CHAPTER - III. metals Ti(Titanium), Zr(Zirconium) and Hf(Hafnium) and its alloys are widely used

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1 CHAPTER - III ELECTRONIC BAND STRUCTURE, DENSITY OF STATES, PHASE TRANSITION AND SUPER CONDUCTIVITY OF Ti(TITANIUM), Zr(ZIRCONIUM) AND Hf(HAFNIUM) UNDER HIGH PRESSURE 3.1 PREFACE Among the various transition metals and its alloys, group IVB transition metals Ti(Titanium), Zr(Zirconium) and Hf(Hafnium) and its alloys are widely used as high temperature structural engineering materials in the aerospace industry [20]. At ambient conditions Titanium (Z=22), Zirconium (Z=40) and Hafnium(Z= 72) crystallize in a hcp structure [28]. When the temperature is raised above 1155K, its crystal structure changes to bcc. The band structure of Ti, Zr and Hf in hcp structure was first obtained by Hugh et al [29] and then by Jepsen[30] using LMTO method. Later Bakonyi et al [31] calculated the electronic structure and superconducting transition temperature T c for Ti, Zr and Hf using LMTO method. There are many investigations both theoretical and experimental on phase transitions of Ti, Zr and Hf under pressure[32]. Recently, the pressure induced structural transformation in Ti, Zr and Hf has received extensive experimental and theoretical attention. [33]. However the investigation limited only to the two phases (hcp and bcc ) of Zr and Hf. Here hcp is the normal pressure phase and bcc is the high temperature and high pressure phase[34]. The present investigation is aimed at getting detailed information about the high pressure band structure (bcc - Ti, Zr and 44

2 Hf), density of states and pressure variation of superconducting transition temperature [12]. We have used the full potential linearized muffin tin orbital method (FP-LMTO) and the details of the calculation are given in Chapter.1 and here we will give the calculation details. The atomic number of Ti, Zr and Hf are 22, 40 and 72 respectively. The electronic configuration of Ti, Zr and Hf are 18 [Ar] 4s 2 3d 2, 36 [Kr] 5s 2 4d 2 and 54 [Xe] 4f 14 6s 2 5d 2 respectively. The valence electronic configuration chosen in our calculations is 4s 2 3d 2 for Ti, 5s 2 4d 2 for Zr and 6s 2 5d 2 for Hf. The calculated total energies of Ti, Zr and Hf were fitted to Murnaghan s equation of state and the determined pressure values [17]. 3.2 REASSESS OF LITERATURE At ambient conditions titanium (Z=22) crystallizes in a hcp structure ( phase) [35]. Similarly zirconium and Hafnium also crystallizes in a hcp structure ( phase)[36]. When the temperature is raised above 1155K, their crystal structure changes to bcc ( phase)[37]. The band structure of Ti, Zr and Hf in hcp structure was first obtained by Hugh et al [29] and then by Jepsen [30] using LMTO method. Later Bakonyi et al [31] calculated the electronic structure and superconducting transition temperature Tc for hcp-ti and high temperature bcc-ti using LMTO method. There are many investigations both theoretical and experimental on phase transitions of Ti, Zr and Hf under pressure [6]. Xia et al [38] observed a (hcp) (hexagonal) transition in Ti above 0.08 Mbar and found the phase to be stable up to 0.87 Mbar[39]. On the theoretical side, the transition of Ti, Zr and Hf was predicted, using LMTO method, by Gyanchandani et al [13] and then by Ahuja et al [40]. Similarly Ostanin et al [41] predicted the formation of high pressure (bcc) 45

3 phase in Ti at 0.95 Mbar. Recently, Vohra et al [42] adopting X-ray diffraction study, observed a transition from phase to a new (distorted hcp) phase at 1.16 Mbar. Akahama et al [20], in their experimental study on Ti, found that the phase was transformed into an orthorhombic phase with a distorted bcc structure via an intermediate phase. This high pressure (distorted bcc) phase was stable up to 2.16 Mbar. Joshi et al [43] and Mehl et al [44] presented the structural phase stability of Ti, Zr and Hf as a function of pressure obtained using FP LAPW and TB LAPW methods respectively. In that Joshi et al [43] predicted that under pressure Ti, Zr and Hf transforms from phase to phase via an intermediate phase and the phase is unstable. Recently, Kutepov et al [45] have reproduced the experimental trends of a phase sequence: (hcp) (hexagonal) (distorted hcp) (distorted bcc) (bcc) with pressure for Ti, Zr and Hf by considering full geometry optimization. Theoretically calculated pressures corresponding to transition [46] (1.02 Mbar) and transition [45] (1.36 Mbar) are very much lower than the experimental value of ( 2.16 Mbar ) Akahama et al [20]. Regarding the band structure and superconductivity at high pressure no work has been reported for Ti, Zr and Hf. The present investigation is aimed at getting detailed information about the high pressure band structures, pressure induced phase transitions and pressure variation of superconducting transition temperature and its relation to band structure and a revisit to the normal pressure properties. 3.3 CALCULATIONAL PROCEDURE The FP LMTO method correctly predicted the ground state and high pressure band structures and total energies of all the transition and noble metals including hcp 46

