THE CENTER SHIFT IN MOSSBAUER SPECTRA OF MAGHEMITE AND ALUMINUM MAGHEMITES

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Clays and Clay Minerals, Vl. 42, N. 5, 628--633, 1994. THE CENTER SHIFT IN MOSSBAUER SPECTRA OF MAGHEMITE AND ALUMINUM MAGHEMITES G. M. DA COSTA, L2 E. DE GRAVE, ~ L. H. BOWEN, 3 R. E. VANDENBERGHE, 1 AND P.

1 Clays and Clay Minerals, Vl. 42, N. 5, , THE CENTER SHIFT IN MOSSBAUER SPECTRA OF MAGHEMITE AND ALUMINUM MAGHEMITES G. M. DA COSTA, L2 E. DE GRAVE, ~ L. H. BOWEN, 3 R. E. VANDENBERGHE, 1 AND P. M. A. DE BAKKER 1 Labrary f Magneism, Deparmen f Subamic and Radiain Physics Universiy f Gen, B-9000 Gen, Belgium 2 On leave frm Deparamen de Quimica, Universidade Federal de Our Pre Our Pre, MG, Brazil 3 Deparmen f Chemisry, Nrh Carlina Sae Universiy Raleigh, Nrh Carlina 27695, USA Absrac--Synheic, relaively well-crysallized aluminum-subsiued maghemie samples, y-(aly. Fe~_y)203, wih y = 0, 0.032, 0.058, 0.084, and have been sudied by X-ray diffracin and zer-field Mrssbauer specrscpy in he range 8 K 475 K, and als wih an exernal field f 60 koe a 4.2 K and 275 K. I was fund ha here are w differen cnverging mdels fr fiing he zer-field specra f he maghemies wih a superpsiin f w Lrenzian-shaped sexes, bh resuling in incnsisen values fr he hyperfine fields (Hhf) and/r he cener shifs (6) f he erahedral (A) and cahedral (B) ferric ins. Frm he applied-field measuremens i is cncluded ha here is a cnsan difference f 0.12 _ mm/s beween 6B and 6A, regardless f he A1 cnen. Fr he Al-free sample he cener shifs are fund as: 64 = mm/s and 6a = mm/s a 4.2 K and 6n = mm/s and fib = mm/s a 275 K (relaive meallic irn), wih an esimaed errr f mm/s. Bh 6A and fib are bserved decrease wih increasing AI cncenrain. The effecive hyperfine fields fr he nn-subsiued maghemie sample are: Hefr.A = 575 koe and Hfr.B = 471 koe a 4.2 K and He~,A = 562 koe and H~er.B = 449 koe a 275 K, wih an errr f 1 koe. The B-sie hyperfine field remains apprximaely cnsan wih A1 subsiuin, while fr he A sie a sligh decrease wih increasing A1 cnen was bserved. Key Wrds--A1-Maghemie, Cener shif, Maghemie, Mrssbauer effec. INTRODUCTION Maghemie is a cain deficien spinel wih 21 and 1/3 Fe(III) ins disribued amng eigh erahedral (A) and sixeen cahedral (B) sies f he uni cell; 2 and 2/3 sies per uni cell are vacan. I is cmmnly represened by he frmula (Fe)[Fe513[]v~]O4, where ( ) denes erahedral sies, [ ] cahedral sies and [] vacancies. Hwever, sme auhrs als repr he vacancies ener bh cahedral and erahedral sies simulaneusly (Armsrng e al., 1966; Annersen and Hafner, 1973; Haneda and Mrrish, 1977). Besides is echnlgical imprance as a maerial fr recrding devices, maghemie is f cnsiderable gephysical ineres since i ccurs in many rpical magneic sils (Schwermann and Fecher, 1984; Fnes and Weed, 1991) and is believed be a significan cnsiuen f he Marian surface (Cey e al, 1990). The magneically spli Mrssbauer specrum f maghemie is cmpsed f w sexes, hardly disinguishable a rm emperaure (RT) due he fac ha he hyperfine parameers fr he w sies are very similar in magniude. Cmmnly ccurring effecs such as small-paricle size disribuins and parial subsiuin f he irn am by anher cain end cause line bradening and/r a line-shape asymmery, and in many cases he RT specrum has be fi wih a disribuin f unreslved A- and B-sie hyperfine fields Cpyrigh , The Clay Minerals Sciey 628 (de Bakker e al, 1991). On lwering he emperaure i is pssible, depending n he crysalliniy, bain a smewha