Identification of phase components in Zr-Ni and Hf-Ni intermetallic compounds; Investigations by perturbed angular correlation spectroscopy and first principles calculations

TL;DR

This study combines perturbed angular correlation spectroscopy and first principles calculations to identify phase components in Zr-Ni and Hf-Ni intermetallic compounds, revealing detailed phase compositions and site-specific information.

Contribution

It provides a comprehensive analysis of phase components in Zr-Ni and Hf-Ni intermetallics using combined experimental and theoretical methods, which is novel in this context.

Findings

Identification of multiple phase components in ZrNi3 and HfNi3.

Confirmation of experimental results with DFT calculations.

Agreement between TDPAC, XRD, TEM, and theoretical data.

Abstract

Time-differential perturbed angular correlation (TDPAC) measurements have been carried out in stoichiometric ZrNi$_3$ and HfNi$_3$ intermetallic compounds using $^{181}$Ta probe in the temperature range 77-1073 K considering the immense technological applications of Zr-Ni and Hf-Ni intermetallic compounds. In ZrNi$_3$, four components due to the production of Zr$_2$Ni$_7$, Zr$_8$Ni$_{21}$, Zr$_7$Ni$_{10}$ and ZrNi$_3$ have been found at room temperature. The HfNi$_3$ sample produces five electric quadrupole interaction frequencies at room temperature. The phase HfNi$_3$ is strongly produced in stoichiometric sample of HfNi$_3$ where two non-equivalent Hf sites are found to be present. Besides this phase, two other phases due to Hf$_2$Ni$_7$ and Hf$_8$Ni$_{21}$ have been found but, we do not observe any phase due to Hf$_7$Ni$_{10}$. X-ray diffraction, TEM/energy dispersive X-ray spectroscopy (EDX) and TEM-selected area electron diffraction (SAED) measurements were used to further characterize the investigated materials and it was found that these results agree with the TDPAC results. In order to confirm findings from TDPAC measurements, density functional theory (DFT) based calculations of electric field gradients (EFG) and asymmetry parameters at the sites of $^{181}$Ta probe nucleus were performed. Our calculated results are found to be in excellent agreement with the experimental results.

Tables

Table 1. Table 1: Results of temperature dependent variations of ω Q subscript 𝜔 𝑄 \omega_{Q} for different components in the stoichiometric samples of ZrNi 3 and HfNi 3 .

Component phases	${\omega }_Q$(0)	$V_{zz}$(0)	$\alpha$	$\beta$
	(Mrad/s)	($\times$1021) V/m2	($\times$10-4) K-1	($\times$10-6) K-3/2
Zr2Ni7	72.9(4)	8.2(1)		6.2(4)
ZrNi3	76(1)	8.5(1)		4.3(6)
Zr8Ni21	98.2(5)	11.0(2)		6.5(3)
Zr7Ni10	64(2)	7.3(1)	2.3(3)
HfNi${}_{}{}^{\left(1\right)}{}_3{}^{}$	32.9(4)	3.7(1)	1.6(2)
HfNi${}_{}{}^{\left(2\right)}{}_3{}^{}$	65.7(4)	7.3(1)		7.0(4)
Hf2Ni7	72.2(4)	8.1(1)		5.7(2)
Hf8Ni21	99.3(6)	11.1(2)		6.6(4)
Hf	53.3(6)	6.0(1)	0.8(2)

Table 2. Table 2: The parameters of the β 𝛽 \beta -HfNi 3 and ZrNi 3 structure, given in Å.

	Our calculated results	Previous experimental	Earlier calculated	Present experimental
	(WIEN 2k) [42, 47]	results (X-ray diffraction) [26, 20]	results [48]	results (X-ray diffraction)
$\beta$-HfNi3
$a$	5.285	5.2822(2)	5.267	5.282(1)
$c$	21.419	21.3916(18)	21.411	21.392(2)
$B$ [GPa]	186
Hf 2b	0 0 1/4	0 0 1/4	0 0 1/4
Hf2 4f	1/3 2/3 0.3488	1/3 2/3 0.3488	1/3 2/3 0.3489
Hf3 4f	1/3 2/3 0.5458	1/3 2/3 0.5458	1/3 2/3 0.5461
Ni 6h	0.5110 0.022 1/4	0.5117 0.0234 1/4	0.5107 0.0213 1/4
Ni2 12k	0.156 0.312 0.0514	0.156 0.312 0.0514	0.1563 0.3126 0.0512
Ni3 12k	0.8320 0.6640 0.1495	0.8316 0.6632 0.1495	0.8322 0.6645 0.1496
ZrNi3
$a$	5.319	5.3090(8)	5.267	5.308(2)
$c$	4.305	4.3034(12)	21.411	4.303(1)
$B$ [GPa]	177
Zr 2c	1/3 2/3 1/4	1/3 2/3 1/4	1/3 2/3 1/4
Ni 6h	0.8435 0.687 1/4	0.829 0.658 1/4	0.844291 0.688581 1/4