4 metals [27]. The total energies of Ti, Zr and Hf in hcp,,,, bcc and fcc structures as a function of reduced volume and compared the total energies of the above six crystal structures have computed [47]. The hexagonal phase of Ti, Zr and Hf is a fairly close packed one as evidenced by its Ewald constant M = , and hence it could be satisfactorily treated by FP LMTO method [44]. The Brillouin zone integrations are performed for 145 k points for both bcc and fcc structures, 150 k points for hcp structure, 270 k points for structure, 396 k points for structure and 341 k points for structure of Ti, Zr and Hf within the irreducible wedge of the Brillouin zone. In atomic sphere approximation (ASA), the crystal is divided into space filled spheres and therefore with slightly overlapping spheres centered on each of the atomic sites. If S1 and S2 are the neighboring atoms sphere-radii, and D is the distance between their centers, the ratio [(S1+S2-D)/Minimum (S1,S2)] is the measure of overlap, and the upper limit of this ratio for a reliable calculation is 0.3. A simple estimate of the structures taken up by us gives the values of this ratio as 0.34 for - phase, 0.37 for - phase, and 0.39 for - phase. With these large overlaps, exceeded the upper limit of 0.30 substantially, these structure calculations are not at all very reliable. But in order to compare the structural stability of Ti, Zr and Hf, these structures are taken in addition to hcp and bcc structures. The atomic number of Ti, Zr and Hf are 22, 40 and 72 respectively. The electronic configuration of Ti, Zr and Hf are 18 [Ar] 4s 2 3d 2, 36 [Kr] 5s 2 4d 2 and 54 [Xe] 4f 14 6s 2 5d 2 respectively. The valence electronic configuration chosen in our calculations is 4s 2 3d 2 for Ti, 5s 2 4d 2 for Zr and 6s 2 5d 2 for Hf. The calculated total energies of Ti, Zr and Hf were fitted to Murnaghan s equation of state [17] and the determined pressure values. 47

5 3.4 PHYSICAL, CHEMICAL PROPERTIES AND APPLICATIONS OF Ti (TITANIUM), Zr (ZIRCONIUM) AND Hf (HAFNIUM) Table Physical and Chemical properties of Ti (Titanium), Zr Zirconin and Hf(Hafnium) Physical and Chemical properties Element Titanium Zirconium Hafnium Phase solid solid solid Density (near r.t.) g cm 6.52 g cm g cm Liquid density atm.p g cm 5.8 g cm 12 g cm Melting point 1941 K3034 F 1668 C, Boiling point 5949 F 3287 C, 3560 K, 2128 K3371 F 1855 C, 7911 F 4377 C, 4650 K K, 2506 K4051 F 2233 C, Heat of fusion kj mol 14 kj mol 27.2 kj mol 8317 F 4603 C, 4876 K, Heat of vaporization 425 kj mol 591 kj mol 648 kj mol Molar heat capacity J mol K J mol K J mol K Crystal structure hexagonal closepacked hexagonal closepacked hexagonal close-packed Magnetic ordering paramagnetic paramagnetic [2] paramagnetic [1] Electrical resistivity (20 (20 (20 Thermal conductivity 21.9 W m K 22.6 W m K 23.0 W m K Thermal expansion (25 C) 8.6 µm m K (25 C) 5.7 µm m K (25 C) 5.9 µm m K Speed of sound(thin rod) (r.t.) 5,090 m s (20 C) 3800 m s (20 C) 3010 m s Young's modulus 116 GPa 88 GPa 78 GPa Shear modulus 44 GPa 33 GPa 30 GPa Bulk modulus 110 GPa 91.1 GPa 110 GPa Poisson ratio Mohs hardness Vickers hardness 970 MPa 903 MPa 1760 MPa Brinell hardness 716 MPa 650 MPa 1700 MPa Element colour silvery grey-white metallic silvery white steel gray 48

