A rapid method for the measurement of small peak shifts

Nuclear Instruments and Methods 185 (1981) 309-312 North-Holland Publishing Company

309

A RAPID METHOD FOR THE MEASUREMENT OF SMALL PEAK SHIFTS Gerhard BRUNNER

Zentralinstitut ffir Isotopen- und Strahlenforschungder AdW der DDR, Permoserstrasse15, 7050 Leipzig, D.D.R. Received 5 August 1980 and in revise4 form 17 November 1980

For a peak and a nearby reference peak the ratio of the ordinate values approximately gives a straight line as a function of the abscissa values in the neighbourhood of the peak maxima. From its slope and its absolute term the peak shift (and the amplitude ratio) can be easily evaluated. Shifts of less than 10-2 of the half-width can be found for nearly Gaussian shapes if the uncertainty of the individual amplitudes (due to statistics) likewise does not exceed 1%.

1. Introduction The precise definition of a peak centroid is important for purposes of pure measuring techniques (stability investigations) as well as for a lot of physical questions like the energetic dependence of emission lines on parameters influencing the excitation and deexitation processes. Especially the emission energy of charateristic X-ray lines slightly depends on the mode of excitation (see ref. 1). For such cases where the shifts are often considerably smaller than the usual resolving power of the applied spectrometer, a method has been developed. A sensitive possibility to detect very small line shifts was recently published by Hansen [2]. It is based on a bias correction to the centroid of a truncated (X-ray) line. Its relative insensitivity to details of the line shape is very advantageous, but severe restrictions are imposed as far as the background is concerned. The cross correlation method which can also be applied again leads to background problems and to great difficulties for the dependence on the integration limits [3]. Thus a method has been proposed which avoids these problems and which works simply and quickly.

2. Principle of the method The functions f(x) and g(x) shall be analytical. Then a Taylor expansion of their quotient may be performed at x =/l. Together with the assumption of f exhibiting an extremum at this point [f'(/~) = 0], it 0 029-554X/81/0000--0000/$02.50 © North-Holland

leads to

g=l. ]72 g2f" _ .gfg" + 2fg'2 + 2~

ga

+ (ha).

(1)

If g also shows an extremum, say at x =/~ + A, higher orders in h than the first one may be neglected for "small" differences A of the extremum positior/s (that means A .~g"/g"') as far as may be supposed ,g h ~ - g~

and simultaneously ~

2g'y~ g,,f_ g f , ,

.

(2)

For the same sign of f"/f and g"/g and quasi-linear trend of the quotient eq. (1) exists within the interval /J + h under these conditions of eq. (2). Consequently, for positive values o f f and g the quotient fig of the two functions can be represented as a straight line along a certain abscissa interval as long as eq. (2) is fulfilled. This straight line equation is f=l~

(i

+la~llg]l#-If-lg g'L x"

(3)

On account of the neighbouring extremum of g, for the small but finite value o f g ' g' I, ~ - & g " lu ~ -2xg" I,+ a

(4)

may be written with small A (see above!). This means: the displacement A is A~

g' I. g" I~z+~

(s)

310

G. Brunner / Measurement of small peak shifts

Of course, this relation can not be evaluated directly, for g' must be taken at the yet unknown x =/J, whereas the amount of curvature at the extremum which follows from the shape of the extremum can be inserted. But by substituting g' of this relation into the straight-line equation (3) one easily gets the value of displacement as

etc., the following expression will hold true: f = A I e x p [ - 4 In 2(x - ;~)2/p}] ,

(9)

( I f half-width). At x =/~, f = Af and f " = - 8 In 2 A l l P}; correspondingly for g. Determining the validity interval according to eq. (2) it results

p2 A~/lgg,--7 ( 1 - - m f ) ,

(6)

or

A ~ ng--~2 g,'f '

(7)

for g, f and g" at the respective extrema. Here m means the absolute term and n stands for the slope of the straight line expression of the quotient fig = q nx + m in ~t - h ~
for

rg= ri= r,

(10)

(which is practically equi~ealent to the above mentioned condition A,~g"/g'"). For small displacements A ~ P/(8 in 2) ~ P/6 the extension of the linearity range of the quotient q = fig can be chosen equal to the half-width h ~ P to meet the condition of eq. (10). An introduction of the not really significant but useful condition of equal values o f f and g for their maxima (preset control of pulses in the region of interest by multichannel pulse-height analyzer, e.g. in energy dispersive X-ray analysis!) leads to Am

p2 ( l - r e + s ) ~ 8-Tn-2n 2 ~

(ll) '

and

or E .

