Charlton M., Humberston J.W. Positron Physics

Подождите немного. Документ загружается.

5.7 Inner shell ionization 259

100 eV impact energy. A small peak was observed close to 42 eV in the

ejected electron energy spectrum when both the electron and positron

were emitted at small angles to the forward (incident beam) direction.

These authors attributed this to the ECC process (see the discussion

in section 5.2) and therefore provided the ﬁrst clear evidence for this

mechanism in positron impact ionization.

5.7 Inner shell ionization

In this section we describe experiments whose aim has been to determine

explicitly inner shell ionization cross sections (or ratios of electron and

positron cross sections). Section 5.5 contained an account of the inﬂuence

of inner shell processes on multiple ionization of the heavier noble gases.

The ﬁrst experiments to study inner shell processes, and indeed the

ﬁrst investigations of positron impact ionization, were those of Hansen,

Weigmann and Flammersfeld (1964), Hansen and Flammersfeld (1966)

and Seif el Nasr, Ber´enyi and Bibok (1974). These workers used β-

spectrometers to velocity-select positrons and electrons emitted from

radioactive sources, and they measured (by detection of the appropriate

X-rays) the ratio of the K-shell ionization cross sections, σ

, for the

two projectiles scattering from various targets at high energies, typically

hundreds of keV. Within the statistical accuracy of the experiment no

deviation from unity was found for the ratio σ

−

)/σ

). Ito et al.

(1980), also using β-spectroscopy, measured K- and L-shell cross section

ratios for a silver target for a range of positron and electron kinetic

energies above 100 keV. Although they found the ratio to be unity for

the L-shell processes, σ

−

)/σ

) was observed to be greater than

unity below approximately 150 keV, around six times the K-shell binding

energy. A similar trend was observed at relativistic energies by Schneibel

et al. (1976).

Turning to more recent experiments, a vertical section of the target

region used by Schneider, Tobehn and Hippler (1991, 1992) and Schneider

et al. (1993) is shown in Figure 5.22. The positron beam was provided

by the pulsed source at the Giessen electron linear accelerator (see e.g.

Ebel et al., 1990, and references therein). An electron beam could also be

produced, using a heated ﬁlament in close proximity to the target. The

kinetic energy of the projectiles was mainly determined by applying an

electrical potential to the target, which was tilted at 45

◦

to the beam axis.

The targets consisted of various thin silver and gold foils and gold–silver

multilayer arrangements (Schneider et al., 1993). The purpose of the

latter was to use the silver L-shell X-ray as a standard for comparison

of the positron and electron cross sections, since little diﬀerence between

them is expected in the energy range under investigation. X-rays emitted

260 5 Excitation and ionization

Fig. 5.22. Experimental set-up for studies of positron impact inner shell ioniza-

tion (Ebel et al., 1989). Key: (1) target support; (2) lead collimator; (3) ﬂange

with thin window for the transmission of X-rays to (4) a Si(Li) detector; (5)

aluminium window 0.1 mm thick; (6) NaI(Tl) gamma-ra

y detector.

into a small solid angle passed through a thin mylar window before being

registered by a Si(Li) detector. A NaI(Tl) detector was located directly

opposite the Si(Li) counter as a veto to suppress the Compton-scattered

annihilation γ-rays which may be registered by the latter.

Figure 5.23 shows a summary of what has been obtained for the K-shell

ionization of various elements expressed as the ratio σ

−

)/σ

) versus

E/I, the reduced impact energy, where I is the K-shell binding energy.

There is good accord between the various experiments in the energy

range of overlap, with σ

−

)/σ

) varying smoothly throughout the

range E/I = 2–16. Above E/I ∼ 5 the ratio is unity, but below this

value the electron cross section rises above that for positrons, the ratio

reaching a value around eight at the lowest impact energy. This is most

clearly seen in the data of Ebel et al. (1989) for silver. The steep rise

in the ratio at the lower impact energies is reproduced by the results

of a plane wave Born approximation with a correction included to take

account of modiﬁcation of the projectile wave function by the Coulomb

ﬁeld of the target nucleus. It is notable that when this correction is

omitted the calculation completely fails in the low energy region. As

summarized by Knudsen and Reading (1992), the diﬀerence between the

K-shell ionization probabilities for the two projectiles is mainly governed

by the diﬀerent deﬂections experienced by the particles in the Coulomb

ﬁeld of the nucleus. The eﬀect of this is expected to lessen with rising

5.7 Inner shell ionization 261

Fig. 5.23. Ratio of K-shell ionization cross sections for electrons and positrons

scattering

from silver and copper at various impact energies: ——, Born ap-

proximation calculation including a Coulomb correction (see text); – – –, Born

approximation without the Coulomb correction. The data are: •, Ag (Ebel et al.,

1989);

, Cu (Ebel et al., 1989); , Cu (Schultz and Campbell, 1985); , Ag (Ito

et al., 1980).

projectile energy (in accord with approach of the cross section ratio to

unity) and, as observed, with increasing principal quantum number of the

electronic shell.