6 3.4.2 Applications of Ti (Titanium,), Zr(Zirconium) and Hf (Hafnium) Titanium is a lustrous transition metal with a silver colour, low density and high strength [48]. It is highly resistant to corrosion in sea water, aqua regia and chlorine. The element occurs within a number of mineral deposits, principally rutile and ilmenite, which are widely distributed in the earth crust and lithosphere, and it is found in almost all living things, rocks, water bodies, and soils[49]. The metal is extracted from its principal mineral ores via the Kroll process or the Hunter process[50]. Its most common compound, Titanium Dioxide, is a popular photocatalyst and is used in the manufacture of white pigments. [4] Other compounds include Titanium tetrachloride (TiCl 4 ), a component of smoke screens and catalysts; and Titanium trichloride (TiCl 3 ), which is used as a catalyst in the production of polypropylene[51]. Titanium can be alloyed with iron, Aluminium, Vanadium, and Molybdenum, among other elements, to produce strong, lightweight alloys for aerospace (jet engines,missiles, and spacecraft), military, industrial process (chemicals and petro-chemicals, desalination plants, pulp, and paper), automotive, agri-food, medical prostheses, orthopedic implants, dental and endodontic instruments and files, dental implants, sporting goods, jewellery, mobile phones, and other applications. Zirconium is a lustrous, grey-white, strong transition metal that resembles Hafnium and, to a lesser extent, Titanium. Zirconium is mainly used as a refractory and opacifier, although it is used in small amounts as an alloying agent for its strong resistance to corrosion[52]. Zirconium forms a variety of inorganic and organometallic compounds such as Zirconium dioxide and Zirconiumdichloride, respectively. 49

7 Five isotopes occur naturally, three of which are stable. Zirconium compounds have no known biological role. Hafnium is a lustrous, silvery gray, tetravalent transition metal. Hafnium chemically resembles Zirconium and is found in Zirconium minerals[53]. Hafnium is used in filaments and electrodes. Some semiconductor fabrication processes use its Oxide forintegrated circuits at 45 nm and smaller feature lengths. Some superalloys used for special applications contain hafnium in combination with Niobium, Titanium, or Tungsten[54]. Hafnium's large neutron capture cross-section makes it a good material for neutron absorption in control rods in nuclear power plants, but at the same time requires that it be removed from the neutron-transparent corrosion-resistant Zirconium alloys used in nuclear reactors [55]. 3.5 TABLES Table 3.5.1: Electrons in s,p and d shells of hcp-ti (Titanium) at different reduced volumes V/Vo Pressure Mbar Volume Å 3 / atom 4s 2 4p 0 3d 2 1 Normal

8 Table 3.5.2: Ground state properties of Ti (Titanium) Different phases Ground state properties Present study Experiment [20] Previous theories a o a.u [44], [38] hcp - Ti c/a [42], [44] B o Mbar [38], [44] B o [44] - Ti a o a.u c/a [17], [42] [17], [42] B o Mbar [41], 1.115[42] B o [41], [42] a o a.u [20] [42], [41] bcc - Ti B o Mbar [41], 1.18 [42] B o a o a.u [45] fcc - Ti B o Mbar [45] B o

9 Table 3.5.3: Phase transition pressures for Ti (Titanium) Phase transition Transition pressure (P T ) Mbar Present work Experiments [20] Previous theory hcp [42], bcc [41] [43] 1.02 [42] bcc [42] hcp bcc [41] No transition [43] bcc No transition [43] Table: Pressure and lattice constant of Zr (Zirconium) V/Vo a (au) Pressure (Mbar)

10 Table: Ground state properties of Zr (Zirconium) Transition Metal a o c o B o B o ¹ (au) (au) (Mbar) Zirconium Table: Structural phase transition pressure of Zr (Zirconium) Transition Metal Structural Phase Transition (V/Vo) T P T (Mbar) Zr (hcp bcc) Table :3.5.7 Pressure and lattice constant of Hf (Hafnium) V/Vo a (au) Pressure (Mbar)

11 Table: Ground state properties of Hf (Hafnium) Transition Metal a o c o B o B o ¹ (au) (au) (Mbar) Zirconium Table: Structural phase transition pressure of Hf (Hafnium) Transition Metal Structural Phase Transition (V/Vo) T P T (Mbar) Zr (hcp bcc) Table : Tc as a function of pressure for Ti (Titanium) - hcp structure Pressure P Mbar D K * Tc K normal

12 Table : Tc as a function of pressure for Ti (Titanium) - structure Pressure P D * Tc K Mbar K Table : Tc as a function of pressure for Ti(Titanium) - bcc structure Pressure P D * Tc K Mbar K normal

13 Table Superconducting transition temperature of hcp and bcc Zr (Zirconium) Structure of Zirconium D (K) Tc(K) hcp zr (normal pressure) bcc zr (P= Mbar) Table Superconducting transition temperature of hcp and bcc Hf (Hafnium) Structure of Hafnium D (K) Tc(K) hcp Hf (normal pressure) bcc Hf (P= Mbar) 56

14 3.6 FIGURES Fig Band structure of hcp-ti (Titanium ) at V/Vo=1 (normal pressure) K M A L Fig Density of states of hcp-ti at V/Vo=1(normal pressure) 57

15 Fig Band structure of hcp-zr (Zirconium) at V/Vo=1 (normal pressure) K M A L Fig Density of states of hcp-zr (Zirconium) at V/Vo=1 (normal pressure) 58