Table 3. Table 3: Calculated EFG values for HfNi 3 in units of 10 21 V/m 2 and asymmetry parameters

Probe	Lattice Site	EFG	asymmetry
			parameter ($\eta$)
no probe	Hf1 2b 0 0 1/4	-1.3	0
(pure compound)	Hf2 4f 1/3 2/3 0.3488(2)	-5.0	0
	Hf3 4f 1/3 2/3 0.5458(1)	-2.2	0
181Ta	Hf1 2b 0 0 1/4	-1.7	0
	Hf2 4f 1/3 2/3 0.3488(2)	-7.1	0
	Hf3 4f 1/3 2/3 0.5458(1)	-3.5	0

Equations

$$G_{2}(t)=\Big{[}S_{20}(\eta)+\sum^{3}_{i=1}S_{2i}(\eta)cos(\omega_{i}t)exp(-\delta\omega_{i}t)exp\big{[}\frac{-(\omega_{i}\tau_{R})^{2}}{2}\big{]}\Big{]}.$$

$$G_2(t)=\sum _i{f}_i{G}_2^i(t)$$

$${\omega }_Q=\frac{eQ{V}_{zz}}{4I(2I-1)\hslash }.$$

$$\eta =\frac{(V_{xx}-V_{yy})}{V_{zz}}$$

$${\omega }_Q(T)={\omega }_Q(0)[1-\beta T^{3/2}]$$

$${\omega }_Q(T)={\omega }_Q(0)[1-\alpha T].$$

Full text

††journal: Journal of Alloys and Compounds

1 Introduction

The elements zirconium and hafnium are alloyed with cobalt, nickel, titanium, palladium etc. to form many intermetallic compounds which have technological applications. Zirconium-nickel alloys are found to have useful hydrogen storage properties. It was shown that the compounds Zr8Ni21, Zr9Ni11, Zr7Ni10, Zr2Ni7, ZrNi are good hydrogen absorbing materials to form interstitial metal hydrides (MH) which have important application in nickel metal hydride (NiMH) batteries as negative electrode material. The electrochemical properties of several Zr-Ni intermetallic compounds were studied earlier [2, 3, 4, 1, 5, 6] by different workers. In ZrNi3, catalytic hydrogen activity was reported by Wright et al. [7]. The hafnium alloyed with nickel, niobium, and tantalum are also useful and can withstand high temperature and pressure. Hafnium alloys are useful in medical implants and devices due to their bio-compatibility and corrosion resistance [8]. The alloys of Ni-Ti-Hf exhibit shape memory behavior [9]. Intermetallic compounds of Hf and transition metals (Fe, Co, Pd, Pt) have also hydrogen storage properties [10], with high H/M ratio at room temperature.

Time differential perturbed angular correlation (TDPAC) or simply PAC is an important nuclear technique to study the structural properties of compounds that contain hafnium and zirconium. Effects of a $\gamma$-$\gamma$ angular correlation in a crystalline environment are measured by this technique through hyperfine interaction (electric quadrupole and/or magnetic dipole). In electric quadrupole interaction, the quadrupole moment of the probe nucleus interacts with the electric field gradient (EFG) that arises in a crystalline material with noncubic symmetry due to charge distribution of the probe environment. In magnetic interaction, the magnetic dipole moment of the probe nucleus interacts with the internal/external magnetic field. Using this technique, several studies in Hf/Zr-Ni systems were carried out earlier to investigate their EFGs and magnetic properties [11, 12, 13, 14, 15, 16, 17]. Recently we have studied the structural properties of (Zr/Hf)8Ni21, (Zr/Hf)7Ni10 using the PAC technique [18, 19]. However, we do not find any previous PAC studies in (Zr/Hf)Ni3. In the present report, attempts have been made to produce the intermetallic compounds (Zr/Hf)Ni3 by arc melting of the constituent elements taken in stoichiometric ratios and characterize them by PAC spectroscopy. According to Becle et al. [20], the ZrNi3 was formed from Zr2Ni5 and Zr2Ni7 at 940∘C or below by a peritectoid reaction. The stable phase of ZrNi3 was obtained at room temperature by annealing the sample at $\sim$860∘C. They found that the phase ZrNi3 was not stable and decayed at 940∘C following 4ZrNi${}_{3}\rightarrow$Zr2Ni7+Zr2Ni5. The phase ZrNi3, however, was not found by other workers [21, 22, 23]. J. H. N. Van Vucht [24] also failed to produce ZrNi3 by replacing Ti with Zr in TiNi3. From theoretical calculations by density functional theory (DFT), the compounds ZrNi3 and HfNi3 were found to be highly stable alloys. Values of enthalpies of formation of these compounds were reported to be -0.36 eV/atom (ZrNi3) and -0.44 eV/atom (HfNi3) [25]. In order to help identifying the different phases produced in the investigated samples, the electric field gradients were calculated by density functional theory (DFT) and compared with the measured EFGs. The temperature dependent PAC measurements enabled us to find any structural changes in the material and give information on the structural stability of the compound.

