Hyperfine InteractDOI 10.1007/s10751-012-0695-3

Realistic models of stochastically varying hyperfineinteractions caused by vacancy diffusionin L12-structured compounds

J. R. Castle · M. O. Zacate · W. E. Evenson

© Springer Science+Business Media Dordrecht 2012

Abstract Perturbed angular correlation spectroscopy (PAC) is an attractive methodfor fundamental studies of diffusion because of the possibility to observe directlyatomic scale defects involved in a diffusion process. Previous work investigated underwhat experimental conditions one could observe a contribution to a PAC spectrumthat clearly could be attributed to a vacancy in L12-structured compounds for thespecial case of self-diffusion. This has since been extended in the present workto consider the case of impurity diffusion and to explore whether or not distantvacancies or configurations with multiple vacancies affect PAC spectra significantly.

Keywords Perturbed angular correlation · Stochastic models · TDPAC

1 Introduction

There have been extensive measurements of Cd jump rates on the Cu sublattice ofcompounds with the Cu3Au, or L12, structure using perturbed angular correlationspectroscopy (PAC) with the 111In probe [1–4]. The Cu site has 4/mmm symmetryso that in the absence of a neighboring point defect, the probe experiences an axiallysymmetric EFG. A jump of the probe to a first neighbor site results in a reorientationof the EFG by 90◦; therefore, if jumps occur on the same timescale as the lifetimeof the PAC level (120 ns for the 111In probe), relaxation, or damping of the PAC

This work was funded in part by NSF grant DMR 06-06006 (Metals Program) andcomputational resources were provided in part by KY EPSCoR grant RSF 012-03.

J. R. Castle · M. O. Zacate (B)Department of Physics & Geology, Northern Kentucky University,Highland Heights, KY 41099, USAe-mail: zacatem1@nku.edu

W. E. EvensonCollege of Science and Health, Utah Valley University, Orem, UT 84058, USA

J.R. Castle et al.

signal, is observed. Measurement of the degree of relaxation provides the EFGreorientation rate and, in turn, the probe’s jump rate.

In close packed lattice structures such as the Cu3Au structure, there must bea vacancy in a site neighboring the probe in order for the probe to jump to thatsite. Such a vacancy would induce a non-axially symmetric EFG. In previous PACexperiments in Cu3Au-structured compounds, a non-axial signal was not observed.It was asserted that this is to be expected when vacancy concentration is low andvacancies jump much more rapidly than the probes [1]. This was verified later for thespecial case of self-diffusion through computer simulation [5].

Application of PAC in systems where relaxation due to atomic jumps is measur-able has the possibility to enhance fundamental understanding of diffusion whena detectable fraction of the PAC signal originates from a point defect that isparticipating in the diffusion process. For example, in the Cu3Au structure when aPAC probe is diffusing on the Cu sublattice, detection of a Cu vacancy would indicatea Cu vacancy diffusion mechanism whereas detection of a Au vacancy would indicatea more complex mechanism such as the six jump cycle. The Cu3Au structure is ofparticular interest because up to 436 compounds have this structure [6]. A techno-logically important example is Ni3Al, which has an attractive combination of severalunique properties, such as a relatively low density, resistance to corrosion, strengthretention at high temperatures, and high thermal conductivity. Most notably, Ni3Alis used as a coating of turbine blades in aircraft engines [7, 8].

Even though a defect signal has not been measured by PAC in the Cu3Au-structured compounds so far, it was possible to gain information about the operativediffusion mechanism in many of the compounds by measuring jump rates at phaseboundaries [1, 3]. Nevertheless it is desirable to obtain more direct evidence for theoperative diffusion mechanism through the observation of defect signals in PACspectra. The purpose of the present study is to investigate through computer sim-ulation how defect concentration, probe-defect association, and atomic jump ratesaffect the distinguishability of a defect signal in order to help identify experimentalconditions favorable for observation of vacancies on the Cu sublattice in compoundswith the Cu3Au structure.

