Applied Acoustics 72 (2011) 737–741

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Applied Acoustics

journal homepage: www.elsevier .com/locate /apacoust

Propagation of ultrasonic waves in neptunium monochalcogenides

Devraj Singh a,⇑, D.K. Pandey b, D.K. Singh c, R.R. Yadav d

a Department of Applied Sciences, Amity School of Engineering and Technology, Bijwasan, New Delhi 110 061, Indiab Department of Physics, P.P.N. (P.G.) College, Kanpur 208 001, Indiac Department of Physics, Government Inter College, Bangra, Jalaun 285 121, Indiad Department of Physics, University of Allahabad, Allahabad 211 002, India

a r t i c l e i n f o

Article history:Received 4 January 2011Received in revised form 5 April 2011Accepted 6 April 2011Available online 4 May 2011

Keywords:MonochalcogenidesElastic constantsUltrasonic properties

0003-682X/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.apacoust.2011.04.002

⇑ Corresponding author. Tel.: +91 11 2806 2106/14E-mail address: dsingh1@aset.amity.edu (D. Singh

a b s t r a c t

The ultrasonic attenuation and acoustic coupling constants due to phonon–phonon interaction and ther-moelastic relaxation mechanisms have been studied for longitudinal and shear waves in B1 structuredneptunium monochalcogenides NpX (X: S, Se, Te) along h1 0 0i direction in the temperature range100–300 K. The second and third order elastic constants (SOEC and TOEC) of the chosen monochalcoge-nides are also computed for the evaluation of ultrasonic parameters. The ultrasonic attenuation due tophonon–phonon interaction process is predominant over thermoelastic relaxation process in these mate-rials. The ultrasonic attenuation in NpTe has been found lesser than other materials NpS, NpSe and GdY(Y: P, As, Sb and Bi). The semiconducting or semimetallic nature of neptunium monochalcogenides can bewell understood with the study of thermal relaxation time. Total ultrasonic attenuation in these materialsis found to be quadratic function of temperature. The nature of NpTe is very similar to semimetallic GdP.The mechanical and ultrasonic study indicates that NpTe is more reliable, perfect, flawless material.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Ultrasonic characterization of materials is the main theme ofinterest for scientists and engineers for a long time, because ofits wide variety of applications in every field of life. In recent yearsultrasonic attenuation and velocity have been studied in differenttype of materials in different ways [1–5].

Monochalcogenides of rare-earth (ReX, with Re: rare-earth ele-ment and X: S, Se, Te) consist of a large number of materials thatcrystallize in simple B1 structure. ReX exposes interesting electri-cal, optical and magnetic properties. On the basis of resistivity, itcan be understood that NpS and NpSe reveal a semiconductingbehaviour [6], while NpTe associates with semimetallic behaviour[7]. The NpS, NpSe and NpTe are anti-ferromagnetic with a Neeltemperature 23 K, 38 K and 30 K respectively [8]. The study of highpressure behaviour and pressure induced phase transition of NpSeand NpTe using a three body potential approach has been per-formed by Jha and Sanyal [9]. Griveau et al. accounted high pres-sure resistivity measurements at low temperature for NpS andNpSe up to 10 and 25 GPa respectively [10]. Anti-ferromagneticnature in NpX below 23, 36.5 and 45 K has been confirmed by spe-cific heat measurements [11]. Optical reflectivity of neptuniummonochalcogenides under high pressure has been investigated by

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87; fax: +91 11 2806 1821.).

Abraham et al. [12]. Spin polarized electronic structure calculationsby the light binding linear muffin tin orbital (TBLMTO) methodwithin the atomic sphere approximation (ASA) for neptuniumcompounds NpAs and NpTe at ambient pressure and high pressurewere reported in text [13]. Gensini et al. inspected the NpSe trans-formation to a B2 allotrope at 23 GPa [14]. High pressure X-ray dif-fraction studies were performed on some RX compounds (R: Np,Pu; X: Sb, Te) [15]. The monochalcogenides compounds displayan extraordinarily wide range of electronic, magnetic, optical andmagneto-optical properties with potential practical applicationsin the field of spintronics. The assessment of nature, selection ofappropriate material for application, quality control and assurancecan also be well understood with the knowledge of ultrasonicproperties and related parameters under different physicalconditions.