beer resluin f he w sies, bu he nly way have hem sufficienly separaed is by applying srng exernal magneic fields (Armsrng e al., 1966). In surveying he lieraure fr maghemie i was niced ha an incnceivably wide scaer fr he qued cener shifs (6) exiss, wih values a RT ranging frm 0.04 mm/s up 0.30 mm/s fr he A-sie (ha) and frm 0.04 mm/s up 0.40 mm/s fr B-sie (rb), all relaive meallic irn. A 4.2 K values as high as 0.48 mm/s fr ~A and 0.62 fr bb have been repred (Pllard, 1988; Pllard and Mrrish, 1987; Cllyer e al., 1988; Mehner e al., 1990; de Bakker e al., 1991; Nikumbh e al, 1992). In an aemp reslve his discrepancy, variable-emperaure Mrssbauer specra (MS) beween 8 K and 475 K, and in addiin a 4.2 K and 275 K wih he absrber subjeced an exernal field f 60 koe, were clleced fr a number f wellcrysallized maghemie and Al-maghemie samples. The specra have been inerpreed using differen appraches. EXPERIMENTAL PROCEDURES Al-maghemie samples were prepared by Bryan (1993) as fllws: apprximaely 1 g f synheic A1-

$X-ray diffracin paern f samples M0 and M6. Sme supersrucure lines due he rdered arrangemen f vacancies n he eahedral sies f he spinel laice are indicaed by arrws. Peaks a 44.8 ~ and 52.$

2 Vl. 42, N. 5, 1994 Maghemie cener shif 629 l m g c c c c-4 I ~ c) x O c~ 0d (N l I 0.5 AI 0.0 ko 0.5 I L ~ / I I AI 0.0 l 2b 3b 4'0 5b ab 7b " Figure 1. X-ray diffracin paern f samples M0 and M6. Sme supersrucure lines due he rdered arrangemen f vacancies n he eahedral sies f he spinel laice are indicaed by arrws. Peaks a 44.8 ~ and 52.3 ~ (20) are due he sample hlder. hemaie was added an aqueus sluin f sucrse in such a way ha a weigh prprin f 1:4 (hemaie/ sucrse) was bained. Maximum mixing was bained by dispersin using a Snifer, Cell Disruper W-350 (Bransn Snic Pwer C.), fr apprximaely ne minue. Aferwards he mixures were immediaely quick frzen, fllwed by freeze drying. The dried mixures were hen heaed a 450~ fr 30 minues, fllwed by cling, grinding, milling and reheaing a 250~ The samples are hencefrward named as MX (M = 0, 3, 6, 8, 11 and 15; X = 100y), where y sands fr he aluminum cnen in "y-(alyfe~ y)203. Pwder X-ray diffracin paerns (XRD) were bained using a Philips diffracmeer wih CKa radiain and a graphie mnchrmar. The scans were dne in he range f ~ (20) a a speed f V4 ~ rain- ~, and he refleced inensiies were recrded in 1024 channels f a mulichannel analyser. The numerical paerns were fied wih a sum f pseud-lrenzian peak shape funcins, w fr each reflecin in rder accun fr he K~ and Ka2 radiain. Mean crysallie diameers (MCD) were esimaed frm he full widhs a half maximum (FWHM) using Scherrer's equain wih K = 0.9. The insrumenal cnribuin he peak widh was evaluaed frm he diffracgram AI M0 M6 f a well-crysallized cmmercial hemaie sample, annealed a 1000~ in xygen amsphere fr 24 hurs. Laice parameers were deermined by he Nelsn- Riley exraplain mehd (Klug and Alexander, 1974). M6ssbauer specra f all synheic samples were clleced a a emperaure f 8.0 K and beween K a seps f 25 K. A ime-mde specrmeer wih a cnsan-accelerain drive and a riangular reference signal was used. The emperaure sabiliy f he absrber in he flw-crysa/furnace cmbinain was beer han 0.5 K. All absrbers were prepared by mixing an amun f maerial wih very pure carbn in rder achieve a hmgeneus hickness f apprximaely 10 mg Fe/cmL The specrmeer was peridically calibraed using he specrum f a well-crysallized hemaie, prepared afer heaing a cmmercial hemaie sample (Merck) a 1000~ in