by JY PrNc · 1993 · Cited by 145 — We describe thickness effects (i.e., unavoidable spectral distortions arising from nonzero absorber thicknesses) for uniform and nonpolarizing absorbers and

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American Mineralogist, Volume 78, pages 1-7, 1993Miissbauer absorber thicknesses for accurate site populations inFe-bearing mineralsD. G. RaNcounrOttawa-Carleton Geoscience Centre and Ottawa-Carleton Institute for Physics, Department of Physics,University of Ottawa, Ottawa KIN 6N5, CanadaA. M. McDoN.q.LoOttawa-Carleton Geoscience Centre, Department of Earth Sciences, Carleton University,Ottawa KIS 586, CanadaA. E. Llr,oNnBOttawa-Carleton Geoscience Centre, Department of Geology, University of Ottawa,Ottawa KIN 6N5, CanadaJ. Y. PrNc*Ottawa-Carleton Institute for Physics, Department of Physics, University of Ottawa,Ottawa KIN 6N5. CanadaAssrnA,crWe define the ideal absorber thickness, I,o”u,, in the usual way, as the absorber thicknessthat gives the largest signal to noise ratio in a given time, and show by measurements thatit is reliably calculated in real situations by the formulae of Long et al. (1983). We identifyproblem areas where it is essential to use the correct lid””r.We describe thickness effects (i.e., unavoidable spectral distortions arising from nonzeroabsorber thicknesses) for uniform and nonpolarizing absorbers and define the thin absorberthickness, I,n,,, as the largest thickness for which thickness effects are negligible. We presenta graphical method whereby t,hin can be evaluated for real mineral absorbers having spectracomposed of intrinsically broad lines.The limits of our methods are carefully outlined. The necessary background concerningthickness effects in multisite materials has not been developed in the literature and istherefore given in detail.Most often /*,i, ( fia”ur, however, with Fe-poor end-members (or with minerals contain-ing Fe as a trace) one can have dn,” > lid.d at 1,0.”, values corresponding to quite doableexperiments. We recommend that, whenever accurate quantitative results are required,the first concern be to obtain the best possible measured spectrum by using / : /,0″”,. Onecan then reliably obtain the intrinsic absorber cross section (thereby eliminating all thick-ness effects) by deconvoluting the measured data using methods such as those recentlydeveloped by Rancourt (1989).INrnolucrroNIn choosing a Mdssbauer absorber thickness, n” (innumber of Miissbauer nuclei per centimeters squared),one must distinguish two characteristic thicknesses: a thinabsorber thickness, n,,,n,., defined to be just small enoughto reduce thickness effects (i.e., spectral distortions arisingfrom nonzero absorber thickness and leading to incorrectspectral areas, heights, widths, and detailed shapes) tosome tolerable amount (e.9., less than the error in the rawspectral area determinations) and an ideal absorber thick-ness, r?a,id.ur, defined to give the largest signal to noise ratio(S/N) in a given time. These two characteristic thick-* Present address: University ofScience and Technoiogy Bei-jing, 100083 Beijing, P.R. China.0003-004x/93l0 102-000 l s02.00nesses are highly sample dependent and, for a given sam-ple, are generally significantly different.It is common practice to use an absorber thickness thatis an uncomfortable compromise betweeil zu.tnin and n^,,o.u,(Hawthorne, 1989). A rule of thumb that is often invokedis to use 5-10 mg/cm2 of natural Fe. No efort is thenmade to estimate the degree of thickness efects at thischosen thickness. This rule of thumb originates partlyfrom Greenwood and Gibb’s (1971) correct calculationthat for metallic Fe the true ideal absorber thickness cor-responds to -10 mg/cmz of a-Fe absorber and partlyfrom Hawthorne’s (1989) suggestion that for Fe-bearingoxide and oxysalt minerals, a good compromise betweenr?u.,n,n and n”.,0.”, is 5 mg/cm2 of Fe. The accurate n”,id.urvalue for a metallic Fe absorber is 16 mg,/cm’ Fe.

