Proton induced X-ray emission as a tool for trace element analysis

NUCLEAR INSTRUMENTS AND METHODS 116 (1974) 487-499 ; C NORTH-HOLLAND PUBLISHING CO. PROTON INDUCED X-RAY EMISSION AS A TOOL FOR TRACE ELEMENT ANALYSIS F.

FOLKMANN, C. GAARDE, T. HUUS and K. KEMP

The NielsBohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark

Received 30 October 1973 For protons and heavier ions with energies in the MeV range we have studied the production of characteristic X-rays from elements with ZZ 13 using a semiconductor X-ray spectrometer. Various competing background processes have been identified. Theoretical estimates of the background radiation due to bremsstrahlung from secondary electrons and due to proton bremsstrahlung have been evaluated, and on this basis we have cal-

culated lower limits for the sensitivity obtainable for the concentration of the trace element, ranging down it- values of 10-8-10-7 . Different projectiles and incident energies hale been compared, and it is shown that this method of X "r:,y production is about 3 orders of magnitude cleaner than van Ix" obtained by electron bombardment.

1 . Introduction We have investigated the possibility of measuring trace amounts of a great variety of elements employing bombardment with protons, alpha particles or heavy ions for generating the characteristic X-rays, and a cooled Si(Li) crystal for detection . Using this method of excitation has several advantages over the more conventional ones, using electrons or photons (from X-ray tubes or y-sources) . In particular it gives a lower level for the continuous background radiation, which implies a better sensitivity. The development of solid-state detectors with a good resolution has made it possible to measure the presence of many elements simultaneously in one run. We have focussed our attention on the limitations ofthis method, mainly trying to understand the characteristics of the main process, i.e . the emission of the characteristic X-rays, and of the dominating background radiation. Knowing the dependences on various external parameters, such as the type of projectile, its energy and the detector resolution etc ., one should have a better basis for choosing the best. experimental conditions and also for understanding ±he factors limiting the method .

electrons and the energy gained by the transitions may be emitted as the characteristic K or L X-radiation in which we are interested . The energy may also, especially for low-Z elements, convert to kinetic energy of Auger electrons. This branching is characterized by the fluorescence yield co i which represents the probability that X-radiation is emitted when the vacancy in the i-shell 'becomes filled . The characteristic X-radiation is to a good approximation isotropically distributed. Hence an element (Z, A) in a thin sample will give rise to a number of X-ray counts n i in a detector occupying a solid angle of AD steradians, which is given by

2. Production process

The atoms considered are bombarded with protons, alpha particles or heavy ions produced in an accelerator with energies in the range of 1-10 MeV per nucleon . These heavy particles will interact with the atomic electrons, having a substantial chance of ejecting one . The holes in the inner electron shells, e.g. in a K- or an L-shell, are subsequently filled by one of the outer

tz _A n, = ai :"Is Nv

AM 41r

Id Cab;,

where the emission cross section ai"I =al l' a), can be written as a product of the vacancy production cross section ai- and co,, the index i standing for 1:, L . M, etc. The number of incident particles is NF and t=/AM represents the number of atoms of the elerient (Z, A) per unit area, t_ being the equivalent thickness in mass per unit area and M the mass unit, 1 .E6 x 10-1 ' g. The internal efficiency of the detector at the radiation energy is Ed, and Cebs is the absorption reduction factor accounting for the attenuation of the X-rays on their way from the sample to the detector passing through windows, air, external absorbers and perhaps a short distance in the sample. For each shell the characteristic X-rays are normally split into groups, e .g . the transitions to the K-s'aell into Ka and K p and the L-shell transitions into L L,1 and L.. When one measures only one component of

487

488

F. FOLKMANN

the radiation, one has to add a factor to eq (1) giving the probability of the observed emission relative to the total X-rayemission leadingto the shell. Inversely, the emission cross section for a certain shell is the sum of the emission cross sections obtained from eq. (1)foreach of thecomponents of the X-radiation leading to the shell, each calculated with its energy-dependent parameters Ed and C,,. We have measured the emission cross sections ftir 2 and 3 MeV protons for a number of elements, and the results are shown in fig. 1 . For the measurements we used a Si(Li) detector with an area of 30 mm' and a resolution which, for the Mn Ka line, was 160eV fwhm . The detector was placed at 90° relative to the proton beam, and between the thin target and the detector there were two 0.013 mm Be windows and a 0.05 mm Be absorber. The thicknesses of the targets, ranging from 1 to 60 pg,!cm2, were measured simultaneously with the yield of the X-rays by recording the protons scattered elastically through 135° and, normalizing on the Rutherford cross section . In fig. 1 are also shown the theoretical values, which one calculatn for the K X-ray emission cross section QK,r. = 4K" wK, using for aKc either the binary encounter approximation of Garcia et al.') or the Born approximation described by Merzbacher and Lewis?). For the latter calculation we have used the numerical values of ref. 3 with the relativistic screening parameter - z-o .312. = UR(Z) 0. (Z) (2) (Z-0.3)2 13 .6eV ( 274 There UK(Z) is the ionization potential of the K-shell for the element Z and the second term of eq. (2) approximates the relativistic effect for the strongly bound K-electrons of high-Z elements). In both cases we have for tK takenthe "fitted values" of Bambynek et al°). Also for higher shells, i= L or M, one can use the above mentioned approximations for Qit,e and teke a>, values from ref. 4 making an appropriate average over the subshells of interest. Nevertheless we have only plotted the experimental L X-ray cross sections and have restricted ourselves to show the theoretical curves forthe K X-rays, as we will mainly argue from the systematics of the K X-ray emission crosssections in this paper. Theexperimental points in fig. 1 are seen to be close to the two theoretical estimates; which exhibit the correctsystematic variation with theatomic number Z. The binary-encounter model accounts best for the points corresponding to the lower Z-values; while for higher Z-values, where we are on the rapidly varying

et al.