2 Method and models

Following the method of Winkler and Gerdau [9], simulated PAC spectra arecalculated for polycrystalline samples under the assumptions that fluctuations amongEFGs are random and independent of history. In that case the perturbation function,or spectrum, is given by G2 (t) = ∑

q Gq exp[t(−λq + iωq

)]where

(−λq + iωq)

is theqth eigenvalue of the Blume matrix and Gq are time-independent factors that dependon the eigenvectors of the Blume matrix [9]. EFG tensors and rates of transitionamong the different EFGs are input parameters for computation of the Blume matrixas described in Ref. [10].

For PAC probes on the Cu sublattice of Cu3Au, defects jumping near the PACprobes or jumps of the probes themselves cause the EFG fluctuations. It is notpossible to consider all possible arrangements of defects and their resulting EFGswhen calculating the Blume matrix; rather the system is modeled by choosing a subsetof the defect configurations and jump vectors that are most probable and lead to the

Realistic models of dynamic hyperfine interactions in L12 compounds

greatest contributions to EFGs. Factors affecting the choice of subset may includedistance of vacancy from probe, number of vacancies near a probe, and whether ornot the probe is an impurity.

The simplest EFG-fluctuation model that includes at least one vacancy for theCu-vacancy diffusion mechanism was described in detail in Ref. [5]. It, like the morecomplicated models considered in the present paper, assumes that (1) the PAC probestays on the Cu sublattice, (2) the only defects other than the probes in the systemare Cu vacancies, and (3) the initial fraction of probes with a neighboring vacancy isthe equilibrium value. In addition, the simplest model assumes that only vacancies inthe first neighbor shell of the probe disturb the EFG and that there is at most onevacancy in the first neighbor shell. This results in a Blume matrix consisting of 15unique EFGs. Results of simulations using this model were reported previously withthe additional restriction that the probe was a host element, that is, for the case ofself-diffusion [5]. The main goal of the present study is to extend those simulationsto the case of impurity diffusion.

It is not obvious a priori if neglecting contributions to the EFG by vacanciesoutside the first neighbor shell or if considering only one vacancy next to a probeat a time can adequately represent the time dependent hyperfine interactions experi-enced by a PAC probe. Therefore, these effects were investigated in the present workby simulating spectra for two additional models: one that considered disturbancesto the EFG by a single vacancy out to the 3rd neighbor shell and another thatconsidered up to three vacancies simultaneously in the first neighbor shell. Thesemodels require more EFGs and have more complicated interconnections in thetransition rate matrix; therefore, the method described in Ref. [11] was required togenerate the input parameters needed for the stochastic models. In summary, fourmodels have been considered:

Model S1V1 Self-diffusion, up to one vacancy in the first neighbor shell (15 EFGs)

Model I1V1 Impurity diffusion, up to one vacancy in the first neighbor shell

(15 EFGs)Model S1V

3 Self-diffusion, up to one vacancy in the 1st–3rd neighbor shells(48 EFGs)

Model S3V1 Self-diffusion, up to three vacancies in the first neighbor shell (93 EFGs)

There are up to seven adjustable parameters in the above models. In all four models,V0 is the strength of the defect-free EFG, Vc is strength of the EFG contributioncaused by a single vacancy in the first neighbor shell (i.e., the largest magnitudeprincipal component of the diagonalized EFG tensor caused by only the vacancy),and [V] is the fractional vacancy concentration on the Cu sublattice. The rates oftransitions are functions of the vacancy jump rates, which are the remaining modelparameters. In general, all five vacancy jump rates of the five-frequency model [12]are needed. They are w0, for jumps between sites beyond the probe’s 1st neighborshell; w1, for sites within the 1st neighbor shell; w2, for vacancy-probe exchanges; w3,for jumps from a first neighbor site away from the probe (detrapping); and w4, forjumps into the first neighbor shell (trapping).