16 Fig Band structure of hcp-hf (Hafnium) at V/Vo=1 (normal pressure) K M A L Fig Density of states of hcp-hf(hafnium) at V/Vo=1 (normal pressure) 59

17 Fig Band structure of hcp-ti (Titanium) at V/Vo=0.9 (high pressure) K M A L Fig Density of states of hcp-ti (Titanium) at V/Vo=0.9 (high pressure) 60

18 Fig Band structure of hcp-zr (Zirconium) at V/Vo=0.9 (high pressure) K M A L Fig Density of states of hcp-zr(zirconium) at V/Vo=0.9 (high pressure) 61

19 Fig Band structure of hcp-hf(hafnium) at V/Vo=0.9 (high pressure) K M A L Fig Density of states of hcp-hf(hafnium) at V/Vo=0.9 (high pressure) 62

20 Fig Band structure of bcc-ti(titanium) at V/Vo=0.5 (high pressure) H N P N Fig Density of states of bcc-ti (Titanium) at V/Vo=0.5 (high pressure) 63

21 Fig Band structure of bcc-zr(zirconium) at V/Vo=0.8 (high pressure) H N P N Fig Density of states of bcc-zr ( Zirconium) at V/Vo=0.8 (high pressure) 64

22 Fig Band structure of bcc-hf (Hafnium) at V/Vo=0.75 (high pressure) H N P N Fig Density of states of bcc-hf (Hafnium) at V/Vo=0.75 (high pressure) 65

23 Fig Band structure of bcc-ti (Titanium) at V/Vo=0.4 (high pressure) H N P N Fig Density of states of bcc-ti (Titanium) at V/Vo=0.4 (high pressure) 66

24 Fig Band structure of bcc-zr (Zirconium) at V/Vo=0.4 (high pressure) H N P N Fig Density of states of bcc-zr (Zirconium) at V/Vo=0.4 (high pressure) 67

25 Fig Band structure of bcc-hf (Hafnium) at V/Vo=0.4 (high pressure) H N P N Fig Density of states of bcc-hf( Hafnium) at V/Vo=0.4 (high pressure) 68

26 Fig Total energy versus reduced volume curve for Ti(Titanium) (Etot-3409) Ry HCP-Ti Omega-Ti FCC-Ti BCC-Ti Gamma-Ti Delta-Ti V/Vo 69

27 Fig Calculated total energy versus c/a graph for (a). hcp-ti (Titanium) (b). Omega Ti (Titanium) (a) Energy (mry) c/a (b) Energy (mry) c/a 70

28 Fig Total energy versus reduced volume curve of Zr (Zirconium) hcp bcc (Etot-14382) Ry V/Vo 71

29 Fig Total energy versus reduced volume curve of Hf (Hafnium) hcp bcc (Etot-60282) Ry V/Vo 72

30 Fig The relation connecting reduced volume and Lattice constant of Ti(Titanium) Lattice constant (au) V/Vo Fig The relation connecting reduced volume and Pressure of Ti(Titanium) Pressure (Mbar) V/Vo 73

31 Fig The relation connecting lattice constant and Pressure of Ti(Titanium) Pressure (Mbar) Lattice constant (au) Fig The relation connecting reduced volume and Lattice constant of Zr(Zirconium) Lattice constant (au) V/Vo 74

32 Fig The relation connecting reduced volume and Pressure of Zr(Zirconium) Pressure (Mbar) V/Vo Fig The relation connecting lattice constant and Pressure of Zr (Zirconium ) Pressure (Mbar) Lattice constant (au) 75

33 Fig The relation connecting reduced volume and Lattice constant of Hf (Hafnium) Lattice constant (au) V/Vo Fig The relation connecting reduced volume and Pressure of Hf (Hafnium) Pressure (Mbar) V/Vo 76

34 Fig The relation connecting lattice constant and Pressure of Hf (Hafnium) Pressure (Mbar) Lattice constant (au) Fig The relation connecting reduced volume and Lattice constant of Ti (Titanium), Zr (Zirconium) and Hf (Hafnium) Lattice constant (au) Titanium Zirconium Hafinium Reduced volume 77

35 Fig The relation connecting reduced volume and Pressure of Ti (Titanium), Zr (Zirconium) and Hf (Hafnium) Pressure (Mbar) 10 Titanium Zirconium Hafinium Reduced volume Fig The comparison of reduced volume, lattice constant and Pressure of Ti (Titanium), Zr (Zirconium) and Hf (Hafnium ) a (au), P (Mbar) Titanium Zirconium Hafinium Titanium Zirconium Hafinium Reduced Volume 78

