Four-dimensional positron age-momentum correlation

We used a Photek IPD340/Q/BI/RS microchannel plate image intensifier (MCPII) with an active diameter of 40 mm. The MCPII exhibits a microchannel plate (MCP) stack in chevron configuration (10 μm pores, thickness to pore diameter ratio 50:1 and 80:1 for the upper and lower MCP) and a bialkali photocathode evaporated onto the 9 mm thick fused silica entrance window.

By using the high viscosity coupling grease Rhodorsil 47 V 100 000 a Scionix CeBr₃ scintillator pixel array (pixel dimension 2.5 mm × 2.5 mm × 8 mm; each pixel wrapped with polytetrafluoroethylene (PTFE) tape; pixel pitch 3.3 mm) was coupled to the entrance window and covered the whole active area of the MCPII. The CeBr₃ pixels were, due to their high hygroscopicity, hermetically sealed in a metal case with a 1.5 mm thick fused silica window. In this work a total of 104 pixels were utilised for the determination of the position.

The electron cloud produced in the channels of the MCP stack of the MCPII is collected on a resistive anode. To derive the two dimensional position, a 2D-backgammon anode is capacitively coupled from the outside of the MCPII to the resistive anode. At a gamma energy of 511 keV (²²Na) the position resolution of the scintillation detector is limited by the pixel cross section. The best single time resolution at a gamma energy of 511 keV (²²Na) was about 320 ps (FWHM) located around the centre of the MCPII. The single time resolution is about a factor 2 to 3 worse at the edge of the active area of the MCPII. A detailed description of the scintillation detector can be found in [17].

The position sensitive HPGe-detector (Canberra EGPS 48*48*20-36 PIX) consists of a planar germanium crystal with a thickness of 20 mm segmented into 6 × 6 pixel contacts (8 mm × 8 mm each). A gamma ray which hits a pixel induces a charge in the neighbour pixels. Thus, by taking also these induced charges into account, the position sensitivity is smaller than the pixel size. At a gamma energy of 662 keV (¹³⁷Cs) an energy resolution of 1.33 keV and a position resolution of 1.6 mm has been achieved [16].

A schematic diagramm of our detector arrangement can be seen in figures 1 and 2. The two detectors were in a 180° arrangement to measure both annihilation quanta in coincidence. The detector distance to the sample was 21 cm for the scintillation detector and 15.5 cm for the pixelated HPGe-detector in order to have comparable angular resolutions.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

To measure the positron lifetime we fed the timing signal of the scintillation detector into a constant fraction discriminator (CFD) to generate the start signal for the time-to-amplitude converter (TAC). The stop signal was produced by a fast gate logic (FGL) utilising the 50 MHz oscillator from the beam pulsing and the CFD signal [18]. From the measured time differences between the start and stop signals we obtained the positron lifetime spectrum L(t) which can be expressed by:

$$\begin{eqnarray}&&L(t)=\left[\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{I}_{i}}{{\tau }_{i}}\exp (-t/{\tau }_{i})\right]\,\ast \,W(t)+\mathrm{BG},\end{eqnarray} \tag{ 1 }$$

where ${\tau }_{i}$ is the ith positron lifetime and I_i is the ith intensity. W(t) is defined as the instrument resolution function and BG is the background that we assume is mainly caused by the chopper of our pulsing unit which does not completely blank out the remaining DC part of the beam. Background events due to backscattered positrons that annihilate elsewhere in the sample chamber are mostly suppressed by our coincident detector set-up. The term $\exp (-t/{\tau }_{i})$ is implicitly assumed to be 0 for $t\lt 0$.

To deduce the three-dimensional momentum of the electron annihilating with the positron the angular correlation of the annihilation radiation was measured with both detectors and the Doppler broadening of one of the annihilation gamma quanta was acquired with the pixelated HPGe-detector.

The angular correlation of the two gamma quanta was derived by the two-dimensional position of interaction of the gamma quanta for both detectors as described in section 2.2. Angular deviations from the 180° angular correlation in x- and y-direction (${\alpha }_{x}$, ${\alpha }_{y}$) can be deduced by the measured positions. Therefore the electron momenta in x- and y-direction are given by [1]:

$$\begin{eqnarray}&&{p}_{x}={\alpha }_{x}\,{m}_{0}c,\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&{p}_{y}={\alpha }_{y}\,{m}_{0}c.\end{eqnarray} \tag{ 2b }$$

The angular deviations ${\alpha }_{x}$ and ${\alpha }_{y}$ are depicted in figure 2. A measured angular deviation of 1 mrad is equivalent to a momentum of 10⁻³ ${m}_{0}c$.