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RANCOURT ET AL.: MOSSBAUER ABSORBER THICKNESSESTABLE 1 . Calculated and measured ideal Mossbauer absorber thicknesses for three representative micas in the phlogopite-anniteSENCSCalculated thicknessestMeasured’*SampleDensityf(g/cm’)Wt% Fe+ rrmfn t. 4$mg/cm,/mica mg/cm’z/Fe 51Felcm2 f”lltmAnnite M42126Biotite MOC2661Phlogopite (Headly). Calculated by the 1/rr” formula of Long et al. (1983) Errors given are maximum errors.t Calculited {rom chemical composition and refined cell parameters These values are within – 1-2’h ot the measured densities determined on a25-mg Berman balance at room temperature using suspensions in both air and toluene.* Obtained by microprobe analysis.$ Assumes a natural abundance of s’Fe/Fe oI 2.14hll Dimensionless Mdssbauer thickness parameter defined as t^ = f.n”oo, where we assume a typical room-temperature recoilless fraction, f” : 0’7’and use the value oo: 2.56 x 10 ‘8 cm2 for the Mdssbauer cross section at resonance.3.313.062.86156 51.6 15.2 3.51 x 1013 6.3230 70.4 9.93 2.29 x 1016 4.1391 112 3.19 0.736 x 10″ 1’329.514.12.85160(+40, -20)230(+40, -20)370 (+50, -20)One study (Dyar, 1984) has attempted to consider theabsorber thickness problem and concludes that an opti-mum thickness is an Fe concentration of 5-7 mg/cm2 ofnatural Fe that is independent of both the chemical com-position ofthe absorber and the intrinsic spectral shape(single line vs. quadrupole doublet, etc.). Fundamentalproblems with the analysis leading to this optimum con-centration (Dyar, 1984) have been pointed out by Way-chunas (see Waychunas, 1986, 1989; Dyar, 1986, 1989).In this paper we compare calculated and measured ide-al absorber thicknesses for three micas representing theannite-phlogopite series (Table l). The approximale cal-culation described by Long et al. (1983) is found to giveexcellent agreement with measured n.,,o”., values. This im-plies that their approximation is valid for calculating ac-curate n”.id”,r values in many applications (even when nu,,n,,11 flo.ia”urt as with the micas), and establishes their simpleformula as a particularly useful tool. We, therefore, usethis formula to calculate the ideal sTFe Mtissbauer spec-troscopy (MS) absorber thicknesses for representativeclasses of Fe-bearing minerals (Figs. 1, 2).We also produce a graph (Fig. 3) that, for the first time,enables one to estimate both n^*,, and the degree to whichdifferent spectral areas are attenuated by thickness efectswhen n. > 0 in real situations with intrinsically broadlines. Such broad lines are pervasive in the spectra ofminerals, where they arise from hyperfine parameter dis-tributions and dynamic effects (Rancourt, 1988, 1989;Rancourt and Ping, l99l).TrrrcrNnss EFFEcrs AND THE THIN ABSoRBERTHICKNESSIn MS, the quantity that contains the desired (chemi-cal, crystallographic, magnetic, morphological, dynami-cal. etc.) information is the total absorber resonant crosssectron:NEooIIct)EorNEo:\oo.EEct)Eo180O Fe/{Fe + Mg}1.0Fig. 1. Calculated ideal 57Fe M (f^.,n^.,). (4)One goal in applying MS is to obtain oi(E), to resolveit into its site-specific components, o,^,,(E), and, knowingthe f^,,, to obtain the site populations, n.,,, from the areas(Eq. 3).The total absorber resonant cross section oi(E) is, how-ever, never observed directly. In absorption experiments,it gives rise to the measured absorption spectrum, N(E).For uniform and nonpolarizing absorbers, N(E) is givenby the well-known transmission integral:N(E’) : BG + n^. f 2- f ‘- A.t, r3/4\.\u/ -v aMrsTrf-J _,rvg+E)r+frJ4.