Perturbed Angular Correlation Spectroscopy – A Tool for the Study of Defects and Diffusion at the Atomic Scale

This paper provides an introduction to perturbed angular correlation (PAC) spectroscopy in the context of its application in the study of point defects and diffusion. It emphasizes what we anticipate to be of interest to non-PAC specialists who are interested in understanding variations in how PAC results are presented by different research groups and in how physical quantities such as defect formation energies, association energies, and migration barriers can be extracted from analysis of PAC spectra. Numerous citations are included to emphasize the universality of the analysis methods across different classes of materials including metallic, ceramic, and semiconducting compounds.


Introduction
Perturbed angular correlation spectroscopy (PAC) has its origins in the field of nuclear physics; however, it has since had widespread application in solid state systems as described in a number of past review articles [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The most common application of PAC in the study of materials involves the time differential angular correlation between pairs of gamma rays emitted by radioactive nuclei, which serve as probes of the local electric charge distributions and magnetic fields in a material. Often the abbreviation TDPAC is used to emphasize that the method is timedifferential PAC as opposed to time integrated PAC; however, IPAC is seldom used and PAC will be used in this paper to be synonymous with TDPAC. In a PAC experiment, the angular correlation between gamma rays changes in time in response to the hyperfine interaction (HFI) between the PAC probe nuclei and the electromagnetic environments of the material where the probes are located.
The present paper provides a detailed introduction to PAC with an emphasis on how physical quantities of interest to studies of defects and diffusion can be obtained. We have three goals in this paper: (1) to bridge introductory [3,17] and comprehensive [18,19] descriptions of PAC with an aim to help non-specialists understand variations in experimental setups and data analysis; (2) to provide an extended description of how adjustable parameters in fits of PAC spectra can be used in studies of defects and diffusion; and (3) to give a sense of the general applicability of PAC to all classes of materials by describing briefly and providing citations to studies in ceramic, semiconducting, and metallic compounds. Examples have been selected based primarily on their suitability for illustrating the analysis techniques discussed in this paper rather than on their impact in the understanding of defect and diffusion processes. As such, many important studies have not been included in this review, and the reader is encouraged to investigate other published works of the cited authors.
The paper is organized as follows. The first part provides an introduction to the theoretical underpinnings of PAC. The mathematical formalism used may be unfamiliar to some, so explanatory text is included to emphasize key results. A complete understanding of the derivation is not necessary to follow the rest of the paper, which consists of a section describing the most common spectrometer configurations and other experimental considerations and a section of example applications of PAC. This paper emphasizes the physical origins of and use of the quadrupole interaction (QI), one of the two basic HFIs that can be used to study defects. A companion paper by Carbonari, Mestnik-Filho, and Saxena [20] provides comparable information for the other basic HFI: the magnetic interaction (MI).

Theoretical Background
Angular correlation of a γ − γ cascade. Consider a 3-level nucleus as show in Fig. 1. The excited state I i decays by emission of γ 1 to the intermediate state I, followed by decay to the ground state I f by emission of γ 2 . The angular correlation function W( γ 1 , γ 2 ) is a measure of the probability that γ 1 is detected in direction k 1 and γ 2 is detected in direction k 2 . Utilizing the density matrix formalism one finds [17,18,19] W (γ 1 ,γ 2 ) = m mm ' ∑ ρ(γ 2 ,0) m' m' ρ(γ 1 ,0) m . (1) Here ρ(γ i ,0) describes the state of the system immediately after the γ 1 emission (i.e. at t = 0). The matrix elements m ρ(γ i ,0) m are the probabilities of finding the system in an eigenstate m immediately after the i th transition (Fig. 2). The explicit forms of these matrix elements are

Defects and Diffusion Studied Using PAC Spectroscopy
The A k i (γ i ) are radiation parameters that depend on the spins of the levels involved in the transition and on the multipolarity of the radiation. The radiation parameters are normalized to A 0 = 1. The D N i 0 k i (γ i → z) are rotation matrices that transform the z-axis into the detection direction of γ i . The D N i 0 k i can be expressed in terms of spherical harmonics Y N i 0 (θ i ,φ i ) .
Inserting (2) into (1) and performing the mm'-sum collapses the k i and N i sums into k 1 = k 2 = k and N 1 = N 2 = N . The angular correlation function becomes Summation over N reduces the spherical harmonics to Legendre polynomials emission of a first γ -ray followed by (2) emission of the second γ -ray. Bars over magnetic substates represent their populations, which are unequal following the first γdecay because of mutlipole radiation selection rules.

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with θ the angle between the detection directions of γ 1 and γ 2 and A k ≡ A k (γ 1 )A k (γ 2 ). The sum is only over even k because the odd A k vanish as a consequence of parity conservation and the fact that polarization of the γ photons is not measured. The summation is finite due to conservation of angular momentum. For the frequently used I = 5 2 probe nuclei the summation index limit is k max = 4, thus W (θ) = 1+ A 2 P 2 (cosθ) + A 4 P 4 (cosθ) . The A k are called the anisotropy factors.
Angular correlation of a γ − γ cascade with perturbations. Note that in Eq. 1 we have used the state of the system immediately after emission of γ 1 . This is appropriate as long as there is no external field interacting with the nucleus. If we want to include interactions of the nucleus with extra-nuclear fields we must include the evolution of the density matrix from t = 0 to the time t just before emission of γ 2 (Fig. 3). Thus we replace ρ(γ 1 ,0) with ρ(γ 1 ,t) , which describes the state of the system a time t after emission of γ 1 immediately before emission of γ 2 . The angular correlation function (1) now is written and the time dependence is expressed using an evolution operator Λ(t), which transforms the density operator ρ(γ 1 ,0) into ρ(γ 1 ,t) : Inserting (6) into (5) and using (2), the angular correlation function becomes W (γ 1 ,γ 2 ,t) = A k 1 (γ 1 ) A k 2 (γ 2 )G k 1 k 2 N 1 N 2 (t) Y k 1 N 1 * (θ 1 ,φ 1 )Y k 2 N 2 (θ 2 ,φ 2 ) (2k 1 +1)(2k 2 +1) The angles θ i and φ i refer to the detection direction of γ 1 with respect to the z-axis. G k 1 k 2 N 1 N 2 (t) is the so-called perturbation factor which contains all time dependence and effects of the perturbation due to the interaction of the nucleus with external electric or magnetic fields. The perturbation factor is given by The matrix elements of the evolution operator that appear in (8) can be expressed in terms of the evolution operator's eigenvalues and eigenvectors. For time-independent Hamiltonians, the eigenvalues are equal to exp −iE j t / ℏ ( ) where E j are the energies of the stationary nuclear spin states in the hyperfine interaction. It can be shown that (8) can be expressed as a sum of cosines:

Defects and Diffusion Studied Using PAC Spectroscopy
The angular frequencies ω n in Eq. 9 come from differences in eigenvalues and therefore depend in general on the strengths and the symmetries of the hyperfine interactions. The amplitudes S N 1 N 2 k 1 k 2 ,n depend on the projections of eigenvectors along the nuclear spin basis vectors, and these depend in general on the value of the nuclear spin state, symmetry of the hyperfine interaction, and orientation of hyperfine symmetry axes with respect to directions of γ -ray detection.
A common experimental condition is to perform PAC measurements on polycrystalline samples. In such cases, one must average over all orientations of the symmetry axes of hyperfine interactions with respect to the directions of γ -ray detection in (7). The result is an angular correlation function that depends only on the angle between γ detections, θ , and time between detections, t: in which G kk t ( ) is the sum over all terms G k 1 k 2 This can be written as a sum of cosines with a different set of amplitudes: Defect and Diffusion Forum Vol. 311 7 This new set of amplitudes does not depend on orientations of hyperfine interactions relative to detector directions. Magnetic Dipole Interaction. The lowest order magnetic interaction is the magnetic dipole interaction (MI). It takes place between the magnetic dipole moment of a PAC probe and an extranuclear magnetic field. The dipole moment is proportional to the total angular momentum of the nucleus and is conventionally expressed in terms of the dimensionless g-factor. The magnetic field may be present in the form of a field applied from outside a sample or, in the case of ferromagnetic and antiferromagnetic materials, from spontaneous magnetic moments within the sample. In general, the magnetic field at the PAC probe nucleus is different from the external field because of spin polarization of the s electrons near the nuclear probe. Thus, different PAC probes will experience different magnetic interactions even when placed in the same extranuclear field.
Classically, a magnetic field exerts a torque on any nucleus with nonzero magnetic dipole moment, resulting in a precession of the moment about the field as illustrated in Fig. 4a. The frequency of precession is known as the Larmor frequency and is given by where g is the dimensionless g-factor, µ N is the nuclear magneton, and B 0 is the magnitude of the magnetic field at the probe, i.e. the hyperfine magnetic field. Quantum mechanically, the magnetic interaction results in a complete lifting of the degeneracy in the intermediate nuclear spin state of the PAC probe (Fig. 4b). The energy difference between successive m-states is ℏω L , leading to oscillations in populations of m-states with frequencies that are integer multiples of ω L . The amplitudes in (9) depend on the experimental setup; in some cases, experimental configurations can be chosen so that the observed perturbation function τ depends only on ω L or on 2ω L , as described in detail by Butz [21].

