Position sensitive detector technology, data reduction and count rates

Transcription

1 Position sensitive detetor tehnology, data redution and ount rates Introdution... Detetor tehnology with respet to ount rates... 2 Photon ounters... 2 Integrating detetors... 2 Summary of detetor properties... 3 Basi statistis of ounting... 4 Photon ounting detetor... 4 Integrating detetors... 5 Small hanges on a large bakground... 6 Data redution issues... 7 Pixel to pixel sensitivity... 7 Remapping in the detetor plane... 7 Remapping on Ewald sphere... 7 Data noise simulations... 8 Detetor tehnology with respet to spatial resolution... Time resolution... Literature... 2 Introdution Over the years I have found that there exist quite some misunderstandings and misoneptions about position sensitive detetors and data olleting issues. This text is intended to shed some light on different aspets of detetors. It should be lear from the start that I m onvined of the fat that the ideal detetor, suitable for all appliations, does not exist. Also be aware that this is an introdution to detetors and not an every aspet and final word doument. I leave the writing of that doument to the real experts. Also one should keep in mind that tehnology is hanging in time. My aim is to show some issues and let the individual researher determine for him/her/itself what to him/her/it are the important ones. I m aware of the fat that some people will make a areer out of remapping detetor grids and also I m also aware of the tendeny of people to ignore issues that they think are too ompliated or not onsistent with their prejudies. As long as one gets nie reliable results without over- or under-interpreting the data I don t mind. The thing I m really allergi to is to publish nonsense and afterwards use the omment but no one ever told me.. My bakground in X-ray researh is time-resolved Small Angle X-ray Sattering using synhrotron radiation soures, where I developed my own partiular set of prejudies. This doument is intended for students who use this tehnique but it is not forbidden for users of other tehniques to read and think about it (and omment). I also would like to stress that I have detetor developers, seond hand ar dealers and even some pahyderms, but very few smooth talking politiians, among my friends. My mate Chris Hall, who is a detetor developer, has shed a ritial eye on this text 2. To use the phrase of the great and wise, Amsterdam born, football professor and philosopher Johan Cruyff: Ellek foordeel hep se nadeel. (For the Anglosaxons: Effery afuntage hass hiss dissadfuntage ) 2 But should not share the blame for errors and misoneptions on my part. Detetor use version.2 08/02/0 WB

2 Detetor tehnology with respet to ount rates Position sensitive detetors exist in several varieties but they an be subdivided into two broad ategories. The first are photon ounting devies and the seond are the integrating detetors. Photon ounters This type of detetor ounts inoming photons as individual events. The detetor itself generates a signal when a photon arrives and the ontents of a ounting register are inremented with one. This might be done in a slightly onvoluted way like for instane in gas filled wire hambers equipped with a delay line in whih the eletronis generate an address in a multisaler register. The gas ounting tehnology an also be used with small deteting elements eah equipped with it s own individual saler. The strong point of these detetors is that after the time frame is finished one only has to read out the digital numbers stored in the memories. The read-out proess doesn t influene the ontents of the registers. The weak point of these detetors is that the detetion of eah individual photon is a rather time onsuming proess during whih the detetor is not registering other inoming photons. This intrinsially limits the ount rate apaity of these detetors. For instane when one uses a 00 nse long delay line one an only detet 0 x 0 6 ounts/se (in detetor jargon 0 MHz). This is an average rate for randomly arriving events. To be on the safe side we usually say Mhz. In pratie even this figure is not obtained with delay line detetors. When individual pixels are equipped with their own ounters the limiting fator beomes the detetion proess in the gas. This improves this situation dramatially so that a detetor with a similar ative length an handle 400 MHz. The great advantage of these detetors is that it is possible to detet a very high dynami range in the pattern. This is the differene between the highest and lowest sattered intensity in a sattering pattern. This differene an beome :0 6-7 with this type of detetor. Another disadvantage is that the deteting element an not be made too small without risking that the physial proess inside the detetor will not trigger of a single pixel but also it s adjaent ones. The limit for this is roughly miron for a gas detetor. Smaller for solid state detetors. Yet another disadvantage is that gas detetors need some gas depth in order to be able to detet photons effiiently. When a photon hits the detetor at an angle this leads to a parallax problem in whih a photon is deteted at a different position than where it should be. This problem is pratially not very well suitable for mathematial orretion. Integrating detetors This type of detetor uses a solid state physis proess in whih a harge is generated when a piee of semionduting material is being hit by a visible light photon (CCD detetor). The visible light is generated by the interation of the X-ray with a phosphor material. The light is then hanneled to the deteting hip by either a lens or by large bundles of fiber optis. Unavoidably some photons will be lost in this way whih means that the effiieny of the detetor is somewhat lower than a gas detetor 3. The harges are aumulated in a small apaitor whih at the end of the time frame has to be read out. The harge is then onverted 3 However, this is very muh dependent on the energy of the inoming photon. A gas detetor will have diffiulties to effiiently detet photons above 5 kev, unless the gas is pressurised. Detetor use version.2 08/02/0 2 WB

