts is the sound velocity. The aver-

aged sound velocity v can be calculated using the formula

v

v vs l

� �

�

�

�

�

�

�

�

�

�

�

�

�

�

�

1

3

2 1

3 3

1 3

, (7)

where vsand v

lare the lateral and longitudinal components of

the sound velocity, respectively,

v Gs

� �, v B Gl

� �

�

�

�

�

�

1 3

4�

. (8)

The estimate of the parameter v based on formulas (7)

and (8) yields the value v = 7656 m�sec, whereas in an ex-

periment [8] v = 8240 m�sec, i.e., the discrepancy between

these values is not more than 7%.

Heat capacity (as well as thermal conductivity) is a very

significant parameter of BeO ceramics, since it is frequently

used in high-temperature equipment, in particular, as dielec-

tric discharge tubes in high-power optical quantum genera-

First-Principle Quantum-Chemical Calculations of Several Thermomechanical Parameters of Beryllium Ceramics 311

TABLE 2. Some Thermomechanical Parameters of BeO Ceramics

Parameter

Data

calculated experimental

Modulus of, GPa

shear G 145.59 142 [13], 102 [14]

compression B 198.39 210 [16]

Isothermal compression coefficient ÷, 10–12 Pa–1 5.04 4.76 [15], 4.13 [16]

Averaged sound velocity v, m�sec 7656 8240 [17]

Debye temperature È, K 1188 1280 [7]

tors and in other functional electronic devices, as well as in

heat dissipation. Since BeO is a dielectric with a wide

(~10.8 eV) forbidden band, the heat capacity of this material

depends only on lattice oscillations. In this case, the tempera-

ture dependence of the heat capacity is estimated using the

Debye theory [17], according to which the molar heat capac-

ity cM is determined by the formula

c RnDT

M�

�

�

�

�

�

3

�

, (9)

where R is the gas constant; n is the number of atoms in the

formula unit; for BeO, n = 2; D(�T) is the Debye function

given in the reference tables, and È is the Debye tempera-

ture, which can also be expressed via the elastic characteris-

tics of the medium (and indirectly via v):

� �

�

�

�

�

�

�

kn

N

Mv6

2

1 3

�

�

A, (10)

where � is Plank’s constant, NA

is the Avogadro number; k is

the Boltzmann constant; M is the molecular mass; ñ is the

crystal density. The estimate È based on formula (10) yields

È = 1188 K, whereas the experimental value is È = 1280 K

[7, 16], i.e., the discrepancy is ~7%.

Figure 1 shows the temperature dependence of molar

heat capacity calculated based on Eq. (9) correlated to exper-

imental data [18]. It can be seen that estimated and experi-

mental data are close, especially at T � 600 K.

The coefficients of volume and linear (CLTE) tempera-

ture expansion, which are also very significant parameters

for BeO ceramics, are determined as follows:

� �

�

�

�

�

�

1

V

dV

dTp

; � �

�

�

�

�

�

1

l

dl

dTp

, (11)

where, for polycrystalline materials, â = 3á. According to the

Gruneisen law,

â = ã�

Mc

M÷, (12)

where ã is the Gruneisen coefficient, whose value in fact

does not depend on temperature; for BeO ã = 1.4 [19]. The

temperature dependence ë and â, in accordance with Eqs. (9)

and (12), is defined as

� �( ) ( ) . .T T DT

� � �

�

�

�

�

�

�

1

31412 10

6 �

(13)

It can be seen from the dependence á(T) in Fig. 2 that the

CLTE decreases with decreasing temperature. Unfortunately,

it is difficult to compare this result with experiments, since

available data hold only for limited temperature intervals:

á = 5.42 � 10–6 K–1 (at T = 300 – 373 K) and á = 7.08 �

10–6 K–1 (at T = 300 – 673 K) [20].