Defects and Diffusion Studied Using PAC Spectroscopy
Electric Quadrupole Interaction. The lowest order electric interaction is the quadrupole interaction (QI). It takes effect between the electric quadrupole moment of a PAC probe and the electric field gradient (EFG) arising from charges in the crystal surrounding the probe nucleus. The nuclear quadrupole moment Q is a single value that indicates how much the distribution of charge in the nucleus deviates from spherical. The EFG is a traceless, second-order tensor that is essentially the second spatial derivative of the electrostatic potential due to extranuclear charges. The tensor is reported customarily in terms of the principal component with largest magnitude, V zz , and an asymmetry parameter η , defined by η = V xx −V yy ( ) / V zz , where V xx and V yy are the other two principal components of the EFG tensor with the convention that V zz ≥ V yy ≥ V xx .
The electric quadrupole interaction partially lifts the degeneracy in the intermediate nuclear spin state of the PAC probe (Fig. 5). The strength of the interaction is often reported in terms of the where e is the fundamental unit of charge and I is the total spin of the intermediate nuclear state. An alternative and common way to report the strength of the interaction is by the quadrupole coupling frequency ν Q ≡ eQV zz / h . The frequencies that are present in the measured perturbation function (Eq. 9) are proportional to ω Q . The number of frequencies that can be observed and the factors multiplying ω Q depend in detail on the experimental setup, including orientation of EFG with respect to detectors in the case of single crystals, value of the intermediate nuclear spin I, and asymmetry of the EFG. As in the magnetic case, the amplitudes in the perturbation function depend on the experimental configuration. In addition, they depend on the spin of the probe and on the symmetry of the EFG.

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Mixed Electric and Magnetic Interaction. It is worth pointing out that it is possible to have a mixed magnetic and electric hyperfine interaction. Again, the frequencies that will be observable in the perturbation function are proportional to differences in energies of eigenstates of the HFI Hamiltonian. Unlike in the case of the pure magnetic interaction, the energy spacing will not be regular, and in general there will be a sizable number of frequencies. It is outside the scope of the present paper to go further into the case of mixed interactions except to assert that it is possible to calculate such perturbation functions theoretically and analyze experimental spectra containing signals that originate from mixed interactions. See, for example, references 24 and 25.
Hyperfine Interactions in Materials. The atomic displacements near PAC probes that occur because of lattice vibrations result in fluctuations in the MI and the QI experienced by the PAC probes. The timescale of a hyperfine measurement is defined by the reciprocal of the hyperfine interaction frequency: 1/ ω L in the case of the MI and 1/ ω Q in the case of the QI. Lattice vibration frequencies typically are several orders of magnitude larger than hyperfine interaction frequencies, and as a consequence, measurements are not sensitive to the variations in HFI induced by the lattice vibrations. Instead, the measurement can be considered to be of a static HFI, the magnetic field or EFG being the time-average over many lattice vibrations. So, when discussing HFIs in a solid, one usually speaks in terms of "static" HFIs. Exceptions to this include situations where point defects near probes jump from one lattice site to another, when the probes themselves jump, or when there are fluctuations in the charge state of defect-probe complexes at a timescale comparable to the HFI frequency, and these are considered separately below as "dynamic" hyperfine interactions.
PAC probes in materials can experience a range of crystallographic environments, depending on the site occupation of the probes, or where the probes substitute in the material, and these environments in turn can be disturbed by defects. For dilute defect concentrations and when defects are attracted to PAC probes, a distinct set of HFIs arise in addition to the defect-free HFIs. For non-dilute defect concentrations and little attraction to PAC probes, the defect-free or lattice HFIs are disturbed slightly, resulting in a distribution of HFIs, the central, or most probable of which, is near the lattice HFI. For non-dilute concentrations and strong attraction, a combination of these effects is observed.
It remains to consider the physical origins of HFIs in materials. The accompanying review article by Carbonari, Mestnik-Filho, and Saxena [20] provides a thorough introduction to the magnetic interaction. The remainder of the present section examines the physical origin of the electric field gradient, which is responsible for the quadrupole interaction.
The EFG at a nucleus is given by where ρ r ( ) is the charge density of everything except the nucleus-of-interest and consists of the electrons of the remaining solid, including the ion-core electrons of the atom that contains the nucleus of interest, and the positive charges of the other nuclei. The EFG has the same symmetry as the point symmetry at the location of the nucleus through its dependence on ρ r ( ) . The variation with 1/r 3 means that the EFG is especially sensitive to non-spherical or non-cubic electronic density contributions near the nucleus of interest, especially those arising from unfilled valence orbitals and distortions of core electron orbitals. For nuclei located at crystal lattice sites with cubic site symmetry, the EFG experienced by the nuclei is "zero;" that is, V zz = 0 . For sites with a threefold or higher axis of symmetry, V zz ≠ 0 , η = 0 , and the principal axis of the EFG is collinear with the axis of symmetry. For lower symmetry, V zz ≠ 0 and 0 ≤ η ≤1. Arguably one of the most straightforward ways to detect point

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Defects and Diffusion Studied Using PAC Spectroscopy defects using PAC occurs when it is possible to measure the EFG at probe nuclei without nearby defects and compare it to EFGs measured when defects are in first or second neighbor positions, especially when the change in local charge distribution induced by the point defects lowers the local symmetry.
As an example of detection of defects by reduction of symmetry, Fe vacancies in FeAl are considered. Al-lattice sites have m3m point symmetry, so that a PAC probe at an Al site experiences V zz = 0 when no defects are nearby (Fig. 6a). A single vacancy in a near neighbor Fe site results in an axially symmetric EFG ( V zz ≠ 0 and η = 0 ) with principal axis oriented in the direction of the neighboring Fe vacancy (Fig. 6b). A second vacancy in a near neighbor site adjacent to the first vacancy further reduces the local symmetry and a non-axially EFG is observed (Fig. 6c). Other arrangements of vacancies around probes lead to still more distinguishable EFGs [26].
One can determine the atomic structure near a probe based on the EFG the probe experiencesin principle. That is, when there are no defects in the probe's local environment, one could determine the lattice site the probe occupies by the EFG, as long as the sites have unique EFGs. An example for which this does not work is a structure such as the CsCl-structure, as both Cs and Cl sites have m3m point symmetry so that V zz = 0. But even in such a case, if defects are in the probe's local environment, then one expects to be able to determine which types of defects are there and how they are arranged based on observed EFGs.
In some cases, consideration of symmetry alone is not sufficient to identify site occupation or defect configurations. Examination of how observed signals change when varying experimental configurations such as sample composition or measurement temperature can help identify the origins of observed QIs. Such arguments often are used in conjunction with estimates of strengths of QIs based on calculations of EFGs using simple point charge approximations. Here, we use the term point charge approximation generically to refer to any scheme that models the atoms or ions in a solid as point charges, including application of empirical corrections to account for distortions of the PAC probe's atomic orbitals derived either from theoretical calculations such as in references 27 and 28 or deduced from experimental trends such as in references 29 and 30. However, point charge approximations are not always sufficient for identifying origins of QIs, and more rigorous calculations of EFGs within the framework of density functional theory (DFT) must be performed.
Unfortunately, it is not always practicable or even possible to identify the origin of an observed EFG based on predictions from DFT calculations. For reliable prediction of EFGs in the presence of point defects, calculations must allow for relaxation of the crystal structure with atomic displacements also to be predicted in the DFT calculations. Most commonly, a supercell approach is used, in which the PAC probe is located at the center of a supercell that is large enough to accommodate significant relaxation but small enough to be computationally tractable. The sensitivity of EFGs to electronic densities of core electrons means that methods descended from the augmented plane wave approach must be employed rather than the widely used pseudopotential plane wave approach. Even then, calculated EFGs depend on the approximation chosen for the exchange and correlation terms in the energy, which lead to uncertainties in predicted values. In cases where differences among predicted EFGs for candidate crystallographic environments are greater than the uncertainties due to choice of exchange-correlation functional, comparison to experimentally measured values does allow one to determine site substitution and identify defect arrangements.
In general, measured QI frequencies vary with temperature, indicating of course that the corresponding EFGs vary with temperature. In non-cubic metals, the EFG decreases with increasing temperature; however, the degree of variation is too large to be due only to thermal expansion [31][32][33]. It was long suspected that the temperature dependence is linked to phonons, but it was not until only recently that the temperature dependence of the EFG in Cd metal was calculated from first principles to verify this picture [34]. It appears that temperature variations of EFGs in other materials systems can come from alternate sources. For example, calculations of the

