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Continuum X-rays produced by a few MeV proton bombardment
57
Nuclear Instruments and Methods in Physics Research B3 (1984) 57-61 North-Holland, Amsterdam
CONTINUUM
X-RAYS PRODUCED
BY A FEW MeV PROTON BOMBARDMENT
K. ISHII Cyclotron and Radioisotope
Center, Tohoku University, Sendai 980. Japan
S. MORITA Research Center of Ion Beam Technology, Hosei University, Kajinocho, Koganei, I84 Tokyo, Japan
Production cross sections of continuum X-rays for quasi-free electron bremsstrahhmg (QFEB), radiative ionization (RI), secondary electron bremsstrahlung (SEB), radiative elastic scattering (RES), and nuclear bremsstrahlung (NB) have been estimated for carbon and aluminum targets bombarded with 1, 3, and 6 MeV protons. Contributions of these processes to the sensitivity of PIXE are discussed. It is shown that, in the case of PIXE using l-3 MeV protons, RES strongly influences the detection limit for those elements for which the K X-ray energy is larger than the energy T,, which is the maximum energy transferred from the projectile to a free and rest electron.
1. Introduction When a solid target is bombarded with heavy-charged particles, continuum X-rays or broad spectral X-rays are produced besides the characteristic X-rays of target atoms. These continuum X-rays influence the detection limit of PIXE. As origin of these continuum X-rays, several processes have been considered: secondary electron bremsstrahlung (SEB) [1,2], radiative ionization (RI) [3,4], quasifree electron bremsstrahlung (QFEB) [5,6], molecular orbital x-rays (MO) [7], quasimolecular bremsstrahlung (QMB) [8,9], radiative electron capture (REC) [lo], radiative elastic scattering (RES) [ll], and nuclear bremsstrahlung (NB) [l]. In addition to these processes, continuum backgrounds might be produced by the Compton scattering of y-rays due to nuclear reactions [5] and also by electronic noise. Among these process, MO, QMB, and REC are predominant in heavy-ion bombardments and can be neglected in the estimation of detection limits of light-ion-induced PIXE. On the other hand, SEB and QFEB have been wnsidered to be predominant in light-ion bombardments. QFEB is a process where a free or a loosely bound electron in a target atom is scattered by the projectileCoulomb field in the projectile frame and emits bremsstrahlung, whose high-energy limit is given by T, = jm,ui (m, is the electron mass and up is the projectile velocity) (see fig. la). This process becomes appreciable when the projectile velocity is large enough in comparison with the velocity of the orbital electron, for example, above 10 MeV/amu for a Be target [5]. In contrast to this process, a process where the orbital electron is scattered by the projectile field in the target 0168-583X/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
(a)
(b)
(cl
Fig. 1. Schematic representation of (a) quasifree electron bremsstrahhrng (QFEB), (b) secondary electron bremsstrahlung (SEB), and (c) radiative elastic scattering (RES); 2, and 2, represent the atomic numbers of the projectile and the target atom, respectively. In (a), the electron collides with the projectile with kinetic energy T, = fm,Vi in the projectile frame (VP is the projectile velocity). In (b), T,, = 4m,E,/M, is the maximum energy which can be transferred from a projectile of mass MP and energy E, to a free electron of mass m,. I. X-RAY
PRODUCTION
58
K. Ishii, S. Moriia / Continuum X-rays
frame and emits bremsstrahlung can also be considered. It is assumed in this process that the velocity of the orbital electron is much larger than the projectile velocity and the collision is quite localized, i.e., a close collision. It would be better to call this process radiative ionization (RI) while Jakubassa and Kleber [3] have called QFEB radiative ionization. However, RI might be used as a general name including both QFEB and this process. Anholt and Saylor have shown that RI contributes in the region of hw > T, (hw is the energy of emitted photons) 141. SEB means bremsstrahlung produced by scattering of a secondary electron, as is schematically shown in fig. lb, where the projectile ejects an electron from a target atom at point A, the secondary electron moves in the target material losing its energy, and at point B, it is scattered by the Coulomb field of a target atom and produces bremsstrahlung, whose