3 to the approximate number of photons that has hit the deteting pixel. The big advantage of this type of detetor is that the ount rate is nearly unlimited 4. There are some disadvantages. The first of these is that during the read-out proess of the pixel some noise is added due to the read-out proess. This is on top of the noise that is aumulated on top of the signal during data olletion. The seond is that not every pixel is equipped with it s own physial onnetions so that the read-out takes plae by reading out the first pixel in the row, erase this one, shift the ontents of pixel number two to number one, read this et. et. This makes the read-out proess for the total detetor umbersome and slow. One an speed this proess up but this is generally at the expense of an inrease in readout noise. The third problem is that the apaitor used for the harge aumulation only has a limited apaitane. This means that when it is full it has to be read out. This means that all other pixels will have to be read out as well even the ones that are not ompletely full. Sine every read-out yle adds read-out noise the weaker signals will suffer strongly thus limiting the dynami range. Eletroni rosstalk between the pixels and horizontal drift in the phosphor will redue the spatial resolution of this type of detetor somewhat but in general it is better than the spatial resolution of a gas detetor. Summary of detetor properties In the table below one an find a oarse summary of the properties of the two types of detetors. Counting Integrating Countrate Relatively low High Spatial resolution Low ( miron) High (25 miron) Dynami range High :0 6-7 Low :0 4-5 Read-out noise None Weak signals will be influened Read-out speed detetor Fast (miroseonds) Slow (00 miroseonds) Maintenane Not easy easy Parallax problems At wider angles None Effiieny High (5 2 kev) Low > 2 kev Low but muh better than gas detetor > 2 kev Obviously this is very muh a generalization. It is possible to build detetors for speifi purposes that will satisfy the needs for a single appliation. For instane it is possible to onstrut gas detetors with a very high spatial resolution. However, this will not be suitable for high ount rate appliations. The same as it is possible to photon ount with solid state devies if one reads out very arefully (i.e. very slow). The fator of maintenane is rather subjetive but for a beamline sientist on a multi user synhrotron radiation beamline it is definitely not a parameter that an be negleted whih means that for a beamline a detetor system an be hosen whih is sub-optimal for many appliations but whih operates reliably. The purhase and maintenane ost of a single system is in general so high that it will not be possible to equip a beamline with more than one system. (This leaves the station sientist often in the position that he has to defend his 4 There is a rate limit for phosphors/sintillators, where the light output goes non-linear with x-ray flux. However, it is large. Anedotally we have reahed it at 0 Mhz from a 00 miron square beam = 0 9 /mm 2 /se (Chris Hall) Detetor use version.2 08/02/0 3 WB