The ZrNi3 is known to be a hexagonal close-packed compound of the SnNi3 type with space group $P6_{3}/mmc$. The lattice parameters were reported to be $a$=5.309 Å and $c$=4.303 Å [20]. Crystal structure of HfNi3 was also reported by L. Bsenko [26]. It was found [26] that HfNi3 exists in two modifications. A high temperature $\alpha$-HfNi3 phase and a low temperature $\beta$-HfNi3 phase. The crystal parameters for the two phases were reported to be $a$=5.27 Å, $c$=19.2324 Å and $a$=5.2822 Å, $c$=21.3916 Å for the $\alpha$ and $\beta$ phases, respectively.

2 Experimental details

To produce the intermetallic compounds ZrNi3 and HfNi3, stoichiometric amounts of constituent elements procured from M/S Alfa Aesar were taken. The purity of the metals used were : Zr-99.2% (excluding Hf), Hf-99.95%(excluding Zr) and Ni-99.98%. To introduce the 181Hf probe, each sample was remelted by adding an active piece of Hf wire ($\sim$1 mg). Shiny globule samples were formed after melting in the arc furnace. Natural Hf ($\sim$ 30% 180Hf) was pre-activated to 181Hf in Dhruba reactor, Mumbai by thermal neutron capture with a flux $\sim$ 1013/cm2/s for 7 days. Samples were then sealed in evacuated quartz tubes to carry out measurements at high temperatures. Separate inactive stoichiometric samples of ZrNi3 and HfNi3 were also prepared in similar manners for X-ray diffraction and TEM/energy dispersive X-ray spectroscopy (EDX) measurements. XRD measurements were carried out using the Rigaku X-ray diffractometer TTRAX-III and Cu $K_{\alpha}$ radiation. Transmission electron microscopy (TEM) measurements were carried out using FEI, Tecnai G2 F30, S-Twin microscope equipped with a high angle annular dark-field (HAADF) detector, a scanning unit and a energy dispersive X-ray spectroscopy (EDX) unit to perform the scanning transmission electron microscopy (STEM-HAADF-EDX).

The perturbed angular correlation is a nuclear technique to measure the hyperfine interactions between the nuclear moments of the probe nucleus and the hyperfine fields present in the investigated sample. The probe 181Hf emits two successive $\gamma$-rays, 133 and 482 keV, passing through the 482 keV intermediate level ($T_{1/2}$=10.8 ns) with a spin angular momentum $I$=5/2${}^{+}\hbar$ [27]. The extra-nuclear electric field gradients present in the sample interact with the nuclear quadrupole moment of the intermediate level ($Q$=2.35 b [27]). Due to this interaction, the angular correlation of the 133-482 keV $\gamma$-$\gamma$ cascade is perturbed. The perturbation function is given by [28],

[TABLE]

The above expression of perturbation function is valid for a polycrystalline sample and for $I$=5/2+ of intermediate state of the probe nucleus. The frequencies $\omega_{i}$ are the transition frequencies between different $m$-sublevels arising due to hyperfine splitting. A damping of perturbation function (Lorentzian) was considered through the first exponential which can arise due to structural defects in the sample. Here, $\delta$ is the frequency distribution width. The finite time resolution ($\tau_{R}$) of the coincidence set up was considered through the second exponential. If more than one quadrupole interaction is present in the sample due to the presence of different component phases or due to two or more non-equivalent sites of a particular phase, the perturbation function can be written as

[TABLE]

where, $f_{i}$ is the fraction of the $i$-th component and $G^{i}_{2}(t)$ is the corresponding perturbation function. A fitting to expression (1) determines the quadrupole frequency $\omega_{Q}$ through the measured values of $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$. The quadrupole frequency is directly related to the electric field gradient ($V_{zz}$) through the relation

[TABLE]

For an axially symmetric EFG ($\eta$=0), $\omega_{Q}$ is related to $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ by $\omega_{Q}$=$\omega_{1}$/6=$\omega_{2}$/12=$\omega_{3}$/18. The asymmetry parameter is defined as the ratio

[TABLE]

and its value lies between 0 and 1. For $\eta\neq$0, this simple relation between $\omega_{Q}$ and $\omega_{i}$’s does not hold but, produces a more complex relation [29].