part of the curves, the Born approximation results are best, partly due to the relativistic correction term of eq. (2). We conclude that we can employ either of the two theories for surveying the systematic behaviour of the crosssections,but for the quantitative applications one has to rely on measured cross sections and then interpolate between them, guided by the theories. The emission cross section increases with the. energy El of the projectile and reaches for i X-rays and a given target element Z a maximum near the energy Et;z~A 1M/m" Uï(Z), where the velocity of the projectile equals the mean velocity of the i-shell electrons of binding energy UI(Z). A 1 M is the mass of the projectile and m is the electron mass. For higher energies the cross sections are slowly decreasing. Generalizing the proton results to other projectil.°m is, according to thesetheories, very simple . If we consider a projectile of A, nuc!-;ons, charge Z, and energy El, we have thefollowing scaling law 2

-4','A, (Ei) = Zt

"'

"'P .."

(

E All1

e

(3)

i .e. the cross section equals that of a proton of the same velocity (energy per nucleon) multiplied by the charge of the projectile squared. In reality, high-Z projectiles will have cross sections that deviate somewhat from eq. (3), as they cannot be treated as point charges such as protons or alphas, and because they produce multiple vacancies') which introduce a shift

z â m

w N N ô Û

Fig. 1 . Experimental emission cross section for Kand L X-rays produced by proton bombardment . The solid curves represent thetheoretical results for theK X-rays in the Bomapproximation (BA) and in the binary-encounter approximation (BEA), respectively.

TRACE ELEMENT ANALYSIS

in the X-ray energies and changes in the fluorescence yields. Also the molecular orbit excitation mechanism at lower projectile energies can significantly change the cross sections, especially making them greater when the projectile is similar to the target atom . For our applications we will not discuss these complications further and in comparison with the proton results we will only apply the scaling estimate, eq. (3). 3. Background radiation' After having established the characteristics of the production process, relating it to theoretical predictions, the next step is to identify those background radiations, which compete with the main process and thus determine the limitations for the sensitivity. From experiments it is found that the characteristic X-ray peaks are superimposed on a continuous background of electromagnetic radiation . The main reason for the interest in heavy-particle induced X-ray production is that this background radiation is lower in intensity than that encountered by the more conventional methods of X-ray excitation, e.g. by electron bombardment or by exposure to electromagnetic radiation from X-ray tubes or radioactive sources. We will consider the problem where we have a trace element placed on or imbedded in the bulk material of the sample . This host material or carrier substance we will name the matrix . Our aim is to find thesmallest concentrations of the trace element one can detect by treasuring the X-ray signal from it against the continuousbackground from thematrix. We have ascribed the continuous background radiations observed from a typical matrix as due to either 1) bremsstrahlung from secondary electrons, 2) bremsstrahlung from the projectile, or 3) Compton scattering of )-rays from nuclear excited states . Other factors which limit the obtainable sensitivity are 4) half width resolution of the detector and 5) tails from characteristic X-ray peaks from the matrix; 6) limited counting rate of the detector ; 7) heating of the sample ; 8) charging of the sample. The first mentioned type of radiation is the most important. Together with the second type, it is discussed more carefully in the following sections. These two types of radiation are always present when one employs protons for generating the X-radiation, and constitute the limiting factor irrespective of the type of detector used for measuring the X-rays .

489

The level of the limit depends, on the detector, in particular for the semiconductor detector through the factors (3), (4). (5) and (6) listed above. The third type of radiation is as fundamental as thefirst two in the sense that the ion beam certainly will, excite nuclear levels, which through their decay will give rise to y-radiation. But the Compton scattering which converts the y-rays to a continuous background in the X-ray region depends on the material of the detector and its near surroundings. This background can often be of similar importance as the proton bremsstrahlung, which it also much resembles in its weak dependence on the radiation energy. It is of course strongly dependent on concentrations in the matrix of special elements which have particularly high cross sections for excitation of low-energy nuclear levels, such as `F and "Na. As it is sensitive to the detailed matrix composition of the samples, we will not at present go further into a discussion of this source for continuous background radiation. Apart from the typesof radiation we have indicated, the five additional factors listed will influence the sensitivity obtainable . They are all less fundamental in the sense that they can be counteracted by appropriate measures. Regarding (5), the low energy tails from characteristic X-rays, this applies especially to solid-state detectors where insufficient charge collection gives rise to a low-energy counting rate which typically is about 300 to 1200 times less than the peak value. It can be suppressed by shielding the edge region of the detector by means of a smaller window aperture, or by wing a guard ring detector . More radically, one could, of course, use a Bragg spectrometer instead whereby one avoids this problem and in addition reduces the radiation (3) and has a smaller width of the X-ray peaks by, say, one order of magnitude. It does not. however, have the capability of simultaneously measuring several elements . The finite width of strong X-raypeaks from thematrix, (4), will also be aproblem when they mask the characteristic X-ray lines from the trace element, which again calls for detectors with high energy resolution. Concerning (6), the counting rate limitation, this is also determined by the detector and in particular its amplifiers and their pile-up rejection system. Apart from future developments extending the present 20 000c/s to higher values, one can improve the conditions by employing absorbers between the target and the detector in order to reduce the numerous low-energy pulses. Concerning (7), the heating of the sample and its

