Journal of Research of the National Bureau of Standards Vol. 54, No. 2, Februa ry 1955 Research Paper 2571
Scintillation Spectrometry of Low-Energy Bremsstrahlung *
Margarete Ehrlich
A method was devised to measure bremsstrahlung spectra from commercial X-ray t ubes on an absolute scale, using a thallium-activated sodium-iodide crystal-scint illation spectrom eter. The method was used to study bremsstrahlung spect ra between 20- and 100- kilovolt s exciting potentials from a beryllium window tube having a t hick t ungsten t arget. Fully correct.ed, absolute experimental spectra were obtained at exciting potentials of 50 and 100 kilovolt s. In order to compare t he experiment al results with theory, a calculation was made yielding t he t hi ck-target bremsstrahlung spectrum derived from Sommerfeld's theory. Experiment and t heory showed order-of-magnitude agreement. However, a characteristic difference in spectral shape was observed, t he experimental spectra showing a more propounced peak. This peak is near 30 kilo electron volts and gives the impression of being superimposed on the spectral shapes expected from theory. Finall y, a point of interest to t he practical user of the X-radiation from beryllium windolV tubes was brought out, namely, that a considerable portion of the low-energy radiation, at least in t he region between 12 and 30 kilo electron volts, is strongly absorbed in the t ungsten target. of a conventional X-ray t ube.
1. Introduction spectrum. The period bctween 191 5 and 1930 produced the discovery and the experimental proof 1. 1. Objective of the Present Study of the existence of a short wavelength limit of the bremsstrahlung continuum and of its dependence Although high-energy bremsstrahlu ng spectra have on electron velocity [6] . This period also produced recently been studied in great detail, a number of the the general exploratory studies of the spectral more fundamen tal features of low-energy brems distribution as a function of electron energy and strahlung spectra, though under investigation for target material [7], the investigation of the efficiency almost 40 years, have as yet not been satisfactorily of the process of bremsstrahlung production, and the explored. In fact, no absolute, fully corrected experi first angular di tribution studies that confirmed mental data have ever been published, and no Sommerfeld's theoretical predictions [8, 9, 10]. technique has been developed for a routine investiga In 1922, Kulenkampff m easured the truc spectral tion of spectra from commercial X-ray tubes. It is intensity distribution of the bremsstrahlung from a the object of this paper to report on the design and number of different thick targets introduced into a performance of a scintillation spectrometer with a gas-discharge t ube that could be operated between 7 single thallium-activated sodium-iodide crystal that and 12 kv [11 ]. His measurements were fully lends itself to the absolute determination of brems corrected and absolu te except for the conversion of strahlung spectra from commercial X-ray tubes. So the ionization in his detector (an air-ionization far, the instrument has b een used up to lOa-lev chamber) into units of absorbed photon energy. exciting potential only, but it is expected that with This conversion was not carried out because the some modification of design it co uld be useful up to wavelength dependence of the energy expended per 150-lev and possibly to 200-lev exciting potential. ion pair was at the time Ullknown. Kulenkampff represented his corrected spectral-intensity data as a 1.2. History of Bremsstrahlung Spectrometry lin ear function of (vo-v), wher e v represents any bremsstrahlung frequency below )/0, the frequency at The history of the investigation of the nonrelativ the Duane-Hunt limit. This expression is a very istic bremsstrahlung cont inuum is surveyed pri good approximation to his experimental spectral marily by Kulenkampff [I, 2]1 and by Finkelnburg distributions. [31. Shortly thereafter Kramm's published his semi classical theory of the bremsstrahlung spectrum [12] . a . Exploratory Experimental Work, Semiclassical Theory Kramers arrived at the frequency distribution of the total emitted radiation, starting 'with the classical Thc study of bremsstrahlung spectra received its expression for the spectral distribution of the energy first impetus from the experimen tal work of Laue radiated by an electron moving in the Coulomb field and Bragg [4, 5], whose crystal diffraction spectrome of an atomic nucleus. This yielded expressions for ter was until recently the only apparatus that lent the bremsstrahlung intensity as a function of itself to a satisfactory study of the bremsstrahlung frequency. When these expressions are integrated "This article was composed for the fulfillmcnt of the publication requirement over the target thickness, they take essentially the for tbe deg ree of P h. D. in the School of Arts and Sciences of the Catholic Uni ver form of Kulenkampff's empirical thick-target sity of America, Washington, D. C. ~_ 1 Figures in brackets indicate the li tera ture references at the end of this paper. formula. 107 Because of the considerable difficulties associated Kramer's. It is, therefore, not too surpnsmg that with exact experimental crystal spectrometry of the the spectral distributions measured by Kulenkampff bremsstrahlung continuum, most of the experimental and others show a fairly good qualitative agreement work confirming Kramer's' theory was carried out by with both Kramers' and Sommerfeld's theoretical indirect methods. One of these, introduced by expressions. Later, Sommerfeld's theory was refined Webster and Hennings [13], consisted in isolating by N edelesky, who introduced screening corrections narrow spectral bands by means of suitable filtration in order to take accoun t of the presence of the and studying their intensity as a function of electron atomic electrons [20] . energy (I sochromatenmethode) . From the partial The progress in theoretical understanding of the information gained in this way, one can reconstruct problems of the nonrelativistic bremsstrahlung the total spectral distribution. Another method, continuum was not matched by experimental used for thin-target work by Duane [9], Kulenkampff advances, which could have facilitated thorough [14], and Nicholas [15], and recently developed to a checks of the theory. The available experimental considerable extent by others [16, 17], consists in methods were still essentially the same as those deducing the spectral distribution from the shape of used in the early studies. This fact may well be absorption curves. Such an indirect method is considered as one of the reasons why experimental probably adequate for the determination of heavily work centered solely on studies of the angular filtered or of unfiltered X-ray spectra close to the distribution and polarization of the bremsstrahlung,2 Duane-Hunt limit, but breaks down for the long while no further attempts were made to measure wave end of the spectrum. The reason for the use of absolute spectral intensities 01' to extend the studies indirect methods to supplement diffraction spec of the spectral intensity distribution to higher trometry becomes clear if one considers the optimum electron energies. The only recent determinations X-ray target design factors and the nature of th e of true spectra by direct