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Chemical Physics 327 (2006) 291–299

The nuclear quadrupole interaction at inequivalent lattice sitesin ammonium paramolybdate: A TDPAC study

S.K. Das b, F. Heinrich c, T. Butz a,*

a Universitat Leipzig, Fakultat fur Physik and Geowissenschaften, Linnestrasse 5, 04103 Leipzig, Germanyb On leave from Radiochemistry Laboratory, Variable Energy Cyclotron Centre (Bhabha Atomic Research Centre), 1/AF Bidhan Nagar,

Kolkata 700064, Indiac NIST Centre for Neutron Research, 100 Bureau Drive, Gaithersburg, MD 20899-8562, USA

Received 26 April 2006; accepted 3 May 2006Available online 15 June 2006

Abstract

A nuclear quadrupole interaction (NQI) study using the time differential perturbed angular correlation (TDPAC) technique onammonium paramolybdate (APM) has shown three inequivalent molybdenum sites in this compound which consists of seven MoO6

polyhedra connected through edges. In this study the nuclear probe 99Mo was used to measure the c–c perturbed angular correlationof 99Tc on Mo-sites to obtain the quadrupole interaction parameters. The quadrupole interaction frequencies (xQ) for the three sitesare 0.0224, 0.0386 and 0.0434 rad/ns and the asymmetry parameters (g) of the electric field gradient (EFG) are 0.45, 0.18, and 0.58,respectively. The site assignment is based on the population ratios 4:2:1. The Mo atoms with the highest population show the lowestxQ indicating that this set of polyhedra is ‘‘least’’ distorted or condensed. Besides the least squares fit, a cross-correlation algorithmhas been used to analyze the experimental data to corroborate the fitted parameters and quoted errors. The derived NQI-parameterscan be used for site assignments in other compounds built from condensed Mo–O octahedra.� 2006 Elsevier B.V. All rights reserved.

Keywords: Nuclear quadrupole interaction; Ammonium paramolybdate; TDPAC-study

1. Introduction

Elements in group 6 of the periodic table in their higheroxidation state behave more like nonmetals and form oxo-anions [1]. Mo and W oxo-anions, in particular, polymerizeto form iso- and hetero-polyanions which have MO6 octahe-dra connected through edges or corners. These polyanionshave controversial structures. The structures of these polya-nions have been studied by several techniques [2]. In partic-ular, the X-ray technique has been used to study the alkalisalts of these polyanions. Modern NMR techniques havealso been used to study these molecules in solution. Ammo-

0301-0104/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2006.05.024

* Corresponding author. Tel.: +49 341 9732 701; fax: +49 341 9732 748.E-mail address: butz@physik.uni-leipzig.de (T. Butz).

nium paramolybdate, (NH4)6Mo7O24,4H2O (APM), is apolyanion which can be obtained by dissolving MoO3 inaqueous ammonia and drying the solution. This is an impor-tant starting material for preparing different catalystsrequired in many applications including hydrodesulfurisa-tion in petroleum industry [3].

X-ray diffraction studies on single crystals [4,5] revealedthat APM consists of MoO6 octahedral units connectedthrough edges sharing oxygen atoms. The seven Mo atomsare all inequivalent but can roughly be grouped into threeclasses with population ratios of 4:2:1. The unique site isthe ‘‘buried’’ octahedron (see Fig. 1). In this figure, wesee that the 7th polyhedron has no terminal oxygen atomsbut all are shared and thus it is the ‘‘buried’’ octahedron.The other two sets (1, 2,3,4 form one set and 5,6 formanother set) of polyhedra have terminal oxygen atomsand other oxygen atoms that are shared to different extents.

Fig. 1. Structural model of APM.