36 3.7 CONSEQUENCE AND DISCUSSIONS Normal and high pressure Band structure and density of states The band structures and density of states of Ti, Zr and Hf in the hcp and bcc structures are obtained as a function of various reduced volumes (figs to ). For hcp- Ti, Zr and Hf the normal pressure (V/V o =1) band structure and density of states are given in ( Figs to )respectively. High pressure band structure and density of states of hcp- Ti, Zr and Hf (V/V o =0.9) are given in (Figs to ) respectively. The overall topology of the band structure at V/V o =1 is similar to previous calculation [38]. It is seen that the band structure of Ti exhibits characteristic features similar to other hcp transition metals Zr and Hf [45]. At normal pressure (Figs 1, 3, 5), there is a overlapping of valance band and conduction band confirming the metallic nature of Ti, Zr and Hf. The two bands lying well below the Fermi level are due to 4s 2 electrons and the bands near the Fermi energy are due to 3d 2 electrons of Ti, 5s 2 electrons and the bands near the Fermi energy are due to 4d 2 electrons of Zr, 6s 2 electrons and the bands near the Fermi energy are due to 5d 2 electrons of Hf, High pressure band structure and density of states of hcp- Ti, Zr and Hf (V/V o =0.9) are given in Figs. 7 to 12 respectively. As the pressure increases, the entire band structure is slowly shifted up in energy and the conduction band width increases Figs.7 to 24. For bcc- Ti, Zr and Hf the band structure and density of states are given in (Figs ) to 18 respectively. The reduced volumes are V/V o =0.5 for Titanium, V/V o =0.8 for zirconium and V/V o =0.75 for Hafnium. At these reduced volumes Ti, Zr and Hf undergo structural phase transition from hcp to bcc structure. The width of the conduction band increases because of the enhanced overlap of the wave function with the neighboring atoms 79

37 [12]. The electron distribution for titanium is given in Table The increase in d electron number leads to structural phase transition as well as superconducting transition under pressure [27]. Figs to corresponds to the band structure and density of states of BCC- Ti, Zr and Hf at V/V o =0.4 ( very high pressure). The density of states (DOS) histogram constructed from the normal pressure band structure of hcp- Ti, Zr and Hf is given in (Figs ,3.6.4,3.6.6) Like in the previous calculations [44,45], Fermi level E F lies in a minimum of the DOS curve (pseudo gap) (Fig ). The levels arising from 3d electrons give a high narrow peak near E F [51]. The short peak near Ry is due to 4s electrons of Ti and 5s electrons of Zr and 6s electrons of Hf. The peaks with considerable width above the Fermi energy E F are due to empty 4p and 5s states of Ti, empty 5p and 6s states of Zr and empty 6p and 7s states of Hf. When pressure increases, the value of E F increases whereas density of states at E F decreases. It is due to the dispersion of the bands with pressure. At normal pressure, the N(E F ) values of cubic bcc and fcc phases are larger than hcp and phase values. This is again in agreement with the results of previous calculations [52]. The superconducting properties (to be discussed) are primarily determined by the N(E F ) values around the Fermi level E F [12] Ground state properties The total energies as a function of reduced volume (V/V o ) are determined for hcp,,,, bcc and fcc phases of Ti (Fig ). Here V/V o ranges from 1.2 to 0.4 insteps of 0.05, where V o is the experimental equilibrium volume and V is any volume under pressure P. From the Fig , it is confirmed that the ground state structure of Ti is hcp as observed experimentally [20] and reported from other theoretical calculations[25]. For the cubic structures (fcc and bcc) the total energies are 80

38 calculated as a function of reduced volume V/V o. For the hcp and structures total energies are calculated as a function of c/a and V/V o. We performed calculations for different reduced volumes and different c/a ratios in order to optimize both V/V o and c/a (Figs (a) and (b) for hcp and structures respectively). Since orthorhombic and phases are not highly close packed structures, similar to previous studies [38] in the present investigation also we have fixed the lattice constants, atomic positions, ratios of cell constants and y-parameters from experimental values [20]. ie, a = a.u., a/b = 1.468, c/b = 1.381, y = 0.3 for phase and similarly, a = a.u., b/a = 1.873, c/a = 1.627, y = 0.1 for phase. The calculated equilibrium lattice constant (a o ), c/a, bulk modulus (B o ) and its pressure derivative (B 1 o ) for the hcp,, bcc and fcc phases of Ti are given in Table 2 together with previous theoretical and experimental values [20]. The predicted ground state properties are in agreement with the previous TB method and experimental measurements. The calculated B o value is found to be maximum for - Ti (1.212 Mbar) followed by hcp-ti (1.131 Mbar), bcc-ti (1.103 Mbar) and - Ti (1.091 Mbar). This trend is in agreement with the reported experimental observation [1] (Table ). The hardness of any material is related to its bulk modulus [53]. So at normal pressure the hardness is in the following order for Titanium : - Ti hcp - Ti bcc - Ti > - Ti. The pressure derivative of bulk modulus at normal pressure B 1 o is a parameter of great physical significance in high pressure physics and few other thermo-physical properties. The value of B 1 o is related to the electron density. The structure with highest B 1 o will have low electron density and vice versa [47]. 81