490

F. FOLKMANN et al.

presence in vacuum, this may be a problem if volatile compounds are present. Also the danger of simply melting the sample sets a limit to thepermissible beam intensity and forces one to use good heat-conducting backings (e.g. C-foils), to defocus the beam, or to make a coating on the samples. Concerning (9), samples with a poor electrical conductivity maybe chargedduring the bombardment, and thus electrons floating around in the scattering chamber may be accelerated . Hitting the sample or walls of the scattering chamber, their bremsstrahlung can easily be the main contribution to the background radiation . This phenomenon can be avoided by making a better electrical connection to the bombarded area, e,g. by using a C-coating of the sample. One can also make an electrical shield near the target, which is kept at a proper voltage to collect the free electrons. In the following we will consider thin sampleswhere we can apply the initial value of the projectile energy throughout , the samples, rather than thick samples, where one hasto take into accountthe slowingdown of the projectiles. This we do for several reasons. Firstly, the situation is more clear-cut when we have adefinite projectile energy, which makes the calculations simpler and the arguments and conclusions more direct . Secondly, both the measured and the predicted optimum ;peak-to-background ratios turn out to be best for a thin sample, and thirdly, the extrapolation of the results to the case of thicker samples is straightforward and does not add any new features to the picture.

energetic secondary electrons ejected from the matrix atoms by the bombardment with the beam particles. These electrons come from the strongly bound orbits, where before the impact they already have high velocities . The chance dY that an electron ejected with a kinetic energy E6 produces abremsstrahlung photon of energy between E, and E,+dE, is given by dY(E6, Er) =

fEE

da,(E c . E,)

>

(4)

S6t(EeEA6~ M if it is assumed that the electron does not leave the sample . Here da,ldE,(E.,E,) is the bremsstrahlung cross section for an electron of energy E. and S~ the stopping power in energy per mass per unit area. Hence dE,,/[S6r (E,)AMM] is the number of atoms per unit area corresponding to the energy loss dE. of the electrons moving in the matrix of atoms with mass number A M. The probability eq. (4) reflects thebalance between the two competing processes, which the electrons are subject to by their passage through the matrix. First they will be accelerated in the electri-, field of the matrix atoms and thereby emit electromagnetic radiation, bremsstrahlung, ranging in energy from that of the electrons and below. Second the electrons will be slowed down by the various collisions they suffer and in that way loose energy. The effective cross section dab for the projectiles indirect production of bremsstrahlung is given by de(E,)

=

,

E,

dY(E6 > E,)

dE (E6) dE6

>

(5)

4, Brenisstrablang from secondary electrons where da~/dE,(E,) is the cross section for the producExperimentally one finds that the continuous back- tion of secondary electrons in the energy interval ground of electromagnetic radiation is very strong E~ to E,+dE,. If one inserts eq. (4) into eq. (5) and at low radiation energies, butthat it decreases rapidly interchanges theorderof integrations dE~dE6~one can when the energy becomes larger than the energy write eq . (5) in the form ///''°° _ 4mAtM _ 4m El _de T~ E' = 9(Ee> E,) ae(E,) dEe > (6) M Ai ' (A, M+m)z dE, (E,) >l E, which represents the maximum energy that can be where transferred from a projectile with mass AlM and I da, . E,), energy E, to a free electron of mass m. g(E_ E,) = (E) M (E > (7) Ser Am dE, When T. is larger than the energy E, of the characthe background teristic X-rays from thetrace element, and radiation will result in a severe limitation on the °° da * (E6)dE6. (8) ac(Ee) _ concentrations which can be measured. Hence,' it is fE, dE. the energy region E,>T. that is of interest for trace element analysis, and we shall show that the' backThe integrated cross section a.(E,) can be calculated ground radiation in thebegining of that region canbe either from the Born approximation2,3) or by meansof for by the bremsstrahlung emitted by the the binary-encounter approximation' 5,6) which will accounted

49 1

TRACE ELEMENT ANALYSIS

be discussed in the next section. One finds that the differential cross section is a very rapidly decreasing function of E,, i.e . (Ee) x E.- I, dE e where p;t~ 10 for E. >Tm. This implies that themain contribution to the integral eq. (8) comes from energies Eajust above F.',, and hence that the main contribution to the integral eq.(6) comes from energies E. in the region 0-20% above E, . For this reason it is an acceptable approximation to consider the thin sample as thick to the secondary electrons, as indicated by the infinite upper limits for the integrals, and to disregard the angular distribution and scattering of the electrons, even forfoil thicknesses of the order of 30 ug/cm2 . For the stopping power Saf(E.) of the electrons, or rather for the mean energy loss per atom per unit area we have used the value Sef(Ee)AmM = CIZMIEe, with C, -- 6 x 105 b keV 2, which is found to be in a reasonable agreement with the values stated by Kirkpatrick and Wiedmann7) and with the usual stopping power formulal°). The coefficient C, is not quite constant, as it depends on the logarithm of E.1 Z, but in the relevant energy region E, 2-25 keV this additional variation is small, in particular for the lighter matrix atoms. The kind of average that enters in the present problem (7) is somewhat different from the average involved in the mean energy loss, which could justify the use of a 30% lower value of Cl, but the corresponding increase in dabmight b-- balanced by the fact that we disregard that some of the secondary electrons may leave the sample by large-angle scattering. For the bremsstrahlung cross section da,/dE, we have used. the results of ref. 7 and checked with the more recent calculations ofref. 8, which are both based on the Sommerfeld theory. For the cross section integrated over all angles of emission of the photons we have used the approximation 2

