SPANC -- SPlitpole ANalysis Code User Manual

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1 Functonal Descrpton of Code SPANC -- SPltpole ANalyss Code User Manual Author: Dale Vsser Date: 14 January 00 Spanc s a code created by Dale Vsser for easer calbratons of poston spectra from magnetc spectrometer experments. In partcular, t was created for use at Yale Unversty for analyzng spectra from our Enge spectrometer. In Spanc, the user enters descrptons of the targets and reactons used n an experment. Then, they enter nformaton about peaks whch are to be used for calbraton of the spectra. Spanc can then ft the calbraton peaks wth polynomals of order 1 thru 4. ρ s ftted as a functon of spectrum channel. The user can then enter addtonal peaks, along wth whch reacton they are assocated wth. Spanc wll return back estmates of ρ and exctaton energy, based on the calbraton curve. Spanc can save nput data, calbraton nfo, and output data to a fle, whch can then be loaded back n later on. Also, Spanc can save a report to a text fle of the nput data, calbraton, and output data. How to Use Spanc A typcal sesson of Spanc goes as follows: 1) Enter nformaton about targets. ) Enter nformaton about reactons. 3) Enter nformaton about calbraton peaks. 4) Select a polynomal ft order. 5) Enter output peaks, and see results. Detaled nstructons on each of these steps follows. Enterng Target Informaton Select the 'Lst Targets...' opton from the Targets menu. Ths dsplays the 'Target Lst' dalog box. To add a new target, clck the button 'Add' n ths dalog box. A 'Defne Target' dalog box appears. By default, ths dalog box shows nformaton for a 0 µg/cm carbon target. If ths s what s desred, all that s necessary s to gve t a name, and clck 'OK'. The target defnton table n the 'Defne Target' dalog has one row for each layer n a target. Layer 0 s the frst layer the beam encounters, and each successve Layer number s the next layer downstream. Layers may be added by clckng 'Add Layer'. Ths adds a layer to the end of the lst, whch s gven a default value of 0 µg/cm of carbon. If you clck on a row, and then clck 'Remove Layer', that layer gets removed from the table. The 'Components' cell for a layer may be edted to specfy what elements are n the layer n what proportons. The format s smple. For each element, type an element symbol (upper/lower case does matter) followed by a space and a number. The numbers should represent the relatve numbers of

2 atoms of each element n the target. For example, slcon doxde would be entered as "S 1 O ". The 'Thckness' cell for a layer may be edted to gve the thckness n µg/cm for that layer. Clckng 'OK' or 'Apply' creates the target object. It wll now be lsted n the 'Target Lst' dalog box. Selectng one of the targets n the lst, and clckng 'Dsplay' wll dsplay a 'Dsplay Target' dalog. In t, you wll see the target as you defned t. Note that the numbers for each element wll be renormalzed to a total of 1. I recommend that you always dsplay a target after creatng t, n order to double check that t was created properly. Targets may be deleted by selectng them n 'Target Lst', and clckng 'Remove'. Enterng Reacton Informaton A reacton n Spanc s specfed by a beam nuclde, beam energy, spectrometer B-feld, target, nteracton layer, target nuclde, projectle nuclde, projectle charge state, and spectrometer angle. All ths nformaton may be entered for a new reacton by clckng 'Add Reacton' n the 'Calbraton Reactons' secton of the man wndow. Ths dsplays an 'Add Reacton' dalog. The 'Beam' and 'Projectle' felds are for specfyng the nucldes. In them, you must enter the nuclde n the form AAAZZZ, where AA s the mass number of the nucleus, and ZZZ s the element symbol. For example, slcon-9 would be entered as "9S". The beam energy s entered n MeV, and the B-feld s entered n kg. The target s specfed by choosng one of your entered targets from a dropdown lst. The layer that your reacton s assumed to occur n must then be selected from the 'Layer' slder. 'Target Nuclde' s a drop-down lst contanng all known sotopes of elements n your chosen target layer. Select the desred sotope. Q s an nteger representng the charge state of your projectle. Fnally, the angle of the spectrometer must be entered n degrees. For every nuclde partcpatng n the reacton, t s possble to specfy the mass as havng uncertanty. When calbratng an experment nternally,.e. usng well known calbraton peaks from the same reacton as the output peaks, the masses can be treated as exact. However, when dfferent reacton channels are used for calbratng, any dfferent reactants and products should have ther mass uncertantes counted. Clckng 'OK' or 'Apply' creates the reacton, and adds t to the 'Calbraton Reactons' table n the man wndow. Each reacton has a number assocated wth t n the 'Reacton' column, whch you wll use when specfyng calbraton peaks and output peaks. Selectng a row n the reacton table, and clckng 'Remove Reacton' wll delete a reacton from the lst. Enterng Calbraton Peak Informaton Calbraton peaks n Spanc are specfed by a reacton, a projectle exctaton energy, a resdual exctaton energy, and a centrod channel. To add a calbraton peak, clck 'Add Peak' n the 'Calbraton Peaks' secton of the man wndow. Ths dsplays an 'Add Calbraton Peak' dalog. Select whch reacton the peak belongs to usng the 'Reacton' slder. You must do ths at least once, even f t appears that your desred reacton s already selected. Enter an exctaton energy n MeV for the projectle. A default value of 0 s already present. Projectle exctaton energes are assumed to be exact n Spanc at the present tme. Enter the resdual exctaton energy n MeV n the 'Ex Resdual [MeV]' feld, and the uncertanty n kev n the 'Ex Resdual Unc. [kev]' feld. These unts were chosen