Model I1V1 is insensitive to jumps of type zero, so that w1, w2, w3, and w4 are its

rate parameters. In the case of self-diffusion w0 = w1 = w2 = w3 = w4 ≡ w, so thatonly one rate parameter, w, is needed in S1V

1 , S1V3 , and S3V

1 . It is convenient to carryout simulations using dimensionless quantities, so time and jump rates are scaled by

J.R. Castle et al.

the quadrupole interaction frequency of the lattice EFG, ωQ, which is related to V0

by ωQ = eQV0/[

4I (2I − 1) �]. Spectra were calculated for I = 5/2, as is appropriate

for the 111In probe.When a vacancy signal is not visible, the probe experiences a fluctuation among

the three axially symmetric defect-free EFGs, which are orthogonal to one another,and the time-dependent interaction is described exactly by the XYZ model [9, 13].The main purpose of this study is to determine whether or not a vacancy signal canbe seen in a spectrum for a given set of model parameters; that is, to determine if theG2(t) generated by a particular model deviates from the G2(t) of the XYZ model. Todo this, curves simulated for the S1V

1 , I1V1 , S1V

3 , and S3V1 models were fitted to spectra

generated using the XYZ model and goodness-of-fit was evaluated using a scaledreduced chi-square value defined by

χ ′2ν ≡ C2

0

νχ2 = 1

(N − 1)

∑ 1exp

(ti/τ)

[GXYZ

2 (ti) − Gmodel2 (ti)

]2

, (1)

where C0 is the size of the error bar at t = 0, v = N − 1 is the number of degreesof freedom, τ is the lifetime of the PAC level, and N is the number of time valuesused in the fit. The only adjustable parameter in the fits was the EFG reorientationrate in the XYZ model. The interpretation of the scaled reduced chi-square is asfollows: if the value of χ ′2

v is larger than the square of the experimental error at t = 0,then one expects an observable difference in the spectra of the two models. Largervalues of χ ′2

v mean that it is possible to discern a vacancy signal with larger statisticaluncertainties in spectra; that is, the larger the value of χ ′2

v, the easier it is to observea vacancy signal in a spectrum.

3 Results

As shown in the antecedent study [5], there are ranges of parameters where thevacancy signal is invisible and spectra look like those of the XYZ model. Thiswas found in the present study as well, and in those ranges, inspection of spectrarevealed that spectra are invariant to changes in jump rates as long as the diffusioncorrelation factor f [14] and the effective first neighbor vacancy concentrationQ ≡ [V] (1 − [V])−1 w4

/w3 are held constant. The factor Q takes into account probe-

defect association through the ratio w4/w3, and the (1 − [V]) term corrects forlarge vacancy concentrations. With that, it is sufficient to use three rate-dependentparameters in ranges where vacancies are invisible: w2, Q, and f . When a vacancysignal is visible a fourth parameter is required, and it is convenient to use w1. Thefollowing values of parameters have been considered: 10−3 ≤ w2/ωQ ≤ 106; 10−4 ≤Q ≤ 10−1; 0.01 ≤ f ≤ 0.99; w1 = w2/100, w2, 1000 w2; and −4.0 ≤ Vc/V0 ≤ 4.0.

Figure 1 shows a selection of simulated spectra and results of fits to the XYZmodel for indicated values of w2. In all spectra Vc/V0 = 0.5, Q = 10−1.5, w1 = w2,and the values of w3 and w4 were chosen such that f = 0.5 for the spectra on theleft and f = 0.01 for the spectra on the right. For w2/ωQ = 10−2 the simulatedspectrum in each case exhibits a non-axial symmetry component so that the fittedXYZ G2(t) deviates appreciably from the simulation for both values of f . Thenon-axial component arises because a significant fraction of probes have a static or

Realistic models of dynamic hyperfine interactions in L12 compounds

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.2

0.2

0.6

1.0w

2/

Q = 10-2

t Q

w2/

Q = 100

w2/

Q = 102

w2/ω

Q = 104

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.2

0.2

0.6

1.0

Q

w2/

Q = 10-2

w2/

Q = 100

w2/

Q = 102

w2/

Q = 104

ω

ω

ω ω

ω

ω

ω

ω. t ω.

Fig. 1 Selected spectra simulated using Model I1V1 (solid line) and the XYZ model (dashed line).