In the present investigation, the theoretical computations ofultrasonic properties and related parameters have been made inneptunium monochalcogenides NpX (X = S, Se and Te) as a functionof temperature along h1 0 0i direction. The obtained results are dis-cussed for the characterization of chosen materials.

2. Theoretical approach

Theory has been categorized in two phases: in the primaryphase, temperature dependence of second- and third-order elasticconstants (SOEC and TOEC) has been discussed, while the atten-uation of ultrasound due to phonon–phonon interaction and

738 D. Singh et al. / Applied Acoustics 72 (2011) 737–741

thermoelastic relaxation mechanisms has been explained insecondary phase.

2.1. Theory for SOEC and TOEC

SOEC and TOEC at absolute zero have been obtained using Cou-lomb and Born–Mayer potential [16] as per Brugger’s definition ofelastic constant [17]. According to the anharmonic theory of latticedynamics, the lattice energy of the single crystal changes with tem-perature [18,19]. Hence an addition of vibrational contribution tothe elastic constants at absolute zero provides SOEC and TOEC atpreferred temperature.

CIJK... ¼ C0IJK... þ CVib:

IJK... ð1Þ

where C0IJK... and CVib:

IJK... represent static and vibrational elastic con-stants respectively. The expressions for C0

IJK... and CVib:IJK... are given

below.

C011 ¼ 3

2e2

r40

Sð2Þ5 þ 1br0

1r0þ 1

b

� �/ðr0Þ þ 2

br0

ffiffi2p

2r0þ 1

b

� �/

ffiffiffi2p

r0

� �C0

12 ¼ C044 ¼ 3

2e2

r40

Sð1;1Þ5 þ 1br0

ffiffi2p

2r0þ 1

b

� �/ð

ffiffiffi2p

r0Þ

9>=>; ð2Þ

C0111¼�15

2e2

r40Sð3Þ7 � 1

b3r2

0þ 3

br0þ 1

b2

� �/ðr0Þ� 1

2b3ffiffi2p

r20þ 6

br0þ 2

ffiffi2p

b2

� �/ð

ffiffiffi2p

r0Þ

C0112¼C0

166¼�152

e2

r40Sð2;1Þ7 � 1

4b3ffiffi2p

r20þ 6

br0þ 2

ffiffi2p

b2

� �/ð

ffiffiffi2p

r0Þ

C0123¼C0

144¼C0456¼�15

2e2

r40Sð1;1;1Þ7

9>>>>=>>>>;ð3Þ

where r0 is the short range parameter; b is hardness parameter;/(r0) is the Born–Mayer potential given by /(r0) = A exp.(�r0/b)and /ð

ffiffiffi2p

r0Þ ¼ A exp :ð�ffiffiffi2p

r0=bÞ; A is the strength parameter given

by A ¼ �3bSð1Þ3e2

r20

6 exp :ð�q0Þ þ 12ffiffiffi2p

exp :ð�q0

ffiffiffi2ph i�1

: q0 = r0/b.

CVib:11 ¼ f ð1;1ÞG2

1 þ f ð2ÞG2

CVib:12 ¼ f ð1;1ÞG2

1 þ f ð2ÞG1;1

CVib:44 ¼ f ð2ÞG1;1

9>>=>>; ð4Þ

CVib:111 ¼ f ð1;1;1ÞG3

1 þ 3f ð2;1ÞG1G2 þ f ð3ÞG3

CVib:112 ¼ f ð1;1;1ÞG3

1 þ f ð2;1ÞG1ð2G1;1 þ G2Þ þ f ð3ÞG2;1

CVib:123 ¼ f ð1;1;1ÞG3

1 þ 3f ð2;1ÞG1G1;1 þ f ð3ÞG1;1;1

CVib:144 ¼ f ð2;1ÞG1G1;1 þ f ð3ÞG1;1;1

CVib:166 ¼ f ð2;1ÞG1G1;1 þ f ð3ÞG2;1

CVib:456 ¼ f ð3ÞG1;1;1

9>>>>>>>>>>=>>>>>>>>>>;

