Room-Temperature ¹⁸¹Ta(TiO₂): An e-γ TDPAC Study

Abstract

In this work, we report on the hyperfine parameters of the foreign ¹⁸¹Ta probe in the rutile structure of the single crystal TiO₂ using the e−γ and γ−γ time differential perturbed angular correlation (TDPAC) technique. We implanted ¹⁸¹Hf ions into a sample of single crystal rutile TiO₂ in the Bonn Isotope Separator. The implanted sample was then thermally annealed at a temperature of 873 K for 315 min in a vacuum. The ¹⁸¹Hf radioisotopes decayed by β⁻ emission, followed by a cascade to the ground of γ rays or conversion electrons into a stable state ¹⁸¹Ta. The ¹⁸¹Ta probe substitutes the Ti lattice site with a unique nuclear quadrupole interaction, allowing for the precise measurement of the largest electric field gradient (V_zz) and asymmetry parameter (η). The hyperfine parameters obtained from the e−γ TDPAC spectroscopy agree with those of the γ−γ TDPAC spectroscopy at room temperature, apart from a calibration factor, both from our experiments and the literature. This suggests that the electronic recombination following the internal conversion of the L shell electron takes less time (ps) than the intermediate lifetime of the metastable ¹⁸¹Ta state (ns).

1. Introduction

In this work, we investigate the hyperfine parameters obtained by applying the room-temperature spectroscopic technique of e−γ time differential perturbed angular correlation (TDPAC) to ¹⁸¹Hf→¹⁸¹Ta probe nuclei implanted in single crystal rutile TiO₂ as host material (this is called ¹⁸¹Ta(TiO₂) e−γ TDPAC spectroscopy hereinafter).

TiO₂, also known as titanium dioxide in the literature, is a solid inorganic compound with wide technological applications in the industry. Pure TiO₂ crystal is available in three phases: rutile, anatase, and brookite [1]. Anatase and brookite are metastable states and thus convert irreversibly into rutile TiO₂ upon heating to 600–800 °C and beyond in air [2]. Typical uses of TiO₂ include but are not limited to (1) photocatalytic applications [3,4], (2) as an ultraviolet-light blocker in sunscreen materials [5], and (3) as a white pigment or whitening material in various everyday commodities such as paints, papers, foods, medicinal pills, tablets, and toothpaste [6].

1.1. State of the Art (¹⁸¹Ta(TiO₂) γ−γ TDPAC Spectroscopy)

The first ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy was done by Adams & Catchen [7]. They measured the hyperfine parameters of a ¹⁸¹Ta probe in bulk polycrystalline rutile TiO₂ over a wide range of temperatures, acquiring signals corresponding to the presence of the single-site electric-field gradients (EFGs), but no local magnetic ordering in ¹⁸¹Ta(TiO₂). This was confirmed by Darriba et al. in their corresponding temperature-dependent study of bulk single crystal rutile ¹⁸¹Ta(TiO₂), accompanied by their ab initio density-functional theory simulations to explain the effect of the background temperature on the EFG of rutile ¹⁸¹Ta(TiO₂) [8].

Further studies of ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy at room temperature were conducted by a group in the Bhabha Atomic Research Centre. They measured the EFGs of the bulk ¹⁸¹Ta(TiO₂) system in both the rutile and anatase crystalline phases [9] and probed the EFGs of the ¹⁸¹Ta probe in the TiO₂ nanocrystal environment [10]. The room-temperature TDPAC signature(s) of the ¹⁸¹Ta probe located at both the surface and the bulk of the TiO₂ nanocrystal (anatase and rutile) were obtained. They simultaneously implanted both ¹¹¹In→¹¹¹Cd and ¹⁸¹Hf→¹⁸¹Ta probe nuclei into a bulk rutile TiO₂ sample and detected signals corresponding to the EFGs of ¹¹¹Cd and ¹⁸¹Ta, located at the substitutional sites of the TiO₂ sample [11]. Their research applications involved the study of the γ-radiation and the leaching behavior on the ¹⁸¹Ta(TiO₂) crystal matrix [12].

Schell et al. conducted ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy over a range of temperatures, the host TiO₂ material being a thin film instead of the usual bulk phase [13,14]. They collected TDPAC signals corresponding to the presence of the EFGs of the ¹⁸¹Ta probe at two unique sites of the thin film TiO₂. The magnitude of the EFGs at one such site suggests that the local environment of the ¹⁸¹Ta probe nucleus is like that of anatase TiO₂, suggesting that the implantation of the ¹⁸¹Hf nucleus modifies the local structure of the TiO₂ thin film.

Finally, at the 2011 International Nuclear Atlantic Conference held in Belo Horizonte (Brazil), Martucci et al. presented their ¹⁸¹Ta(TiO₂) TDPAC and their corresponding X-ray diffraction (XRD) results on the effects of thermal annealing on polycrystalline samples produced using the sol-gel Pechini method [15]. A literature review of the sol-gel Pechini method [16] on the precipitation of TiO₂ powder reveals the formation of anatase and rutile phases, with their ratio determined by the molar ratio of citric acid/metallic Ti cations as reagents [17]. They have shown that if their TiO₂ powder is subjected to a 10 h thermal annealing of 1273 K in the nitrogenic atmosphere, their XRD and ¹⁸¹Hf→¹⁸¹Ta γ−γ TDPAC spectroscopies demonstrate a complete phase transformation from the anatase phase to the rutile phase, consistent with reference [2].

Further literature on TDPAC spectroscopy applied to TiO₂ is available in reference [18], which gives room-temperature results from various TDPAC probes.

1.2. Physics and Properties of Rutile TiO₂

Bulk rutile TiO₂ is spatiality related to the tetragonal space group P4₂/mnm [19]. The lattice parameters of the unit cell are such that it has lattice translational vectors $\overrightarrow{R}={{\displaystyle \sum }}_{i\in \left\{1,2,3\right\}}^{ }{n}_i{\overrightarrow{a}}_i$, where $\forall i\in \left\{1,2,3\right\}, n_i\in \mathbb{Z}$, alongside ${\overrightarrow{a}}_1=a\widehat{x}, {\overrightarrow{a}}_2=a\widehat{y},{\overrightarrow{a}}_3=c\widehat{z}$, with the lattice parameters a = 4.65922 Å and c = 2.9574 Å [20]. The unit cell comprises of $2$ titanium ions, located at both [0,0,0] and [0.5a,0.5a,0.5c], and $4$ oxygen ions, located at [±xa,±xa,0] and [(0.5 ∓ x)a,(0.5 ± x)a,0.5c], where x is defined as the internal parameter, is dimensionless and is experimentally verified as 0.30496 [20].

Under this perfect lattice configuration, both of the titanium ions in the unit cell are site-equivalent, forming bonds with its 6 nearest neighbor oxygen ions under the distorted octahedral geometry. It is more precisely understood as having a D_2h site symmetry. Each of the oxygen atoms in the unit cell are site-equivalent as well, forming a 3-fold coordinated bond with its 3 nearest neighbor titanium ions in a trigonal planar form, their site symmetry being C_2v.

In practice, high-quality single crystal rutile TiO₂ could be commercially manufactured using, for example, the Verneuil method [21]. However, it is known that even for such a refined method, there exists a non-trivial number density of oxygen vacancies (site defects where the supposing oxygen atoms are missing) for freshly fabricated single crystal rutile TiO₂. This phenomenon has been verified over the course of numerous experiments, including a recent 2018 experiment using the Femtosecond Time Resolved Spectroscopy on bulk rutile TiO₂ to determine its charge carrier dynamics. This revealed that the presence of innate oxygen vacancies as a trap site contributes to a fast recombination/annihilation lifetime (<1 ns) of the electrons (as a charge carrier) [22].

The existence of the oxygen vacancies (if numerous enough) creates a large density of localized states in the band gap of rutile TiO₂. As the name implies, electrons in these localized states are not mobile (compared to the delocalized states at the valence and conduction bands). However, quantum tunneling between localized states is possible. With that principle, an electron can travel from one localized state to another, a phenomenon known as “hopping transport”, “variable range hopping”, or “hopping conduction” [23,24].