In all cases, Vc/V0 = 0.5, Q = 10−1.5, and w1 = w2. The values of w3 and w4 were chosen such thatf = 0.5 for spectra on the left and f = 0.01 for spectra on the right

slow-moving vacancy in the first neighbor shell. The sequences of spectra are notablydifferent for larger values of w2.

For f = 0.5, fitted XYZ perturbation functions and simulated spectra are inclose agreement when w2/ωQ ≥ 1. At w2/ωQ = 1 the vacancy traps and detraps fastenough that its non-axial contribution is motionally averaged away, and the spectrumclosely resembles the XYZ G2(t) in the slow fluctuation regime1 as probes jumpamong Cu sites. For w2/ωQ = 102 and 104 simulated spectra resemble the XYZ G2(t)in the rapid fluctuation regime as the vacancy and probe jump faster.

For f = 0.01 close inspection of the simulated spectrum at w2/ωQ = 1 reveals asuperposition of an axially symmetric signal due to probes without first neighbordefects and a quickly decaying signal due to probes with a vacancy jumping rapidlyamong its first neighbor sites. These two signals are described well accidentally byan XYZ G2(t), with deviations between the XYZ and I1V

1 G2(t)s within the statisticaluncertainty typical of experimental spectra. At w2/ωQ = 102 the simulated spectrumconsists of contributions from probes with or without initially trapped vacanciesand damping arises from trapping and detrapping of vacancies. The XYZ G2(t),however, does not fit this situation well. At w2/ωQ = 104 trapping and detrappingis fast enough that the simulated spectrum is described well by the XYZ G2(t) nearthe critical relaxation rate. For larger values of w2 simulated spectra correspond tothe fast fluctuation regime of the XYZ model.

1The reader is referred to Refs. [9], [13], and [16] for more information regarding characteristics ofthe XYZ model in the the slow, critical, and fast fluctuation regimes.

J.R. Castle et al.

Fig. 2 Contour plots of thescaled reduced chi-squaresobtained fitting model I1V

1with the XYZ model. Withinthe colored regions, the largerthe value of χ ′2

v, the easier it isto distinguish a vacancy signalin a spectrum

-4

-3

-2

-1

log(

Q)

1351030501003001000

f

w1w2/100 w

21000w

2

-4

-3

-2

-1

log(

Q)

-2 0 2 4-4

-3

-2

-1

log(

Q)

log(w2/

Q)

Vc /V0 = 0.5

-2 0 2 4log(w

2/

Q)

-2 0 2 4log(w

2/ω

Q)

0.99

0.5

0.01

χ 'ν2

10-4

10-4

ωω

To display the results of all simulations conveniently, contour plots of χ ′2v as a

function of Q and w2 were generated for combinations of Vc/V0, f , and w1. Resultsfor Vc/V0 = 0.5, which are characteristic for most values of Vc/V0, are shown inFig. 2. The extents of the colored regions

(χ ′2

v ≥ 10−4)

are largely independent ofVc/V0 for |Vc/V0| > 0.1; however, the steepness of the contours increases as |Vc/V0|increases. At |Vc/V0| ∼ 0.1 the extents shrink toward the upper left of the plots sothat for |Vc/V0| < 0.1, χ ′2

v < 10−4 throughout the simulation range and a vacancysignal cannot be observed.

Approximate boundaries for the visibility of vacancies can be established byinspection of Fig. 2. As can be seen, χ ′2

v is small and therefore there is no visiblevacancy signal for Q � 10−2. It appears in Fig. 2 that there is a boundary in theparameter w2, which depends on w1 and on f ; however, the condition for obtainingsmall values of χ ′2

v is most sensitive to the value of w3 through a complicatedinterdependence of w1, w2, w3, and w4 induced by the constraint that the correlationfactor be constant within the contour plot. It should be noted that the dependence ofthe boundary on f is not linear so that the f = 0.5 plots represent contour plots wellfor a wide range, approximately 0.1 < f < 0.9.

