Fitting techniques for K X-ray multiplets in Ge(Li) spectra

Nuclear Instruments and Methods 174 (1980) 549-553 © North-Holland Publishing Company

FITTING TECHNIQUES FOR K X-RAY MULTIPLETS IN Ge(Li) SPECTRA C.W. SCHULTE, H.H. JORCH and J.L. CAMPBELL

Department of Physics, Universityof Guelph, Guelph, Ontarto, CanadaN1G 2W1 Received 21 January 1980

Two approaches are used to fit Kc~X-ray doublets at high Z recorded in Ge(L1) spectra They are based respectwely on the use of analytic approximations to the Volgt integral and on dtrect numerical convolution of the gamma-ray response functmn with a Lorentzmn Excellent fits are obtained m both cases Atomic physics applications of the techmque are indicated.

1. Introduction

The main component G(x) is a Gausslan centred at Xo viz

There IS now an extensive literature on the analytic representation of gamma ray peaks in spectra from semiconductor detectors, most of it published in this journal. Since gamma ray natural linewidths are negligible compared with the typical Ge(Li) detector resolution, the observed hneshapes are a direct measure of the resolution function. Much less effort has been devoted to the fitting of X-ray peaks, where the natural width is not negligible and the observed peak is a convolution of the natural Lorentzlan hneshape with detector resolution. In this paper we compare various means of fitting the doublets observed In Ge(L1) or Ge(I) spectra due to the Ka X-rays of heavy atoms. Part o f our motivation derives from attempts to measure intrinsic X-ray hnewldths; .another aim is to achieve accurate separation of the K a l and Ka2 components for studies of subsequent cascades via coincidence techniques. A third application relates to efficiency curves for X-ray detectors ( 5 - 1 0 0 keV), which are usually determined using various gamma and X-ray emitting radionuclldes. Derivation of gamma peak areas through analytic fitting is straight forward, but many workers derive X-ray peak areas without attention to their different hneshapes; this can introduce several percent scatter into efficiency data generally cited to be accurate to 1 - 3 % . In most detectors excellent fits to single gamma ray peaks can be obtained [ 1 - 3 ] using an analytic function introduced by Phillips and Marlowe [1]. This is

G(x) -- HG exp [--(x - xo)2/2o 2]

F(x) = G(x) + S(x) + D(x) + B(x).

(2)

and S(x) is a corresponding step function

I1-

I-

The low energy distortion D(x) consists o f long-term and short-term exponential tails convoluted with the Gausslan to yield DL(X)= ~HDL exp(X - Xo~ e r r c[x ~ ° - Xo +

o ) (4)

\I3L ] and a similar expression for Ds(x). Thus the distortion adds 4 parameters (HDL, t3L, HDS,/3s) to the 4 parameters o f the Gaussian and the step. The linear background B(x) brings the total number of parameters to describe one gamma ray peak to 10. Many alternatives for D(x) exist but those used here appear to be among the most successful [2,3]. Two complications arise when one switches attention to a K X-ray doublet. The trivial one is the increased number of parameters to be varied in the fit. We have shown [2] that for gamma peaks the dlstorhon and step parameters vary only slowly with energy if expressed as multiples of the Gaussian parameters VlZ Hs/HG, HDL/Hc, (3L/O etc. So for a multiplet, at most 3 new parameters must be introduced per peak, viz those o f the Gaussian. The 5 parameters of D and S are used as multipliers to generate successively from each set of Gausslan parameters the corresponding tail function. A doublet fit then needs at most 13 parameters. The non-trivial comphcatlon is the need to fold

(1) 549

C.ht Schulte et al/ Fitting techniques

550

the peak components of eq. (1) with the intrinsic Xray lineshape. Approaches to this are described below.