At the surface and interfaces, the phenomenon of hopping transport becomes a lot more important due to the existence of surface states (another set of localized electronic states) caused by the sharp transition (surface reconstructions) between atoms of both materials located near the interface (air and vacuum can be understood as material in this sense) [23].

Numerous experimental studies have been conducted on the nature of the electronic surface states of rutile TiO₂, which (1) includes the longer vacuum decay length of the Ti(3d) surface electronic states over the O(2p) surface electronic states (discovered using Scanning Tunnel Microscopy) [25], (2) the nature of the oxygen adatoms at the surface (using sensitive Fast Atomic Force Microscopy), which could exist as single quantum dots [26], and (3) a paper that explores on the electronic structures of rutile (110), anatase (101) and anatase (001) single crystal surfaces using Resonant Photoemission and X-ray Absorption Spectroscopy [27]. They have discovered that the rutile single crystal surfaces, prepared under cycles of bombarding and annealing in a vacuum, show a higher concentration of oxygen vacancies over the anatase variant, which is consistent with the findings of [22]. Moreover, they have shown that the method of surface preparation, and thus, the concentration of the oxygen vacancies, is proportional to the $\sigma$-antibonding hybridization between the Ti(3d-e_g) and the O(2p) states (using X-ray Absorption Spectroscopy), and that the $\pi$-bonds between the Ti(3d-t_2g) and the O(2p) states are increasingly disrupted as the amount of surface defects increases (using Resonant Photoemission).

Theoretically, Density Functional Theory (DFT) techniques have been used to study the electronic structure of bulk rutile TiO₂ with various point defects (including oxygen vacancies) [28], alongside the electronic structure of surface rutile TiO₂ in various surface orientations [29].

To encapsulate the discussion in this section, semiconductor oxides like TiO₂ present very interesting electronic properties in the hopping transport (a form of quantum-mechanical tunneling), which, once well-understood, can increase the number of high technological applications of the material, including the enhancement of its photocatalysis capability [24,30].

1.3. Motivation for Applying the e−γ TDPAC Spectroscopy to the TiO₂ Material

The e−γ TDPAC spectroscopy could probe physics typically inaccessible from its much more experimentally established γ−γ TDPAC counterpart. The degree to which the e−γ and γ−γ TDPAC signals differ from each other depends on the timescale (and hence speed) of the host material to provide its recombination electrons to the TDPAC nuclear probe following internal conversion of the excited ¹⁸¹Ta to ground state ¹⁸¹Ta as a TDPAC probe (Further details are contained in Section 2).

This is wonderfully demonstrated in the recent ¹⁸¹Ta(GaN) TDPAC experiment by Barbosa et al. [31], where they have performed a temperature- and “Si- and Zn- dopant” concentration-dependent study. In Figure 1 of the corresponding paper, they had shown that their e−γ signals differ from those of the corresponding γ−γ signals. Furthermore, with the help of their in-house dynamic fitting program PACme^TM and DFT, they can assign electronic charges to the metastable states related to their e−γ signals. These metastable states represent the various stages (around the nanosecond scale) in which the recombination electrons will fill the holes following the internal conversion process of the excited ¹⁸¹Ta to ground state ¹⁸¹Ta as a TDPAC probe (Further details are contained in Section 2 as well).

On a related note on the topic of electronic recombination, Schell et al. have conducted and compared the signals acquired from both single crystals (¹¹¹In→¹¹¹Cd)TiO₂ and (^111mCd→¹¹¹Cd)TiO₂ TDPAC spectroscopies [32]. While both TDPAC spectroscopies exhibit a common set of hyperfine parameters (largest EFG and asymmetry parameter, see Section 2) relating to the Ti-substitutional site, an additional set of hyperfine parameters (with a high asymmetry parameter) is manifested in the (^111mCd→¹¹¹Cd)TiO₂ TDPAC spectroscopy (colloquially referred to as the second site). DFT simulations accompanying this study also revealed that the individual EFGs (and hence the largest EFG and the asymmetry parameter, see Section 2) of a single charge-deficient Cd atom resting in the Ti-substitutional site is consistent with the second site. Noting that the valence number of ¹¹¹In and ^111mCd in TiO₂ are +3 and +2, respectively, (Ti has a valence number of +4), and the fact that the half-life of the intermediate ¹¹¹Cd state is 85ns, ¹¹¹In→¹¹¹Cd requires only one recombination electron from the host TiO₂ material, whereas ^111mCd→¹¹¹Cd needs two. This implies that, at least qualitatively, the time needed to fill up two holes in a single ¹¹¹Cd atom sitting in the Ti-substitutional site is around the order of tens of nanoseconds.

Thus, e−γ TDPAC spectroscopy, complementing γ−γ TDPAC spectroscopy, can probe even further into the electronic structure in TiO₂, both in the bulk and on the surface. For bulk rutile TiO₂, we can compare the degree to which their e−γ and γ−γ ¹⁸¹Ta(TiO₂) TDPAC signals differ, to verify that the recombination lifetime of the electrons is indeed on the sub-ns scale according to [22]. Further possible applications of e−γ TDPAC on TiO₂ could be done on its surfaces and interfaces, where we can further probe the local electronic characters of the Ti and O ions around our TDPAC probe in a more precise manner as compared to other typical experimental methods.

2. Time Differential Perturbed Angular Correlations

TDPAC spectroscopy is a technique used in solid-state nuclear physics. It is sufficiently sensitive to observe the hyperfine interactions between the nuclear moments of the incorporated TDPAC probe and the local electric and magnetic fields of the immediate surrounding host material. This implies that precise measurements of the local EFGs and the local hyperfine magnetic field of the host material can be determined (see references [33,34,35] for an overview). This allows for the study of the structural and electronic properties of the host material applicable to the relevant fields of physics [36,37], chemistry [38], and even biology [39]. One advantage of TDPAC spectroscopy is that it requires only parts per billion number density of TDPAC probe atoms for effective characterization, keeping the electronic structure of the host material virtually intact.

In a typical TDPAC experiment, a suitable TDPAC probe is introduced into a host material. It can occupy an interstitial (in between the native ions in the lattice of the host material) and/or substitutional (directly occupying the space in which the native ions should sit) position, depending on the annealing conditions and probe-incorporation methods. The typical TDPAC probe then undergoes a consecutive two-stage decay (Figure 1a); with each stage of decay, a particular particle is released into the environment and then captured by the relevant detectors for signal acquisition. In the intermediate state of the TDPAC probe, which is defined as the state after the first-stage decay but before the second-stage decay, its nuclear moments can be coupled to the surrounding local electric or magnetic fields of the host material. This process can alter the direction of the emission of the gamma ray or conversion electron associated with the second-stage decay relative to the first-stage decay.

For the more traditional gamma-gamma (γ−γ) TDPAC spectroscopy, γ rays of various energies are released at each stage and acquired by the compatible photomultiplier tubes of the detectors (Figure 1b). Both γ rays are associated with the consecutive de-excitation of the nucleus of the TDPAC probe.

ISOLDE-CERN [41] is equipped to perform electron-gamma (e−γ) TDPAC spectroscopy [42,43] with Siegbahn-type magnetic lenses [44] to detect the conversion electrons arising from the first stage of the decay (Figure 1c), and the corresponding γ rays arising from the second stage of ¹⁸¹Hf decay.

Thus, the subtle difference between γ−γ TDPAC spectroscopy and e−γ TDPAC spectroscopy comes from the change in the local electronic environment of the host material around the vicinity of the implanted TDPAC probe after the first stage of decay. This is especially applicable to e−γ TDPAC spectroscopy because, in the first-stage decay, the excited nucleus interacts with specific inner-shell electron(s), resulting in the ejection of the given electron from the TDPAC probe nucleus. This process is called “internal conversion” in the literature, and the ejected electron is called a “conversion electron” (See left side of Figure 2).

