Ultrasonic wave propagation in ternary intermetallic CeCuGe compound

Accepted Manuscript Ultrasonic Wave Propagation in Ternary Intermetallic CeCuGe Compound Aashit Kumar Jaiswal, Pramod Kumar Yadawa, R.R. Yadav PII: DOI: Reference:

S0041-624X(17)30357-8 https://doi.org/10.1016/j.ultras.2018.04.009 ULTRAS 5738

To appear in:

Ultrasonics

Received Date: Revised Date: Accepted Date:

9 May 2017 26 February 2018 22 April 2018

Please cite this article as: A. Kumar Jaiswal, P. Kumar Yadawa, R.R. Yadav, Ultrasonic Wave Propagation in Ternary Intermetallic CeCuGe Compound, Ultrasonics (2018), doi: https://doi.org/10.1016/j.ultras.2018.04.009

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Ultrasonic Wave Propagation in Ternary Intermetallic CeCuGe Compound Aashit Kumar Jaiswal*1, Pramod Kumar Yadawa2, and R.R. Yadav1 1

Department of Physics, University of Allahabad, Allahabad-211002, U.P., India

2

Department of Applied Physics, AMITY School of Engineering and Technology, New Delhi-110 061, India ‫٭‬Email address: [email protected]

Abstract The elastic and ultrasonic properties of the hexagonal intermetallic CeCuGe ternary compound have been evaluated along unique axis at room temperature. Higher order elastic constants have been calculated for CeCuGe compound using Lennard–Jones interaction potential model. The ultrasonic velocity, Debye average velocity, thermal relaxation time and acoustic coupling constant are evaluated along unique axis of the crystal and the temperature dependent ultrasonic velocities are evaluated along different angles with unique axis of the crystal. Finally temperature dependent ultrasonic attenuation is evaluated to extract important information about the material. Keywords: Elastic properties; Thermal conductivity; Ultrasonic velocity; Ultrasonic attenuation; Intermetallic.

1. Introduction Over the last few decades, ternary CeCuGe intermetallic compound has been extensively investigated for the magnetic, electrical and many other properties [1-3]. CeCuGe has the hexagonal AlB2-type structure and have showed ferromagnetic behaviour at Curie temperature, TC = 10 K [2]. A distinguished behaviour of thermal conductivity of CeCuGe at 30 K has been reported by Y. Oner et.al. [1]. However, experimental or theoretical study on ultrasonic propagation behaviour in CeCuGe compound has not been reported in the literature. Study of ultrasonic attenuation at different temperatures along with the other associated properties are prerequisite for any material characterization scheme to extract the important thermophysical properties based on the structure of the material and other physical conditions like temperature [4-8]. Therefore, we have made the theoretical investigations of ultrasonic attenuation, ultrasonic velocity and other associated parameters like thermal relaxation time in the temperature range 50-300 K. Study is intended to know the effect of thermal conductivity and microstructural phenomena on ultrasonic properties to establish the important correlations for the material characterization. For the evaluations of ultrasonic properties we have calculated the nonlinear elastic constants using very simple interaction potential model.

2. Theory In the present investigation, the theory for calculations is divided into three parts: 2.1 Second-and third order elastic constants The second (CIJ) and third (CIJK) order elastic constants of material are defined by following expressions. C IJ 

 2U ; e I e J

I or J  1,......6

(1)

C IJK 

 3U ; I or J or K  1,......6 e I e J e K

(2)

where, U is elastic energy density , eI=eij (i or j = x, y, z, I=1, …6) is component of strain tensor. Eqs (1) and (2) leads six second and ten third order elastic constants (SOEC and TOEC) for the hexagonal structure materials [9,10]. C11  24.1 p 4 C  C13  1.925 p 6 C  C 44  2.309 p C  4

C12  5.918 p 4 C    C 33  3.464 p 8 C   C66  9.851 p 4 C  

C112  19.168 p 2 B  1.61 p 4 C    C113  1.924 p 4 B  1.155 p 6 C  C123  1.617 p 4 B  1.155 p 6 C   C133  3.695 p 6 B C155  1.539 p 4 B   4 6 C144  2.309 p B C 344  3.464 p B  2 4 8  C 222  101.039 p B  9.007 p C  C 333  5.196 p B 

(3a)

C111  126.9 p 2 B  8.853 p 4 C 

(3b)

where p = c/a: axial ratio; C    a / p 5 ; B   a 3 / p 3 ;   (1 / 8)[{nb 0 (n  m)}/{ a n 4 }]    /{6 a 2 (m  n  6)} ; m, n=integer quantity; b0=Lennard Jones parameter.