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EFG experienced by Ta dopants in TiO 2 indicate that the primary cause of temperature dependence comes from a structural relaxation around the dopant, the degree of which depends strongly on lattice parameter, which in turn varies due to thermal expansion [35]; thus, thermal expansion can contribute significantly to the temperature variation of EFGs, at least indirectly. A complete description of measured QI frequencies therefore must include information about their dependence on temperature. Such information is not always reported because it is not easily measured or is superfluous to the main focus of a particular study.
Perturbation functions when probes are in multiple environments. The perturbation functions given by Eqs. 9 and 11 assume all probes are in the same crystallographic environment. There are many interesting applications when probes are in more than one environment. In this case, Eqs. 7 and 10 can be written and where the bar over the perturbation function indicates an average of the single-interaction perturbation functions of the individual environments. Three situations are described in more detail below: (1) probes are in a few discrete static environments, or sites, (2) probes are in an inhomogeneous environment that results effectively in a continuous distribution of static HFIs, and (3) probes experience fluctuating HFIs as the environment around a probe changes.

Fig. 6.
Examples of defects causing EFGs at the location of the radioactive probe atom.

Defects and Diffusion Studied Using PAC Spectroscopy
Discrete set of HFIs -site fractions. If there are N distinct hyperfine environments, then the average perturbation function is given as a weighted sum of the perturbation functions of the individual environments: where G kk , j t () is the perturbation function that corresponds to the hyperfine interaction of the jth environment and f j is the site fraction, or fraction of probes, in environment j. The f j are normalized; that is, Continuous distribution of HFIs. Sample inhomogeneity, due for instance to randomly distributed defects outside the first couple of near neighbor shells of the probes, results in a distribution of quadrupole interactions. This leads to a damping of the PAC signal. The effect is known as inhomogeneous broadening because it appears as a broadening of frequency distributions when viewing Fourier transforms of PAC spectra. The more defects there are in a system, the larger the disturbance to a HFI will be and the greater the inhomogeneous broadening.
There have been several theoretical studies done to clarify the physical sources of the broadening and to explore lineshapes for special cases [36][37][38][39][40][41][42][43][44]; however, a universal expression for lineshape that is appropriate for broadening caused by charged defects distributed randomly on a crystal lattice has not yet been established. The usual approach when analyzing PAC spectra is to assume that the distribution of EFGs leads to a Gaussian or Lorentzian distribution of frequencies.
For example, this leads to for 5/2 spin probes in a polycrystalline sample. Here, p determines the type of distribution: 1 for Lorentzian and 2 for Gaussian. The full width half maximum (FWHM) of the Lorentzian distribution is given by 2δ , whereas the FWHM of the Gaussian distribution is 2δ 2ln 2 .
Simulations of PAC spectra in the presence of randomly distributed defects have shown that there is an interdependence among empirical fitting parameters so that ω n = ω n δ ( ) , or alternatively, [45,46].
There is some variation in how the width of the distribution is reported in the literature. Many authors use an exponential term of the form where σ is an absolute measure of distribution width* and the g n factors are proportional to the interaction frequencies ω n . The relations g n = ω n / ν Q , g n = ω n / ω Q , and g n = ω n / ω 1 all have appeared in the literature.
*σ is not universally accepted as the symbol to use for absolute width and, for example, some authors use δ for absolute width.

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Fluctuating HFIs. When fluctuations occur among multiple HFIs at a time scale comparable to 1/ ω L or 1/ ω Q , the perturbation function can no longer be described simply by a superposition of static HFIs, as in Eq. 15. The method for modeling stochastic fluctuations of HFIs by Winkler and Gerdau [47] allows derivation of an appropriate expression for the angular correlation function by employing the Liouville formalism, so that Eq. 5 becomes where r is an index that spans all possible HFIs, b and b' are sums over m-states of the intermediate nuclear state, and ρ γ 2 , 0 ( ) and ρ γ 1 ,t ( ) are vectors in the Liouville space, defined here by is given by Eq. 2 and prob(r) denotes the probability of the system being found in state r at the instant of the γ -ray emission.
For random fluctuations that are independent of history among static HFIs, the evolution operator can be expressed as Ω t ( )= expBt ( ) , where B is known as the Blume matrix [47]. The B is constructed from the Hamiltonians of all HFIs through which the system evolves and the rates of transition among the accessible HFIs as described in detail in refs. 47 and 48. The evolution operator matrix elements are written rb ′ b where p 0,q is the probability that the system is in HFI state q at the time of the first gamma emission. Within the Liouville formalism, Eq. 7 does not change and Eq. 8 becomes For most interesting physical cases, the evolution operator can be diagonalized so that the where M m ( and N m ) are the m th left and right eigenvectors of the Blume matrix and −λ m + iω m is the m th eigenvalue.
In the presence of fluctuating HFIs, an equation similar to Eq. 9 can be obtained: where the ω n come from the imaginary parts of the eigenvalues of the Blume matrix, the exponential decay constants λ n come from the real parts of the eigenvalues, and the amplitude factors S k 1 k 2 ,n N 1 N 2 depend on projections of the eigenvectors of the Blume matrix along the state vectors