In response to the vacancy left in the electronic inner shell of the TDPAC probe by the conversion electron, an electron cascade (or recombination process) occurs, where electrons from the higher-energy shells fill the lower-energy shells emitting characteristic X-rays, Auger electrons, or both. The process of electron cascade terminates when the vacancy (hole) induced by the ejection of the conversion electron is transferred to the valence shell of the TDPAC probe and filled by “recombination electrons” provided by the host material (See right side of Figure 2).

For the first-stage process of an isomer probe in γ−γ TDPAC spectroscopy, the TDPAC probe nucleus is de-excited, releasing γ rays of a specific energy that thus do not interact with the electron shells of the TDPAC probe. This is not the case with probes decaying via electron capture, such as ¹¹¹In(¹¹¹Cd). However, ¹⁸¹Hf decays via the beta-minus emission to the 1/2⁺ excited nuclear level in ¹⁸¹Ta. This level has a half-life of 17.6 μs [45], which is long enough for electronic rearrangements. Thus, no after-effects [46] from the primary ¹⁸¹Hf decay are expected in the subsequent TDPAC cascades proceeding via the 5/2⁺ state with a 10.8 ns half-life (See Figure 1a for the ¹⁸¹Hf→¹⁸¹Ta decay path).

This scenario implies that if the electron recombination process is at least on the order of the half-life t_1/2 of the intermediate stage of the ¹⁸¹Hf(¹⁸¹Ta) probe or above, we should observe, for e−γ TDPAC spectroscopy, a characteristic dynamic signal determined by the hyperfine parameters that differs significantly from the characteristic signal of the corresponding γ−γ TDPAC spectroscopy. Conversely, if the recombination process is much less than t_1/2 of the intermediate stage of the TDPAC probe, then both e−γ and γ−γ TDPAC spectroscopy will produce the same hyperfine parameters.

The use of e−γ TDPAC spectroscopy dates back to the 1970s [47,48,49,50] with the experimental group HISKP from the University of Bonn. Their e−γ TDPAC spectrometer was significantly upgraded later and brought to its full capacity at the Solid-State Physics ISOLDE-CERN facility in the early 1990s [42,43,44]. Recent experiments using the modern e−γ TDPAC spectrometer include a temperature- and “Si- and Zn-” dopant concentration-dependent study on ¹⁸¹Ta(GaN) [31], which provides signals related to the recombination process and with the help of simulations based on DFT. On this basis, researchers ascribe this phenomenon to variable-range hopping in the low-temperature regime and thermally mobile electrons (electrons in the conduction band) in the high-temperature regime.

Signals from bulk ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy (single crystal and polycrystalline) reveal only significant electric quadrupole interactions but not magnetic dipole interactions. Sufficient annealing allows the detection of γ−γ TDPAC signals corresponding to the presence of the EFGs of a single unique site (See Appendix A for more details). Thus, we expect ¹⁸¹Ta(TiO₂) e−γ TDPAC spectroscopy to produce a signal like that of EFGs corresponding to a single unique site. The EFG magnitudes detected by ¹⁸¹Ta(TiO₂) e−γ TDPAC spectroscopy may differ from that detected by the corresponding γ−γ TDPAC spectroscopy if the recombination time is on the order of the half-life t_1/2 of the intermediate stage of ¹⁸¹Ta. Otherwise, we would expect the same EFG magnitudes from both e−γ and γ−γ TDPAC spectroscopy.

For ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy, the energy of the first (second) γ emission (γ₁ and γ₂ emission) has an energy of 133 keV (482 keV). The perturbation angular correlation function $W\left(\theta ,t\right)$ that describes the time-dependent modification of the direction of the γ₂ emission with respect to that of the γ₁ emission may be expressed as:

$$W\left(\theta ,t\right)={{\displaystyle \sum }}_{k=0, \mathrm{even}}^{k_{\max }}{A}_{kk}{G}_{kk}\left(t\right) P_k\left(\cos \theta \right),$$

(1)

where $A_{k_1{k}_2}$ is the γ₁−γ₂ anisotropy term defined as $A_{k_1{k}_2}=A_k\left({\gamma }_1\right)A_k\left({\gamma }_2\right)$, with $A_k\left({\gamma }_1\right)$ and $A_k\left({\gamma }_2\right)$ called the orientation and correlation anisotropy coefficients [35], $\theta$ is the angle between the γ₁ and γ₂ emission, and $G_{kk}\left(t\right)$ is the time-dependent perturbation factor related to the internal dynamics of the ¹⁸¹Ta probe in the TiO₂ environment and is not related to the external angle $\theta$. It is given explicitly in Equation (4) reference [35]. Finally, $P_k\left(x\right)$ is the Legendre polynomial of order $k$.

We used a four/six-detector setup with planar geometry, implying that each detector forms a 90° angle to its adjacent neighbors. Therefore, the possible values of $\theta$ are 90° and 180°. In a coincidence search, we can express the coincidence count rate $N\left(\theta ,t\right)$ as

$$N\left(\theta ,t\right)=N_0\exp \left(-\frac{t}{{\tau }_N}\right)W\left(\theta ,t\right)+C,$$

(2)

where ${\tau }_N=\left(\ln 2\right)/t_{1/2}$ is the lifetime of the intermediate state, $N_0$ is the coincidence count rate at $t=0$, and $C$ is a constant that describes a time-independent background count rate that arises from random coincidences and is subtracted.

Note that (1) $A_0\left({\gamma }_1\right)=A_0\left({\gamma }_2\right)=1 \mathrm{and} G_{00}\left(t\right)=1$ and (2) the lower-order $k$ terms contribute much more than the higher-order $k$ terms. Setting the cutoff at $k_{\max }=2$ gives the following terms:

$$N\left(90°,t\right)\approx N_0\exp \left(-\frac{t}{{\tau }_N}\right)\left(1-\frac{1}{2}{A}_{22}{G}_{22}\left(t\right)\right),$$

(3)

$$\left(180°,t\right)\approx N_0\exp \left(-\frac{t}{{\tau }_N}\right)\left(1+A_{22}{G}_{22}\left(t\right)\right).$$

(4)

Thus, by combining Equations (3) and (4), we obtain the count rate ratio R(t) as

$$R\left(t\right)=2\left(\frac{N\left(180°,t\right)-N\left(90°,t\right)}{N\left(180°,t\right)+2N\left(90°,t\right)}\right)\approx A_{22}{G}_{22}\left(t\right).$$

(5)

One implicit assumption in the derivation of Equation (5) is that the terms $N\left(90°,t\right)$ and $N\left(180°,t\right)$ are the arithmetic mean of all the possible start-stop coincidences from the different possible combinations of the detectors from the entire detector setup. For example, in standard six-detector γ−γ TDPAC spectroscopy, one counts 24 unique $90°$ start-stop coincidences and six unique $180°$ start-stop coincidences. In the standard four-detector γ−γ TDPAC spectroscopy, one counts eight unique $90°$ start-stop coincidences and four unique $180°$ start-stop coincidences.

However, in four-detector e−γ TDPAC spectroscopy, the electron detector only detects conversion electrons and the γ ray detectors only detect γ rays. In addition, the detection of the conversion electrons serves as the start signal, and the detection of the relevant γ rays serves as the stop signal, so one counts only two unique $90°$ and $180°$ start-stop coincidences each. This reduces the statistics of the e−γ TDPAC spectroscopy with respect to its γ−γ cousin.