xygen amsphere fr 24 hurs. Exernal field (HexO f 60 koe (Oersed = cm -~/:-g~/2. s-~) parallel he direcin f he inciden 3,-rays was applied a 4.2 K and 275 K. Fr hese measuremens he calibrain specra were recrded simulaneusly using anher surce and cuning elecrnics a he ppsie end f he ransducer. Cener shifs are qued relaive meallic irn. Deails abu he fiing prcedures are given belw. RESULTS AND DISCUSSION Figure 1 shws he XRD paerns fr synheic samples M0 and M6. All bserved peaks can be assigned he spinel phase. Sme supersrucure lines are clearly visible, indicaing an rdered arrangemen f he vacancies n he B sies (Haneda and Mrrish, 1977). The diffracgrams fr he her samples all resemble hse shwn in Figure 1. Mean crysallie diameer values alng he (220) and (311) laice direcins are 44 and 35 nm fr M0, 39 and 41 nm fr M6 and 26 and 23 nm fr M 15 samples. The rdered arrangemen f vacancies seems be relaed he MCD values since fr samples f paricle size less han 20 nm he supersrucure lines are n bserved (Haneda and Mrrish, 1993). The laice parameers are nm fr M0, nm fr M6 and fr M15 sampies, wih an esimaed errr f nm. The value fr he nn-subsiued maghemie sample is smewha larger han expeced due he presence f a small amun f magneie in he sample (see M6ssbauer resuls belw). Mre srucural and hermal deails fr he varius samples can be fund elsewhere (Bryan, 1993; Bwen e al., 1994). In he range f emperaures beween 80 K and 475 K all zer-field MS cnsis f a magneically spli sexe and a weak duble (usually less han 2%). This laer cmpnen was fied as a symmeric duble wih all parameers fixed a: AEQ = 0.60 mm/s; 6 = 0.36 mm/s and I' = 0.35 mm/s, AEQ and r being he quadruple spliing and line widh (FWHM) respecively. These

3 630 Da Csa e al Clays and Clay Minerals values were chsen n he basis f several rial fis a few seleced specra. This duble is n due any impuriy in he cuner r in he crysa windws, bu, cnsidering is lw inensiy and he apparen lack f any effec f is M6ssbauer parameers n hse fr he maghemie cmpnen, n furher aenin was paid is rigin. The magneic cmpnens culd be fied wih a superpsiin f w Lrenzian-shaped, symmerical sexes, bu w differen cnverging mdels appears be applicable, bh leading similar gdness-f-fi (X 2) values. In all cases he fllwing parameer cnsrains were used: quadruple shif EQ = 0 fixed fr bh sies; hree widh parameers, equal fr bh subspecra; line-area rais fixed a 3:2:1 and als fixed B- A-sie area rai f 62.5:37.5, i.e., assuming randm A1 disribuin and vacancies in cahedral sies nly. Allwing he quadruple shifs vary did n imprve he fis significanly. A ypical experimenal specrum (M6 a 8 K) is shwn in Figure 2a, geher wih he fied subspecra and heir superpsiin (full lines). This fi, in which he firs line (i.e., lef-ms absrpin) f he B-sie cmpnen is a higher abslue velciy han he firs line f he A-sie cmpnen, will be called Mdel 1. Sme relevan parameers fr M6, as bained frm his mdel, are given in Table 1 fr a few seleced emperaures. The her samples exhibi very similar resuls, wih he hyperfine fields and cener shifs shwing sligh variains wih he A1 subsiuin. As seen frm Table 1, Mdel 1 yields a nearly cnsan difference AHhf = Hhf, B -- Hhf, A f apprximaely 15 koe, increasing smewha when he emperaure is raised 475 K. In cnras, he difference A6 = 6a -- ~g is 0.10 mm/s a 8 K, 0.04 mm/s a 275 K, and mm/s a 475 K, i.e., a a cerain emperaure in he viciniy f rm emperaure, ~g seems becme larger han 6B. This laer phenmenn is ally unexpeced and hard