4 hoie for a ertain detetor system and therefore will slak of other types of detetors. This is a sin whih normally fades away after the age of 35 or 0 years of experiene on a beamline, whatever omes first). Basi statistis of ounting 5 In the following text no attention is paid to the effets that eletroni rosstalk between hannels will have and every pixel is treated individually. Also Gaussian statistis is being used while the more orret proedure for the integrating ase would be to use Poisson statistis. The main reason for this is that I think it more important to get a feel for what is possible and whih parameters are important rather than writing the ultimate text on detetors. I m not an expert in error propagation through alulations. For more expert views take a look at the book of Rabinovih. The basi equation for primary data redution reads: Is ( q) = I s As ( q) ( q) F( q) I A ( q) ( q) I s (q) sattered intensity due to sample I s (q) sattered intensity due to sample and ell/beamline windows/air et. I (q) sattered intensity due to ell/beamline windows/air et. A s (q) intensity of transmitted beam with filled sample ell A (q) intensity of transmitted beam with empty sample ell F(q) pattern produed by even illumination of detetor (flood field/detetor response) In this q atually should be expressed in the Cartesian oordinates q(x,y) but for simpliity sake I will refrain from doing that. Now make some simplifying assumptions. - the detetor response urve is muh better than the atual data. This is a valid assumption in the ase of dealing with proper experimenters. 2 - the saling fators for the intensity have a zero error. This again is not without basis sine this is a parameter integrated over time. This redues the equation to be onsidered for the error propagation to: I q = I q I q s ( ) ( ) ( ) s Photon ounting detetor Defining error margins and dropping the q to redue the writing: I ± I Is ± Is With a photon ounting detetor one an basially assume Gaussian statistis. This gives for the error margin: I s = ( I ) + ( I ) 2 s 2 5 There are lies, bigger lies and statistis. (insription found on the wall of a toilet in Herulaneum, next to the insription Kilroy was here ) Detetor use version.2 08/02/0 4 WB

5 Integrating detetors With an integrating detetor one finds two noise terms on top of the statistial noise intrinsi to any measurement. These are the dark-urrent and the read-out noise. The first is a parameter that is linear with time. The read-out noise is a funtion of the read-out speed of the detetor. The faster the read-out is the higher this parameter is. I ± I ± D Is ± Is ± D The dark urrent and read-out noise are independent of the ountrate. Now we an onsider two regimes. The first where the ountrate is suffiiently high that the intrinsi experimental error is muh larger than the dark urrent and read-out noise: I s >> D In this ase the integrating detetor performs equally well as a photon ounting detetor when onsidering equal ountrates. However, integrating detetors have in general a muh higher ountrate ompared with photon ounters so it is likely that an integrator will outperform a ounter here. The seond regime is the opposite ase where the detetor noise is higher than the experiment noise: I s << D In this ase the statistial error an be approximated by: I s = 2D It will be lear that a ounter outperforms an integrator here. One ould argue that with enough flux this is not too muh of a problem sine one will never enter into this regime. However, the limitation on integrating detetors is the full well depth of the integrating element. If this is over-filled the detetor either saturates or has to be readout again. Now, keeping in mind that a lot of SAXS patterns fall of in intensity as 4 ( q) ~ / q I at wider angles it is not unlikely that at some point in the data range one will enounter this problem. In my opinion this is therefore losely related to what q-range one wants to over in a single experiment. As an example take the ase where at q = 0.0 Å - the intensity reahes the full well depth of a 6 bit detetor. This is 2 6 = Where do we reah the point, q = z, then where the ounts, N, fall off to a real low level? I ( ) ( q = 0) I q = z = = N 4 q 0.0 N = z = 4 ( ) N = 0 z =.3 N = 00 z = 0.4 For N = 00 the statistial error margin due to ounting statistis is 0%. Now if the detetor reads out fast, one an sum several frames to artifiially inrease the dynami range. At the high ounting end this is no problem but as soon as I s D one does not make any large Detetor use version.2 08/02/0 5 WB

