Energy dissipation of para-positronium in polymers and silica glass

An energetic positron may pick up an electron from the surrounding media to form the electron-positron bound state, positronium (Ps). ^1–3) Ps can exist either as spin antiparallel para-positronium (p-Ps) or spin parallel ortho-positronium (o-Ps). Owing to spin statistics, the relative formation probability of p-Ps to o-Ps is one to three. The intrinsic lifetime of p-Ps for self-annihilation into two photons of 511 keV is 125 ps and that of o-Ps into three photons is 142 ns. In condensed matter, the positron in o-Ps may pick-off one of the spin opposite electrons from the surrounding media, resulting in two photon annihilation with a shortened lifetime, a process called pick-off annihilation.

Ps favorably forms in a number of polymers ^4–8) and silica. ^9,10) When formed in these substances, both p-Ps and o-Ps get trapped in free volume before annihilation. The pick-off annihilation lifetime of o-Ps can be directly related to the free volume size, so positron annihilation lifetime spectroscopy (PALS) is widely applied to probe the amorphous structure of these substances. ^11–22) In the Tao-Eldrup model, ^23,24) Ps is regarded as a single particle without internal structure and the free volume is approximated as a spherical potential well with a uniform electron layer. The pick-off annihilation lifetime of thermalized o-Ps is inversely proportional to the probability of o-Ps being inside the electron layer and can be expressed as follows:

$$\begin{eqnarray}&&{\tau }_{{\rm{o}}-{\rm{Ps}}}=0.5{\left\{1-\displaystyle \frac{R}{{R}_{0}}+\displaystyle \frac{1}{2\pi }\,\sin \left(\displaystyle \frac{2\pi R}{{R}_{0}}\right)\right\}}^{-1}\left[{\rm{ns}}\right],\end{eqnarray} \tag{ 1 }$$

where ${\tau }_{{\rm{o}}-{\rm{Ps}}}$ denotes the o-Ps pick-off annihilation lifetime, 0.5 ns is the spin averaged Ps lifetime, $R$ and ${R}_{0}$ denote the radii of the free volume and the spherical potential well, respectively, and ${R}_{0}=R+0.166\,{\rm{nm}},$ where 0.166 nm is the thickness of the electron layer.

As is formed from an energetic positron, Ps possesses certain excess energy at the instant of its formation, which is gradually dissipated upon collisions with the surrounding atoms and molecules. ^25,26) However, the energy dissipation may not be completed, and Ps, especially short-lived p-Ps, may still have some energy at the time of annihilation. Understanding the mechanism of Ps slowing down is crucial from the viewpoint of Ps Bose–Einstein condensation (BEC). ^27,28) Ps BEC is expected to enable many innovations that are yet to be achieved, such as the development of $\gamma$-ray lasers. ²⁹⁾ As BEC requires cooling of Ps to an extremely low temperature within a sufficiently short period, seeking a suitable medium for efficient cooling of Ps is important.

In this letter, we discuss the energy dissipation of short-lived p-Ps in polymers and silica glass. One can give insight into the excess energy of p-Ps at the time of annihilation from the Doppler broadening of p-Ps annihilation photons, characterized by the S parameter. The S parameter is defined as the ratio of positron annihilation photon counts in a predetermined window centered at 511 keV to the total counts contained in the entire two photon annihilation peak. A smaller S parameter signifies a larger extent of Doppler broadening due to higher p-Ps momentum (larger kinetic energy). To extract the S parameter specific to p-Ps, we use systematic positron annihilation age momentum correlation (AMOC) data. ³⁰⁾ The deduced S parameters of p-Ps for fluorinated polymers such as poly(tetrafluoroethylene) (PTFE), and silica glass, are significantly less than the S parameters expected for thermalized p-Ps localized in a free volume, indicating the difficulty for Ps to thermalize in a substance containing heavy elements such as fluorine or silicon.