A four detector LaBr3(Ce)-BaF2 set up was used for present TDPAC measurements. The crystal sizes were 38$\times$25 mm2 and 51$\times$51 mm2 for LaBr3(Ce) and BaF2, respectively. The 133 keV $\gamma$-rays were detected in the LaBr3(Ce) detector and the 482 keV $\gamma$-rays were detected in the BaF2 detector. Standard slow-fast coincidence assemblies were employed to acquire four coincidence spectra at 180o and 90o [30]. A typical prompt time resolution (FWHM) of $\sim$800 ps has been obtained for the energy window settings of 181Ta $\gamma$-rays. The perturbation function $G_{2}$($t$) was obtained from the ratio of coincidence counts at 180o and 90o. Details on the experimental set up and data acquisition can be found in our earlier report [30].

3 PAC Results

3.1 Stoichiometric ZrNi3 sample

The X-ray powder diffraction pattern found in the stoichiometric sample of ZrNi3 is shown in Figure 1. The peaks in the spectrum were identified using ICDD database, 2009. Presence of Zr2Ni7 [PDF Card No.: 01-071-0543], ZrNi3 [PDF Card No.: 01-029-0946], Zr8Ni21 [PDF Card No.: 01-071-2622] and Zr7Ni10 [PDF Card No.: 03-066-0045] phases have been found in the XRD spectrum. Analysis of the X-ray powder pattern was carried out by FULLPROF software package [31] using the known crystallographic parameters of Zr2Ni7 [32], ZrNi3 [20], Zr8Ni21 [33] and Zr7Ni10 [34]. From present XRD analysis, refined values of lattice parameters obtained are shown in Table 3.1. The presence of ZrNi3 phase in this stoichiometric sample of ZrNi3 has been observed from TEM/EDX measurement (Figure 2) also. The atomic percentages for Zr and Ni at the indicated spot have been found to be 24.4(3) and 75.6(2), respectively. Selected area electron diffraction (SAED) pattern obtained from a region marked by a dotted circle for the same particle in the stoichiometric sample of ZrNi3 is shown in Fig. 2. Some of the measured interplaner spacings ($d$-spacing) from the SAED pattern are 2.07(4) Å, 2.86(4) Å and 3.60(4) Å. These measured $d$-spacings are very close to the (201), (110) and (101) inter-planer spacing of hexagonal ZrNi3 (JCPDS $\#$ 29-0946), respectively. This further confirms the presence of ZrNi3 phase in the sample.

The PAC spectrum in the stoichiometric sample of ZrNi3 at room temperature is shown in Figure 3. The spectrum was best fitted by considering four electric quadrupole interactions. The sample produced was found to have non-random orientation of microcrystals and the spectrum was fitted by considering free $S_{2n}$ coefficients. The results of different components found are shown in Figure 4. The main frequency component ($\sim$48%) produces values of $\omega_{Q}$=70.9(2) Mrad/s, $\eta$=0.25(2). This component can be assigned to Zr2Ni7 by comparing the values of $\omega_{Q}$ and $\eta$ with the earlier reported results in Zr2Ni7 [12, 17]. From previous PAC measurements in ZrNi5 also, a similar component to this was obtained and attributed to Zr2Ni7 [11, 35]. The ZrNi5 has a cubic crystal structure and no EFG at the probe site is expected due to ZrNi5. The component 2 with a symmetric EFG ($\eta\sim$0) can be attributed to ZrNi3 by comparing with our calculated results from density functional theory (discussed later). The crystal structure of ZrNi3 is hexagonal close-packed and, therefore, a value of $\eta$=0 is expected for this compound. However, this is found to be a minor phase compared to other phases produced in this sample. The results of component 3 can be compared with our recent results in Zr8Ni21 [18]. This component is found to be similar to one component of Zr8Ni21 found from our previous measurements [18] and can, therefore, be attributed to Zr8Ni21. The component 4 can be attributed to Zr7Ni10 by comparing with the results found in Zr7Ni10 [19]. A similar component to this was found in Zr8Ni21 also where it was attributed to Zr7Ni10 [18].