ð5Þ

where f(n) and Gn are given as:

f ð2Þ ¼ f ð3Þ ¼ hx0

8r30

coth x

f ð1;1Þ ¼ f ð2;1Þ ¼ hx0

96r30

hx0

2kTsinh2xþ coth x

� �

f ð1;1;1Þ ¼ hx0

384r30

ðhx0Þ2 coth x

6ðkTÞ2sinh2xþ hx0

2kTsinh2xþ coth x

!

G1 ¼ 2 2þ 2q0 � q20

� �/ðr0Þ þ 2

ffiffiffi2pþ 2q0 � 2q2

0

� �/

ffiffiffi2p

r0

� �n oH

G2 ¼ 2 �6� 6q0 � q20 þ q3

0

� �/ðr0Þ

þ �3ffiffiffi2p� 6q0 �

ffiffiffi2p

q20 þ 2q3

0

� �/

ffiffiffi2p

r0

� �H

G3 ¼ 2 30þ 30q0 þ 9q20 � q3

0 � q40

� �/ðr0Þ

�þ ð15=2Þ

ffiffiffi2pþ 15q0 þ ð9=2Þ

ffiffiffi2p

q20 � q3

0 �ffiffiffi2p

q40Þ/

ffiffiffi2p

r0

� �n o

G1;1 ¼ �3ffiffiffi2p� 6q0 �

ffiffiffi2p

q20 þ 2q3

0

� �/

ffiffiffi2p

r0

� �n oH

G2;1 ¼ðð15=2Þ

ffiffiffi2pþ 15q0 þ ð9=2Þ

ffiffiffi2p

q20 � q3

0 �ffiffiffi2p

q40Þ/

ffiffiffi2p

r0

� �H

G1;1;1 ¼ 0

Here, x = hx0/2KBT, x20 ¼ 1

Mþþ 1

M�

� �1

Hbr0

� �; KB is Boltzmann’s con-

stant, H ¼ q0 � 2ð Þ/ðr0Þ þ 2 q0 �ffiffiffi2p� �

/ffiffiffi2p

r0

� �n o�1and values of

lattice sums are:

Sð1Þ3 ¼ �0:58252; Sð2Þ5 ¼ �1:04622; Sð1;1Þ5 ¼ 0:23185

Sð3Þ7 ¼ �1:36852; Sð2;1Þ7 ¼ 0:16115; Sð1;1;1Þ7 ¼ �0:09045

2.2. Theory for evaluation of ultrasonic attenuation

In the secondary phase, we have developed theoretical ap-proach for finding ultrasonic attenuation in these materials. Themain causes of ultrasonic attenuation for a perfect crystal aredue to phonon–phonon interaction, the thermoelastic relaxationand electron–phonon interaction mechanisms. At the temperaturerange 100–300 K, electron mean free path is not equal to phononmean free path, so no coupling takes place. Thus the ultrasonicattenuation due to electron–phonon interaction is absent. Hencethe ultrasonic attenuation due to phonon–phonon interactionand thermoelastic relaxation mechanisms are governing processesat this temperature range. Both types of attenuation have beenmonitored in these materials. The modified Mason–Bateman The-ory [20] is still widely used theory to study the ultrasonic attenu-ation at higher temperature (�300 K) in solids. It is more reliabletheory to study anharmonicity of the crystals as it involves elasticconstants directly through acoustic coupling constant ‘D’ in theevaluation of ultrasonic attenuation ðAÞ.

