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Curve fitting complex Mössbauer spectra; application to HoBa2(Cu0.95Fe0.05)3O7.02
Nuclear Instruments and Methods in Physics Research B 93 (1994) 521-529 North-Holland
IHOMI B
Bum Interactions wlth Materials 8 Atoms
Curve fitting complex ~~ssbauer spectra; application to HoBa 2(Cu 0_95 FeOm05) @ 7,02 M.J. Durst a, C.E. Violet b**, N.W. Winter b, Z. Mei ’ a University of ~al~omia, Lawrence Berkeley ~~ra~o~, Berkeley, CA 94720, USA b Universi@ of California, Lawrence Livemore National La~rato~, ’ Hewlett-Packard Company, Palo Alto, CA 94304, USA
Livermore, CA 945.70, USA
Received 7 December 1993; revised form received 11 April 1994
A method for determining the best fit of a complex M&batter spectrum is presented and applied to the room temperature 57Fe Mossbatter spectrum of HoBa,(Cu a,ssFea,as)sO,,,,. The method generates many ‘“good fits” as do previous methods. The “best fit”, which is the goal of curve fitting, is selected from the class of good fits by means of a statistical test using the “F” function. The best fit for this compound represents 7 doublets. This is significantly larger than the previously reported number (3 or 4) of subspectra associated with REBa,(Cu,_,Fe,),O, compounds. Assuming no mixed valence, this implies 7 Fe sites. This result is also significantly larger than the previously reported number of Fe sites (l-4) in the REBa&Cul_nFe,)sO, compounds. For 6 of the 7 doublets, the isomer shift, quadrupole splitting and line width are strongly correlated. The valence state corresponding to these 6 doublets is high spin + 4. The seventh doublet is high spin + 3. The sum of Mossbatter site multiplicities is related to the number of accidental degeneracies by a sum rule.
1. Introduction
This paper is a report, in part, of our study of the thermogravimetry and “Fe MSssbauer spectrometry of the high T, superconductor HoBa,(Cu,,Fe.,),O,,. The original report of this work [l] was judged to be too lengthy to be published as a single paper. It was then divided into two parts. The first part consisted of a complete description of the the~ogravimet~ and a partial description of the Miissbauer spectromet~ which included the curve fitting results but omitted the description of the curve fitting processes itself. This first part has already been published [2]. The second part consists of the description of the curve fitting process. This is the subject of the present paper. (The present paper is referred to as Ref. 1311 in our first paper RI). In our first paper [2] we concluded that although the number of Fe crystalline sites was most Iikely 6, we could not reject the possibility of 7 Fe sites. Subsequently, since we were faced with the task of rewriting the second part as a second paper, we re-analyzed the Mdssbauer spectrum with a higher degree of accuracy in an attempt to determine whether 6 or 7 was the most likely number of Fe sites.
* Corresponding
author, fax + 1 510 422 2851.
Substances used as Mossbauer absorbers can be divided into two categories depending on whether the Miissbauer element crystallographic sites are a priori known or unknown. The first category would include chemical compounds or alloys in which the Miissbauer element is one of the known constituents and occupies known or determinable sites. In the second category the MSssbauer element is an impurity or dopant occupying unknown sites. This category would also include those compounds that undergo o~gen-vacant ordering or disording such as occurs in the high T, superconductors 131.This process can generate subs&es of a single site. These subsites depend on the number of nearest neighbor oxygen ions and their spatial orientations about the site. If a Mossbauer isotope can substitute for a site it may also substitute for its subsites. The number and types of subsites is generally unknown a priori. Since each site gives rise to unique hyperfine (hf) interactions it is generally assumed that the number of subspectra in a complex Mossbauer spectrum equals the number of Miissbauer isotope sites and vice versa. However, because of the possibility of mixed valence and accidental degeneracy [4] this simple one-to-one correspondence is not necessarily true. Even when the crystal structure is known the number of subspectra in the Mijssbauer spectrum is a priori unknown. The Mijssbauer hyperfine spectra (hfs) depend on
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M.J. Dust et al. / Nucl. Instr. and Meth. in Phys.Res. B 93 (19941 521-529
the hf fields generated primarily by the atomic electrons surrounding the M&batter nucleus and to a lesser extent by the surrounding ions in the crystal [S]. Thus, the site of a Mossbauer nucleus is a “nuclear site environment”. Similarly, a ~~ssbauer ion experiences an “ionic site environment” defined by the fields due to the surrounding ions. Mixed valence does not change the ionic site environment but it can produce changes in the nuclear site environment. Jn the case of slow relaxation the two valence states associated with mixed valence can produce two distinct nuclear site environments, or in case of fast relaxation, a single environment. Therefore, the multiplicity of a ~~ssbauer site with mixed valence is one or two in the limits of fast or slow relaxation respectively. Mixed valence could aiso complicate the comparisons of experimental results with theoretical calculations as well as the results from different experimental techniques which have different characteristic measurement times. A large class of complex ~~ssbauer spectra are su~e~os~tions of pure hfs. These may consist of magnetic dipole hfs, electric quadruple hfs, or a combination of both. These types of spectra are well known and well characterized for most of the M&shatter isotopes [6]. In order to curve-fit complex Miissbauer spectra a fitting function representing the actual number and types of hfs is required. The fitting begins with a first “trial fitting function” (TFF) consisting of a sum of subspectra. (Because of the possibility of accidental degeneracy, we consider the TFF to be composed of subspectra rather than hfs.) The number and types of subspectra in the first TFF are determined by inspection. After fitting this function, an improved fit is attempted using a second TFF consisting of the first TFF, in which one of the subspectra is replaced by two subspectra. A third or higher order TFF can be constructed using this same procedure. The curve fitting then proceeds as follows: 1) Determine whether, (a) the reduced x2 is close to unity, and (b) a plot of the function superimposed on the data results in a visually satisfactory fit. 2) If these two criteria are not met, replace one of the subspect~~ with two subspectra and repeat the curve fitting. If the two criteria of step 1) are still not met, continue adding subspectra and curve fitting until they are. 31 When the two criteria of step 1) are met the result is a good fitting function. 4) Continue to generate good fitting functions by adding subspectra to the fitting function. 51 From this class of good fitting functions determine the best fitting function by statistical test. The “good fitting functions” and “best fitting function” have the following properties: 1) good fitting function - results in (a1 physically meaningful parameters, (b) a visually satisfactory fit
and (c) a reduced x2 that is approximately unity. (Conditions (b) and (c) are highly correlated). 2) best fitting function - this is the good fitting function which has the highest probabili~ of being the true fitting ~~t~~.
The parameters of the true fitting function have their lrrce r_&ues. The parameters of the best fitting function have their best u&es. The parameters’ best values are the best estimates of their true values. From the analysis presented herein it will become evident that the conventional curve fitting approach does not necessarily provide the best fitting function. We shall also see that in the process of dete~ining the best fit, accidental degeneracy is necessarily removed. Of course any multiplicities associated with the crystallographic sites are not removed. The procedure described in this paper is tantamount to treating the number of subspectra, N, as an additional adjustable parameter. It is to be determined, along with the parameters of the resonance lines, by means of curve fitting and statistical testing. From the correct number of subspectra one can determine, 1) the correct number of nuclear site environments (but not necessarily the number of sites) and 2) the hyperfine interactions associated with each of these environments. The purpose of this paper is to describe a curve fitting procedure that secures the best fitting function and apply this to the room temperature 57Fe Miissbauer spectrum of HoBa,(Cu,,,,Fe,,05)30~.~~. In so doing a general procedure for obtaining the best fitting function of a complex Mossbauer spectrum is demonstrated. We also obtain information regarding the na. . ture of the Fe sites m HoBa,(Cu,,, %.&~7.0~.
