Study of the electron-positron annihilation coincidence peak two-dimensional pofile

Study of the electron-positron annihilation coincidence peak two-dimensional pofile

E. do Nascimento^I; O. Helene^I; V. R. Vanin^I; C. Takiya^II

^IInstituto de Física, Universidade de São Paulo, C.P. 66318, 05389-970, São Paulo, SP, Brazil

^IIDepartamento de Ciências Exatas, Universidade Estadual do Sudoeste da Bahia, Vitória da Conquista, BA, Brazil

ABSTRACT

Positron annihilation radiation profi le in aluminum was observed with a pair of Ge detectors in coincidence. ²²Na was used as a source of positron and the two-dimensional gamma energy spectrum wasfi tted using a model function. Annihilation components of positron at rest with conduction band, 1s, 2s, and 2p electrons were observed. The in-flight positron annihilation was also observed. The model function also took into account the detector response function, relative effi ciency corrections and the gamma backscattering. Coincidences involving a combination of Compton effect, pileup, ballistic defi cit, and pulse shaping problems were treated as well.

1 Introduction

This study aimed to understand the shape of the electron-positron annihilation peak measured in coincidence by two photon detectors (Fig. 1).

The proper fit of the 511 keV-511 keV peak requires many analytical functions, giving information about the electron momentum distribution in the analyzed material. This technique is known as Coincidence Doppler Broadening (CDB) of the electron-positron annihilation radiation [1]and is used in studies of the electronic and atomic strutures of defects in solids [2, 3, 4]].

2 Experimental Setup

The profile of the annihilation peak of positrons from a ²²Na source in metallic Al was measured with the Linear Accelerator Laboratory residual radioactivity multi-detector array (MULTI) [5]. The two annihilation gamma-rays were measured with a pair of Ge detectors in coincidence, placed in diametrically opposed positions, separated by 15 cm, and with a 3.7 × 10⁵ Bq (10 mCi) ²²Na source. This source was placed between two 2 mm thick aluminum sheets (99.999% pure). An ¹⁹²Ir source was simultaneously measured to provide references for detector calibration and follow any energy calibration drift during the experiment. The measurement run lasted for 200 h, when 1.5 × 10⁷ events in the peak region were accumulated.

3 Model Function

Usually the results of Doppler broadening measurement are analyzed comparing the calculated annihilation probability density with the experimental data. In this work we opted for another procedure. The convolution of the detector response function with empirical functions to represent the gamma-rays emitted after positron annihilation with 1s, 2s, 2p and conduction electrons were calculated. All these functions were parametrized. This procedure avoids the dificult problem of deconvolution of the Doppler broadening spectrum [6] .

The function model was determined from a qualitative analysis of the experimental data and published theoretical results [7, 8], Ghosh2000. Positron annihilation with band electrons was fitted by three arcs of parabola and one gaussian along the line E₁ + E₂ = 1022 keV:

where E₁ and E₂ are energies in detectors 1 and 2 respectively, and a_i are the cutoff parameters (C_i = 0 when |E₁–E₂| > a_i). Positron annihilation with 1s electrons was fitted by one gaussian along the line E₁ + E₂ + B_1s = 1022 keV:

where B_1s is the binding energy of the 1s electrons. Positron annihilattion with 2s electrons was fitted by two gaussians along the line E₁ + E₂ + B_2s = 1022 keV:

where B_2s is the binding energy of the 2s electrons. Positron annihilation with 2p electrons was fitted by one gaussian along the line E₁ + E₂ + B_2p = 1022 keV:

where B_2p is the 2p electron binding energy. Finally, in-flight positron annihilation was fitted by:

where

(Fig. 2). The A's and s's are the areas and widths of the gaussians respectively. Detection effects due to ballistic deficit, pile-up and Compton scattering (Fig. 3) were considered in the fit. Coincidences involving a combination of Compton effect, pileup, ballistic deficit, and pulse shaping problems (Fig. 4), backscatering (Fig. 5) and efficiency corrections of the detectors in the fitting region, were taken into account. The model functions were fitted to the experimental data and the result is shown in Fig. 6.

The fit was done by the least-squeares method (see, for instance [9]) with the Gauss-Marquardt algorithm due to the non-linearity in the parameters [10]. The chi-squared value was calculated by

where n_ij is the number of observed events in channel (i, j) of the coincidence spectrum (Fig. 1), and F_ij is the fitted function (Fig. 6).

4 Conclusion

The reduced c² obtained in this study, 1.1 with about 62448 degrees of freedom, does not show a disagreement between the data and the model, suggesting that a complete statistical analysis of the coincidence Doppler broadening annihilation radiation is possible. Better model functions can be considered in order to improve the c² value [1].

Acknowledgments

We wish to acknowledge the support of Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP, and the Data Section of the International Atomic Energy Agency.

Received on 20 October, 2003

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Publication Dates

• Publication in this collection
  26 Oct 2004
• Date of issue
  Sept 2004

History

• Received
  20 Oct 2003