Applied Radiation and Isotopes 69 (2011) 756–761

Contents lists available at ScienceDirect

Applied Radiation and Isotopes

0969-80

doi:10.1

n Corr

E-m

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

High energy Compton scattering study of TiC and TiN

Ritu Joshi, K.C. Bhamu, Alpa Dashora, B.L. Ahuja n

Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur 313001, Rajasthan, India

a r t i c l e i n f o

Article history:

Received 31 December 2010

Received in revised form

1 February 2011

Accepted 1 February 2011

Keywords:

Compton scattering

Transition metal compounds

Density functional theory

Fermi surface

43/$ - see front matter & 2011 Elsevier Ltd. A

016/j.apradiso.2011.02.002

esponding author. Tel.: +91 9414317048; fax

ail address: blahuja@yahoo.com (B.L. Ahuja).

a b s t r a c t

We present the experimental Compton profiles of TiC and TiN using 661.65 keV g-ray from 20 Ci 137Cs

source. To explain our experimental data on momentum densities, we have computed the theoretical

profiles, energy bands and density of states using linear combination of atomic orbitals scheme within

the framework of density functional theory. In addition the energy bands, density of states and Fermi

surfaces using full potential linearised augmented plane wave method have also been computed.

Energy bands and density of states obtained from both the theoretical models show metallic character

of TiC and TiN. The anisotropies in Compton line shapes and the Fermi surface topology are discussed in

term of energy bands.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Compton scattering is a well known technique to obtain themomentum density of variety of materials (see, for example,Cooper, 1985; Cooper et al., 2004; Heda and Ahuja, 2010). Comptonprofile, J(pz), which is the projection of electron momentum densityr(p) along the scattering vector (z-axis of a Cartesian coordinatesystem) is defined as

JðpzÞ ¼

Zpx

Zpy

rðpÞdpx dpy ð1Þ

An electron with momentum component pz along the scatter-ing vector shifts the photon energy from o1 (incident) to o2

(scattered). The pz can be related to o1, o2 and scattering angle yusing the following relation:

pz

m0c¼ o2�o1þo1o2ð1�cosyÞ=m0c2� �

=ðo21þo

22�2o1o2 cosyÞ1=2

ð2Þ

Titanium based refractory compounds like TiC and TiN havedefinite technological importance due to their high melting tem-perature, wear resistance and extreme hardness (Zhang, 1993).Due to high melting point they are used in crucibles, nuclearreactors, aircraft shielding etc. In addition to being very hard, theserefractory compounds may also exhibit metallic conductivity.Among earlier theoretical calculations, several authors (Neckelet al., 1976; Ahuja et al., 1996; Delin et al., 1996; Guemmazet al., 1997; Vines et al., 2005; Upadhyaya et al., 2005) havereported electronic band structure calculations and optical proper-ties of these compounds. Mahapatra and Padhi (1983) have

ll rights reserved.

: +91 294 2411950.

reported Compton profile of low purity (99%) TiN sample at poorresolution (0.6 a.u.) using 59.54 keV g-ray. The authors havecompared their experiment with a crude calculation, namely,renormalized free atom model.

The purpose of the present paper is many folds

(a)

To measure accurate Compton profiles of TiN and TiC at anintermediate resolution (0.39 a.u.).

(b)

To compute the Compton profiles of both compounds usinglinear combination of atomic orbitals (LCAO) with densityfunctional theory (DFT) and to compare them with theexperimentally measured momentum densities.

(c)

To derive energy bands, density of states (DOS) and Fermisurface topology using LCAO and full potential linearisedaugmented plane wave (FP-LAPW) calculations.

Accordingly, in the present paper, we report the isotropic Comp-ton profiles of TiC and TiN using our 20 Ci 137Cs g-ray Comptonspectrometer. Due to non-availability of large size single crystals(diameter 15 mm and thickness 2 mm), we could not measure thedirectional Compton profiles. For the band structure calculations, wehave employed the CRYSTAL03 (LCAO) and Wien2k (FP-LAPW)codes. Since Wien2k code does not include the computation of themomentum densities, we have deduced the Compton profiles usingDFT approach as embodied in the LCAO method.