Defects and Diffusion Studied Using PAC Spectroscopy
in Liouville space. The primary result of the fluctuating HFIs is that the perturbation function becomes damped through the presence of the exponential decay factors. A secondary effect is that the interaction frequencies can deviate from the frequencies that would be observed in a superposition of static HFIs; however, this deviation may or may not be as pronounced as the damping, depending on the underlying HFI fluctuation rates.
A rigorous calculation of the perturbation function can be made when the changes in HFIs are connected to changes in the physical origins of the HFIs within the stochastic model. In cases involving fluctuations among charge states, the HFI fluctuation rates in the stochastic model are simply the charge fluctuation rates. In cases involving atomic jumps of defects, changes in HFIs are connected to defect jump paths and jump rates. Measurement of the damping parameters λ n then allows one to determine the charge fluctuation rates or atomic jump rates.
For fluctuations caused by atomic jumps, the accessible HFIs and rates of transition in the stochastic model of hyperfine interactions depend on crystal structure, defect concentrations, and atomic jump vectors. In general, stochastic models and their corresponding theoretical expressions for PAC perturbation functions must be considered in detail on a case-by-case basis. There have been several detailed investigations of how damping constants and frequencies depend on fluctuation rates [49,50,51] for a variety of models. The results of these investigations can be used to analyze experimental spectra obtained from physical situations that can be described by those models. Unfortunately, though, many physical situations examined in experiments are not included in those previous analyses, and alternate means of analyzing experimental data must be used.
Often, empirical expressions for fluctuating HFIs based on the results of Baudry and Boyer [52] and on Forker et al. [53] are used when analyzing PAC spectra. At least for "simple" fluctuation models, one can to a good approximation express the average perturbation function in two different fluctuation regimes: a slow fluctuation regime in which HFI fluctuation rates are less than ω L or ω Q and a fast fluctuation regime in which the rates are greater than ω L or ω Q . Then provides a reasonable approximation in the two regimes. Here, G kk ,Static t ( ) is the superposition of perturbation function that originates from probes distributed among the attainable static HFIs and G kk ,M.A. t ( ) is the perturbation function that results from the average of attainable HFIs in the motionally averaged limit. The damping rate λ slow is proportional to HFI fluctuation rate (and the underlying defect jump rate) and λ fast is inversely proportional to fluctuation rate.
It is worth pointing out that for many fluctuating systems, the λ n is nonzero even when ω n is zero so that the non-oscillating contribution to the perturbation factor can be damped. This is in contrast to the case of inhomogenous broadening, for which zero frequency terms do not have an exponential damping factor. Thus, observation that the non-oscillating portion of a perturbation function decays to zero is evidence for dynamic effects.
Experimental Considerations PAC Probes. The PAC probe nucleus acts as a foreign agent, a spy so-to-speak, that scopes out its environment and reports its observations in code by emitting correlated γ -ray pairs. To be suitable as a PAC probe a nucleus must decay from an excited state via emission of two γ -rays.
The intermediate state must have a reasonably long lifetime so that the probe has time to "feel" the extra-nuclear fields and to interact with them. Ideally, the γ -ray energies are sufficiently different so that γ 1 and γ 2 can be distinguished. Moreover, the nuclear moments of the intermediate state, as well as the anisotropy of the angular correlation between γ -rays have to be large enough to allow for measurement of the perturbation function. Last but not least the parent nucleus must be readily available or produced, and it must have a long enough lifetime to be practical to work with. There are about 10-20 nuclei that can be used as PAC probes; however, in practice most of the PAC work reported in the literature is done with two probes: 111 Cd and 181 Ta. Table 1 lists some of the properties of these two nuclei, as well as those of 140 Ce and 44 Sc, two probes utilized by authors in this volume. 111 Cd is the decay product of electron-capture-decay of 111 In with a half-life of 2.83 days. The parent nuclei are produced in an accelerator either by 110 Cd (d,n) 111 In or by alpha-bombardment of silver, 109 Ag( α, 2n) 111 In. As 111 In is frequently used in biomedical and pharmaceutical research, it is readily available and usually comes dissolved in dilute aqueous HCl. Usually, 111 In-activity is either diffused into the material under study or implanted with a heavy-ion accelerator. Because of the relatively short parent half-life, an 111 In-sample retains adequate activity to be used for about two weeks. 181 Ta is the product of βdecay of 181 Hf which is easily prepared by irradiating natural Hf in a flux of thermal neutrons, 180 Hf( n, γ ) 181 Hf. Because of the large absorption cross section of 180 Hf for thermal neutrons, typical irradiation times are a few minutes to few hours. 181 Hf-samples retain sufficient activity to perform experiments for 3-4 months.
Most PAC probes, and all the ones in Table 1, have two distinct chemical identities: one determined by the parent nucleus and one by its daughter nucleus. Sometimes researchers identify the PAC isotope used according to its parent and sometimes by its daughter. To add to the confusion, different physical quantities are measured when the probe is in its parent form versus its daughter form. Equilibrium distributions of probe-defect complexes, which determine site fractions, are established by the parent state. In general, the time between transmutation to the emission of the first gamma ray is short enough that these complexes remain intact during the double-γ cascade, at least at room temperature. But, the time is long enough for the newly formed daughter and the atoms around it to have a chance to relax to the optimal configuration. Thus, interaction frequencies (and amplitudes in the case of non-polycrystalline samples) depend on the daughter state. Fluctuating HFIs are detected when their rates of change are comparable to the inverse lifetime of the intermediate state of the daughter probe, so dynamic effects are correctly expressed in terms of behavior associated with the daughter state of the probe.
PAC Spectrometer. Measurement of the (unperturbed) angular correlation of two γ -rays is usually performed with one fixed and one movable detector. The γγ coincidences are recorded for equal time intervals with varying angles between the detectors.
Hyperfine interactions between the PAC probe's moments and extra-nuclear fields result in a modulation of the coincidence count rate, in which case it is not necessary to measure count rate as a function of angle. Instead, a PAC spectrometer is configured with 3 or 4 detectors arranged in fixed angular intervals. The most common arrangement uses 4 detectors at 90˚ intervals, and

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Defects and Diffusion Studied Using PAC Spectroscopy delayed coincidences between the 90˚ and 180˚ detector pairs are recorded and stored as histograms in a multichannel analyzer (MCA) or a computer acting as an MCA. Schematics of two common fixed detector spectrometer designs are shown in Fig. 7.
Most PAC spectrometers use scintillation detectors, such as BaF 2 or NaI mounted on photomultiplier tubes (PMT). These scintillator materials combine high detection efficiency with both reasonable energy and excellent time resolution. More recently Lutetium-Oxyorthosilicate (LSO) scintillators have been used successfully. LSO combines a time resolution almost as high as BaF2 with a greater detection probability and better light output [54]. Scintillators can be of cylindrical shape, 1-2" in height and diameter; however, to obtain the best solid-angle properties they are frequently in the shape of a truncated cone.
Detectors usually provide two output signals; the dynode, or "slow signal" is typically taken off the 7 th to 10 th dynode of the PMT, and its pulse height is proportional to the energy of the absorbed γ -ray. The anode signal, also referred to as the "fast signal", comes from the PMT's anode and provides a relatively precise indicator of the actual time of detection of the γ -ray. The dynode signal output is shaped by an on-board RC network and provides a signal with relatively good energy resolution.
Most PAC spectrometers in use today are based on the fast-slow coincidence method, which uses the fast signals for timing information and the slow signals for energy information. The timing information is derived from the fast signal by so-called constant fraction discrimination (CFD). Compared to simple leading edge triggering this provides superior timing properties as it eliminates amplitude-dependent walk for signals with variations in rise times. The slow signal is amplified and then sent to two energy discriminators*, which generate a logic pulse if the amplified slow signal falls into the energy window defined by a lower and upper discriminator level.
Of the fast-slow coincidence PAC spectrometers, there are two basic designs that are in use, labeled below and in Fig. 7 as type I and type II.
Type I Spectrometer. The logic signals from all energy discriminators associated with the various detectors are fed into the so-called routing logic, which provides a digital code indicating which γ was absorbed in which detector. The routing code is then used to address a specific storage area in the multichannel analyzer that is used to record the delayed coincidence events.
Moreover, the routing logic also determines if a particular event is valid. A valid event means a γ 1 was detected in one detector followed by a γ 2 in another detector, while the other detectors did not register a γ 1 or γ 2 .
Meanwhile all CFD outputs are logically ORed and connected to the start and stop inputs of a time-to-analog converter (TAC). The TAC functions as a clock and provides an analog output with a pulse height proportional to the time between the start and the stop signal. The TAC output is converted into a digital representation of the time difference in an analog-to-digital converter (ADC), which is connected to the multichannel analyzer and is gated by the routing logic. For valid events the ADC output is recorded in the appropriate memory bank of the MCA, while the routing logic inhibits the TAC for all invalid events. Note that in some spectrometers, the ADC and MCA are contained in a single module.
Type II Spectrometer. The energy discriminator output signal tuned to γ 1 for a specific detector is used to gate the CFD signal of the same detector. The gated CFD signals are then logically ORed to provide the start signal for the TAC. Similarly the CFD signal is gated by the energy discriminator output signal tuned to γ 2 and logically ORed to generate the stop signal for the TAC. In this way the TAC is started and stopped only by signals that satisfy the proper energy relationship, i.e. the TAC will not be started or stopped by a signal that does not correspond to a γ 1 or a γ 2 as is possible in a type I spectrometer. A routing logic similar to that used in a type I *Also known as Single Channel Analyzers.