If one measures the radiation at 90° with respect to the beam one should take into account the angular distribution of the bremsstrahlung e ) that has a higher probability of being emitted perpendicular to the direction of motion for the electron, which is forwardpeaked') relative to the beam . For the use in yield expressions assuming isotropy one can take this effect into account by increasing C2 with some 50%. Although, from the above arguments, it would be an advantage to measure at small angles with respect to the beam, it is not likely to be any large effect, because the forward peaking is rather moderate, and because of multiple scattering in the sample. In total we obtain for g(E,, E,) the estimate

F,

Znf fJ(E~, E)=C 3

with C3 ;:t: 2 .8 x 10 -6 (keV)' 1 , from whichis seen thatg has avery weak dependence on E, (and E.) as compared to that of ae(E,) on E.. The constant C3 may well bcs uncertain to within a factor of 1 .5, but the aim of our calculation is not to make a very accurate evaluation of the background radiation from the matrix, but to establish that the process considered can account for the observed radiation and in particular that it can reproduce the experimental dependences on E, and Zaf . For these reasons we have also confined the calculations to the case where one matrix element is dominating, but the generalization to more comrlex samples is relatively straightforward. 5. Production of secondary electrons In the binary-encounter approximation one disregards during the collision all other forces than, the Coulomb interaction between the nuclear projectile (1), which has charge Z,e, mass .4,M, velocity t', and energy El and the struck electron (2) of mass tit, which at the moment of the collision has the velocity e2 in theatom . If v2is assumed to be isotropically distributed, one can from the Rutherford cross section for the binary encounter') calculate the average cross section for adding the energy AE to the kinetic energy E, of the electron, which for the cases of interest gives

_ da nZ ; e° x (AE, ul, a2) Er EE e =* 3a 1 2 u2 AE 3 dE2 with C2 1 .6 b keV, since for the relevant energies Ee 0 <_ dE <- b, v2(vi+3v2), if just above E, the coefficient C2 is approximately constant . This is true in the classical approximation if b 5 .2E _< a, I(v1+17,)3+2(V2-ll2)3, where the collision index n=Lnfe2/f1v. Z 0.5, as is the 0, if a -< AE and 2mv2 5 (A 1 M- 111) u1, case for the experimental conditions discussed in the (10) present paper. du,r

(Ee , Er) =

C2

492

F. FOLKMANN et al .

where

all contributions come from electrons with

E2 ~ F2o(Et I AE = U+Es) ,: )2 AE \2 dE J AE -VI + ~tn lm + 2mv 2dEl 2mvt t At Mvi u2 . CU2 mJ which means that the result is only sensitive to the tail of the velocity distribution, eq . (11), with V2> 4A, Min [El -EZ-#(AtM-m)vtv2] , I-4El2rnvt-v t 1 . For the binding energy U in eq . (11) (A IM+m) 2 we have employed the values in accordance with the Slater rules, which seem , ) to give a satisfactory 4A,Mnt -E 2 a _ [El +J(A l M-m)v t v2 ] . representation of the atomic states. An improvement (A,M+m) 2 in the calculation would involve more detailed velocity Considering an electron with the binding energy U, distributions for the subshells and for higher energies and requiring that it leaves the atom with a kinetic Ee relativistic wave functions for the K-electrons . Fig. 2 shows the result of the binary-encounter enema E., one has AE= U+ E, . In order to calculate the energy spectrum for E, one has to integrate over calculations outlined above. For energies E.>Tm it is the normalized velocity distribution f(v2) in the atom, evident that the magnitude of the contributions from for which we use the Fock distribution as an average the different shells is strongly dependent on the binding energies, mainly throughrelation (11) and AE = U+ E,. over a shell In addition to the ejected electrons arising from direct s 2 32vo t) 2 impact of the projectile on the atom there will, Î(v2) = especially for light matrix atoms, come a considerable (v2+v.)4 number of Auger-electrons in connection with the where vo-(? Ulm)}. subsequent readjustment of the atoms. The energy The resulting cross section for emission of an electron of these electrons will in the cases considered be low with energy E, is therefore compared to T., so they are of no importance for the background radiation, and hence they have not been = I°° du da = (E~) -La (dE U+Ee , VI 1 V2).Î(v2)dv2 . included in the figure. dE, Jnz=o

vt

(12)

This result for an electron with binding energy U has to be summed over the atomic electrons to give the total cross section dtr,/dEJE,) in eq. (8). The integration, eq . (12), can be carried out explicitly, but we leave out the rather elaborate formula, because a similar result has been published by Rudd et al.'). When one introduces the electron energy E2 = J MV2 as a variable instead of v2, the integration eq. (12) will be divided into two parts by the energy E2 = Ego. for which either a(E2)=dE in the case dET.. E2o

+

.4' M~1 = Et J l2m

Cl _AE Et /

+

1 + 1 --l} _ AE (AlM+m)2 ' 2A,M[

E,

in KeV

Fig. 2. Binary-encounter calculation of the energy distribution ( EE/ 4A, Mrn of electrons ejected by impact of protons. The binding energies arecalculated from the Slater rules and in parentheses is given In the importantcase, where AE>T. the first integral the numberofelectrons in each shell. Tm is themaximum energy corresponding to OS E,.5 E20 gives zero, and hence transferred to a free electron.