2.2 Ultrasonic velocities The anisotropic behaviour of the material can be understood with the knowledge of ultrasonic velocity because the velocity is related to the second order elastic constants [11]. On the basis of mode of atomic vibration, there are three types of velocities (longitudinal, quasi shear and shear) in acoustical region [12]. These velocities vary with the direction of propagation of wave from the unique axis of hexagonal structured crystal. The ultrasonic velocities as a function of angle between direction of propagation and unique axis for hexagonal structured materials are [11,13]: VL2  {C33 cos 2   C11 sin 2   C44  {[C11 sin 2   C33 cos 2   C44 (cos 2   sin 2  )]2

 4 cos 2 sin 2 (C13  C44 ) 2 }1 / 2 }/2  VS21  {C33 cos 2   C11 sin 2   C44  {[C11 sin 2   C33 cos 2   C44 (cos 2   sin 2  )]2

(4)

 4 cos 2 sin 2  (C13  C44 ) 2 }1 / 2 }/2 

(5)

VS22  {C44 cos 2   C66 sin 2  } / 

(6)

where VL, VS1 and VS2 are longitudinal, quasi shear and pure shear wave ultrasonic velocities. Variables  and  represent the density of the material and angle with the unique axis of the crystal respectively. The Debye temperature (TD) is an important physical parameter for the characterization of materials, which is well related to the Debye average velocity (VD) as:

TD 

 VD (6  2 n a )1/3 kB

(7) -1/3

 1  1 1 1   VD    3  3  3    3  VL VS1 VS 2  

here

(8)

where  is Planck’s constant divided by 2 ; kB is Boltzmann Constant; na is atom concentration. The above formulae have been used for the evaluation of ultrasonic velocity and related parameters for our compound. 2.3 Ultrasonic attenuation and allied parameters The predominant causes for the ultrasonic attenuation in a solid at room temperature are phonon-phonon interaction (Akhieser loss) and thermoelastic relaxation mechanisms. The ultrasonic attenuation coefficient due to phonon-phonon interaction (α)Akh and thermoelastic relaxation mechanisms (α)Th is given by the following expression [11]:





(α /f2)Akh  4 2 3E 0  ( ij ) 2     ij  2 C V T  / 2  V 3 (α /f2)Th  4 2   ij  2 kT / 2VL5

(9) (10)

where, f: frequency of the ultrasonic wave;  : density of the material V: ultrasonic velocity for longitudinal and shear wave; VL: longitudinal ultrasonic velocity; E 0: thermal energy density;  i j : Grüneisen number (i, j are the mode and direction of propagation) [14-16].

The Grüneisen number for hexagonal structured crystal along <001> orientation or =0ᵒ

is

direct

consequence



of

second

and

third

order

elastic

constants.



D  3 3E 0  ( ij ) 2     ij  2 C V T / E 0 is known as acoustic coupling constant, which is the

measure of acoustic energy converted to thermal energy. When the ultrasonic wave propagates through crystalline material, the equilibrium of phonon distribution is disturbed. The time for re-establishment of equilibrium of the thermal phonon distribution is called thermal relaxation time () and is given by following expression:   S   L / 2  3 k / C V VD2

(11)

Here L and S are the thermal relaxation time for longitudinal and shear wave. k and CV are the thermal conductivity and specific heat per unit volume of the material respectively [17]. Table 1. Second and third order elastic constants (x10 11 N m-2) of CeCuGe compound at ambient 300 K T (K)

C11

C12

C13

C33

C44

C66

300

4.603

1.13

0.311

0.472

0.373

1.805

T (K)

C111

C112

C113

C123

C133

C344

C144

C155

C222

C333

300

-75.064

-11.901

-0.797

-1.012

-1.594

-1.495

-1.179

-0.786

-59.393

-1.894

3. Results and discussion The unit cell parameters ‘a’ (basal plane parameter) and ‘p’ (axial ratio) for CeCuGe compound are 4.316 Å and 0.9191, respectively [1]. The value of m and n for CeCuGe are taken 6 and 7. The value of b0 for CeCuGe is 4.5x 10-84 J m7. SOECs and TOECs at ambient temperature (= 300 K) have been calculated using Eq. (3) and are presented in Table-1.