C-group polymers [high-density polyethylene (HDPE), polypropylene (PP), polystyrene (PS)], O-group polymers [polycarbonate (PC), poly(ester carbonate) (PEC), poly(ether sulfone) (PES)], and F-group polymers [poly(tetrafluoroethylene) (PTFE), ethylene-tetrafluoroethylene copolymer (ETFE), poly(vinyl fluoride) (PVF)], and silica glass are studied. ³⁰⁾ Notably, the C-group polymers are molecules comprising carbon and hydrogen, the O-group polymers comprise carbon, oxygen and hydrogen (PES also contains sulfur), F-group polymers contain carbon, fluorine and hydrogen (except for fully fluorinated PTFE), and silica glass comprises the network of silicon and oxygen.

The S parameters specific to p-Ps for various polymers and silica glass are determined from the positron age dependent overall $S\left(t\right)$ parameters obtained from previous AMOC experiments. ³⁰⁾ Assuming that the positron annihilates from three states, p-Ps, non-Ps positron, and o-Ps, and each state has a definite lifetime, we obtain ⁸⁾

$$\begin{eqnarray}&&\begin{array}{l}S\left(t\right)=\,\displaystyle \frac{\displaystyle \frac{1}{3}{I}_{{\rm{o}}-{\rm{Ps}}}{S}_{{\rm{p}}-{\rm{Ps}}}\left(\displaystyle \frac{1}{{\tau }_{{\rm{p}}-{\rm{Ps}}}}\right){e}^{-\left(\displaystyle \frac{t}{{\tau }_{{\rm{p}}-{\rm{Ps}}}}\right)}+\left(1-\displaystyle \frac{4}{3}{I}_{{\rm{o}}-{\rm{Ps}}}\right){S}_{{e}^{+}}\left(\displaystyle \frac{1}{{\tau }_{{e}^{+}}}\right){e}^{-\left(\displaystyle \frac{t}{{\tau }_{{e}^{+}}}\right)}+{I}_{{\rm{o}}-{\rm{Ps}}}{S}_{{\rm{o}}-{\rm{Ps}}}\left(\displaystyle \frac{1}{{\tau }_{{\rm{o}}-{\rm{Ps}}}}\right){e}^{-\left(\displaystyle \frac{t}{{\tau }_{{\rm{o}}-{\rm{Ps}}}}\right)}}{\displaystyle \frac{1}{3}{I}_{{\rm{o}}-{\rm{Ps}}}\left(\displaystyle \frac{1}{{\tau }_{{\rm{p}}-{\rm{Ps}}}}\right){e}^{-\left(\displaystyle \frac{t}{{\tau }_{{\rm{p}}-{\rm{Ps}}}}\right)}+\left(1-\displaystyle \frac{4}{3}{I}_{{\rm{o}}-{\rm{Ps}}}\right)\left(\displaystyle \frac{1}{{\tau }_{{e}^{+}}}\right){e}^{-\left(\displaystyle \frac{t}{{\tau }_{{e}^{+}}}\right)}+{I}_{{\rm{o}}-{\rm{Ps}}}\left(\displaystyle \frac{1}{{\tau }_{{\rm{o}}-{\rm{Ps}}}}\right){e}^{-\left(\displaystyle \frac{t}{{\tau }_{{\rm{o}}-{\rm{Ps}}}}\right)}},\end{array}\end{eqnarray} \tag{ 2 }$$

where ${S}_{{\rm{p}}-{\rm{Ps}}},$ ${S}_{{{\rm{e}}}^{+}}$ and ${S}_{{\rm{o}}-{\rm{Ps}}}$ denote the S parameters of p-Ps, non-Ps positron and o-Ps, respectively, ${\tau }_{{\rm{p}}-{\rm{Ps}}},$ ${\tau }_{{{\rm{e}}}^{+}},$ and ${\tau }_{{\rm{o}}-{\rm{Ps}}}$ denote their corresponding lifetimes; ${I}_{{\rm{o}}-{\rm{Ps}}}$ denotes the yield of o-Ps. The p-Ps yield is equated to one third of the o-Ps yield. To reduce the number of unknown parameters in data analysis we make the following simplifications.

