| GSI | Biophysics | SATAN long write-up |

**SATAN** provides means for special processing of analyzer
spectra. Among these are *spectrum deconvolution* and
automatic *peak search*.

anlin = anlres X anloutThe spectrum 'anlout' is calculated by an iterative procedure. The region of the analyzer 'anlin' which is to be deconvoluted must be displayed; the names of 'anlres' and 'anlout' are given within the com- mand. A 'check-analyzer' may be specified to contain the reconvoluted resulting spectrum for comparison with the input spectrum

DELTAE = E(k+1) - E(k)one expects an ideal spectrum f(k) with the observed counts in one channel:

f(k) | +-+ (counts) | | | | | | | | | | | | | | | | | | +------------------------------ channels k EIn practice the spectrum y(k) (e.g. gamma-rays) is measured:_{0}(energy E(k))

tail peak y(k) | (counts) | +-+ | | | | | | |---------+ +-+ +-+ | +-+ | | | +-----+ +-+ +------------------------------- channels k EMathematically this can be expressed as:_{0}(energy E'(k))

y(k) = SIGMAwhere the summation is performed over index l. r(k,l) is the response matrix. In case the ideal spectrum f contains one discrete energy E_{l}(r(k,l) X f(l))

y(k) = r(k,lHence to construct the matrix the detection system has to be calibrated with monoenergetic sources; a calibration spectrum y(k) measured at the energy E(l) is stored in column l of the response matrix r(k,l) as illustrated in the two-dimensional representation below._{0})

+--------------------------- E(l) |====-- --------------- | -====-- ------------- | -====-- ----------- <- tails | -====-- --------- | -====-- ------- | -====-- ----- | -====-- --- | -====-- - | -====-- | -====-- | -====-- | -====- E'(k) | -=== <- peaksGenerally a sufficient number of sources of different energies to fill the whole matrix is not available. This means that the measured response spectra have to be interpolated. Finally the columns of the response matrix must be normalized to unity, because only on this condition the detector efficiency can be mathematically separated from the total response function. Generating the response function it should be considered that a matrix needs a lot of storage (4 bytes per element).

x(k) = xof the input spectrum always corresponds to channel numbers k. If a non-zero offset x_{0}+ x_{1}X k

fwhere k labels the elements of spectrum y and i is the number of the current approximation to the exact solution. Each element is corrected individually by a correction factor c(k). As zeroeth approximation the input spectrum is taken:_{i}(k) = c(k) X f_{i-1}(k)

fIn every iteration step i a check spectrum_{0}(k) = y(k)

dis computed to test the quality of the solution f_{i}(k) = SIGMA_{l}(r(k,l) X f_{i}(l))

chi(DELTAy = errors of input spectrum, N = number of spectrum elements) which should be of the order of 1 or less. If the relative change of chi-square becomes less than the value in the accuracy parameter the iteration process is stopped even if the maximum number of iteration steps defined by the ITER-parameter is not executed . The algorithm to evaluate the correction factor is based on the formula_{2 = SIGMAk ((di(k)-y(k)) / DELTAy(k))2 / N }

c(k) = SIGMAwhere the summation is performed over the index l extending from l_{l}(r(l,k) X y(l) / d_{i}(l)) / SIGMA_{l}r(l,k)

l(l_{min}= k-n , l_{max}= k+n

c(k) = y(k) / dThis is known as the 'quotient method'. It is useful for spectra with a lot of peaks where the intensity from the tail contributions shall be stored into separate peaks. It is possible that even small 'hidden' peaks hooked on the shoulder of dominating peaks (e.g. conversion electron spectra where an L-line may be located very close to the K-line) appear after applying this deconvoluting technique. On the other hand the quotient method may interprete statistical fluctuations in the spectrum as nearly hidden peaks and emphasize them during deconvolution . If the conformity between refolded and measured spectrum is found to be too poor (chi_{i}(k)