undersand because he cainxygen bnd-lengh is less fr A sies han fr B sies and hence he cener shif a any emperaure is expeced be larger fr he cahedral ins (Armsrng e al., 1966), as is cmmnly bserved fr spinel ferries (Vandenberghe and De Grave, 1989). Mrever, he apparenly large emperaure variain f 6A cann be explained by any reasnable value fr he characerisic M6ssbauer emperaure deermining he secnd-rder Dppler shif in erms f he Debye mdel fr he laice vibrains (De Grave and Van Albm, 1991). Depending n he iniial values f he adjusable parameers, he fiing ruine was fen fund cnverge a sluin in which he firs line f he B-sie cmpnen is a lwer abslue velciy han he firs line f he A-sie cmpnen, as shwn in Figure 2b. This ype f fi will hereafer be referred as Mdel 2. The as-such bained X 2 is, in many cases, smewha smaller as cmpared he values calculaed frm I00 -~ ' 'B~A ~ "~ 96 c 2 ~_ 9r i' / i 192 [i}i 98 E E 96 ~_ 94 I 92 IO0 99 "~ 98 c ~" 98 2 I 94 i i i i i i i i i r i i i i i i i 12 lo - f Ibl i i I Figure 2. M6ssbauer specra f sme seleced samples: a) M6 a 8 K fied wih Mdel 1; b) M6 a 8 K fied wih Mdel 2; c) M0 a 4.2 K and applied field f 60 koe; d) M6 a 4.2 K and applied field f 60 koe. Crsses represen he experimenal daa, and full lines represen he calculaed subspecra and heir sum. The psiin f he firs lines f A and B sies is indicaed in Figures 2a and 2b. The arrws in Figures 2c and 2d shw he psiin f lines 2 and 5. d

4 Vl. 42, N. 5, 1994 Maghemie cener shif 631 Table 1. Hyperfine fields H~f.B, Hhf, A and heir differences AHhf in koe and cener shifs 6B, 6/, and heir differences A6 in mm/s bained frm fiing he specra f sample M6 wih Mdel 1 and Mdel 2. The esimaed saisical errrs are 2 koe fr Hhf and 0.02 mm/s fr & Temp. (K) H~f.a H~f,A AH~r fia ~ia A~ Mdel Mdel Mdel 1, bu n appreciably s, making i difficul judge which fi mdel is be preferred as far as reprducing he experimenal line shape is cncerned. The parameers fr sample M6 as bained frm Mdel 2 are als lised in Table 1. As can be seen frm Figures 2a and 2b bh mdels describe he experimenal specra in an equally adequae way. Furhermre, each f he w adjused parameer ses are in accrdance wih sme f he lieraure daa, alhugh cnsiderable differences beween he hyperfine parameers are bained by he w mdels. Sme represenaive cener-shif and hyperfine-field daa fr maghemies a rm emperaure and 4.2 K as qued in he lieraure are lised in Table 2. There is clearly a wide scaer in hese values, and alhugh sme auhrs did n specifically shw he calculaed subspecra in heir papers, i is believed ha his scaer is relaed he w pssible mdels f fiing as re- pred in he presen paper. I shuld be sressed a his pin ha cnvergence any f he w mdels seems be ally arbirary in he sense ha he upu resuls are deermined by he inpu parameer values. I was necessary ry several differen inpu-parameer ses in rder arrive a he ieraed daa as repred in Table 1, because n sraighfrward crrelain beween he inpu parameers and he resuling fi culd be esablished. In an aemp recncile he apparen discrepancies cncerning he maghemie hyperfine parameers, exernal-field measuremens (60 koe) a 4.2 K fr all samples and a 275 K (his was he highes emperaure which culd be applied wih he available sysem) fr samples M0 and M6 have been perfrmed. Wih such a srng field i was pssible separae he w sies, allwing he deerminain f he hyperfine parameers wih a relaively high precisin. The specra were fied wih w main asymmeric sexe cmpnens. The asymmery is due he cmbined