2. Anaiysis of the s7Fe ~~ss~auer spectrum 2.1. Cwve fitting strategy The analysis of our 57Fe Mossbauer spectrum was accomplished by means of nonlinear least squares fitting {7]. As dictated by the spectrum itself all of the subspectra are doublets. The line depths as well as line widths were constrained to be equal. All line shapes were approximated with the Lorentzian, A room temperature magic angle spectrum was also measured and the effect of texture was found to be negligible [2]. Therefore, these assumptions regarding the resonance line characteristics are reasonable. The fitting ~uctiou can be written,
M.J. Durst et al. /Nucl.
Instr. and Meth. in Phys. Res. B 93 (1994) 521-529
where Yhi is the depth of the spectrum at velocity, u, yi is the depth of the ith doublet, and f(u; k, 1, m) is the parabolic baseline. Since we approximate line shapes with Lorentzians we have, yi(U; e,i, 1 la
54)
_i[[(;(“-(61-;)))2+l]-1
523
smaller reduced x2, and hence would be an improved fit. The question is; is this improved fit significant? The answer can be provided by a statistical test as described herein. If the answer is affirmative we would then construct a third TFF in which the previously selected doublet is replaced by three superposed doublets of equal depths and line widths giving, N-l
i#j
+[(;(“-(8;+;))~+l]-1]. (2)
Four adjustable parameters are required for the case of line widths and depths constrained to be equal. Therefore, we specify (arbitrarily) the I?,, as, @ii= Si (doublet splitting), OZi= 6, (isomer shift), OSi= c (line width), @,,= li (maximum resonance absorption), 19s)= the union of all Oai; i.e., f#) = (Oai, 1 I (YI 4, 1
f(u; k, 1, m) = k + lv + mu2 N = the number of doublets, Curve fitting begins with the first TFF, Y$), Yhi) = EE1yi + f(u; k, 1, m). The starting values of the parameters, Oai, are determined by inspection. The fitted first TFF, rYhl), is then examined for doublets with significant excess line broadening, i.e., broadening in excess of that which can be attributed to absorber thickness. We presume that a significant component of this broadening is due to accidental degeneracy (nonhomogeneous broadening). If the data is good enough, accidental degeneracy can be removed by curve fitting. Any significant excess line broadening remaining after this process must then be homogeneous broadening. (Disorder in distant neighbor shells might produce line separations so small as to be, for a given set of data, unseparable by curve fitting. This residual nonhomogeneous broadening would then be a small component of the homogeneous broadening.) Now we suppose that rY$‘) consists of N doublets some or all of which have significant excess line broadening. We first select the doublet with the broadest lines and replace it with two superposed doublets of equal depths and widths. The result is a second TFF, N-l Y$?l=
C
Yi+Yj,+YjZ+f(U;k,f,m),
and proceed as before. In this process the starting values of the parameters, other than those of the added doublets, are the values resulting from the previous fit. This procedure is continued until the statistical test indicates that the last TFF, is not a significant improvement over the next-to-last TFF. After the jth doublet has been analyzed in this manner we apply the same procedure to the other significantly broadened doublets. The “final fitting function” (FFF) consists of these as well as any unbroadened original doublets. Finally, we do a global fit of the FFF in which the parameters of all of the doublets are free. The result is the best fitting function in which all adjustable parameters have their best values. These are also the quoted values. The global fit of the FFF represents pure quadrupole doublets and these refer to actual Fe sites. The FFF can be represented as,
e$), k, 1, m) = f
Y,(u,
i-1 +f(u;
i
yij( O; e,,,,
1 I (YI 4)
jz1
k,
1, m),
(3)
where Yij(
UT
emi,, 1 I (Y54)
=~~j~[(~(~-(~ij-~))i2tl] +[(+(&-;j+~)))*+l]-l\. yij(v, e,,,, 1 I (YI 4) is the depth of the embedded in the i-th doublet of the first the set of parameters analogous to Bui. f@ ofall IX, i.e.,8~)={e,,,,lI(yI4,1IiIN,lljI Ji). Ji = number of doublets embedded in blet. The number of doublets in the Mijssbauer
(4) j-th doublet TFF. Oati, is is the union the ith douspectrum is
i#j
where j is the index of the selected doublet. We then fit this function using parameter starting values obtained from rYA1)and from the two doublets just constructed. The resulting fit, rYdyl, would necessarily have a smaller residual sum of squares (RSS), a
M=
f.Ii. i=l
In some cases the acceptance or rejection of additional doublets is made by the least squares fitting process itself. For example, an added doublet may
524
M.J. Durst et al. /‘Nucl. Ins@. and Metk in Fhys. Res. B 93 f1994 521-529
0.96
Fig. 1. 57Fe Miissbauer spectrum of HoBaz(Cu,,Fe,os)307,, at RT and normal incidence fitted to 3 doublets. The source was “Co: Pd at RT. The residuals plot runs between +5000 counts; the curve was determined by numerical averaging.
monotonically decrease in size with each iteration of the fitting process until it becomes comparable to the base Iine noise. Or, the final value of one or more of the parameters may be unreasonable. The latter case would include any negative values irrespective of whether the parameters do or do not appear only as the square In these two cases the added doublet should be rejected. Such pathological behavior is frequently symptomatic of an over determined least squares fitting process and indicates that the number of subspectra should be decreased. 2.2. Detailed fitting
of the Miissbauer
The “Fe Miissbauer spectrum Fe0.0S~~~7.Q~~at room temperature
spectrum
of HoBa,(Cu,,, and normal inci-
dence is shown in Fig. 1. It is clear, by inspection, that the first TFF for this spectrum should consist of the sum of three doublets, D-l, D-2, and R-3 131.A least squares fit of this TFF appears in Fig. 1. The parameters resulting from this fit are given in Table 1. These are consistent with previously published three-doublet fits of REBa,(Cu,_,Fe,),O,_, spectra @-lOI. However, this TFF does not satisfy the conditions for a good fitting function. This is because (1) the fit is not visually satisfactory, i.e. there are several regions where the curve deviates significantly from the experimental points, and (2) the reduced x2 is significantly larger than unity. Also the resonance lines of D-l and D-2 are much broader than can be accounted for by the absorber thickness [ll. We therefore proceeded to analyze the spectrum as prescribed above.
Table I Results of curve fitting with tbe first trial fitting function Doublet
Doublet splitting, S
Isomer shift, 6 [mm/s] b
Line width, r [mm/s1
tmm/sl a 1 2 3
1.944 1.063 0.265
0.0485 0.~89 0.2370
0.366 0.491 0.305
Maximum absorption,
Relative intensity,
E[%Ib
R [%I
3.42 4.65 0.923 Total: 8.99
x2 = 2.21 RSS = 5.48 X 10” (residual sum of squares), a to convert (mm/s) to MHz multiply by 11.624 MHz/(mm/s). b uncorrected for background (background correction: 30 + 3%). ’ c/ch - counts/channel.