18
Defects and Diffusion Studied Using PAC Spectroscopy spectrometer is employed to determine which detector pair was involved with the event, so it can be stored in the appropriate memory bank of the MCA. A type II spectrometer performs a preselection of pulses before they are used to start and stop the TAC, while in type I valid as well as invalid events, or even γ -rays that are neither γ 1 nor γ 2 , will start and stop the TAC. This limits the source activity that can be used with a type I spectrometer to lower values than can be used with a type II spectrometer. However, the type II spectrometer requires additional circuitry to accomplish this and thus is more complex. In particular a long delay of the CFD signals is required in order to gate them by the slower energy discriminator signals. The long delay is avoided in the type I spectrometer as the evaluation of the event validity goes on while the TAC processes the timing information. By the time the TAC is finished and makes available the output pulse, the slow signals have been processed by the routing logic and a decision can be made as to forward the timing signal to the MCA or discard it altogether.
There are alternatives to the fast-slow spectrometer design. One notable design is a fast-fast system that reduces the amount of nuclear instrumentation needed to construct a complete spectrometer [55]. Recently, there has been a trend to develop spectrometers that process outputs of the scintillation detectors digitally. Once the detector signals are digitized, further processing is done entirely in software, thus significantly reducing the amount of hardware needed [56,57].
Data Reduction. The goal in a PAC experiment is to analyze the modulations in coincidence count rate stored in the multichannel analyzer as histograms for each pair of detectors. Figure 8 illustrates the process used to arrive at the PAC spectrum known as the counting rate ratio R(t) from the raw counting rate data.
The number of counts that correspond to a time t between events in detectors i and j can be

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is the number of true coincidences at angle θ ij and time t, and B ij θ ij ,t ( ) is the number of accidental coincidences. True coincidences occur when both detected gamma rays came from the same nucleus, and N ij θ ij ,t ( ) is proportional to the angular correlation function W θ ij ,t ( ) . Accidental coincidences occur when detected gamma rays come from different nuclei; therefore, B ij θ ij ,t ( ) is independent of W θ ij ,t ( ) . The first step in the data reduction process usually involves subtraction of the background contributions from the total count rates, leaving N ij θ ij ,t ( ) .

The number of true coincidences is given by
where detector pairs 1-3 and 2-4 are 180° apart and detector pairs 1-4 and 2-3 are 90° apart. The R(t) can be considered the PAC spectrum, and an example is shown in Fig. 9a. For a more general discussion of counting rate ratios, particularly those used in other experimental configurations, the reader is referred to the work of Arends et al. [58] and of Butz [21]. For a polycrystalline sample, it can be shown that Eq. 22 reduces to where Q kk are attenuation factors that arise because of finite detector size [59]. It is common to report results in terms of effective anisotropy parameters, A kk eff. ≡ Q kk A kk .
For spectra obtained from polycrystalline samples, the second subscript on Q kk , A kk , and G kk is redundant, and it usually is dropped. When using probes for which anisotropy terms of order higher than A 22 can be neglected, the counting rate ratio is simply R t () when the anisotropy is negative (e.g., in Fig. 9b). Other authors display spectra after dividing R(t) by A 2 eff. and label it G 2 (t) in order to provide a closer connection to the underlying physics of the hyperfine interactions (e.g., in Fig. 9c). Often, authors display a Fourier transform of the perturbation spectrum to better illustrate the frequency multiplets -three frequencies in the case of spin-5/2 probes, (see Fig. 9d).
As can be seen in Fig. 9a-c, error bars increase with time between γ -ray emissions.
Estimated uncertainty in PAC experiments is based on the statistics of nuclear counting. For the channel corresponding to time interval t in the i-j histogram, N ij θ ij ,t ( ) , the uncertainty is estimated to be the standard deviation of a Gaussian distribution (assuming a large number of observed counts) with mean given by N ij θ ij ,t ( ) so that the uncertainty is N ij θ ij ,t ( ) . Hence, absolute uncertainties in collected data decrease with time as exp −t / 2τ ( ) . The counting rate ratio of Eq. 22 is a normalized difference in counting rates at 180° and 90°, so the uncertainty in a data point of the counting rate ratio is effectively a relative counting uncertainty, N / N , which is proportional to exp t / 2τ ( ) . Thus, the error bars in the counting rate ratio increase in time.
Fitting. Following data reduction, PAC spectra are fitted to a theoretical expression of the perturbation function by varying parameters in the model function until a χ 2 goodness of fit parameter is minimized. Generally, an empirical form of the model perturbation function is used*: with for polycrystalline samples, which simultaneously takes into account multiple discrete static sites, inhomogeneous broadening**, and dynamic effects. Best-fit values obtained from fits can provide information about defects in a material. Interaction frequencies ω n, j often can be used to determine location(s) of probes, and what defects are near the probes by symmetry considerations, by comparison with results of DFT calculations, or by a combination of both. Variation in site fractions f j can yield information about defect formation energies, defect association energies, and can help in identification of defect configurations by examining their dependence on temperature and sample composition. The degree of inhomogeneous broadening, δ j , provides a qualitative measure of the concentration of randomly distributed defects in a material, and the dynamic damping parameter λ j provides a measure of the rate at which defects jump near the probes. For non-polycrystalline samples, it is often possible to allow the s kn parameters in Eq. 24b to vary as adjustable parameters, in which case Eq. 24 can be * Conventionally, the model perturbation function is written without the bar: . We have included the bar here to distinguish the measured perturbation function from the single-interaction perturbation functions in Eqs. 8,9,11. ** Again, there is some variation in Eq. 24b because of how the linewidth of the inhomogeneous broadening term is described. used to analyze spectra obtained from single crystal samples provided that anisotropy terms of higher order than A 22 can be neglected or included by further adjusting amplitudes.