The results of temperature dependent PAC measurements in the stoichiometric ZrNi3 are shown in the Figure 4. The corresponding TDPAC spectra are shown in Figure 3. It is found that the component fraction due to ZrNi3 is present in the temperature range (77-973 K). At 77 K, the site fraction of ZrNi3 was found to be maximum ($\sim$30%). In the temperature range 77-873 K, four component fractions are found to be present with no appreciable change in parameters. The fractional variations of different components are shown in Figure 4. At 973 K, a distinct change in PAC spectrum has been observed. At this temperature, the component due to Zr7Ni10 disappears. On the other hand, a new frequency component with values of $\omega_{Q}$=53.0(4) Mrad/s, $\eta$=0 appears. This component is similar to that found in Zr8Ni21 at 1073 K [18]. The site percentage of this new component is found to be $\sim$39% at this temperature and it enhances abruptly at 1073 K ($\sim$90%). At 1073 K, only two components are found to be present. The minor component found at this temperature is due to Zr2Ni7. At this temperature, ZrNi3 and Zr8Ni21 phases completely disappear. We have repeated the measurement at room temperature after the measurement at 1073 K. The remeasured spectrum at room temperature produces a strong electric quadrupole interaction ($\sim$81%) with values of $\omega_{Q}$=56.5(1) Mrad/s, $\eta$=0. This component can be recognized as the same component that appeared at 973 and 1073 K. Besides this, two other minor components are found here. The component due to Zr8Ni21 reappears with a small fraction ($\sim$8%) and the component due to Zr2Ni7 is also found to be present ($\sim$10%). No component of ZrNi3 at room temperature after heating the sample to 1073 K has been observed. This indicates a decomposition of ZrNi3 at 1073 K.

From previous XRD measurements [36], ZrNi3 was found to be produced after heating ZrCrNi-H2 system to 1024 K and a decomposition of ZrNi3 to Zr2Ni7 was observed [36] after heating the sample to 1083 K. The results of present PAC measurements, therefore, support the results of previous XRD measurements [36]. On the other hand, decomposition of ZrNi3 at 1213 K as reported by Becle et al. [20] is not supported.

In the present Zr-Ni sample, the predominant component found at 1073 K and subsequently at room temperature can probably be assigned to Hf. At temperatures below 973 K, the probe atoms which were settled at various lattice sites come out from the lattice positions after gaining sufficient energy at high temperature. It seems that at 973 K, probe atoms have detached partially ($\sim$39%) from the Zr-Ni compounds and at 1073 K, only a small fraction of the probe nucleus ($\sim$10%) is attached with the compounds. A similar phenomenon was observed from our recent PAC investigation in Zr8Ni21 [18].

The evolution of quadrupole frequency, asymmetry parameter and site fraction with temperature for different components observed are shown in Figure 4. It is found that quadrupole frequencies for the components Zr2Ni7, ZrNi3 and Zr8Ni21 vary with temperature following $T^{3/2}$ relationship. For the Zr8Ni21 component, a similar temperature dependent behavior was observed from our previous PAC investigation in Zr8Ni21 [18]. For these three components, values of $\omega_{Q}$ have been fitted using the relation

[TABLE]

where, $\omega_{Q}(0)$ is the extrapolated value at 0 K. The results of $\eta$ and site fraction for different components are also plotted (Figure 4). These results do not show large variations except the site fraction of Zr2Ni7 which decreases drastically at 1073 K ($\sim$10%) compared to the fraction found at 973 K ($\sim$41%). The fitted results are listed in Table 2. Contrary to these, $\omega_{Q}$ for the component Zr7Ni10 is found to obey a linear temperature dependent behavior. In this case, we have fitted the results of $\omega_{Q}(T)$ using the relation

[TABLE]

Both $T^{3/2}$ and $T$ variations of EFG (proportional to $\omega_{Q}$) for metallic and intermetallic systems are found in literature [37].