TRACE ELEMENT ANALYSIS

493

A calculation using the Born approximation') has electrons, which increases rapidly with E approxialso been made for the K- and L-shells and for a few mately corresponding to the power 4. The term (Z,/A,-ZIA)' arises from interference examples of interest it gave the same result in the region E. >Tm. between the radiation of the projectile and of the An important feature of the cross section for pro- recoiling nucleus, and implies that one can make the duction of secondary electrons is its scaling with the electric dipole bremsstrahlung, eq. (14), vanish by projectile parameters Z,, A1 and E,, similar to that of employing projectiles of the same charge-to-mass the vacancy production cross section eq. (3) ratio as that of the nuclei in the matrix. In that case higher multipolarities will be important, but of much du~(Z, A1, El) du . (proton, E,/A1) lower intensity than given by eq . (14) without the (EJ, (EJ= Zi dEe dE . cancellation. factor. As typically Z/A x 1, it is from (13) this point of view an advantage to use '2He or 1.10 as projectiles. We have tested this experimentally by which is understandable, as it is a part of the same bombarding the same targets with 3 MeV alpha physical process. The scaling is evident from the above equations, particles and JMeV protons, which have the same velocity or energy per nucleon. According to our when one uses the following approximations discussion above the alpha particles should give a 4m E, factor 4 more indirect bremsstrahlung, whereas the 6 z :, 2mv, v2 , v; v, , direct bremsstrahlung of the projectile should only M A, be found with protons. This is verified by the results E, for a pure carbon target, where the dominant lowTm 4m a ~ ~ Ll +2mv,VZ, energy shape due to the electrons was the same, while M A 1 the level at higher energies for the alphas was at least and 3 orders of magnitude lower than for protons. This clearly demonstrates the effect, but the bombardment of more complex matrices or with higher a-energies gives comparable amounts of continuous high energy bar'.-.ground for alpha particles and protons. Actually, 6. Bremsstrahlung from projectile the -high-energy level seemed even higher for alphas, Thesame was The large accelerations which occur during close and new peaks of nuclear origin grew up.a2S projectiles, collisions between the projectile (Z,, A E,) and the found for 30-80 MeV 16 0 and 25 MeV nuclei (Z, A) of the matrix atoms result in a direct as compared to the result obtaine,.i with 1-3 MeV production of bremsstrahlung which becomes of protons. Thus it seems that other processes often take interest at higher radiation energies. The cross section over the role of the projectile bremsstrahlung and. like it, have a very weak dependence on the spectral for the process is given by the formula") energy. ' duB =C Il ZÎZ Z_, __Z ' The angulardistribution of the dipole bremsstrahlung (14) 4 from the projectile is given by dE, E, E, (A, A) W(0) = 1+Ja2 -J(3 COS' 0-1), where C4 = 4.3 x 10 -4 In

4E E b keV (Z,ZE, lA, x 100 keV) )

where 0 is the angle between the incoming projectile and the quanta observed . The deviation from .,-) isotropy is determined by the coefficient 21(q given in refs . 11 and 12 and for our case one should use either the Born approximation12,13) result _ 3 for q Z 0.5 and S < 0.03, az - 1 In [4E,/E,]

is approximately constant and we have taken the classical approximation value for C4 which is relevant under the experimental conditions. The cross section eq. (14) is a slowly decreasing function of E, and E, . As a function of E, it is nearly flat as compared to the high power of -8 from the indirect bremsstrahlung or the classical result produced by thesecondaryelectrons. Also the decrease 3 , for q> 1 and ; < 0.03, with the projectile energy E, is in contrast with the aZ = 1 In ['/(1.78g)] behaviour of the contribution from the secondary_.

F. FOLKMANN et al. '

wherethe collision index n(Et) - Izt Z

CAt " 100 keVl ~ E,

J

and the adiabaticity parameter g

y(El

Er)-n(E,)

-

n(E,)

E

2Et .

In the cases of interest W(90') ;:0.8-0.95, which means that it is an advantage to have the detector at 90°, in contrast to the case for the two-step process where the bremsstrahlung should be peaked near 90°. However, the deviation of W(8) from isotropy is so small that it is insignificant in the present context. 7. Comparison with background measurements The theoretical estimate ofthebackground radiation outlined in the previous sections has been compared with experimental spectra. We bombarded relatively pure targetswith protons of an energy of 2 and 3 MeV. With the detector placed perpendicular to the beam direction, we found the cross sections for production of continuous background radiation shown in figs. 3-6 for a -Eig/cm2 plastic foil, consisting essentially of carbon, andfor a 200 pg/cm2 Al foil, respectively. The theoretical values are those obtained for the bremsstrahlung eq. (14) from the protons and [eq. (6)] from the' secondary electrons [eq. (8)] found from the

binary-encounter model eq. (12) and with a value C3=4.1 x 10'6 (keV)-' in eq. (9) to account for the angular distribution of the radiation. For the binding energy U in eq. (11) we have used the Slater rule values . From thefigures it is seen that the theoretical approx imations areacceptable . Alltheimportant experimental features are reproduced, andin particular it is possible to account for the dominant low-energy contributions by means of the bremsstrahlung from the secondary

~ toy to' 9~9 to` 10 to t0

i-

2

s

to

Er

20 in KeV

so

Fig. 4. Experimental and theoretical background radiation cross sections fora thin sample.

6~ekgraxd rad'v~tion .in 90, cross .ti for 2 MeV pratms on At - te~ ___ ~xguimm2

Er in KeV Fig. 3. Experimental and theoretical background radiation cross sections fora thin sample :

Fig. 5. Experimental andtheoretical, background raliation cross sections for athin sample.