Examples
This section examines how information about the properties and the processes related to defects can be obtained from parameters in Eq. 24 when fitting PAC spectra. Information obtained from interaction frequencies, site fractions, amplitudes, inhomogeneous broadening, and dynamic damping are considered separately. The discussion here is left largely in general terms with references provided to specific studies, many of which go into more detail regarding derivation of equations or methodology. Where possible, examples from multiple classes of materials are provided. Interaction frequencies.
The measured interaction frequencies provide information about the strength of the magnetic field present at the nucleus or of the distribution of charge surrounding the nucleus. Depending on the experiment, this can allow one to determine the location(s) of probes, how defects are arranged around probes, or charge states of defects. When different probe environments have distinct point symmetries, a simple consideration of symmetries of the observed EFGs can allow one to determine probe site occupation or arrangements of defects near the probes. Usually, more than just a symmetry consideration is needed, though. Examination of how site fractions change when varying experimental configurations such as sample composition, measurement temperature, annealing temperature, and sample orientation can help identify the origins of observed EFGs. In some cases, comparisons of strengths of measured QIs to predictions based on a simple point charge model or to QIs observed in analogous systems can allow identification of the sources of EFGs. In other cases, identifications can be made by comparison to predictions based on calculations using density functional theory.
The EFG underlying the QI is the quantity of interest in a study of materials properties, but the strength of the QI usually is reported (in terms of the quadrupole interaction frequency ω Q or the quadrupole coupling frequency ν Q ) rather than the largest magnitude principle component of the EFG. This is because nuclear quadrupole moments, which are needed to calculate EFG components from interaction frequencies, generally are known with less precision than the measured frequencies. The EFG asymmetry parameter, on the other hand, does not depend on quadrupole moment, so that EFG asymmetries routinely are reported along with the QI strengths.
There have been a number of cases where site occupation could be determined by symmetries of EFGs alone. For example, the site occupation of indium in GdAl 2 was determined this way. The two primary EFGs observed had ω Q = 0 and ω Q = 7.5 Mrad/s with η = 0, which correspond to probes occupying the Gd site (with 43m site symmetry) and the Al site (with 3m site symmetry) respectively [60]. As a second example, site occupation of indium in Zr 2 Rh was based on observed EFG symmetries. The two EFGs observed had ω Q = 13 Mrad/s with η = 0 and ω Q = 11 Mrad/s and η = 0.93, which correspond to probes on the Rh-site (with 6m2 site symmetry) and on the Zr site (with mm symmetry) respectively [61]. Other examples include indium site occupation in the HfAl 2 and ZrAl 2 Laves phases [62], and indium and hafnium in bixbyites [63,64].
It is common for more than one lattice site in a compound to have the same site symmetry, in which case simple symmetry arguments cannot be used to determine the site occupation of a probe. In such cases, it is sometimes possible to determine site occupation by comparing signals to those obtained in isostructural compounds where site occupation of probes can be determined. Indium site occupation in Ga 7 Pd 3 is a good example of this [65]. Ga 7 Pd 3 and In 7 Pt 3 have the Ge 7 Ir 3 structure, in which there are two inequivalent Ge sites: Ge(1) with 4m.2 site symmetry and Ge(2) with .3m site symmetry, both of which lead to axially symmetric EFGs. Two axially symmetric signals for indium in each system were observed:  [65]. Not all systems are as simple as Ga 7 Pd 3 , and identification of EFGs by analogy alone may not be possible because more EFGs are observed in the system-of-interest than in the isostructural reference compound. In such cases, it may be possible to determine origins of EFGs by using the point charge approximation. An example of this was identification of sites occupied by indium in Ni 2 Al 3 -structured compounds [66]. Another example involved picking out which of (several) observed EFGs corresponded to the defect-free EFGs in bixbyites [67], for which an empirical correction could be applied to allow comparison between different PAC probes and to compensate for shortcomings in the point charge model [30].
When results from analogous systems are not available, it may still be possible to identify occupation sites in complex compounds by comparison to predictions of EFGs calculated using density functional theory. A good example of this was the recent determination of indium site occupation in (Hf/Zr) 3 Al 2 and (Hf/Zr) 4 Al 3 [68].
Interaction frequencies provide a means of detecting defects near probes. When there is an attractive interaction between the PAC probe and defects, significant fractions of probes can be found in complexes with one or more defects. Measured interaction frequencies provide a "fingerprint" of sorts for each defect complex; however, the interaction frequencies alone typically do not allow one to identify the defect complex because multiple defect arrangements produce the same point symmetry at the PAC probe site.
In compounds where PAC probes sit at lattice sites with cubic point symmetry, detection of non-zero interaction frequencies indicates the presence of nearby point defects. It often is possible to obtain attributions for the origins of the EFGs causing the QIs by varying experimental conditions such as sample composition, measurement temperature, or oxygen partial pressure. For example, the farther a sample is from its stoichiometric composition, the more point defects will be present. Thus, defect complexes involving first one defect, then two defects, etc. would be observed as one increases the deviation from stoichiometry. This was used to identify indium, oxygen-vacancy complexes in CeO 2 [69] and indium, transition-metal-vacancy complexes in B2structured compounds [26 and references therein]. As another example, monitoring the appearance and the disappearance of signals when varying oxygen partial pressure in CoO allowed attribution of different indium, cobalt-vacancy complexes [70].
As with site occupation, it is not always possible to identify which defect or defects induce an observed EFG by varying sample composition, measurement temperature and by using symmetry with simple arguments based on a point-charge approximation. In such cases, it may be possible to make a positive identification by comparing observed interaction frequencies to those predicted from calculations using density functional theory. For example, the charge states of acceptors and Te-vacancies in CdTe were determined by calculating EFGs induced by these defects when next to PAC probes using DFT calculations assuming neutral and ionized states for the defects and comparing the predictions with experiment [71].
Site fractions. Qualitative examination of how site fractions vary with composition or temperature can aid in identifying origins of HFIs, as mentioned in the previous section. Moreover, quantitative analysis of these variations can yield information about defect thermodynamics. Depending on experimental conditions, it is possible to determine site occupation transfer energies, defect association energies, effective defect formation energies, and migration energies.
Composition and temperature dependences of defect concentrations in a material are coupled to changes in the free energy of the material accompanying the introduction of defects. Theoretical

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expressions for equilibrium concentrations of defects can be obtained, for example, through minimization of a material's Gibbs free energy subject to structural and compositional constraints within the framework of the canonical ensemble. At the stoichiometric composition, the concentration of an intrinsic defect, defect       , is given simply by , where S F eff. is the effective formation entropy and H F eff. is the effective formation enthalpy.
For non-stoichiometric compositions or in situations where extrinsic defects are non-negligible, [defect] depends in general on the deviation from stoichiometry and on concentrations of extrinsic defects. Fig. 10 illustrates the equilibrium concentration of an A antisite defect in a generic binary alloy A 1+x B 1-x at three compositions (solid curves). At high enough temperature, [defect] is given by the stoichiometric value, and this is referred to below as the thermally activated regime. At low temperature for x > 0, in which case the A antisite is a constitutional defect, [defect] is independent of T to a good approximation. (For x < 0 at low T, [defect] also shows a thermally activated behavior with a different activation energy than H F eff. , but this is not considered further here.) The dashed curve in Fig. 10 gives a schematic representation of the defect concentration actually measured in a material. At some temperature intrinsic defect concentrations are effectively "frozen in" when defects move too slow to allow equilibrium to be maintained. A rough estimate for the "freeze-in" temperature T f can be made based on diffusivity. In an isotropic medium, the root-mean-square displacement r rms over a time period τ is related to the diffusion coefficient by D = r rms 2 / 6τ , and the diffusion coefficient is proportional to the defect jump rate: where w 0 is the attempt frequency and Q is the diffusion barrier in the potential-barrier model of diffusion. The estimate for T f is found by solving for the temperature at which r rms is equal to the average grain size in a sample. In Fig. 10, T f was shown to lie in the "thermally activated" regime; however, there is nothing to prevent it from being in the nonthermally activated regime.
PAC is particularly sensitive to point defects in materials when there is an attractive interaction between the probe and defects so that probe-defect complexes form. The fraction of probes complexed with a defect is a function of defect concentration and temperature, and measurement of the site fraction as a function of temperature allows one to determine thermodynamic quantities such as the defect association energy. The connection between PAC site fractions and thermodynamic quantities is derived for a simple system in which probes are found either in a defect-free environment or as a probe-defect pair. The extension to larger clusters involving multiple defects is straightforward. Below, f 0 denotes the fraction of defect-free probes and f p denotes the fraction of probes that are part of defect pairs.
The principle of detailed balance relates the probe fractions: f p r d = f 0 r t where r t is the total rate at which the defect "traps," or jumps from a distant site to a near neighbor site of the probe, and r d is the total detrap, or disassociation, rate. Under the usual assumptions of a potential-barrier model and of low defect concentrations, where z d is the number of detrapping pathways and z t is the number of trap pathways. Note that z t = z d z where z is the coordination number of the probe site. S D and H D are the entropy and the enthalpy of dissociation while S M and H M are for the barrier that the defect must overcome to jump from a distant site to a near neighbor site of the probe. The S M and H M may or may not be equivalent to comparable bulk quantities (that is, for defects far away from the PAC probe); however, it usually is assumed that any differences are negligible. Finally, ν d and ν t are the vibrational attempt frequencies.
The ratio of fractions is given by Arguably, the most direct way of obtaining values for H A and, when possible S A , experimentally is by taking measurements at multiple temperatures, in which case f p and f 0 correspond to the PAC site fractions. This method was used to study indium-transition metal vacancy association in NiAl [73], FeAl [26], indium-cadmium vacancy association in CdTe [74], indium-samarium vacancy association in SmNi 2 [75], and a wide range of probe-solute pairs in metals [9].
In the non-thermally activated regime or for temperatures below T f , [defect] is essentially independent of temperature so that the temperature-dependence of f p / f 0 enters only through the it is the temperature where defect complex formation freezes, so that there is no variation in site fractions below this temperature. T c is expected to be lower than the T f of intrinsic defect formation as explained below. The above expressions for r d and r t are rigorous when the defect is an interstitially diffusing defect or a vacancy, aside possibly from a correlation factor which generally is not included in derivations based on the principle of detailed balance. For substitutional defects diffusing via a simple vacancy mechanism, the above expressions are satisfactory as long as the vacancy jump rates are appreciably larger than the defect jump rates; otherwise correlation effects will be significant and will need to be included.
It is not always possible or practical to perform PAC measurements at elevated temperature, and many PAC studies of defect association, or rather disassociation, have been performed by collecting PAC spectra at room temperature following a sequence of anneals at elevated temperature, usually as an isochronal sequence. Here, two common types of experiments are described: (1) formation and dissociation of complexes formed between probes and intrinsic defects and (2) dissociation of complexes formed between probes and extrinsic defects.
In annealing studies of probe-intrinsic defect complexes, equilibrium concentrations of intrinsic defects initially are established at elevated temperature and then quenched into the sample by cooling to room temperature rapidly enough that equilibrium defect concentrations cannot be established. The cooling rate also is fast enough that defects do not have sufficient time to establish equilibrium concentrations of probe-defect complexes. So, initially there is a non-equilibrium distribution of defects and probe-defect complexes: an excess of intrinsic defects and a deficiency of probe-defect complexes. The first PAC measurement establishes a starting set of site fractions. Then, an anneal step, which allows distributions of defects to partly shift toward equilibrium, is performed. A subsequent PAC measurement will reveal a change in site fractions. The annealmeasurement sequence can be repeated as many times as needed.