2. Methods of convolution To illustrate the problem at hand, fig. 1 shows the Ka X-ray doublet of platinum (Z = 78) recorded with a typical, small planar Ge(L1) detector. The nonGaussian nature of the two peaks is clearly seen in the Lorentzian flaring at fight and left. The intrinsic X-ray lineshape is the Lorentzian P/2rr

L(x) = (x - xo) 2 + (F/2) =

(5)

which is here normalized to unit area. A numerical convolution of L(x) with the non-background components of F(x) is in principle straightforward; in practice a least-squares fit based on this approach is very expensive since the convolution procedure is repeated many times. An approximate approach is based on the analytic convolution of a Lorentzian L(x') by a Gaussian

G(x, x') = H o exp ( - (x 2o - _x')a'~ 2 ]

(6)

which yields

K(x)= f

H o P exp [ - ( x - x')=/2o =]

2.

v(r-

(7)

displayed fits to the uranmm K~ X-ray doublet. (The context was isotopic analysis of plutonium-bearing materials via Ge(Li) spectroscopy). Although no chisquared data were given, the fit appeared visually good. While the Wilkinson-Gunnink approximation is simple and fast, it is of limited accuracy. Several more accurate computational procedures e.g. refs. [6,7] have been published in the astrophysical literature; here we use one due to Armstrong [7] and accurate to 1 - 2 parts in 106. As noted, a possible flaw here is the neglect of explicit convolution of D and S. Roberts et al. [8] dealt with this by substituting a Gaussian for DL(X) + Ds(x ) in eq. (1); then in the application to X-rays (the context was K--mesic X-ray studies) they replaced the Gausslan by the corresponding Voigt function. This afforded good fits (X~ = 2.24) to mesic X-ray lines where the Lorentzian width P was roughly twice the fwhm of the detector resolution. The case of ordinary K X-rays is much more difficult since P is much smaller than the detector resolution (at Z - 8 0 , l P ~ 5 0 eV, f w h m = 2 . 3 5 0 370 eV). Moreover in our experience replacement of D by a subsidiary Gaussian leads in the gamma ray case to infenor fits. In the following we compare the results of (a) The full convolution approach where the nonbackground components of eq. (1) are convoluted numerically with a Lorentzian; (b) the approximate approach where in eq. (1) G(x) is replaced by K(o, A).

--oo

Setting A = I'/2oX/2, then v = ( x - Xo)/Ox/2 and t = (x - x')]ox/2, this reduces to K(x) = K(v, A) where e -t2 dt

K(o'A)=HG(A) /

( v - t)2 +A 2"

(8)

There are various piece-wise series approximations available for evaluation of the Voigt function K(o, A). If G(x) in eq. (1) is replaced by K(v, A), then the major part of the fitting function has been convoluted. The neglect of the convolution of D and S will be most serious in detectors exhibiting strong lowenergy tailing. However one might expect the Phillips-Marlowe tail function DL + Ds, whose form is very flexible, to modify its parameters during the fit in order to reflect the physical convolution of the low-energy tail. This approach was taken recently by Gunnink [4] who modified an approximation of Wilkinson [5] and

3. Experimental The Ge(Li) detector employed was an Ortec LEPS of dimensions 5 mm X 80 mm 2 and energy resolution 500 eV at 122 keV. It was used with a TC205A amplifier (time constant 3/as) and an ND2200 pulse height analyser of conversion gain 4096. A slow-risetime rejection system [2] was incorporated to reduce low energy tailing effects, and a pileup inspector (Ortec Model 404A) minimized distortion at high energies. Digital gain stabilization, adjusted for minimum peak broadening, was employed. Pulse height spectra were recorded at rates of ~500 cps until 5 10 million counts were accumulated m the peaks of interest. We have fitted K~ doublets at various energies (Z ranging from 70 to 82) and here we simply discuss a representative case, viz Z = 78 where the Pt X-rays

C W. Schulte et al. / Ftttmg techmques

551

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are derived from a 19SAu source of a few microcunes activity. Since the Ka2 and Kal energies are 65.12 and 66.83 keV, we also fitted the 66.7 keV gamma ray of 17aTm to examine the validity of the basic response function of eqs. (1)-(4). We have already reported [2] good fits over the range 4 6 - 8 4 keV for this response function. The sources were prepared by droplet evaporation on 0.05 mm beryllium foils.