The influence of background changes of the peaks or lines in spectroscopic applications presents a special problem. It can be treated by substituting fl = f + uf and gl = g + Us, with the second terms meaning the backgrounds. By eqs. (6) and (7) the new/xl can be calculated. For instance, if one restricts oneself t o uf = ax + b, that is a linear background added to f, and keeps ug = 0, a reference line without any background, A 1 ~-, ,~k q- a / j et'

h A ~ 8 ln2

(g)

results as the true line shift.

3. The influence of some types of functions for peak modelling In energy spectroscopic applications the division of absorbed energy into really detectable primary processes, like ionization, and in excitation of lattice vibrations of the detection substance (Fano factor), causes a distribution of the number of the registered primary processes of a Gaussian character. If one does not take into consideration such modifying secondary processes like imperfect collection, summing effects

/Xn ~" 8 in 2

'

(12)

for the line displacement. The ~ expresses the relation of a previously and arbitrarily adopted background uf = ( A f X s/IJ ) X x , proportional to the value of x and to the amplitude Af of the investigated peak function f. Such a rising background could be caused by peaks of higher energy. Of course, an instrumental line may differ from the Gaussian shape (see ref. 4), and the question arises which is the right line shape e.g. in energy dispersive X-ray spectrometry, and what could be expected with regard to sensitivity of the proposed method to line shape variations. In fig. 1 the experimental shape is given for an ionexcited Si K X-ray peak of an energy dispersive spectrometer together with a Gaussian model, with a peak of a Breit-Wigner shape (Cauchy distribution) and with a sum of centered Gaussians (since no asymmetric tailing is noticeable !)

adapted to the special apparatus by the parameter

G. Brunner I Measurement of small peak shifts

311

1.1

10 4,

1.0

Cl

0.9 lo 3

,

",,,,,,,,

?'~.

l

0,8

•

l

i

t

I

I

I

I

I

1

.

.

1.1 1.0

q

0.9 06

20

40

go 80 chonnet number

100

Fig. 1. Experimental line shape in comparison with a n adapted Gaussian, a sum of (six) Gaussians (see text) and a Cauchy distribution.

1.1

I

I

I

I

I

I

I

I

I

,[

~"~r

C

1.o 0.9

0B values p = 3.8 and k = 5. The last expression approximates the experimental peak quite well as can be precisely examined b y regarding the valley to the more energetic and nearby sulfur peak. But, first o f all, this concerns the outer parts of the peak, whereas the method discussed is restricted to the inner region [see eq. (10)]. To answer the question for the best fitting function, near the peak centre a background of 80 counts was subtracted from the peak counts in 56 ~< x ~< 64 (the integer values of x are the ,'channel numbers"), and the obtained values as f = f ( x ) w e r e divided b y the calculated g = g(x), channel by channel for g being (a) a Gaussian, (b) a sum o f Gaussians like in eq. (14), and (c) a Cauchydistribution, all of them centered for lag = 59.5 and exhibiting an amplitude o f A = 104. The half-widths in all cases equal F = 7.4. In fig. 2a, b and c, the results can be seen. They agree with the relations in the outer regions of the peak: the sum o f Gaussians fits best. The respective correlation coefficients as a measure of fit amount to rGauss_E = 0.97, rGauss = --0.95 and rCauchy = --0.91. Using eqs. (11) or (12) the values o f A are --0.195, --0.191, --0.283 in units of the channel width (the background has been already corrected for ). The last value can be ignored because o f the results of fig. 2: too poor fit.

b

,

I i i I f , i 57 58 59 60 61 62 63

~

i

channel number Fig. 2. Least-squares •straight lines of the quotient of the experimental peak and the three types of model peaks of fig. 1: (a) sum of six Gaussians;q = -0.01969x + 2.1650; (b) single Gaussian; q = -0.01934x + 2.1287; (c) Cauchy distribution;q = 7-0.02862x + 2.7013.