effecs f randm rienain f he hyperfine field wih respec he elecric-field-gradien's principal axis and a nn-zer diplar cnribuin he hyperfine field (De Grave e al, 1993). Fr all maghemies a 4.2 K a small hemaie fracin (less han 3 mle %) culd be deeced in he exernal-field specra (Bwen e al, 1994). The M0 specra a 275 K, bh wih and wihu he exernal field, shw ha his sample in addiin cnains apprximaely 4 mle % f a magneie impuriy. This magneie cmpnen was included in he fis, is hyper fine parameers being fixed a heir respecive values recenly fund by De Grave e a. (1993) fr pure magneie. Sme represenaive experimenal and calculaed specra are shwn in Figures 2c and 2d, and relevan evaluaed parameers are lised in Table 3. Sme f Table 2. Cener shifs (relaive meallic irn, in mm/s) and hyperfine fields (koe) fr nn-subsiued maghemie as repred in lieraure. Fr he exernal-field resuls, Hhr.A and Hhf.B are he effecive hyperfine fields (see ex). Temp. ~B 6A A6 Hh~A H~f,B Ref. Cmmens RT 0, mdel 2 0, Hex = 16 koe * mdel K H~ = 61 koe H,x = 50 koe C: 3,-Fe I~, = 42 koe Pllard (1988). 2 Mehner e al. (1990). 3 CUyer e a. (1988). 4 Nikumbh e al. (1992). de Bakker e al. (1991). Pllard and Mrrish (987). * The auhrs did n menin if he qued values are reaive he surce r meallic irn.

5 632 Da Csa e al Clays and Clay Minerals Table 3. Hyperfine fields (koe), cener shifs (mm/s) and caning angles 0 (0) f samples M0, M3, M6, M8, M11 and M15 derived frm he 60 koe applied-field measuremens a 4.2 K. Samples M0 and M6 were measured a 275 K as well. The errrs are 1 koe fr H~, 4 koe fr Hhf, and mm/s fr & Sample T (K) HemB Hefr.A 08 0A Hhr.a Hh~.A AHhf ~a ~A Af M II l M M M Mll M hese daa have been repred briefly in an earlier paper (Bwen e al., 1994), bu are repeaed here fr he ease f he discussin. Frm Table 3 i can be seen ha he B-sie effecive hyperfine field, H~e,a, a 4.2 K remains apprximaely cnsan wih A1 subsiuin, while fr he A-sies a sligh decrease wih increasing A1 cnen is fund. As argued by Bwen e al., his variain (r lack f variain in he case f B-sie ferric ins) is cnsisen wih he A1 ins ccupying he B sies exclusively. Bh 6A and 6B seem decrease wih increasing A1 cncenrain. The variains are weak, bu believed be significan. An explanain fr i has n been fund ye. Finally, he slubiliy limi fr he A1 species in he maghemie srucure, as repred earlier by Wlska and Schwermann (1989) be 10 a %, culd n be cnfirmed. Anher impran feaure f he in-field specra fr he maghemie cmpnens is he presence f lines 2 and 5. These absrpins are expeced be absen if he hyperfine fields are cmpleely aligned alng he exernal field. Frm he relaive areas A~.6/A2, 5 ne can deermine hecaning angle 0, defined as he angle beween I2Ihf and H,x (Vandenberghe and De Grave, 1989). On he her hand, he measured effecive hyperfine fields are given in he fllwing, well-knwn equain: H~r = HZf + He2x - 2Hex, HhfCOS(Tr - 0). Thus, knwing he area rais AI.6/A2. s ne can calculae 0 and Hhf. Since he A- and B-sie cmpnens are well separaed in he exernal-field specra, he adjused Herr values are n affeced by he varius resricins impsed he fi prcedure. On he her hand, he adjused values fr he specral areas f lines 2 and 5 were bserved be dependen sme exen n he impsed fiing cndiins, s ha he caning angles 0, and hence he hyperfine fields, cann be deermined wih very high precisin. Fr he presen experimens, an uncerainy f abu 4 koe fr he Hhf values given in Table 3 is esimaed based n he resuls f numerus rial fis. In a recen