(e .:r – l) (5)where ,BG is the measured background level, 4, is thepart ofthe BG that is from both recoilless and nonrecoil-less Mdssbauer y-rays, and I is the recoilless fraction ofthe single-line thin source. An absorber is uniform if allvariations (on a scale of the Mdssbauer cross section atresonance, oo, or larger) in n^ arc small (6n^/n^ << l) ev-erywhere on its exposed surface.Until recently, only the case of a single site had beendiscussed in detail in the literature (Rancourt, 1989). Inthat case, the relevant cross section can be written aso:(E) : t"o^(E)/oo: f^n"o^(E) (6)where.a - Jat.awo(7)is the usual dimensionless thickness parameter. This fac-torization (Eq. 6) is again possible in the multisite case,if we define o,(E) as012345678910toFig. 3. Lines of equal spectral area attenuation (equal A/A,n ")in the plane 2,.,n*. - I0 vs. /". This graph allows the thin ab-sorber thickness to be evaluated. See text for details.?f^,,n^,,o^r(E) (8)vE (10)'6irl, (l l)Io^(E)= = " _ZJ J u''rtu'isuch that Equations 6 and 7 are valid, where nowr:-! ),n r,'o4'^'l^'' (9)and, obviously, n^ = 2 n^.,.Few practitioners use the full transmission integral inanalyzing their spectral data. Instead, the thin absorberconditiono'^(E) = 4 ri.,(D = 2 .f^.,r^,,o^.,(E) << |is usually assumed. This assumption leads to the thinabsorber expression for the measured spectrumI0.80.60.4o.20AcE,Ioc!c=t:*(E'): AG_ n.f4TL oorN,n(E) : BG - n.. f.2- ), f ,',.t , oln a,l J ^.t..a.]' f** '' ll,/4J - d9OT6'* r'l+'""''l' (12)where one has performed a Taylor expansron of the expo-nential term in the integrand ofthe transmission integral.This expression (Eq. l2) has relatively desirable prop-erties that have motivated its overuse. These are (1) eachsite-specific absorption (i.e., spectral area) is equal to PAGE - 4 ============ RANCOURT ET AL.: MOSSBAUER ABSORBER THICKNESSESrooloq-fJ),,n^,,/2 and is therefore directly proportional ton^.,, (2) subspectral areas are additive, such that the totalspectral area isl-*-I anlnc - N,,(E)l : %ootonMf,2 f^,,r^., (13)""U O, if oL@)is a sum of Lorentzian lines (or a contin-uous distribution ofLorentzian lines, as in the case oftheVoigt line shape) then the corresponding measured spec-trum consists of the same sum (or distribution) but witheach Lorentzian FWHM increased by the source linewidth Io.If the thin absorber condition (Eq. l0) is not sufficientlysatisfied, then one cannot take advantage ofthe thin ab-sorber expression (Eqs. ll, 12) or make use ofany ofitsproperties. Instead, one must either fit directly with thetransmission integral or deconvolute out d:(E) from themeasured data (Rancourt, 1989; Rancourt and Ping,l99l).We may now describe how nu,,n,, can be evaluated fora particular absorber. Any such criterion is necessarilysomewhat arbitrary.Each line in the spectrum will have its depth, area,width, and detailed spectral shape affected by differentamounts, depending on the extent to which the thin ab-sorber condition (Eq. l0) is satisfied for that line. Oncomparing the observed line given by Eq. 5 with thatpredicted by the thin absorber limit (Eq. I l), one notesthat the depth is affected more than the area, which isaffected more than the width (see Ping and Rancourt,1992, for explicit demonstrations of these points). Sincearea is often of primary concern, we define n".,n,n withreference to the ratio (A/A,n) of the observed area to thethin limit line area. For any n^ + 0, A I A,n, and for athin absorber A/A,n = l.We choose a tolerable value of A/A,n, say 0.98 or 0.95(spectral areas can often be evaluated with precisions ofa few percent), that defines n".,n," for a given intrinsic linewidlh, W,,, [i.e., a true FWHM of the corresponding linein ol@)1. This is shown in Figure 3 where lines of con-stant A/A,n are drawn in the plane of W,", - fo vs. 1".Figure 3 is calculated by the methods of Ping and Ran-court (1992). Since widths are affected much less thanareas, the observed FWHM, W"o", is related ro W,", asWov"= Wi^r+lo.