The frequency dependent part of the boundary can be described adequatelythroughout the full range of f considered in the present study by the approximationw3/ωQ ∼1; that is, no vacancy signal is visible for w3 � ωQ. This can be understoodas follows. If w3 and w4 are small, then the PAC spectrum consists of a superpositionof two signals: one from probes initially with a neighboring vacancy and one fromprobes without. For Q � 0.01, the fraction of probes with an initially trapped vacancybecomes large enough to be visible in the spectrum. For f > 0.1, w3 and w4 arelarger than w1 and w2 so that trapping and detrapping provide the dominant sourceof relaxation. In this case spectra look like the XYZ model’s G2(t) in the slow,critical and rapid fluctuation regimes when w3 � ωQ. For f � 0.1, w1 and w2 arelarger than w3 and w4, trapping and detrapping is reduced, and the resulting spectraare comprised of two distinct signals up until w3 ∼ 10wQ, which is the origin of thepeninsula-like feature in the contour plots. For w3 � 10 ωQ, spectra resemble theXYZ model in the critical and rapid regimes.

Realistic models of dynamic hyperfine interactions in L12 compounds

Except for cases with very small correlation factors, these results indicate that avacancy signal can be observed only when |Vc/V0| > 0.1, Q � 10−2, and w3 � ωQ. Itis worth noting that small values of f usually do not attract much interest in diffusionstudies because they arise when vacancies are tightly bound to probes, reducing thelong-range movement of the probes.

4 Discussion

There are two potential shortcomings in the I1V1 model. First, concentrations for

which a vacancy signal is visible are large enough that one expects a significantfraction of probes to have more than one vacancy next to a probe, but the modelincludes only one vacancy at a time. Second, it was assumed without justificationthat vacancies more distant than the first neighbor will not appreciably disturb theperturbation function. The validity of these approximations will now be addressed.

4.1 Effect of multiple vacancies in first neighbor shell

For the case of self-diffusion or when there is a negligible interaction betweenvacancies and a PAC probe, the probability PN of N vacancies being in first neighborsites of a probe is given by

PN = (ZN

)[V]N (1 − [V])Z−N (2)

where Z is the number of first neighbor sites (8 for the L12 structure) and(

ZN

)is

the binomial coefficient. Values of PN for N > 1 are less than 1 % up to [V] ∼ 0.02,about the concentration at which values of χ ′2

v are large enough for a vacancy signalto be just visible. The I1V

1 model therefore should be adequate as long as vacanciesare not visible; however, it may not be adequate when a vacancy signal is visible. Toinvestigate this, simulations including up to three vacancies in the first neighbor shellof the probe for the case of self-diffusion (model S3V

1 ) were performed.To judge whether or not a defect signal is present in a spectrum generated by the

S3V1 model, the spectrum was fitted to the XYZ model and goodness-of-fit assessed

using χ ′2v. Figure 3 shows a contour plot of the results for Vc/V0 = 0.5, which is

representative of most other values for Vc/V0. For comparison, the result for the caseof self-diffusion with just one vacancy also is shown in Fig. 3. As can be seen, inclusionof more than one vacancy in the model has a minimal effect on the boundaries forobserving a vacancy signal.

Although not the main topic of the present study, it is of interest to comparespectra of the two models: (1) model S1V

1 for self-diffusion with just one vacancy inthe first neighbor shell and (2) model S3V

1 for self-diffusion with up to three vacanciesin the first neighbor shell. The jump rate of the probe, and by extension the degreeof relaxation, is proportional to the number of vacancies NV within the first neighborshell of the probe: NV = ∑Z

i=1 iPi, where Pi is given by (2). When summing over thefull range of i, NV/Z is equal to the vacancy concentration input into the model, [V];however, when the summation is truncated (e.g., at i = 1 for model S1V

1 and i = 3 formodel S3V

1 ),[V

]eff ≡ NV

/Z is less than the input value [V]. Therefore, to compare

models S1V1 and S3V

1 , different values for [V] were input to achieve the same valuesof

[V

]eff.

J.R. Castle et al.