3 because they are the standard unts used n the Table of Isotopes. In the next two felds, enter the centrod channel for your peak, and ts uncertanty. Clckng 'OK' or 'Apply' creates the peak, and adds t to the 'Calbraton Peaks' table n the man wndow. ρ s calculated automatcally for the peak, along wth an uncertanty based on entered resdual exctaton energy. If any reacton partcpant nucldes have been specfed as havng mass uncertanty (see the secton on enterng reacton nformaton), ther mass uncertantes are also ncluded n ρ s uncertanty. Spanc, lke all of my nuclear physcs codes, gets ts nuclear masses from the 1995 evaluaton by Aud and Wapstra 1 (as provded by the fle mass_rmd.mas95 avalable at Spanc assumes that the nuclear nteracton pont n the target s exactly halfway through the specfed nteracton layer. Beam energy loss s calculated up to ths pont (for normal ncdence), and used as the beam energy n the -body knematcs calculaton. The projectle has ts energy loss through the remander of the target. The projectle energy loss uses the ext angle out of the target nto the spectrometer. The fnal projectle energy after extng the target s reflected n the reported ρ value. Energy losses are calculated usng the formulas gven n Zegler. Performng a Ft Spanc performs lnear regressons to ft ρ as a functon of channel usng polynomal. The polynomals may be 1st order thru 4th order. The dependence ft s as follows: ρ = a0 + a1 * (Channel - Channel[0]) + a * (Channel - Channel[0])^ +... Channel[0] s smply the unweghted mean of all the calbraton peak channels. It s exact, and not really a parameter of the ft. Rather, t s a constant shft of the data made n order to reduce the covarance between the a0 and a1 terms. (In an unweghted 1st-order lnear regresson, ths transformaton actually elmnates the covarance.) The value of Channel[0] s dsplayed all the way to the rght n the 'Ft' secton.. To perform a ft, smply slde the 'Ft Order' slder to the desred order of polynomal you would lke to use. Spanc dsplays the degrees of freedom n the 'd.o.f.' box, and requres that there be at least one degree of freedom before t wll perform a ft. When performng a ft, Spanc frst assumes the channels (x-axs) to be exact, and performs a weghted ft usng the ρ error bars (y-axs). Ths s a standard lnear regresson wth error bars n the y-axs. It then uses ths ft as an approxmaton to determne an effectve contrbuton of the channel error bars n the y-axs (ft slope at channel * channel error bar). Ths gets added n quadrature wth the gven y-error bar to gve an effectve y-error bar. A second ft s performed wth these effectve y-error bars. 'ChSq/d.o.f.' contans the calculated value of the χ statstc for the ft, dvded by degrees of freedom. The expectaton value for the χ dstrbuton s the number of degrees of freedom. Therefore, for good fts, you expect to see a value for 'ChSq/d.o.f.' near 1. The 'p-value' feld contans the tal probablty of the χ dstrbuton for ths ft. It s the probablty that 'ChSq/d.o.f.' would be equal to or greater than the present calculated value f the ft does, n fact, represent the true calbraton curve. Ideally, t should be greater than 0.90 or 0.95, but anythng above 0.5 should be OK, n my opnon. The true check of the goodness-of-ft s to take a look at the resduals, whch are lsted n the second table n the 'Ft' secton. The 'Resd./Sgma' column uses the effectve y-error bars calculated for the ft. If the ft s