Given that the total angular momentum $I$ of the intermediate state of the ¹⁸¹Ta probe is 5/2, a pure electric quadrupole interaction associated with the presence of non-zero EFGs causes the intermediate state $E_0$ to hyperfine split into three sub-energy levels denoted $E_{1/2},E_{3/2},E_{5/2}$. Reference [35] gives

$$E_{\frac{5}{2}}=E_0-2r\cos \left(\frac{\phi }{3}\right)\hslash {\omega }_Q,$$

(6)

$$E_{\frac{3}{2}}=E_0+\left[r\cos \left(\frac{\phi }{3}\right)-\sqrt{3}r\sin \left(\frac{\phi }{3}\right)\right]\hslash {\omega }_Q,$$

(7)

$$E_{\frac{1}{2}}=E_0+\left[r\cos \left(\frac{\phi }{3}\right)+\sqrt{3}r\sin \left(\frac{\phi }{3}\right)\right] \hslash {\omega }_Q.$$

(8)

$\phi$ and the quadrupole frequency ${\omega }_Q$ are given by

$$\cos \left(\phi \right)=\frac{q}{r^3},$$

(9)

$$r=-\sqrt{\left|p\right|},$$

(10)

$$p=-28\left(1+\frac{{\eta }^2}{3}\right),$$

(11)

$$q=-80\left(1-{\eta }^2\right),$$

(12)

$${\omega }_Q=\frac{eQ{V}_{zz}}{40\hslash },$$

(13)

where $Q$ is the quadrupole moment of the intermediate state of the ¹⁸¹Ta probe, which is $2.36\left(5\right)$ barns according to reference [51]. The EFGs of the local TiO₂ environment that surrounds the ¹⁸¹Ta probe can be expressed by using two independent parameters: the largest magnitude of the EFGs, denoted $V_{zz}$, and the asymmetry parameter $\eta \in \left[0,1\right]$, which is given as

$$\eta =\frac{\left|V_{yy}\right|-\left|V_{xx}\right|}{\left|V_{zz}\right|}; \left|V_{xx}\right|\le \left|V_{yy}\right|\le \left|V_{zz}\right|.$$

(14)

Thus, the perturbation function $G_{22}\left(t\right)$ is written as

$$G_{22}\left(t\right)=s_0\left(\eta ,\theta ,\phi \right)+{{\displaystyle \sum }}_{i=1}^3{s}_i\left(\eta ,\theta ,\phi \right)\exp \left(-\delta {\omega }_{i}t\right)\cos \left({\omega }_{i}t\right).$$

(15)

All three precession frequencies ${\omega }_1,{\omega }_2,{\omega }_3$ depend on $V_{zz}$ and $\eta$ and are written as

$${\omega }_1=\left(E_{3/2}-E_{1/2}\right)/\hslash =-2\sqrt{3}r\sin \left(\frac{\phi }{3}\right){\omega }_Q,$$

(16)

$${\omega }_2=\left(E_{5/2}-E_{3/2}\right)/\hslash =-\left[3r\cos \left(\frac{\phi }{3}\right)-\sqrt{3}r\sin \left(\frac{\phi }{3}\right)\right]{\omega }_Q,$$

(17)

$${\omega }_3=\left(E_{5/2}-E_{1/2}\right)/\hslash =-\left[3r\cos \left(\frac{\phi }{3}\right)+\sqrt{3}r\sin \left(\frac{\phi }{3}\right)\right] {\omega }_Q.$$

(18)

Also, $s_i\left(\eta ,\theta ,\phi \right)$ are the coefficients that, for the polycrystalline case, depend only on the asymmetry parameter $\eta$, and for the single crystal case, depend on both $\eta$ and the orientation of the $V_{zz}$ with respect to the axes of the detector plane, and is represented (in our experimental fitting) by two independent parameters of the spherical coordinate system $\theta ,\phi$ [8,52]. $s_0\left(\eta ,\theta ,\phi \right)$ is often called the “hard-core value” in the literature. The values for $s_i\left(\eta ,\theta ,\phi \right)$ for a single crystal host material are derived in references [34,35].

In practice, we also define as follows the fundamental frequency ${\omega }_0$ that frequently appears in the fitting parameters:

$${\omega }_0=6{\omega }_Q.$$

(19)

Experimentally, the lattice of the host material is not perfect, and the local environment (and thus the EFGs) may differ slightly for different implanted probes. The degree of difference between the EFG of different probes is described by the Lorentzian damping factor $\delta$, which causes the exponential damping of $G_{22}\left(t\right)$. In an ensemble of ¹⁸¹Ta probes that contributes to $G_{22}\left(t\right)$, each of the individual precession frequencies follow a Lorentzian distribution with the corresponding mean ${\omega }_i$. For a sufficiently annealed ¹⁸¹Ta(TiO₂) sample, we expect most implantation damage to be repaired, and thus that $\delta$ will be small. This was demonstrated by Darriba et al. [8].

Therefore, the final equation of our experimental fit for ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy is

$$R\left(t\right)\approx A_{22}\left(s_0\left(\eta ,\theta ,\phi \right)+{{\displaystyle \sum }}_{i=1}^3{s}_i\left(\eta ,\theta ,\phi \right)\exp \left(-\delta {\omega }_{i}t\right)\cos \left({\omega }_{i}t\right)\right).$$

(20)

For ¹⁸¹Ta(TiO₂) e−γ TDPAC spectroscopy, we must incorporate $b_k$, the so-called particle parameter correction factor, into $A_{k_1{k}_2}$ defined in Equation (1) [53,54]. Thus, the equation used for our experimental fit is modified from Equation (20) into the following similar form:

$$R\left(t\right)\approx A_{22}^{\prime }\left(s_0\left(\eta ,\theta ,\phi \right)+{{\displaystyle \sum }}_{i=1}^3{s}_i\left(\eta ,\theta ,\phi \right)\exp \left(-\delta {\omega }_{i}t\right)\cos \left({\omega }_{i}t\right)\right); A_{22}^{\prime }=b_2{A}_{22}.$$

(21)

3. Experimental Setup

3.1. Implantation of ¹⁸¹Hf Isotope at the Bonn Radioisotope Separator

The radioactive isotope ¹⁸¹Hf (half-life of 42.4 days) was produced by irradiating 1.5 mg of hafnium oxide (HfO₂) powder for 6 days in a thermal-neutron flux of ${10}^{15} n {\mathrm{cm}}^{-2} {\mathrm{s}}^{-1}$ in the V4 beam tube of the high-flux reactor of Institut Laue-Langevin (ILL) in Grenoble, France. The ¹⁸⁰Hf isotope in the hafnium oxide undergoes neutron capture [¹⁸⁰Hf(n,γ)] to produce the desired ¹⁸¹Hf isotope.

Single crystal rutile TiO₂ substrates measuring 10 × 10 × 1.0 mm³ were commercially purchased from Crystal GmbH in Germany [55] (See Figure 3 for a photograph of the crystal(s)). The samples were then brought to the Bonn Radioisotope Separator, where a long side of the sample was ion-implanted with ¹⁸¹Hf isotope at 80 keV and an incidence angle of 10° with respect to the normal of the sample. The beam-sweeping technique was employed, ensuring that the implanted face of the sample was homogeneously covered with the implanted ¹⁸¹Hf isotope.

Technical details of the implantation process and the Bonn Radioisotope Separator are available in references [13,56].

A quick and naive calculation of the stopping and range of ions in matter (SRIM) for the implantation of ¹⁸¹Hf in rutile TiO₂ using the stoichiometric composition and with a density of 4.25 g/cm³ [57] reveals a peak of 233 Å with a standard deviation of 60 Å (Figure 4). This implies that the ¹⁸¹Hf isotope is deeply embedded in the bulk region of the rutile TiO₂ substrate. Thus, short of thermal diffusion at sufficiently high temperatures, we would expect our resulting TDPAC signals of ¹⁸¹Ta to probe the bulk region of TiO₂.

3.2. Data Accquisition of ¹⁸¹Ta(TiO₂) γ−γ and e−γ TDPAC Spectroscopy

The resulting samples were then safely transported to the SSP lab at ISOLDE, CERN, where they were sufficiently annealed at a temperature of 873 K in a vacuum for 5.25 h to repair the damage and the distortion to the TiO₂ lattice caused by ¹⁸¹Hf-ion implantation and to allow ¹⁸¹Hf to diffuse to a preferable site.