(14)This implies that if an absorber has known values of l,(or some representative value) aud nu (known from ele-mental analysis) and has a fraction, a, ofits spectral areain its deepest line of observed width, Vy"o",d, then its ap-proximate Figure 3 coordinates areW^ - lo = W"o,o - 2llo and t^ = af^n^oo (15)such that one easily determines whether it is thin or thick.The crossover occurs al nu: /4".,n," and thus defines nu.,n,".Figure 3 can also be used to estimate thickness conec-tions to spectral areas of separate lines in a given ob-served spectrum. It gives a quantitative estimate of howindividual lines in a spectrum are affected differently. Wedo not recommend these uses of Figure 3 as substitutesfor using either the correct transmission integral or anappropriate deconvolution. For partially overlapping lines,spectral distortions arising from finite thickness are suchthat fitted areas in raw spectra can be significantly wrongby more than just the expected attenuation predicted inFigure 3, because of the effects that the spectral distor-tions have on fitted line tradeoffs (e.g., Hargraves et al.,l 989).IoT,c.L AND THIN ABSORBER THICKNESSES IN THEPHLOGOPITE.ANNITE SERIESAbsorber thickness can be expressed in various ways:n^ (in s7Fe/cm2) and t^ (dimensionless) have already beendefined and are related by Equation 7; t^ = oo 2 f^,,n^,,corresponds to the average number of Mdssbauer nucleiencountered by a y-ray traversing the sample at any point.Given the concentration of sTFe in the sample, these canbe related to a thickness, lF", expressed in milligrams ofFe per centimeters squared. Given the mean absorberstoichiometry, this in turn corresponds to a thickness, /",expressed in grams (or milligrams) of absorber materialper centimeters squared. Finally, if the material density,p, is known, an actual physical thickness, I (in microme-ters, say), can be calculated by t: tr/p.The ideal absorber thickness arises because too thin anabsorber has too little sTFe to give an appreciable reso-nance absorption (i.e., sigrral), and too thick an absorbercauses too much ordinary mass absorption of the 7-raysfor significant statistics to be accumulated. It depends onthe material from which the absorber is made, on theabsorber resonance cross section lo'^(E)1, and on some ofthe experimental circumstances (Long et al', 1983; Sarmaet al., 1980; Blamey, 1977; Shimony, 1965).Long et al. (1983) have pointed out that, with the as-sumption that resonance absorption is proportional toabsorber thickness, an ideal absorber thickness is calcu-lated that is both independent of o"(E) and predomi-nantly determined by the ordinary mass absorption of theabsorber. Their result is that (l) when the non-Mdssbauerbackground is small,tr,,a,^r:2/Fo (BG - ni/BG << 1(16)and (2) when the non-Mdssbauer background is large,tr.ia.^r: l/P., (BG - q*1)/BG >> | (17)where p. is the electronic (i.e., ordinary, nonresonance)mass absorption coefficient of the absorber material forthe Mdssbauer (14.4 keV) ‘y-rays.The first situation (Eq. l6) is expected when, as is com-mon practice, a narrow counting window is set on theMiissbauer T-rays and the p” is small. The other limitingcase (Eq. 17) occurs when either no window is used,thereby allowing many non-Mdssbauer “y-ray counts, orpr. is large because of the presence of relatively heavyelements, or both (Long et al., 1983).