Defects and Diffusion Studied Using PAC Spectroscopy
The rest of the discussion will consider again the simplified two-state case consisting only of defect-free probes and probes with one neighboring defect. Following the initial quench, where T Q is the quench temperature, which is equal to the T f as determined by the rate of cooling. The site fraction ratio will be temperature. For simplicity, it will be assumed that subsequent anneals take place below T f so that [defect] does not change; however, f p and f 0 will as the system evolves toward reaching an equilibrium number of probe-defect complexes.
In an anneal step, the sample is held at the target temperature for time t A and the temperature is adjusted up or down rapidly enough so that to a good approximation, one can analyze changes in site fractions as occurring entirely at the anneal temperature T A . At this temperature, the dissociation rate r d and trap rate r t are given by the same expressions as above, with T = T A and similar qualifiers for validity. In the equation for r t , [defect] is the concentration of unpaired defects. As long as the concentration of probes is much smaller than [defect], so that the decrease in [defect] is negligible as pairing occurs, the trapping rate will be independent of time. This corresponds to the random telegraph process, which is a stationary Markov process, that is described by the Master equation(s) The solution to these equations is (25)   where f p (0) is the fraction of probes with a near neighbor defect before the anneal, established by either the quench or a previous anneal step. Eq. 25 describes an exponential decay or growth from an initial fraction f p (0) to the equilibrium fraction r t / r d + r t ( ) with a lifetime given by 1 / r d + r t ( ) .
A simulation of f p for a sequence of anneals is shown in Fig. 12.
Typically, an equation like Eq. 25 is not used to analyze site fractions in an anneal sequence. Rather, expressions derived assuming a single step, irreversible process for trapping (the left side of the peak in Fig. 12) or for disassociation (the right side of the peak in Fig. 12). For example, when trapping rates dominate over dissociation, the fraction of paired probes f p can be found from the differential equation that describes association, df 0 / dt = −r t f 0 , and the normalization condition The solid curve in Fig. 12 is a plot of Eq. 26, and as can be seen, it agrees well with the data generated using Eq. 25, at least up until about the anneal temperature where dissociation becomes important, and it can provide a good estimate of migration barrier energy. This type of analysis, or variations of it, have been used to study probe-defect formation and disassociation in semiconductors and metals. For example, this was used to analyze the formation and the dissociation of indium/cadmium-vacancy pairs in CdS [76]. A variation of this method, in which the excess vacancies were residuals of the radiation damage resulting from implantation of probes or from irradiation of samples rather than those frozen in after a quench from high temperature, was used to study association and disassociation of Pd-vacancy pairs in Si [77] and of In-vacancy complexes in fcc metals [4] and bcc metals [78,79,80,81]. For the studies involving Si and the metals, vacancy concentrations were not constant so that Eq. 25 was not applicable, and analyses were based on an alternative equation (see ref. 4 for more information).  In annealing studies of probe-extrinsic-defect dissociation, the first step is to establish an equilibrium, or near-equilibrium, number of probe-defect complexes. The lower the temperature at which this is done, the larger the fraction of probes will be found in complexes; however, the temperature needs to be above the complex formation freeze-in temperature. Anneal steps are then performed at higher temperatures. At each anneal temperature, the initial distribution of probedefect complexes is not in equilibrium, and the anneal allows a partial shift towards equilibrium. As in the probe-intrinsic defect case, a PAC measurement after each anneal step reveals a change in site fractions.
Again, the simple two-state case consisting of defect-free probes and probes with one neighboring defect is considered. As long as anneals are done at temperatures below T f , so that the concentration of extrinsic defects does not change, and as long the concentration of extrinsic defects is significantly larger than the concentration of probes, Eq. 25 is valid. A simulation of f p for a sequence of anneals is shown in Fig. 13.
An equation derived assuming an irreversible single step process usually is used. So, under the assumption that retrapping rates are negligible, the relevant differential equation for dissociation is The solid curve in Fig. 13 is a plot of Eq. 27, and as can be seen it represents the data well. Most notably, dissociation of Cd-H complexes in III-V semiconductors and of In-acceptor complexes in Si and Ge were studied by monitoring changes in site fractions of probe-defect complexes with anneal temperature and calculating dissociation energies using Eq. 27, as summarized in the review by Deicher [11]. As a final note, Eq. 25 suggests a means of estimating T c , the complex formation freeze-in temperature for in situ measurements. If activation enthalpies and entropies are known, an  approximate value for T c is obtained by solving for the value of temperature such that 1 / r d + r t ( ) equals the time needed to collect a PAC spectrum, which is typically 12 -48 hours. Amplitudes. The amplitudes in Eq. 24 deviate from their polycrystalline values when measurements are made on single crystal samples. The amplitudes depend in detail on orientation(s) of principal axes of EFGs and/or the directions of magnetic fields with respect to detector locations. Thus it is possible, within some limitations, to determine the orientation of HFI axes based on signal amplitudes. In crystals with high symmetry -especially those with cubic symmetry -a unique determination of EFG axis orientation will be possible only if the main principal axis is directed along one of the symmetry axes of the crystal because otherwise there is a large number of crystallographically equivalent orientations of the EFGs and the deviation in amplitudes from their polycrystalline values is diminished.
It is possible to take into account sample orientation with respect to detector placement, calculate perturbation functions, and fit spectra for single crystal samples rigorously -for example, by using the DEPACK software [82]. It also is possible to obtain information about orientations of hyperfine fields through a more qualitative examination of amplitude variation with sample orientation and detector configuration -illustrated nicely for the case of EFGs in cubic crystals in Fig. 3 of ref. 83.
It is reasonable to expect that determining the orientation of EFG principal axes would help in the identification of what types of defects are near a PAC probe or in determining the conformation of a defect complex. For example, in the zincblende structure, a substitutional probe without nearby defects would experience zero EFG. Nominally, a defect at the first neighbor site would lie in the <111> crystal direction whereas a defect in the second neighbor site would lie in the <110> crystal direction, each producing an axially symmetric EFG. Thus, measuring whether the axis of symmetry were in the <111> or the <110> direction would allow one to determine where the defect is located.
Lattice relaxations can reduce further the symmetry of the local crystal structure and the induced EFG, thereby affecting the orientation of the EFG principal axes as well as the values of the components (and the measured interaction frequencies). In samples with the zincblende structure, this would mean that the main principal axis of the EFG is along neither the <111> nor the <110> direction. For example, the main principal axes of EFGs experienced by substitutional indium probes with a second neighbor vacancy in II-VI compounds were not in the <110> direction as expected by symmetry considerations [13]. Lattice relaxation, and possibly an additional effect caused by a shift in electronic density near probes, was so strong in the cases of ZnSe and CdTe that the main principal axes were in the <111> direction [13].
Another good example of the undependability of EFG axis orientation can be seen in the EFGs experienced by Ta and Cd in TiO 2 . The main principal axis of the EFG experienced by Hf is along the c-unit cell direction [35] whereas the main axis in the case of Cd was observed by PAC to be perpendicular to the c-axis [84]. In this case, the "undependability" originates from the inadequacy of symmetry arguments, which are based on a point charge picture, because contributions to the EFG by valence electrons dominate [35].
Observed orientations of EFGs are not always contrary to expectations based on symmetry. For example, orientations of most PAC probe/acceptor pairs in Si, Ge, and several III-V semiconductors are oriented along the <111> directions [11]; although, the orientation of the In-Te pair and of one of the In-Fe complexes in Si deviate [85,86]. As another example, the indiumoxygen vacancy pair and the collinear arrangement of indium with two oxygen vacancies in CeO 2 also resulted in the main principal axis of the EFG along the <111> crystal direction [87].
Inhomogeneous broadening. Because inhomogeneous broadening is treated empirically, it currently is not possible to relate degree of inhomogeneous broadening to defect concentration or measures of other physical sources of broadening such as grain size. Nevertheless, the degree of inhomogeneous broadening does offer qualitative information about disorder in a sample that can be useful.