4. Least squares fits to data

The pulse height spectra had dispersions of ~18 eV/channel. Our fitting techniques, based on the Marquardt algorithm [9], have been described in detail elsewhere [2]. For the 66.7 keV gamma ray a 300 channel region was fitted and for the Pt Ks doublet a region of 480 channels. The reduced chisquared for the fit o f e q . (1) to the 66.7 keV line was ×rz = 1.26, an excellent value judged by the criteria prevaihng in this field. We therefore regard 1.26 rather than 1.0 as an ultimate value to be aimed at for the doublet, but given its greater length a rather larger Xr2 e.g. 1.5 would be satisfactory. In doublet fits, the same Lorentzian parameter [' was used for both the Ka~ and Ka2 X-rays. This is justified by the fact that P(Kal,2) = P(K) + P(L3,2)

(9)

where P(K) and P(Li) are the natural atomic level

widths. P(K) is responsible for ~90% of either X-ray width and so the ~0.5 eV difference between P(L2) and F(L3) is negligible compared with the ~ 5 0 eV value of P(K). The 480-channel doublet spectrum was fitted by

F(x) = B ( x ) + PI (x) + P2 (x)

(1 O)

where P,(x) = Ki(A , o) + Si(x ) + Di(x ) i = 1,2. This is the conventional doublet fit with Gaussians replaced by Voigt functions. Gunnink's approximation [4] gave a poor fit, principally because of a discontinuity in K(A, o) at the junction of the two piece-wise approximations. With Armstrong's approximation, an excellent fit was obtained, with ×rz = 1.39; the residuals are shown in fig. 2(a). The preceding work was done with the Fortran program HALFIT. For the direct numerical convolution an APL version of HALFIT was written since this permits elegant and efficient handling of the necessary matrix arithmetic. The function fitted was

F(x) =B(x) + L * [Ql(x) + Q2(x)]

(11)

where

Qi(x)=Gi(x)+Si(x)+D,(x)

z = 1, 2 ,

and the asterisk denotes numerical convolution. Since the Lorentzian falls very slowly towards zero, it was necessary to disperse each channel content of Qi(x) over a region at least 1.5 times the original spectrum length. However truncation effects still worsened the

C. W. Sehulte et al. / Fittmg techniques

552

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fit at the left and right extremities. These were dealt with by extending the fitted regmn (by 50 channels at the left and 20 at the right), repeating the fit and then calculating X~ for only the basic 480 channels. An excellent fit resulted, with Xr2 = 1.66; the residue is shown in fig. 2(b).

4. Discussion Both approaches used here provide good fits and there appears little reason to favour one rather than the other. The l'(Ka,) values are compared with predictions of relativistic H a r t r e e - F o c k theory [10] in table 1; since they bracket the prediction and are both close to it they are again equally acceptable. This conclusion is supported by work on several Ks doublets in the high Z region. In either method, a very slight improvement is achievable by introducing the forbidden Ka3 X-ray. This becomes rapidly more important as Z increases above 78.

The numerical convolution approach is extremely expenmve in terms of computer time. This is because with the 14 free parameters needed to describe the doublet, 29 numerical convolutions are performed for each ×r2 test in the Marquardt algorithm. One convolution occurs in calculating the function and the remainder occur in calculating its 14 numerical derivatives with respect to parameters. To minimize CPU time, excellent starting parameters must be provided, and of course these are obtainable from the Voigt function fit. With these parameters three or four loops were needed (each 150 c p u s on an Amdahl V5 with Sharp APL) to optimize the Xr2.