3.2 Stoichiometric HfNi3 sample

The XRD measurement in the stoichiometric sample of HfNi3 has been carried out at room temperature (Figure 5). The peaks in the spectrum were identified using ICDD database, 2009. Presence of $\beta$-HfNi3 [PDF Card No.: 01-071-0476], $\alpha$-HfNi3 [PDF Card No.: 01-071-0474], Hf2Ni7 [PDF Card No.: 01-074-6880] and Hf8Ni21 [PDF Card No.: 01-074-0476] phases have been found in the XRD spectrum. Analysis of the X-ray powder pattern was carried out by FULLPROF software package [31] using the known crystallographic data of $\beta$-HfNi3 [26], $\alpha$-HfNi3 [26], Hf2Ni7 [38] and Hf8Ni21 [39]. From present XRD analysis, refined values of lattice parameters found are shown in Table 3.1. It was reported [40] that high temperature phase of HfNi3 was formed from the melt and Hf2Ni7 by peritectic reaction. Apart from HfNi3, there are phases due to Hf2Ni7 and Hf8Ni21. But, no prominent peak due to Hf7Ni10 was found in the XRD spectrum. The phase HfNi3 has been observed from our TEM/EDX measurement also (Figure 6). The atomic percentages for Hf and Ni at the indicated spot have been found to be 24.4(2) and 74.3(2), respectively. The SAED pattern obtained from a region marked by a dotted circle for the same particle in the stoichiometric sample of HfNi3 is shown in Fig. 6. Some of the measured interplaner spacings ($d$-spacing) from the SAED pattern are 2.06(4) Å, 2.31(4) Å and 3.41(4) Å. These measured $d$-spacings are very close to the (205), (201) and (104) inter-planer spacing of hexagonal $\beta$-HfNi3 (JCPDS $\#$ 71-0475), respectively. This further confirms the presence of $\beta$-HfNi3 phase in the sample.

The PAC spectrum observed in HfNi3 at room temperature after preparing the sample in argon arc furnace is shown in Figure 7. It is found that five electric quadrupole frequencies are required to fit the time spectrum. Analysis was done by considering free $S_{2n}$ coefficients. The major component ($\sim$32%) found with values of $\omega_{Q}$=32.0(3) Mrad/s, $\eta$=0 can be attributed to HfNi3 by comparing with our calculated results from DFT (discussed later). From our previous studies in Hf8Ni21 [18] and Hf7Ni10 [19], a component similar to this was observed and tentatively assigned to HfNi3. Component 2 can also be attributed to HfNi3 because the values of EFG and asymmetry parameter for this component are found to be in good agreement with the calculated results from DFT (discussed later). The component 3 has been attributed to Hf2Ni7 by comparing with the earlier reported results in Hf2Ni7 [12, 11]. From previous PAC measurement in HfNi5, a similar component to this was also found and attributed to Hf2Ni7 [11]. The component 4 ($\sim$14%) with values of $\omega_{Q}$=94.8(6) Mrad/s and $\eta$=0.67(2) can be assigned to Hf8Ni21. From our recent investigation in Hf8Ni21 [18], similar values of quadrupole frequency and asymmetry parameter were found. In Hf8Ni21, however, two non-equivalent Hf sites were found. But, in this stoichiometric HfNi3 sample, we have found only one site of Hf8Ni21. The other non-equivalent site of Hf8Ni21 is not observed here. Besides these, a component with symmetric EFG (component 5) was found which can be attributed to pure hcp Hf by comparing the values of $\omega_{Q}$ and $\eta$ with earlier reported results [41]. Probably, this component arises due to unreacted Hf with Ni. Decrease of this component at 973 and 1073 K indicates that more Hf reacts with Ni to form compounds at high temperatures.

Temperature dependent PAC results are shown in Figure 8. The corresponding PAC spectra are shown in Figure 7. In the temperature range 77-873 K, there are no appreciable changes in the PAC spectra. At 973 K, the component Hf8Ni21 disappears. The component due to Hf8Ni21 does not appear at 1073 K also. But other four components are found to exist at 1073 K. However, unlike stoichiometric ZrNi3, no additional component is observed at 1073 K. The PAC measurement was then repeated at room temperature. At this temperature, two components of HfNi3 reappear which indicates that HfNi3 is a stable phase. The component Hf8Ni21 does not appear when remeasured at room temperature. The components of Hf2Ni7 and Hf are found to be present when remeasured at room temperature.

The evolution of the quadrupole frequency, $\eta$ and site fraction with temperature for the different components observed in stoichiometric HfNi3 are shown in Figure 8. The components Hf8Ni21 and Hf2Ni7 follow the same $T^{3/2}$ temperature dependence as found for Zr8Ni21 and Zr2Ni7 in ZrNi3. The EFG for the two components of HfNi3 are found to vary different manner. The quadrupole frequency of HfNi${}_{3}^{(2)}$ varies with temperature following $T^{3/2}$ relationship (Eqn. 5). On the other hand, a linear temperature dependent behavior (Eqn. 6) was found for the HfNi${}_{3}^{(1)}$ component. Variation of the quadrupole frequency for the hexagonal Hf was also found to be linear. The fitted results are shown in Table 2. The variations of $\eta$ and site fractions for different components do not show large changes (Figure 8).