32.8 56.X 7‘38
Parabolic baseline k I
k,‘chl = 2804193
k/W,’
;;c/ch),
(mm/s11
(mm./s)2l
- 1I .97097
- 891.6922
M.J. Durst et al. / Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 521-529 2.2.1. Analysis of D-2
We first fitted the broadest-line doublet, D-2, with two superposed doublets. We next tried three doublets. In this case one doublet always decreased monotonically with each iteration of the least squares fitting process while the other two increased and became virtually identical to the original two doublets. We repeated this computation several times with different parameter starting values. Each attempt exhibited the same behavior while the reduced x2 remained close to unity. We concluded that the third doublet presented an over determined problem to the least squares fitting computation. Therefore, two doublets were the best fit to doublet D-2. 2.2.2. Analysis of D-l We tried fitting 2, 3, 4, and 5 doublets to the next broadest-line doublet, D-l. In each case the parame-
525
ters stabilized at reasonable values and the reduced x2 steadily decreased. The best value for the number of doublets in D-l was obtained by the statistical test described below. 2.2.3. Analysis of D-3 We attempted to fit two doublets into the remaining doublet, D-3. This resulted in one doublet shrinking and eventually becoming unphysically small while the other grew until it became virtually identical to D-3. Choosing a range of starting values for the parameters did not alter the outcome. We concluded that D-3 is a pure quadrupole doublet. In the foregoing analysis the curve fitting was stopped in two instances (D-2 and D-3) by the fitting process itself. Only the curve fitting of D-l was stopped by the statistical test. As a result of curve fitting we obtained a total of five “good” FFFs consisting of 3, 5,
Table 2 Curve fitting results Total fitting function
Doublet
Doublet splitting, S [mm/s] a
Isomer shift 6 [mm/s]
Line width l[mm/s]
Doublet maximum absorption, E [%I b
Relative intensity R [%]
((EFG), [v/,Q,21
I’
Parabolic baseline parameters k 1 m Kc/ch)/ lc/chl d Kc/ch)/
(mm/s)1
(mm/s)*1
Five doublet x* = 1.35 RSS = 3.32 x lo9
1 2 3 4 5
2.068 1.827 1.219 0.933 0.260
0.0551 0.0403 0.0085 0.0083 0.2079
0.274 0.325 0.428 0.379 0.350
1.84 2.02 2.40 2.67 0.96
14.3 18.6 29.1 28.6 9.5
241.9 213.6 142.9 109.3 30.5
2798443
- 157.9907
- 119.6289
Six doublet x2 = 1.25 RSS = 3.06 x lo9
1 2 3 4 5 6
2.138 1.921 1.659 1.183 0.903 0.261
0.0552 0.0467 0.0297 0.0088 0.0077 0.2052
0.256 0.264 0.297 0.393 0.366 0.362
1.26 2.14 0.98 2.69 2.36 0.96
9.4 16.4 8.4 30.7 25.1 10.1
250.1 224.8 194.1 138.5 105.8 30.5
2796594
- 188.7096
118.3776
Seven doublet x2 = 1.23 RSS = 3.00 x lo9
1 2 3 4 5 6 7
2.138 1.929 1.688 1.326 1.088 0.859 0.266
0.0545 0.0465 0.0317 0.0069 0.0101 0.0055 0.2007
0.263 0.255 0.299 0.339 0.323 0.363 0.380
1.31 1.99 1.13 1.21 2.16 2.03 0.98
10.1 14.9 9.9 12.0 20.5 21.6 10.9
249.6 255.9 198.1 155.1 128.5 100.9 31.4
2795 835
- 188.1470
203.0215
Eight doublet x2 = 1.22 RSS = 2.96 x lo9
1 2 3 4 5 6 7 8
2.139 1.929 1.686 1.307 1.069 0.840 0.269 0.776
0.0546 0.0467 0.0319 0.0063 0.0107 0.0030 0.2009 0.0781
0.262 0.255 0.302 0.350 0.332 0.355 0.386 0.007
1.30 1.99 1.13 1.33 2.31 1.71 0.98 0.16
10.0 14.9 10.0 13.7 22.5 17.8 11.1 0.03
250.6 226.4 197.3 153.2 125.1 98.7 31.8 91.1
2795 903
- 185.5649
195.2873
a b ’ d
to convert mm/s to MHz multiply by 11.624 MHz/(mm/s). uncorrected for background (background correction: 30f 3%). Total EFG magnitude (see Ref. [4]). c/ch - counts/channel.
526
0.98
~_“---_
---
-1
-2
0
2
1
3
Y (mm/secf
Fig. 2. “Fe M~ssbauer spectrum of HoBaz(Cu,,,Fe0,053307,02 at RT and normal incidence fitted to 7 doublets. The source was “Co: Fd at RT, The residuals plot runs between ~5000 counts; the curve was determined by numerical averaging.
6, 7, and 8 doublets. The parameters for these fits are given in Tables 1 and 2. The fitted FFF for the 7 doublet fit is shown in Fig. 2.
We now proceed to answer the question: which of these five “good” fitting functions is the “best” fitting functions We used nonlinear Ieast squares fitting to arrive at parameter values for the fitting functians with M doublets. That is, we minimized the residual sum of squares
FE-1
doublets M’
over all possible choices of the vector of parameters
8$), where y,, is the number of counts in the nth channel and “‘m” is the m~imum number of channels, In our spectrum M = 923. For each possible number of doublets M, we have a best residual sum of squares, RSSM .
If we choose an incrementally
larger number
of
- RSSw),‘v, -
RSS,J/‘vr
Here V~is the number of degrees of freedom of the “full” (i.e., larger) model, which is the number of channels minus the number of parameters fitted. From Eq. (1) this is m - (3 + 4 x IM’). V~ is the so-called “extra” degrees of freedom; if vf is as above and vp is
Table 3 Goodness of fit results Doublets
RSS
3 5 6 7 8
5.48 x 3.33 x 3.06X 3.00x 2.97 x
10” 10” 10” 10” log
Parameters
F ratio
dfl
df2
Pr
15 23 27 31 35
72.87 19.33 4.28 2.60
8 4 4 4
900 8% 892 888
2.4~30-~~ 2.9x10-‘5 0.002 0.035
(Fdfl,dfZ'F)
M.J. Durst et al. / Nucl. In&. and Meth. in Phys. Res. B 93 (1994) 521-529
defined similarly for the “partial” (i.e., smaller) model, then v, = vP - vf = 4(M’ -M). The interpretation of this F ratio is as follows. If we were really observing a spectrum with only M doublets, but we overfitted using a function with M’ doublets, then the ratio given above would have the F distribution with v, and vf degrees of freedom. An F distribution is derived as the distribution of the ratio of two independent variables, both of which have the chisquared distribution. It is commonly used in statistics to compare two models, particularly when one model is a subset of the other. The degrees of freedom defining the specific F distribution are the numerator and denominator degrees of freedom of the chi-squared variables [12]. If, in our actual fitting, we obtain an F ratio so large as to be very unlikely under this distribution, then we have obtained strong evidence against there being only M doublets present, even if the fit with M doublets was satisfactory. Conversely, if our actual fitting obtains an F ratio which is not very unlikely under the given distribution, then we must accept that the evidence for more than M doublets is not compelling. We made fits with FFFs consisting of 3, 5, 6, 7 and 8 doublets; Table 3 gives the F statistics comparing each model with the next larger one. The column headed “Doublets” gives the number of doublets in the fitting function, “RSS” gives the residual sum of squares for that fit, “Fobs” gives the observed F ratio, “dfl” and “df2” give the numerator and denominator (respectively) degrees of freedom of the F ratio, and the last column gives the tail probability associated with the observed F value. This value, Rr( Fdrl ,df2> Foss1 Y tells us the probability of observing an F ratio as large or larger under what statisticians call the “null hypoth-
527
esis”: that the number of doublets used is an overfit, with the true number of doublets being the next smallest number in the table. The improvement in fit is unquestionable as we move from 3 to 5 doublets, as when we move from 5 to 6 or from 6 to 7. In moving from 7 to 8 doublets, we observe a tail probability of 0.035, which is low (below the frequently used 0.05 threshold for significance) but not unquestionable. To amplify this slightly, a recent conference of Mossbauer spectroscopists (ICAME-93) attracted 365 attendees. If the true number of doublets present were 7, and if a copy of our sample were given to each of 365 spectroscopists, we would expect that by chance 12 of them would obtain an F ratio as large as the one we observed. Furthermore, on examining the parameter values for the &doublet fit, we see that doublet 8 has a line width of 0.007 mm/s, which is unphysically small. Our estimation process always started with physically reasonable values, but never lowered the residual of squares significantly without resulting in physically unreasonable parameters. We are satisfied by the above that there is no compelling evidence for more than 7 doublets. No fit with 8 doublets that was physically reasonable produced any significant decrease in residual sum of squares over the 7-doublet fit. By contrast, the progression from 3 up to 7 doublets was accompanied by unquestionable improvement at each stage. We therefore conclude that the best fitting function is the one with 7 doublets. 2.4. Correlation of 6, S and r For the 7-doublet fit, visual examination of the 7 (S, S, r> points in 3-space reveals that 6 of the 7 points
Fig. 3. A plot of S, 6, and r for the 7 doublets clustered about the best fitting straight line.