2. Methodology

2.1. Experiment

The Compton measurements were performed using indigen-ous 740 G Bq (20 Ci) 137Cs g-ray Comptometer (Compton

R. Joshi et al. / Applied Radiation and Isotopes 69 (2011) 756–761 757

spectrometer). The details of experimental set-up are alreadypublished by Ahuja et al. (2006) and Heda and Ahuja (2010). Thehigh energy g-rays of 661.65 keV were scattered at an angle of16070.61. High purity (99.99%) polycrystalline samples (pelletsof thickness 3.2 mm and diameter 25 mm) of TiC and TiN wereused in the present measurements. The scattered g-rays wereenergy-analysed using a high purity Ge detector (Canberra, modelGL0210P) and associated electronics like spectroscopy amplifier,analogue to digital converter and Accuspec 4K-channel analyser.An overall momentum-resolution of the spectrometer (Gaussian,full width at half maximum), mainly dominated by detectorresolution, was found to be 0.39 a.u. The calibration of thespectrometer was checked from time-to-time using weak 57Coand 133Ba g-ray source. To deduce the true Compton profiles, theraw data were corrected for background, instrumental resolution,sample absorption, detector efficiency, Compton cross-section,etc. (see Cooper, 1985; Cooper et al., 2004; Timms, 1986). Theeffect of multiple scattering within the sample was eradicatedusing Monte Carlo simulation (Felsteiner et al., 1974). Thecorrected profiles of TiC and TiN were normalised to 12.69 and13.34 e� (in the pz range 0–7 a.u.), respectively, following the freeatom Compton profiles of Biggs et al. (1975).

2.2. Theory

2.2.1. LCAO

To derive theoretical Compton profiles, energy bands and DOS,the LCAO with DFT method (Saunders et al., 2003; Towler et al.,1996) and also the hybridisation of DFT with Hartree–Fock (HF)have been used. According to the DFT formalism, the electronicenergy E is regarded as a functional of the electron density r. Thepresent LCAO package so called CRYSTAL03 facilitates theexchange and correlation effects and treats them approximatelywithin the local density and generalised gradient approximation(LDA and GGA, respectively). The LDA functionals depend only onr(r), while the GGA functionals depend on both r(r) and itsgradient rr(r).

In case of DFT-LDA the exchange potential of Dirac–Slater(Towler et al., 1996) and the correlation potential of Perdew andZunger (1981) have been adopted, while in the DFT-GGA, theexchange and correlation potentials as prescribed by Becke(1988) and Perdew and Wang (1986), respectively, have beenused.

In the hybrid (HF+DFT) Hamiltonian so called B3LYP (Beckethree-parameter hybrid functionals), EXC can be calculated asfollows:

EXC ¼ ELDAX þ0:72DEBECKE

X þ0:20ðEHFX �ELDA

X Þþ0:19EVWNC þ0:81ELYP

C

ð3Þ

Here EHFX , ELDA

X and EBECKEX are the exchange energies of HF,

Dirac–Slater (Saunders et al., 2003) and Becke (Becke, 1988),respectively. ELYP

C and EVMNC represent the correlation energies of

Lee–Yang–Parr (Lee et al., 1988) and Vosko–Wilk–Nusair (Voskoet al., 1980), respectively. Therefore, the B3LYP functionalinvolves exact exchange (through HF) with local and gradient-corrected exchange and correlation terms (through DFT).

Using the LCAO approach, r(r) can be calculated as follows:

rðpÞ ¼Xn,occ

XBZ

k

XG

dp,kþG½wnðk, pÞ�2=N ð4Þ

wn(k,p) is the Fourier transform of wave function derived from theab initio calculations, n is the band index, k is the wave vector, andN is the normalisation constant.

Since the Compton profile has a directional property, thedifference between each pair of directional Compton profilesgives anisotropy in the momentum density.

The self-consistent field calculations were performed at 413(24, 24, 24) k points in the irreducible Brillouin zone (BZ) for bothcompounds. The all electron Gaussian basis sets used for thepresent computations consist of five s, four p and one d-typeshells for Ti (86411/6411/3 set) and three s and two p-type shellsfor N (731/31 set). Three s, two p-type and one d shells (731/31/1set) were used for C. All the basis sets for TiC and TiN have beenenergy optimised up to standard tolerance values using BILLYsoftware (Saunders et al., 2003).