4. Some remarks for application From the example just discussed the excellent agreement o f the peak position estimated b y a sum of Gaussians [/~f =(/ag + A)Gauss_ ~ = 59.5 -- 0.195 = 5 9 . 3 0 5 ] and b y a single Gaussian [ / l f , = 0 a g + A)oauss = 59.5 -- 0.191 = 59.309] points to the high accuracy o f the method. But what about the precision with respect to the statistical variance of the channel counts? In principle one may adopt a smooth reference peak being calculated e.g. as a sum of Gaussians, and thus have the influence o f statistics only b y the measured peak under investigation. This is the way o f treatment used above. But the half-width must be adapted carefully, because a 1% error in P leads to about a 2.5% error in A . On the other hand, one can avoid the accurate adaptation of P b y using a measured

G. Brunner / Measurement of small peak shifts

312

Table i The maximum positions (near channel number x = 59) of two Gausslan peaks (104 counts in maximum channel and 12.5% tel. half-width each) differ by zX~. These displacements were re-calculated by eq. (12) for smooth and statistical rough peaks, respectively, in 55 ~
A/~calc.smooth

AX#calc.rough

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.159 0.179 0.199

0.053 0.052 0.039 0.111 0.094 0.137 0.129 0.187 0.168 0.212 0.231

± 0.000 ± 0.000 -+ 0.000 ± 0.000 ± 0.001 +- 0.001 ± 0.002 -+ 0.003 ± 0.004 ± 0.004

± 0.045 ± 0.051 ± 0.029 ± 0.058 ± 0.046 ± 0.015 ± 0.051 -+ 0.034 ± 0.048 ± 0.054 ± 0.028

peak as a reference. The latter procedure also involves a statistical influence f r o m the reference peak. To d e m o n s t r a t e the effect o f statistics, two simple Gaussians, the one for /~ = 59 and the o t h e r w i t h a k n o w n displacement 2x/~ around x = 59 were taken ( b o t h having an amplitude o f A = 104 and a P = 7.4), and the A/a was calculated again as the A o f e q . (11) or (12). In table 1 the results are presented. In the second c o l u m n s m o o t h calculated functions f and g

0.3 /

0.2

6o

}

--c

tin.

time

f,oate

{~ 48 h }

Fig. 4. Line positions (Si K X-ray peak) vs. time; 20 errors.

are used, b u t for the third c o l u m n the values o f f and g are calculated having a statistical roughness: t h e y are n o r m a l l y distributed like in measured peaks. The error values given in the table correspond to the 95% significance level ( 2 a ) according to the errors o f n and m in f / g = q (see e.g. ref. 5). The m e a n error o f a single m e a s u r e m e n t o f displacement thus a m o u n t s to 2 a = 0.042 channel units, as can also be seen f r o m fig. 3. So one can settle a precision o f 0 . 0 4 2 / 7 . 4 = 0.57% o f the peak half-width within the given conditions for the definition o f the peak shift o f f ( x ) relative to g(x). It seems interesting.to apply the described m e t h o d for testing the s t a b i l i t y o f the energy scale in an energy dispersive X - r a y spectrometer. Fig. 4 gives the drift o f a Si K X-ray peak recorded by an Ortec device for about 48 h. A f t e r four m e a s u r e m e n t s [relative to a g(x) = k exp {L(x - 59)2/19.5} m o d e l peak] the s p e c t r o m e t e r was switched o f f for some hours. T h e n an u n i n t e r r u p t e d working period followed. This demonstrates the possibilities o f the m e t h o d for very sensitive c o m p o n e n t testing.

References d

i

O

04 true disptacement

02

Fig. 3. Calculated displacements (channel contents of f as well as of g are normally distributed) vs. given displacements (see also table 1).

[1] K.-C. Wang, A.A. Hahn, F. Boehm and P. Vogel, Phys. Rev. A 18 (1978) 2580. [2] P.G. Hansen, Nucl. Instr. and Meth. 154 (1978) 321. [3] M. Braune, Zfl Leipzig, private communication. [4] L.A. McNelles and J.L. Campbell, Nucl. Instr. and Meth. 127 (1975) 73. [5] L. Sachs, Statistisehe Methoden (Springer-Verlag, Berlin, Heidelberg, New York, 3rd ed. 1976) p. 85.