paper f de Jesus Filh e al. (1993) i was prpsed ha in Al-subsiued maghemies, cmparable he samples used in his sudy, he A-sie mmens are cmpleely aligned wih Hex, while hse frm he B sies are caned by abu 30 ~. Hwever, n deails abu he applied fiing prcedures were indicaed by he auhrs. The presen resuls disagree wih ha cnclusin. Accrding he daa f Table 3, which refer he bes fis fr any f he given samples, he sum 0 A ~- 0 B is clse 180 ~ regardless f he A1 subsiuin, meaning ha wihin experimenal and saisical uncerainies he A- and B-sie magneic mmens are cllinear fr all invlved maghemies. In summary, he resuls lised in Table 3 fr he synheic maghemies clearly shw ha he hyperfine parameers bained frm fiing he zer-field MS wih Mdel 2 are in serius discrepancy wih he mre reliable and accepable exernal-field resuls, in paricular as far as he difference f abu 0.20 mm/s beween he B- and A-sie cener shifs is cncerned. This discrepancy persiss even in he daa referring 8 K. On he her hand Mdel 1 gives accepable resuls a lw emperaures, bu cann be used fi high-emperaure measuremens. Fr bh mdels he disagreemens are mre significan fr he A-sie, prbably due he lwer inensiy f his cmpnen. An impran general cnclusin ha can be drawn frm hese resuls is ha he applicain f an exernal magneic field is absluely essenial if ne wans exrac reliable infrmain frm he M/Sssbauer specra fmaghemies. As exernal-field specrmeers are n ruinely accessible fr he vas majriy f he Mrssbauer labs, an alernaive way derive accepable parameer values frm zer-field specra migh be keep he difference beween 68 and 6A fixed in fiing he specra. This prcedure will be esed sysemaically by fuure wrk in his labrary n her synheic, and als n varius envirnmenal maghemie sysems. The principal quesin in ha respec is wheher he value f 0.12 _ mm/s fr his difference in cener shifs is unique r wheher i changes depending n he mrphlgy and cmpsiin f he samples. A his sage, an impran finding in favur he abve suggesed prcedure shuld be menined. I cncerns he resuls bained fr he maghemie phase

Vl. 42, N. 5, 1994 Maghemie cener shif 633 presen in sme Brazilian sils, in which he difference beween he B- and A-sie cener shifs were als fund be clse 0.12 mm/s.

6 Vl. 42, N. 5, 1994 Maghemie cener shif 633 presen in sme Brazilian sils, in which he difference beween he B- and A-sie cener shifs were als fund be clse 0.12 mm/s. CONCLUSIONS The Mrssbauer parameers exraced frm he exernal-field specra are beynd any dub he ms precise ha can be bained fr synheic and naural maghemies and subsiued-maghemies. The wide scaer exising fr cener-shif and hyperfine-field lieraure daa as bained frm zer-field specra, is cncluded be an arifac f he fi prcedures. Using a 60 koe applied-field i was deermined ha he cener shifs fr he A- and B-sies are differen by a cnsan amun f 0.12 mm/s n he average, a leas fr he range f emperaures beween 4.2 K-275 K. Sligh variains f sme f he hyperfine parameers wih he A1 cnen have been bserved fr he synheic A1 maghemies. ACKNOWLEDGMENTS This wrk was suppred in par by he Fund fr Jin Basic Research (Belgium), by a NATO cllabraive Research Gran 91/0116 and by CNPq (Brazil). The auhrs wish hank A. M. Bryan (Nrh Carlina Sae Universiy) fr he sample preparains. REFERENCES Annersen, H. and Hafner, S.S. (1973) Vacancy disribuin in synheic spinels f he series Fe304-'y-Fe203: Z. Krisallg. 