528
M.J. Durst et al. /Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 521-529
are nearly collinear. We can find the best fitting straight line in space to these 6 points by calculating their “principal components.” The principal components are the eigenvectors of the 3 X 3 covariance matrix of the 6 points. The “first principal component,” which is the eigenvector corresponding to the largest eigenvalue, lies in the direction of the best-fitting line, and the line passes through the mean of the 6 points in 3-space. The equation of this line is, in units of mm/s,
Therefore we have the sum rule: R
s
CM,= CDS. r=l
s=l
ZZf=lDs equals the number of accidental degeneracies in the Mossbauer spectrum. The sum rule can be stated: the sum of the Mossbauer site multiplicities over all Mijssbauer sites equals the number of accidental degeneracies in the Mossbauer spectrum.
S = 24.206 + 0.878, r = - 1.956 + 0.357, and this line explains 99.9% of the variance. A view demonstrating the wide extent of the 7 (S, S, r> points in 3-space and how tightly clustered 6 of them are about their best fitting straight line is shown in Fig. 3. This analysis suggests that 6 doublets are associated with similar nuclear site environments while the nuclear site environment associated with the 7th doublet differs significantly.
3. A sum rule When Fe is substituted into REBa,(Cu,_,Fe,), O,_, one would expect Fe to occupy to some extent all of the cation sites. However experimental evidence has shown that the Fe dopant occupies predominantly the Cu(l) sites in the basal plane. In addition to the Cu(l1 sites, there exists a number of Cu(l) subsites due to oxygen disorder [3]. Therefore, the potential number of nonequivalent basal plane sites available to an Fe substituent can be greater than the number of Cu(1) sites. The one-to-one correspondence between a Mossbauer nuclear site environment and a crystallographic site environment can be broken by site multiplicity due to mixed Fe valence or accidental degeneracy. These two mechanisms are not independent; they must obey a sum rule. Let us define M, as the multiplicity of the rth Fe site and D, as the degeneracy of the sth subspectrum. If we define the multiplicity of a site with no mixed valence to be unity and the degeneracy of a pure hf spectrum to be unity then, M, r 1, and D, 2 1. The number of nuclear site environments, N,, can be written either as:
or N,=
?Q, s=l
where: R = number of Mossbauer sites, S = number of subspectra in the best fitting function.
4. Valence states and numbers of Fe sites We have determined that the most likely number of Fe sites in room temperature HoBa,(Cuo,ssFeo,os)307,02 is 7. If there were no mixed Fe valence, the number of Fe crystallographic sites would also be 7. Based on magnetic susceptibility measurements made on Fe-substituted YBa,Cu,O,_, compounds the Fe is in a high spin state. If the same is true for our Fe-substituted Ho-based compound then this result combined with the isomer shift and quadrupole splitting results, lead to the conclusion that Fe is present in the Ho-based compound only in the high spin + 3 and + 4 states [2]. The quadrupole splitting and isomer shift results strongly suggest that one of the 7 doublets is high spin + 3 and the remaining 6 are high spin + 4 [2]. Therefore, if mixed valence were not operative we would have a total of 7 nonequivalent Fe sites, 6 sites are high spin +4 and the 7-th site is high spin +3.
5. Conclusions
A method for obtaining the best fit to a complex Miissbauer spectrum has been demonstrated. The application of this method to the room temperature s7Fe Mijssbauer spectrum of HoBa,(Cuo,ssFe,.,,),O,,,, results in a best fitting function of 7 doublets. The possibility of 8 or more or 6 or less doublets can be rejected. This is significantly larger than the previously reported number (3 or 4) of subspectra associated with REBa,(Cu, _xFe,),O, compounds. Assuming no mixed valence this implies 7 Fe sites. Seven Fe sites is a significantly greater number than the previously reported number of Fe sites (l-4) in the REBa,(Cu,_, Fe,130, compounds. This discrepancy may stem from the fact that the Mossbauer spectrum depends critically on the oxygen stoichiometry of the sample [3]. For 6 of the 7 doublets, S, S, and r are strongly correlated. The valence state of these 6 Fe sites is high spin f4; the seventh Fe site is high spin +3. A sum rule was derived which relates the sum of the Miissbauer site multiplicities to the sum of the accidental degeneracies in the Miissbauer spectrum.
M.J. Durst et al. /Nucl. instr. and Meth. in P&s. Res. B 93 (1994) 521-529
We acknowledge many enlightening discussion with, R.D. Taylor, Los Alamos National Laboratory, M.G. Smith, Los Alamos National Laboratory and University of Texas, Austin, M.P. Pasternak, Los Alamos National Laboratory and Tel Aviv University, J.J. Spijkerman, Ranger Scientific, Inc., Prof. L. May, Catholic University of America, and S.P. Verrili, Forest Products Laboratory. We also acknowledge the support of M. Ross and M.J. Fluss, LLNL, and the hospitaiity of Prof. J.W. Morris Jr., Lawrence Berkeley Laboratory. This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract no. W-7405-Eng-48.
References [I] C.E. Violet, R.G. Bedford, N.W. Winter, G.S. Smith, Z. Mei, M.J. Durst and J.K. Wu, UCRL-JC-107783 Rev. (1991).
529
[Z] C.E. Violet, R.G. Bedford, N.W, Winter, G.S. Smith, Z. Mei, M.J. Durst and J.K. Wu, Physica C 207 (1993) 347. [3] C.E. Violet, R.G. Bedford, P.A. Hahn, N.W. Winter and Z. Mei, Physica C 162-164 (1989) 1291. [4] D.G. Rancourt, Nucl. Instr. and Meth. B 44 (1989) 199. [5] N.W. Winter, C.I. Merzbacher and C.E. Violet, Appl. Spec. Rev. 28 (1993) 123. [6] See for example, N.N. Greenwood, Mossbauer Spectroscopy (Chapman and Hall, 1971). [7] G. Longworth, in: Miissbauer Spectroscopy Applied to Inorganic Chemistry, vol. I, ed. G.J. Long (Plenum, 19841 p. 43. IS] C.E. Violet, P.A. Hahn, Z. Mei, T. Datta, C. Almasan and J. Bstrada, Extended Abstracts, High-Temperature Superconductors II (Materials Research Society, 198X) p. 335. 191 C.E. Violet, P.A. Hahn, T. Datta, C. Almasan, and J. Estrada, J. Appt. Phys. 64 (1988) 5911. [lo] P. Boolchand and D. McDaniel, Studies on High-Temperature Superconductors, vol. 4, ed. A.V. Narlikar (Nova Scientific, 19901 p. 143. [ll] D.M. Bates and D.G. Watts, Nonlinear Regression Analysis and Its Applications (Wiley, 1988) 53.10.1. [12] N.R. Draper and H. Smith, Applied Regression Analysis, 2nd ed. (Wiley, 1981).