39 3.7.3 Structural phase transition At ambient pressure, Ti has hcp structure similar to other group IVA metals Zr and Hf [43-45]. Under pressure Ti undergoes hcp bcc phase transitions [20]. Based upon the recent experimental study, Akahama et al [20] opinioned that there could be a possibility of bcc structure to exist above 2.16 Mbar and suggested that further experimental and theoretical investigations are needed in this direction. To ensure the high pressure structure of Ti above 2.16 Mbar, we have calculated the total energies corresponding to hcp,,,, bcc and fcc structures and the results are shown in Fig Further we have analyzed the structural phase transition from hcp, from, from bcc and from bcc structures. From this we get the phase transition pressures as Mbar ( hcp ), 0.42 Mbar ( ), 3.16 Mbar ( bcc) and Mbar ( bcc). These values are given in Table. 3 together with experimental and other theoretical results. From this, we observe that Ti phase changes from to bcc with an intermediate phase and this phase is stable in between 0.42 to 3.16 Mbar. This leads to the fact that titanium is in distorted hcp phase up to 3.16 Mbar. But phase is obtained as an unstable structure at all compressions[53-55]. This feature is similar to the one observed in the previous calculation of Joshi et al [43]. For hcp transition, our calculated phase transition pressure is in good agreement with the previous theory and experiments whereas considerable difference exists in the case of, bcc and bcc transitions (Table.3.5.3). Due to the limitation of the method (for non-cubical structures), fixed ratios of cell constants and y - parameters are used for and phase calculations. Another reason is that, we 82

40 have not performed geometric optimization for each volume considered in the and phase calculations. Our calculated value for bcc transition is higher than the experimental value of Xia et al [38] but our results support the hypothesis formulated by Akahama et al [20], viz. the phase transition pressure for obtaining bcc structure is greater than 2.16 Mbar. In the calculation the total energies as a function of reduced volume (V/V o ) are determined for hcp and bcc phases of Zr (Fig ). In this V/V o ranges from 1.2 to 0.4 insteps of 0.05, where V o is the experimental equilibrium volume and V is any volume under pressure P. From the ( Fig ), it is confirmed that the ground state structure of Zr is hcp as observed experimentally and reported from other theoretical calculations. The calculated total energies of hcp-zr were fitted to Murnaghan s equation of state[17] and the determined pressure values are given in Table 4. For the cubic structures (bcc) the total energies were calculated as a function of reduced volume V/V o. For the hcp structures total energies were calculated as a function of c/a and V/V o. We performed calculations for different reduced volumes and different c/a ratios in order to optimize both V/V o and c/a. In this calculation, the lattice constant and pressure of hcp Zr at different reduced volume are calculated is shown in Table The calculated equilibrium lattice constants (a o & c o ), bulk modulus (B o ) and its pressure derivative (B 1 o ) of Zr are given in Table In this investigation, the total energies as a function of reduced volume (V/Vo) are determined for hcp, and the bcc phases of Zirconium are shown in (Fig ). The predicted ground state properties are in agreement with the previous TB model and experimental measurements. The pressure derivative of bulk modulus at normal pressure B 1 o is a 83

41 parameter of great physical significance in high pressure physics and few other thermo-physical properties. The value of B 1 o is related to the electron density. The structure with highest B 1 o will have low electron density and vice versa. At ambient conditions, Zr has hcp structure similar to other group IVA metals Ti and Hf [47]. experimental study, Akahama et al [20] opinioned that there could be a possibility of bcc structure to exist above 0.33 Mbar and suggested that further experimental and theoretical investigations are needed in this direction. To ensure the high pressure structure of Zr above 0.33 Mbar, we have calculated the total energies corresponding to hcp and bcc structures and the results are shown in (Fig ). Further from the enthalpy calculation, similar to Titanium it is analyzed the structural phase transition from hcp bcc structure. Further from the enthalpy calculation, we have analyzed the structural phase transition from hcp bcc structure is shown in Table 6. The calculated values from this we get the phase transition pressure as 0.32 Mbar. Our calculated value for hcp bcc transition is good agreement with the experimental value of Akahama et al[20], the phase transition pressure for obtaining bcc structure is greater than 0.33 Mbar. The pressure and lattice constant under reduced volume of Hf is given in Table The calculated equilibrium lattice constants (a o & c o ), bulk modulus (B o ) and its pressure derivative (B 1 o ) of Hf are given in Table In this investigation, the total energies as a function of reduced volume (V/Vo) are determined for hcp, and the bcc phases of Zirconium are shown in (Fig ). The predicted ground state properties are in agreement with the previous TB model and experimental measurements. The pressure derivative of bulk modulus at normal pressure B 1 o is a 84

