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CHAPTER 3

HIGH PRESSURE STRUCTURAL STUDY AND

ELECTRONIC STRUCTURE CALCULATION ON UGa3

3.1 INTRODUCTION

The 2-D structural stability maps for AB3 type compounds indicate

that the cubic AuCu3 type compounds occupy larger island than other

structures at ambient pressure as shown in Figure 3.9. This might indicate that

the AB3 type compounds adopting the AuCu3 type structure (space group:

Pm3m) have large range of structural stability under pressure. The UGa3

system also stabilizes in cubic AuCu3 type structure at NTP. In order to

validate the 2-D stability maps for AB3 type compounds, high pressure X-ray

diffraction experiments have been carried out on the UGa3 system. The

electronic structure calculation has also been carried out as a function of

reduced volume to correlate the structural stability and the electronic structure

of UGa3. Most of the UX3 type compounds stabilize in cubic AuCu3 type

structure at NTP, where X is an element from d or p block. The Hill limit

(~ 3.4 Å) separates the localized f-states of the compound from the

delocalized f-states. UX3 type compounds with cubic AuCu3 type structure

have large U-U distances which are located far away from the Hill limit,

showing varieties of magnetic properties, such as Pauli paramagnetism,

antiferromagnetism, spin fluctuation and superconductivity (Kaczrowski

2006, Kambe et al 2008, Aoki 2000, Koelling et al 1985). This is because of

the hybridization effect between the 5f states and the s, p and d electronic

states of neighboring atoms.

53

UGa3 system has been studied extensively to comprehend the

behavior of 5f states. It shows itinerant antiferromagnetic behavior at ~ 67 K

and also exhibits metallic conductivity. Though the U-U distance in UGa3 is

far from the Hill limit, it shows itinerant 5f – electron magnetism (Koelling

et al 1985, Lawson et al 1985). This arises due to the strong hybridization

between the uranium 5f states and gallium 2p states. Earlier, Murasik et al

(1974) have carried out the neutron diffraction studies on UGa3. This result

confirms itinerant antiferromagnetic ordering in UGa3. Following this study,

results from the studies on heat capacity (Kaczorowski et al 1993) and the

magnetic properties as a function of pressure (Grechnev et al 1996) also

confirm the itinerant 5f magnetism in UGa3. Several UX3 compounds (X= Al,

Si, Ga, Ge, In, Sn) have already been studied under high pressure using

Energy Dispersive X-ray Diffraction (Bihan et al 1995). These compounds do

not reveal any structural phase transformation by the application of pressure.

