The ratio of total double to total single ionisation cross sections, R, by high velocity projectiles containing electrons is examined in the first Born approximation. For ionisation of helium by H+, H0 and He0 impact, it is found that t filt H+ \u3c filt Ho \u3c filt Heo . Connection with Bohr\u27s free-collision model for collisions with projectile electrons is given

DuBois, R. D.

McGuire, James H.

Straton, Jack C.

Wang, Y. D.

English

PDXScholar (Portland State University)

Portland State University PDXScholar Physics Faculty Publications and Presentations Physics 1990 High-velocity Limits for the Ratio of Double to Single Ionisation of Helium by Projectiles with Electrons Y. D. Wang Jack C. Straton Portland State University, straton@pdx.edu James H. McGuire R. D. DuBois Follow this and additional works at: https://pdxscholar.library.pdx.edu/phy_fac  Part of the Atomic, Molecular and Optical Physics Commons Let us know how access to this document benefits you. Citation Details Wang, Y. D., Straton, J. C., McGuire, J. H., & DuBois, R. D. (1990). High-velocity limits for the ratio of double to single ionisation of helium by projectiles with electrons. Journal of Physics B: Atomic, Molecular and Optical Physics, 23(9), L133. This Letter to the Editor is brought to you for free and open access. It has been accepted for inclusion in Physics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: pdxscholar@pdx.edu. Journal of Physics B: Atomic, Molecular and Optical PhysicsHigh-velocity limits for the ratio of double to singleionisation of helium by projectiles with electronsTo cite this article: Y D Wang et al 1990 J. Phys. B: At. Mol. Opt. Phys. 23 L133 View the article online for updates and enhancements.Recent citationsThe ratio of cross sections for double tosingle ionization of helium by high energyphotons and charged particlesJ H McGuire et al-Theory of projectile ionization by molecularhydrogen targetsWang, Y. D. et al-This content was downloaded from IP address 131.252.181.103 on 21/07/2018 at 00:44J. Phys. B: At. Mol. Opt. Phys. 23 (1990) L133-L138. Printed in the UK LE'lTER TO THE EDITOR High-velocity limits for the ratio of double to single ionisation of helium by projectiles with electrons Y D Wangt, Jack C Stratont, J H McGuiret and R D DuBoisS t J R McDonald Laboratory, Department of Physics, Kansas State University, Manhattan, KS 66506, USA $. Pacific Northwest Laboratory, Richland, WA 99352, USA Received 26 September 1989, in final form 21 February 1990 Abstract. The ratio of total double to total single ionisation cross sections, W, by high velocity projectiles containing electrons is examined in the first Born approximation. For ionisation of helium by H+, Ho and Heo impact, it is found that % Z H + < % H ~ < % H e ~ .  Connection with Bohr's free-collision model for collisions with projectile electrons is given. At high projectile velocities much of the physics of single and multiple ionisation can be understood within the framework of a first Born approximation (Bates and Griffing 1955, Bell er a1 1969, Inokuti 1971, Manson et a1 1975, McGuire 1984, 1987). For a bare projectile, observed cross sections for both single and double ionisation vary as Z 2 ,  where Z is the charge of projectile, in accord with the first Born approximation (Haugen er a1 1982). For a projectile with bound electrons, it has been shown for single ionisation (Briggs and Taulbjerg 1978, Inokuti 1979, McGuire et a1 1981, Anholt 1986) that Z 2  is replaced by Zf,( Q )  which describes the screening and antiscreening of the projectile nucleus by the electrons. (Here Q is the momentum transfer.) Thus the single ionisation cross sections for bare projectiles and projectiles with electrons can be interrelated through the effective charges. Given the first Born inter-relationship for single ionisation for bare and charged projectiles, it is possible to begin to consider the effect of projectile electrons on the ratio of double to single ionisation at high collision velocities. In this letter the ratios between both differential double to single ionisation cross sections R and total