Mechanical and thermophysical properties of actinide

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Modern Physics Letters B 1850248 (9 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217984918502482

Mechanical and thermophysical properties of actinide monocarbides

Devraj Singh∗,‡ , Amit Kumar†,§ , Vyoma Bhalla∗,¶ and Ram Krishna Thakur†,k ∗ Department

of Applied Physics, Amity School of Engineering and Technology, Bijwasan, New Delhi 110061, India † Amity School of Applied Sciences, Amity University, Manesar 122413, India ‡ [email protected] § [email protected] ¶ [email protected] k [email protected] Received 5 April 2018 Accepted 25 May 2018 Published 9 July 2018 This paper describes the mechanical and thermophysical properties of actinide monocarbides AnCs (An=Np and Cm) as a function of temperature and crystallographic direction. The temperature-dependent second- and third-order elastic constant (SOECs and TOECs) have been computed first using Coulomb and Born–Mayer potential up to second nearest neighbor. SOECs have been applied to find out mechanical constant such as bulk modulus, shear modulus, tetragonal modulus, Poisson’s ratio and Zener anisotropy for the prediction of futuristic performance of the NpC and CmC. We also found the value of G/B > 0.59 for the chosen materials, which indicates that NpC and CmC have brittle nature. The computed elastic constants are further applied directly to indirectly find out the ultrasonic velocity, Gr¨ uneisen parameters, pressure derivative, Debye temperature, micro-hardness, Breazeale’s nonlinearity parameter, thermal relaxation time and thermal conductivity. These evaluated parameters were finally used to compute ultrasonic attenuation of the NpC and CmC along h100i, h110i and h111i directions at room temperature. The behavior of the obtained results of this investigation has been compared with similar type of materials. Keywords: Actinide monocarbides; elastic properties; ultrasonic properties; thermal properties.

1. Introduction Ultrasonics has been used to inspect objects, measure distances, cleaning, detect invisible flaws, mixing and to accelerate chemical processes. The actinides play an important role in nuclear fission technology, medical diagnostics and treatments. ¶ Corresponding

author. 1850248-1


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The structural, magnetic, electronic and thermal characteristic of actinide monocarbides, AnCs (An=Np and Cm) have been investigated by Soni et al.1 Petit et al.2 performed detailed study of the ground-state valency configuration of the actinide ions in the actinide mononitrides and the actinide monocarbides by using self-interaction corrected local spin-density approximation. Quasi-self-consistent relativistic energy band structures have been investigated for the stoichiometric actinide monocarbides by Mallett.3 The ground states of ThC, UC and PuC have two doubly occupied π orbitals at origin with a bond distance between 1.8 ˚ A and 2.0 ˚ A. A study of four representative actinide monocarbides has been performed with relativistic quantum chemical calculations by Pogany et al.4 Charles et al.5 have found the theoretical observation of thermodynamical characteristic of actinide carbides. Adiabatic elastic constants at room temperature of uranium carbide single crystals grown by the radio-frequency, induction heated floating-zone technique investigated as a function of carbon composition with the help of phase comparison technique by Routbort.6 Vigier et al.7 have investigated uranium monocarbide phase with the help of correlative X-ray spectroscopy and quantum chemical computation. Mechanical characteristic of PuC have been investigated by Tokar et al.8 Haire9 provided the summary of selected methods in consequence of converting µg to kg amounts of the actinide material and investigated the characteristic of these materials. As per our knowledge, very few studies have been done for actinide monocarbides. The present aim is to extend the study of ultrasonic properties of actinide monocarbides in temperature range 0–300 K along h100i, h110i and h111i directions by the application of second-order and third-order elastic constants (SOECs and TOECs). 2. Theory The SOECs and TOECs at particular temperature are computed by adding the vibrational contribution to the static elastic constants10 0 vib CIJ (T ) = CIJ + CIJ ;

0 vib CIJK (T ) = CIJK + CIJK ,

(1)

where superscript “0” denotes static elastic constant at 0 K and superscript “vib” denotes vibrational part of elastic constant at particular temperature. The interionic potential is represented by the aggregate of electrostatic/Coulomb and repulsive/Born–Mayer potentials.11 φ(r) = φ(C) + φ(B) ,

(2)

where φ(C) represents long-range electrostatic/Coulomb potential and φ(B) represents short-range repulsive/Born–Mayer potential −r0 e2 and φ(B) = A exp , (3) φ(C) = ± r0 b 1850248-2



where “e” is charge of electron, “r0 ” is the nearest neighbor distance, “b” is the hardness parameter and “A” is the strength parameter given as A = −3b

e2 (1) 1 √ √ S . r0 3 6 exp(−ρ0 ) + 12 2 exp(− 2ρ0 )