2.2.2. FP-LAPW

To calculate the energy bands, DOS and Fermi surfaces wehave also employed FP-LAPW scheme (Blaha et al., 2001). Thisscheme combines the choice of the LAPW basis sets with thetreatment of full potential and charge density without any shapeapproximation in the interstitial and the muffin-tins (MT) regions.The latest version of gradient-corrected exchange and correlationapproach as suggested by Wu and Cohen (2006) has beenconsidered. The value of RMTKMax (multiplication of MT radiusand largest amplitude of reciprocal lattice vector) was kept equalto 7 for both compounds. In case of TiC, the MT radius for Ti waskept equal to 2.15 A while for C this was 1.91 A. In case of TiN, thevalues of MT radii were 2.11 and 1.87 A for Ti and N, respectively.The k points in the present calculations were about 364 forboth TiC and TiN. The convergence criterion for total energy wasset to 10�5 Ry and maximum radial expansion lmax was keptequal to 10.

In both computational techniques, the lattice parameters forTiC and TiN (rock salt structure) were taken to be a¼4.328 and4.242 A, respectively (Neckel et al., 1976).

3. Results and discussion

3.1. Isotropic compton profiles

In the inset of Fig. 1(a), we have shown the absolute experi-mental Compton profiles of TiC and TiN, normalised to therespective free atom Compton profile areas 12.69 and 13.34 e�

in the momentum range 0–7 a.u. It is seen that near pz¼0, theCompton profile of TiN is higher than that of TiC. To check the roleof 2p electrons in electronic properties of TiC and TiN, in Fig. 1(a),we have compared the equally normalised (EN) isotropic experi-mental Compton profiles of TiC and TiN. It is seen that both the EN(to 13.34 e�) profiles almost overlap, particularly in the lowmomentum (pzo0.5 a.u.) region. A similarity in both EN profilesin low momentum side shows a similar type of behaviour of 2pelectrons of C and N. This is in contradiction to other hardmaterials like TaC and TaN (Fig. 1b), where N-2p electrons inTaN are more localised in momentum space leading to sharperCompton profiles (Dashora, 2010).

The difference between the convoluted spherically averagedtheoretical and the experimental Compton data along withstatistical accuracy (7s) of the experiment are plotted inFig. 2(a) and (b) for TiC and TiN, respectively. It is observed thatin the high momentum region (pzZ4 a.u.), all LCAO calculationsshow a close agreement with the experiment. It is expectedbecause this region mainly consists of contribution from coreelectrons, which are well defined by the free atom profiles. In thevicinity of the Compton peak, different types of exchange andcorrelation energies show almost similar differences. In the lowmomentum region, the differences between theoretical andexperimental Compton profiles may be due to (a) non-relativistic

R. Joshi et al. / Applied Radiation and Isotopes 69 (2011) 756–761758

nature of LCAO calculations, (b) limitations of LDA approximationand (c) quality of basis sets used in LCAO calculations. It is worthmentioning that within LDA approximation, the diagonal elements

2

3

4

5

6

7

TiC (13.34 e-)

TiN (13.34 e-)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

5

6

7

8

9

10

11

pz (a.u.)

J (p

z) (e

/a.u

.)

TaC (30.33 e-)

TaN (30.33 e-)

Fig. 1. (a) Compton profiles of TiC and TiN normalised to the same area (13.34 e� ,

area of free atom Compton profiles of TiN within 0–7 a.u.). In the inset, absolute

experimental Compton profiles of TiC and TiN normalised to individual areas are

shown. (b) Equally normalised (at area of TaN within 0–7 a.u. i.e. 30.33 e�)

experimental Compton profiles of TaC and TaN. 137Cs source based experimental

Compton profiles of TaC and TaN are taken from Dashora (2010).

0-0.4

-0.2

0.0

0.2

0.4

0.6

pz (a.u.)

DFT - LDADFT - GGAB3LYP

Ι ERROR

ΔJ (p

z) (e

/a.u

.)