Table 1 Measured and calculated Ks X-ray hnewldths for platinum r (eV) Relativistic Hartree-Fock [ 10] Voigt fit Convolution fit

55.0 61.7 5 2.1

C W. Schulte et al. / Fitting techmques

In general then the present methods provide excellent fits to X-ray multiplets, affording a great improvement over the generally adopted use of Gaussians to represent the peaks. Finally we comment on the specific application which was the primary motivation for this work, viz the measurement of the Coster-Kronig yield f2a by X-ray coincidence techniques. By gating successively on the Ka 1 and Kct2 X-rays in a Ge(L1) detector, we can record in a coincident Si(L1) detector the subsequent L X-ray spectra due to filling of La and L2 vacancies respectively. The L2 case shows La X-rays in the spectrum due to Coster-Kronig transfer of L2 vacancies to the L3 shell with probability f~3 (typically 0.15). But contamination of the Ka2 gate by Kcq X-rays yields L3 X-rays with probability 1.0 relative to the 0.15 figure. Thus measurements of f2a are particularly susceptible to error; any error made in estimating the Kt~l tail under Kct2 is magnified by a factor 1/f23 "" 7 in its effect on the derived ]'23 value. This accounts for the great scatter in measured values of this widely-studied atomic phenomenon [11]. In our two fits to the Pt Ka doublet we placed a gate on Ka2 spanning its fwhm and then determined the Kal intensity contribution. In the Voight case it was 1.98% and in the convolution case 2.06%. This excellent agreement suggests that our uncertainty in estimating the gate contamination would be approximately +0.04%, yielding an uncertainty of ~+0.3% in f23. Typical quoted errors [11] in f23 are +-10%, reflecting the graphical methods used hitherto for Kal tail interpolation; the present techniques certainly reduce these to less than 1%.

5. Conclusions We have demonstrated two approaches to fitting K X-ray multiplets in Ge(I) or Ge(Li) spectra; both afford excellent fits. Either means of incorporating

553

Lorentzian broadening is preferable to the widely encountered representation of X-ray peaks by Gaussians. These methods enable determination of intrinsic linewidths F(K~) with a preosion comparable to or better than the classical crystal spectroscopy methods. They also provide very precise relative intensities for the major K X-ray lines, and enable study of the 1-forbidden Ka3 line the relative intensity of which is disputed [12]. Exploitation of the methods to obtain these quantities over a range of atomic number will be reported elswhere. Measurements of the f23 Coster-Kronig rate are also in progress. This work was supported by the Ontario Ministry of Colleges and Universities and by the Natural Sciences and Engineering Research Council of Canada. We thank Dr. J. Law for useful discussions.

References [1] G.W. Phillips and K.W. Marlow, Nucl. Instr. and Meth. 137 (1976) 525. [2] J.L. Campbell and H.H. Jorch, Nucl. Instr. and Meth. 159 (1979) 163. [3] R.G. Helmer, private communication. [4] R. Gunnlnk, Nucl. Instr. and Meth. 143 (1977) 145. [5] D.L. Wilkinson, Nucl. Instr. and Meth. 95 (1971) 259. [6] S.R. Drayson, J. Quant. Spectrosc. Radiat. Trans. 16 (1976) 611. [7] B.H. Armstrong, J. Quant. Spectr. Radiat. Trans. 7 (1976) 61. [8] B.L. Roberts, R.A.J. Riddle and G.T.A. Squter, Nucl. Instr. and Meth. 130 (1975) 559. [9] D.W. Marquardt, J. Soc. Ind. Appl. Math. 11 (1963) 431. [10] M.O. Krause and J.H. Oliver, J. Phys. Chem. Ref. Data 8 (1979) 329. [11] W. Bambynek, B. Crasemann, R.W. Fink, H.U. Freund, H. Mark, C.D. Swift, R.E. Price and P.V. Rao, Rev. Mod. Phys. 44 (1972) 716. [12] J.H. Scofield, in. Atomic Inner-Shell Processes, ed. B. Crasemann (Academic, New York, 1975) p. 265.