IHOMI B
Bum Interactions wlth Materials 8 Atoms
Curve fitting complex ~~ssbauer spectra; application to HoBa 2(Cu 0_95 FeOm05) @ 7,02 M.J. Durst a, C.E. Violet b**, N.W. Winter b, Z. Mei ’ a University of ~al~omia, Lawrence Berkeley ~~ra~o~, Berkeley, CA 94720, USA b Universi@ of California, Lawrence Livemore National La~rato~, ’ Hewlett-Packard Company, Palo Alto, CA 94304, USA
Livermore, CA 945.70, USA
Received 7 December 1993; revised form received 11 April 1994
A method for determining the best fit of a complex M&batter spectrum is presented and applied to the room temperature 57Fe Mossbatter spectrum of HoBa,(Cu a,ssFea,as)sO,,,,. The method generates many ‘“good fits” as do previous methods. The “best fit”, which is the goal of curve fitting, is selected from the class of good fits by means of a statistical test using the “F” function. The best fit for this compound represents 7 doublets. This is significantly larger than the previously reported number (3 or 4) of subspectra associated with REBa,(Cu,_,Fe,),O, compounds. Assuming no mixed valence, this implies 7 Fe sites. This result is also significantly larger than the previously reported number of Fe sites (l-4) in the REBa&Cul_nFe,)sO, compounds. For 6 of the 7 doublets, the isomer shift, quadrupole splitting and line width are strongly correlated. The valence state corresponding to these 6 doublets is high spin + 4. The seventh doublet is high spin + 3. The sum of Mossbatter site multiplicities is related to the number of accidental degeneracies by a sum rule.
1. Introduction
This paper is a report, in part, of our study of the thermogravimetry and “Fe MSssbauer spectrometry of the high T, superconductor HoBa,(Cu,,Fe.,),O,,. The original report of this work [l] was judged to be too lengthy to be published as a single paper. It was then divided into two parts. The first part consisted of a complete description of the the~ogravimet~ and a partial description of the Miissbauer spectromet~ which included the curve fitting results but omitted the description of the curve fitting processes itself. This first part has already been published [2]. The second part consists of the description of the curve fitting process. This is the subject of the present paper. (The present paper is referred to as Ref. 1311 in our first paper RI). In our first paper [2] we concluded that although the number of Fe crystalline sites was most Iikely 6, we could not reject the possibility of 7 Fe sites. Subsequently, since we were faced with the task of rewriting the second part as a second paper, we re-analyzed the Mdssbauer spectrum with a higher degree of accuracy in an attempt to determine whether 6 or 7 was the most likely number of Fe sites.
* Corresponding
author, fax + 1 510 422 2851.
Substances used as Mossbauer absorbers can be divided into two categories depending on whether the Miissbauer element crystallographic sites are a priori known or unknown. The first category would include chemical compounds or alloys in which the Miissbauer element is one of the known constituents and occupies known or determinable sites. In the second category the MSssbauer element is an impurity or dopant occupying unknown sites. This category would also include those compounds that undergo o~gen-vacant ordering or disording such as occurs in the high T, superconductors 131.This process can generate subs&es of a single site. These subsites depend on the number of nearest neighbor oxygen ions and their spatial orientations about the site. If a Mossbauer isotope can substitute for a site it may also substitute for its subsites. The number and types of subsites is generally unknown a priori. Since each site gives rise to unique hyperfine (hf) interactions it is generally assumed that the number of subspectra in a complex Mossbauer spectrum equals the number of Miissbauer isotope sites and vice versa. However, because of the possibility of mixed valence and accidental degeneracy [4] this simple one-to-one correspondence is not necessarily true. Even when the crystal structure is known the number of subspectra in the Mijssbauer spectrum is a priori unknown. The Mijssbauer hyperfine spectra (hfs) depend on
0168-583X/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0168-583X(94)00204-9
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M.J. Dust et al. / Nucl. Instr. and Meth. in Phys.Res. B 93 (19941 521-529
the hf fields generated primarily by the atomic electrons surrounding the M&batter nucleus and to a lesser extent by the surrounding ions in the crystal [S]. Thus, the site of a Mossbauer nucleus is a “nuclear site environment”. Similarly, a ~~ssbauer ion experiences an “ionic site environment” defined by the fields due to the surrounding ions. Mixed valence does not change the ionic site environment but it can produce changes in the nuclear site environment. Jn the case of slow relaxation the two valence states associated with mixed valence can produce two distinct nuclear site environments, or in case of fast relaxation, a single environment. Therefore, the multiplicity of a ~~ssbauer site with mixed valence is one or two in the limits of fast or slow relaxation respectively. Mixed valence could aiso complicate the comparisons of experimental results with theoretical calculations as well as the results from different experimental techniques which have different characteristic measurement times. A large class of complex ~~ssbauer spectra are su~e~os~tions of pure hfs. These may consist of magnetic dipole hfs, electric quadruple hfs, or a combination of both. These types of spectra are well known and well characterized for most of the M&shatter isotopes [6]. In order to curve-fit complex Miissbauer spectra a fitting function representing the actual number and types of hfs is required. The fitting begins with a first “trial fitting function” (TFF) consisting of a sum of subspectra. (Because of the possibility of accidental degeneracy, we consider the TFF to be composed of subspectra rather than hfs.) The number and types of subspectra in the first TFF are determined by inspection. After fitting this function, an improved fit is attempted using a second TFF consisting of the first TFF, in which one of the subspectra is replaced by two subspectra. A third or higher order TFF can be constructed using this same procedure. The curve fitting then proceeds as follows: 1) Determine whether, (a) the reduced x2 is close to unity, and (b) a plot of the function superimposed on the data results in a visually satisfactory fit. 2) If these two criteria are not met, replace one of the subspect~~ with two subspectra and repeat the curve fitting. If the two criteria of step 1) are still not met, continue adding subspectra and curve fitting until they are. 31 When the two criteria of step 1) are met the result is a good fitting function. 4) Continue to generate good fitting functions by adding subspectra to the fitting function. 51 From this class of good fitting functions determine the best fitting function by statistical test. The “good fitting functions” and “best fitting function” have the following properties: 1) good fitting function - results in (a1 physically meaningful parameters, (b) a visually satisfactory fit
and (c) a reduced x2 that is approximately unity. (Conditions (b) and (c) are highly correlated). 2) best fitting function - this is the good fitting function which has the highest probabili~ of being the true fitting ~~t~~.
The parameters of the true fitting function have their lrrce r_&ues. The parameters of the best fitting function have their best u&es. The parameters’ best values are the best estimates of their true values. From the analysis presented herein it will become evident that the conventional curve fitting approach does not necessarily provide the best fitting function. We shall also see that in the process of dete~ining the best fit, accidental degeneracy is necessarily removed. Of course any multiplicities associated with the crystallographic sites are not removed. The procedure described in this paper is tantamount to treating the number of subspectra, N, as an additional adjustable parameter. It is to be determined, along with the parameters of the resonance lines, by means of curve fitting and statistical testing. From the correct number of subspectra one can determine, 1) the correct number of nuclear site environments (but not necessarily the number of sites) and 2) the hyperfine interactions associated with each of these environments. The purpose of this paper is to describe a curve fitting procedure that secures the best fitting function and apply this to the room temperature 57Fe Miissbauer spectrum of HoBa,(Cu,,,,Fe,,05)30~.~~. In so doing a general procedure for obtaining the best fitting function of a complex Mossbauer spectrum is demonstrated. We also obtain information regarding the na. . ture of the Fe sites m HoBa,(Cu,,, %.&~7.0~.