Magnetic excitation spectrum from neutron diffraction experiments

on a single crystal of UGa3 reveal that there is rapid change in intensity at

about 40 K, and the moments are along [111] direction (Dervenagas et al

1999). This may be due to either a small distortion or can be of magnetic

orientation. Magnetic susceptibility experiment has been carried out up to

2 kbar pressure in the temperature range of 64 K – 300 K (Grechnev et al

1999). The measured pressure derivative of the Neel temperature is found to

be dTN/dP = - 1.1 K/kbar. A full potential LMTO calculation was also

performed along with the experiment to analyze the magnetovolume effect of

UGa3 system. These results show that the magnetic properties of these UX3

compounds are mainly determined by the U-U spacing variation. In order to

understand the itinerancy of 5f states in UGa3, a number of studies have been

carried out along with electronic structure calculations (Kaczorowski et al

1999, Kaczorowski et al 1998, Cornelius et al 1999, Schoenes et al 2000,

Sanchez et al 2000, Hiess et al 2001, Gouder et al 2001, Aoki et al 2001,

54

Nakashima et al 2001, Mannix et al 2001, Ikushima et al 2001, Ikushima et al

2001, Kambe et al 2002, Kambe et al 2003, Nakamura et al 2002, Hotta 2004,

Biasini et al 2005, Muraoka et al 2005, Kambe et al 2005). All these studies

are fully based on itinerant 5f electron model. Divis (1994) investigated the

reliability of non-self–consistent tight binding method by a comparison with

the results of a more accurate LDA method. This work reveals that the

electronic structure of UGa3 calculated by TB method is able to provide only

gross features and the itinerant 5f-electron magnetism of UGa3 is in good

agreement with LDA total energy analysis. The recent electronic structure

calculations show the itinerancy of 5f electrons in UGa3, which is supported

by the experiments based on the measurement of angular correlation of

electron-positron annihilation radiation (Rusz et al 2004). Although, the

results of numerous experiments and calculations lead to clear perception on

the itinerancy of 5f magnetism in UGa3, still there is a need to have a unique

model for f-electron based systems, because, itinerant model could not explain

the reason for the observation of large orbital moments. The hybridization

strength can be changed by applying external pressure with which we can

understand the nature of bonding in f-electron systems more precisely. A

recent review on f -electron based binary intermetallic compounds gives the

clear idea about the structural sequences observed among various homologues

under pressure and correlations with their electronic structure as a function of

reduced volume (Chandra Shekar and Sahu 2006).

3.2 EXPERIMENT AND RESULTS

3.2.1 Sample Preparation and Characterization

UGa3 was prepared by using a standard arc melting technique.

Stoichiometric measure of U (99.98% pure) and Ga (99.999% pure) was

melted in a tri-arc furnace in He atmosphere and the melted button was

flipped 2-3 times during remelting to obtain a homogeneous compound. The

55

ingot was then vacuum sealed in Ar atmosphere in silica tube and annealed

for about 4 weeks at temperature of 1100 K. In order to remove any oxide

layer, the annealed ingot was etched in 1:1 mixture of nitric and sulphuric

acid for about 2 minutes. It was then washed with acetone and stored in

hexane medium. This ingot was crushed into fine powder in hexane medium

using pestle and mortar. The powdered sample was used for characterization

and subsequent high pressure experiments.

The powdered UGa3 sample was characterized by X-ray diffraction

using a high – resolution Guinier diffractometer with scintillation detector

described in chapter 2. The UGa3 sample was found to be in single phase with

a AuCu3 type cubic structure having lattice parameter a = 4.251 ± 0.001 Å.

3.2.2 High Pressure X-ray Diffraction

High pressure X-ray diffraction studies on UGa3 were carried out

using a Mao-Bell type diamond anvil cell (DAC) in the angle dispersive mode

with a custom-built Guinier diffractometer system described in chapter 2

(Sahu et al 1995). Finely powdered sample was loaded into a 250 -µm-

diameter hole drilled at the centre of pre-indented stainless steel gasket.

A mixture of methanol, ethanol and water in the volume ratio 16:3:1 was used

as the pressure-transmitting medium. Pressure calibration was carried out

using ruby fluorescence technique, for which a small ruby chip (Al2O3 + 0.5%

Cr3+) was placed in intimate contact with sample. The Cr3+ ions in the ruby

crystal close to the sample are excited by an argon laser of wavelength 514.5 nm

to emit fluorescence radiation at 692.7 nm and 694.24 nm (the R1 and R2

lines at ambient conditions). The pressure inside the sample chamber is

measured from the shift in the fluorescent lines (Forman et al 1972,

Piermarini et al 1975).

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3.2.3 Compressibility Behavior

The high pressure X-ray diffraction pattern of UGa3 is shown in

Figure 3.1. Pressure was increased stepwise up to 30 GPa and X-ray

diffraction data were collected at each pressure step. Figure 3.1 shows the

multiplot of high pressure X-ray diffraction pattern for 3, 12 and 30 GPa

respectively. The し value in the X-ray diffraction pattern was limited to about

18º because of mechanical restriction of the exit slit in the diamond-anvil cell.

Furthermore, at higher pressures, the signal-to-noise ratio for the diffraction

peaks decreased and the pattern beyond 15º was noisy, and hence in Figure 3.1,

data beyond 15º is not shown. Six resolvable peaks (100), (110), (111), (200),

(210) and (211) within the limit 5° - 15° were obtained. The signature of

gasket and ruby peaks appear in-between (200) and (210) sample peaks.

However, as the pressure was increased, the peaks move towards higher し

values. There were no appearances of new peaks or disappearance of old

peaks even up to 30 GPa. It is evident from X-ray diffraction pattern that

there is no change in the structure of UGa3 up to 30 GPa.