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RANCOURT ET AL.: MOSSBAUER ABSORBER THICKNESSESThe mass absorption coefficient of an absorber is cal-culated using its mass fractions, B,, of the element i, as.. -Sp–lLe – Zl PlFe).(18)For example, at 14.4 keV, p.,o” : 64 cmr/g such that l/p”: 16 mg/cm2 for a metallic Fe foil. This corresponds toa 20-pm foil. Long et al. (1983) have tabulated the p. ofthe elements at the important Miissbauer transition 7-rayenergres.We have compared the approximate predictions of Longet al. (1983) (above, Eqs. 16, 17) with measured idealthicknesses of three micas belonging to the phlogopite-annite series. These consisted of specimens of single-crys-tal near-end-member annite (M42126, Mont St. Hilaire,Qu6bec), biotite (MOC266I, Silver Crater mine, Ban-croft, Ontario) and near-end-member phlogopite (Head-Iey mine, Qu6bec), all of which have been extensivelystudied and well characterized (Rancourt et al., in prep-aration; Hargraves et al., 1989).The S/N ratios for the two strongest mica lines at ap-proximately -0. I and approximately +2.3 mm/s (withrespect to a-Fe at room temperature) were measured forabsorbers of various thicknesses from the three samples.The 7-ray incidence was normal to the cleavage plane.Graphs of S/N for spectra obtained in equal times (typ-ically several minutes to several hours) vs. wafer thick-ness showed distinct maxima occurring at the same idealthicknesses for the two absorption lines of a given sample.The resulting measured ideal thicknesses for the annite,biotite, and phlogopite are, respectively, 160 (+40, -20),230 (+40, -20), and 370 (+50, -20) pm, where maxi-mum errors are indicated. These are in excellent agree-ment with the predicted values for the case of large non-Mdssbauer background (Eq. l7), as seen in Table l.The experiments were performed with a relatively broadcounting window (i.e., single-channel analyzer windowlarger than the FWHM of the 14.4-keV line in the pulse-height analysis spectrum). The small non-Mossbauerbackground case (Eq. 16) or some intermediate case mighthold for an optimized counter window or a differentcounter, etc. Each spectroscopist must determine whichcase applies to his or her particular operating conditions.We expect that in most instances, with the most commonFe-bearing minerals (next section), the large backgroundcase will apply.The important point here is that the approximateEquations 16 and l7 give the correct bounds for idealthicknesses in real situations. This is further supportedby the fact that, for each mica studied, the two absorptionlines (with significantly different intensities at normal in-cidence, a = 0.65) gave the same ideal thicknesses. Thisis true even though za.trrin ( n.,id”.r in our samples.Using Figure 3, the line A/A,n:0.98 (not shown), andW^, – lo = 0.3-0.4 mm/s, we estimate that in all micast^.rn^ = 0.2, corresponding to r?^.rh,. = 0.2 x l0t8 57Fe/cm2at normal incidence. This gives thin absorber wafer thick-nesses of approximately 8, 20, and 90 pm, respectively,for our annite, biotite, and phlogopite. These are clearlysmaller than the ideal thicknesses.By comparison, adopting the rule of thumb using 5-10mg/cm2 Fe in micas causes an area attenuation of thestrongest line (a = 0.65) of -10-20o/o (A/A,h = 0.8-0.9)and a relative attenuation difference for the two strongestlines of -4-l60/o (i.e., area ratios between the two stron-gest lines will be incorrect and closer to I by this amount-for a single crystal wafer at normal incidence).In conclusion, our mica samples have a wide range ofideal thicknesses that are accurately given by the largenon-Mdssbauer background expression (Eq. l7) of Longet al. (1983). This implies that the expressions of Long etal. (Eqs. 16, 17) give the correct bounds for real situa-tions. In addition, for all of our mica samples, z”,tr,in ((nu.,o.u,, and using either the correct ru.,.””, values or a rule-of-thumb value results in spectra that are significantlyaltered by thickness effects.Note that single-crystal wafers such as our mica sam-ples are not nonpolarizing absorbers. Their spectra