-4-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

log[w]

log[

V]

135103050100200205

' 2

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

log[w]

χν

-3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

Fig. 3 Contour plots of the scaled reduced chi-squares obtained fitting model S3V1 with the XYZ

model (left) and model S1V1 with the XYZ model (right). In both cases, Vc/V0 = 0.5

0.0-0.2

0.2

0.6

1.0[V]

eff = 10

-3

[V]eff

= 10-2

[V]eff

= 10-1.5

[V]eff

= 10-1

-0.2

0.2

0.6

1.0

[V]eff

= 10-3

[V]eff

= 10-2

[V]eff

= 10-1.5

[V]eff

= 10-1

t Q Qω. t ω.0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig. 4 Selected spectra simulated using model S3V1 (solid line) and model S1V

1 (dashed line) atindicated values of [V]eff. In all cases, Vc/V0 = 0.5. On the left, w[V]eff = 0.01 ωQ and on the rightw[V]eff = 100 ωQ

Figure 4 shows sequences of spectra for Vc/V0 = 0.5 at four effective vacancyconcentrations [V]eff and two different probe jump rates, w

[V

]eff = 0.01ωQ for the

slow fluctuation regime and w[V

]eff = 100 ωQ for the rapid fluctuation regime. As

can be seen, these agree with the results shown in the contour plots of Fig. 3: spectralook like the XYZ G2(t) as long as

[V

]eff � 0.01 or w

[V

]eff >∼ wQ. For

[V

]eff >

0.01 and w[V

]eff < ωQ, not only do the simulated spectra deviate from the XYZ

G2(t)s, the shapes of spectra from models S1V1 and S3V

1 deviate from one another,

Realistic models of dynamic hyperfine interactions in L12 compounds

-0.2

0.2

0.6

1.0[V] = 10-3

[V] = 10-2.5

[V] = 10-2

[V] = 10-1.5

0.0-0.2

0.2

0.6

1.0

[V] = 10-3

[V] = 10-2.5

[V] = 10-2

[V] = 10-1.5

t Q Qω. t ω.

0.5 1.0 1.5 2.0 2.5 3.00.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig. 5 Selected spectra simulated using the S1V3 model with vacancy contributions to EFG out to the

third neighbor (solid line) and with vacancy contributions from the 2nd and 3rd neighbors suppressed(dashed line) at indicated values of [V]eff. In all cases, Vc/V0 = 0.5. On the left, w[V]eff = 0.01 ωQand on the right, w[V]eff = 100 ωQ

with the deviation increasing with increasing vacancy concentration. This is becausethe fraction of probes with 2 and 3 vacancies in the S3V

1 model grows with vacancyconcentration. For

[V

]eff > 0.01 and w

[V

]eff � ωQ, spectra can be described by the

XYZ model; however, the S1V1 and S3V

1 models exhibit substantially different degreesof relaxation. Therefore, if the S1V

1 model is used in this region, one must apply acorrection to obtain a degree of relaxation, and thus the probe jump rate, that is inagreement with the S3V

1 model.To summarize, the conditions for observing a clear signature of a defect signal are

the same in both models (S1V1 and S3V

1 ), but when analyzing jump rates by degreeof relaxation, the S1V

1 model and, by extension, the I1V1 model are insufficient for

Q � 0.01, as the additional vacancies increase relaxation.

4.2 Effect of vacancies beyond the first neighbor shell

There has been limited investigation as to whether or not defects beyond thefirst neighbor of a probe affect relaxation. Abromeit and Gabriel [15] performedsimulations for EFG fluctuations induced by vacancy motion in the first and secondneighbor shells of a body centered lattice and found that contributions from thesecond neighbor shell were significant, at least for large vacancy concentrations.Therefore, it is prudent to investigate the effect of vacancies in more distant shells inthe Cu3Au system.

Figure 5 shows simulated spectra using the S1V3 model, which considers vacancies

out to the 3rd neighbor shell. Because only one vacancy is permitted within the

J.R. Castle et al.

first three neighbors, the effective vacancy concentration[V

]eff is less than the

input vacancy concentration [V]. The vacancy concentrations indicated in Fig. 5correspond to

[V

]eff.