137: Armsrng, J. R., Mrrish, A. H., and Sawazky, G.A. (1966) Mrssbauer sudy f ferric ins in he erahedral and cahedral sies f a spinel: Phys. Le. 23: Bwen, L. H., De Grave, E., and Bryan, A.M. (1994) MiSssbauer sudies in exernal field f well crysallized Al-maghemies made frm hemaie: be published in Hyperfine Ineracins. Bryan, A. M. (1993) The hermal ransfrmain f A1- subsiued hemaie and lepidcrcie maghemie sudied by SVFe Mrssbauer specrscpy: Ph.D. hesis, Nrh Carlina Sae Universiy, USA. Cey, J. M. D., Mrup, S., Madsen, M. B., and Knudsen, J. M. (1990) Tianmaghemie in magneic sils n earh and Mars: J. Gephys. Res. 95: Cllyer, S., Grimes, N. W., Vaugham, D. J., and Lngwrh, G. (1988) Sudies fhe crysal srucure and crysal chemisry f ianmaghemie: Amer. Mineral.73: de Bakker, P. M. A., De Grave, E., Vandenberghe, R. E., Bwen, L. H., Pllard, R. J., and Persns, R. M. (1991) M6ssbauer sudy f he hermal decmpsiin f lepidcrcie and characerisain f he decmpsiin prducs: Phys. Chem. Minerals 18: De Grave, E. and Van Albm, A. (1991) Evaluain f ferrus and ferric Mrssbauer fracins: Phys. Chem. Minerals 18: De Grave, E., de Bakker, P. M. A., Bwen, L. H., and Vandenberghe, R. E. (1992) Effec fcrysalliniy and A1 subsiuin n he applied-field Mrssbauer specra f irn xides and xyhydrxides: Z. Pflanzenerniihr. Bdenk. 155: De Grave, E., Persns, R. M., Vandenberghe, R. E., and de Bakker, P. M. A. (1993) M6ssbauer sudy f he high emperaure phase f C-subsiued magneies, C~Fe3_~. 04. I. x -< 0.04: Phys. Rev. B 47: de Jesus Filh, M. F., da Nva Mussel, W., Qi, Q., and Cey, J. M. D. (1993) Magneic prperies f aluminum-dped yfe203: Prc. 6h In. Cnf. n Ferries, Tky 1992 (in press). Ericssn, T., Krisnhamarhy, A., and Srivasava, B.K. (1986) Mrin ransiin in Ti-subsiued hemaie: A Mrssbauer sudy: Phys. Scripa 33: Fnes, M. P. F. and Weed, S. B. (1991) Irn xides in seleced Brazilian Oxysls: I. Mineralgy: SilSci. Sc. Am. J. 55: Haneda, K. and Mrrish, A. H. (1977) Vacancy rdering in 3'-Fe203 small paricles: SlidSae Cmmun. 22: Haneda, K. and Mrrish, A. H. (1993) Srucural peculiariies in magneic small paricles: Nuclear Insrumens and Mehds in Physics Research B76: Klug, H. P. and Alexander, L. E. (1974) X-ray Diffracin Prcedures fr Plycrysalline and Amrphus Maerials: Jhn Wiley and Sns, New Yrk. Mehner, H., Kppe, H. J., and M6rke, W. (1990) Miissbauer invesigain f y-fe203 paricles: Hyperfine Ineracins 54: Nikumbh, A. H., Aware, A. D., and Sayanekar, P.L. (1992) Elecrical and magneic prperies f "y-fe203 synhesized frm ferrus ararae ne and half hydrae: J. Magn. Maer. 114: Pllard, R.J. (1988) On he Mrssbauer specrum fy-fezo3: Hyperfine Ineracins 41: Pllard, R. J. and Mrrish, A. H. (1987) High-field magneism in nn-plar y-fe203 recrding paricles: IEEE Trans. Magn. MAG(23): Schwermann, U. and Fecher, H. (1984) The influence f aluminum n irn xides: XI. Aluminum-subsiued maghemie in sils and is frmain: Sil Sci. Sc. Am. J. 48: Vandenberghe, R. E. and De Grave, E. (1989) Mrssbauer effec sudies f xidic spinels: in M6ssbauer Specrscpy Applied Inrganic Chemisry, Vl. 3, G. J. Lng and F. Grandjean, eds., Plenum, New Yrk, Wlska, E. and Schwermann, U. (1989) The vacancy rdering and disribuin f aluminium ins in 3'-(Fe,A1)203: Slid Sae Inics 32/33: (Received 14 July 1993; acceped 11 March 1994; Ms. 2404)

PRINCE SULTAN UNIVERSITY Department of Mathematical Sciences Final Examination Second Semester (072) STAT 271.

PRINCE SULTAN UNIVERSITY Deparmen f Mahemaical Sciences Final Examinain Secnd Semeser 007 008 (07) STAT 7 Suden Name Suden Number Secin Number Teacher Name Aendance Number Time allwed is ½ hurs. Wrie dwn