42 parameter of great physical significance in high pressure physics and few other thermo-physical properties. The value of B 1 o is related to the electron density. The structure with highest B 1 o will have low electron density and vice versa. At ambient conditions, Hf has hcp structure similar to other group IVA metals Ti and Zr [27]. experimental study, Akahama et al[20] opinioned that there could be a possibility of bcc structure to exist above 0.50 Mbar and suggested that further experimental and theoretical investigations are needed in this direction. To ensure the high pressure structure of Hf above 0.50 Mbar, the total energies corresponding to hcp and bcc structures are calculated and the results are shown in (Fig ). Further from the enthalpy calculation, similar to Titanium the structural phase transition from hcp bcc structure is analysed. Further from the enthalpy calculation, we have analyzed the structural phase transition from hcp bcc structure is shown in Table 9. The calculated values from this we get the phase transition pressure as 0.48 Mbar. The calculated value for hcp bcc transition is good agreement with the experimental value of Akahama et al[20], the phase transition pressure for obtaining bcc structure is greater than 0.50 Mbar. The relation connecting reduced volume and Lattice constant of Titanium is presented in (Fig ). When reduced volume decreases lattice constant also decreases. Fig contains the relation connecting reduced volume and Pressure of Titanium. When reduced volume decreases pressure increases. Fig contains the relation connecting lattice constant and Pressure of Titanium. When lattice constant decreases pressure increases.in Fig we presented the relation connecting reduced volume and Lattice constant of Zirconium. When reduced volume decreases lattice constant also decreases. Fig contains the relation connecting reduced volume and Pressure of Zirconium. When reduced 85

43 volume decreases pressure increases. Fig contains the relation connecting lattice constant and Pressure of Zirconium. When lattice constant decreases pressure increases. In Fig we presented the relation connecting reduced volume and Lattice constant of Hafnium. When reduced volume decreases lattice constant also decreases. Fig contains the relation connecting reduced volume and Pressure of Hafnium. When reduced volume decreases pressure increases. Fig contains the relation connecting lattice constant and Pressure of Hafnium. When lattice constant decreases pressure increases. Fig contains the comparison of the relation connecting reduced volume and Lattice constant of Titanium, Zirconium and Hafnium. Fig contains the comparison of the relation connecting reduced volume and Pressure of Titanium, Zirconium and Hafnium. Fig contains the comparison of the reduced volume, lattice constant and Pressure of Titanium, Zirconium and Hafnium Superconductivity The promotion of s electron to d shell in solids is one of the factors which will induce superconductivity [12]. The manner in which the d electron number is increasing as a function of pressure in hcp-ti is given in Table 10. At normal pressure itself the contribution of 3d states is large. This clearly shows the possibility of a superconducting transition in hcp - Ti under normal pressure. With the results obtained from the self-consistent calculation, we have computed D,, * and Tc as a function of pressure. The calculated values of Tc at normal and various high pressures are given in Table 10 for hcp-ti. At normal pressure the Tc is 0.36 K. This 86

44 is in agreement with the experimental observation of 0.4 K [20]. The electron-phonon mass enhancement factor and electron-electron interaction parameter * at normal pressure are and respectively (Table.10), which are in agreement with the values of 0.38 and 0.13 respectively reported by McMillan [12]. From Table 10, it is seen that Tc increases with increase of pressure and reaches a maximum, there after Tc starts to decrease. It is found, the highest Tc obtained for hcp Ti is K at 1.5 Mbar pressure with a pressure coefficient of K/Mbar. On further increasing of pressure, the value of Tc goes on decreasing and reaches K at Mbar with a pressure coefficient of K/Mbar. The increase of Tc up to 1.5 Mbar in hcp-ti is due to the softening of phonon modes which arises because of lattice softening induced by s dtransition [12]. The increase in the d-electron number is high up to 1.5 Mbar, above this pressure the rate of increase of d-electron number is not appreciable when compared with low pressure values (Table 1). We found that hcp and bcc transitions are obtained at and Mbar respectively. The variation of Tc with pressure is computed for and bcc structures and the results are given in Tables 11 and 12 respectively. From Table , it is seen that the onset of superconductivity in -Ti occurs at Mbar and there after Tc increases with increase of pressure and reaches a maximum value of K at Mbar with a pressure coefficient of K/Mbar. On further increasing of pressure, the value of Tc decreases in -Ti. In order to compare the variation of Tc with pressure in the high temperature phase (bcc), we have calculated the values of D,, * and Tc corresponding to a range of pressure from normal to 4.52 Mbar in the bcc phase of Ti and the values are given in Table The onset of superconductivity in bcc-ti occurs at normal pressure. The corresponding Tc value 87