TRACE ELEMENT ANALYSIS

electrons. Theoretically the proton bremsstrahlung at 3 MeV should be lower than at 2 MeV. This is more or less true forthe carbon target, but for Al the experiments show the opposite trend, which is due to contributions from Compton scattering of gamma rays . For the 2 MeV protons of fig . 5, the high-energy contribution can be explained qualitatively as due to proton bremsstrahlung, whereas for the 3 MeV pro- .. tons in fig. 6 the Compton contributions are dominating, giving rise to a slightly different Eidependence. On the basis of the qualitative agreement found for these examples, we conclude that the two bremsstrahlung processes through the indirect and direct production, represent a lower limit to the background whichunavoidably will be encountered in trace element analysis . Forheavyions with Z,/A, 1/2 we can in principle disregard the projectile bremsstrahlung, but in practice one finds comparable or even larger contributions from Compton scattering of nuclear de-excitation gamma rays. They behave in a similar manner, since both processes depend on the close collisions between the nuclei of the projectile andthe nuclei of the atoms in the matrix material. There is very little which can be done to reduce the bremsstrahlung background . The angular distributions are not pronounced enough to be exploited and the sample thicknesses which would allow the secondary electrons to escape from the matrix before they can emit radiation, areimpracticaily small. Forthicknesses

Y E B a olui vv

5

10

20

50

Er in KeV Fig. 6. Experimental and theoretical background radiation cross sections fora thin sample.

495

down to 20 pg/cm' of carbon we have not found any reduction due to a splitting of *c twc-step process. However, even though the secondary electrons are produced by the same type of process as that responsible forthe characteristic X-rays of the trace element, they are produced in different elements and moreover it is only the high-energy tail of the electron distribution which is relevant for the background radiation, whereas it is the low-energy electrons which give the largest contribution to the total inner-shell ionization . It is therefore possible to optimize the yield of the characteristic X-rays from the trace element relative to the background from the secondary electrons, by employing proton energies so high that the energy 7. is just below, say, half the X-rayenergy to be measured and hence the tail of the electron distribution is still kept relatively low. If one goes to even lower proton energies, the absolute intensities become so small that the other background processes, such as the bremsstrahlung from the projectile, become significant. In the following section we shall study the optimum conditions more quantitatively. 8. Sensitivity of the method With an understanding of the production processfor charyAeristic X-rays from thetraceelement(section 2) and with a method to calculate the background radiation from the matrix (sections 4-6) we can give a quantitative estimate for the limits to the sensitivity of the method . As a criterion forbeingable to measure a peak of the characteristic X-rays from the trace element on the background from the matrix, one can for a given detector demand that the signal-to-noise ratio S> l . i .e. that the number of counts in the X-ray peak is equal to, or larger than the number of background counts within an interval equal to the width of the peak . Forthebackground radiation dueto bremsstrahlung from secondary electrons and the proton bremsstrahlung caused by a thin carbon matrix, we show in fig. 7 the concentrations of the various trace elements which give S= 1 for a variety of proton energies . Amounts of material of this order of magnitude and higher can easily be detected when only a few different trace elements are present. The calculations refer to thin targets and include only the K-radiation from the traceelement, for which the yield of the K X-ray peaks has been computed from the Born approximation cross sections') with the screening ecl. (2) and the fluorescence yields of

496

F. FOLKMANN

ref. 4, as described in section 2 . From fig. 1 it is seen that these values are better than the binary-encounter results in the region Z Z 40, As spectrometer we have assumed a good commercial solid-state detector with a resolution of 150 eV fwhm at Er = 5 .9 keV, and scaled with a factor (Er/5.9 keV)'I' to account for the larger width at the higher energies. Within this width for Er equal to the K- energy of the trace element we have calculated the yield of bremsstrahlung quanta from the considered carbon matrix by means of the total cross sections eqs, (6), (12) and (14), using eqs . (8), (9) and the Slater rule values for U in eq. (11) . The details of the angular distributions are not taken into account. From fig . 7 it is seen that the smallest concentrations which are measurable according to the criterion S=1 are of the order 10 - ° for all trace elements, provided the bombarding energy is adjusted to give the optimum condition for the trace element considered. The optimum region corresponds to elements which have their K- lines between 1 .4 Tm and 4 Tm . 'With projectiles heavier than protons, one will get exactly the same result for the same energy per nucleon as far as the effect of the bremsstrahlung from the

et al.

secondary electrons is concerned, because the signal and the background have the same Zr-dependence, cf. the scaling eqs. (3) and (13). When Z,/AI & 1/2 one should expect the curves of fig. 7 with the dashed continuation at higher energies to be relevant. However, as stated in section 7, the Compton scattering of nuclear gamma rays becomes normally as important as the proton bremsstrahlung, and for practical applications it is also in this case safer to use the fulldrawn proton curves at all energies as the lower limit, even to expect the actual values to be somewhat hidher. In fig . 8 we have made the same calculations fce a fixed proton energy, but extended to other matrix atoms, and with the results for the characteristic X-rays of the L-shell included. It is advantageous to have light elements as the matrix because of the way the atomic number Z i enters in eq. (11) through the binding energy U of the atomic shells, and in the electron bremsstrahlung contribution through eq . (9) and the proton bremsstrahlung cross section eq . (14) . The yields for the L-lines, taken from fig. 1, should be compared to the same background radiation. In the diagram the Incurves are simply shifted in Z according to the energy of the L-lines, here represented by the Lp, energies. The criterion of a peak-to-background ratio S of 1 used in figs 7 and 8 is very convenient, as it is independent of external parameters, e .g. sample size, detection efficiency, including absorbers used, and the integrated beam intensity. It does, however, not take into account the rapid variation of the absolute yields which the 10Z _ Cmreentrati- giving a peck to baokgroend ratés _ of 1 for 3 M.V protons with different -lemmts as the matrix. 1~3 c .ft ô