2. Anaiysis of the s7Fe ~~ss~auer spectrum 2.1. Cwve fitting strategy The analysis of our 57Fe Mossbauer spectrum was accomplished by means of nonlinear least squares fitting {7]. As dictated by the spectrum itself all of the subspectra are doublets. The line depths as well as line widths were constrained to be equal. All line shapes were approximated with the Lorentzian, A room temperature magic angle spectrum was also measured and the effect of texture was found to be negligible [2]. Therefore, these assumptions regarding the resonance line characteristics are reasonable. The fitting ~uctiou can be written,
M.J. Durst et al. /Nucl.
Instr. and Meth. in Phys. Res. B 93 (1994) 521-529
where Yhi is the depth of the spectrum at velocity, u, yi is the depth of the ith doublet, and f(u; k, 1, m) is the parabolic baseline. Since we approximate line shapes with Lorentzians we have, yi(U; e,i, 1 la
54)
_i[[(;(“-(61-;)))2+l]-1
523
smaller reduced x2, and hence would be an improved fit. The question is; is this improved fit significant? The answer can be provided by a statistical test as described herein. If the answer is affirmative we would then construct a third TFF in which the previously selected doublet is replaced by three superposed doublets of equal depths and line widths giving, N-l
i#j
+[(;(“-(8;+;))~+l]-1]. (2)
Four adjustable parameters are required for the case of line widths and depths constrained to be equal. Therefore, we specify (arbitrarily) the I?,, as, @ii= Si (doublet splitting), OZi= 6, (isomer shift), OSi= c (line width), @,,= li (maximum resonance absorption), 19s)= the union of all Oai; i.e., f#) = (Oai, 1 I (YI 4, 1
f(u; k, 1, m) = k + lv + mu2 N = the number of doublets, Curve fitting begins with the first TFF, Y$), Yhi) = EE1yi + f(u; k, 1, m). The starting values of the parameters, Oai, are determined by inspection. The fitted first TFF, rYhl), is then examined for doublets with significant excess line broadening, i.e., broadening in excess of that which can be attributed to absorber thickness. We presume that a significant component of this broadening is due to accidental degeneracy (nonhomogeneous broadening). If the data is good enough, accidental degeneracy can be removed by curve fitting. Any significant excess line broadening remaining after this process must then be homogeneous broadening. (Disorder in distant neighbor shells might produce line separations so small as to be, for a given set of data, unseparable by curve fitting. This residual nonhomogeneous broadening would then be a small component of the homogeneous broadening.) Now we suppose that rY$‘) consists of N doublets some or all of which have significant excess line broadening. We first select the doublet with the broadest lines and replace it with two superposed doublets of equal depths and widths. The result is a second TFF, N-l Y$?l=
C
Yi+Yj,+YjZ+f(U;k,f,m),
and proceed as before. In this process the starting values of the parameters, other than those of the added doublets, are the values resulting from the previous fit. This procedure is continued until the statistical test indicates that the last TFF, is not a significant improvement over the next-to-last TFF. After the jth doublet has been analyzed in this manner we apply the same procedure to the other significantly broadened doublets. The “final fitting function” (FFF) consists of these as well as any unbroadened original doublets. Finally, we do a global fit of the FFF in which the parameters of all of the doublets are free. The result is the best fitting function in which all adjustable parameters have their best values. These are also the quoted values. The global fit of the FFF represents pure quadrupole doublets and these refer to actual Fe sites. The FFF can be represented as,
e$), k, 1, m) = f
Y,(u,
i-1 +f(u;
i
yij( O; e,,,,
1 I (YI 4)
jz1
k,
1, m),
(3)
where Yij(
UT
emi,, 1 I (Y54)
=~~j~[(~(~-(~ij-~))i2tl] +[(+(&-;j+~)))*+l]-l\. yij(v, e,,,, 1 I (YI 4) is the depth of the embedded in the i-th doublet of the first the set of parameters analogous to Bui. f@ ofall IX, i.e.,8~)={e,,,,lI(yI4,1IiIN,lljI Ji). Ji = number of doublets embedded in blet. The number of doublets in the Mijssbauer
(4) j-th doublet TFF. Oati, is is the union the ith douspectrum is
i#j
where j is the index of the selected doublet. We then fit this function using parameter starting values obtained from rYA1)and from the two doublets just constructed. The resulting fit, rYdyl, would necessarily have a smaller residual sum of squares (RSS), a
M=
f.Ii. i=l
In some cases the acceptance or rejection of additional doublets is made by the least squares fitting process itself. For example, an added doublet may
524
M.J. Durst et al. /‘Nucl. Ins@. and Metk in Fhys. Res. B 93 f1994 521-529
0.96
Fig. 1. 57Fe Miissbauer spectrum of HoBaz(Cu,,Fe,os)307,, at RT and normal incidence fitted to 3 doublets. The source was “Co: Pd at RT. The residuals plot runs between +5000 counts; the curve was determined by numerical averaging.
monotonically decrease in size with each iteration of the fitting process until it becomes comparable to the base Iine noise. Or, the final value of one or more of the parameters may be unreasonable. The latter case would include any negative values irrespective of whether the parameters do or do not appear only as the square In these two cases the added doublet should be rejected. Such pathological behavior is frequently symptomatic of an over determined least squares fitting process and indicates that the number of subspectra should be decreased. 2.2. Detailed fitting
of the Miissbauer
The “Fe Miissbauer spectrum Fe0.0S~~~7.Q~~at room temperature
spectrum
of HoBa,(Cu,,, and normal inci-
dence is shown in Fig. 1. It is clear, by inspection, that the first TFF for this spectrum should consist of the sum of three doublets, D-l, D-2, and R-3 131.A least squares fit of this TFF appears in Fig. 1. The parameters resulting from this fit are given in Table 1. These are consistent with previously published three-doublet fits of REBa,(Cu,_,Fe,),O,_, spectra @-lOI. However, this TFF does not satisfy the conditions for a good fitting function. This is because (1) the fit is not visually satisfactory, i.e. there are several regions where the curve deviates significantly from the experimental points, and (2) the reduced x2 is significantly larger than unity. Also the resonance lines of D-l and D-2 are much broader than can be accounted for by the absorber thickness [ll. We therefore proceeded to analyze the spectrum as prescribed above.
Table I Results of curve fitting with tbe first trial fitting function Doublet
Doublet splitting, S
Isomer shift, 6 [mm/s] b
Line width, r [mm/s1
tmm/sl a 1 2 3
1.944 1.063 0.265
0.0485 0.~89 0.2370
0.366 0.491 0.305
Maximum absorption,
Relative intensity,
E[%Ib
R [%I
3.42 4.65 0.923 Total: 8.99
x2 = 2.21 RSS = 5.48 X 10” (residual sum of squares), a to convert (mm/s) to MHz multiply by 11.624 MHz/(mm/s). b uncorrected for background (background correction: 30 + 3%). ’ c/ch - counts/channel.