45 is 2.17 K. There is no direct experimental measurement of Tc. But the Tc value obtained for bcc - Ti, by extrapolation of the experimental value of Ti-Mo alloy is 6.4 K [21]. The value of electron-electron interaction parameter * is which is in agreement with value of 0.17 reported by Bakonyi [31] for the bcc-ti. Similar to hcp and phases, in bcc-ti also the Tc increases with increase of pressure and reaches a maximum, there after Tc starts to decrease. The highest Tc obtained for bcc-ti is 4.85 K at Mbar pressure with a pressure coefficient of K/Mbar. On further increasing of pressure, the value of Tc goes on decreasing and reaches K at 4.52 Mbar with a pressure coefficient of K/Mbar. The decrease of Tc above Mbar pressure is due to decreasing value of followed by the density of states at Fermi energy (N(E F )) [40]. At normal pressure the Debye temperature for the bcc phase (250 K) is less than that of hcp phase (420 K) [45]. Thus there is a softening of phonon in bcc phase. This phonon frequency softening (decrease of 2 ) contributes to a large and hence a high value of Tc at normal pressure in bcc-ti when compared to hcp-ti (Tables. 10 and 12). Under the application of pressure phonon softening is in the following order for titanium : bcc-ti -Ti hcp-ti. The Tc - max values (highest value of Tc(P)) estimated are (Tables 10,11,12) K for hcp Ti, 4.538K for -Ti and K for bcc - Ti. From this, it is inferred that the maximum value of Tc is rather insensitive to the crystal structure of Ti. The path to higher Tc lies in the direction of higher D (P). But under very high pressure, a higher Debye temperature can also lower Tc. That is because can decrease if the phonon frequencies are large [12]. 88

46 In Zirconium, at normal pressure the Tc is 0.53 K. This is in agreement with the experimental observation of 0.55 K. The electron-phonon mass enhancement factor and electron-electron interaction parameter * at normal pressure are 0.41and 0.13 respectively (Table.13), which are in agreement with the values of 0.40 and 0.13 respectively reported by McMillan[12]. The increase of Tc in bcc-zr at 0.32 Mbar (5.807 K) is due to the softening of phonon modes which arises because of lattice softening induced by s dtransition. In our calculation, hcp bcc transition of Zr are obtained at 0.32 Mbar. The variation of Tc with pressure is computed for hcp and bcc structures and the results are given in Table This increasing trend of Tc is similar to other group IVB transition metal Ti. At normal pressure the Debye temperature for the hcp phase (291 K) is higher than that of bcc phase (200 K) at 0.32 Mbar. Thus there is a softening of phonon in bcc phase. This phonon frequency softening (decrease of 2 ) contribute to a large (0.88 in Table 13) and hence a high value of Tc. The path to higher Tc lies in the direction of higher D (P). But under very high pressure, a higher Debye temperature can also lower Tc. That is because can decrease if the phonon frequencies are large. The calculated Tc values depend more sensitively on changes in and * as these quantities appear with in the exponential term [27]. In Hafnium, at normal pressure the Tc is K. This is in agreement with the experimental observation of 0.12 K. (Table ). The increase of Tc in bcc-hf at 0.48Mbar (2.38 K) is due to the softening of phonon modes which arises because of lattice softening induced by s dtransition. 89

47 3.8 CONCLUSION In summary, the pressure dependent band structure, density of states, structural phase transition and superconductivity of Titanium, Zirconium and Hafnium are investigated using FP-LMTO method. The total energies of Titanium, Zirconium and Hafnium are computed and the results are used to study the structural phase transition and superconductivity under pressure. In titanium, we could find a phase transformation sequence of (hcp) (hexagonal) (distorted hcp) (bcc) under pressure. From our analysis we predict a (distorted bcc) phase which is not stable at any high pressures. At normal pressure the hardness of Ti is in the following order: - Ti hcp - Ti bcc - Ti > -Ti. When the pressure is increased, it is predicted that, Tc increases and reaches a maximum value thereafter Tc starts to decrease. The highest value of Tc(P) estimated is K for hcp - Ti, K for - Ti and 4.85 K for bcc - Ti. From this, it is inferred that the maximum value of Tc(P) is rather insensitive to the crystal structure of Ti. The structural phase transition from hcp to bcc is found to occur at 0.32 Mbar for Zirconium and 0.48Mbar for hafnium. Experimental phase transition pressure is 0.33 Mbar for Zirconium and 0.5Mbar for hafnium. Our calculated value is good agreement with the experimental value. At normal pressure the Tc is 0.53 K for Zirconium and K for hafnium.. This is in agreement with the experimental observation of 0.55 K for Zirconium and 0.12 K for hafnium. The increase of Tc in bcc-zr at 0.32 Mbar (5.807 K) and bcc-hf is due to the softening of phonon modes which arises because of lattice softening induced by s d transition. 90