cd U Ô

Fig. 7. Calculated concentration for which the K X-ray in a Si(Li) detector is as intense as the background bremsstrahlung from secondary electrons and protons. The X-ray production yields are found from the Horn approximation and the bremsstrahlung of the protons and the secondaryelectrons, estimated in the binary-encounter approach, is calculated in an interval 150 eV x(Exl(5 .9 kcV)jk around the energy,Ex of the Ktt line. For heavy ions as projectiles simple scaling gives the same curves for the same energy per nucleon (Er/Al) as for the secondary-electron part (with dashed continuation), whereas the projectile bremsstrahlung is lower.-

cJ

10~

lu, lu , 1ff 10

-

80 90 60 70 Element Fig. 8. Calculated concentration for which the characteristic X-ray intensity in a Si(Li) detector equals the background radiation for different elements as the matrix. The calculations are the same as in fig. 7 with the L X-ray yields taken from fig. 1 and at the energy of the Lpi lines. 10

20

30

40 50 Zof Trace

TRACE ELEMENT ANALYSIS

49 7

cross sections in figs. 1 and 2 give rise to for X-ray practical importance in measuring small concenenergies E.(Z)>T. . In order to secure that the X-ray trations. For the same number of incident projectiles peak is statistically significant, we have calculated the and the same energy per nucleon as protons one can, concentrations of the trace elements which correspond due to eqs . (3) and (13), divide the concentrations in to the condition : fig . 9 by Z t. This applies for a thin sample, but as the Counts in peak >_ 2 (background counts)t, where the stopping cross section also increases with Z;, we background counts are taken in the interval of twice the same result for samples with thicknesses adjusted tine fwhm of the X-ray peak . This criterion is, ofcourse, to give the same loss in energy per nucleon . For a thick dependent on the external parameters, and the con- san .ple, where the projectiles are stopped, it means that centration obtainable will be inversely proporüonal the sensitivity in concentration will be independent of to the square root of the number of counts in the peak. whether protons or other heavy ions of the same energy This number includes a product of the detector solid per nucleon are used. angle AD, the matrix thickness tM, the number of The sensitivity of the method for thick samples can incident particles, Np, the detector efficiency Ed and easily be calculated from the same computer code by an absorption reduction factor Cabs, cf. eq . (1). In the varying the proton energy down through the sample example shown in fig. 9, calculated on the same basis and adding the contributions. As the cross sections for as fig. 7, we have chosen AD =0.003 x4tr sr, t M = both the X-ray production froln the trace element and 0.1 mg/cm, an accumulated proton charge X = 10 pC the electron bremsstrahlung from tLe matrix decrease (corresponding to Np = 6 x 1013) and Ed = Cabs = 1 . with decreasing proton energy, the, dominant contribution comes from the surface where the energies are Choosing otter external parameters the result scales with (A9 tM g Ed Cabs) -* as long as the protons are not not much lower than the incident energy. The peak-tosignificantly slowed down in the sample. background ratio due to secondary electrons is thereThe G`1-times higher cross sections for heavy ions fore practically the same as for the incident energy on of the same energy per nucleon, eq. (3), are of little a thin sample. In some cases it can be even better, since the background is decreasing faster with E, than the X-ray production when E > T.. The proton bremsstrahlung eq. (14), however, increases with decreasing energy so in this respect the situation is worse than for a thin sample and one will get a smaller peak-to-background ratio by as much as a factor of 10. Hence, the result f^r a thick sample is roughly that of fig. 7 for the indicated bombarding energies, but with the lesel determined by proton bremsstrahlung raised somewhat, say, one order of magnitude . For light trace elements the absorption of the low-energy X-rad,ation in the matrix will of course be an additione.l problem which will make the results approach those of the thin sample values. One should expect the fundamental limitations of the technique considered according to the conclusions Fig. 9 . Calculated concentration for which the K X-ray peak in of this section to be in the concentration range from a Si(Li) detector is statistically significant, i.e. when counts in 10 -6-10 -7 . This is not exceptionally low, so after peak = 2(2-background counts)I . The calculations are per- surveying the problem of concentration sensitivity, formed with the same procedure as in fig. 7 and the result is there is reason to draw attention to tht: fact that one proportional to (proton charge-sample thickness " do-ed-Cabs)-} can analyze and handle with this technique very small for thin samples. For the curves shown the solid angle of the detector AQ = 0.003 x 4a sr, the internal efficiency sd = I and absolute amounts of material. Also the fact that the the absorption reduction Cabs = 1 . For heavy ions one can, semiconductor detector can record many trace :dements as the secondary-electron contribution (dashed line), use the in one run is important. As an illustration it can be curves with Ei/A1 for Eproton and divide by Zi for the same mentioned that from practical applications of the number of incident projectiles. The direct bremsstrahlung level technique based on proton-induced X-ray emission is lower than for protons, but Compton scattering from nuclear deexcitation gamma rays may often be higher than the proton one can normally measure concentrations of trace elements down to 10 ppm = 10 -5 without trouble. bremsstrahlung level .