(4)

The pressure derivatives (dCij /dp) for the first-order in terms of Lagrangian strains are given in the literature.11 The value of dCij /dp can be investigated for the chosen materials by using the computed values of SOECs and TOECs. The value of Debye temperature,12 θD , is dependent on Debye average velocity VD and is given by 1/3 h 3n NA ρ VD , (5) θD = kB 4π M where “n” is the number of atoms per molecule, “ρ” is the density, “NA ” is the Avogadro’s number, “kB ” is the Boltzmann’s constant and “M ” is the molecular weight. The mechanical properties such as bulk modulus, shear modulus, tetragonal modulus, Poisson’s ratio, Zener anisotropy and micro-hardness parameters are computed using the formulae given in the literature.13,14 The formula for Breazeale’s nonlinearity parameter (β) is given below12 β = −(3K2 + K3 )/K2 ,

(6)

where K2 and K3 are linear combinations of SOECs and TOECs along h100i, h110i and h111i direction of a cubic crystal. The common application of thermal relaxation time is the time prescribed for heat to conduct away from a directly-heated material zone. The thermal relaxation time for longitudinal and shear modes of wave propagation is given below 3κ 1 τL = τS = τth = , (7) 2 Cv Vm2 where κ is thermal conductivity, CV is the specific heat per unit volume and VD is the Debye average velocity of ultrasonic wave. The thermal conductivity has been evaluated using Slack approach.12 Ultrasonic attenuation represents the decrease of wave amplitude or signal strength in case of all mechanisms such as absorption and scattering. The ultrasonic attenuation is dependent on thermo-physical parameters like energy density, thermal relaxation, ultrasonic velocities and specific heat. The main causes of ultrasonic attenuations are thermo-elastic relaxation mechanisms, phonon–phonon interaction at high temperature and electron–phonon interaction at low temperatures for perfect crystals.15,16 (α/f 2 )L =

2π 2 τth E0 DL ; 3ρVL3

(α/f 2 )S =

2π 2 τth E0 DS , 3ρVS3

(8)

where α denotes the ultrasonic attenuation constant, f denotes frequency of the ultrasonic wave, VL and VS denote the ultrasonic velocities for longitudinal and shear modes of wave propagation and E0 denotes the thermal energy density. 1850248-3


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3. Results and Discussion Elastic properties of solids play an important role in the investigation of mechanical and thermal behavior of crystals. The SOECs and TOECs have been calculated with the help of two primary parameters, viz. nearest neighbor distance (r0 ) and hardness parameter (b) in the temperature range of 0–300 K. We have taken the lattice parameter 2.37 ˚ A and 2.44 ˚ A for NpC and CmC, respectively1 and the hardness parameter 0.29 ˚ A for NpC and CmC materials. The computed value of temperature dependent SOECs and TOECs are listed in Table 1. From Table 1, we observed that the elastic constants of actinide monocarbides material changes linearly with temperature. The value of elastic constants: C11 , C44 , C112 and C144 increases and C12 , C111 , C166 and C123 decreases with rise in temperature. Thus, elastic constant are temperature sensitive. But the magnitude of C456 remains same which are observed in metallic, semiconductor and intermetallic materials.16 The computed values of SOECs and TOECs are used to calculate the bulk modulus (B), shear modulus (G), tetragonal moduli (CS ), Poisson’s ratio (σ), Zener anisotropy factor (A), G/B and micro-hardness (H) have been listed in Table 2. It is observed from Table 2 that the computed value of B is smaller as compared to values available in literature.1,2 The bulk modulus provides the information about the hardness of material. The value of B for NpC is greater than the value for CmC. This represents the inter-atomic bonding stability of NpC to be the strongest compare to CmC. The toughness to fracture ratio, G/B > 0.59, indicates brittle nature Table 1. SOECs and TOECs of AnCs (An=Np and Cm) at the temperature 0–300 K range in the unit GPa. Material

Temp.