The elastic constants of the material are important, since they are related to hardness and, therefore, are of interest in applications where mechanical strength and durability are important. Also, the second order elastic constants are used for the determination of the ultrasonic attenuation and related parameters. The comparison can be made with the value of Debye temperature of CeCuGe compound. Y. Oner et.al. determined the Debye temperature of CeCuGe as 240 K [1]. We have calculated the Debye temperature for CeCuGe using Eq. (7) as 243 K. Here, the Debye temperature has been evaluated with the ultrasonic velocities and the ultrasonic velocities are dependent on second order elastic constants. Thus our theoretical approach for the calculation of second order elastic constants for hexagonal structured compound at ambient temperature is well justified. However, third order elastic constants are not compared due to lack of data in the literature but the negative third order elastic constants are found in our previous papers for hexagonal structure materials [9,10]. Hence applied theory for the evaluation of higher order elastic constants at room temperature is justified. The temperature dependent density and thermal conductivity have been taken from the literature [1]. The value of CV and E0 are evaluated using tables of physical constants and Debye temperature. The Longitudinal, shear and Debye Average Velocities of ultrasonic wave propagating along the unique axis of the CeCuGe are calculated using Eqs. (4-6,8) and are shown in Table 2. Specific heat per unit volume ‘C V’ and thermal energy density ‘E0’ are determined with the help of VD and represented in Table 2. The value of temperature dependent density ‘’ and calculated acoustic coupling constants ‘DL’ & ‘DS’ are also presented in Table 2. The computed orientation dependent ultrasonic wave velocities and Debye average velocities at different temperatures are shown in Figs 1–4.

Table 2. Density (: in 103 kg m-3), specific heat per unit volume (C V: in 105Jm-3 K-1), thermal energy density (E0: in 106Jm-3), thermal conductivity (k: in Wm-1K-1), ultrasonic velocities (VL, VS1, VS2) along =0º, and acoustic coupling constant (DL, DS) of CeCuGe compound. T



CV

E0

k

VL

VS (=VS1=VS2) −1

2

(K)

2

(×10 m s )

VD

−1

(×10 m s )

DL

DS

−1

2

(×10 m s )

50

7.34 2.62

4.27

11.3

25.36

22.53

23.05

17.01 21.74

100

7.33 5.15

25.6

10.26

25.38

22.54

23.06

17.02 21.74

150

7.32 5.96

52.9

10.43

25.40

22.56

23.08

17.03 21.74

200

7.31 6.28

83.6

10.78

25.41

22.58

23.09

17.03 21.74

250

7.3

117

10.87

25.43

22.59

23.11

17.03 21.74

300

7.29 6.52 147.5

10.96

25.45

22.61

23.12

17.03 21.74

50 K 100 K 150 K 200 K 250 K 300 K

80

70

VL ( x 102 m/s)

6.44

60

50

40

30

20 0

20

40

60

80

Angle (degree Figure 1: VL versus angle with unique axis of crystal

100

22.65

VS1 (x102 m /s)

22.60 22.55 22.50 22.45

50 K 100 K 150 K 200 K 250 K 300 K

22.40 22.35 22.30 22.25 0

20

40

60

80

100

Angle (degree Figure 2: VS1 versus angle with unique axis of crystal

50

VS2

2 ( x 10 m/s)

45 40 35

50 K 100 K 150 K 200 K 250 K 300 K

30 25 20 0

20

40

60

80

Angle (degree Figure 3: VS2 versus angle with unique axis of crystal

100

32

28

2

VD (x10 m/s)

30

at 50 K at 100 K at 150 K at 200 K at 250 K at 300 K

26

24

22

0

20

40

60

80

100

Angle (degree Figure 4: VD versus angle with unique axis of crystal The plots of thermal conductivity and the calculated thermal relaxation time are shown in Fig. 5(a) & 5(b), respectively. The angle dependent thermal relaxation time curves (Fig. 6) follow the reciprocal nature of VD as   3K / C V VD . This implies that ‘’ for chosen 2

compound is mainly affected by the thermal conductivity. The thermal relaxation time () for hexagonal structured materials is the order of pico second as in case of heavy rare-earth metals studied by P.K Yadawa and co-workers [18]. Hence the calculated thermal relaxation time justifies the hexagonal structure of chosen compound. In the evaluation of ultrasonic attenuation, it is supposed that wave is propagating along the unique axis of CeCuGe compound. The attenuation coefficient over frequency square (α /f2)Akh due to phonon-phonon interaction for longitudinal (α /f2)L and shear wave (α /f2)S are calculated using Eq. (9) under the condition <<1 at different temperature. The thermoelastic loss over frequency square (α /f2)Th is calculated with the Eq. (10). The values of temperature dependent (α /f2)L, (α /f2)S, (α /f2)Th and total attenuation (α /f2)Total are presented in Fig. 7.