498

F. . FOLKMANN et al.

With . some care choosing appropriate bombarding energies or elements of interest one can go down to 1 ppm and under favourable conditions with a clean matrix and few trace: elements present in not too high concentrations we have measured concentrations of the order of 0.1 ppm ---10' ? of, say Fe, Zn and Pb . We have used matrix thicknesses from 3x 10-3 to 2x 10 - ' g/cm', and with a beam size of 1/4 cm' we have detected absolute amounts oftrace elements down to 10 -tz g. This absolute value might even be significantly improved by focussing the beam. 9. C with electbombardment Electrons are widely used as an easy means of exciting X-rays, and they are in some respects similar to the nuclear projectiles considered here, i .e. protons and heavy ions of energies 1-10 MeV per nucleon. Because they have found a fruitful application in the determination 4 small quantities of elements via the electron micro-probe or scanning electron microscopes, we shall in this section compare our results with similar considerations for electrons. Both the nuclear projectiles and the electrons are produced by accelerators and the beams are in both eases handled in vacuum and focussed with electrornagnetic lenses. Typical electron energies are 2050 keV, produced in small-size generators . Electrons of energy E. have a cross section for ionization of the K-shell of the bombarded atom, which according to Gryzinskyt°) is given by 1 eas® U.X(x+1.) K 11

.7+(X-l)']J, +23X11n[2 (15)

with C s =1 .61 x 105 b keV2 (ref. 15), where UK is the K-shell ionization potential for the atom and X= Ee/UK.This cross sectionhas, for 20-50 keV electrons, roughly the same magnitude and the same variation with the atomic number Zof thebombarded' atoms as that for 2-4 MeV protons, so in this respect they are much alike. The continuous background radiation, however, is much more intense for -lectrons where it is' mainly due to bremsstrahlung from the incident beam . The cross section for the direct bremsstrahlung from the bombarding electrons is given! in closed form in ref.` which in a simple approximation leads to the result stated in section 4 for- drs /dE,(E.,Ej). This cross sectioninconsiderably largerthan theresult in section6 for protons, mainly due to the greater charge-to-mass ratio of the electron.

An easy wayto compare electron bombardmentwith proton or heavy ion bombardment is to compute the corresponding concentrations which give a peak-tobackground ratio S=1. As for the: protons in fig. 8, we have in fig. 10 shown the result for an electron bombardment of a thin sample; for which the energy loss of theincident 50 keV electrons is small compared to their energy. In this estimate is included the fluorescence yields of ref. 4 together with eq. (15) to get the X-ray production yield, and the background yield from the formula of ref. 7 integrated over all angles of the emitted quanta is takenwithin the same energy interval as employed for the proton calculations in section 8, fig. 8. Comparing fig . 10 with fig. 8 it is evident that the concentrationfoundfrom the criterion S=1 is, at least three orders of magnitudehigher than forproton bombardment under optimum conditions, i .e. EKQ (Z) ;-> 1.4 T.. This comparison is not dependent on 'the energy resolution assumed for a good Si(Li) spectrometer, as a better resolution will improve both estimates to the same degree . We have not gone further and have not calculated the limit of detection for electron bombardment of thicker samples, where one will have to integrate over the electron energies, as the electrons areslowed down in the sample . Because the electrons do not dissipate as much energy in the sample as protons which have been stopped, the acceptable electron beam intensity may be higher than for protons or heavy ions, if 1 .

.

,

, ,

i

concentration giving o peak to background ratio of t for 50 KW electrons with diff~rant t-ts At as a tMn matri . cadic

Fig. 10. Electron bombardment calculation of the concentration forwhich theK X-raypeak from atrace elementis as intense as the electron bremsstrahlung from the thin matrix . The back. 1(es/ ground }diction is aken within an interval (5.9 keV)l corresponding to the energy resolution, fwhm, of a Si(Li) detector at the energy Ex of the K g line.

TRACE ELEMENT ANALYSIS

heatingis a problem. If, however, the counting rate it the limiting factor on thebeam intensity of the electron beam, the situation is not much different from that for protons of a few Me V energy, because the X-ray production cross sections are of the same order of magnitude for the two beams, in which case it is relevant to argue on the basis of the peak-to-background ratio. References

1) J. D. Garcia, R . J. Fortner and T . M . Kavanagh, Rev . Mod. Phys . 45 (1973) 111 . 3) E. Merzbacher and H . W. Lewis, Handbuch der Physik 34, ed . S. FIGgge (Springer Verlag, Berlin, 1958) p. 167 . a) G . S. Khandelwal, B. H. Choi and E. Merzbacher, Atomic Data 1 (1969) 103. 4) W. Bambynek, B. Crasemann, R . W. Fink, H . U. Freund,

499

H. Mark, C. D. Swift, R . E. Price and P. V. Rao, Rev . Mod. Phys . 44 (1972) 716 . 5 ) E . Gerjuoy, Phys . Rev. 148 (1966) 54. 6) M . E. Rudd, D. Gregoire and J . B . Crooks, Phys. Rev . A3 (1971) 1635 . 1) P . Kirkpatrick and L. Wiedmann, Phys . Rev . 67 (1945) 321 . s) H . K. Tseng and R . H . Pratt, Phys . Rev. A3 (1971) 100. 0) L. H . Toburen, Phys . Rev . A3 (1971) 216. 10) R . D . Evans, The atomic nucleus (McGraw-Hill Book Co., Inc., New York, 1955). 11) K . Alder, A . Bohr, T . Haas, B . Mottelson and A . Wint1wr, Rev. Mod. Phys. 28 (1956) 432. 18) R . M. Thaler, M . Goldstein, J . L. McHale and L . C. Biadenharn, Phys . Rev . 102 (1956) 1567 . 13) R . M . Eisberg, D . R. Yennie and D . 11 . Wilkinson, Nucl. Phys. 18 (1960) 338. 14) M. Gryzinski, Phys. Rev. 138 (1965) A336 . 1s) W . Hink and Z. Ziegler, Z. Physik 226 (1969) 222 .