The solid curves in Fig. 5 take into account EFGs induced by vacancies inthe second and third neighbor shells assuming that the strengths of the EFGsdecrease as expected for an unscreened point charge. For the Cu3Au structure,strengths of EFGs induced by vacancies in the 1st, 2nd and 3rd neighbor shellsare Vc (1n) = 4

√2qVa−3 ≡ Vc, Vc (2n) = 2qVa−3, and Vc (3n) = 3

√6qVa−3

/4 where

qV is the effective charge of the vacancy and a is the lattice parameter. Therefore,Vc (2n) = Vc

√2/

4 and Vc (3n) = Vc3√

3/

16.Dashed curves in Fig. 5 were generated using the S1V

3 model with Vc(2n) =Vc(3n) = 0; that is, with the assumption that contributions of vacancies beyondthe first neighbor shell to the EFG are negligible. As can be seen, the agreementbetween solid and dashed curves is good except for the region of parameters forwhich a vacancy signal is visible: for

[V

]eff � 10−2 and w

[V

]eff in the slow fluctuation

regime. Thus, including contributions beyond the first neighbor shell does not affectthe range over which a vacancy signal is visible, and inclusion of second and thirdneighbor contributions does not affect spectra in the rapid fluctuation regime.

4.3 Combined effects from multiple vacancies and neighbor shells

Based on the above results, it appears likely that the I1V1 model is adequate for

describing relaxation as long as Q < 0.01. In fact, because the vacancy signal isnot visible for Q < 0.01, the XYZ model is sufficient. However, for Q > 0.01and especially when w < ωQ (self-diffusion) and by extension w3 < ωQ (impuritydiffusion), the I1V

1 model does not adequately reproduce the G2(t) that results frommultiple vacancies jumping near PAC probes.

Unfortunately, it is not trivial to simulate spectra under the condition of multiplevacancies beyond the first neighbor shell. For example, one requires a model with2475 EFGs to simulate up to three vacancies within the first three neighbor shells.As another example, a model with two vacancies out to the third neighbor shellrequires only 408 EFGs, which is still beyond what is practicable computationallyusing a standard desktop workstation. Only 108 EFGs are required for a model withtwo vacancies out to the second neighbor shell; however, it is not clear if such amodel incorporates enough vacancies to a sufficient distance to adequately modelthe full effect. Given these difficulties, further investigation of the region Q > 0.01and w3 < ωQ will be postponed until experimental data from that region has beenobtained and there is therefore a clear need to accurately characterize the shape ofG2(t).

5 Conclusion

Computer simulations of perturbed angular correlation spectra based on the sto-chastic modeling method of Winkler and Gerdau were carried out to determine theexperimental conditions under which one can observe direct evidence for vacancydefects participating in the diffusion process in Cu3Au-structured compounds. Inputparameters were the strengths of EFGs (denoted V0 for the defect-free EFG and

Realistic models of dynamic hyperfine interactions in L12 compounds

Vc for the additional contribution from a vacancy in a first neighbor position), jumprates w1, w2, w3, and w4 of the five frequency model for impurity diffusion, and defectconcentration [V].

It was found that a detectable contribution to the PAC signal that originates froma vacancy only will be visible when |Vc/V0| > 0.1, i.e. when the disturbance to theEFG caused by the vacancy is large enough, when [V] (1 − [V])−1 w4

/w3 � 0.01, i.e.

for large vacancy concentration [V] or defect association w4/w3, and when w3/ωQ �

1, i.e. for small trapping and detrapping rates. In such a case, it is necessary to usea stochastic model that considers more than one vacancy and more than the firstneighbor shell in order to calculate an accurate perturbation function. Otherwise,the XYZ model can be used to calculate accurate perturbation functions in orderto fit spectra and determine EFG reorientation rates and, in turn, PAC probe jumprates.

Attention can now be directed toward the identification of chemical systemswith the Cu3Au structure in which PAC probes occupy the Cu sublattice, vacancyconcentrations are large, and probe-vacancy association is large. These conditionscan be predicted through calculation of site occupation, defect formation, anddefect association energies through computer simulation using, for example, densityfunctional theory.

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