292 S.K. Das et al. / Chemical Physics 327 (2006) 291–299

The motivation for the present work is twofold:

(i) from an applied nuclear physics point of view itwould be desirable to determine the quadrupolemoment Q of the I = 5/2 excited state at 181.1 keVfor future applications. A natural way to proceedwould be to perform an ab initio calculation, e.g.using the Wien-code and supercells, of the electricfield gradient at Tc on Mo-sites in MoS2 which hasa simple structure and accurate NQI-data are avail-able [6]. The disadvantage of this approach is thatthe calculation will yield a single number (the electricfield gradient is axially symmetric) and there is nointernal control for validation. A better approach isbased on a Mo-compound which exhibits inequiva-lent lattice sites and lower symmetry such as APM.In addition, the preparation of APM is straightfor-ward. It is a molecular compound consisting of hep-tamers and counterions, ideally suited for codes likethe Amsterdam density functional (ADF) code. Here,the calculation has to reproduce correctly three asym-metry parameters and two NQI-ratios, all indepen-dent of Q. This poses a very stringent test on thevalidity of such a calculation. Hence, the derived Q

should be reliable. In addition, comparing the Wienand the ADF results would be interesting.

(ii) From the solid state chemistry point of view, it wouldbe interesting to see whether there is a sort of finger-print for the various types of distorted oxygen poly-hedra around Mo which exist in nature. Corner andedge sharing octahedra occur, e.g. in MoO3 orAPM with a formal Mo valence of 6+, whereas e.g.c-Mo4O11 with a lower formal valence is built fromoctahedra and tetrahedra sharing corners only.Finally, MoO2 with a formal valence of 4+ consistsof condensed trigonal prisms.

In the present work, we describe a precision time differ-ential perturbed angular correlation (TDPAC) study ofAPM using 99Mo(b�)99Tc as the nuclear probe in order

to determine the nuclear quadrupole interaction (NQI)parameters for these three different classes of Mo-polyhe-dra. Despite the poor frequency resolution of the 99Moprobe it turned out that these three classes of Mo atomscan be identified and we argue that they can possibly beused for site assignments for other polymolybdates.

2. Experimental

2.1. Sample preparation

99Mo labeled APM was prepared through differentchemical routes described below:

(i) Sample 1: inactive APM is dissolved in water andthen 99Mo tracer in 0.002 M NaOH is added to thissolution which is then dried under an infrared lamp.

(ii) Sample 2: as sample 1, with the difference that the99Mo tracer was added in neutral medium.

(iii) Sample 3: MoO3 (Aldrich, >99.5%) was subjected toa thermal neutron flux of 1013 n/cm2/s at the Hahn–Meitner Institute Research Reactor in order to pro-duce 99Mo. APM was then prepared by dissolvingradioactive MoO3 in ammonia solution and drying.

2.2. X-ray analysis

Inactive APM crystals were prepared by the same meth-ods mentioned above without adding the 99Mo tracer. AnXRD study of all these inactive samples was carried outto examine the quality of the samples. The data for samplepreparation 3 gave the best match with the literature data.

2.3. TDPAC setup

The TDPAC setup consists of six detectors in a cubearrangement (TDPAC camera) [6]. A routing fast–slowcoincidence logic is used to record 30 time spectra (6 at180� and 24 at 90�). For details of the data acquisitionand handling see Ref. [7].

All samples were investigated simultaneously with twocameras consisting of six BaF2 and six NaI(Tl) detectors,respectively. Since the BaF2 detectors have a poor energyresolution, the 740(40)-141 cascade of 99Tc was measuredtaking only the lower half of the (141 + 181) keV unre-solved peak. This selects preferentially the 141 keV in thestop channel. On the other hand, since the NaI(Tl) detectorresolves the 141 and 181 keV lines, the 181 keV line waschosen as the stop. Both cascades have their advantagesand drawbacks. The 740(40)-141 cascade has a better coin-cidence rate and larger anisotropy but the admixture of the143 keV line of the daughter 99Tc yields more chance coin-cidence in the background and this is less suited for longerobservation times. On the other hand, the 740–181 keVcascade is not subjected to the same degree of chance coin-cidence background and thus is more valuable at longerobservation times. The perturbation function for both cas-

S.K. Das et al. / Chemical Physics 327 (2006) 291–299 293

cades is the same since the same intermediate level isinvolved. It should be noted that both cascades have oppo-site signs of their anisotropies and thus must not mix.

The observed A2G2 data obtained from the two cameraswere then subtracted with weights according to the statisti-cal accuracy of each data point. The A2G2 spectra were thenleast squares fitted with appropriate theoretical functionsincluding the effect of the finite time resolution. APM insolution was also recorded in regular intervals in order todetermine the t = 0 channels for the 30 subspectra. Initially,we used 0.1 ns/channel to obtain t = 0 data as accurately aspossible but later compressed the data to 0.4 ns/channel.