The thermal conductivity ë of dielectrics with a wide for-

bidden band (i.e., their phonon thermal conductivity) can be

determined using the following equation:

� =�

3Mc

Mvl, (14)

here l is the length of the phonon free path, and its determina-

tion is the main problem in computing ë. Under temperatures

perceptibly lower than the Debye temperature, the length of

the free path of the phonon is [21]:

l = l0e

�2T, (15)

i.e., as temperature decreases, the value l grows. At suffi-

ciently low temperatures, the value l may reach the size of

the ceramic grain; then, if T continues to decrease, the pho-

non free path remains constant. The grain size in BeO ceram-

ics varies from 2 – 4 to 50 – 70 ìm, depending on the pro-

duction technology. Furthermore, one should take into ac-

count the scattering of phonon on the impurities, i.e., the

maximum possible value l = lcr

can be achieved with a free

path length l significantly smaller than the grain size. The

value l0

in Eq. (15) can be determined from the relation

lim .T

l l� �

�

0(16)

Then, for the case of a single-crystal BeO sample,

l V0

1 3�

e.c

/, where Ve.c is the elementary cell volume, i.e.,

l ~ 0.3 nm.

312 Yu. N. Makurin et al.

60

40

20

0

cM

, J (mole · K)�

0 500 1000 1500 2000

T, K

12

Fig. 1. Temperature dependence of molar heat capacity cM

of beryl-

lium oxide: 1, calculation; 2, experiment [18].

16

12

8

4

0

á, 10 K–6 –1

0 500 1000 1500 2000 2500

T, K

Fig. 2. Temperature dependence of CLTE of beryllium oxide.

Figure 3a gives the temperature dependence of the ther-

mal conductivity of BeO ceramics at lcr equal to 50, 10, and

1 ìm. It can be seen that all dependences ë(T) have

extremums at certain temperatures Tcr. The growth of ë at

T < Tcr is due to the increased heat capacity and the subse-

quent decrease in ë at T > Tcr is caused by a decreased

phonon free path length. It is essential that, according to our

calculations, as lcr decreases (i.e., as the ceramic grain size

decreases and the concentration of impurity increases), the

maximum ë(T) is shifted toward higher temperatures and its

height (ëcr) decreases. Unfortunately, we are not familiar

with relevant experimental data. Figure 3b for reference pur-

poses gives the experimental temperature dependence of the

thermal conductivity of single-crystal beryllium oxide [22].

It can be seen that the specified regularities of ë(T) variations

established for BeO ceramics are validated: for the single

crystal the maximum l(T) is located in the lowest-tempera-

ture range (Tcr ~ 40 K) and the value ëcr is maximal

[13,500 W�(m · K)].

Thus, the above numerical estimates based on the theo-

retically calculated elastic constants of BeO make it possible

to satisfactorily reproduce a wide range of significant physi-

cochemical characteristics of BeO ceramics. It has been

noted that BeO and affined oxides are especially interesting

for developing various polyfunctional ceramics materials. A

common method for modifying their properties is alloying

them with different impurities. First-principle zonal calcula-

tions are widely used in simulating the effect of alloying

crystals; therefore, this approach can be very useful for pre-

dicting elastic properties of alloyed polycrystalline materials

and their physicochemical parameters responsible for the

service properties required from these materials.

This work has been performed with support from the

Russian Fund for Basic Research (project No. 05-08-01279)

and grant NSh-5138.2006.3.

REFERENCES

1. R. A. Belyaev, Beryllium Oxide, Atomizdat, Moscow (1980) [in

Russian].

2. S. J. Pearton, D. R. Norton, K. Ip, et al., “Recent advances in pro-

cessing of ZnO,” Vacuum Sci. Technol., B22(3), 932 – 948

(2004).

3. A. N. Banerjee and K. K. Chattopadhyay, “Recent developments

in the emerging field of crystalline p-type transparent conducting

oxide thin films,” Progr. Crystal Growth Charact. Mater.,

50(1 – 3), 52 – 105 (2005).