C11

C12

C44

C111

C112

C123

C144

C166

C456

NpC

0 100 200 300

75.2 79.9 81.4 83.4

31.2 30.2 29.2 28.2

31.2 31.4 31.5 31.6

−1162 −1194 −1196 −1204

−128.2 −124.6 −121.7 −118.3

49.6 44.5 39.4 34.3

49.9 50.2 50.6

−129.2 −129.5 −129.9

49.6 49.6 49.6

0 100 200 300

71.8 76.3 77.8 79.7

27.5 26.5 25.5 24.5

27.5 27.6 27.7 27.8

−1127 −1159 −1162 −1169

112.7 −109.1 −106.0 −102.7

44.1 38.9 33.7 28.6

44.4 44.7 45.0

−113.6 −113.9 −114.2

44.1 44.1 44.1

CmC

Table 2.

B, G, CS , σ, A, G/B and H for AnCs (An=Np and Cm) materials at 300 K.

Material

B (GPa)

G (GPa)

Cs (GPa)

σ

A

G/B

H

NpC

46.60 208.7 (Ref. 1) 179 (Ref. 2)

28.93

27.6

0.76

1.14

0.62

4.73

CmC

42.90 150.3 (Ref. 1) 151 (Ref. 1)

27.67

27.6

0.71

1.01

0.64

4.88

1850248-4


Mechanical and thermophysical properties of actinide monocarbides Table 3. Ultrasonic velocities (in 103 m/s) of AnCs (An=Np and Cm) along h100i, h110i and h111i directions in the temperature range 100–300 K. Material

Direction

Velocity

100 K

200 K

300 K

h100i

VL ∗V S1 = VS2 VL $V S1 #V S2 VL @V S1 = VS2

2.27 1.42 2.39 1.32 2.36 1.42 1.79

2.29 1.43 2.39 1.34 2.37 1.43 1.83

2.32 1.43 2.40 1.36 2.38 1.43 1.89

VL ∗V S1 = VS2 VL $V S1 #V S2 VL @V S1 =VS2

2.27 1.37 2.33 1.32 2.31 1.37 1.84

2.29 1.37 2.33 1.34 2.32 1.37 1.88

2.32 1.37 2.33 1.37 2.33 1.37 1.93

h110i NpC h111i

h100i h110i CmC h111i

*, $, # and @ polarized along h100i, h001i, h1¯ 10i and h¯ 110i directions, respectively.

of actinide monocarbides. It is obvious from Table 1 that the actinide monocarbides satisfies the Born stability criterion17 : BT = (C11 + 2C12 )/3 > 0,

C44 > 0,

CS = (C11 − C12 )/2 > 0 .

(9)

The ultrasonic velocities of chosen materials can be investigated using SOECs and density of the materials for longitudinal and shear modes of propagation.13 The magnitude of shear velocity is highest along h110i orientation, which is polarized along h110i orientation in NpC and lowest for CmC along h110i orientation, which is polarized along h001i orientation. By using the computed values of ultrasonic velocities, we have calculated Debye average velocities along different directions and are plotted in Fig. 1. From Fig. 1, we inspected that the Debye average velocity is highest along h100i orientation. So the direction h100i will be most convenient for wave propagation for the given material, AnCs. It indicates that the Debye average velocity increases with temperature and thus, is temperature sensitive. Gr¨ uneisen parameters provide the information regarding the physical characteristic of the given materials like temperature variation of elastic constants, thermal expansion and thermal conductivity. The values of Gr¨ uneisen parameters have been tabulated in Table 4. The obtained values of Gr¨ uneisen parameters are more or less as the values of these for rare-earth monoarsenides.16 The pressure derivatives have been computed using the obtained values of SOECs and TOECs for AnCs. The investigated magnitude of SOECs and TOECs and its pressure derivatives show important characteristics of the anharmonic 1850248-5

D. Singh et al.

NpC CmC

1.80

Debye average velocity, VD (m/s)


1.82

1.78

1.76

1.74 1.72

1.70 1.68 100

150

200

250

300

Temperature (K) Fig. 1.

Debye velocities versus temperature along different directions.

Table 4. Gruneisen parameters of AnCs (An=Np and Cm) along h100i, h110i and h111i direction at room temperature. Material