The samples were then analyzed by both γ−γ and e−γ TDPAC spectroscopies. For the γ−γ TDPAC spectroscopy, six CeBr₃ scintillators in the standard geometry [58] were used in a coincidence setup of the 133−482 keV γ cascade of the ¹⁸¹Ta isotope (Figure 1a,b).

For the e−γ TDPAC spectroscopy, the coincidence setup consists of two electron detectors and two γ-ray detectors. The electron detectors consist of magnetic lenses of the Siegbahn type [43] that focus the 122 keV conversion electrons arising from the internal conversion onto fast plastic scintillator detectors (Figure 1a,c). In contrast, the corresponding γ-ray detectors consist of BaF₂ scintillators that detect the 482 keV γ rays arising from the decay of the 5/2⁺ level (Figure 1a,b). Note that due to the magnetic dispersive elements, the electron detectors have a better energy resolution than the corresponding γ-ray detectors [43,44].

For e−γ TDPAC spectroscopy, we placed one such sample in the preferred Raghavan geometry configuration [59], where the long side of the ¹⁸¹Ta(TiO₂) sample (the 10 × 10 mm² side) makes a 45° angle with the detectors in the x-y plane of the detector setup. Another such sample was used for the γ−γ TDPAC spectroscopy. However, the sample could not be mounted in the Raghavan geometry due to vertical constraints. Hence the sample was additionally tilted at an angle with respect to the z-axis.

The corresponding raw TDPAC signals were then processed, first through Interlude^TM [60] to extract out R(t) and then through Gfit19 ^TM, which is a modified version of the NNfit ^TM software package [61,62,63]. For the six-detector γ−γ TDPAC spectroscopy setup, we produce four R(t) graphs, three of which correspond to the four-detector setup in the x-y, x-z, and y-z plane, while the last R(t) graph corresponds to the full six-detector setup. With the four-detector e−γ TDPAC spectroscopy setup, we produce only a single R(t) graph. Given that the concept of angles is ill-defined for R(t) corresponding to the six-detector setup, we use the polycrystalline fit instead of the single crystal fit, unlike the other R(t) graphs. In addition, we verified with Nightmare ^TM that R(t) fits independently [64].

Based on private communication with Correia [53,54,65,66,67,68] and alongside the geometry of both our TDPAC setups, we use the following parameters for the effective A_kk values for the γ−γ TDPAC spectroscopy:

$$A_{22}=-0.2213, A_{24}=-0.0276, A_{42}=-0.1491, A_{44}=-0.0186 .$$

(22)

For our e−γ TPDAC spectroscopy, we use

$$A_{22}=-0.3077, A_{24}=-0.0494, A_{42}=-0.0803, A_{44}=-0.0129.$$

(23)

Finally, our time calibration is 0.101 ns/channel for our γ−γ TDPAC spectroscopy and 0.195 ns/channel for our e−γ TDPAC spectroscopy.

3.3. X-ray Diffraction of the ¹⁸¹Ta(TiO₂) Sample after the γ−γ and e−γ TDPAC Spectroscopy

After the completion of both ¹⁸¹Ta(TiO₂) γ−γ and e−γ TDPAC spectroscopies, both samples, as well as another fresh and commercially manufactured rutile TiO₂ sample bought from Crystals GmbH [55] (for reference), were sent to the University of São Paulo, Brazil. X-ray diffraction (XRD) using the Empyrean diffractometer (Malvern Panalytical, [69]) was conducted on the samples to determine their global crystalline structure unambiguously. We have also compared our diffractogram(s) to the corresponding results produced in the paper of Guo et al. [70], in which the mentioned researchers fabricated their own single crystal rutile TiO₂ from powdered anatase TiO₂ using the floating zone method (See [71] for a detailed explanation of the floating zone method).

In the Empyrean diffractometer, the crystal is, with its long side (the 10 × 10 mm² side), placed face up in the goniometer and was subjected to a beam of X-rays ($K_{\alpha }$, $K_{\beta }$ lines) produced by the deceleration of the electrons in the copper plate target. The voltage and current of the electrons are 45 kV and 40 mA. The sample is then rotated through an angle of 5 to 80 degrees, with a step size of 0.013 degrees and a step time of 2 s. X-rays (both background and crystal-reflected) in the direction of the angle are hence captured by the detectors in the Empyrean diffractometer. To generate the diffractogram, we subtract the background intensity (measurements with the absence of the crystal) from the observed intensity, which is succinctly referred to as “Icalc” in the outputs of the Empyrean diffractometer. Considering the kinetic energy of the electrons, and that we are analyzing a commercially manufactured single crystal rutile TiO₂, we expect our diffractogram to mainly reflect a single dominant peak in a specified crystalline direction.

The peaks of the diffractogram are fitted with the corresponding analysis software graciously provided onboard the Empyrean diffractometer, giving various information such as the full width at half maximum (FWHM), crystalline size, and micro strain. With this XRD analysis, we could investigate, in more detail, the effects of the implantation of the ¹⁸¹Hf isotopes and the subsequent annealing with respect to our single crystal rutile TiO₂ sample.

As this discussion detracts us from the main TDPAC study, we would only mention and validate the results of the XRD diffraction in the main section of the paper. More details on the effects of ion implantation and thermal annealing are elaborated in Appendix A.

4. Results

4.1. γ−γ TDPAC Spectroscopy Setup

The left half of Figure 5 shows R(t) (blue data points) in different geometries after the corresponding raw TDPAC signal was processed with Interlude. The right half of Figure 5 shows the corresponding cosine fast Fourier transform (FFT), which displays the frequencies ${\omega }_i,i\in \left\{1,2,3\right\}$ that drive R(t) (Equation (20)), the peak amplitudes $[A_{22}{s}_i\left(\eta \right), i\in \left\{1,2,3\right\}$], and their spread (delta Lorentzian $\delta$).

Information on the hyperfine parameters contained in R(t), along with the corresponding FFT, are then extracted in the fitting process with Gfit19^TM (red curves in Figure 5).

The peaks of the FFT were marked with magenta arrows, with the values of the hyperfine parameters and the relevant experimental information tabulated in

Table 1. Table of fitted hyperfine and experimental parameters for the single γ−γ TDPAC measurement of ¹⁸¹Ta(TiO₂) with four possible geometries, as given by the Gfit19^TM program. For the individual values of $V_{zz}$, the fit uncertainty is separated into (a) the statistical uncertainty caused by our instruments, and (b) the (dominant) systematic uncertainty of the quadrupole moment (±2.1%) as quoted in reference [51]. Note (1) the large ${\chi }^2$ value of the six-detector measurement and (2) the large multiplicative constant of the four-detector in the x-y plane TDPAC measurement vis-à-vis the two other four-detector TDPAC measurements. The averaged values of ${\omega }_0$, $\eta$, and $V_{zz}$ from the four different geometries were calculated, with (in the case of $V_{zz}$, (c)) the uncertainty assigned the square root of the sum of squares of their statistical uncertainty (in the case of $V_{zz}$, (a)).

Geometry	Fundamental Frequency ${\mathit{\omega }}_0$ (Mrad/s)	Asymmetry Parameter $\mathit{\eta }$	${\mathit{V}}_{\mathit{z}\mathit{z}}$ (1020 V/m2)	Delta Lorentzian $\mathit{\delta }$ (%)	Chi Squared ${\mathit{\chi }}^2$	Theta $\mathit{\theta }$ (Deg)	Phi $\mathit{\varphi }$ (Deg)	Multiplicative Constant	Additive Constant
Six-detectors	762.0 (4)	0.540 (1)	141.68 ± 0.07 a ± 3.00 b	0.94 (5)	3.5	N.A.	N.A.	0.692 (5)	−0.1111 (4)
Four-detector x-y plane	761.8 (2)	0.542 (1)	141.64 ± 0.04 a ± 3.00 b	0.17 (3)	0.77	64.4 (5)	47.1 (3)	0.719 (5)	−0.0817 (5)
Four-detector x-z plane	761.9 (7)	0.545 (2)	141.7 ± 0.1 a ± 3.0 b	0.49 (9)	0.79	28.6 (8)	45.8 (3)	0.314 (7)	−0.1795 (5)
Four-detector y-z plane	761.5 (6)	0.544 (2)	141.6 ± 0.1 a ± 3.0 b	0.46 (8)	0.96	29.5 (7)	46.3 (4)	0.300 (6)	−0.1843 (4)
Average	762 (1)	0.543 (3)	141.6 ± 0.2 c	N.A.	N.A.	N.A.	N.A.	N.A.	N.A.