Defects and Diffusion Studied Using PAC Spectroscopy
In studies that introduce PAC probes to samples via implantation techniques, radiation damage results in large concentrations of randomly distributed point defects in the region of the sample where the PAC probes are located. Observed PAC signals exhibit significant inhomogeneous broadening in such cases. Depending on the material and on the probe, the observed PAC signals could be broadened signals that arise from probes without defects in the first or second neighbor shell, as was the case for indium implanted in ZnO [88], in GaP [89], and in Nb [90], or that arise from probes with defects in the first or second neighbor shell, as was the case for Pd in Nb [90].
Following implantation, one expects to be able to reduce the number of defects by annealing at a temperature high enough for defects to be sufficiently mobile to recombine, at least partially. Indeed, isochronal anneal sequences with PAC measurements in between anneal steps typically show decreases in the degree of inhomogeneous broadening that is correlated to the reduction of defects. This is illustrated nicely in the two recent papers on indium implanted in ZnO [88] and GaP [89].
The sensitivity of the PAC signal linewidth to lattice defects suggests that PAC could be used as an atomic-scale probe of crystalline structure in highly disordered material. However, it has been shown that care must be taken when drawing conclusions regarding disorder throughout a sample based on the PAC signal. For example, two types of environments of Hf probes were observed in metamict zircon: (1) a moderately defective crystalline environment and (2) a nearly, if not completely, amorphous environment [91]. Annealing resulted in an increase of the site fraction of probes found in the crystalline environment and a decrease of the site fraction of probes found in the amorphous environment; however, the fraction of probes in the crystalline environment was smaller than what was expected based on the volume fractions of the two environments determined using conventional X-ray diffraction (XRD). The discrepancy was attributed to the difference in length scales probed by PAC and by XRD [91].
In PAC studies of Cu-Hf alloys, a significant site fraction of probes were in a crystalline state resembling Cu 3 Hf 2 [92] even though the sample composition was Cu 57 Hf 43 and XRD indicated that the sample was amorphous [93]. This discrepancy may again be attributable to the difference in length scales probed by the two methods. As pointed out by Kanazawa, the Cu-Hf result suggests that PAC could provide a better measure of amorphization quality than XRD [93], which certainly looks to be the case in Cu-Hf. However, the zircon results show that PAC alone is not reliable for such a characterization in general. Hence, it looks like a combination of PAC and techniques that are sensitive to long-range order, such as XRD, is required for complete characterization.
Dynamic damping. The most commonly observed source of dynamic damping occurs when the jump of a defect near the probe results in a change of the HFI at the probe or when a jump of the probe itself results in a change in the HFI. For simplicity, we shall assume that there is a one-toone correspondence between a defect jump and a change in the HFI so that the rate at which the HFI changes is equal to the jump of a defect.
In the usual potential-barrier model of atomic jumps, the jump rate w of a point defect is given by w = cw 0 exp S m / k B ( ) exp −H m / k B T ( ) where w 0 is the attempt frequency, S m is the migration entropy, H m is the migration enthalpy, or migration barrier, and c is a factor that describes the availability of neighboring sites to which the defect can jump. For diffusion of a tracer in a metallic system via a simple vacancy diffusion mechanism, the availability is given by c = Z [V] where Z is the number of near neighbors on the solute's sublattice and [V] is the concentration of vacancies on that sublattice.
When using the empirical form of Eq. 24b, the damping factor λ is proportional to the HFI fluctuation frequency, and in turn to the underlying defect jump rate, in the slow fluctuation regime, and λ is inversely proportional to the fluctuation frequency in the rapid fluctuation regime. Thus a plot of ln( λ ) versus 1 / k B T in the slow regime would yield a straight line, the slope of which is equal to −H m and a plot of ln( λ ) versus 1 / k B T in the rapid regime would yield a slope equal to +H m .
A number of studies have utilized an empirical analysis to study defect jump rates near probes. These include the observation of H jumps near Hf in yttrium, hafnium, and dysprosium hydrides [94,95,96], of H jumps near Cd in Si [97] and HfV 2 H x [98], of Cu jumps near Cd in Si [99], of rare-earth (R) vacancy jumps near Cd in R 1-x Ni 2 [75], and of oxygen vacancies in R 2 M 2 O 7 (R = Nd, Sm, Eu, and Gd; M = Zr and Hf) pyrochlores [100] and in CeO 2 [69]. Examples of how such an empirical analysis has been used to study the motion of the PAC probe itself include studies of Cd jump rates in -Mn [101], Pd 3 Ga 7 [102], Pt 3 Ga 7 [103], Al 11 R 3 (R = La, Ce, Pr) [104], and LaSn 3 [105].
An alternative way to determine defect jump rates from PAC spectra is to fit spectra with test functions generated from numerical solutions to an appropriate stochastic model. This method was used to determine Cd jump rates in RIn 3 (R = La, Ce, Pr, Er, Nd, and Y) [106,107] and RGa 3 (R = Dy, Er, Lu) [108].
Another important physical origin of fluctuating HFIs arises from a fluctuation in charge states of defect complexes involving the PAC probe. This was reported to occur for Cd-H and Cd-D (D = P, As, Sb) complexes in Si [109]. In this case, it was possible to find an analytical solution to Eq. 20 [110], which is a rare occurrence. Charge fluctuations in these studies were in the rapid fluctuation regime, and it was possible to obtain fluctuation rates and relative populations of charge states.
For very large fluctuation rates, damping factors become too small to measure and the perturbation function is equal to simply G kk ,M.A. t ( ) . It is possible to express the time average of accessible HFIs in terms of relative fractions of probes in the accessible HFIs so that one can determine relative populations of defect configurations. For example, this was the method employed in the original analysis of Cd-H charge states in Si [111]. It also was applicable to the case of rapid trapping and detrapping of oxygen vacancies to Cd probes in tetragonal ZrO 2 so that it was possible to determine equilibrium occupation probabilities of vacancies in sites next to probes as a function of temperature, which allowed determination of the cadmium/oxygen vacancy binding energy [112].

Summary
An introduction to perturbed angular correlation spectroscopy with an emphasis on its application in the study of point defects and diffusion was presented. This included explanations of the theoretical underpinnings of the technique, of the origin of the quadrupole interaction in solids, of probe selection and spectrometer configuration, and of a tutorial on how different physical parameters of interest to the study of defects and diffusion can be obtained from the analysis of PAC spectra. Brief explanations of a number of example studies were provided, arranged based on the analysis methods used. These are summarized in Table 2, arranged by the type of material and the general physical phenomenon considered.