Calculations by Gryzi´nski and Kowalski (1993) for inner shell ionization

by positrons also conﬁrmed the general trend. Theirs was essentially

a classical formulation based upon the binary-encounter approximation

and a so-called atomic free-fall model, the latter representing the internal

structure of the atom. The model allowed for the change in kinetic energy

experienced by the positrons and electrons during their interactions with

the screened ﬁeld of the nucleus.

Lennard et al. (1988) studied L-shell ionization ratios for gold, where it

was hoped that the Coulomb eﬀect would be so reduced as to permit the

observation of certain basic diﬀerences between positron–electron scatter-

ing (Bhabha, 1936) and electron–electron scattering (Møller, 1932), for

large energy transfers. Later Schneider, Tobehn and Hippler (1991, 1992),

using the apparatus described above, measured the same quantities. As

262 5 Excitation and ionization

Fig. 5.24. Ratio of L-shell ionization cross sections for electrons and positrons

at various impact energies: ——, Born approximation calculation including the

Coulomb correction (see text). The data are: •, Schneider, Tobehn and Hippler

(1991) for a gold target; ◦, Lennard et al. (1988) for a silver target.

shown in Figure 5.24, a major discrepancy exists between the two sets

of measurements, which increases as the threshold is approached, with

−

)/σ

) dipping below unity in the data of Lennard et al. (1988)

whilst exhibiting the opposite trend in the work of Schneider, Tobehn

and Hippler (1991). The solid line in this ﬁgure, which is in qualitative

accord with the Schneider data, was obtained using the plane wave Born

approximation, including exchange and allowance for Coulomb eﬀects.

The Møller–Bhabha eﬀects mentioned above were not included in this

calculation.

Although the general trend of the theories tends to support the work

of the Giessen–Bielefeld group, Gryzi´nski and Kowalski (1993) pointed

out that the diﬀerences between the results from the two experiments

may be explained by diﬀerent target arrangements. They noted that the

orientation of the latter with respect to the beam (Ebel et al., 1989, target

at 45

◦

; Lennard et al., 1988, target perpendicular) may play a role if there

is some anisotropic orientation of the electron orbitals in the foil targets.

5.7 Inner shell ionization 263

This could lead to anisotropies in the emission of the X-rays. Further

experimental work is needed to clarify this issue.

Schneider et al. (1993) developed their group’s technique further in

order to assign an absolute scale to the ionization cross sections by electron

and positron impact for the K-shell of silver and the L

-shell of gold.

Magnitudes were of the order of 10

−23

and 10

−22

respectively,

though strongly energy dependent as the relevant threshold is approached.

Positron annihilation

6.1 Introduction and theoretical considerations

Before the advent of low energy beams, the only means of investigating

positron interactions with atoms and molecules was to study their annihi-

lation. Information could thereby be obtained directly on the annihilation

cross section but only indirectly for other processes such as elastic scatter-

ing. In this chapter we consider the annihilation of so-called free positrons

in gases. The fate of positrons which have formed positronium prior to

annihilation is treated in Chapter 7.

The basic physical principles governing positron annihilation were de-

scribed in subsection 1.2.1, where the non-relativistic limit of the Dirac

cross section for the annihilation of a free positron–electron pair into two

gamma-rays was given as σ

2γ

=4πr

c/v, equation (1.3). If the electron

density in the vicinity of the positron is n

, then the annihilation rate is

2γ

=4πr

. However, as shown in subsection 1.2.2, annihilation into

two gamma-rays requires the positron–electron pair to be in a singlet spin

state, but only one quarter of the electrons in an unpolarized ensemble

would form such a state with the positron. The remaining electrons would

form a triplet spin state with the positron, for which annihilation is into

three gamma-rays at a much lower rate (less than 1% of the two-gamma

rate). Thus, the total free-positron annihilation rate is

 πr

. (6.1)

If the electrons are bound in atoms or molecules, each having Z elec-

trons, and the number density of atoms or molecules is n, the electron

density is n

= nZ. Therefore, if there were no distortions of the positron–

atom system, the free annihilation rate would be given by

= πr

cnZ. (6.2)