1 2 3 4 5 6 7

Fig. 2. Difference between the isotropic experimental and convoluted theoretical (s

(a) TiC and (b) TiN. The statistical error corresponds to 7s.

of the occupation number density nnnuð Þ have only the values one orzero. In fact nnnu reads as

nnnuðkÞ ¼ p�1

Z m

�1

ImGnnuðk,EÞdE ð5Þ

where m is chemical potential and Gnnuðk,EÞ is spectral densityfunction (see, for other details, Schulke, 2007). The differencebetween theory and experiment may also be attributed to thenon-inclusion of the Lam–Platzman (LP) correction, which shiftsthe momentum density below the Fermi momentum to above theFermi momentum. Therefore, this correction is expected to reducethe amplitude of theoretical Compton line shapes near pz¼0.

3.2. Energy bands and DOS

In Figs. 3 and 4, we have shown the energy bands and DOSusing FP-LAPW-GGA scheme for TiC and TiN, respectively.The overall shape of the energy bands and DOS computed usingLCAO-DFT schemes is found to be similar to FP-LAPW-GGA;therefore we have not shown the LCAO-DFT based energy bandsand DOS. Except some fine structures, our energy bands and DOSare seen in agreement with the earlier data (Neckel et al., 1976;Ahuja et al., 1996). Due to cross-over of Fermi level (EF) by theenergy bands as can be seen in Figs. 3 and 4, both the LCAO-DFTand FP-LAPW schemes confirm the metallic-like behaviour of TiCand TiN. It is observed that band structures of TiC and TiN closelyresemble each other. The important features of energy bands(Figs. 3 and 4) are summarised below:

(1)

ΔJ (p

z) (e

/a.u

.)

-0

-0

-0

0

0

0

0

pher

The lowest energy band with symmetry G1 (Figs. 3a and 4a)disperses within the energy range �12.39–�8.87 eV (�17.05–�14.57 eV) for TiC (TiN). This band is mainly due to the 2s stateof C(N) atom. Since this band is well below the valence bands, itis inferred that the contribution of C/N 2s electrons is very smallin deciding the electronic properties of both of the hardmaterials.

(2)

In case of TiC (Fig. 3a), the next three bands (6th, 7th and 8th)are degenerated at Gu

25 (0.66 eV) after crossing the EF while inTiN (Fig. 4a) these bands are degenerated at G15 (–2.03 eV)which lies below EF. These bands mainly originate from the 2pstates of C/N and 3d states of Ti.

pz (a.u.)

.3

.2

.1

.0

.1

.2

.3 DFT - LDADFT - GGAB3LYP

Ι ERROR

0 1 2 3 4 5 6 7

ically averaged) Compton profiles using various schemes of LCAO-DFT for

Fig. 3. Energy bands and DOS of TiC along high symmetry directions of the first Brillouin zone calculated using FP-LAPW method. Here X (1/2, 0, 1/2), L (1/2, 1/2, 1/2),

G (0, 0, 0), W (1/2, 1/4, 3/4), K (3/8, 3/8, 3/4) are the featured k points in the BZ.

Fig. 4. Same as Fig. 3 except for the sample, which is TiN.

R. Joshi et al. / Applied Radiation and Isotopes 69 (2011) 756–761 759

(3)

In both the hard materials, the degeneracy at G12 mainlyconsists of contribution from the d states of Ti atoms withsmall contribution of 2p states of C/N atom.

(4)

The value of DOS at EF of TiN (0.80 states/eV) is higher thanthat of TiC (0.30 states/eV).

3.3. Anisotropy in Compton profiles

The theoretical anisotropies (unconvoluted) in the Comptonprofiles of TiC and TiN are shown in Fig. 5(a) and (b), respectively.Now we explain the general trend of oscillations in the aniso-tropies in terms of degenerate states and cross-overs in theenergy bands. For example, oscillations in J111� J110 can beexplained on the basis of degenerate states in G�L [1 1 1] andG–X [1 1 0] branches. In case of G–X [1 1 0] branch of TiC,degenerate states at G point and cross-over of bands at EF are

responsible for high momentum density in this direction incomparison to G-L branch. It leads to the negative amplitude ofJ111–J110 between 0 and 0.7 a.u. (within G–X distance). In a similarway, we can also explain other positive and negative oscillationsin different anisotropies.