intensity decreases rapidly for the photon energy larger than T,,, = 4m, E,/M, ( Mp and E, are mass and energy of the projectile). SEB has been considered to be a major factor in the estimation of the detection limit of PIXE 1121. We have reported in a previous paper [2] that the calculation of SEB based on the BEA theory gives an excellent agreement with the experimental results on an Al target for bombarding energies of 3 and 4 MeV/amu, while the calculation underestimates experimental results at lower bombarding energies. This underestimation in the lower projectile-energy region is quite considerable, especially at the high-energy part of the Xrays. Such a discrepancy has also been reported in heavy-ion impact [4,9]. Anholt and Saylor [4] and Anholt and Salin [9] have tried to interpret these excess continuum X-rays in terms of RI and QMB, respectively. However, the predicted production cross sections of these processes were found to be smaller than the experimental ones by one order of magnitude. In order to solve this discrepancy, we [ll] have recently calculated a new process, where an orbital electron of the target atom is excited to continuum states by the projectile. In getting back to its original bound state, the electron emits continuum X-rays (see fig. lc), and it is found that the prediction from the PWBA theory can now well explain the experimental results on an Al target bombarded with 1 MeV protons. We call this process radiative elastic scattering (RES). In accordance with our calculation, RES is predominant in the region of high-energy ho and for low-energy light-ion bombardment. Accordingly, the effect of RES on the sensitivity of PIXE is expected to be large. We have here calculated the production cross section of continuum X-rays from QFEB, RI, RES, SEB, and NB for Al and C targets bombarded with 1, 3, and 6 MeV protons, and discuss the sensitivity of PIXE taking account of the contributions of these processes.
2. Theoretical 2.1. Secondary electron bremsstrahlung (SEB) In accordance with the BEA theory, we [5] have previously obtained a formula for the differential cross section of SEB. On the basis of this formula, the cross section of SEB daSEB/dh W is expressed by
imdkh(
k)irnfi(
u2)du21)tl ‘O(‘) O* (k+/c,)
3 x,
(I)
with
+m>,
h(k)fdr21n(fi
ln( yhwt)
qo(s)
I
=--&[8+(s-x)~]
for li-ll
=+-(s’+
for 1 - $2
3x2)
= 0
a), s > 0,
fors<$-1,
k,+=(k,+k)x+
$ r
x’=s*+yand
W,=hw/(Z$Ry),
where N,, Vi, and f, ( u2) are the number of electrons, the ionization energy, and the velocity distribution function of the i th shell, respectively; a is the fine structure constant, Ry is the Rydberg constant, a, is the Bohr radius, and r is the average ionization potential. In the case of very thin targets, eq. (1) must take into account the escape effect of the secondary electron, which means the escape of the electron from the target before producing the bremsstrahlung. This effect reduces the production of high-energy X-rays [5]. Moreover, for the case of high-energy ion bombardment, SEB should be calculated on the basis of the relativistic theory, since the spectrum of SEB is affected by the Doppler effect [6]. 2.2. Quasifree electron bremsstrahlung (QFEB) and radiative ionization (RI) Jakubassa and Kleber [3] have calculated QFEB on the basis of the PWBA theory, and Anholt and Saylor [4] have estimated RI in terms of BEA. In accordance with the second Born approximation, we [ll] have recently obtained a formula including both QFEB and RI: hw daQFEB __.._-=y&$ 2 2 3 dho zpaOa
,
,’
1
j;.dW _,”
K. Ishii, S. Morita / Continuum
X
/
,f2,4q 21F,.i(Q)12 I
(2)
,
where R’Ln = W(
59
X-rays
with
CC.n(QJ= tQK:l)i+ 2(Q;;;--f;-1) n + (3Q,
Z?RY) 79, = W( Z,ZRY)7
l)(Q,
- 3)( Qi4 - ~QM + 3, (QM + I)6
and
+ ...,