4 good, about /3 of the values n ths column should be less than 1. One wouldn't expect more than about one n twenty values to be greater than. The ftted parameters, along wth ther assocated error bars are lsted n the frst table n the 'Ft' secton. The error bars are smply the square of the approprate dagonal element n the covarance matrx. The covarance matrx s dsplayed to the rght n the same table. Enterng Output Peaks Output peaks n Spanc are specfed by a reacton, a projectle exctaton energy, and a channel. Output peaks are dsplayed n the bottom secton of the man wndow, and are entered n a very smlar fashon to the calbraton peaks. Clckng 'Add Peak' n the 'Output Peaks' secton dsplays an 'Add Output Peak' dalog. Agan, you must slde the 'Reacton' slder to the desred reacton number, even f t appears to already be selected. The projectle exctaton energy s agan presumed to be exact and gven a default value of zero, whch you may change. Channel number and uncertanty are entered just as wth the calbraton peaks. Clckng 'OK' or 'Apply' creates the output peak. If a calbraton ft has already been performed, the peak s added to the table n the 'Output Peaks' secton. In ths table, a ftted value for ρ s dsplayed. The exctaton energy for the resdual nucleus s calculated for ths ρ by addng back the energy loss to the nteracton pont n the target, then workng the two-body knematcs n reverse. The exctaton energy error bar s determned from the ρ, whch has ts error bar calculated as descrbed below. Two sources of error are added n quadrature: the channel uncertanty n the output peak, and the predcton nterval one gets assumng the output peak channel were known exactly. The former smply comes from multplyng the channel error bar by the slope of the ft functon evaluated at the channel. The second s the Predcton Interval, calculated by the followng formula: Channelout Channel[0] P. I. = ± t α ; dof MSE InputChannel Channel[0] where SSE MSE = and d. o. f. effectve ( resdual σ ) effectve ( 1 σ ) MSE = For a t-dstrbuton wth r degrees of freedom, the probablty for values greater than t(α;r) s equal to α. Ths t-dstrbuton s the approprate samplng dstrbuton for the calculated ft values. α n the above expresson for the predcton nterval s chosen to be , where s the probablty nsde ±1σ for the standard normal dstrbuton. Notce that the Predcton Interval has a parabolc form where the predcton uncertanty s lowest at the center of the calbraton data.

5 Adjusted Error Bars There s a checkbox n the output peaks secton for adjustng error bars. If ths box s checked and χ /d.o.f. s greater than 1, the predcton nterval error bars get multpled by the square root of χ /d.o.f. Ths s to attempt to compensate for a poor qualty calbraton by amplfyng the uncertanty. It s equvalent to assumng that your calbraton peak error bars were under-estmated by a factor whch would gve the expected value of χ /d.o.f.=1. Ths s not a statstcally justfable procedure, but t can come n handy when one s tryng to get a rough calbraton of a data set. Savng Data and Creatng Reports In the 'Fle' menu, you may select 'Save Data...' to save your entered targets, calbraton reactons, calbraton peaks, and output peaks to a fle. 'Load Data' may then be used to load ths nformaton back nto Spanc. You may also select the 'Text Export...' menu tem to create a text fle showng all your entered nformaton and calbraton results n a human-readable format. The text fle gves Adjusted error bars for output peaks, whch are equal to the predcton nterval for good fts (χ /d.o.f. 1). For χ /d.o.f.>1, t s modfed as dscussed n the above secton. References 1 G. Aud and A. H. Wapstra, Nuclear Physcs A 595 (4), 409 (1995). J.F. Zegler H.H. Andersen, The Stoppng and Ranges of Ions n Matter 3 (1977); J.F. Zegler H.H. Andersen, The Stoppng and Ranges of Ions n Matter 5 (1977).