methods seem to have been various distorting influences of the diffraction spec those by Kulenkampff and his coworkers [21 , 22]. trometer : On th e one hand, it is desirable to work They used a hot-filament glass-bulb X-ray tube, with thin targets in order to minimize electron with a massive tungsten target, and arrived at diffusion and energy loss through multiple non relative bremsstrahlung spectra obtained with excit radiative collisions in the target proper and to meas ing potentials up to 50 kv. After applying sui table ure the bremsstrahlung spectrum produced by a corrections, they showed that their experimental truly monodirectiona.l and monoenergetic electron data followed the empirical law found by Kulen beam. It is further desirable to have low-atomic kampff in 1922. They mention , however, that the number targets in order to preven t the absorption conditions for targets of low atomic number seem to of a portion of the bremsstrahlung by the target be more complex. Diffraction spectrometry was itself. On the other hand, because of the low never attempted by direct methods on thin targets efficiency of the diffraction spectrometer, it is of low atomic number, probably in part because of important to produce high radiation fiuxes-a re the lack in efficiency of the old-type diffraction quirement that called for target-design factors spectrometer and also because of the apparent exactly opposite to those outlin ed before as optimum shift in general in.terest in basic physics from low for the determination of an undistorted brems energy atomic to high-energy nuclear physics; strahlung spectrum. Further complications are the latter may also have been one of the most introduced by the fact that the reflectivity of the important reasons for the fact that the curved crystal varies with the wavelength of the incident crystal spectrometer, which has proved so useful in radiation and that th e factors entering into th e gamma-line sp ectroscopy up to 1 YIev an.d higher determination of beam intensity from the measure [23], has never been used to study bremsstrahlung ment of ionization curren ts in air-ionization cham continua. However, the interest in the field seems bers are quite complex, and 'were at the time of the to have been k ept alive, if not for fundamental then early bremsstrahlung studies not at all well under at least for practical reasons, as shown by the re stood. newed attempts to find indirect methods to arrive at spectral distributions of bremsstrahlung con tinua b . Quantum-Mechanical Theory of the Nonrelativistic Brems by way of absorption measurements [16 , 17] . strahlung Spectrum, Experimental Verifications The impetus for this phase came clearly from th e c . Scintillation Spectrometry theoretical side. In 1929 and 1931 , Sommerfeld Around 1945 it became apparent that by co upling published the results of his quantum-mechanical Rutherford's scintillating screen (first used for the calculations. Limiting his calculations to non detection of individual alpha particles) with a relativistic velocities, to dipole transitions, and photomultiplier tube, one could produce an extremely initially to unscreened fields, he arrived at an exact sensitive system for the measurement of radiation expression for the production of photons of given energies and intensities [24]. Application of photo energy and direction by incident electrons of given multiplier and electronic techniques, in conjunction energy [18, 19]. Finkelnburg [2] points out that with newly developed scin tillator systems, eventually when integrated over all photon. directions, Sommer transformed this simple device in to one of the mos feld's expression for the X-ray spectrum shows a considerable formal similarity to th at derived by 2 See the literature given in W. Finkelnburg [3] and Kulenkampff's article [2].
108 '"'1 ! I powerful tools of modern physical research [25, 26, ubseq uenLly, the N aI (Tl) scin tilla Lion specLrometer 27, 28]. In prin ciple, the method is based on estab was used on everal occasions for Lhe measurement lishing simple relations between the r eadily measur of "internal" bremsstrahlung, associaLed with Lhe able intensity of the ligh t pulses within the particular beta deca:v of a number of radioacLive clem.en t [35, scintillator and the energy of the inciden t photons 36]. In Lhese studies the corrections for tbe change (or particles), as well as be tween the number of of resolution with energy were carried ou t in a Jightpulses in the scintillator and the number of manner similar to those applied in the case of mag inciden t photons (or particles.) netic spectrometry [37 , 38, 39], and additional Hofstadter was the first to use thallium-activated corrections for the escape of radiation energy in the sodium iodide, N aI (Tl), for scintillation spectrometry form of iodine K fluorescence from the N aI (Tl) and to show that this material is particularly suited crystal surface were calculated [36] . R ecently, for spectrometry of low-intensity photon sources R amm and Stein [40] used a scintillation spectrom [29 ,30]. Subsequently, he and Johansson independ eter to determine rough qualitative spectral distri ently observed the "photolines" caused by the butions of heavily filtered X-ray beams from com strong photoelectric absorp tion of monoenergetic mercial t ungsten target X -ray tubes, again without photons in the iodine of the crystal [3 1, 32]. Hof any further corrections or attempts of a comparison stadter also established that the height of the light with the true, unfiltered bremsstrahlung spectrum. pulses ensuin g from the impinging photons was No 0 ther work on scin till a tion spec trome tery of proportional to the energy lost by the secondary bremsstrahlung continua hfts so far been reported. electrons in the crystal [33]; also, that in a NaI(Tl) However, it may be noteworthy that the use of a crystal large enough to facilitate total absorption of N aI Crl) crystal counLer as detecLor of th e low the radiation, the number of light pulses is equal to intensity X-radiation scattered off a quartz erystal the number of impinging photons, which, in turn, enabled Beekman [40] to m easure the spectrum of a is proportional to the total radiation flux [31 , 331. commercial X-ray tube operated at 125 kv constant These facts established the usefuln ess of the NaI (Tl) potential. Beekman's main interest lay in the scin tillation counter- when co upled with a suitable determination of the influence of current waveform pulse-height discriminator- for X- and gamma-ray on spectral distribution, and he therefore made no spectrometry. eHort Lo obLain absolute experimental data, to apply Up to now the scintillation spectrometer has been corrections, or to compare his results wi th theory. mainl:IT used for gamma-line spectroscopy. How ever, in spite of its rather poor resolution (as com 2 . The Experiment pared with a diffraction sprectometer) the advantages of the scintillatiOll spectrometer over the diffraction 2 .1. Source of Bremsstrahlung spec trometer for the measurement of con tinuous The most basic experimental study on bremsstrah spectra are quite apparen t. The diffraction spec lung would be that carried out on th e pecLra from a trometer is complicated to operate, h as a low .yield number of thin targets of differen t compositions. that depends on photon energy in a complicated way, 'iVork on 1-~ ev thin-target spectra was recently and is limi ted to compara tivcly low energies. No performed at the National Bureau of tandards by such inherent limitations exist for th e NaI (Tl) Motz and Miller [42]. A thick-target X-rfty tube scintillation spectrometer. It is of near-portable wa chosen for the present study, mainly because it character, which makes its adap tation to a number of was readily accessible and a st ud~T of its radiaLion :lifferent problem s of difficult geometries possible. could therefore be carried out with the expenditure Provided that th e crystal is large enough to absorb of a minimum in funds, machine time, and personnel the incident radiation in i ts entirety, its efficiency time. However, the theoretical problem presented is equal to unity. At present, the high-energy limit by the radiation spectrum from such a thick-target of sll ch a total-absorption spectrometer is given only tube is much more complicated than that from thin by th e maximum size of the available N aI(Tl) targets, and it can, in fact, not as yet be solved in crystals. Its efficiency is so high that the problem great detail. is usually not h ow to obtain sufficiently strong In order to facilitate a calculation of the absolute radiation sources, but how to make th em weak intensity distribution emerging from the tungsten enough. target proper, an experimental X-ray tube with a The first to use the N aI (Tl) scintillation spectrom beryllium window of known thickness and approxi eter for a study of th e bremsstrahlung spectrum wa mately known purity was chosen for thi s study. Johansson, who made a qualitative study of the The tube was held to constant poten tial with a specLrum from a commercial, thick-tungsten-target, ripple of less than 0.05 percent per milliampere. glass-walled X -raj' tube in the energy range from The potential across th e tube was determined by about 10 to 100 kev [34]. H e corrected his experi means of measurements of the voltage drop across m ental data for th e change in instrument resolution a calibrated 250-megohm resistor in series, and was with energy, but made no attempts to assess other maintained to an accuracy of less than 1 percent. distorting factors, such as absorption of the radiation Because of the high sensitivity of the spectrometer, in the tube window, the counter wall, or the inter it was necessary to operate the X-ray tube at vening air. These corrections would be n eeded to extremely low currents. Such a procedure is / allow a comparison of experiment and th eory. justified under th e assumption that a change in
326198- 55--4 109 tube current merely produces a change in X-ray intensity but not in spectral distribution. The DURAL RETAINING RING
tube currents were too low for conventional dynam SILICONE RUBBER GASKET ical measurements, and for this reason the relation between tube current and the integral number of pulses counted in the detector system was established L FOIL by the following indirect method: The tube was first operated on a relatively high current (order of DURAL SHIELD magnitude of milliamperes) and a relation was established between tube current and ionization in a PHOTOMULTIPLIER cavity chamber placed in the path of the radiation.3 SILICONE RUBBER The tube current was then lowered to fractions of a O'RING microampere (about 0.01 I.ta) and the ionization in the cavity chamber was measured once more, this time simultaneously with the total number of pulses above a given height. The low-ionization measurements were carried out by a static method. The relation between tube current and total number I of pulses above a given height was determined concurrently with the spectral measurement, which 2 . 6" PIA .1 established the relation between the number of I. counts per minute per pulse-height interval and the FIr: URE 1. Photomultiplier- sodium iodide crystal assembly. total number of counts pel' minute above a given pulse height. The bias setting, which fixed the at 100 key. The cube was sanded on the four side lower limit of pulse heigh t for the total counts, was faces, which were then covered with aluminum foil maintained constant to within 0.25 v throughout the for higher light-collecting efficiency. The upper entire experiment (which barely lasted more than 1 and lower surfaces remained unsanclecl. The crystal hour per spectrum). was cemented to a Dumont K- ll77 photomultiplier tube. The tube-crystal assembly was placed within 2.2. Spectrometer Design Factors an air-and-light-tight Dural can, filled with dry It was essential to construct a crystal-pho to carbon dioxide of slightly higher than atmospheric multiplier assembly that would have good resolution, pressure. Figure 1 shows the details of the assembly. facilitate total absorption in the crystal, and have a The X-ray beam entered the assembly through an window of considerable transparency to the incident aluminum window, 0.0006 in. thick, after having radiation. A NaI(Tl) crystal lends itself especially been collimated by a 1-in.-thick brass diaphragm, well to this task in the low-energy region, because of a diameter of approximately 6 MM. The practically all photon absorption in the crystal diaphragm confined the radiation to the cen tral takes place by photoelectric effect. In this case, portion of the crystal, without contributing any the pulse-height distribution resulting from the characteristic radiation within the measured energy absorption of monoenergetic photons consists of an range. essentially Gaussian curve, withou t any further 2.3. Experimental Setup disturbing distributions due to absorption by Comp ton effect or pair production. The broadening of The essentials of the experimental setup are monoenergetic lines into Gaussian distributions is sketched in figure 2. The electrons impinged upon mainly due to statistical fluctuations in pulse the target of thc X-ray tube uncleI' an angle of 22 de height originating at the emission from the cathode grees, producing an X-ray beam whose central ray and th e first few dynodes of the photomultiplier. emerged from the X-ray tube in a direction perpen Pulse-height resolu tion is customarily defined as a dicular to the incident electrons. The X-ray beam, quantity proportional to the fractional mean-square initially collimated by a I-in. brass diaphragm elose deviation of pulse height for identical scintillations. to the X-ray tube, reached the detector in a distance Thus, as long as photon energy and pulsc hcight arc of 1 m from the target; the portion of the beam that proportional, resolution should thus be inversely entered the crystal sub tended at the targct a solid proportional to the square root of the photon angle of 2.92 X I0- 5 steradian. (The target dimell energy. sions were neglected in this estimate.) The electric A freshly -cleaved N aI (Tl) crys tal, 0.5 in. high by pulses from the photomultiplier, corresponding to the 0.5 in. by 0.7 in., was chosen as th e core of the light pulses in the N aI (Tl) crystal, were led through a detector. Calculation had shown that a crystal of cathode follower to a linear nonoverloading amplifier these dimensions absorbs more than 99.99 percent [43], giving pulses of a maximum height of 60 v, of all incident radiation up to 80 kev in single clipped by means of a delay-line to approximately photoelectric absorption events and about 98 percent square shape, and of about 10,usec in duration.