Wi= W+t,,/(Z,zRy). Here, Z, is an effective nuclear charge for the i th shell and FW,i(Q) should be referred to ref. 13. Eq. (2) provides the cross section of QFEB for the case of Aw < T, and that of RI for the case of A w > q. Since the condition Ao % T, has been assumed in the derivation of eq. (2), this equation may not well predict the cross section of QFEB. In the case of Z,/Z, < 1 and qi > 1, however, this formula corresponds well to a simple theory of QFEB which agrees well with the prediction of Jakubassa and Kleber [3]. In the calculation of QFEB, Jakubassa and Kleber have used the Coulomb wave function in the projectile frame for the final state of ejected electrons, while we have used it in the target frame. This difference in the calculations appears in the projectile-charge dependence of QFEB in the region of tiw = T, and becomes large with increase in the value of Z,/Z,. The behavior of QFEB for a change in the projectile charge is interesting for the research of the velocity-distribution function of orbital electrons as well as for REC, especially in high-energy bombardments where the relativistic treatment is required [6]. During RI, when fiw > T,, bremsstrahlung is mainly produced by the orbital electrons which have a high velocity and are strongly bound by the target nucleus. Therefore, eq. (2) should be applied to this case, although the formula of Jakubassa and Kleber also predicts the cross section of QFEB in this region. 2.3. Radiative elastic scattering (RES) This process has been quite recently considered among the various radiative processes mentioned in the previous section. RES has been introduced on the basis of the idea that bremsstrahlung can be produced by a particle with internal structure (e.g., an atom) scattered from a structureless particle, but the internal structure of the particle is not changed after and before emissions. The cross section of RES can be obtained in the same manner as that for the derivation of eq. (2) and is expressed by -.-=tiw ~‘~2~3 P
-
0
daRES dtio
where n is the principal quantum number. It has been assumed that Ao % T, in eq. (3) as well as in eq. (2). As seen from eq. (3), RES has large cross sections for the where n’ is the case of tiw < TRW [ = (4Z,./n’),/m, principal quantum number of the shell whose number of electrons is largest]. The energy of TREs, therefore, characterizes RES in the region of low-photon energy, though eq. (3) may not give a good approximation for RES in this region. 2.4. Nuclear bremsstrahlung (NB) In the case that the energy of the emitted photon can be neglected in comparison with the projectile energy, the cross section of nuclear bremsstrahlung can be estimated by using a classical method [1,2] and is expressed by
(4)
where M, is the mass of the target nucleus. Eq. (4) gives the dipole radiation and vanishes under the condition of Z,M,/( ZpMT) = 1 as in the case of a carbon target bombarded with a-particles. For such a case, it is necessary to consider a quadrupole radiation for the bremsstrahlung. Nuclear bremsstrahlung is emitted over the range of photon energy from keV to MeV. However, the intensity of this radiation is negligibly small in comparison with other radiation processes owing to the factor (m,/Mp)‘.
3. Results and discussion The cross sections of QFEB, RI, SEB, RES, and NB for Al and C targets bombarded with 1, 3 and 6 MeV protons have been calculated from eqs. (l)-(4), and are shown in figs. 2 and 3. Here, the cross sections are multiplied by a factor Aw/(a&x3) and the total cross sections of NB, NB + RES, NB + RES + SEB, and NB + RES + SEB + QFEB (or RI) are presented in this I. X-RAY PRODUCPION
K. Ishii, S. Morita / Continuum X-rays
60
i I
2
I
3
4
1
/
I
/
1
5
6
7
8
9
IO
hw (keVl Fig. 2. Cumulative spectra (each curve includes contributions from the curves below) taking into account the processes mentioned in the introduction. The three cases considered are 1, 3, and 6 MeV bombardments of a carbon target.
order from the bottom to the top for each projectile energy. For the bombarding energies of 3 and 6 MeV, the cross sections of NB are not shown in the figures since the contribution of NB is quite negligible in comparison with those of QFEB, RI, SEB, and RES. It is found from the figures that RES is predominant in the region of 2-8 keV for 1 MeV bombardment. It has been confirmed [ll) that the theoreticai predictions of RES are in agreement with the experimental results on an Al target bombarded with 1 MeV protons (see also fig. 3). Thus, the detection limit of PIXE with low-energy projectiles is affected by RES. For 3 MeV proton ~mb~~ents, SEB and RES are predominant in the region of, respectively, ho G 6.5 keV (= T,) and tto > 6.5 keV. This result shows that for the analysis of heavier elements than Fe (K X ray = 6.4 keV), RES should affect the sensitivity. For 6 MeV proton bombardments, QFEB and SEB
I
2
3
4
5 6 hw (kev)
7
8
9
10
Fig. 3. Same as fig. 2 but for an aluminum target. The data are from ref. 2.