264

6.1 Introduction and theoretical considerations 265

In practice, the positron does inﬂuence the charge distribution in the

atom or molecule, in such a way as to enhance the electron density in its

vicinity. Allowance for this can be made by replacing Z by an eﬀective

number of electrons, Z

eﬀ

. Thus, the annihilation rate may be expressed

= πr

cnZ

eﬀ

=0.201ρZ

eﬀ

(µs

−1

), (6.3)

where ρ is the gas density in amagat (1 amagat ≡ 2.69 × 10

−3

), and

the annihilation cross section is therefore

2γ

= λ

/(vn)=πr

eﬀ

/v. (6.4)

Distortion of the target atoms or molecules is particularly pronounced

at very low positron speeds, and Z

eﬀ

may then be considerably larger

than Z; however, as the speed increases and the electrons have less time

to react to the perturbing ﬁeld of the positron, Z

eﬀ

initially decreases.

The value of Z

eﬀ

is a measure of the probability that the positron is

at essentially the same position as any of the electrons in the target, and

it can be calculated from the wave function representing elastic positron

scattering by the target. If the wave function is Ψ(r

, r

,...,r

Z+1

), where

is the positron coordinate and r

,...,r

Z+1

are the coordinates of the

Z electrons in the target system, then

eﬀ

Z+1



i=2



|Ψ(r

, r

,...,r

Z+1

δ(r

− r

) dr

···dr

Z+1

= Z



|Ψ(r

, r

,...,r

Z+1

δ(r

− r

) dr

···dr

Z+1

, (6.5)

since the wave function is antisymmetric in all the electron coordinates.

In this calculation the positron wave function must be normalized in such

a way that its asymptotic form has unit amplitude, i.e.

Ψ(r

, r

,...,r

Z+1

) ∼

→∞

exp(ik · r

)Φ(r

,...,r

Z+1

), (6.6)

where Φ(r

,...,r

Z+1

) is the wave function of the target atom.

Whereas the error in the calculated value of the elastic scattering phase

shift is usually of second order in the error in Ψ, the error in Z

eﬀ

is of

ﬁrst order; the values of Z

eﬀ

therefore tend to be rather less accurate than

the corresponding phase shifts. Consequently, the value obtained for Z

eﬀ

provides a sensitive test of the accuracy of a wave function, although

admittedly in a very restricted region of conﬁguration space where the

positron is close to one of the electrons. Drachman and Sucher (1979)

developed an alternative method of calculating Z

eﬀ

in which the delta

function δ(r

− r

) is replaced by a global operator but, because it is

266 6 Positron annihilation

more diﬃcult to implement, the method has had very limited use (Ujc

and Stauﬀer, 1985).

In the ﬁrst Born approximation the total wave function is taken to

be a plane wave for the positron multiplied by the undistorted target

wave function, and consequently Z

eﬀ

= Z. This approximation is valid at

suﬃciently high energies, and one might expect the calculated value of Z

eﬀ

to tend to Z as the positron energy increases. However, the calculation

should only be carried out for positron energies below the positronium

formation threshold, E

. At higher energies, the explicit representation

of the open positronium channel in the total wave function, when inserted

into equation (6.5), yields an inﬁnite value for Z

eﬀ

, the interpretation of

which is as follows. Above E

, the cross section for positronium formation

is several orders of magnitude larger than the annihilation cross section.

Once formed, positronium certainly undergoes annihilation, and therefore

the positronium formation and positron annihilation cross sections can be

considered to be equivalent, implying a very large value of Z

eﬀ

. At ener-

gies just below the positronium formation threshold, the positron tends

to form virtual positronium with one of the atomic electrons, resulting

in an enhanced electron density in its vicinity and, consequently, a very

rapid increase in the value of Z

eﬀ

as the threshold is approached (Van

Reeth and Humberston, 1998).

Examples of the energy dependence of Z

eﬀ

for atomic hydrogen and

helium are given in Figure 6.1. These results were obtained using the

very accurate elastic scattering wave functions described in detail in sub-

sections 3.2.1 and 3.2.2. The only molecule for which reasonably accurate

calculations of Z

eﬀ

have been made is H

, where Armour, Baker and

Plummer (1990) used the elaborate variational wave functions obtained

from their studies of low energy scattering (see subsection 3.2.4). How-

ever, such is the sensitivity of the value of Z

eﬀ

to the quality of the wave

function that even this calculation only yielded the value 10.2, compared

to the experimental result of 14.8 at room temperature. Nevertheless, this

is much closer to the measured value than any other theoretical result for

this molecule.