.

$V_{zz}$ is written in a different convention because the error from the quadrupole moment $Q$ dominates the error for $V_{zz}$ in Equation (13) by several orders of magnitude.

To compare with the e−γ TDPAC spectroscopy setup, we use averaged values for the fundamental frequency ${\omega }_0$, the asymmetry parameter $\eta$, and the largest component of the electric-field gradient $V_{zz}$ arising from a set of single γ−γ TDPAC measurements in all four geometries. The quoted uncertainty for the averaged hyperfine parameters is the square root of the sum of squares of each of the measurement’s statistical uncertainty.

4.2. e−γ TDPAC Spectroscopy Setup

In fitting the R(t) spectrum from our e−γ TDPAC experiment, we realized that the multiplicative constant does not deviate significantly from unity. Thus, we fixed the multiplicative constant to unity. Since the R(t) signals are generated from the same sample in a unique sample holder, they are expected to have the same additive constant if the experimental setup is not altered or disturbed in any form. In the measurement process, we changed the sample holder once, so R(t) signals 1–6 include the fixed additive constant of 0.0187, while R(t) signals 7–13 include the fixed additive constant of 0.0085.

Note also that the angle $\theta$ is robust against fitting and does not deviate significantly from zero (with the error included). Thus, for all R(t) signals, we use $\theta$ = $0°$.

Like Figure 5, the left halves of Figure 6 and Figure 7 show R(t) (in blue) for different geometries after the corresponding raw TDPAC signal was processed with Interlude. The right halves of Figure 6 and Figure 7 show the corresponding FFT, which displays the frequencies ${\omega }_i,i\in \left\{1,2,3\right\}$ that drive R(t) [as seen in Equation (21)], as well as their peak amplitudes $[A_{22}{s}_i\left(\eta \right), i\in \left\{1,2,3\right\}$] and their spread (delta Lorentzian $\delta$). Information on the hyperfine parameters contained in R(t) along with the corresponding FFT transform are then extracted in the fitting process with Gfit19 (see red curves in Figure 6 and Figure 7).

The peaks of the FFT were marked with magenta arrows, with the values of the hyperfine parameters and the relevant experimental information tabulated in

Table 2. Table of fitted hyperfine and experimental parameters for the set of 13 different e−γ TDPAC measurements of ¹⁸¹Ta(TiO₂), returned by the Gfit19^TM program. For the individual values of $V_{zz}$, the fit uncertainty is separated into (a) the statistical uncertainty caused by our instruments, and (b) the (dominant) systematic uncertainty of the quadrupole moment (±2.1%) as quoted in reference [51]. Compared with the single γ−γ TDPAC measurement of ¹⁸¹Ta(TiO₂) with four different geometries (Table 1), the multiplicative constant and additive constant of this set of e−γ TDPAC measurements is fixed at 1 and 0.0187 or 0.0085, respectively. The averaged values of ${\omega }_0$, $\eta$, and $V_{zz}$ from this set of e−γ TDPAC measurements were calculated, with (in the case of $V_{zz}$, (c)) the uncertainty assigned the square root of the sum of squares of their statistical uncertainty (in the case of $V_{zz}$, (a)). (d) The averaged $\delta$ is written in this form, as its uncertainty exceeds its value.

R(t) Signal	Fundamental Frequency ${\mathit{\omega }}_0$ (Mrad/s)	Asymmetry Parameter $\mathit{\eta }$	${\mathit{V}}_{\mathit{z}\mathit{z}}$ (1020 V/m2)	Delta Lorentzian $\mathit{\delta }$ (%)	Chi Squared ${\mathit{\chi }}^2$	Phi $\mathit{\varphi }$ (Deg)	Additive Constant (Fixed)
1	765.0 (5)	0.549 (3)	142.23 ± 0.09 a ± 3.01 b	0.03 (6)	0.72	37.8 (4)	0.0187
2	765.6 (3)	0.546 (2)	142.35 ± 0.06 a ± 3.02 b	0.04 (6)	0.99	37.9 (2)	0.0187
3	765.6 (3)	0.546 (2)	142.35 ± 0.06 a ± 3.02 b	0.04 (6)	0.99	37.9 (2)	0.0187
4	764.4 (3)	0.543 (2)	142.12 ± 0.06 a ± 3.01 b	0.02 (2)	0.89	37.5 (2)	0.0187
5	764.4 (3)	0.543 (2)	142.12 ± 0.06 a ± 3.01 b	0.02 (2)	0.89	37.5 (2)	0.0187
6	764.4 (3)	0.543 (2)	142.12 ± 0.06 a ± 3.01 b	0.02 (2)	0.89	37.5 (2)	0.0187
7	763.8 (7)	0.536 (4)	142.0 ± 0.1 a ± 3.0 b	0.04 (9)	0.73	38.6 (5)	0.0085
8	764.6 (4)	0.539 (2)	142.17 ± 0.08 a ± 3.01 b	0.09 (6)	0.85	38.8 (3)	0.0085
9	764.6 (4)	0.539 (2)	142.17 ± 0.08 a ± 3.01 b	0.09 (6)	0.85	38.8 (3)	0.0085
10	765 (1)	0.552 (5)	142.2 ± 0.2 a ± 3.0 b	0.07 (1)	0.60	37.2 (7)	0.0085
11	765 (1)	0.552 (5)	142.2 ± 0.2 a ± 3.0 b	0.07 (1)	0.60	37.2 (7)	0.0085
12	765.1 (5)	0.544 (3)	142.3 ± 0.1 a ± 3.0 b	0.06 (7)	0.71	36.3 (3)	0.0085
13	765.1 (5)	0.544 (3)	142.3 ± 0.1 a ± 3.0 b	0.06 (7)	0.71	36.3 (3)	0.0085
Average	765 (2)	0.54 (1)	142.2 ± 0.4 c	0.05 ± 0.19 d	N.A.	38 (1)	0.0085

.

We compute the average and the standard deviation of the hyperfine parameters of this set of e−γ TDPAC measurements, which are given in the final row of Table 2. The quoted uncertainty for the averaged hyperfine parameters is the square root of the sum of squares of each measurement’s statistical uncertainty.

4.3. X-ray Diffraction of the ¹⁸¹Ta(TiO₂) Sample after the γ−γ and e−γ TDPAC Spectroscopy

Figure 8 shows the XRD diffractogram of both our (80 keV ion-implanted ¹⁸¹Hf complimented with 5.25 h of thermal annealing) TiO₂ samples alongside a corresponding reference sample (fresh sample from Crystals GmbH [55]) for comparison. Note that on the left side of the graphs, we show a log-10 plot of the intensity (“Icalc”) between 0–80°, and on the right-hand side, we show a linear plot of the intensity between 38–40°. We present the graphs in this way as (1) the peaks in the domain of 39–40° dominate all other peaks and (2) to allow for the visualization of the successful resolution of the two most dominant peaks as shown in the right-sided graphs contained in Figure 8.

For ease of comparison, at the bottom of Figure 8, we present 2 corresponding graphs in which all 3 diffractograms are normalized (with respect to the highest recorded intensity for each diffractogram) and plotted in the same window(s).

Table 3. The relevant parameters of the fit produced by Empyrean diffractometer’s complimentary analysis software. We note here that the splitting between the two peaks in the same sample is on the order of 0.1°, and there is a systematic shift of the two most dominant peaks across all 3 samples, within a small range of 0.22°. This shift could be attributed to the individual micro-stress of each sample.