3. Results

Since the XRD data for the sample 3 resembled very clo-sely that described in the literature [4] more emphasis hasbeen attached to the sample 3.

The 181 keV-level of 99Tc has a nuclear spin of I = 5/2,hence three frequencies corresponding to the energy leveldifferences �hx1 = E±3/2 � E±1/2, �hx2 = E±5/2 � E±3/2 and�hx3 = E±5/2 � E±1/2 = �h(x1 + x2) are expected.

3.1. Least squares fitting

The half-life of the 181 keV state is only 3.5 ns, hence theperturbed angular correlation can be observed for 30–35 nsonly. For each class of inequivalent Mo-atoms the follow-ing function was used:

A2G2ðtÞ ¼X3

n¼0

an e�0:09x2nr

2

cos xnt ð1Þ

Here, an are intensities of the power perturbation functionand xn are the corresponding frequencies which are directlyobservable in the Fourier- or cosine-transformed time spec-tra. The frequencies xn depend on the electric field gradienttensor component Vzz and the asymmetry parameter g =j (Vxx � Vyy)j/Vzz in the following way [8,9]:

xQ ¼eQV zz

�h4Ið2I � 1Þ ð2Þ

In the present case of I = 5/2,

xQ ¼eQV zz

40�hð3Þ

where

x1 ¼ xQ 4f7ð3þ g2Þg12 sin ðu=3Þ

h ið4Þ

and

cos u ¼ 80ð1� g2Þ283ð3þ g2Þ

� �32

ð5Þ

The other two frequencies are related to the first onethrough

x2 ¼ x1

1

2

ffiffiffi3p

cotanðu=3Þ � 1n o

ð6Þ

and

x3 ¼ x1 þ x2 ð7ÞQ denotes the nuclear quadrupole moment of the I = 5/2excited state which is not known.

For more than one site, the total attenuation factor isgiven by the weighted sum of the individuals with fractionsfi:

A2G2ðtÞ ¼X3

i¼1

fiðA2G2ðtÞÞi ð8Þ

The amplitudes an in Eq. (1) depend on g but they are notadjustable. They are theoretically given [8,9]. The exponen-tial factors in Eq. (1) account for the time resolution(r � 0.5 ns and 1.2 ns for the BaF2 and NaI(Tl) setup,respectively). We used a mean value of r � 0.85 for the com-bined spectra. In principle, we should allow for a line broad-ening for the two classes of polyhedra which consist of 4 and2 inequivalent sites, respectively. However, considering thepoor frequency resolution of about 0.150 rad/ns (FWHM),we did not allow for additional line broadening in ordernot to overload the fitting procedure. In fact, none of the sub-spectra, as described below, required any additional freedomfor line broadening within error limits. As free parameters,we fitted xi

1, gi and the corresponding fraction fi, i = 1,2,3.Undoubtedly, we required extremely good statistics to doso. Compared to the spectrum shown in Ref. [10], we col-lected about 50 times more coincidences. Thus, the error barsfor the first few data points in the present spectrum are about4 · 10�4, too small to be visible in the following figures. Thefit started at t = 0.4 ns and stopped at 35 ns.

Since the spectral components for the three sites are notresolved we used the following strategy: First, a fit with asingle site was attempted to get a rough idea for the NQI-parameters of the prominent class, i.e. the one with popula-tion 4. Clearly, this did not give a satisfactory fit, as shownin Fig. 2, top left, a fact which has been noted previously[10], but gave a constraint, nevertheless. The value for v2

was 7.3. We also show in Fig. 2, top right, the residual,i.e. the difference between the experimental spectrum andthe theoretical perturbation function. We also plot theresidual with error bars, as is common practice, to demon-strate the significance of the deviations. It is interesting tonote that the theory is even unable to correctly reproducethe initial part of the spectrum, say the first 10 ns,let alone the later part between 20 and 30 ns, where it isout of phase with the data. The usual attempt to fit in addi-tion frequency distributions (in the present case assumedLorentzian) helps in reproducing the initial part but doesnot ameliorate the problems at later times (see Fig. 2, bot-tom). The value for v2 was now reduced to 3.4. This demon-strates the necessity to observe the spin precession for timesas long as possible, about 10 half lives in the present case.