3 The ionization chamber had polystyrene walls, O.OOl in. thick, and although These pulses were monitored with an oscilloscope, placed between the X-ray sOUl'ce and the spectrometer, its presence did not reduce and were also counted in a scaler (scaler I in figure 2). the integral counting rate in tbe spectrometer, which was biased to record only pulses in the npper half of t he spectrum. Simultaneously, the pulses were fed through a single-
110 1 METER
BRASS COLLIMATOR PHOTOMULTIPLI ER LEAD LINING X- RAY CATHODE TUBE FOLLOWER LEAD RADIATOR D.. COLLIMATOR 'f(~~ o CATHOOE I flAY· f iLTER SLOT ~\:'tL~E~A~O~C~OL!!2Li!'MiJJLIAT OR OSCILLOSCOPE ... ..ri. INSfRT I FIGURE 2. D etails of til e e:tperimental setup. Insert at lower left shows details of geometry of bremsstrahlnng prodnction. I channel differential pulse-h eight discriminator, and =l~RA SS DIAPHRAGM the number of pulses per channel width was detcr TO mined by another scaler (scaler II in figure 2) . SPECTROMETER caler II was biased to discriminate against low-noise FIGU RB 3. Fluorescence radiator chambe1" . pulses, and the bias on scaler I was adjusted so as to count only pulses above a certain height. This pro of the entire K serics. 4 Table 1 gives a li st and cedure allowed comparatively high differential coun t description of th e rad iators used and of the filter ing rates in scaler II, at the same time keeping the matcrials that were fou nd to be necessary to isolate in tegral counting rates in scaler I wi thin reasonable the K-fluorescc nce lines, along with the exciting limits. The differential counting rates varied be volLage for Lhe bremsstrahlung used to produce the tween about 100 and 6,000 counts per minute; the particular K fluorescence. Each pulse-height distri integral counting rates did not exceed 3,500 counts bu tion was determined several times, each time with per minute. In the case of the 50-kev and the a different gain. The average resolution (i n percent) 100-kev spectra, which were the only ones obtained was obtained for each line by computing Lhe line on an absolute scale, the integral counting rates width at h alf height in volts, ex pressed in percent ,,-ere used to tic in wi lh the previously discussed of Lhe pulse h eigh t at the fluorescence peak, also curren t measuremen ts. in volts.
2.4. Spectrometer Calibration T A B LID l. Fllwrescence radiators and absorbers used to isolate jllwrescen ce peaks A calibration of resolution as a function of energy was obtained by determining th e width of the pulse Hadiator Exciting -.-----,--- bremsstrahltmg A bsorber element height distributions at half hcigh t due to thc K I a nd t hickness fluorescence lines from nine different radiators. Element K a l line potential A radiator chamber similar to that described by kev kev mm eemann [44] and a number of radiators supplied Strontinm ______14.21 50 ______• None. Sil ver ______---- through the courtesy of Dr. Seemann were used for 22. 2;l 50 ------Do. Cadminm ______23. 24 50 ______. Do. this purpose. The experimental setup is sketched in Antill1011Y ______26. 43 55 to 65 ______])0. figure 3. Bremsstrahlung from a commercial tung Lanthanum_. ___ 33. 53 55L065 ____ -- Lead , 0.075. sten-target X-ray tube operated at suitable vol~ages Samarium ______40. 23 80 ______Lead , 0.075, pIns gold, 0.050. is made to impinge upon a sheet of the radIator 'I' ungsLc IL ______59 .•7 l()(L ______Silver, 0.57. Gold ______68. 94 115 ______Lead.0.95. material. The fluorescent radiation emerging under Lead ______75.12 120 to 110 ___ . __ Do. and angle of ninety degrees is analyzed i.n the NaI(TI) crvstal spectrometer. In order to Isolate the fluorescence from high Z materials, suitably Samples of pulse-height distributions obtained fo r chosen filters have to be introduced in the path of four of the fluorescent radiators were used to check the fluorescent radiation, in order to prevent the the lineari ty of the en ergy scale, that is, the pro portionality of the voltage at pulse-height maximum gO-degree Compton-scattered bremsstrahlung sp~c trum from being registered in the detector along WIth with the energy of the X-ray photons. Figure 4 the K-fluorescence lines. No isolation of the Ka lines shows the results of this check. The importan ce of wa attempted, as experiments h ad shown that, the proper choice of filtration becomes apparent within the limit of the resolution of the instrument, • This is not a t ali surprising, because the othel' lines of the K series are much less than one- half as intense as the K . lines and can therefore not mflnence the the wiel th of the Ka lines alo ne was the same as that K . line width at half height. III 60 I
f-"' Z :::> f ~1 50 ~ I 0 t5 40 0 '"0: Q f- a: a; w 0: Q. 0 q 100 0 Z 30 o ~0 2 ~ f ::::> ~ 50 z o-' ~~ ::> IJ) 8 0 I ~ ~ 2 '-... > 'a PULSE HE IGHT , VOLTS a: 0 w'" z w FIGURE 4. Energy-scale c!1libration of spectmmeter. 0 0
0 (j) 10 20 30 40 60 80 100 f PHOTON ENERGY, Key Z ::::> 400 FIGU R E 6. P ercentage energy resohition of spectrum as a func >- a: tion of photon energy.