The thermoelastic loss and ATh Akhieser loss AAkh (phonon–phonon loss) [20] under condition xs� 1 is given by:

ATh ¼ x2hcjii

2kTh i

=2qV5L ð6Þ

AAkh ¼ E0Dx2s=ð6qV3Þh i

ð7Þ

where q is the density; x is angular frequency of ultrasonic wave; kis the thermal conductivity; E0 is the thermal energy; V is velocity oflongitudinal and shear waves; T is the temperature in Kelvin scale; Land S represent longitudinal and shear waves. Grüneisen number cj

i

(i is the mode of vibration and j is the direction of propagation) isrelated to SOEC and TOEC. The thermal relaxation time (s) for lon-gitudinal wave is twice that of shear wave.

s ¼ 0:5sL ¼ sS ¼ ð3k=CV V2DÞ ð8Þ

where CV is the specific heat per unit volume and VD is the Debyeaverage velocity. The acoustic coupling constant, which is measureof conversion of ultrasonic energy into thermal energy, can be ob-tained by:

D ¼ 9hðcjiÞi

2 � 3hcjii

2qCV Th i

=E0 ð9Þ

3. Results and discussion

The SOEC and TOEC of the chosen monochalcogenides have beenevaluated in temperature range 100–300 K using Eqs. (1)–(5) and

Fig. 1. D vs. temperature.

D. Singh et al. / Applied Acoustics 72 (2011) 737–741 739

taking nearest neighbour distance [21] r0 = 2.780 Å, 2.921 Åand 3.125 Å for NpS, NpSe and NpTe respectively and hardnessparameter [22] b = 0.311 Å for all the three materials. The evalu-ated values of SOEC and TOEC are presented in Table 1.

The specific heat per unit volume (CV) and crystal energy den-sity (E0) are evaluated using AIP Handbook [23]. The thermal con-ductivity of Neptunium monochalcogenides is taken fromliterature [24]. The thermal conductivity (k) and specific heat perunit volume (CV) are used to evaluate thermal relaxation time (s)from Eq. (8). SOEC and TOEC have been used to evaluate Grüneisenparameters along h1 0 0i direction for longitudinal and shear waveusing Mason’s Grüneisen table [20]. The acoustic coupling con-stants (DL and DS) have been computed with Eq. (9) and is shownin Fig. 1.

The ultrasonic attenuation has been evaluated using Eqs. (6)and (7) and are listed in Table 2. For each direction of propagationin solids, there are three types of vibration as one longitudinal andtwo transverse / shear. For the both shear wave, velocities are sameðffiffiffiffiffiffiffiffiffiffiffiffiffiC44=q

pÞ along h1 0 0i direction of propagation in B1 structured

materials. Thus both the shear waves have same attenuation coef-ficient over frequency square. Hence, the total shear loss due tophonon–phonon interaction ðA=f 2ÞS will be sum of both. Obtainedtotal attenuation A ¼ ðA=f 2Þtotal ¼ ðA=f 2ÞL þ 2ðA=f 2ÞS þ ðA=f 2ÞTh

� �in NpX is shown in Fig. 2.

It is clear from the Table 1 that, out of nine elastic constants,four (i.e. C11, C44, C112 and C144) are increasing and four (i.e. C12,C111, C166 and C123) are decreasing with increasing the temperature

Table 1SOEC and TOEC (all in 1010 Nm�2) of NpX.

Materials T (K) C11 C12 C44 C111

NpS 100 5.483 1.495 1.586 �87200 5.622 1.415 1.592 �87300 5.794 1.332 1.598 �88

NpSe 100 4.626 1.215 1.294 �74200 4.772 1.141 1.299 �75300 4.935 1.068 1.304 �76

NpTe 100 4.084 0.884 0.957 �68200 4.231 0.813 0.962 �69300 4.389 0.743 0.966 �70

while C456 is found to be untouched. The increase or decrease instiffness constants is due increase or decrease in atomic interactionwith temperature. If inter-atomic distance increases or decreaseswith temperature then interaction potential decreases/increases,which causes decrease or increase in stiffness constants. This typeof behaviour has been found already in other NaCl-type materialslike lanthanum, praseodymium and thulium monochalcogenides[25–27]. The calculated bulk modulus of NpS, NpSe and NpTe are28.2 GPa, 23.5 GPa and 19.6 GPa respectively. The negative valuesof C111, C112, and C166 are justified with literature [19,25–27]. Theratio |C111|/C11 vary roughly from 15 to 16 for the chosen materials,which is similar to B1 structured materials [28–30]. A direct com-parison has been made for C44 of NpTe with experimental result[24]. The value of C44 for NpTe at 300 K is found equal to be9.6 GPa and 10.6 GPa in present evaluation and literature (experi-mental) respectively. Thus, our calculated higher elastic constantsfor the chosen materials are justified.