double to single ionisation cross sections 92 are examined. A first-order interrelation of 92 for projectiles with and without electrons is given. At high projectile velocities, v,,  it has been shown (Mittleman 1966, Byron and Joachain 1967, Carlson 1967, Aberg 1973, Carlson and Nextor 1973, Quarles 1973, Carter and Kelly 1981, McGuire 1984, 1987) that the total double ionisation cross section (a++) is related to the total single ionisation cross section (a+) by 92 = lim a++/u+  constant. up-'os The ratio of double to single ionisation is sensitive to electron correlation. Let us now consider the differential single ionisation cross section, d a + ,  and differential double ionisation cross section, da++.  By integrating over one of the ejected 0953-4075/90/090133 +06$03.50 @ 1990 IOP Publishing Ltd L133 L134 Letter to the Editor electron momenta, e.g. the slower electron, one may write, in accord with equation ( 1 )  where E is the energy of the other outgoing electron. Here the 2' or Z%,( Q )  dependence comes from the first Born approximation for single ionisation due to bare projectiles or projectiles with electrons. The proportionality factor R can be taken as a ratio between differential double and single ionisation cross sections. Generally R depends on both Q and E. A relation between R and 3 can be obtained from equations ( 1 )  and (2) for bare projectiles and for projectiles with electrons. First, for bare projectiles (such as e-, H+, HeZ+, etc),  we have The minimum value of Q depends on the ionisation potential, I, which is different for single and double ionisation. For fast heavy projectiles, E ranges from threshold to infinity. For projectiles carrying N electrons (such as Ho, Heo, etc),  in the first Born approximation, the single ionisation differential cross section (da+  = d a & ,  dw&, etc) can be expressed (McGuire et a1 1981) as a product of two separate parts: d a + = Z & ( Q )  da:. (4) The first part is an effective charge squared Z:ff( Q )  which is independent of the target; the second part, which depends on the target and is independent of the projectile, is the cross section da: for a bare projectile with Z = 1. Here Zf,(Q) can be expressed as IZeff(Q)IZ=Z2+2-4Z(l  +(Q/2Z)21-z+211+(Q/22)2)-4 ( 5 a )  Izeff(Q)12 = z2+ 1-2211 + (Q/22)z1-2 (5b) I Zeff(Q )I2 = z2 ( 5 c )  for two projectile electrons (helium-like), as for one projectile electron (hydrogen-like), and as for 0 projectile electrons (bare nucleus), where Z refers to the projectile charge. It is interesting to note that, at large Q, the antiscreening part of 2:,( Q )  yields an incoherent limit ( Z 2 +  N )  and total cross section is a sum of independent cross sections from the bare projectile (- 2') and the N electrons. Thus at large Q this first Born ionisation cross section reduces to Bohr's free-collision model (Bohr 1948, Andersen et a1 1987) where cross sections for the projectile nucleus and electrons simply add. Now by using equations ( l ) ,  (2), and the effective charge, the ratio of total double to total single ionisation cross sections can be written as Letter to the Editor L135 This central result makes it possible to study 92 for projectiles with electrons in the first Born approximation and sets up a connection with bare projectiles. The first Born cross sections for single ionisation by bare projectiles in (3) and (6) are well known (McDowell and Coleman 1970, Rudd et al 1985). However, R ( Q ,  E )  which includes information about two correlated continuum electrons is not known. Nevertheless, recent experimental observations (Shah and Gilbody 1985, Andersen et a1 1987, DuBois 1987, DuBois and Toburen 1988, Giese and Horsdal 1988, Kamber et a1 1988, Dubois and Kover 1989) can be used to make some guesses about the form of R. First, let us consider a recent experiment (Kamber et a1 1988) of double ionisation of helium by proton impact at energies of 3 MeV and 6 MeV. It has been found that although R depends on both variables, the Q dependence can be regarded as weak compared with the E dependence. The R values for protons and photons differ by about a factor of two