are pr~o~n~t in the region of tiw < 3.2 keV (= T,), and SEB is predominant in the region of 3.2 keV Q hw < 10 keV. This means that SEB and QFEB, especially SEB, play a major role in determining the detection limit of PIXE in the case of high-energy bombardment. It is seen from the figures that, in the region of Ao > T,, SEB, RI, and NB are quite small in comparison with RES. This result is in contrast with the fact that high sensitivity of PIXE has been expected in this region from the theories of SEB, RI, and QFEB [1,2]. Whether RES actually affects the sensitivity of PIXE in the region of hw 3 T, or not depends on the relative intensity of the Compton scattering of y-rays or of the electronic noise, which is not directly estimated because of many ambiguous factors. However, electronic noise can be generally neglected in the region above 1 keV and cross sections of nuclear reactions are considered to be quite small for the bombardment of low-energy charged particles. We have here estimated the cross sections of continuum X-rays for proton bombardments. As seen from
K. Ishii, S. Mori’ra / Coniinuum X-rays
eqs. (l)-(4), the cross section of QFEB, SEB, and RES have a Z,’ dependence. On the other hand, the inner shell ionization cross section depends also on the square of the projectile charge. The detection limit of PIXE, therefore, is proportional to l/Z,, since it is usually where Nr( a Zp’) is the counts of given by 3/6/N,, a characteristic X-ray peak and Na( a Zz) is the background counts under the peak. Therefore, the use of heavier ions (at the same MeV/amu) is not likely to produce higher backgrounds from processes that are considered here.
4. Summary The formulae of the cross sections for QFEB (or RI), SEB, RES, and NB have been derived. The cross sections of continuum X-rays from these processes have been evaluated for the cases of Al and C targets bombarded with 1, 3, and 6 MeV protons. The results of calculations have shown. that QFEB, SEB, and RES contribute predominantly to the continuum background of PIXE in regions of, respectively, trw < T,, ho Q T,, and tto > T,. RI and NB do not seriously contribute. It was pointed out that RES as well as SEB is an important factor to determine the detection limit of PIXE using low-energy light-ion bombardment, and further, the detection limit is inversely proportional to the projectile charge (i.e., to l/Z,). The present formulae for QFEB, SEB, and RES are very useful to evaluate the sensitivity of PIXE. The calculations of the detection
61
limit of PIXE based on these formulae and further measurements of continuum X-rays produced by low-energy protons are now in progress in our laboratory.
References [l] F. Folkmann, G. Gaarde, T. Huus and K. Kemp, Nucl.
Ins&. and Meth. 116 (1974) 487. [2] K. Ishii, S. Morita and H. Tawara, Phys. Rev. A 13 (1976) 131. (31 D.H. Jakubassa and M. Kleber, 2. Physik A 273 (1975) 29. [4] R. Anholt and T.K. Saylor, Phys. Lett. 56A (1976) 455. [5] A. Yamadera, K. I&ii, K. Sera, M. Sebata and S. Morita, Phys. Rev. A 23 (1981) 24. [6] T.C. Chu, K. Ishii, A. Yamadera, M. Sebata and S. Morita, Phys. Rev. A 24 (1981) 1720. [7] F.W. Saris, W.F. van der Weg, H. Tawara and R. Laubert, Phys. Rev. Lett. 28 (1972) 717. [S] J.C.Y. Chen, T. Ishihara and K.M. Watson, Phys. Rev. Lett. 35 (1975) 1574. [9] R. Anholt and A. Salin, Phys. Rev. Al6 (1977) 799. [lo] H.W. Schopper, J.P. Delvaille, K. KaIata, A.R. Sohval, M. Atidulwahab, K.W. Jones and H.E. wegner, Phys. Lett. 47A (1974) 61. (111 K. Ishii and S. Morita, to be published. (121 A. Yamadera, K. Ishii, K. Sera, S. Morita and T.C. Chu, Nucl. Instr. and Meth. 181 (1981) 15. [13] E. Merzbacher and H.W. Lewis, Encyclopedia of physics, ed., S. Fhigge (Springer, Berlin, 1958) vol. 34, p. 166.