4. U. Ozgur, Ya. I. Alivov, C. Liu, et al., “A comprehensive review

of ZnO materials and devices,” J. Appl. Phys., 98(4) (2005).

5. P. Blaha, K. Shwarz, G. K. H. Madsen, D. Kvasnicka, and

J. Luitz, “An augmented plane wave plus local orbitals pro-

gram for calculating crystal properties,” in: K. Schwarz (ed.),

WIEN2K, Techn. Universität Wien, Austria (2001).

6. J. P. Perdew, S. Burce, and M. Erzerhof, “Generalized gradient

approximation made simple,” Phys. Rev. Lett., 77(18),

3865 – 3868 (1996).

7. C. F. Cline, H. L. Dunegan, G. Henderson, “Elastic constants of

hexagonal BeO, ZnS, and CdSe,” J. Appl. Phys., 38(4), 1944 –

1952 (1967).

8. G. G. Bentle, “Elastic constants of single-crystal BeO at room

temperature,” J. Am. Ceram. Soc., 49(3), 125 – 133 (1966).

9. N. N. Sirota, A. M. Kuz’mina, and N. S. Orlova, “Elastic-modu-

li of beryllium oxide at 10 – 270 K from the x-ray data,” Dokl.

AN SSSR, 314(4), 856 – 859 (1990).

10. W. Voigt, Lehrbuch der Kristallphysic, Teubner, Leipzig

(1928), pp. 716 – 761.

11. A. Reuss and Z. Angew, Calculation of flow limits of mixed

crystals on the basis of plasticity of single crystals,” Math.

Mech., 9, 49 – 57 (1929).

12. R. Hill, “The elastic behaviour of a crystalline aggregate,” Proc.

Phys. Soc. London, A65, 349 – 357 (1952).

13. R. E. Fryxell and B. A. Chandler, “Creep, strength, expansion

and elastic modulus of sintered BeO as a function of grain size,

porosity, and grain orientation,” J. Am. Ceram. Soc., 47(6),

283 – 291 (1964).

14. E. Ryshkewitch, “Rigidity modulus of some pure oxides and

bodies,” J. Am. Ceram. Soc., 34(10), 322 – 326 (1951).

15. K. J. Chang and M. L. Cohen, “Theoretical study of BeO: struc-

tural and electronic properties,” Solid State Comm., 50(6), 487 –

491 (1984).

16. C. F. Cline and D. R. Stephens, “Volume compression of BeO

and other 2–6 compounds,” J. Appl. Phys, 36(9), 2869 – 2873

(1965).

17. L. Girifalco, Statistical Solid-State Physics [Russian transla-

tion], Mir, Moscow (1995).

18. H. Landolt and R. Bornstein, Zahlenwerte und Funktionen aus

Physik, Chemie, Astronomie, Geophysik und Technik. Bd. 2.

T. 4 – 6, Springer, Berlin (1960).

19. H. Iwanaga, A. Kunishige, and S. Takeushi, Anisotropic ther-

mal expansion in wurtzite-type crystals,” J. Mater. Sci., 35(10),

2451 – 2454 (2000).

20. G. V. Samsonov and I. M. Vinnitskii, High-Melting Compounds

[in Russian], Metallurgiya, Moscow (1976).

21. Ch. Kitel’, Introduction to Solid-State Physics [in Russian],

Nauka, Moscow (1978).

22. G. A. Slack, “Nonmetallic crystals with high thermal conductiv-

ity,” J. Phys. Chem. Solids, 34(2), 321 – 335 (1973).

First-Principle Quantum-Chemical Calculations of Several Thermomechanical Parameters of Beryllium Ceramics 313

4

3

2

1

0

ë, 10 W (m · K)3 �

1

10

50

14

10

6

2

0 50 100 150 200

a

b

T, K

Fig. 3. Temperature dependence of thermal conductivity ë of BeO

ceramics (a) under different values of lcr

(indicated on the curves,

ìm) and dependence ë(T) for single-crystal BeO (b) [22].