while the corresponding Q values differ by about 50. Thus at sufficiently high velocities, one can use R ( Q ,  E )  -- R ( E )  if Q does not vary too much. Between H+ and Ho we expect that Q varies by only a factor of two or less, so that the variation of R may be small. Using R ( Q ,  E )  = R ( E )  and equation (4), it is obvious that after integrating over Q, equations ( 3 )  and (6) will have a simple form, namely, 92 - j  dER(E)(l/u+) da+/d&. Introducing a density distribution, P ( E )  = (l/a+) dai /ds ,  we have 92 = R( E ) P (  E )  ds. (7) 5 This derivation is more general than that of McGuire (1984) where R was taken as the photon ratio, R,, for all charged projectiles. Also our derivation is not restricted by the validity of the Bethe approximation. By fitting experimental data, McGuire used a rather simple monotonic relation for R ( E ) ,  namely E R ( E ) = -  R ,  E1/2+ E where is a fitting parameter, R ,  is the limiting value of R (  E )  when E + 03. This monotonic relation R ( E )  is shown in figure 1 (full curve), where R ,  is taken as 3%. Figure 1. Phenomenological ratio (in YO) for differential double to differential single ionisation as a function of ejected electron energy (in eV). Full curve, monotonic ratio R = [ & / ( E , / * +  E ) ] & ,  where E , / *  is taken as 40 eV, Rm as 3%. Broken curve, a peak is added to the monotonic ratio at E = 250 eV with a width of 20 eV. L136 Letter to the Editor A qualitative understanding of 3 for projectiles with and without electrons can be givenusing(5) and(6) withp(E)= (l/cr+) I:,,,,,, .Z:,(Q)(da:/dQ de)  dQ. Forthesmall Q region, equations ( s a )  and (5b) give IZee(Q)IZ= ( Z -  N ) ' ,  which is zero for Z = N, whereas in (5c) it is 2'. Thus, in this region where da:/dQ de is large, the contribution to the integral for bare projectiles will be larger than for projectiles with electrons. Because Qmin = ( E  + I ) /2up,  where I is the target ionisation potential (and therefore I/2up is small), small values of Q correspond to small values of E so that enhancement of P(E)  for bare projectiles relative to projectiles with electrons occurs for small E. The distribution p(  E )  is normalised to one. For projectiles with electrons, p(  E )  is small at small E and is therefore peaked at larger E. And this is the same region of E in which R(E) ,  from equation (7) and figure 1, is appreciable. In contrast, for bare projectiles p ( e )  is peaked at small E where R ( E )  is small. Consequently 92 is greater for Ho than for H+. Our qualitative understanding is consistent with our quantitative calculation in table 1. The ratio 92 is even bigger for Heo than for Ho since Z:,(Q) is twice as large in the small Q region for Heo than for Ho. This suggests that for H+, Ho and Heo: 3 H +  < 9 2 H ~  < 9 2 H e 0 .  (9) The monotonic relation assumed in (8) conflicts with recent studies by Giese and Horsdal (1988), Reading et a1 (1989) and Vegh (1989). Giese and Horsdal observed a peak in the ratio of double to single ionisation of He by proton impact as a function of scattering angle 0 (i.e. Q ) ,  suggesting a possible non-monotonic R ( E ) .  We use the binary encounter model (Kamber et a1 1988) to relate E and Q to obtain a double binary encounter peak in R (  E )  consistent with the observations of Giese and Horsdal. We assume that the projectile has a binary encounter with each of the two electrons in helium. In figure 1, a peak at E = 250 eV which corresponds to binary encounter electrons from 500 keV amu-' projectiles is added to the monotonic curve. The width of the peak is taken as 20 eV, approximately corresponding to the width of the electron momentum distribution in helium. The height of 3% is taken to fit experiment. In table 1, we present calculations for 3 in two cases. The first (denoted as 3') is for the calculation using the monotonic expression in equation (8). The second (denoted as 92') is for the calculation using the monotonic relation together with a peak centred at E = 250 eV since a double binary collision could account for the peak observed by Giese and Horsdal. We have varied the position