musttherefore suffer from more severe thickness effects thanthose predicted by both Equation 5 and Figure 3; bothassume nonpolarizing absorbers. In such cases, the thinabsorber thickness obtained from Figure 3, as explainedabove, must be viewed as an overestimate. The correct1,n,” for polarizing absorbers (single crystals, mosaic sam-ples, nonrandom powders, magnetized ferromagnets, etc.)is smaller than the /,hi” predicted for nonpolarizing ab-sorbers. Methods for thickness-correcting spectra with in-trinsically broad lines from polarizing absorbers have notbeen developed.IOBaT, ABSoRBER THICKNESSES oF Fe.BEARINGMINERALSValues of /rio.u, calculated using Equation l7 are shownvs. composition in Figure I for several mineral groupshaving Mg-Fe solid solutions. Although more material isrequired as the Fe content decreases, when the Fe contentis very small, far less material is required than wouldresult from applying the rule of thumb of 5-10 mg/cm2Fe. This is seen in Figure 2, where the ideal /.” values areshown for the same mineral groups as in Figure l. Theseall go monotonically to zero as the Mg end-member isapproached.Figures 1 and 2, therefore, illustrate a major break-down in the usual rule of thumb: it suggests thicknessesthat are orders of magnitude too large when Fe contentsare low. Indeed, one sees that, with such Fe-poor solidsolutions, it is quite possible to have za,trin ) n,.,u”u,. It isalso injust such cases that having the correct ideal thick-ness will often make the difference between the detect-ability of the signal and the impossibility of obtaining aspectrum. With more Fe-rich minerals, the longer timesrequired by not optimizing absorber thicknesses will of-ten not be prohibitive, but they can easily be with Fe-poor samples.On the other hand, it is correct to conclude (Fig. 2)

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RANCOURT ET AL.: MOSSBAUER ABSORBER THICKNESSESthat, when Fe/(Fe + Mg) > 0.2 in the major Fe-Mg solidsolutions, the ideal thicknesses range from low values of-2-5 mg/cm2 Fe to high values [at Fe/(Fe + Mg) : l]of -6-15 mg/cm’?Fe-not far from the rule of thumb(5-10 mg/cm’zFe). We expect, however, that in most ofthese cases, zu.mi. (( n”.,0″., such that thickness effects aresignificant at these ideal thicknesses.As a concrete example, consider near end-memberphlogopite: KMg, nrFeo orSi3AlO,o(OH)r. This particularmineral has nu.,o.u, = flutnin = 0.2 x l0t8 57Felcm2, wherenu.,n,, is approximately the same for all micas and wascalculated in the previous section. This value of n.,,u””‘ forthis phlogopite, corresponds to -0.8 mg/cm2 Fe or to- 120 mg/cm’zof the sample. On the other hand, the ruleof thumb would require 5-10 mg/cm2 Fe or 750-1500mg/crn2 of the sample. This would cause an ordinary massattenuation of the incident y-radiation that would be-10′?-l0o times too large, meaning that the Mdssbauerradiation getting through the sample and being countedwould be only – 1-0.010/o of the amount that gets throughat the correct ideal thickness of 0.8 mg/cmz Fe. Althoughthe experiment is comfortably doable with the correctnu.,o.ur, it becomes impossible at l0 mg/cm’z Fe.In addition to Fe-poor minerals, another area of diffi-culty will correspond to less common Fe-bearing miner-als containing large amounts of elements with large elec-tronic (i.e., ordinary) mass absorption coefrcients. Wemust anticipate problems with minerals containing largemass fractions of elements with p” at 14.4 keV larger thanapproximately 100 cm’?/g. Such elements (see table givenby Long et al., 1983) are, with few exceptions, those withatomic numbers larger than -60, plus the particular ab-sorption edge group Ga, Ge, As, Se, Br, and Kr, withrelatively small atomic numbers. With such minerals, es-pecially if they also contain only small amounts of Fe, itwill again be essential to use the correct ideal thicknesses.As a final warning, we remind readers that all the re-sults of the present paper (calculations of both thin andideal thicknesses) are for uniform absorbers in which thedepthwise average distributions of 5’Fe and of all the el-ements are uniform on every length scale of r,6″ = 0. 