At around 30 GPa, the intensity of ruby peak increased and the

disappearance of gasket peak was observed. All the patterns for different

pressures were indexed and the lattice parameters were calculated by using

the software NBS-AIDS83 (Mighell et al 1981).

The calculated d-spacings are plotted as a function of pressure in

Figure 3.2. It is observed that these values decrease smoothly by the

application of pressure. The fitting of P-V data has been carried out using

three different equations of state (EOS), namely, Murnaghan (1937),

Birch-Murnaghan (1947) and Vinet (1989). Table 3.1 lists the bulk modulus

and its derivative computed by fitting the three equations of state and it shows

the consistency of the fitted values. It is observed that the bulk modulus

obtained by using the Murnaghan equation of state is always overestimated as

57

compared to the other equations of state and the same trend has been reported

in previous investigation also (Menoni et al 1983). The data are comparable

with the earlier reported values (Bihan et al 1995).

Table 3.1 Values of bulk modulus B0 and its pressure derivative B0ガ

for UGa3

Form of equation

of states B0 (GPa) B0'

Birch-Murnaghan 73 ± 8 5.8 ± 1.4

Vinet 72 ± 8 6.0 ± 1.2

Murnaghan 76 ± 8 5.0 ± 0.9

Figure 3.1 High-pressure X-ray diffraction patterns of UGa3 at 3, 12

and 30 GPa (The diffraction peak from ruby is marked ‘R’

and that from gasket as ‘G’)

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Figure 3.2 d-Spacing of UGa3 system as a function of pressure upto 30 GPa

3.3 ELECTRONIC STRUCTURE CALCULATION

3.3.1 Calculation Details

The electronic structure calculations were carried out using

WIEN2K code (Blaha et al 2001) implemented with Full Potential – Linear

Augmented Plane Wave (FP-LAPW) method. The exchange-correlation

interaction was treated within GGA approximation of Perdew et al (1996).

Radii of non-overlapping atomic spheres RMT, plane wave with a cut-off

parameter RMT*Kmax (where Kmax is the magnitude of the largest k-vector in

the plane wave expansion) and number of points in irreducible wedge of

Brillouin Zone (IBZ) have been optimized and chosen in order to achieve a

satisfying convergence of calculated parameters.

The radius of Muffin Tin was chosen at 2.5 au for both the

magnetic and non-magnetic calculations. The band structure and the density

of states were calculated using 286 k-points in the irreducible Brillouin zone

(IBZ) for both the calculations. The plane-waves with a cut off of RMT*Kmax = 9

59

were used. The value of Gmax was chosen to be 12 bohr-1 (where, Gmax is the

magnitude of the largest vector in the charge density fourier expansion).

Figure 3.3 shows the crystal structure of UGa3, it consists of the U atom at the

unit cell corners and Ga atoms at the face centres. As all the atoms occupy

special positions in the unit cell, only the volume minimization of the total

energy was carried out to obtain the equilibrium lattice parameter. The

relaxed lattice parameter was found to be 4.241 Å in close agreement with the

experimental value for the ambient structure. Band structure and DOS for

non-magnetic and magnetic states were calculated using above mentioned

parameters as function of reduced volume. The relaxed lattice parameter value

was used as reference for all these calculations. The electronic structure of

UGa3 at 0 K up to ~22 GPa was calculated in order to confirm the presence of

magnetism at high pressures. In this case, the indicated pressures are obtained

using the equation of state fitting. The unit-cell volumes of all the structures

were fitted to Birch-Murnaghan equation of state to obtain the theoretical

value of the bulk modulus and its derivative (Birch 1947). The calculated total

energy values were fitted to the Birch-Murnaghan equation of states to obtain

theoretical pressure - volume data (Figure 3.4). Table 3.2 lists the bulk

modulus B0 and its pressure derivative B0’ for UGa3 determined by fitting

various forms of equation of state to the P-V curve. The calculated bulk

modulus was found to be 79 GPa, in good agreement with the experimental

value of 73 GPa.