For each of these systems the value of Z

eﬀ

exceeds the corresponding

value of Z by a signiﬁcant factor, particularly in the case of atomic

hydrogen, for which it is almost nine times greater at very low energies,

whereas for helium the factor is only two. In a quite highly polarizable

atom, such as hydrogen, the outer electrons are readily attracted towards

the incident positron, enhancing the probability for annihilation. In an

early study of the correlation between the value of Z

eﬀ

and the dipole

polarizability of the target, α, Osmon (1965) found that, for many simple

atoms and molecules, a reasonably good ﬁt to the experimental data was

6.1 Introduction and theoretical considerations 267

Fig. 6.1. Variation of Z

eﬀ

with positron momentum. (a) Atomic hydrogen,

Humberston and Wallace (1972). (b) Helium: ——, Campeanu and Humberston

(1977b) for helium model H5; – – –, Campeanu and Humberston (1977b) for

helium model H1; — · —, McEachran et al. (1977); ×, Roellig and Kelly (1965)

(Fraser, 1968); •, Coleman et al. (1975b).

provided by the relationship Z

eﬀ

∝α

1.25

. However, more recent data

are consistent with Z

eﬀ

∝α (Wright et al., 1983). Some molecules,

particularly large organic molecules (e.g. Iwata et al., 1995) do not ﬁt this

pattern and have values of Z

eﬀ

several orders of magnitude greater than

the number of electrons in the molecule; for example, Z

eﬀ

is around 18 000

for benzene, and greater than 10

for anthracene. One theory is that these

very large values arise because the positron forms a pseudo-bound-state

or resonance with the molecule, in which the positron is trapped in its

vicinity for much longer than the usual collision time. Molecules with

very large values of Z

eﬀ

 also have low threshold energies for positronium

formation, E

, the relationship being reasonably well represented by

lnZ

eﬀ

≈A/E

+ B,

where A and B are constants (Murphy and Surko, 1991). This fact

prompted Laricchia and Wilkin (1997) to develop an alternative positron-

trapping mechanism based on the increasing signiﬁcance of virtual positro-

nium formation as the threshold energy for real positronium formation

268 6 Positron annihilation

is lowered. Exploiting the time–energy uncertainty relationship, they

assumed that a positron with an incident energy an amount ∆E below

the positronium formation threshold will form virtual positronium and

be trapped in the vicinity of the target system for a time ∆t  /∆E,

during which it might annihilate with one of the electrons. The value of

eﬀ

is then expressed as

eﬀ

πr



γ[1−exp(−λ

)]+(1−γ){1−exp[−∆t(λ

+λ

)]}



, (6.7)

where σ

is the elastic scattering cross section, λ

and λ

are the direct

and pick-oﬀ annihilation rates, λ

is the spin-averaged annihilation rate of

positronium and γ = exp(−∆t/t

), where t

is the collision time. The pre-

dictions of this model are in qualitative agreement with the experimental

data and also with the theoretical results of Van Reeth and Humberston

(1998) mentioned above. Further discussion of positron annihilation on

large molecules has been given by Iwata et al. (2000).

The wave function of the ion that remains after annihilation is a super-

position of eigenstates of the Hamiltonian of the ion, the relative proba-

bilities of which may be determined from the wave function used in the

calculation of Z

eﬀ

. The annihilation process takes place so rapidly, com-

pared with normal atomic processes, that it is reasonable to assume the

validity of the sudden approximation. Consequently, the wave function

of the residual ion when the positron has annihilated with electron 2 at

the position r

= r

F (r

; r

,...,r

Z+1

)=Ψ(r

= r

, r

,...,r

Z+1

), (6.8)

where we have used the same nomenclature as in the positron–atom wave

function. The relative probability that the residual ion is in its n

ion

eigenstate, with wave function Φ

ion

,...,r

Z+1

), is then obtained by

projecting the function F (r

; r

,...,r

Z+1

) onto this state, squaring the

result and integrating over the positron coordinate to give

P (n

ion

) ∝



|F (r

; r

,...,r

Z+1

)Φ

ion

,...,r

Z+1

) dr

···dr

Z+1

(6.9)

By exploiting closure in the summation over all the states of the ion,

it may be shown that the normalized probability of the ion being in the

state n

ion

P (n

ion

eﬀ



F (r

; r

,...,r

Z+1

)

×Φ

ion

,...,r

Z+1

) dr

...dr

Z+1

, (6.10)