16A or larger on the sample surface. This means that onlysmall variations in all the depth-wise average numbers ofintersections with the relevant specific cross sections canbe tolerated, as steps of0.16 A or larger are taken on thesample’s exposed surface. As the variations become com-parable to the numbers themselves (6N – N) the resultspresented here become invalid. Also, polarization effectshave not been considered. With textured, nonrandom,mosaic, magnetized, or single-crystal absorbers, these canbe significant effects and should be included in the trans-mission integral: see Housley et al. (1968, 1969) for thecase of a simple compound having elemental absorptionlines.Nonuniformity occurs in particular with granular ab-sorbers consisting of large grains with comparable spacesbetween the grains. Here using average thicknesses canlead to calculated n.,,n,” and nu,ro”u, values that are off byan order of magnitude or more. Such granular absorberswill exhibit much more severe thickness spectral distor-tions than would be expected from their mean thickness-es. These difficulties can often be avoided by using uni-formly spread finely powdered absorbers (small grainscompared with the sample thickness in micrometers). Inaddition, such fine powders having random orientationsdo not give rise to polarization efects.Zoning of Fe content in mineral samples and whole-rock spectra ofrocks containing several phases with dif-ferent Fe contents are two more obvious problem areas.With such cases, one must resort to trial and error firstto find thicknesses that give acceptable spectra and thento explore changes in thickness distortions arising fromdifferent nominal thicknesses. Thickness distortions arepresently virtually impossible to correct in these cases.Different spectral components from different positions inthe samples can have very different degrees ofboth spec-tral distortions and observability.RncoutmNDATIoNsBefore choosing an 5?Fe Mdssbauer absorber thicknessfor a particular material, spectroscopists should know (l)which thickness (/,o”.’) will give the largest S/N ratio, (2)which thickness (t,hi”) will ensure that, as long as t = tth,n,thickness-effect spectral distortions will not be significant,and (3) in considering a compromise between /,o.u’ and 1,n,,(when 1,0″,, ) l,r,i,), what the thickness effects will be inthe measured spectrum collected at the compromisethickness. Alternatively, if the absorber thickness cannotbe imposed or if the spectroscopist does not care to op-timize it, then the key question is: what are the thicknesseffects in this situation?In this paper, the above points are addressed for realsituations involving spectra with intrinsically broad lines.Using the figures and methods described here, spectros-copists can evaluate /,n'” and the degree of thickness at-tenuation ofpeak areas for given thicknesses oftheir par-ticular materials. They can also confidently use theexpressions oflong et al. (1983) (Eqs. 16, l7) to calculate/,o.., for the particular absorber.In all cases, the largest thickness one would ever use is1,0.”,. When tnio ) tia.ot, one uses t : t,u., and one is certainof collecting the best possible spectrum (largest S/N in agiven time) that also has only negligible thickness effects.When /,n,” ( lia”ur, either one uses / : /,nr”‘ thereby sacri-ficing spectrum quality in order to reduce thickness effectsto some predetermined tolerable level, or one uses I :tid”d to obtain a high-quality spectrum that contains sig-nificant thickness effects that one rigorously takes intoaccount, either by fitting with the full transmission inte-gral (e.g., Eq. 5 in the absence of polarization effects) orby deconvoluting out the total absorber-resonant crosssection (Rancourt, 1989; Rancourt and Ping, l99l).No compromise is needed when /,o,^ ) /ia”a. In the casewhere 1,n,” ( fia”ur, most routine work will use I : dn,n,which is a compromise of known consequence, given thetolerance level chosen by the user in using Figure 3 to

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