6 gains in the low ounting end of the spetrum. One should also keep in mind that D inreases when one is reading out fast. For a photon ounting detetor no suh problems exist but it might take a longer time to reah the 2 6 level due to the fat that the maximum ountrate is muh lower than with an integrator. So it is a matter of hoosing between Sylla and Charybdis (or the devil and the deep blue sea). The message in this is that if one just needs a pattern to study orientation effets, phase transformations et. an integrating detetor will be a good hoie. This is the ase in experiments where one is mainly interested in phase reognition like for instane high pressure experiments on polymers. If one needs a large dynami range then one is muh better of with a photon ounter. This is the ase in for instane solution sattering experiments. Some people will laim that photon ounting detetors are 'best' for time resolved experiments as well. I prefer to leave this issue to be determined by the individual experimenter. Small hanges on a large bakground When onsidering small hanges in a time-resolved experiment there are some triks that an be used. Consider the time-dependent version of the bakground subtration at time t : I q, t = I q, t I q, t s Is ( ) ( ) ( ) 2 t ) = ( I t )) + ( I t )) 2 s s A similar formula holds for t 2. Now one is interested in the differene: Id t2 t ) = I s t2 ) Is t ) Conventionally this is obtained by subtrating two urves like in the equation above. Now the error margin in this differene is: Id t2 t ) = ( I s t2 )) + ( I t2 )) + ( I s t )) + ( I t )) 2 I t ) s 2 However, if one assumes that the bakground doesn t hange, whih is not ompletely unrealisti sine we re only dealing with small hanges, one skips two mathematial operations and the error in the data beomes: Id t2 t ) 2 Is t ) This is by definition smaller than the previous error. One should keep in mind that the redution in the bakground due to the inrease in the signal is not taken into aount so that the total hange in the atual signal is probably slightly smaller but this is offset by a muh smaller error margin. This method looks partiular favorable in the ase of for instane the growth of a rystalline peak above an amorphous bakground. It should be noted that as long as the ount rate on the detetor is high enough the read-out noise of an integrating detetor will not play a role. Detetor use version.2 08/02/0 6 WB

7 Data redution issues Detetor pitures are not always what they appear to be. Before one an start with data analysis several artifats introdued by the detetor should be taken are of. These problems are not the same for every type of detetor. Even with supposedly idential detetors variations an our. The order in whih this happens is sometimes important. Also one should be areful to keep the amount of mathematial operations, in whih experimental sattering patterns are used, to a minimum sine errors on the experimental data will propagate. For instane if one divides one pattern, with a statistial error margin of 5%, by another pattern with the same error margin, the resulting pattern will have an error margin of about 7%. Pixel to pixel sensitivity It is very unlikely that every pixel, or deteting element, in a detetor has the same sensitivity. Some will ount 9 out of 0 photons another will only detet 7. One has to ompensate for this effet. See figure. The experimental proedure for this is to illuminate the detetor with a uniform sattering pattern and divide the original pattern by this uniform pattern. Remapping in the detetor plane Ideally a detetor is divided up in a grid with square pixels. This is often not the ase in real life. In a CCD the deteting phosphor is oupled by fiber optis to the deteting hip. The individual fibers annot be bonded with mathematial preision. In a wire hamber, equipped with a delay line, the delay line an introdue errors sine the thikness of the line influenes the speed of the eletrons whih means that sometimes 00 pixels are mapped on m line and at another position maybe 95. See figure. One way of orreting this is to use a metal grid, with regular arrays of holes, whih is plaed before the detetor. After this the detetor is illuminated with a fairly uniform soure. The dark stripes in the pattern should be visible as straight lines. If this is not the ase the pixels have to be shifted and readjusted in dimensions. Sometimes there is software available to do this. However, one should try to asses how important this issue is in the total data redution. Remapping on Ewald sphere Generally position sensitive detetors are flat. However, diffration theory tells us that in priniple one shouldn t measure in a flat plane but everywhere the distane from the sample Detetor use version.2 08/02/0 7 WB

8 to the detetor should be the same. This means that one has to use, in priniple, a detetor with a spherial surfae. For linear detetors this is possible but for area detetors this is not pratial. This means that one has to remap the square detetor grid onto a spherial surfae. In fat this is the inverse problem that produers of atlases fae. This requires speialized software. See for instane the CCP3 software suite. In small angle sattering this problem is muh less important than in wide angle sattering. A ynial remark is that researhers working with biologial fibers apparently have disovered this problem many years ago but that syntheti polymer researhers hardly see this as a problem. Another ynial remark is that people looking at biologial fibers apparently have so little data that they an afford to spend years on the analysis of a single pattern. Data noise simulations In ombined experiments, like for instane SAXS ombined with WAXS or Raman spetrosopy, the issue sometimes rops up what the time-orrelation is between parameters derived from the different tehniques. In the following the results of a simulation of noisy data are shown. The first urve (blue) represents the funtion: y = 40 ± 2 t 40 y = t ± 2 t > 40 The seond urve (red) is: y = 0 ± 6 t 40 y = 0 + 2t ± 6 t > 40 The error margins represent the intrinsi statistial noise levels assoiated with experiments. The offsets of 40 and 0 respetively are purely for graphial reasons in this ase. The green urve is y = 0 ± 6 + (3 ± ) t 40 y = 0 + 2t ± 6 + (3 ± ) t > 40 This is the same data as above but now with a bakground and noise level added. This ould for instant represent the read-out noise of an integrating detetor. The blak urve is again with extra bakground and noise added: y = 0 ± 6 + (6 ± 3) t 40 y = 0 + 2t ± 6 + (6 ± 3) t > 40 Detetor use version.2 08/02/0 8 WB