Secondly, an additional site was included with adjust-able parameters x2

1, g2, f2 while keeping the previouslydetermined values x1

1, g1, f1 fixed. The initial guesses forthese parameters are obtained as follows: from the

A2G

2(t)

-0.04

0.00

0.04

0.08

-0.015

0.000

0.015

Time(ns)

0 10 20 30

-0.04

0.00

0.04

0.08

Time(ns)

0 10 20 30

-0.015

0.000

0.015

(a)

(b)

Residual

Residual

Fig. 2. TDPAC spectrum fitted with one component. Residual is shown on the right: (a) is the fit with a line broadening parameter d = 0; (b) with d as freeparameter.

294 S.K. Das et al. / Chemical Physics 327 (2006) 291–299

cosine-transformed spectrum Fig. 5, top right, it is evidentthat the intensities of the two prominent resolved peaks arein conflict with theoretical intensities; the sum peak is notvisible with adequate intensity (cf. Fig. 5 second fromtop, right), hence a second component is required whichhas a low value for g but roughly the same line positions(cf. Fig. 5 third from top, right). Then, x1

1, g1, f1 and x21,

g2, f2 were allowed to adjust freely simultaneously. Thisalready gave a reasonably good fit with minor discrepan-cies only as shown in Fig. 3, top. The value for v2 was fur-ther reduced to 2.0. Similarly, allowing in addition for twofreely adjustable frequency distribution parameters helpssomewhat in reducing v2 to 1.7. However, there are still dis-crepancies around 15 ns and 30 ns (see Fig. 3 bottom).

Then, x11, g1, f1 and x2

1, g2, f2 were fixed and a third sitewith x3

1, g3, f3 was fitted. The initial guess of these param-eters was based on the residual shown in Fig. 3. Finally, allparameters were allowed to adjust freely simultaneously.Essentially, this corresponds to a successive approxima-tion. The final result is shown in Fig. 4 with a v2 of 1.3,a significant improvement compared to a two-componentfit. Allowing again for three additional freely adjustabledistribution parameters reduces v2 to 1.2. The fitted valuesfor these distribution parameters were all compatible with

zero within error limits. However, these error limits becameexcessively large, certainly a consequence of overloadingthe fit. It is actually against the spirit of untangling discretesites and still using distribution parameters; however, wewanted to see whether there is a need for allowing for fur-ther heterogeneities within classes 1 and 2. In order to besure to have found the global minimum, the experimentalspectrum was decomposed into three subspectra whichare displayed for sample 3 in Fig. 5, left. Again, we ploterror bars for the subspectra which could be interpretedas residuals derived from the experimental spectrum withtwo components each being subtracted. We also plot theircosine transforms in Fig. 5, right. Here, we used a Kaiser–Bessel window with parameter 4. These cosine transformsare the basis for the subsequent cross-correlation analysis.

Table 1 lists all hyperfine parameters for samples 1, 2and 3. The values for the x’s and g’s are essentially thesame for all three samples. For sample 1 and sample 2,the fractions f1 deviate slightly from the expected4/7 = 0.571 whereas they are in good agreement for sample3. Since the quadrupole moment (Q) of the 181 keV state of99Tc is not known, we could not quote the EFG values inthe table. Also in the present measurement it is not possibleto obtain the sign of the EFG.

A2G

2(t) -0.04

0.00

0.04

0.08

-0.015

0.000

0.015

0 10 20 30

-0.04

0.00

0.04

0.08

Time(ns)

0 10 20 30

-0.015

0.000

0.015

(a)

(b)

Residual

Residual

Fig. 3. TDPAC spectrum fitted with two components. Residual is shown on the right: (a) is the fit with d = 0; (b) with d as free parameter.

S.K. Das et al. / Chemical Physics 327 (2006) 291–299 295

The quoted error limits appear rather small and couldpossibly be a result of rather elongated ellipsoids of confi-dence, i.e. a cut through the osculating paraboloid of thev2-surface, in parameter space oriented around 45�. Even-tually, such correlations, e.g. between xQ and g, are notproperly taken into account in the error calculation of leastsquares fitting routines.