Direction

hγij il

h(γij )2 il

h(γij )2 is1

h(γij )2 is2

NpC

h100i h110i h111i

0.43 −0.73 −0.62

1.52 1.98 1.83

0.11 0.13 1.69

0.11 2.47 1.69

CmC

h100i h110i h111i

0.43 −0.72 −0.62

1.66 1.99 1.83

0.11 0.12 1.69

0.11 2.58 1.69

properties of crystalline solids, inter-atomic forces and inter-ionic potentials. The magnitude of pressure-induced change in the longitudinal elastic constants i.e. dC 11 /dp are highest compared to dC 12 /dp and dC 44 /dp. This process of elasticity– pressure dependence will be recreated in the corresponding change of wave velocities with pressure. Figure 2 indicates that the magnitude of pressure derivative (dC 11 /dP ) undergoes minor change with temperature, (dC 12 /dP ) decreases with increase in temperature and the magnitude of pressure derivative (dC 44 /dP ) does not change for NpC and increases with increase in temperature for CmC materials. We compare our results with cerium monopnictides12 and we found justified results. The Debye temperature (θD ) can be calculated by using the magnitude of Debye average velocity and molecular weight. From Eq. (5), it is clear that if the value of M increases then the value of θD will decrease and if Vm will increase so θD will increase. Thus, θD is found to be highest for NpC. It is an important parameter and provides the information regarding the solids representing the temperature at 1850248-6

CmC NpC

9 8 dC11/dp

7

Pressure derivatives



dC12/dp dC44/dp

6 5 4 3

NpC CmC

2 1

NpC CmC

0 100

150

200

250

300

Temperature (K) Fig. 2.

Pressure derivative versus temperature.

Table 5. Debye temperature, θD of AnCs (An=Np and Cm) along h100i, h110i and h111i at temperature range 100–300 K. Materials

Directions

100 K

200 K

300 K

NpC

h100i h111i h110i

226.00 217.48 215.51

226.87 219.91 217.18

228.03 222.95 219.18

CmC

h100i h111i h110i

213.62 209.85 205.20

214.42 212.22 206.62

215.49 215.15 208.31

which all the vibrational modes are agitated and also used for further computation of specific heat and thermal energy density by normalizing temperature. Micro-hardness parameter (H) has the important role in investigation of microscopic contact area because the magnitude of micro-hardness decreases with increase of size. The value of H for NpC and CmC are 4.73 and 4.88, respectively at room temperature. It was also observed that the magnitude of micro-hardness parameter decreases with increase of the atomic mass number. So NpC is stronger compare to CmC. The computed value of Breazeale’s nonlinearity parameter at room temperature 300 K is listed in Table 6 along h100i, h110i and h111i orientations. We observed that the maximum numerical value of nonlinearity parameter is along h111i and lowest along h110i direction. The thermal conductivity of NpC and CmC has been computed using Slack’s approach.18 The specific heat per unit volume (CV ) and crystal energy density (E0 ) have been obtained by applying θD /T tables of AIP Handbook19 using the 1850248-7


D. Singh et al. Table 6. Breazeale’s nonlinearity parameter (β) of AnCs (An=Np and Cm) at room temperature 300 K. Material

h100i

h110i

h111i

NpC CmC

−11.44 −11.67

−27.72 −27.90

2.29 2.16

Table 7. Thermal conductivity (κ in W/mK), specific heat per unit volume (Cv in 108 erg/cc.degree), crystal energy density (E0 in 1010 erg/cc.degree), thermal relaxation time (τth in ps), acoustical coupling constant D and ultrasonic attenuation (α/f 2 )L , (α/f 2 )S1 and (α/f 2 )S2 (in 10−16 Np · s2 /m) along h100i, h110i and h111i orientation at room temperature. Material Direction

K

Cv

E0

DL

Ds1

Ds2

τth

(α/f 2 )L (α/f 2 )s1 (α/f 2 )s2

NpC

h100i h110i h111i

0.32 5.77 5.58 7.74 0.38 5.81 5.62 10.09 0.32 5.81 5.64 9.35

0.56 0.56 5.07 0.66 12.58 6.40 8.63 8.63 5.27

2.24 3.35 12.16

0.69 1.15 4.86

0.69 4.29 4.86

CmC

h100i h110i h111i

0.35 5.81 5.69 13.41 0.88 0.88 5.83 0.35 5.81 5.69 16.08 0.96 20.85 6.18 0.31 5.81 5.73 14.80 13.67 13.67 5.16