9 The results are shown below y =2x+40+/-2 y =2x+0+/-6 noise 3+/- noise 6+/ time Now we want to determine from this data the moment at whih the data points start to inrease. This an be done by fitting the straight parts to linear funtions and determining the interept. The result is shown in the following figure: noise and bakground simulation interepts should be at t = lowest noise 30 I(q) 20 high noise low noise time From this figure we find values of 30.5, and 30.3 seonds while the underlying mathematis tells us that it should be 30.0 seonds. The absene of any systemati tendeny is somewhat ounterintuitive. In this simulation we see that noisy data doesn t always have to inhibit the finding of the interept. It should be noted that only point to point noise is taken into aount here. To be learer the differenes between I(t) and I(t+) and I(t-). In a real experiment one has to take into aount the intrinsi error margin as well. This is the margin that one an determine by repeating the experiment several times and then alulating the average. This is the variation between I n (t) and I m (t) with n and m indiating different repeats of the same experiment. The reason that one still finds the proper interept is mathematial explainable. The horizontal part an be represented by: f(t) = the other part by: g(t) = + at For the interept f(t) = g(t) t = 0 Detetor use version.2 08/02/0 9 WB

10 If now a bakground or noise level is added it reads like: f(t) = ± f ; g(t) = +a t ± g ; f(t) = g(t); ± f = +a t ± g ; 2 t = a Assuming the noise levels in both urves to be the same at the moment that the inrease in intensity starts we an apply onventional error analysis to this and estimate that the expeted error margin is 2 times a single error margin. So variations in the position of the interept are only due to the higher noise levels whih influenes that auray of the linear fits and thus the position of the interept. This ultimately leads to a larger error margin but not to a systemati tendeny. This is easily verified by numerial simulations. So variations in the position of the interept are only due to the higher noise levels whih influenes that auray of the linear fits and thus the position of the interept. This ultimately leads to a larger error margin but not to a systemati tendeny. This is illustrated in the following graph. In this ten fits are shown to different (repeat-) data sets and from one of the data sets all data points. The data-sets are reated by taking a onstant value of f(x) = 20 and add to that random noise with as maximum.5 x time time In the right hand panel the same data set is shown. The onneted line is the average of the ten data sets and the line the fit to this data set. It will be lear that the variation in the position of an interept with a sloped line with similar error margins due to a high noise levels an vary onsiderably. These problems are quite important at low detetor ountrates. (This is off ourse something that we all know and generally prefer to forget sine it means that for taking aurate data we need either to repeat the experiments several times and take an average or that the data quality should be high). The method of determination of the exat point where the intensity starts to rise also an have an influene. In the ase that one uses the method of two linear interepts we mathematially get out the orret answer but it might not be ompatible with the underlying physis. It should be noted that when the hanges in sattered intensity are less than the experimental statistial noise level any attempt to detet a hange is useless. Detetor use version.2 08/02/0 0 WB