3.4. Fermi surface

Since TiC and TiN show a metallic character, the mapping of FSthat arises from electron ordering phenomena is quite interesting.The FP-LAPW based FS topology of TiC and TiN are shown inFigs. 6 and 7, respectively. In case of TiC (TiN) there are 4 (3) bandsthat cross EF. These bands are also marked in Figs. 3 and 4.

In Fig. 6(a), the standard BZ for cubic (rock salt) structure isshown. The lower three bands 6–8th in TiC (Fig. 3a), which aredegenerated at Gu

25 lead to deformed structures. The shapes ofdeformed spherical structures centred at G point (Fig. 6b–d) are

-0.2

0.0

0.2

0.4

-0.15-0.10-0.050.000.050.100.15

ΔJ (p

z) (i

n e/

a.u.

)

ΔJ (p

z) (i

n e/

a.u.

)-0.2

0.0

0.2

0.4

0.6

DFT-GGADFT-LDAB3LYP

-0.2

0.0

0.2

pz (in a.u.)pz (in a.u.)

-0.08

-0.04

0.00

0.04

0.08

0-0.4

-0.2

0.0

0.2J111-J100

J110-J100

J111-J100

J111-J100

J110-J100

J111-J100

DFT-GGADFT-LDAB3LYP

1 2 3 4 5 6 70 1 2 3 4 5 6 7

Fig. 5. Anisotropy in the unconvoluted theoretical directional Compton profiles of (a) TiC and (b) TiN, derived from different schemes of LCAO-DFT. Solid lines are drawn to

guide eyes.

Fig. 6. (a) Standard structure of BZ for rock salt structure (space group fm3m).

FS of TiC arising from (b) 6th, (c) 7th, (d) 8th and (e) 9th bands using the

FP-LAPW scheme. In (f) the overall mapping of FS as contributed by all the bands

are shown.

Fig. 7. FS of TiN arising from (a) 9th, (b) 10th and (c) 11th bands using the

FP-LAPW scheme. Complete mapping of FS contributed by the 9th, 10th and 11th

bands (a+b+c) is shown in part (d).

R. Joshi et al. / Applied Radiation and Isotopes 69 (2011) 756–761760

found to be identical, except for a difference in their size. The 9thband, which passes through G15 point, is found to be dispersedwithin the energy range �0.42 �2.92 eV. This band leads toelongated egg like electron pockets between G and X points(Fig. 6e). In Fig. 6(f), an overall FS structure arising from 6th to9th bands of TiC is also shown.

In case of FS of TiN, a large multiply connected ‘‘jungle-gym’’structure that touches the surface of the BZ around the X point (and

R. Joshi et al. / Applied Radiation and Isotopes 69 (2011) 756–761 761

extends towards W) arises due to the 9th band within the energyrange –3.10–1.70 eV (Fig. 7a). The 10th (–0.63–2.28 eV) and 11th(–0.63–2.44 eV) bands give small structures at the G point (Fig. 7band c). A total mapping of FS due to 9–11th bands is shownin Fig. 7(d).

4. Conclusions

Experimental Compton profiles of TiC and TiN are measured at anintermediate resolution of 0.39 a.u. The experimental profiles arecompared with the theoretically computed Compton profiles usingDFT schemes of LCAO approach. It is seen that various approxima-tions within LCAO-DFT show almost a similar agreement with theexperimental Compton data. In both the compounds, the behaviourof 2p electrons of C and N seems to be almost similar. The energybands and density of states indicate a metallic character of TiC andTiN, showing their utility as high melting point conducting materials.An analysis of energy bands shows that the valence band mainlyconsists of overlapped Ti-3d and C- or N-2p states. It is interesting tonote that overall mapping of Fermi surface in TiC and TiN differssignificantly, although overall trend of energy bands looks similar. Todiscuss more about the Fermi surface topology, high resolutiondirectional Compton profiles may be helpful.

Acknowledgements

The authors would like to thank Prof. R. Dovesi and CRYSTALsupport team for providing the CRYSTAL03 code. The authors arealso grateful to Prof. P. Blaha for permitting the use of WIEN2kcode. We are grateful to UGC, New Delhi, for financial support.

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