Slowing-down of positronium: analysis of the age–momentum correlation

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Abstract

The problem of deconvoluting measurements of the slowing-down of positronium by means of the age–momentum correlation technique based on the Doppler broadening of the annihilation-photon line is reduced to the solution of systems of linear equations. The solution gives directly the kinetic energy of positronium as function of the positronium age (=time passed since its formation). A novel time-resolution function consisting of one Gaussian times an arbitrary number of Hermite polynomials is proposed.

Introduction

The age–momentum correlation technique (AMOC) (Mackenzie and McKee, 1976, Stoll et al., 1992) determines simultaneously two pieces of information on the annihilation of positrons, viz. the positron age, i.e. the time interval between the implantation of a positron and its annihilation with electrons of the target material, and the Doppler shifts, ±ΔEγ, of the 511 keV photons resulting from the annihilation of the positron with an electron of opposite spin. In the terminology of nuclear physics, AMOC may be described as time-differential Doppler-broadening measurement of the 2γ-annihilation radiation. It has been demonstrated that the AMOC technique allows the gradual loss of the kinetic energy of positronium “atoms” Ps = (e+e) to be investigated in a wide range of materials (Stoll et al., 1997).

Typically, Ps is formed with kinetic energies of less than 101eV. As will be shown presently, this means that the Doppler shifts to be studied are comparable with the resolution of the ΔEγ measurements. In the majority of the investigated condensed phases the time required for Ps to slow down to thermal velocities is of the order of a few 10−11s (Stoll et al., 1995, Stoll et al., 1997), hence much shorter than the time resolution of even the best positron lifetime spectrometers. The present paper is devoted to the quantitative analysis of AMOC measurements of the Ps slowing-down and, specifically, to developing procedures capable of coping with the limited momentum and time resolutions of the available AMOC equipment.

According to Einstein’s special theory of relativity, the energy of a photon emitted by a particle with velocity ν relative to the detector, if observed in the direction of ν, is increased byΔEγ=E01−ν/c2−1/21+ν/c−1=E0ν/c1+Oν/c,where E0 is the photon energy as seen by an observer moving at the same velocity as the emitting particle and c the speed of light in vacuum. For a photon generated by the 2γ-annihilation of an e+e-pair with longitudinal momentum p1 (= momentum component in the direction of the observer), to first order in ν/c the so-called longitudinal Doppler shift is thus given byΔEγ=plc/2.

In the annihilation of “free” e+ (i.e., positrons not bound in Ps) and in the pick-off annihilation of e+ bound in orthopositronium (o-Ps), the main contributions to the momenta p of the annihilating e+e-pairs come from the momenta of the annihilating electrons. The information on the slowing-down of Ps, however, is provided by the self-annihilation of para-positronium (p-Ps). In this case p is the momentum of the Ps “atoms” and therefore expressible in terms of the Ps kinetic energyEkin=p2/4me,where me denotes the electron (and positron) mass.

In the special case plp, insertion of Eq. (3) into Eq. (2) gives usΔEγmec2Ekin1/2.Thus, in the self-annihilation of p-Ps the maximal longitudinal Doppler shift (corresponding to the detection of one of the annihilation photons in the ν-direction) is the geometric mean of the Ps kinetic energy and the positron rest energy, mec2=511 keV. For p-Ps starting out with, say, Ekin = 6.8 eV (i.e., equal to the Ps binding energy in vacuum1) the maximum shift following from Eq. (4) is 1.68 keV. This means that the Doppler shifts to be determined in AMOC studies of the Ps slowing-down are comparable with or smaller than the resolution of the Ge detector of the most advanced AMOC set-up existing (Stoll et al., 1992).

From the preceding remarks it follows that in order to obtain quantitative information on the slowing-down of Ps in condensed matter, the AMOC relief (=number of “good” counts per recording channel as a function of the positron age and the Doppler shift of one of the annihilation photons) has to be deconvoluted with respect to both positron age and longitudinal Ps momentum. Thus, full use of the information obtainable by AMOC requires two-dimensional data fitting with two deconvolutions2. This is far from being a trivial task, particularly so since at best one of the resolution functions (that of the ΔEγ measurements) may be obtained by independent measurements (cf. Section 2). In 3 The momentum distributions, 4 The AMOC relief we develop a procedure that reduces, in a rigorous manner, the two-dimensional data-fitting problem to a one-dimensional problem, viz. to the task of solving linear Fredholm integral equations of the first kind, with the lifetime resolution function as kernel. Physical reasoning and computational efficiency both lead us to propose a novel form of the time resolution function. For this class of resolution functions a general, model-independent algorithm for obtaining the solutions of the integral equations is developed (5 Solution of the integral equations for, 6 Generalized time-resolution functions). Once the algorithm has been implemented, information such as the age dependence of the Ps energy may be obtained routinely from the measured AMOC data. One of the strengths of the procedure is that the computational effort may be quite readily adjusted to the amount and quality of the experimental data.