I. X-RAY
PRODUCTION
Nuclear Instruments and Methods in Physics Research B3 (1984) 57-61 North-Holland, Amsterdam
CONTINUUM
X-RAYS PRODUCED
BY A FEW MeV PROTON BOMBARDMENT
K. ISHII Cyclotron and Radioisotope
Center, Tohoku University, Sendai 980. Japan
S. MORITA Research Center of Ion Beam Technology, Hosei University, Kajinocho, Koganei, I84 Tokyo, Japan
Production cross sections of continuum X-rays for quasi-free electron bremsstrahhmg (QFEB), radiative ionization (RI), secondary electron bremsstrahlung (SEB), radiative elastic scattering (RES), and nuclear bremsstrahlung (NB) have been estimated for carbon and aluminum targets bombarded with 1, 3, and 6 MeV protons. Contributions of these processes to the sensitivity of PIXE are discussed. It is shown that, in the case of PIXE using l-3 MeV protons, RES strongly influences the detection limit for those elements for which the K X-ray energy is larger than the energy T,, which is the maximum energy transferred from the projectile to a free and rest electron.
1. Introduction When a solid target is bombarded with heavy-charged particles, continuum X-rays or broad spectral X-rays are produced besides the characteristic X-rays of target atoms. These continuum X-rays influence the detection limit of PIXE. As origin of these continuum X-rays, several processes have been considered: secondary electron bremsstrahlung (SEB) [1,2], radiative ionization (RI) [3,4], quasifree electron bremsstrahlung (QFEB) [5,6], molecular orbital x-rays (MO) [7], quasimolecular bremsstrahlung (QMB) [8,9], radiative electron capture (REC) [lo], radiative elastic scattering (RES) [ll], and nuclear bremsstrahlung (NB) [l]. In addition to these processes, continuum backgrounds might be produced by the Compton scattering of y-rays due to nuclear reactions [5] and also by electronic noise. Among these process, MO, QMB, and REC are predominant in heavy-ion bombardments and can be neglected in the estimation of detection limits of light-ion-induced PIXE. On the other hand, SEB and QFEB have been wnsidered to be predominant in light-ion bombardments. QFEB is a process where a free or a loosely bound electron in a target atom is scattered by the projectileCoulomb field in the projectile frame and emits bremsstrahlung, whose high-energy limit is given by T, = jm,ui (m, is the electron mass and up is the projectile velocity) (see fig. la). This process becomes appreciable when the projectile velocity is large enough in comparison with the velocity of the orbital electron, for example, above 10 MeV/amu for a Be target [5]. In contrast to this process, a process where the orbital electron is scattered by the projectile field in the target 0168-583X/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
(a)
(b)
(cl
Fig. 1. Schematic representation of (a) quasifree electron bremsstrahhrng (QFEB), (b) secondary electron bremsstrahlung (SEB), and (c) radiative elastic scattering (RES); 2, and 2, represent the atomic numbers of the projectile and the target atom, respectively. In (a), the electron collides with the projectile with kinetic energy T, = fm,Vi in the projectile frame (VP is the projectile velocity). In (b), T,, = 4m,E,/M, is the maximum energy which can be transferred from a projectile of mass MP and energy E, to a free electron of mass m,. I. X-RAY
PRODUCTION
58
K. Ishii, S. Moriia / Continuum X-rays
frame and emits bremsstrahlung can also be considered. It is assumed in this process that the velocity of the orbital electron is much larger than the projectile velocity and the collision is quite localized, i.e., a close collision. It would be better to call this process radiative ionization (RI) while Jakubassa and Kleber [3] have called QFEB radiative ionization. However, RI might be used as a general name including both QFEB and this process. Anholt and Saylor have shown that RI contributes in the region of hw > T, (hw is the energy of emitted photons) 141. SEB means bremsstrahlung produced by scattering of a secondary electron, as is schematically shown in fig. lb, where the projectile ejects an electron from a target atom at point A, the secondary electron moves in the target material losing its energy, and at point B, it is scattered by the Coulomb field of a target atom and produces bremsstrahlung, whose intensity decreases rapidly for the photon energy larger