32.8 56.X 7‘38
Parabolic baseline k I
k,‘chl = 2804193
k/W,’
;;c/ch),
(mm/s11
(mm./s)2l
- 1I .97097
- 891.6922
M.J. Durst et al. / Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 521-529 2.2.1. Analysis of D-2
We first fitted the broadest-line doublet, D-2, with two superposed doublets. We next tried three doublets. In this case one doublet always decreased monotonically with each iteration of the least squares fitting process while the other two increased and became virtually identical to the original two doublets. We repeated this computation several times with different parameter starting values. Each attempt exhibited the same behavior while the reduced x2 remained close to unity. We concluded that the third doublet presented an over determined problem to the least squares fitting computation. Therefore, two doublets were the best fit to doublet D-2. 2.2.2. Analysis of D-l We tried fitting 2, 3, 4, and 5 doublets to the next broadest-line doublet, D-l. In each case the parame-
525
ters stabilized at reasonable values and the reduced x2 steadily decreased. The best value for the number of doublets in D-l was obtained by the statistical test described below. 2.2.3. Analysis of D-3 We attempted to fit two doublets into the remaining doublet, D-3. This resulted in one doublet shrinking and eventually becoming unphysically small while the other grew until it became virtually identical to D-3. Choosing a range of starting values for the parameters did not alter the outcome. We concluded that D-3 is a pure quadrupole doublet. In the foregoing analysis the curve fitting was stopped in two instances (D-2 and D-3) by the fitting process itself. Only the curve fitting of D-l was stopped by the statistical test. As a result of curve fitting we obtained a total of five “good” FFFs consisting of 3, 5,
Table 2 Curve fitting results Total fitting function
Doublet
Doublet splitting, S [mm/s] a
Isomer shift 6 [mm/s]
Line width l[mm/s]
Doublet maximum absorption, E [%I b
Relative intensity R [%]
((EFG), [v/,Q,21
I’
Parabolic baseline parameters k 1 m Kc/ch)/ lc/chl d Kc/ch)/
(mm/s)1
(mm/s)*1
Five doublet x* = 1.35 RSS = 3.32 x lo9
1 2 3 4 5
2.068 1.827 1.219 0.933 0.260
0.0551 0.0403 0.0085 0.0083 0.2079
0.274 0.325 0.428 0.379 0.350
1.84 2.02 2.40 2.67 0.96
14.3 18.6 29.1 28.6 9.5
241.9 213.6 142.9 109.3 30.5
2798443
- 157.9907
- 119.6289
Six doublet x2 = 1.25 RSS = 3.06 x lo9
1 2 3 4 5 6
2.138 1.921 1.659 1.183 0.903 0.261
0.0552 0.0467 0.0297 0.0088 0.0077 0.2052
0.256 0.264 0.297 0.393 0.366 0.362
1.26 2.14 0.98 2.69 2.36 0.96
9.4 16.4 8.4 30.7 25.1 10.1
250.1 224.8 194.1 138.5 105.8 30.5
2796594
- 188.7096
118.3776
Seven doublet x2 = 1.23 RSS = 3.00 x lo9
1 2 3 4 5 6 7
2.138 1.929 1.688 1.326 1.088 0.859 0.266
0.0545 0.0465 0.0317 0.0069 0.0101 0.0055 0.2007
0.263 0.255 0.299 0.339 0.323 0.363 0.380
1.31 1.99 1.13 1.21 2.16 2.03 0.98
10.1 14.9 9.9 12.0 20.5 21.6 10.9
249.6 255.9 198.1 155.1 128.5 100.9 31.4
2795 835
- 188.1470
203.0215
Eight doublet x2 = 1.22 RSS = 2.96 x lo9
1 2 3 4 5 6 7 8
2.139 1.929 1.686 1.307 1.069 0.840 0.269 0.776
0.0546 0.0467 0.0319 0.0063 0.0107 0.0030 0.2009 0.0781
0.262 0.255 0.302 0.350 0.332 0.355 0.386 0.007
1.30 1.99 1.13 1.33 2.31 1.71 0.98 0.16
10.0 14.9 10.0 13.7 22.5 17.8 11.1 0.03
250.6 226.4 197.3 153.2 125.1 98.7 31.8 91.1
2795 903
- 185.5649
195.2873
a b ’ d
to convert mm/s to MHz multiply by 11.624 MHz/(mm/s). uncorrected for background (background correction: 30f 3%). Total EFG magnitude (see Ref. [4]). c/ch - counts/channel.
526
0.98
~_“---_
---
-1
-2
0
2
1
3
Y (mm/secf
Fig. 2. “Fe M~ssbauer spectrum of HoBaz(Cu,,,Fe0,053307,02 at RT and normal incidence fitted to 7 doublets. The source was “Co: Fd at RT, The residuals plot runs between ~5000 counts; the curve was determined by numerical averaging.
6, 7, and 8 doublets. The parameters for these fits are given in Tables 1 and 2. The fitted FFF for the 7 doublet fit is shown in Fig. 2.
We now proceed to answer the question: which of these five “good” fitting functions is the “best” fitting functions We used nonlinear Ieast squares fitting to arrive at parameter values for the fitting functians with M doublets. That is, we minimized the residual sum of squares
FE-1
doublets M’
over all possible choices of the vector of parameters
8$), where y,, is the number of counts in the nth channel and “‘m” is the m~imum number of channels, In our spectrum M = 923. For each possible number of doublets M, we have a best residual sum of squares, RSSM .
If we choose an incrementally
larger number
of
- RSSw),‘v, -
RSS,J/‘vr
Here V~is the number of degrees of freedom of the “full” (i.e., larger) model, which is the number of channels minus the number of parameters fitted. From Eq. (1) this is m - (3 + 4 x IM’). V~ is the so-called “extra” degrees of freedom; if vf is as above and vp is
Table 3 Goodness of fit results Doublets
RSS
3 5 6 7 8
5.48 x 3.33 x 3.06X 3.00x 2.97 x
10” 10” 10” 10” log
Parameters
F ratio
dfl
df2
Pr
15 23 27 31 35
72.87 19.33 4.28 2.60
8 4 4 4
900 8% 892 888
2.4~30-~~ 2.9x10-‘5 0.002 0.035
(Fdfl,dfZ'F)
M.J. Durst et al. / Nucl. In&. and Meth. in Phys. Res. B 93 (1994) 521-529
defined similarly for the “partial” (i.e., smaller) model, then v, = vP - vf = 4(M’ -M). The interpretation of this F ratio is as follows. If we were really observing a spectrum with only M doublets, but we overfitted using a function with M’ doublets, then the ratio given above would have the F distribution with v, and vf degrees of freedom. An F distribution is derived as the distribution of the ratio of two independent variables, both of which have the chisquared distribution. It is commonly used in statistics to compare two models, particularly when one model is a subset of the other. The degrees of freedom defining the specific F distribution are the numerator and denominator degrees of freedom of the chi-squared variables [12]. If, in our actual fitting, we obtain an F ratio so large as to be very unlikely under this distribution, then we have obtained strong evidence against there being only M doublets present, even if the fit with M doublets was satisfactory. Conversely, if our actual fitting obtains an F ratio which is not very unlikely under the given distribution, then we must accept that the evidence for more than M doublets is not compelling. We made fits with FFFs consisting of 3, 5, 6, 7 and 8 doublets; Table 3 gives the F statistics comparing each model with the next larger one. The column headed “Doublets” gives the number of doublets in the fitting function, “RSS” gives the residual sum of squares for that fit, “Fobs” gives the observed F ratio, “dfl” and “df2” give the numerator and denominator (respectively) degrees of freedom of the F ratio, and the last column gives the tail probability associated with the observed F value. This value, Rr( Fdrl ,df2> Foss1 Y tells us the probability of observing an F ratio as large or larger under what statisticians call the “null hypoth-
527
esis”: that the number of doublets used is an overfit, with the true number of doublets being the next smallest number in the table. The improvement in fit is unquestionable as we move from 3 to 5 doublets, as when we move from 5 to 6 or from 6 to 7. In moving from 7 to 8 doublets, we observe a tail probability of 0.035, which is low (below the frequently used 0.05 threshold for significance) but not unquestionable. To amplify this slightly, a recent conference of Mossbauer spectroscopists (ICAME-93) attracted 365 attendees. If the true number of doublets present were 7, and if a copy of our sample were given to each of 365 spectroscopists, we would expect that by chance 12 of them would obtain an F ratio as large as the one we observed. Furthermore, on examining the parameter values for the &doublet fit, we see that doublet 8 has a line width of 0.007 mm/s, which is unphysically small. Our estimation process always started with physically reasonable values, but never lowered the residual of squares significantly without resulting in physically unreasonable parameters. We are satisfied by the above that there is no compelling evidence for more than 7 doublets. No fit with 8 doublets that was physically reasonable produced any significant decrease in residual sum of squares over the 7-doublet fit. By contrast, the progression from 3 up to 7 doublets was accompanied by unquestionable improvement at each stage. We therefore conclude that the best fitting function is the one with 7 doublets. 2.4. Correlation of 6, S and r For the 7-doublet fit, visual examination of the 7 (S, S, r> points in 3-space reveals that 6 of the 7 points
Fig. 3. A plot of S, 6, and r for the 7 doublets clustered about the best fitting straight line.