FIG URE 5. Pulse height distribution of fluore scence radiation from Lead, irradiated with bremsstrah 2.5. Raw Experimental Results lung and obsaved per pendicular to ihe direciion of bremsstrahlung incidence. Fio'ure 7 shows the differential co unting rates ob Cu rve I. No filter betweon radiator and detector; A. peak of Compton·scattered bremsstrahlung spectrum; B, Compton· taine"'d for the bremsstrahlung continua from the scattered K fluorescence peak from tun gsten target; C J fluores cence peak fro m lead radiator. beryllium-window X-ray tube, operated at 50 and Curve 2. 0.95 mm lead filter betwee n radiator and detector. at 100 kv, as a function of photon energy. Due to the extremely low current levels used for this study, in the pulse-height distribution shown in figure 4 fluctuations in the bremsstrahlung intensity could for gold. This distribution was obtained with not be en tirely eliminated. For this reason, the insufficient filtration and is therefore markedly quantity plotted on the ordinate is the following skewed. The effect of filtration on the K-fluores ratio: cence pulse-height distributions is seen even better in the case of lead. Figure 5 shows the distribution (2) obtained with a lead radiator, with and without filtration. ' Vhen no filter is used, the entire brems difI is the differential counting rate per discriminator strahlung spectrum, scattered through an angle of e channel width; Ctota], the integral counting rate; and 90 degrees, is superposed on the lead K fluorescence. TIll the width of the discriminator channel, expressed A 0.95-mm lead filter- although transparent to its in 'J,;:ilo electron volts. The plotted quantity is thus own K fluorescence- strongly absorbs all lower in units of number of differential co unts per minute energies, and thus isolates the lead K-fluorescence per milliampere pCI' kilo electron volt. . distribution. (The distribution below 60 kev, not The two spectral distributions of figure 7 WIll be shown in the graph, contained no points appreciably used in section 4 for an absolu te comparison between above background.) 5 theory and experimen t. In order to give a better 5 rI'hc resolutions shown for Lhe lead-fluorescence pulse-height di st ribll t i o n s~ picture of how the spectral distributions change wi th w ith and without added absorbers. arc not quite comparable, as the tw·o cur ves wore obtained with different Na1('1' l) crystals. exciting potential, fi ve more spectra obtamed at 112 - . ~.
6 r-,---,---.--.--,---, exciting poten Lials beLwccn 20 and 100 kv arc hown in figure 8. T he ordin ates of these spectra arc not 10 on a comparable scale . The energy scales shown on the abscis as of figu re 7 and 8 were obtained by a calibration of Lhe spec j4 trometer with the known gamma and X-ray lines of Cd 109• The differential counting rates are ac ~ 3 6 -'" curate to approximately ± 5 percen t, the inaccuracy ... stemming from the lim ited coun ting statistics, as .~ 2 4 well as from the flu ctuations in the discriminator ...~ channel width . There is an additional inaccuracv a . Variable Resolution The p roblem of correcLin o' the pulse-heigh t distri b utions fo r vari a bl e fini te resolut ion consists essen- . t ially in finding the relation between C(E') , tho measured p ulse-heigh t distribu tion (n urn bel' of counts per minu Le per discriminator window width)- cor 10 responding to a particul ar m easured photon energy in terval between E' and (E' + dE')- and N (E ) , the true pulse-heigh t distribu tion . It involves the solu 9 tion of the following inhomogeneous F redholm equation of th e fir st kind : 8 ('(E') = Sa '" K (E,E')N (E )dE, (3) where K (E ,E'), the kernel' of th e eq uation, repre sen ts a distor tion function expressing the probability K (E, E')=N e-(E- E')2/ 2u 2. 3 where the mean-square deviation is given by 0- = [W(E)]/ (2 ... 'Zln2), and N represen ts the normaliza 2 tion factor. An equ ation of the type (3) lends itself readily to numerical evaluation wi th the aid of automatic computers. The method was first worked out by the Naval R esearch L aboratory group in connection with a somewh at differen t problem [47]. The method consists essen tially in red ucing the integral equation to an equation involving discrete sum s. T he function K (E ,E') becomes a matrix (K EiEj) whose rank: r (here chosen as 19) is equal FIGURE 8. Change of spectral distribution with to the number of selected discrete energy points, and exciting potential. the fun ctions N (E ) and C(E') becom e the r-climcn- 113 sional vectors (NE t) and (GE }) of the matrix equation aluminum and berylli um were estimated to be less than 2 percent. (5) 2,7 . Corrected Experimental Spectra Its solution for (NE i) , the true pulse-height distri bution, is readily obtained by matrix inversion of The pulse-height distributions shown in figure 7 were corrected in the man.ner outlined in section 2.6. (KE,E') after the sum of the elements of each column Figures 9 and 10 show the uncol'rected distributions, of the matrix (K) has been normalized- a procedure taking the place of a normalization of the Gaussian function K (E,E'). 4.