of the peak from 1 keV (corresponding to binary encounter electrons from 2 MeV amu-' projectiles) down to 40 eV. At 40 eV the peak increases all the ratios by about a factor of two. As the peak moves to larger E, its influence on 92 decreases. Although the various forms of R ( E )  give somewhat different values for 3, all of those we tried agree with (9). In the development above, several points are worth further discussion. First, the expression for Z$( Q )  is derived using a closure approximation. We expect that the closure correction of Anholt (1986) to (5a)-(5c) is small in our case since the collision velocity is well above the threshold velocity for ionisation by electrons sensitive to Q. For charged projectiles where contributions from small Q and small E are significant in ( 6 ) ,  3 is somewhat sensitive to the difference in the single and double ionisation potential (McGuire 1984). However, for neutral projectiles 92 is not sensitive since Q = Qmin contributions are damped by Z:e( Q). Calculations confirmed that the order- ing of the various ratios 92 for different projectiles does not depend on how ionisation potentials are used. Another interesting aspect is that the ratios obtained for the various projectiles are all smaller than either the differential R ,  limit of 3% or the observed Letter to the Editor L137 Table 1. Ratio (a ) of total double to total single ionisation of helium H+ and Ha impact at 1 MeV, 2 MeV and 5 MeV and by Heo impact at 2 MeV a m - '  and 5 MeV a m - ' .  U+ denotes the calculated first Born single ionisation cross section. 3' denotes the calculation using the monotonic R ;  Wz denotes the calculation using the modified R with a peak added at E = 250 eV. At or below 1 MeV amu-', Z 2  dependence may be altered by higher order terms in Z. Energy U+ 9' W 2  Projectile (MeV a m - ' )  ( n a 3  (%I (%I Experiment (%) H+ 1 0.073 2 0.040 5 0.018 ~~ 0.35 0.33 0.31 ~ 0.41 0.38 0.35 0.33" 0.28" 0.26h Ha 1 2 5 Heo 2 5 0.065 0.033 0.013 0.041 0.016 0.69 0.69 0.69 1.04 1.03 0.80 0.8 1 0.81 1.23 1.23 0.83' - - a Shah and Gilbody (1988). Andersen et a1 (1987). DuBois and Kover (1989). photon ratio of 5%. This implies that %, could set an upper limit on both (which could differ for various projectiles) and the right-hand side of (9). Our results also imply inequalities for other systems, e.g. gH+ = $ B H e 2 +  < $TiHe+ < %Heo.  This work was supported by the Division of Chemical Sciences, Office of Energy Research, US Department of Energy. Reference Aberg T 1973 Proc. Int. Con$ on Inner Shell Ionisation Phenomena and Future Applications (Atlanta, G A )  Andersen L H, Hvelplund P, Knudsen H, Moller S P, Sorensen A H, Elsener K, Rensfelt K-G and Uggerhoj Andersen L H, Nielsen L B and Sorensen J 1988 J. Phys. B :  At .  Mol. Opt. Phys. 21 1587-1602 Anholt R 1986 Phys. Lett. 114A 126 Bates D R and Griffing G W 1955 Proc. R. Soc. A 68 831 Bell K L, Dose V and Kingston A E 1969 J.  Phys. B :  A t .  Mol. Phys. 2 653 Bohr N 1948 K. Dansk Vidensk Selsk. Mat. Fys. Meddr. 18 No 8 Briggs J S and Taulbjerg K 1978 Topics in Current Physics vol 5 ed I Sellin (New York: Springer) p 105 Byron F W and Joachain C J 1967 Phys. Rev. 164 1 Carlson T A 1967 Phys. Rev. 156 142 Carlson T A and Nextor C W 1973 Phys. Rev. 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Rev. Lett. 16 498 Quarles C A 1973 Proc. Int. Conf: on Inner Shell Ionisation Phenomena and Future Applications (Atlanta, Reading J F, Ford A L and Fang X 1989 Phys. Rev. Lett. 62 245 Rudd M E, Kim Y-K, Madison D H and Gallagher J W 1985 Rev. Mod. Phys. 57 965 Shah M B and Gilbody H B 1985 J. Phys. E :  At .  Mol. Phys. IS 899 Vegh L 1989 J. Phys. E :  At .  Mol. Opt. Phys. 22 L35 North-Holland) ch 7 pp 337-9 C A )  ed R W Finks er a1 (Oak Ridge, TN: US Atomic Energy Commission) p 1617 

High-velocity Limits for the Ratio of Double to Single Ionisation of Helium by Projectiles with Electrons

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High-velocity Limits for the Ratio of Double to Single Ionisation of Helium by Projectiles with Electrons

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