By measuring the Doppler broadening ${\rm{\Delta }}E$ of the 511 keV photopeak with the HPGe-detector one receives the electron momentum in z-direction (longitudinal electron momentum p_l) [1]:

$$\begin{eqnarray}&&{p}_{z}={p}_{{\rm{l}}}=\displaystyle \frac{2{\rm{\Delta }}E}{c}.\end{eqnarray} \tag{ 3 }$$

The absolute value of the three-dimensional momentum $| \vec{p}|$ of the electron is given by:

$$\begin{eqnarray}&&| \vec{p}| =\sqrt{{p}_{x}^{2}+{p}_{y}^{2}+{p}_{z}^{2}}=\sqrt{{p}_{{\rm{t}}}^{2}+{p}_{{\rm{l}}}^{2}},\end{eqnarray} \tag{ 4 }$$

where the transversal electron momentum p_t is defined as:

$$\begin{eqnarray}&&{p}_{{\rm{t}}}=\sqrt{{p}_{x}^{2}+{p}_{y}^{2}}.\end{eqnarray} \tag{ 5 }$$

By histogramming only the positron age of each event in the 4D-AMOC spectra we obtained two coincident positron lifetime spectra. In figure 8 the spectra as well as the fit of the positron lifetime components to the data are shown. Formula 1 from section 2.3.1 was used for the fit so that the absolute values of the residuals of the fit were as small as possible (figure 8).

Zoom In Zoom Out Reset image size

Our instrument function W(t) was determined from the gold foil measurement by fixing the positron lifetimes ${\tau }_{1}$ and ${\tau }_{2}$ to the values obtained by the PLEPS measurement (section 3). Additionally the ratio ${I}_{1}/{I}_{2}$ was kept constant as given by the measurements at PLEPS. With these boundary conditions we fitted through the positron lifetime spectrum of the gold foil. We received ${\tau }_{3}$, I₁, I₂ and I₃ and the instrument function W(t) consisting of two Gaussian distributions with an overall width of 540 ps (FWHM).

For the fitting procedure of the positron lifetimes and intensities of the carbon tape the above mentioned instrument function W(t) from the gold foil measurement was used. Additionally ${\tau }_{1}$, ${\tau }_{2}$, ${\tau }_{3}$ and ${I}_{1}/{I}_{2}$ were fixed as determined by the PLEPS measurements.

In table 1 the positron lifetimes and their respective intensities of the gold foil and the carbon tape are summarised for the measurements done at PLEPS and the SPM interface. The PLEPS values could be well reproduced at the 4D-AMOC measurements done at the SPM interface. The absolute values of the residuals of the fit of the positron lifetime spectra were for the gold foil and the carbon tape less than a value of about three (figure 8). ${\tau }_{4}$ (carbon tape) of the measurement done at PLEPS could not be reproduced with the 4D-AMOC measurement. This can be explained on the one hand by the worse statistics at the SPM interface measurements as the positron lifetime spectra at PLEPS contained 100 times more events. On the other hand the peak to background ratio was a factor 100 better at the PLEPS measurements. Additionally the fact that not exactly the same piece of the carbon tape was taken for the measurements at PLEPS and at the SPM interface could have influenced the positron lifetime ${\tau }_{4}$.

Table 1.  Positron lifetimes and their intensities from the gold foil and the carbon tape measured at PLEPS (see section 3) and the SPM interface. Italic bold values were taken for the fit procedure of the positron lifetimes and intensities for the SPM interface measurements (for more details see text).

Gold foil	${\tau }_{1}$ (ps)	I1(%)	${\tau }_{2}$ (ps)	I2 (%)	${\tau }_{3}$ (ps)	I3 (%)
PLEPS	${\boldsymbol{175}}$	${\boldsymbol{22.4}}$	${\boldsymbol{347}}$	${\boldsymbol{77.3}}$	2454	0.3
SPM intf.	175	22.1	347	76.1	2550	1.8

Carbon tape	τ1 (ps)	I1 (%)	τ2 (ps)	I2 (%)	τ3 (ps)	I3 (%)	τ4 (ps)	I4 (%)
PLEPS	${\boldsymbol{162}}$	${\boldsymbol{13.9}}$	${\boldsymbol{380}}$	${\boldsymbol{52.2}}$	${\boldsymbol{1029}}$	5.9	3013	28.0
SPM intf.	162	13.7	380	51.5	1029	7.8	2650	27.0

The measured positron lifetime spectra from figure 8 are given by the different possible states in which the positrons can annihilate. Equation (1) from section 2.3.1 can also be written as:

$$\begin{eqnarray}L(t) & = & \displaystyle \sum _{i=1}^{N}\left[\displaystyle \frac{{I}_{i}}{{\tau }_{i}}\exp (-t/{\tau }_{i})\,\ast \,W(t)\right]+\mathrm{BG}\\ & = & \displaystyle \sum _{i=1}^{N}{\xi }_{i}(t)+\mathrm{BG}.\end{eqnarray} \tag{ 6 }$$