Table 3.2 Values of bulk modulus B0 and its pressure derivative B0’ for

UGa3 determined by fitting various forms of equation of state

to the P-V curve

From of equation of states B0 (GPa) B0

Birch-Murnaghan 73 8 5.8 1.4

Vinet 72 8 6.0 1.2

Murnaghan 76 8 5.0 0.9

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Figure 3.3 Structure of UGa3 (Cubic AuCu3 type structure)

Figure 3.4 Experimental and theoretical P-V data are compared

Experimental:

B0=73.3, B0’=5.3

Theoretical

B0=79.5, B0’=5.3

Uranium atom

Gallium atom

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3.3.2 Band Dispersion Curves and DOS of UGa3

3.3.2.1 Non-magnetic state of UGa3

Band structure calculation on UGa3 has been carried out for cubic

AuCu3 type structure. Here, band dispersion curves and DOS were

constructed in between the energy values -8 eV to +8 eV. Band structure was

calculated along the Brillouin zone path R-d-X-M-d at ambient pressure.

High pressure calculations were also carried out up to 28 GPa and compared

with ambient pressure. Figures 3.5 (a) and (b) show the band dispersion

curves of UGa3 at ambient and high pressure. The position of narrow f like

band lies around the Fermi level indicates the itinerancy of 5f state in UGa3.

At higher pressure, the f like bands around the Fermi level start to move

towards the higher energy values and p like bands below the Fermi level start

to move towards lower energies. This band shifting and broadening in UGa3

system does not show any phase transition which is confirmed by

experiments. Partial DOS of non-magnetic state of UGa3 in Figure 3.6 shows

that around EF, the contributions are from uranium 5f states and gallium 4p-

states and a strong hybridization between them exists. This is the foremost

reason behind itinerancy of UGa3.

Moreover, it is also seen that the Fermi level lies near a valley in

the DOS at ambient pressure. DOS for various pressures have been plotted in

Figure 3.7. The EF lies near a valley in the DOS and with the application of

pressure, it remains immobile. Also, it is observed that as a function of

pressure, the total DOS decreases. In DOS the occupied states show some

changes in the s, p, d state electrons with respect to pressure, but these

changes are insignificant and do not affect the structural stability of the

system. At near by EF, the f-electrons did not show any change even at high

pressures.

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Figure 3.5 (a) Band dispersion curves for non-magnetic state at ambient

pressure

Figure 3.5 (b) Band dispersion curves for non-magnetic state at 28 GPa

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Figure 3.6 The partial density of states of UGa3 in the non-magnetic state

Figure 3.7 The total density of states in UGa3 at various pressures for

non-magnetic state upto 28 GPa (enlarged portion near the

Fermi level is shown as an inset)

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3.3.2.2 Antiferromagnetic state of UGa3

UGa3 exhibits itinerant antiferromagnetism at around 67 K. Band

structure calculations for magnetic state helps to understand the behavior of

UGa3 under high pressure. Total DOS for magnetic states at various pressures

are plotted in Figure 3.8. Due to the antiferromagnetic order, the number of

atoms in the unit cell is doubled and DOS for the two spin directions are

identical. In DOS, Fermi level lies in the deep valley and the movement of EF

under higher pressure is hardly visible, but the regions slightly away from the

Fermi level show small changes. However, this does not get reflected in the

total magnetic moment of the system. Since the contribution towards the total

DOS is mainly due to the uranium 5f states, the band around EF is narrow

with a width of ~3 eV and it seems that the contributions of other bands are

very less. The f-band width and EF-Ef influence the structural as well as the

transport properties of a material (here Ef is the middle of the f band)

(Chandra Shekar et al 2005).

Magnetic calculations show that the calculated magnetic moment

for UGa3 is 1.72 µB per unit cell and the value is good in agreement with

earlier reported values (Murasik et al 1974). In order to find out the magnetic

phase transition, magnetic moments are calculated as a function of pressure.

The effect of pressure on the magnetic moment shows that the total magnetic

moment decreases linearly at the rate of dµ/dP = -0.027 µB/GPa with

increasing pressure. It decreases from 1.72 µB at atmospheric pressure to

1.12 µB at 23 GPa. This large magnetic moment arises mainly due to the large

atomic 5f orbital and spin moments. It mainly affects the spin moment rather

than orbital moment. At ~23 GPa, the magnetic moment has finite value and

the extrapolation of data indicates that it might disappear completely at

around ~ 81 GPa. Results of our calculations deviate from the extrapolated

65

experimental result of a previous report wherein it is predicted that magnetism

in this system might disappear at ~ 4.5 GPa (Kaczorowski et al 1997). This

may be due to limitations of the calculation. However, the decrease in the

magnetic moment does not seem to have any effect on the stability of the

crystal structure of UGa3.