Sample and Peak Number	Angle $2\mathit{\theta }$ (2 × Deg)	d-Spacing (Angstrom)	Observed Intensity (Counts)	$\mathit{F}\mathit{W}\mathit{H}\mathit{M}$ (2 × Deg)	Background Intensity (Counts)	Crystallite Size (Angstrom)	Micro-Strain (%)
Reference Sample, Peak 1	39.4731	2.28104	1,264,663	0.0468	4616.40	29,652	0.003846
Reference Sample, Peak 2	39.5760	2.28100	612,150	0.0468	4647.93	29,735	0.003846
Sample 1, Peak 1	39.6912	2.26901	152,133	0.0624	1421.86	29,672	0.003823
Sample 1, Peak 2	39.7943	2.26900	76,443	0.0468	1427.01	29,756	0.003813
Sample 2, Peak 1	39.5332	2.27771	378,807	0.0468	2202.72	29,658	0.003840
Sample 2, Peak 2	39.6359	2.27769	182,428	0.0468	2227.21	29,741	0.003829

shows the 2 most dominant peaks (right side of Figure 8) produced by the analysis software (that comes complimentary to the Empyrean diffractometer), alongside the corresponding information (d-spacing, full width at half maximum [FWHM], crystalline size, and micro-strain).

We note that up to a systematic shift of 0.22°, all 3 samples exhibit the same double peaks, and within the same sample, the split between the two peaks is in the order of 0.1°. The ratio of the intensity between the left peak and the right peak is 2.07 for the reference sample, 1.99 for our first sample, and 2.08 for our second sample.

5. Discussion

5.1. γ−γ TDPAC Spectroscopy Setup

Table 1 shows that the multiplicative constant for the y-z and x-z plane measurements of the four-detector γ−γ TDPAC spectroscopy is less than half of the multiplicative constant of the x-y plane. This suggests qualitatively that the effective anisotropy constant A_kk of the y-z and x-z plane geometry is much less than that of the x-y plane. This may be attributed to the reduced solid angle of the detectors lying on the z axis, which in turn is because the distance between the sample and the z-axis detectors is greater than the distances between the sample and the x- and y-axis detectors.

This result implies that, in the analysis of the full six-detector R(t) spectra, the signal from the x-y plane is more pronounced than that of the x-z or y-z plane. This would invalidate the assumption of random distribution of the size and direction of the individual crystallites as per the definition of a polycrystalline solid. Thus, the high ${\chi }^2$ of the six-detector is expected from the relative ill-fitting.

From geometric considerations of the three planes in the four-detector measurements, as elaborated in Appendix B, we note that

$$\begin{gathered} \left|{\theta }_{x-y}\right|+\left|{\theta }_{x-z}\right|=\left|{\theta }_{x-y}\right|+\left|{\theta }_{y-z}\right|=90°, \\ \left|{\phi }_{x-y}\right|+\left|{\phi }_{x-z}\right|=\left|{\phi }_{x-y}\right|+\left|{\phi }_{y-z}\right|=90°. \end{gathered}$$

Experimentally, we have

$$\begin{gathered} \left|{\theta }_{x-y}\right|+\left|{\theta }_{x-z}\right|=93.0\left(9\right)°, \\ \left|{\theta }_{x-y}\right|+\left|{\theta }_{x-z}\right|=93.9\left(9\right)°, \\ \left|{\phi }_{x-y}\right|+\left|{\phi }_{x-z}\right|=92.9\left(4\right)°, \\ \left|{\phi }_{x-y}\right|+\left|{\phi }_{y-z}\right|=93.4\left(5\right)°. \end{gathered}$$

In our derivation based on geometric considerations, we assume that the crystal is an infinitely thin sheet centered in the middle of the detector setup. The small offset of the angles from the expected values may be attributed to (1) the sample not being a perfect single crystal; (2) the crystal not being infinitesimally thin, so the average depth of the implantation may vary statistically; and (3) the ¹⁸¹Ta implantation being done only on one long side of the sample, as discussed in the experimental setup section of the paper.

Regardless, note that the effective anisotropy constant A_kk and the angles $\theta$ and $\phi$ only affect the amplitude of the individual cosine components of R(t) but not the delta Lorentzian $\delta$ or the individual frequencies ${\omega }_i,i\in \left\{1,2,3\right\}$ (Equations (20,21)). Consequently, Figure 5 shows that R(t) from the different geometrical setups exhibits roughly the same FFT peaks ${\omega }_i,i\in \left\{1,2,3\right\}$. Based on Equations (16–20), this implies that quantitatively, both the measured fundamental frequencies ${\omega }_0$ and the largest-magnitude EFG $V_{zz}$ must be robust against the R(t) from the different geometrical setups.

5.2. e−γ TDPAC Spectroscopy Setup Compared with γ−γ TDPAC Spectroscopy Setup

Comparing R(t) acquired by the γ−γ TDPAC spectroscopy setup (see Figure 5) with R(t) acquired by the e−γ TDPAC spectroscopy setup (Figure 6 and Figure 7) shows that the periodicity of the γ−γ R(t) signals are roughly twice that of the e−γ R(t) signals. This effect is qualitatively shown in Figure 9, which compares R(t) between the x-y plane of the γ−γ TDPAC spectroscopy setup with R(t) from sample 8 of the e−γ TDPAC spectroscopy setup.

This discrepancy is attributed to the difference in the time calibration (0.101 ns/channel for the γ−γ TDPAC spectroscopy and 0.195 ns/channel for the e−γ TDPAC spectroscopy). Given that the Gfit19^TM fitting process considers the time calibration for each TDPAC spectroscopy setup as a parameter, we note that the average fundamental frequencies and the asymmetry parameters for the γ−γ TDPAC spectroscopy setup (762(1) Mrad/s, 0.543(3), [Table 1]) lie very close to those of the e−γ TDPAC spectroscopy setup (765(2) Mrad/s, 0.54(1), [Table 2]).

After correcting for the quadrupole moment of ¹⁸¹Ta according to reference [51], $V_{zz}$ (and ${\omega }_0$ in extension) obtained from both our γ−γ and e−γ ¹⁸¹Ta(TiO₂) TDPAC spectroscopies are consistent with published values up to an order of magnitude, and our asymmetry parameter $\eta$ is consistent with the quoted values up to 0.05. This is shown in

Table 4. Published literature values (after correcting for the quadrupole moment of ¹⁸¹Ta according to reference [51]) of the relevant hyperfine parameters for the ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy compared with the hyperfine parameters obtained from our γ−γ and e−γ experiments. We note that across all the papers mentioned below, they did not consider the dominant systematic uncertainty of the quadrupole moment as quoted in reference [51]. Hence, for our values, we only consider the statistical uncertainty caused by our instruments. Note that (a) Martucci et al. did not include the uncertainty in their report, (b) they mentioned that their asymmetry parameters agree with the reference [10], (c) they also mentioned that discrepancy of $V_{zz}$ of their sample 4 concerning the literature values is due to the presence of the fluorine atoms (A byproduct caused by their preparation method) near the ¹⁸¹Ta probe, and (d) the averaged $\delta$ is written in this form, as its uncertainty exceeds its value.