4.77 5.99 22.84

1.52 1.76 7.54

1.52 0.86 7.54

computed values of ρ and θD . The strength of thermal phonons to consume energy from sound wave have been found along h100i, h110i and h111i directions by using the acoustic coupling constants DL and DS (for longitudinal and shear waves). From Table 7, it was observed that the magnitude of τth is of picoseconds order which indicated that the actinide monocarbides AnCs (An=Np and Cm) are semimetallic in nature. Ultrasonic attenuation due to phonon–phonon interaction (α/f 2 )p–p and thermo-elastic attenuation (α/f 2 )th have been investigated at room temperature along h100i, h110i and h111i orientation and listed in Table 7 along with the values of thermal conductivity, specific heat, energy, acoustic coupling constant and thermal relaxation time. The values of ultrasonic attenuation, (α/f 2 )L is found lowest for NpC along h100i direction and highest along h111i orientation for CmC. The ultrasonic attenuation due to p–p interaction for longitudinal wave is predominant over total thermal loss. It clearly indicates that CmC is more relevant than NpC. 4. Conclusions The investigated value of SOECs and TOECs imply to be authentic for actinide monocarbides AnCs (An=Np and Cm) in the temperature range 0–300 K. The magnitude of bulk modulus is higher for NpC material compare to CmC material, this represents the NpC is stronger compare to CmC material. The toughness/fracture, G/B > 0.59 which indicates brittle nature of actinide monocarbides. Actinide monocarbides AnCs (An=Np and Cm) fulfilled the Born stability criterion, so the actinide monocarbides are mechanical stable in nature. The magnitude of the Debye average velocity is higher along h111i orientation. So h111i will be most 1850248-8



convenient for wave propagation for actinide monocarbides. The magnitude of Debye average velocity is increase with increase of the value of temperature. The magnitude of Gr¨ uneisen parameters decreases with increase in temperature. The ultrasonic velocity in NpC is greater than that of CmC, so NpC is more suitable material for ultrasonic wave propagation. The magnitude of micro-hardness parameter decreases with increase of atomic mass number. Thus, NpC is stronger compare to CmC. We observed that the highest value of nonlinearity parameter is along h111i and lowest along h110i direction. The magnitude of τth has order in picoseconds which indicate that the actinide monocarbides AnCs (An=Np and Cm) are semimetallic in nature. Ultrasonic attenuation due to phonon–phonon interaction (α/f 2 )p–p and thermo-elastic attenuation (α/f 2 )th has been investigated at room temperature 300 K along h100i, h110i and h111i directions successfully and confirm the validity of Mason’s theory for the NpC and CmC. The results obtained in this paper can be further used for the computation of higher parameters such as edge and screw dislocation damping. These parameters are useful for enhancement of their future performance and further studies. References 1. P. Soni, G. Pagare, M. Rajagopalan and S. P. Sanyal, AIP Conf. Proc. 1447 (2012) 829. 2. L. Petit, A. Svane, Z. Szotek, W. M. Temmerman and G. M. Stocks, Phys. Rev. B 80 (2009) 045124. 3. C. P. Mallett, J. Phys. C: Solid State 15 (1982) 6361. 4. P. Pogany, A. Kavocs, L. Visscher and R. J. M. Konings, J. Chem. Phys. 145 (2016) 244310. 5. C. E. Holley Jr., J. Nucl. Mater. 51 (1974) 36. 6. J. L. Routbort, J. Nucl. Mater. 40 (1971) 17. 7. N. Vigier, C. D. Auwer, C. Fillaux and P. Moisy, Chem. Mater. 20 (2008). 8. M. Tokar, A. W. Nutt and J. A. Leary, Mechanical properties of carbide and nitride reactor fuels LA-4452, UC-80, Reactor Technology, TID-4500 (1970). 9. R. G. Haire, J. Less-Common Met. 121 (1985) 379. 10. K. Brugger, Phys. Rev. 133 (1964) A1611. 11. A. Kumar, D. Singh, R. K. Thakur and R. Kumar, J. Pure Appl. Ultrason. 39 (2017) 1. 12. V. Bhalla, D. Singh and S. K. Jain, Int. J. Thermophys. 37 (2016) 33. 13. D. Singh, V. Bhalla, J. Bala and S. Wadhwa, Z. Naturforsch. A 72 (2017) 977. 14. K. A. Matori, M. H. M. Zaid, H. A. A. Sidek, M. K. Halimah, Z. A. Wahab and M. G. M. Sabri, Int. J. Phys. Sci. 5 (2010) 2212. 15. W. P. Mason, Physical Acoustics (Academic Press, New York, 1965). 16. V. Bhalla, D. Singh, S. K. Jain and R. Kumar, Pramana J. Phys. 86 (2016) 1355. 17. M. Born, Math. Proc. Cambridge 36 (1940) 160. 18. G. A. Slack, Solid State Phys. 34 (1979) 1. 19. D. E. Gray, American Institute of Physics Handbook (McGraw-Hill Book Company, Inc., New York, 1981).

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