4 DFT calculations

The first-principles density functional theory (DFT) calculations were performed with the WIEN2k simulation package [42] based on the full potential (linearized) augmented plane waves method (FP (L)APW). Electronic exchange-correlation energy was treated with generalized gradient approximation (GGA) parametrized by Perdew-Burke-Ernzerhof (PBE) [43]. In our calculations the muffin-tin radii for Hf, Ni, Zr and Ta were 2.3, 2.1, 2.3 and 2.3 a. u., respectively. The cut-off parameter $R_{mt}K_{max}$ for limiting the number of plane waves was set to 7.0, where $R_{mt}$ is the smallest value of all atomic sphere radii and $K_{max}$ is the largest reciprocal lattice vector used in the plane wave expansion.

The Brillouin zone integrations within the self-consistency cycles were performed via a tetrahedron method [44], using 6-50 $k$ points in the irreducible wedge of the Brillouin zone (4$\times$4$\times$2 and 8$\times$8$\times$8 meshes for Ta doped HfNi3 and ZrNi3, respectively) for the supercell calculations. The atomic positions were relaxed according to Hellmann-Feynman forces calculated at the end of each self-consistent cycle, with the force minimization criterion 2 mRy/a.u.. In our calculations the self-consistency was achieved by demanding the convergence of the integrated charge difference between last two iterations to be smaller than 10*-5* $e$. All the calculations refer to zero temperature.

4.1 HfNi3

HfNi3 at the temperatures below 1200∘C, has the $\gamma$-Ta(Pd,Rh)3-type structure, with a stacking of ten AB3 layers in the sequence ABCBCACBCB. The space group is $P6_{3}/mmc$ and the unit cell dimensions are $a$=5.2822(2) Å, $c$=21.3916(18) Å at room temperature [26]. This structure contains 40 atoms in the unit cell, distributed at 6 non-equivalent crystallographic positions, 3 for Hf atoms and 3 for Ni atoms (Table 3.1).

After obtaining the optimized structural parameters, we constructed 2$\times$2$\times$1 supercell from periodically repeating unit cells of the host crystals. To simulate PAC measurements at Hf1 position, we replaced one Hf atom in the supercell at the position (0 0 1/4) with Ta (Figure 9a [45]). In the case of Ta at the Hf2 and Hf3 positions due to the complexity of the calculations, we had to replace two Hf atoms at the corresponding position with Ta, thus obtaining the cell with 50 non-equivalent atoms (Figure 9b and c). We checked that the two Ta atoms are sufficiently far from each other (11.1 Å) to avoid significant impurity-impurity interactions. After determing the self-consistent charge density we obtain the electric field gradient (EFG) tensor $V_{ij}$ using the method developed in reference [46]. The usual convention is to designate the largest component of the EFG tensor as $V_{zz}$. The asymmetry parameter $\eta$ is then given by $\eta$= ($V_{xx}$-$V_{yy}$)/$V_{zz}$, where $|V_{zz}|\geq|V_{yy}|\geq|V_{xx}|$. All the calculations refer to zero temperature.

The theoretically determined cell and structure parameters for the investigated structure, along with the experimental values obtained from X-ray diffraction measurements are given in Table 3.1. The theoretical volume slightly overestimates the experimental one. The bulk modulus $B_{0}$, obtained by fitting the data to the Murnaghan’s equation of state [47] is also given in Table 3.1. The calculated formation enthalpy, -0.50 eV/atom, is in good agreement with the earlier measured (-0.52 eV/atom [49]) and calculated (-0.44 eV/atom [25], -0.54 eV/atom [50]) values.

The calculated EFGs in the pure compound as well as at Ta probe position in the $\beta$-HfNi3 are given in Table 3.1. It can be observed that EFG is smallest at Hf1 position and the largest at Hf2 position. This trend preserves also for the electric field gradients calculated at corresponding Ta positions, but the EFGs are now larger from 30% to 60%. We see that the calculated result for EFG at the Ta probe site replacing Hf3 atom (3.5$\times$1021 V/m2) is in excellent agreement with the measured value of EFG=3.7(1)$\times$1021 V/m2 ($\omega_{Q}$(0)=32.9(4) Mrad/s) for the component HfNi${}_{3}^{(1)}$, thus confirming that the mentioned component of the measuered PAC spectra originates from HfNi3. Similarly, the calculated results at the Ta probe site replacing Hf2 atom (Vzz=7.1$\times$1021 V/m2 and $\eta$=0) are in excellent aggrement with our measured values of HfNi${}_{3}^{(2)}$ component (Vzz(0)=7.3(1)$\times$1021 V/m2 and $\eta$=0) which confirms that this component also originates from HfNi3.