than T,,, = 4m, E,/M, ( Mp and E, are mass and energy of the projectile). SEB has been considered to be a major factor in the estimation of the detection limit of PIXE 1121. We have reported in a previous paper [2] that the calculation of SEB based on the BEA theory gives an excellent agreement with the experimental results on an Al target for bombarding energies of 3 and 4 MeV/amu, while the calculation underestimates experimental results at lower bombarding energies. This underestimation in the lower projectile-energy region is quite considerable, especially at the high-energy part of the Xrays. Such a discrepancy has also been reported in heavy-ion impact [4,9]. Anholt and Saylor [4] and Anholt and Salin [9] have tried to interpret these excess continuum X-rays in terms of RI and QMB, respectively. However, the predicted production cross sections of these processes were found to be smaller than the experimental ones by one order of magnitude. In order to solve this discrepancy, we [ll] have recently calculated a new process, where an orbital electron of the target atom is excited to continuum states by the projectile. In getting back to its original bound state, the electron emits continuum X-rays (see fig. lc), and it is found that the prediction from the PWBA theory can now well explain the experimental results on an Al target bombarded with 1 MeV protons. We call this process radiative elastic scattering (RES). In accordance with our calculation, RES is predominant in the region of high-energy ho and for low-energy light-ion bombardment. Accordingly, the effect of RES on the sensitivity of PIXE is expected to be large. We have here calculated the production cross section of continuum X-rays from QFEB, RI, RES, SEB, and NB for Al and C targets bombarded with 1, 3, and 6 MeV protons, and discuss the sensitivity of PIXE taking account of the contributions of these processes.
2. Theoretical 2.1. Secondary electron bremsstrahlung (SEB) In accordance with the BEA theory, we [5] have previously obtained a formula for the differential cross section of SEB. On the basis of this formula, the cross section of SEB daSEB/dh W is expressed by
imdkh(
k)irnfi(
u2)du21)tl ‘O(‘) O* (k+/c,)
3 x,
(I)
with
+m>,
h(k)fdr21n(fi
ln( yhwt)
qo(s)
I
=--&[8+(s-x)~]
for li-ll
=+-(s’+
for 1 - $2
3x2)
= 0
a), s > 0,
fors<$-1,
k,+=(k,+k)x+
$ r
x’=s*+yand
W,=hw/(Z$Ry),
where N,, Vi, and f, ( u2) are the number of electrons, the ionization energy, and the velocity distribution function of the i th shell, respectively; a is the fine structure constant, Ry is the Rydberg constant, a, is the Bohr radius, and r is the average ionization potential. In the case of very thin targets, eq. (1) must take into account the escape effect of the secondary electron, which means the escape of the electron from the target before producing the bremsstrahlung. This effect reduces the production of high-energy X-rays [5]. Moreover, for the case of high-energy ion bombardment, SEB should be calculated on the basis of the relativistic theory, since the spectrum of SEB is affected by the Doppler effect [6]. 2.2. Quasifree electron bremsstrahlung (QFEB) and radiative ionization (RI) Jakubassa and Kleber [3] have calculated QFEB on the basis of the PWBA theory, and Anholt and Saylor [4] have estimated RI in terms of BEA. In accordance with the second Born approximation, we [ll] have recently obtained a formula including both QFEB and RI: hw daQFEB __.._-=y&$ 2 2 3 dho zpaOa
,
,’
1
j;.dW _,”
K. Ishii, S. Morita / Continuum
X
/
,f2,4q 21F,.i(Q)12 I
(2)
,
where R’Ln = W(
59
X-rays
with
CC.n(QJ= tQK:l)i+ 2(Q;;;--f;-1) n + (3Q,
Z?RY) 79, = W( Z,ZRY)7
l)(Q,
- 3)( Qi4 - ~QM + 3, (QM + I)6
and
+ ...,