The variation of ‘D’ with temperature can be seen from Fig. 1.Ratio of acoustic coupling constant (DL/DS) along h1 0 0i directionlies between 8 and 16, which are comparable with the experimen-tal value of B1 structured material LiF [31]. The thermal relaxationtime (s) for NpS, NpSe and NpTe at 300 K is found 0.82 ps, 1.86 psand 6.02 ps respectively. The order of these s is similar to that ofother semiconductors [28,32] and semimetallics [33]. Thus, it isvery clear that NpS and NpSe materials have semiconductingbehaviour and NpTe having semimetallic behaviour.

It can be seen from Table 2 that the thermoelastic loss is verysmall in comparison to Akhieser loss and ultrasonic attenuationfor longitudinal wave (A/f2)L is greater than that of shear wave(A/f2)S for the chosen materials. This reveals that ultrasonic atten-uation due to phonon–phonon interaction for longitudinal wave ispredominant to total attenuation. This type of characteristic is ob-served in semiconductors [32]. Hence ultrasonic attenuation alsojustifies the semiconducting nature of NpX.

From Eq. (7), it can be found that the ratio of AL and AS is equalto ð2DLV3

S Þ=ðDSV3L Þ. The AL=AS is 3.76, 3.61 and 3.11 for NpS, NpSe

and NpTe respectively along h1 0 0i direction. The AL=AS for Ge andSi are observed 3.14, 3.45 respectively [20]. The present results arein good agreement with experimental results of semiconductors.

The total attenuation at 300 K is found 4.168 � 10�16 Np s2 m�1,11.046 � 10�16 Np s2 m�1 and 39.66 � 10�16 Np s2 m�1 for NpS,NpSe and NpTe respectively (Fig. 2). The total ultrasonic attenua-tion for AlN, GaN and InN at 300 K has been observed 2.291 �10�16 Np s2 m�1, 8.423 � 10�16 Np s2 m�1 and 20.485 � 10�16

Np s2 m�1 respectively [32]. Yet the structure of neptunium mon-ochalcogenides and 3rd group nitrides is different but the totalattenuation is very close to each other. Thus, the chosen materialsmay replace the 3rd group nitrides in future. The total attenuationis found lowest in NpS in comparison to other two monochalcoge-nides; hence its applicability will be most in industry.

At room temperature, the ultrasonic attenuation coefficientover frequency square for longitudinal and shear waves {(A/f2)L

C112 C123 C144 C166 C456

.4 �6.11 2.145 2.641 �6.48 2.620

.9 �5.82 1.671 2.660 �6.50 2.620

.7 �5.51 1.196 2.679 �6.52 2.620

.7 �4.96 1.719 2.170 �5.27 2.153

.3 �4.68 1.285 2.186 �5.29 2.153

.1 �4.40 0.852 2.203 �5.31 2.153

.2 �3.55 1.186 1.654 �3.87 1.641

.5 �3.27 0.962 1.666 �3.88 1.641

.3 �2.98 0.276 1.679 �3.90 1.641

Table 2Ultrasonic attenuation coefficient of the NpX.

Material T(K)

(A/f2)Th

(10�16 Np s2 m�1)(A/f2)L

(10�16 Np s2 m�1)(A/f2)S1 = (A/f2)S2

(10�16 Np s2 m�1)

NpS 100 0.001 0.228 0.053200 0.004 1.135 0.28300 0.008 2.716 0.722

NpSe 100 0.002 0.57 0.134200 0.009 2.852 0.725300 0.019 7.097 1.965

NpTe 100 0.006 1.875 0.506200 0.022 9.149 2.692300 0.05 24.092 7.759

Fig. 2. (A/f2)total vs. temperature.

Table 3Comparative values of ultrasonic attenuation coefficient of the NpTe and GdY at roomtemperature.