4.2 ZrNi3

ZrNi3 crystallizes in the hexagonal Ni3Sn type structure, which possesses two non-equivalent crystallographic positions, Zr 2c and Ni 6h [20]. The optimized lattice constants, which slightly overestimate the experimental values, are given in Table 3.1. The calculated formation enthalpy, -0.41 eV/atom, agrees well with the earlier calculated (-0.36 eV/atom [25], -0.46 eV/atom [51]) values. The calculated EFG at Zr position is -3.0$\times$1021 V/m2, with zero asymmetry parameter. In order to simulate PAC measurement, we constructed 2$\times$2$\times$2 supercell from periodically repeating unit cell and then replaced one of the Zr atoms by Ta (Figure 10 [45]). The point group symmetry around the impurity Ta atom remained the same as around the original Zr atom, but the number of non-equivalent positions increased. The calculated EFG at the Ta probe atom -8.4$\times$1021 V/m2 is in excellent agreement with two mutually similar EFG values from measured PAC spectra (8.2 and 8.48$\times$1021 V/m2, corresponding to $\omega_{Q}$(0)= 72.9 and 76 Mrad/s, respectively). The fact that the corresponding calculated asymmetry parameter is zero, enables us to assign the 76 Mrad/s component to ZrNi3 and thus definitely confirm the presence of this phase in our stoichiometric sample.

5 Conclusion

From TDPAC and XRD measurements, multiple phases have been found in the stoichiometric samples of ZrNi3 and HfNi3. The presence of ZrNi3 and HfNi3 in these stoichiometric samples have been confirmed from TDPAC, XRD and TEM/EDX measurements. From PAC studies, it is found that ZrNi3 is produced as a minor phase while the phase HfNi3 is found to be largely produced. Also, our temperature dependent PAC studies show that HfNi3 is a very stable phase. In the stoichiometric samples of ZrNi3 and HfNi3, secondary phases due to (Zr/Hf)8Ni21 and (Zr/Hf)2Ni7 are found to be produced. In ZrNi3, the phase due to Zr7Ni10 is observed while no phase due to Hf7Ni10 is found in HfNi3 sample. Only one and the same crystallographic site of (Zr/Hf)8Ni21 is found in present stoichiometric samples of ZrNi3 and HfNi3 although two non-equivalent sites were found in (Zr/Hf)8Ni21 [18]. The experimental values of EFG and $\eta$ for ZrNi3 and HfNi3 are found to be in excellent agreement with the theoretically calculated values of EFG and $\eta$ at 181Ta impurity sites by the first-principles density functional theory based on the FP (L)APW. From our calculation, three non-equivalent Hf sites in HfNi3 have been found whereas two of these have been observed from PAC measurements. In ZrNi3, on the other hand, the present DFT calculation produces one EFG corresponding to a single Zr site. From our PAC measurements in ZrNi3 also, a single frequency component has been found.

The solubility of Hf in Ni is found to be less compared to Zr in Ni. The Hf solubility in Ni is found to increase with temperature and it decreases again when the temperature is lowered. In Zr-Ni compounds, the binding energy of Hf probe to the lattice sites is not strong enough at high temperature and, probably, the probe atoms are detached from the compound at 1073 K.

The TDPAC is found to be an useful nuclear technique to detect weak component phases that are produced in a material. Particularly, a LaBr3(Ce)-BaF2 set up is found to be very useful for separating the minor component phases when multiple components are present in the sample.

Acknowledgement

The help of Prof. Dr. T. Butz, University of Leipzig, Germany in data analysis is gratefully acknowledged with thanks. We are grateful to Dr. B. Satpati of SINP, Kolkata for TEM/EDX measurements and analysis. We would like to thank A. Karmahapatra and S. Pakhira of SINP, Kolkata for their helps in XRD measurements and data analysis. The present work is supported by the Department of Atomic Energy, Government of India through the Grant no. 12-R&D-SIN-5.02-0102. J. Belo$\check{\text{s}}$ević-$\check{\text{C}}$avor and D. Toprek acknowledge support by The Ministry of Education, Science and Technological Department of the Republic of Serbia through the Grant no. 171001.