Thermal conductivity (W/mK)

11.4 11.2 11.0 10.8 10.6 10.4 10.2 50

100

150

200

250

300

T (K) Figure 5(a): Thermal conductivity versus temperature

26

Relaxation time (ps)

24 22 20 18 16 14 12 10 8

50

100

150

200

250

T (K) Figure 5(b): Relaxation time versus temperature

300

Thermal relaxation time (ps)

26

at 50 K at 100 K at 150 K at 200 K at 250 K at 300 K

24 22 20 18 16 14 12 10 8 6 4

0

20

40

60

80

100

Angle (degree Figure 6: Relaxation time versus angle with unique axis of crystal

2

(/f )Th

/f2 (x10-15 Np s2 /m)

10

2

9

(/f )L

8

(/f )S

7

(/f )Total

2 2

6 5 4 3 2 1 0 50

100

150

200

250

300

T (K)

Figure 7: Ultrasonic attenuation coefficients over frequency square versus angle with unique axis of crystal

Table 3. Ultrasonic attenuation coefficients over frequency square (α/f2) (Np s2/m) in the temperare range 50-300 K T

(α/f2)Th

(α/f2)L

(α/f2)S

(α/f2)Total

(K)

(x 10-19)

(x 10-15)

(x 10-15)

(x 10-15)

50

0.525

0.3888

0.3545

0.7434

100

0.951

1.0763

0.9807

2.0571

150

1.45

1.9467

1.7734

3.7202

200

1.99

3.0120

2.7436

5.7558

250

2.50

4.1177

3.7504

7.8684

300

3.02

5.1816

4.7192

9.9011

When the ultrasonic wave propagates in the crystal, thermoelastic loss and Akhieser loss for longitudinal and shear wave increases with temperature (Table 3). Thus the total ultrasonic attenuation increases with temperature. Table 3 indicates that thermoelastic loss is very small in comparison to Akhieser type of loss. Although the thermal conductivity at 50 K is high and minimum at 100 K but the ultrasonic attenuation is minimum at 50 K and then it increases with temperature. It reveals that the thermal conduction does not very much affect ultrasonic attenuation. Further, at the temperature lower than 100 K phonon–phonon interaction goes down, thus minimum attenuation at 50 K confirm this fact. Electron-phonon interaction is prominent at the temperature lower than 100 K. Thus we can conclude that the phonon-phonon interaction (interaction of lattice phonon with acoustical phonon) is relevant and prominent mechanism for the temperature dependent ultrasonic attenuation at higher temperatures (>100 K). Thus the microstructure of the intermetallic is important for the ultrasonic propagation behaviour.

4. Conclusions In summary, the elastic and ultrasonic properties of the hexagonal intermetallic CeCuGe ternary compound have been evaluated along unique axis at room temperature. The

ultrasonic velocity, Debye average velocity, thermal relaxation time and acoustic coupling constant are evaluated along unique axis of the crystal. Anomalous behavior of orientation dependent ultrasonic velocities in CeCuGe is observed due to combined effect of second order elastic constants. Our theory based on simple interaction potential model for calculation of higher order elastic constants is justified for the hexagonal structured ternary compound. The order of thermal relaxation time for CeCuGe is found in picoseconds, which justifies their hexagonal structure. Discussion of calculated parameters provides a detailed understanding about structural information, mechanism responsible for loss, factor affecting to the ultrasonic loss and nature of the material. Finally temperature dependent ultrasonic attenuation is evaluated to extract important information about the material. The phononphonon interaction mechanism is the major dominating cause for ultrasonic attenuation at higher temperature in the CeCuGe compound. The present study along with known properties of CeCuGe opens a new dimension for further research. The investigated significant data can be utilized in online characterization of the materials in applications.

5. Acknowledgement Aashit Kumar Jaiswal acknowledges the financial support provided by the University Grants Commission, India.

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Research highlights  The elastic and ultrasonic properties of the hexagonal intermetallic CeCuGe ternary compound have been evaluated at room temperature.  Temperature dependent ultrasonic attenuation is evaluated to extract important information about the material.  Anomalous behavior of orientation dependent ultrasonic velocities in CeCuGe is observed.