The annihilation probability w_i(t) or rather w_b(t) that a positron is annihilating at the time t in the ith state or rather contributes to the background is given by:

$$\begin{eqnarray}&&{w}_{i}(t)=\displaystyle \frac{{\xi }_{i}(t)}{L(t)},\ \ {w}_{b}(t)=\displaystyle \frac{\mathrm{BG}}{L(t)}.\end{eqnarray} \tag{ 7 }$$

In figure 9 the positron age dependent annihilation probabilities from the gold foil and carbon tape measurements are depicted. For the calculation of the annihilation probabilities the positron lifetimes ${\tau }_{i}$ and intensities I_i were taken from figure 8 (table 1).

Zoom In Zoom Out Reset image size

By projecting the counts of the 4D-AMOC spectra from figure 7 onto the electron momentum axis we obtained the spectra of the absolute value of the three-dimensional electron momentum (figure 10(a)).

Zoom In Zoom Out Reset image size

In figure 10(b) the ratio of the gold foil spectrum to the carbon tape spectrum is depicted. The ratio is smaller than 1 for momenta between 0 × 10⁻³ ${m}_{0}c$ and 15 × 10⁻³ ${m}_{0}c$ due to a more elevated valence electron contribution to the positron annihilation in the carbon tape (minimum ratio value: about 0.8). For momenta from 15 × 10⁻³ ${m}_{0}c$ to 40 × 10⁻³ ${m}_{0}c$ the ratio is larger than 1 (maximum ratio value: about 1.2) because of an increased core electron contribution of the gold foil. The statistical noise is larger than the mean ratios for momenta higher than 40 × 10⁻³ ${m}_{0}c$.

The mean absolute values of the three-dimensional electron momenta versus the positron age are plotted in figure 11. For each data point at least 1000 events were taken from the respective 4D-AMOC spectrum (figure 7).

Zoom In Zoom Out Reset image size

The mean momenta of the gold foil (figure 11(a)) up to a value of 1.3 ns can be assigned to annihilation in vacancies and in surface states (see section 3). The maximum mean momentum value at about 3.9 ns is probably due to contaminants on the surface.

For the carbon tape (figure 11(b)) the mean momentum is at its lowest value at around 0 ns due to the contribution of para-positronium annihilation. The maximum at about 0.9 ns can be assigned to positrons that do not form positronium. Above circa 2 ns ortho-positronium annihilation gets dominant and contributes the most to the mean momentum. Up to a positron age of approximately 2.5 ns the shape of the mean momentum data of the carbon tape is comparable to 2D-AMOC data from e.g. [37] where the S parameter is plotted as a function of the positron age. (A high S parameter corresponds to a low mean momentum.) In our measurements the mean momentum rises again for positron ages above 2.5 ns while in [37] the S parameter stays constant. Due to the low statistics at higher positron ages this rise might not be significant.

By utilising the annihilation probabilities from figure 9 the discrete mean momentum states P_i could be derived by fitting the equation

$$\begin{eqnarray}&&P(t)=\displaystyle \sum _{i=1}^{N}{P}_{i}{w}_{i}(t)\end{eqnarray} \tag{ 8 }$$

to the data points from figure 11. To reduce the influence of the background (see figure 9), the fits were only performed in a range that is indicated by the solid lines. The sum over all w_i was normalised for each time thus one could write

$$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{N}{w}_{i}(t)=1.\end{eqnarray} \tag{ 9 }$$

The discretized mean momentum states are summarised in table 2. These were not corrected for the three-dimensional momentum resolution of the 4D-AMOC setup. Due to their low annihilation probabilities (see figure 9), the states P₁ of the gold foil and P₁ and P₃ of the carbon tape have a higher confidence interval of the fit.

Table 2.  Discrete momentum states P_i derived from the gold foil and the carbon tape measurements. For each P_i the annihilation channel is specified.

	P1 [10−3 ${m}_{0}c]$	P2 [10−3 ${m}_{0}c]$	P3 [10−3 ${m}_{0}c]$	P4 [10−3 ${m}_{0}c$]
Gold foil	13.9 ± 1.0	17.3 ± 0.3	18.1 ± 0.5	—
	Vacancies	Surface states	Contaminants on surf.	—
Carbon tape	6.9 ± 1.5	19.4 ± 0.6	8.7 ± 2.2	16.8 ± 0.3
	Para-positronium	Various defect sites	Ortho-positronium	Ortho-positronium