Figure 3.8 The total density of states in UGa3 at various pressures for

both up and down spin of the antiferromagnetic state is upto

~23 GPa (enlarged portion near the Fermi level is shown as

an inset)

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3.4 SUMMARY AND CONCLUSIONS

High pressure X-ray diffraction studies on UGa3 were carried out

up to 30 GPa. UGa3 does not show any structural phase transition and retains

its AuCu3 type cubic structure up to the maximum pressure studied. The 2-D

structural stability map for AB3 type compounds is generated using the

following parameters: average electronegativity (〉X), average Zunger

pseudopotential radii (〉R) and average valence electron (VE) at NTP (Villars

et al 1989). To generate the map, average Zunger pseudopotential radii versus

electronegativity have been plotted in Figure 3.9. The making of 2-D

structural stability maps and importance in predicting the high pressure phases

using these maps are already discussed in the first chapter. In 2-D structural

stability map the cluster belongs to cubic AuCu3 type structure and occupies

more space than other structures. The calculated value for the following

parameters 〉X, 〉R and VE of UGa3 are 0.00, 1.51 and 3.00 respectively. The

position of UGa3 in 2-D map is shown in Figure 3.9 using red colored circle.

The cluster belongs to VE and 〉R ranging from 2.75 to 3.24 and ~ 0.5 to ~1.5

respectively, occupies a vast area in 2-D map. It is believed that the 〉R

decreases, 〉X and VE increases as pressure increases. In order to check the

structural stability of UGa3, one can see whether UGa3 (AuCu3 type structure)

falls in some other cluster which is also having similar values of 〉X, 〉R and

VE of UGa3 under pressure.

Although, VE increases with pressure, UGa3 system will not move

to other clusters from the parent cluster. Because, when the pressure

increases, 〉X also increases, the cluster belongs to AuCu3 type structure

where as the UGa3 is located at very high 〉X values. Due to this reason UGa3

remains in the same cluster even at higher pressures. Moreover, the cluster

belonging to UGa3 system looks vast in area, even at higher pressures, the

position of UGa3 remains in the same cluster which confirms the structural

67

stability of the system. Here, the information obtained from the 2-D structural

stability map supports the experimental result on UGa3.

In order to understand the structural stability of UGa3, band

structure calculations were performed for both magnetic and non-magnetic

states. The total DOS of non-magnetic state shows that the Fermi level lies

near a valley in DOS curve, whereas in the case of magnetic system, the

Fermi level lies in a deep valley in DOS curve. Both the states do not show

any deviation in the position of EF even at higher pressures. Moreover, there

is no sign of any change in the magnetic phase even up to ~ 23 GPa. Band

dispersion curves also do not reveal any significant change. A criterion on

stability proposed by Yamashita et al (1972, 1972 a) states that the stable

phases with given crystal structure are formed when the Fermi level coincides

with density of states minimum. They investigated this criterion on B2-type

intermetallics of transition metals and they found that the criterion is validated

well for those systems. Recently, that criterion is applied and evaluated for f-

electron based intermetallics and it is ensured that the criterion can be

applicable for f-electron based intermetallics also. Moreover, the additional

reasons for the stability of f-electron based intermetallics were analysed based

on position of Fermi level in the DOS curve (Ravindran and Asokamani

1997). It is found that stability of f-electron based intermetallics not only

depends on the position of the Fermi level, but also on the position of f-band

with respect to the Fermi level. The criterion for structural stability from

electronic structure is discussed elaborately in chapter 1. Here the electronic

structure of UGa3 is correlated to the structural stability through those criteria.

Total DOS of UGa3 system exhibits that the f-band is delocalized and the

Fermi level also lies close to deep valley indicating the large structural

stability of the system. The criteria which are based on the position of the EF

and position of f band width with respect to the EF in the total DOS curve

agree with the experimental results of UGa3 under pressure.

68

Figure 3.9 Position of UGa3 compound in 2-D structural maps