Wi= W+t,,/(Z,zRy). Here, Z, is an effective nuclear charge for the i th shell and FW,i(Q) should be referred to ref. 13. Eq. (2) provides the cross section of QFEB for the case of Aw < T, and that of RI for the case of A w > q. Since the condition Ao % T, has been assumed in the derivation of eq. (2), this equation may not well predict the cross section of QFEB. In the case of Z,/Z, < 1 and qi > 1, however, this formula corresponds well to a simple theory of QFEB which agrees well with the prediction of Jakubassa and Kleber [3]. In the calculation of QFEB, Jakubassa and Kleber have used the Coulomb wave function in the projectile frame for the final state of ejected electrons, while we have used it in the target frame. This difference in the calculations appears in the projectile-charge dependence of QFEB in the region of tiw = T, and becomes large with increase in the value of Z,/Z,. The behavior of QFEB for a change in the projectile charge is interesting for the research of the velocity-distribution function of orbital electrons as well as for REC, especially in high-energy bombardments where the relativistic treatment is required [6]. During RI, when fiw > T,, bremsstrahlung is mainly produced by the orbital electrons which have a high velocity and are strongly bound by the target nucleus. Therefore, eq. (2) should be applied to this case, although the formula of Jakubassa and Kleber also predicts the cross section of QFEB in this region. 2.3. Radiative elastic scattering (RES) This process has been quite recently considered among the various radiative processes mentioned in the previous section. RES has been introduced on the basis of the idea that bremsstrahlung can be produced by a particle with internal structure (e.g., an atom) scattered from a structureless particle, but the internal structure of the particle is not changed after and before emissions. The cross section of RES can be obtained in the same manner as that for the derivation of eq. (2) and is expressed by -.-=tiw ~‘~2~3 P
-
0
daRES dtio
where n is the principal quantum number. It has been assumed that Ao % T, in eq. (3) as well as in eq. (2). As seen from eq. (3), RES has large cross sections for the where n’ is the case of tiw < TRW [ = (4Z,./n’),/m, principal quantum number of the shell whose number of electrons is largest]. The energy of TREs, therefore, characterizes RES in the region of low-photon energy, though eq. (3) may not give a good approximation for RES in this region. 2.4. Nuclear bremsstrahlung (NB) In the case that the energy of the emitted photon can be neglected in comparison with the projectile energy, the cross section of nuclear bremsstrahlung can be estimated by using a classical method [1,2] and is expressed by
(4)
where M, is the mass of the target nucleus. Eq. (4) gives the dipole radiation and vanishes under the condition of Z,M,/( ZpMT) = 1 as in the case of a carbon target bombarded with a-particles. For such a case, it is necessary to consider a quadrupole radiation for the bremsstrahlung. Nuclear bremsstrahlung is emitted over the range of photon energy from keV to MeV. However, the intensity of this radiation is negligibly small in comparison with other radiation processes owing to the factor (m,/Mp)‘.
3. Results and discussion The cross sections of QFEB, RI, SEB, RES, and NB for Al and C targets bombarded with 1, 3 and 6 MeV protons have been calculated from eqs. (l)-(4), and are shown in figs. 2 and 3. Here, the cross sections are multiplied by a factor Aw/(a&x3) and the total cross sections of NB, NB + RES, NB + RES + SEB, and NB + RES + SEB + QFEB (or RI) are presented in this I. X-RAY PRODUCPION
K. Ishii, S. Morita / Continuum X-rays
60
i I
2
I
3
4
1
/
I
/
1
5
6
7
8
9
IO
hw (keVl Fig. 2. Cumulative spectra (each curve includes contributions from the curves below) taking into account the processes mentioned in the introduction. The three cases considered are 1, 3, and 6 MeV bombardments of a carbon target.
order from the bottom to the top for each projectile energy. For the bombarding energies of 3 and 6 MeV, the cross sections of NB are not shown in the figures since the contribution of NB is quite negligible in comparison with those of QFEB, RI, SEB, and RES. It is found from the figures that RES is predominant in the region of 2-8 keV for 1 MeV bombardment. It has been confirmed [ll) that the theoreticai predictions of RES are in agreement with the experimental results on an Al target bombarded with 1 MeV protons (see also fig. 3). Thus, the detection limit of PIXE with low-energy projectiles is affected by RES. For 3 MeV proton ~mb~~ents, SEB and RES are predominant in the region of, respectively, ho G 6.5 keV (= T,) and tto > 6.5 keV. This result shows that for the analysis of heavier elements than Fe (K X ray = 6.4 keV), RES should affect the sensitivity. For 6 MeV proton bombardments, QFEB and SEB
I
2
3
4
5 6 hw (kev)
7
8
9
10
Fig. 3. Same as fig. 2 but for an aluminum target. The data are from ref. 2.