Material (A/f2)Th

(10�16 Np s2 m�1)(A/f2)L

(10�16 Np s2 m�1)(A/f2)S1 = (A/f2)S2

(10�16 Np s2 m�1)

NpTe 0.050 24.092 7.759GdP [33] 0.082 30.012 8.467GdAs [33] 0.113 46.312 13.383GdSb [33] 0.285 198.584 101.145GdBi [33] 0.624 350.736 120.695

740 D. Singh et al. / Applied Acoustics 72 (2011) 737–741

and (A/f2)S} of pure germanium have been reported in literature,which are 3.44 � 10�16 Np s2 m�1 and 1.02 � 10�16 Np s2 m�1

respectively along h1 0 0i direction [20]. The total attenuation[(A/f2)L + 2(A/f2)S + (A/f2)Th] of semiconducting materials GaAs[34,35] and InSb [36] are 1.85 � 10�16 Np s2 m�1 (along h1 1 0idirection) and 1.598 � 10�16 Np s2 m�1 (along h1 1 1i direction)respectively. The value of {(A/f2)L and (A/f2)S} for NpS at 300 K isfound equal to be 2.716 � 10�16 Np s2 m�1 and 1.444 �10�16 Np s2 m�1 respectively that are quite similar to that of puregermanium material. Thus, the NpS may replace the material Gein future. Although most of the experimental values for semicon-ducting materials are positioned lower than the obtained resultsfor present semiconducting materials, yet quantum of attenuationis found approximately. We compare semimetallic NpTe at roomtemperature with our previous work on semimetallics GdY (Y: P,

As, Sb and Bi) in Table 3. It depicts that NpTe having less attenua-tion in comparison to GdY, so we can anticipate for NpTe that itwill be more ductile, stable and contain fewer defects in its crystalstructure than others. NpTe will be more suitable for industries aswell as common purposes. It is seen that NpTe and GdP have sim-ilar trend of attenuation, so we can say that these two will be morevaluable in comparison to other semimetallics.

In Fig. 2, it is visualized that the total attenuation (A) increaseswith temperature. The fit curve for total attenuation implies thatthe total attenuation in these materials can be written as functionof temperature as A ¼

Pn¼2n¼0AnTn. The value of An depends on spe-

cific heat per unit volume, energy density, thermal relaxation time,thermal conductivity, elastic constants and density.

4. Conclusions

On the basis of above discussion, the following conclusions canbe drawn.

(1) The theory for evaluation of second- and third-order elasticconstants is also valid for neptunium monochalcogenides.

(2) The study of thermal relaxation time and ultrasonic attenu-ation validate the semiconducting behaviour of NpS andNpSe, while semimetallic behaviour of NpTe.

(3) Total attenuation in these materials follows the expressionA ¼

Pn¼2n¼0AnTn. The value of An depends on specific heat

per unit volume, energy density, thermal relaxation time,thermal conductivity, elastic constants and density.

(4) The total attenuation of the chosen monochalcogenides isfound very close to 3rd group nitrides even though they havedifferent structures.

(5) The nature of semimetallic NpTe is very similar to semime-tallic GdP. Due to less attenuation, NpTe is more reliable,perfect, flawless material.

The obtained results of present investigation may be theoreti-cally related to many other physical properties of solids, such asspecific heat, infrared lattice resonance frequency, X-ray scatteringand various transport properties. These results can be used for fur-ther investigations and industrial purposes.

Acknowledgements

Authors are especially grateful to Dr. A.K. Tiwari (BSNVPGC,Lucknow), Dr. Priyanka Awasthi (DMO, Lucknow), Dr. AkhileshMishra (DST, New Delhi), Dr. A.K. Gupta (NIOS, Noida), Dr. A.K.Yadav (SPMIT, Kaushambi), Dr. Giridhar Mishra (AU, Allahabad),Mr. S.K. Verma (AU, Allahabad), Mr. Meher Waan (AU, Allahabad)and Mrs. Neera Bhutani (ASET) for many useful helps like readingthe manuscript, discussion and knowledgeable suggestions duringthe preparation and revision of the manuscript.

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