528
M.J. Durst et al. /Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 521-529
are nearly collinear. We can find the best fitting straight line in space to these 6 points by calculating their “principal components.” The principal components are the eigenvectors of the 3 X 3 covariance matrix of the 6 points. The “first principal component,” which is the eigenvector corresponding to the largest eigenvalue, lies in the direction of the best-fitting line, and the line passes through the mean of the 6 points in 3-space. The equation of this line is, in units of mm/s,
Therefore we have the sum rule: R
s
CM,= CDS. r=l
s=l
ZZf=lDs equals the number of accidental degeneracies in the Mossbauer spectrum. The sum rule can be stated: the sum of the Mossbauer site multiplicities over all Mijssbauer sites equals the number of accidental degeneracies in the Mossbauer spectrum.
S = 24.206 + 0.878, r = - 1.956 + 0.357, and this line explains 99.9% of the variance. A view demonstrating the wide extent of the 7 (S, S, r> points in 3-space and how tightly clustered 6 of them are about their best fitting straight line is shown in Fig. 3. This analysis suggests that 6 doublets are associated with similar nuclear site environments while the nuclear site environment associated with the 7th doublet differs significantly.
3. A sum rule When Fe is substituted into REBa,(Cu,_,Fe,), O,_, one would expect Fe to occupy to some extent all of the cation sites. However experimental evidence has shown that the Fe dopant occupies predominantly the Cu(l) sites in the basal plane. In addition to the Cu(l1 sites, there exists a number of Cu(l) subsites due to oxygen disorder [3]. Therefore, the potential number of nonequivalent basal plane sites available to an Fe substituent can be greater than the number of Cu(1) sites. The one-to-one correspondence between a Mossbauer nuclear site environment and a crystallographic site environment can be broken by site multiplicity due to mixed Fe valence or accidental degeneracy. These two mechanisms are not independent; they must obey a sum rule. Let us define M, as the multiplicity of the rth Fe site and D, as the degeneracy of the sth subspectrum. If we define the multiplicity of a site with no mixed valence to be unity and the degeneracy of a pure hf spectrum to be unity then, M, r 1, and D, 2 1. The number of nuclear site environments, N,, can be written either as:
or N,=
?Q, s=l
where: R = number of Mossbauer sites, S = number of subspectra in the best fitting function.
4. Valence states and numbers of Fe sites We have determined that the most likely number of Fe sites in room temperature HoBa,(Cuo,ssFeo,os)307,02 is 7. If there were no mixed Fe valence, the number of Fe crystallographic sites would also be 7. Based on magnetic susceptibility measurements made on Fe-substituted YBa,Cu,O,_, compounds the Fe is in a high spin state. If the same is true for our Fe-substituted Ho-based compound then this result combined with the isomer shift and quadrupole splitting results, lead to the conclusion that Fe is present in the Ho-based compound only in the high spin + 3 and + 4 states [2]. The quadrupole splitting and isomer shift results strongly suggest that one of the 7 doublets is high spin + 3 and the remaining 6 are high spin + 4 [2]. Therefore, if mixed valence were not operative we would have a total of 7 nonequivalent Fe sites, 6 sites are high spin +4 and the 7-th site is high spin +3.
5. Conclusions
A method for obtaining the best fit to a complex Miissbauer spectrum has been demonstrated. The application of this method to the room temperature s7Fe Mijssbauer spectrum of HoBa,(Cuo,ssFe,.,,),O,,,, results in a best fitting function of 7 doublets. The possibility of 8 or more or 6 or less doublets can be rejected. This is significantly larger than the previously reported number (3 or 4) of subspectra associated with REBa,(Cu, _xFe,),O, compounds. Assuming no mixed valence this implies 7 Fe sites. Seven Fe sites is a significantly greater number than the previously reported number of Fe sites (l-4) in the REBa,(Cu,_, Fe,130, compounds. This discrepancy may stem from the fact that the Mossbauer spectrum depends critically on the oxygen stoichiometry of the sample [3]. For 6 of the 7 doublets, S, S, and r are strongly correlated. The valence state of these 6 Fe sites is high spin f4; the seventh Fe site is high spin +3. A sum rule was derived which relates the sum of the Miissbauer site multiplicities to the sum of the accidental degeneracies in the Miissbauer spectrum.
M.J. Durst et al. /Nucl. instr. and Meth. in P&s. Res. B 93 (1994) 521-529
We acknowledge many enlightening discussion with, R.D. Taylor, Los Alamos National Laboratory, M.G. Smith, Los Alamos National Laboratory and University of Texas, Austin, M.P. Pasternak, Los Alamos National Laboratory and Tel Aviv University, J.J. Spijkerman, Ranger Scientific, Inc., Prof. L. May, Catholic University of America, and S.P. Verrili, Forest Products Laboratory. We also acknowledge the support of M. Ross and M.J. Fluss, LLNL, and the hospitaiity of Prof. J.W. Morris Jr., Lawrence Berkeley Laboratory. This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract no. W-7405-Eng-48.
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[Z] C.E. Violet, R.G. Bedford, N.W, Winter, G.S. Smith, Z. Mei, M.J. Durst and J.K. Wu, Physica C 207 (1993) 347. [3] C.E. Violet, R.G. Bedford, P.A. Hahn, N.W. Winter and Z. Mei, Physica C 162-164 (1989) 1291. [4] D.G. Rancourt, Nucl. Instr. and Meth. B 44 (1989) 199. [5] N.W. Winter, C.I. Merzbacher and C.E. Violet, Appl. Spec. Rev. 28 (1993) 123. [6] See for example, N.N. Greenwood, Mossbauer Spectroscopy (Chapman and Hall, 1971). [7] G. Longworth, in: Miissbauer Spectroscopy Applied to Inorganic Chemistry, vol. I, ed. G.J. Long (Plenum, 19841 p. 43. IS] C.E. Violet, P.A. Hahn, Z. Mei, T. Datta, C. Almasan and J. Bstrada, Extended Abstracts, High-Temperature Superconductors II (Materials Research Society, 198X) p. 335. 191 C.E. Violet, P.A. Hahn, T. Datta, C. Almasan, and J. Estrada, J. Appt. Phys. 64 (1988) 5911. [lo] P. Boolchand and D. McDaniel, Studies on High-Temperature Superconductors, vol. 4, ed. A.V. Narlikar (Nova Scientific, 19901 p. 143. [ll] D.M. Bates and D.G. Watts, Nonlinear Regression Analysis and Its Applications (Wiley, 1988) 53.10.1. [12] N.R. Draper and H. Smith, Applied Regression Analysis, 2nd ed. (Wiley, 1981).