Therefore, as a sort of control, we performed in additiona cross-correlation analysis – which is closely related to thev2-surface – of the subspectra which will be described.

3.2. Cross-correlation

This is a calculational approach [11] to improve the sig-nal to noise ratio in the hyperfine spectra. This is done bycross-correlating the experimental cosine-transform spectrawith the theoretical ones for a wide parameter space of xQ

and g. The algorithm essentially piles up the three spectrallines in a single spot in a two-dimensional parameter-plotcalled Czjzek-plot [12]. Fig. 6 shows the results for the threesubspectra. The position of the admittedly rather broadspots allows to read off xQ and g directly. The lines startingfrom the origin (the origin of the abscissa is suppressed fortechnical reasons) are g = constant lines, those nearly per-

pendicular to them are Vzz = const. or xQ = const. lines. Inthese spectra a cutoff at 50% of the maxima was chosen. Inorder to read off g, one has to go to the nearest g = con-stant line and g can be read off at the right hand scale; inorder to read off xQ, follow the closest xQ = constant lineupwards to the g = 0 boundary, then go horizontally to theleft hand scale and read off xQ.

All three spots in the parameter landscape are welldefined with negligible satellites. The absence of relevantside maxima demonstrates that the decomposition intothree subspectra was properly done.

The maximum for class 2 is an elongated hill, i.e. there isa certain correlation between the abscissa and the ordinatewhich in turn transforms into a certain correlation betweenxQ and g. However, in the present case this is a conse-quence of the fact that the eigenvalues of the quadrupoleHamiltonian for half-integer spins depend on g quadrati-cally in the vicinity of g = 0.

4. Discussion

Contrary to the previous data [10] the present data havesufficient statistical accuracy to identify three inequivalentclasses or sites. The amplitude ratios for sample 3 are

A2G

2(t) -0.04

0.00

0.04

0.08

-0.015

0.000

0.015

0 10 20 30

-0.04

0.00

0.04

0.08

0 10 20 30

-0.015

0.000

0.015

Time(ns)Time(ns)

(a)

(b)

Residual

Residual

Fig. 4. TDPAC spectrum fitted with three components. Residual is shown on the right: (a) is the fit with d = 0; (b) with d as free parameter.

296 S.K. Das et al. / Chemical Physics 327 (2006) 291–299

0.58:0.26:0.16, i.e. close to the expected 0.57:0.29:0.14within error limits. This is essentially true also for the othersamples. This is also in agreement with the structurededuced from the X-ray analysis [4]. For the first two sam-ples, the present TDPAC data show that the interactionfrequencies are about the same but their amplitudes deviatesomewhat from the expected ratios. The reason may beattributed to the fact that in the first two samples, the start-ing material (inactive APM) might have not been depoly-merized completely upon dissolution and the exchange ofthe tracer 99Mo with the inactive Mo sites is incomplete.We also noticed that upon redissolution, a small undis-solved white fraction remained which certainly is notAPM but another Mo–O species. Its xQ apparently inter-feres with that for the prominent site in APM. In the caseof the third sample, the APM is recrystallized from themolybdate solution consisting of monomers of inactiveMoO2�

4 and 99MoO2�4 and, hence, no labeling preferences

can occur.A comparison of the present result with other Mo-oxi-

des viz. MoO3, c-Mo4O11, and MoO2 can be made in termsof a distortion parameter. A distortion parameter d of anoctahedron or a tetrahedron can be defined as the distanceof the NQI spot in the Czjzek-plot from the origin which

corresponds to a perfect octahedron or tetrahedron,respectively, with zero NQI:

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p¼ 2xQ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2

3

rð9Þ

where x ¼ xQðffiffiffi3pþ g=

ffiffiffi3pÞ and y = xQ(1 � g) are the ab-

scissa and ordinate of the Czjzek-plot, respectively. Thisdistortion parameter compresses all available informationinto a single number and is of particular use in cases wherea simultaneous determination of xQ and g is impossible, aswas the case for c-Mo4O11 [13]. In this case, d turns out tobe a robust quantity.