are pr~o~n~t in the region of tiw < 3.2 keV (= T,), and SEB is predominant in the region of 3.2 keV Q hw < 10 keV. This means that SEB and QFEB, especially SEB, play a major role in determining the detection limit of PIXE in the case of high-energy bombardment. It is seen from the figures that, in the region of Ao > T,, SEB, RI, and NB are quite small in comparison with RES. This result is in contrast with the fact that high sensitivity of PIXE has been expected in this region from the theories of SEB, RI, and QFEB [1,2]. Whether RES actually affects the sensitivity of PIXE in the region of hw 3 T, or not depends on the relative intensity of the Compton scattering of y-rays or of the electronic noise, which is not directly estimated because of many ambiguous factors. However, electronic noise can be generally neglected in the region above 1 keV and cross sections of nuclear reactions are considered to be quite small for the bombardment of low-energy charged particles. We have here estimated the cross sections of continuum X-rays for proton bombardments. As seen from
K. Ishii, S. Mori’ra / Coniinuum X-rays
eqs. (l)-(4), the cross section of QFEB, SEB, and RES have a Z,’ dependence. On the other hand, the inner shell ionization cross section depends also on the square of the projectile charge. The detection limit of PIXE, therefore, is proportional to l/Z,, since it is usually where Nr( a Zp’) is the counts of given by 3/6/N,, a characteristic X-ray peak and Na( a Zz) is the background counts under the peak. Therefore, the use of heavier ions (at the same MeV/amu) is not likely to produce higher backgrounds from processes that are considered here.
4. Summary The formulae of the cross sections for QFEB (or RI), SEB, RES, and NB have been derived. The cross sections of continuum X-rays from these processes have been evaluated for the cases of Al and C targets bombarded with 1, 3, and 6 MeV protons. The results of calculations have shown. that QFEB, SEB, and RES contribute predominantly to the continuum background of PIXE in regions of, respectively, trw < T,, ho Q T,, and tto > T,. RI and NB do not seriously contribute. It was pointed out that RES as well as SEB is an important factor to determine the detection limit of PIXE using low-energy light-ion bombardment, and further, the detection limit is inversely proportional to the projectile charge (i.e., to l/Z,). The present formulae for QFEB, SEB, and RES are very useful to evaluate the sensitivity of PIXE. The calculations of the detection
61
limit of PIXE based on these formulae and further measurements of continuum X-rays produced by low-energy protons are now in progress in our laboratory.
References [l] F. Folkmann, G. Gaarde, T. Huus and K. Kemp, Nucl.
Ins&. and Meth. 116 (1974) 487. [2] K. Ishii, S. Morita and H. Tawara, Phys. Rev. A 13 (1976) 131. (31 D.H. Jakubassa and M. Kleber, 2. Physik A 273 (1975) 29. [4] R. Anholt and T.K. Saylor, Phys. Lett. 56A (1976) 455. [5] A. Yamadera, K. I&ii, K. Sera, M. Sebata and S. Morita, Phys. Rev. A 23 (1981) 24. [6] T.C. Chu, K. Ishii, A. Yamadera, M. Sebata and S. Morita, Phys. Rev. A 24 (1981) 1720. [7] F.W. Saris, W.F. van der Weg, H. Tawara and R. Laubert, Phys. Rev. Lett. 28 (1972) 717. [S] J.C.Y. Chen, T. Ishihara and K.M. Watson, Phys. Rev. Lett. 35 (1975) 1574. [9] R. Anholt and A. Salin, Phys. Rev. Al6 (1977) 799. [lo] H.W. Schopper, J.P. Delvaille, K. KaIata, A.R. Sohval, M. Atidulwahab, K.W. Jones and H.E. wegner, Phys. Lett. 47A (1974) 61. (111 K. Ishii and S. Morita, to be published. (121 A. Yamadera, K. Ishii, K. Sera, S. Morita and T.C. Chu, Nucl. Instr. and Meth. 181 (1981) 15. [13] E. Merzbacher and H.W. Lewis, Encyclopedia of physics, ed., S. Fhigge (Springer, Berlin, 1958) vol. 34, p. 166.
I. X-RAY
PRODUCTION