----,----,----,----,----,---, b. Crystal. Conversion Efficiency l"- I Equation (3) actually contains another factor, ~ 3 namely, the efficiency of the spectrometer for the ~ conversion of incident photons into electric pulses. :;;,. As discussed earlier in this paper, at least 98 percent c of all the impinging radia tion are pho toelec trically "c 0 ~ absorbed by the crystal. Furthermore, as no K u 2 escape peaks [36] were ever observed on pulse > height distributions obtained with the particular '"" spectrometer, it was concluded that the geometry c E of the detector was such that the escape of the iodine 0 E K fluorescence from the crys tal surface was negligible. "- (f) It was therefore considered legitimate to set the >- z 1.6 :::> conversion-efficiency factor equal to 0 u c, Absorption of Bremsstrahlung During Passage From Target to Detector 0 0 Table 2 lists the thicknesses and compositions of the absorbers in the path of the X-radiation and FIGU RE 9. Correction of experimental gives the values for the absorption corrections for data, 50-kev brem sstrahlu11!J spectrum. -----, Uncorrec ted experimental data; --- 18 points between 10 and 100 kev. These values experimental data corrected [or fin ite spectrometer were computed with the aid of current absol'ption resolution; ------, experimental data corrected coefficient tables [48]. The inaccuracies introduced also for absorption. in the absorption COl'rection due to impurities in the TABLE 2. Absorption of bremsstmhlung during passage from target to detector ""'0 12 Composit io n a nd thickness of absorbers )( .s;;; Beryllium wiudow,· 0.272 g/cm ': 99.27 % Be: 0.25% BeO:O.lO% Be,C; 0.15% ~ 10 AI; 0.1 5% Pe; 0.080/, Si. :J Aluminum window, 0.004267 ~/c m ': 99.5% AI ; 0.1 to 1% Fe (considered as 0.5% in calc ulation) ; traces of a; c Cu. I\'i. P b, Si, Ti, V, Zn, were c n eglected. 0 8 .s;;; Air, 0.1077 g/cm': 78% N ,; 21% 0 2; 1% A. u > T otal absorption correction as a function of quantum energy Q) -'" 6 ..... c Quan- Correction QlIHn- Correction turn tum E energy factor energy factor 0 4 E kev kev .... 13 I. 52 58 1. 06 III I- 18 L 29 63 I. 06 2 23 I. J9 68 1.06 z ~ 28 1. 12 73 l. 05 0 33 l. 09 78 I. 05 U 38 l. 08 83 1. 05 43 1. 07 88 1. 05 48 I. 07 U3 I. 05 53 I. 03 98 I. 05 key & '"1'110 composit ion given [or the beryllium window is that of the a,'eragc stock F I GU RE 10. COTrection of exp erimenial data, 100-kev brems oCthe Brush Beryllium Co., whofllm jshed the window to the tube manufact urer. strahlung spectrum. 6 Since this paper went to press, the spectrometer efficieucy belo\\' 20 key has -----, U nco rrected experimen tal data; ------, experimental data decreas ed over the m onths, Some of the remarks belo w, about the lo\\'er por co rrected for finite spectrometer rcsohltio n; ---, experimental da ta cor tion of the spectrum, m ust accordingly be !'egarded with some caut ion . rected also lor a hsorption. 114 together with the distribu tions obtained in the For the present. tudy, a sliohtly different approach energy interval between 13 and 100 kev after cor was chosen. Because of the primary interest in the recting for finite spectrometer resolution and after spectrum as it emerges from the target of a com~ additional correction. fol' absorption. mercial X-ray tube, it was decided to apply the It is interesting to compare the spectra of figure 9, necessary corrections not to the experimental data obtain.ed 'with the beryllium-window tube, with the but to Sommerfeld 's expression for the cross section Compton-scattered-bremsstrahlung distribution of for thin-target bremsstrahlung. In this way, one fi gure 5 from a tube with a window equivalent to arrives at an expression for the theoretical thick ~-mm aluminum. The spectral peaks faU in both target spectrum, and can compare it directly with cases between 20 and 30 kev (note that the change the experimental spectrum emerging from the thick in energy due to 90-degree Compton scattering of a target (that is, after correcting the experimental data 30-kev photon is negligible). Even after all the for absorption in air and in the other intervening corrections for absorption in the materials between materials). This procedure seems to be especially target and detector have been taken into consider appropriate to this case, as only a rough es timate of ation, the spectra from the beryllium-window tube the theoretical corrections has been made. still show a sharp intensity decline for energies below A precise calculation of the bremsstrahlung spec 25 to 30 kev. There is, furthermore, a quite dis trum emerging from a massive target would require tinct irregularity in the 50-kev spectrum near the the following: tungsten L-absorption edge e ee the uncorrected l. Integration of the thin-target spectrum over data) and only a comparatively small shift in the the path length of the electrons in the target, con location of the ensuing pectral pea.k with an increase sidering that, as the exciting electrons penetrate into of the exciting energy from 50 to 100 kev. These the target, they soon lose their monoenergetic and fa cts make one suspect that the shape of the low monodirectional character due to nonradiative colli energy section of the spectrum is a property of the sions within the target material. In consequence, target rather than of the tube window. the observed brems trahlung spectrum is not the one The low-energy peak of the 100-kev spectrum is produced by a monodirectional, monoenergetic elec considerably more pronounced than that found by tron beam, as would be the case for a thin target, Johansson by scintillation spectrometry [34] or that but is produced by a stream of electrons distributed found by Beekman by crystal spectrometry [4 1]. in energy over a wide specLrum and having a com Experiments wi th two other N aI (1'1) detectors and plicated spatial distribution. an X-ray tube with higher inherent filtration con 2. Correction of this spectrum for the losses in firmed that this difference is only in part due to the exciting electrons due to backscattering fl'Om the fact that an X-ray tube of lower inherent filtration target. was used for the present study. Also, it was found 3. Correction of this spectrum for the losses in