11 Detetor tehnology with respet to spatial resolution The onventional attitude towards the spatial resolution of detetors is that the higher the better. A slightly modified attitude ould save quite a bit of work and improve results and ease of data analysis. In the ase that one wants to drill a small hole in a wall to hang a frame for a photograph one uses a small eletri drill. In the ase that one wants to drill a hole in the ground to look for oil hidden 5 kilometers (3 miles for the Anglosaxons) below the surfae of the earth, one uses a drill head meter wide (. yard) equipped with speial drill bits. Most people understand this logi but forget it as soon as position sensitive X-ray detetors are being disussed. Assuming that one is working on a beamline where the size of the X-ray beam in the foal plane of the beamline is approximately 300 mirons there is very little to be gained by using a position sensitive detetor with a spatial resolution muh better than 300 mirons. The mathematial theory of this was developed some sores of years ago by a guy alled Nyquist (928) and formalized mathematially by Shannon in 949 (see referene). Mind you, it does not hurt to oversample your data but your data files will be larger and might give you the impression that your spatial resolution is better than it atually is. Time resolution When performing time-resolved experiments there are many experimental aspets that one has to onsider. These range from the suitability of the X-ray equipment, like detetors, to the sample environment that one is using. The latter espeially deserves attention sine in a time resolved experiment one generally varies a physial or hemial parameter in order to study the response of the sample to this perturbation. The ruial aspet here is that this parameter should be varied in a ontrolled and uniform way. Before starting a time-resolved experiment it is important to hek if it is possible to obtain the information that one needs with the experimental equipment that is available. Generally the real limits an only be determined empirially but by arefully assessing the fators that influene the time resolution it might be possible to see beforehand if there is not a parameter that inhibits the experimenter to perform the experiment at the time-resolution needed in order to render relevant results. These parameters are: - X-ray flux With synhrotron radiation soures the flux is dependent on the photon energy. The higher the flux the faster the experiment an be performed. 2 - detetor effiieny Different types of detetors have different effiienies whih are also dependent on the used energy. Also one should keep in mind that some types of detetors, like CCD s, add dark urrent and read-out noise to the signal whih influenes the dynami range. A higher effiieny and low noise level will have a positive effet on the ahievable time resolution. 3 - maximum ountrate a detetor an handle This is dependent on the physial proess by whih the photons are deteted and on the assoiated eletronis. If this value is high more photons an be deteted in a single frame and will therefore have a positive effet on the time resolution. 4 - read-out time of the detetor Detetor use version.2 08/02/0 WB

12 This depends on the number of pixels to be read out and on the physial proess by whih the detetor is read-out. The shorter the better (noise being equal) 5 - attenuation of X-rays by windows around sample and extra bakground satter In some ases this fator an be eliminated by plaing both sample and detetor in a vauum hamber. The smaller this number is the faster the experiment an be performed. However, even when X-ray flux is abundant extra bakground satter will have a negative impat on the quality of the data. 6 - X-ray ontrast In an X-ray sattering experiment this is the eletron density ontrast giving rise to the sattering. If the ontrast is high one needs less time to obtain a statistial signifiant result. 7 - statistial auray needed This parameter depends heavily on whih parameters one wants to derive from the experiment but in general this is also a human fator. Some people have the eye of faith and others don t. The higher this value is the better the data quality but the slower the time resolution. 8 - available memory to store the different time frames This is a ruial onsideration and might be a limiting fator in appliations where it is ruial to use two-dimensional detetors. (Although omputer memory is heap nowadays it is still not trivial to store large data volumes rapidly) 9 dynami range The maximum intensity a detetor an detet in omparison with the minimum is alled the dynami range. In sattering experiments where the intensity generally falls of at wider angles as the fourth power of the sattering vetor this is not a trivial issue. All these parameters an be summarized in a symboli formula as follows: read - out statistis attenuatio n Time resolution = memory dynami range flux detetor effiieny maximum ountrate ontrast The higher the value of the parameter time resolution the longer the length of a time frame and onsequently the slower the experiment. This value should, as mentioned in the previous setion, be onvoluted with the time sale on whih it is possible to perturb the sample uniformly. The parameters memory and dynami range at as a type of delta funtions. If these onditions are satisfied the experiment, regarding this aspet, is possible. If this is not the ase the experiment is not possible with the equipment under srutiny. The fastest experiments, using onventional methods and non-yli data olletion, are at the moment limited to roughly 0. se/frame. When dealing with a proess that an be aurately yled one an play different triks and the time resolution an be pushed towards the milliseond/frame resolution at the expense of a very ompliated experimental protool. My personal opinion is that for most X-ray sattering experiments at the moment the ahievable time-resolution is not neessarily limited by the X-ray beamlines and detetors but by the speed with whih a sample an be perturbed in a ontrolled and uniform fashion. Literature Semyon G. Rabinovih, Measurement errors and unertainties, Springer 995, ISBN C. E. Shannon, Proeedings IRE, 37, 949, 0-2. Detetor use version.2 08/02/0 2 WB