For visualization purposes and qualitative interpretations, one-dimensional data reductions such as age-dependent lineshape parameters and momentum-resolved mean positron lifetimes may be used; in Section 7 they are discussed in terms of the general theory. Should the available data not allow a model-free evaluation, it is still possible to derive the parameters of models for the slowing-down mechanism by various fitting procedures (Section 8).

Section snippets

The resolution functions

The resolution function of the lineshape measurements may be obtained fairly directly by measuring the lineshape of one of the γ-ray emission lines in the neighbourhood of 511 keV, e.g. of the 514 keV line of 85Sr. It is represented very well by a GaussianRppl−pl−1/2σpexp−σ2ppl−pl2,where pl is the longitudinal momentum of an e+e-pair at the instant of its annihilation and plpl the deviation of the observed value from the true one. Thus, the parameter σp may be considered to be quite

The momentum distributions

Let wpl;pdpl denote the conditional probability that the 2γ-annihilation of an e+e-pair with momentum p corresponds to a longitudinal momentum between pl and pl+dpl. Owing to the absence of any preferred spatial direction in the 2γ-annihilation of positrons — a direct consequence of the electromagnetic nature of the annihilation process — the distribution of the photon emission directions is isotropic. From this it follows immediately that the conditional probability density is given by (

The AMOC relief

In deriving an expression for the AMOC relief, we make the assumption that the modulus of the Ps momentum, p, is a unique function p(t) of the positronium age t with a unique inverse t=fp. This assumption is well justified if the principal slowing-down mechanism is the scattering of Ps by optical phonons (Seeger, 1995, Seeger, 1998). For Ps in condensed rare gases, in which owing to the absence of optical phonons the dominant slowing-down mechanism is presumably the scattering by acoustical

Solution of the integral equations for p2m

For given amt, Eq. (17) are Fredholm integral equations of the first kind of the Wiener–Hopf type with kernel Rtt−t′,t′=∞t′=0ymt′Rtt−t′dt′=12m+1!amt,for the unknown functionsymt′σppt′2mexp−λt′m=1,2,3,….As Eq. (3) shows, y1t is proportional to the Ps kinetic energy; hence by solving Eq. (19) we may obtain Ekin as a function of the Ps age t, provided a1t is known from Eq. (18) with sufficient statistical accuracy. (We assume that by extrapolation from larger t the contribution of “free” e+ and

Generalized time-resolution functions

As a rule, the representation of the time resolution function by one Gaussian is not fully satisfactory. As mentioned in Section 2, in such cases Rtt−t′ is customarily represented by a sum of Gaussians. Quite common is the representation by three Gaussians. However, this is clearly not the optimal procedure to handle the time-resolution problem, as may be seen as follows.

In a sum of Gaussians there is necessarily one term that dominates at large |tt′|. Therefore, the Gaussian behaviour of the

One-dimensional visualization

Experience has shown that the two-dimensional AMOC reliefs, an example of which is shown in Fig. 1, are not very amenable to interpretation by inspection. For the visualization of ΔEγ-based AMOC data, two one-dimensional data reductions are in use. These are

    (i) momentum-resolved mean lifetimes (Kishimoto and Tanigawa, 1982)
τ̄ppl+∞−∞tNAMOCpl,tdt+∞−∞NAMOCpl,tdtand
    (ii) age-resolved lineshape parameters
Sttpl=pSpl=−pSNAMOCpl,tdpl−∞−∞NAMOCpl,tdpl.In Eq. (49) the choice of the integration

Parametrization of p(t)

The procedures to be presented in this section are based on the assumption that the general form of p(t) is known, so that our original task, namely having to solve the integral equation (11) for f(p), is reduced to determining the adjustable parameters in the ansatz for p(t). As is easy to see, no physically reasonable functional form of p(t) permits the evaluation of the right-hand side of Eq. (11) in closed form. It is, therefore, better to start not from Eq. (11) but from the set (19) of

Conclusions

  • (i) Quantitative and reliable AMOC studies of Ps slowing-down based on Doppler-broadening measurements are feasible but require careful deconvolution procedures both with respect to Doppler broadening and positron age.

  • (ii) A key step is the data reduction by computing from the AMOC raw data the moments of the age-resolved annihilation-photon lineshapes.

  • (iii) Provided the statistics are good enough, the slowing-down of Ps may be studied without presupposing a model. Initial and final momenta may

Acknowledgements

The author acknowledges gratefully the help and advice of Dr. Hermann Stoll during many years of joint work on positron annihilation and the assistance of Mrs. D. Stammler in preparing the typescript. He also wishes to express his appreciation for the efforts of the referees and of Prof. J. Krystiak and Dr. M. Morhàč (both Bratislava), who helped to eliminate inaccuracies, slips, and typing errors from the original text.

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