not to be a characteristic inherent in one particular emerging photons due to their absorption within the J aI eTl) detector. target m aterial, taking into consideration the com The second , higher-energy peak appearing in the plicated spatial distribution of their origins, which is 100-kev spectrum and in all other spectra excited at the result of the penetration and diffusion of the potentials above 60 kev, is no t a part of the brems exciting electrons within the target. In spite of strahlung, but represe nts the tungsten line spectrum recent advances [49], the general theory of electron (K at line at 59.5 kev). In spite of the correction , penetration and diffusion has not as yet progressed this line spectrum- although approximately of the to a point where a detailed calculation of this type righ t over-all width- still appears entirely unre would be feasible. What can be done at the present solved. This in part is probably due to the limited time is, at best, in the nature of a rough estimate. stability of the electronic equipment, and in part to the coarseness of the grid used in the discontinuous 3.2. Present Approach correc ting procedure. The dotted line in figure 10 indi cates the estimated course of the bremsstrahlung The present estimate followed the steps outlined spec(,rum after the subtraction of the tungsten in section 3.1 . As some of the corrections are known fluorescence peak and before correcting for absorp only by order of magnitude, it was considered ade tion. quate to approximate the bremsstrahlung cross sec tion by a simple analytical expression, and to carry 3. Theoretical Considerations out all integrations by graphical means. Following is a brief review of the procedure. 3.1. General Remarks a . Bremsstrahlung Cross Section Kulenkampff, in his original experimental work on the thick-target bremsstrahlung spectrum [11], According to theoretical considerations [50] elec applied a number of corrections. These corrections trons penetrating matter diffuse considerably before stemmed in part from experimental and in part from losing a substantial fraction of their energy. In fact, theoretical considerations. He arrived in this way in a heavy element such as tungsten, they become at a distribution for the bremsstrahlung spectrum practically completely isotropic before losing 10 per from a thin target, and compared it with the then cent of their initial energy. accepted theoretical expression. It thus seemed reasonable to assume in this first 115 approximation th at the bremsstrahlung was emitted c. Photon Absorption Correction by an isotropic distribu tion of electrons. Further more, even though the energy losses within the target If jk(x)dx represents the fraction of the radiation are discontinuous, it is for the present purpose justi of energ.v k produced at a depth x in the target fied to carry out the integration as if the energ~' were measured in the direction of electron incidence, the a smooth function of path length. Then, if i(E,k)dk is fraction of the photons of energy k leaving the th e nonrelativistic rate of bremsstrahlung emission target is given by f o'" jk(X) dx exp [-fJ.(k)x/tan eJ, per unit path length per unit solid angle, averaged where x/ tan (J represen ts the layer that the photons over all directions, in the photon-energy interval be have to traverse ,,,hen leaving the target, as seen by tween k and k + dk (E being the exciting electron inspection of the insert in fig. 2. The integral can be energ ~T ), the integral over the electron tracks in the written in the form exp[-fJ.(k)(xlk/tan eJ, where target takes the form (xlk is a suitable average of x over the function j k(X ). Although this function is not well known, the average valu e (X)k has been estimated, utilizing some general I (Eo ,k)= J ~Sk i(E,k)ds= J<;E~" i(E,k) (ds/dE)dE, (6) knowledge recently gained about the actual depth of electron penetration and bremsstrahlung produc where s is the depth of penetration, E is a function of tion [49]. For the present approximate calculations, s, and Eo is the energy of the electrons before their the value used for this average is one-fifth of the penetration and diffusion into the target. For the total electron range p = f °'" (ds /dE) dE, for all values actual calculations, values for i(E,k)dk as obtained of k. from the numerical computations of Kirkpatrick and The fullv corrected spectrum was th erefore Wiedmann [51] (which are based on the Sommerfeld represented 'by: formula) were used. They could be approximated by a function of the form 1" (E o,k)=exp[- }J.(k)p /5 tane] i(E,k) =~ [a - b In(k/E)]. lEo i(E,k) (ds /dE) [l-{3(E/E o)] dE. (6b) If i(E,k) is in units of 10 t4 kev per steradian per milli Essentially, this amounts to th e assumption of ampere per minute per unit kilo electron volt range j k(x) = o(x - p/5). This approximation is only ade per cen timeter , and k and E are in kilo electron volts quate for }J.(k )p«l. For }J.