Paper	${\mathit{V}}_{\mathit{z}\mathit{z}}$ (1020 V/m2)	Asymmetry Parameter $\mathit{\eta }$	Delta Lorentzian $\mathit{\delta }$ (%)	Crystal Type	Sample Preparation	Reference
James 1994	141.6 (7)	0.57 (1)	≈0.01	Polycrystal	Thermal Annealing at 1570 K for several hrs	[7]
Darriba 2011	143.2 (7)	0.555 (8)	0.4 (2)	Single Crystal	Thermal Annealing at 1073 K for 6 h	[8]
Banerjee 2016	141.9 (2)	0.56 (1)	0.6 (1)	Polycrystal	Thermal Annealing at 1273 K for 10 h	[11]
Satyendra 2009	145 (1)	0.56 (1)	Unknown	Polycrystal	Thermal Annealing at 1273 K for 10 h	[9]
Banerjee 2010 {S4}	149.5 (8)	0.51 (1)	5.2 (5)	Polycrystal	Thermal Annealing at 1123 K for 4 h	[10]
Banerjee 2010 {S7}	145.6 (1)	0.55 (1)	1.0 (1)	Polycrystal	Thermal Annealing Anatase TiO2 at 1223 K for 8 h (Convert to Rutile TiO2)	[10]
Martucci 2011 {3}	142.06 a	0.55 a,b	Unknown	Polycrystal	Thermal Annealing Anatase TiO2 at 973 K for 10 h under Nitrogen (Convert to Rutile TiO2)	[15]
Martucci 2011 {4}	139.00 c	0.55 a,b	Unknown	Polycrystal	Thermal Annealing Anatase TiO2 at 1223 K for 10 h under Nitrogen (Convert to Rutile TiO2)	[15]
This study (γ–γ)	141.6 (2)	0.543 (3)	N.A.	Single Crystal	Thermal Annealing at 873 K for 5.25 h	N.A.
This study (e–γ)	142.2 (4)	0.54 (1)	0.05 ± 0.19 d	Single Crystal	Thermal Annealing at 873 K for 5.25 h	N.A.

in numerical details of the relevant hyperfine parameters and Figure 10 and Figure 11 for the corresponding graphical outputs for the values of $V_{zz}$ and $\eta$, respectively.

These results show that the hyperfine parameters determined from the ¹⁸¹Ta(TiO₂) γ−γ and e−γ TDPAC experiments are consistent with the literature values for the corresponding ¹⁸¹Ta(TiO₂) γ−γ TDPAC results.

Following the understanding of the electron-recombination process (see Figure 2 and Section 2), we conclude that the timescale of the electron-recombination process is at least one order of magnitude faster than nanoseconds, which is the timescale of the second-stage decay (10.8 ns) of the ¹⁸¹Hf→¹⁸¹Ta probe nucleus. This also implies that we cannot detect the recombination process with our present setup for ¹⁸¹Ta(TiO₂) e−γ TDPAC spectroscopy.

This is consistent with the implications of the findings with regards to the Femtosecond Time Resolved Spectroscopy conducted by a group from the KAUST Solar Centre, in which an induced hole (as an electron trap) in the bulk rutile TiO₂ will capture an electron in the order of picoseconds [22].

5.3. X-ray Diffraction of the ¹⁸¹Ta(TiO₂) Sample after the γ−γ and e−γ TDPAC Spectroscopy

From the results in Table 3, we can see that up to a systematic shift of 0.22°, the corresponding peaks of our (80 keV ¹⁸¹Hf ion-implanted and complimented with sufficient thermal annealing) TiO₂, as well as the reference sample, agree with each other. The cause of such a systematic shift could be due to the individual micro-stress of each sample. We also note that the X-rays produced from the Empyrean diffractometer are not monochromatic, which explains the double peaks observed on the right side of Figure 8. We especially note that the micro-strain of our samples, along with the reference, is about 0.0038%, which is very low. The micro-strain is a measure of the amorphization of a solid sample, and thus, both our TiO₂ samples, as well as the reference, could be said to exhibit single crystal-like properties.

The presence of the other small peaks, as seen on the left side of Figure 8, could be attributed to the existence of the oxygen vacancies (see Section 1.2). We make no further attempts to characterize these peaks at the time of dissemination, as our main result is on the TDPAC study of (¹⁸¹Ta)TiO₂. Our ¹⁸¹Ta probe (which sits on a well-defined Ti-substitutional site) does not explicitly detect such oxygen deficiencies.

The dominant peaks in our result correspond to the (200) Miller index direction of the rutile TiO₂ (regarding the polycrystalline rutile TiO₂ sample taken from the RRUFF™ Project website [72,73]).

In addition to our reference sample, we refer to a paper by Guo et al. [70], in which they have fabricated their own single crystal rutile TiO₂, and their corresponding X-ray diffractograms have revealed a similar (single peak-like) behavior. The position of their peaks and sample preparation are elaborated in

Table 5. An XRD comparison of the 2 dominant peaks of our single crystal rutile TiO₂ samples to the (1) reference sample produced by Crystal GmbH [55] and (2) the samples produced via the floating zone method as described in the paper of Guo et al. [70]. We note that in their corresponding XRD diffractogram (Figure 2 of [70]), regardless of the direction of the cut, we could see multiple small peaks that are too fine to be distinguished with their domain of 20–50°, like our case as seen in Figure 8, in which we must restrict our domain from 0–80° to 38–40° to allow for sufficient resolution. (a) The peak identification of the Miller indices for our samples (and the reference sample) concerns reference [72]. For samples from paper [70], the peaks are already identified in the corresponding paper and cross-verified with reference [72].

Sample and Peak Number	Angle $2\mathit{\theta }$ (2 × Deg)	Peak Identity (Miller Indices) a	Sample Preparation
Sample from [70], cut along an arbitrary direction.	62.6	(002)	Sample is produced in the laboratory setting via the floating zone method. The resulting sample is then cut into 5 × 2 mm2 thin slices alongside an arbitrary direction and the direction of the crystal growth, respectively.
Sample from [70], cut perpendicular to the c-axis.	27.5	(110)
Reference Sample, Peak 1.	39.5	(200)	Sample (10 × 10 × 1.0 mm3) is commercially manufactured by Crystal GmbH [55]. XRD measurement is conducted face-(10 × 1.0 mm2) side up.
Reference Sample, Peak 2.	39.6
Sample 1, Peak 1, This study.	39.7	(200)	Sample (10 × 10 × 1.0 mm3) is commercially manufactured by Crystal GmbH [55], then 80 keV 181Hf ion-implanted and thermally annealed at 873 K for 5.25 h. XRD measurement is conducted face-(10 × 1.0 mm2) side up.
Sample 1, Peak 2, This study.	39.8
Sample 2, Peak 1, This study.	39.5
Sample 2, Peak 2, This study.	39.6

below.

From the information obtained in the XRD diffraction of our TiO₂ sample, we can conclude that the treatment of our parts per billion ¹⁸¹Hf ion-implanted TiO₂ sample, complimented with 5 hr 15 min of thermal annealing at 873K, does not significantly alter its global structure, as the low value of the micro-strain shows in all 3 of our XRD diffractograms.

6. Conclusions and Future Works

We use ¹⁸¹Ta(TiO₂) e−γ TDPAC spectroscopy in combination with ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy to obtain a wealth of information on TiO₂, which is a prominent industrial compound. Although both spectroscopies appear the same at first glance, subtle differences appear in the first stage of the two-stage decay that highlights how e−γ TDPAC spectroscopy is complementary to γ−γ TDPAC spectroscopy. In particular, for γ−γ TDPAC spectroscopy, the starting ¹⁸¹Ta nucleus does not interact with its electron cloud, leading to a direct de-excitation. This contrasts with e−γ TDPAC spectroscopy, where the ¹⁸¹Ta nucleus interacts with some of its electron shells, leading to the ejection of the conversion electron. This subtle difference permits the dynamic observation of the recombination process, a multistep process where the electrons provided by the host material fill the hole left by the expelled conversion electron.

The hyperfine parameters obtained from ¹⁸¹Ta(TiO₂) e−γ TDPAC spectroscopy are consistent with those obtained from γ−γ TDPAC spectroscopy, which suggests that the electron-recombination process is at least an order of magnitude less than 10 nanoseconds and thus could not be detected with our current setup. This compliments the recent results and implications of a group from the KAUST Solar Centre [22], in which an induced hole in bulk rutile will capture an electron in the order of picoseconds.

Upon reviewing the literature on ¹⁸¹Ta(TiO₂) γ−γ TDPAC spectroscopy, we realized the dearth of information pertaining to the hyperfine parameters in the low-temperature regime (0–500 K). References [7,8] assume that $V_{zz}$ is linear in temperature, which we wish to test with our e−γ TDPAC spectroscopy setup.