The distortion parameter d for c-Mo4O11 [13] is about0.019 rad/ns, a rather low value which is certainly a resultof polyhedra sharing corners only. In this situation, thepolyhedra are weakly distorted, as is the case for frozenmolbydate solutions at elevated pH, where isolated tetrahe-dral MoO2�

4 -species prevail [14]. The distortion parameterfor MoO2 is about 0.024 rad/ns [10]. We include MoO2

in our discussion although the formal valence of Mo is4+ and it consists of layers of Mo surrounded by oxygenatoms in a trigonal prismatic configuration, i.e. we haveneither tetrahedra nor octahedra (2H-modification). Thepolymorph with octahedral coordination (1T-modification)

0.00

0.04

Inte

nsity

(arb

itrar

y un

it)

0

1

2

0.00

0.04

0

1

2

A2G

2(t)

0.00

0.04

0

1

2

Time(ns)

0 10 20 30

0.00

0.04

Frequency(rad/ns)0 1 2

0

1

2

(a)

(b)

(c)

(d)

Fig. 5. Left: TDPAC spectra of APM: (a) total A2G2, (b), (c) and (d) are the individual components for the three sites. Right: corresponding cosinetransforms (circles: experimental; solid line: fitted function).

Table 1Fitted parameters for the three Mo sites in APM for the three samples

Group xQ (rad/ns) g Fraction

Sample 1

1 0.0222(5) 0.45(2) 0.61(6)2 0.0393(13) 0.15(5) 0.23(5)3 0.0400(12) 0.65(5) 0.16(4)

Sample 2

1 0.0213(6) 0.35(3) 0.63(4)2 0.0393(11) 0.14(5) 0.21(4)3 0.0443(12) 0.70(5) 0.16(3)

Sample 3

1 0.0224(5) 0.45(3) 0.58(7)2 0.0386(11) 0.18(9) 0.26(6)3 0.0434(14) 0.58(6) 0.16(6)

S.K. Das et al. / Chemical Physics 327 (2006) 291–299 297

does not exist for MoO2, contrary to the dichalcogenides ofTa. Here, the NQI – and consequently the distortionparameters – for the 1T- and 2H-modifications for TaS2

and TaSe2 are rather similar [15]. Hence, we would expecta similar distortion parameter for 1T-MoO2, if it existed. Itis interesting to note that the distortion parameter forMoO3 and the site I in APM are rather similar:0.051 rad/ns and 0.047 rad/ns, respectively. This meansthat the 4 octahedra of APM denoted by 1–4 in Fig. 1and those in MoO3 with edge and corner sharing are sim-ilar, unless the distortion parameters are so close bychance. Finally, the sites II and III in APM appear to besimilar as far as the distortion parameter is concerned:d = 0.079 rad/ns and 0.091 rad/ns, respectively. Clearly, itremains to be seen whether a thorough study of furtherMo-oxides will yield a continuous distribution of distortionparameters or whether, as we speculate, separate classesexist indeed.

Calculations of the electric field gradient using theAmsterdam density functional code are under way. Thedifficulty in these calculations arises due to the presenceof the Tc impurity on a Mo site plus the associated lattice

Fig. 6. Czjzek-plot for the cross-correlation analysis of the three subspectra of Fig. 4. They are normalized individually and a cutoff at 50% of themaximum was chosen. Maximum: violet; threshold: dark blue.

298 S.K. Das et al. / Chemical Physics 327 (2006) 291–299

relaxation. Moreover, the hydrogen coordinates of theNHþ4 ion and H2O are unknown. Such calculations wouldshow whether the NQI’s within the classes consisting of 4and 2 inequivalent sites, respectively, are close enough suchas to justify the neglect of any additional line broadening inthe fitting procedure.

5. Conclusion

The quadrupole interaction study clearly indicates thatthe seven MoO6 polyhedra in the APM molecule consist

of three classes of Mo atoms. This conclusion has beenobtained from both a least squares fitting and a cross-cor-relation analysis of the experimental data. Since the MoO6

polyhedra can be classified by their quadrupole interactionparameters, they can possibly be used for identificationpurposes for other polyhedra of similar Mo–O compounds.

Acknowledgements

The authors sincerely thank Dr. H.C. Semmelhack ofthe Division of Superconductivity and Magnetism, Insti-

S.K. Das et al. / Chemical Physics 327 (2006) 291–299 299

tute for Experimental Physics II, University of Leipzig, forcarrying out the X-ray analysis of the APM samples.

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