(k )p»l this exponential the constant a is approximately equal to 16.7 and b correction factor no longer holds, and the photon is approximately 4.64. The reciprocal stopping correction is approximated by [f(O) tan e] / }J.. power ds/dE was calculated with the aid of the Bethe For the present calculations, eq (6b) was used. Bloch formula [52] . The absorption-correction term was evaluated by setting the average depth of penetration (p /5) equal b. Correction for Backscaltering of Electrons to 7.53 X 10- 4 em for 100-kev electrons and to The electron backscatter from the target was esti 2.29 X 10- 4 cm for 50-kev electrons. The total mated, usin g Bothe's experimental results that give absorption coefficient for tungsten was obtained the probability (3' (E/Eo)d(E/Eo) for an electron of from curren t tables [48]. known ini tial energy Eo and of known direction, to be backscattered with a known fraction E/Eo of its 4. Comparison of Experiment and Theory primary energy [53]. For scattering materials of high atomic number and for a scattering angle of 50 ± 15°, Bothe obtained for this probability a func In order to facilitate a comparison between theory tion that increases first slowly with increasing values and experiment, the theoretical spectra correspond of E /Eo , then rises sharply to a maximum for E/E~ ing to eq (6), (6a) , and (6b) are shown in figures 11 somewh at above 0.9, and finally falls off to zero for and 12 , along with the experimental photon-energ:v E /E o= O.1. A graphical integration of this function spectra. The experimental spectra were obtained yields (3 (E /Eo), the fraction of the total number of from the fully corrected number-of-photons spectr n, electrons, averaged ove rall angles, which is back of figures 9 and 10 by weighting the ordinates by the scattered with energies between a given energy E and respective photon energies and dividing b~~ 2.92 X the maximum energy Eo . According to Bothe, 10- 5, which is the solid angle in steraclians sub about 75 percent of the electrons incident on a block: tended by the spectrometer. of tungsten are scattered bacle This agrees very Considering the roughness of the theoretical esti well with the observations of Seliger [54]. It was mate, the over-all order-of-magnitude agreement is th erefore decided to usc for the present estimate quite good. There is, however, a remarkable differ Bothe's backscattering results b:v appl~~ing a correc ence in shape between the calculated and the tlOn term l - {3(E/E o) to the integrand of eq (6). measured spectra, the peak of the measured spectra The equation then becomes in the neighborhood of 30 kev giving the impression of being superposed upon a continuous spectrum I'(Eo, k)= i(E,k) (ds/dE) [l - {3 (E /E )]dE. (6a) similar in shape to that of the calculated one. l Eo o Figure 8 shows clearly how the 30-kev peak becomes 11 6 4,----,-----,-----,----,-----, more and more intense as co mpared with the re t of the spectrum a the exciting poten tial i increased. As pointed out in section. 2.7, the pronounced 30-1-:e\T )( peak was found with two different detectors and two X -ray tubes of differen t makes and inherent filtra tions. Even if iodine K escape llad not been found negligible with the detectors employed, the peak > could not have been attribu ted to iodine K escape """ from the tungsten-fluorescence lines for two reasons: Z a for the iodine K escape from the tungsten K-fluo E rescence lines, is too pronounced. Second, and quite 2 convincingly, spectra determined experimentally behind varying thicknesses of different absorbers , :3 showed that a sufficient amount of filtration removed the spectral peak in the neighborhood of 30 key almost completely, although it absorbed only a .. comparatively small fraction of the tungsten K 4 . ... fluorescence . 10 20 Another point of interest is the steep decrease of PHOTON the experimental spectral intensity immediately below the 30-kev peak. An inspection of the FIGURE 11. Comparison of expeTiment and theory, 50-kev bremsstmhlung. theoretical curves of figures 11 and 12 shows that the I , No correction for electron backscatter 0 1' photon absorption of the bremsstrahlung within the target absorption ; 2, theoretical thiCk-target spectrum corrccled [or electron backscatter; 3, theoretical tbiek-target spec is predominan tly responsible for the peaks in the trum, full y corrected; 4, experim ental spectru m. fully corrected theoretical distributions. This strengthens the suspicion that the shape of the low 8,---,---,---,---,----,---,---,---,----,---, energy portion of thick-target bremsstrahlung spectra is strongly influenced by the choice of the target material. Thus, in the case of a beryllium-windo\\ )( 7 tube with a tungsten targeL, the effectiveness of the ...J low-absorp tion window for passing low-energy ~ a: X-radiation seems to be o ffset--at least above ~6 z 12 .12 kev, the energy of the tungsten L-absorption edge--by the strong absorption of these radiations within the tungsten target. The present measure ments were not carried to low enough voltages to determine conclusively wheLher or not there is an appreciable rise in intensity below the L-absorption edge of tungsten. c ·11 a :3 E The author is indebted to Professor L. F. Talbott > x.. of the Catholic University of America for his interest l;"2 in the work, and to Evans Hayward of the National a:.... Bureau of Standards, whose vast experience wa z.... invaluable for the accomplishment of the experiment. ...JI Thanks are also due to U. Fano, W. Miller, L. V. 117 [29] R. Hofstadter, Phys. Rev. U, 100 (1948). 5 . References [30] R. Hofstadter, Phys. Re v. 75, 796 (1948). [3 1] J . A. McIntyre and R. Hofstadter, Phys. Rev. 78, 61 7 [1] Handbuch der Physik 23, 458 (Julius Springer, Berlin, (1950) . 1926) . [32] S. A. E . Johansson, Nature 165, 396 (1950). [2] H . Kulenkampff, article in Physics of the Electron Shell, (33] R. Hofstadter and J . A. McIntyre, Phys. Rev. 79, 389 H. Kopfermann, editor, Fiat Review of German (1950) . Science (1939-1946), Office of Military Government [34] S. A. E. Johansson, Arkiv Fysik 3, 533 (1951). for Germany, Field Information Agencies T echnical [35] L. Madansky and F. Rasetti, Phys. Rev. 83, 187 (1951). (1948) . [36] T. B. Novey, Phys. R ev. 89,672 (1953). [3] W. Finkelnburg, Kontinuierliche Spektren (Julius [37] G. E . Owen and H . Primakoff, Phys. Rev. 7 U, ::i. GOVERNMENT PRINTING OfFICE: 1955 118