• (1)  
  The lifetime (${\tau }_{{\rm{o}}-{\rm{Ps}}}$) and yield (${I}_{{\rm{o}}-{\rm{Ps}}}$) of o-Ps are fixed to the lifetime and relative intensity of the longest-lived component obtained from three component analysis of the positron lifetime spectrum, recorded by PALS, for the same sample.
• (2)  
  The S parameter (${S}_{{\rm{o}}-{\rm{Ps}}}$) of o-Ps is fixed to the asymptotic value of $S\left(t\right)$ at the longest times (positron ages) studied.
• (3)  
  The lifetime (${\tau }_{{{\rm{e}}}^{+}}$) of non-Ps positron is fixed to the lifetime of the intermediate second component of the positron lifetime spectrum for the same sample.
• (4)  
  The lifetime of p-Ps (${\tau }_{{\rm{p}}-{\rm{Ps}}}$) is fixed to 160 ps rather than the intrinsic p-Ps lifetime of 125 ps, taking account of the reduced contact density of Ps in polymers. ³¹⁾

Figure 1 shows the fitting result for $S\left(t\right)$ of polycarbonate (PC), which gives 0.885 and 0.683 for ${S}_{{\rm{p}}-{\rm{Ps}}}$ and ${S}_{{{\rm{e}}}^{+}},$ respectively. Table I lists ${S}_{{\rm{p}}-{\rm{Ps}}}$ for C-, O-, and F-group polymers and silica glass, determined from the $S\left(t\right)$ data. The table also presents the free volume radii $R$ estimated from the o-Ps pick-off annihilation lifetimes based on the Tao-Eldrup model, Eq. (1).

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Table I. Doppler broadening S parameter for p-Ps (${S}_{{\rm{p}}-{\rm{Ps}}}$) and free volume radius ($R$) for C-, O-, and F-group polymers, and silica glass. For the determination of ${S}_{{\rm{p}}-{\rm{Ps}}}$ and $R,$ see text.

Group	Substance	${S}_{{\rm{p}}-{\rm{Ps}}}$	$R$/nm
C-group polymer	PS	0.841	0.29
	HDPE	0.881	0.32
	PP	0.862	0.31
O-group polymer	PC	0.885	0.29
	PEC	0.865	0.30
	PES	0.871	0.28
F-group polymer	PTFE	0.735	0.42
	ETFE	0.746	0.36
	PVF	0.776	0.30
Silica glass	SiO2	0.765	0.25

The zero-point energy ${E}_{0}$ of Ps trapped in a free volume is given by

$$\begin{eqnarray}&&{E}_{0}=\displaystyle \frac{{\pi }^{2}{\hslash }^{2}}{2{m}_{{\rm{Ps}}}{{R}_{0}}^{2}},\end{eqnarray} \tag{ 3 }$$

where ${m}_{{\rm{Ps}}}$ denotes the mass of Ps (twice the electron mass) and $\hslash$ denotes the reduced Planck constant. If p-Ps were thermalized before annihilation, Doppler broadening and hence the ${S}_{{\rm{p}}-{\rm{Ps}}}$ parameter would depend solely on the free volume; in a smaller hole, p-Ps is more strongly confined with a larger zero-point energy, yielding a smaller S parameter, and vice versa. Therefore, it is expected for thermalized p-Ps that ${S}_{{\rm{p}}-{\rm{Ps}}}$ monotonically increases with increasing free volume hole size. Contrary to this expectation, a comparison of ${S}_{{\rm{p}}-{\rm{Ps}}}$ with the free volume radius ($R$) for the polymers and silica glass in Table I reveals that the two parameters are not simply correlated with each other. For example, the ${S}_{{\rm{p}}-{\rm{Ps}}}$ of PTFE with $R$ of 0.42 nm is 0.735, which is much smaller than the ${S}_{{\rm{p}}-{\rm{Ps}}}$ of HDPE with $R$ of 0.32 nm (0.881).

The Doppler broadening spectrum of thermalized p-Ps trapped in a spherical potential well, $DBR\left({\rm{\Delta }}{E}_{\gamma }\right),$ can be calculated as follows: ³²⁾

$$\begin{eqnarray}&&{DBR}\left({\rm{\Delta }}{E}_{\gamma }\right)\propto {\int }_{\frac{2{R}_{0}{\rm{\Delta }}{E}_{\gamma }}{c\hslash }}^{\infty }\frac{1}{x}{\left(\frac{\sin (x)}{{\pi }^{2}-{x}^{2}}\right)}^{2}dx\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&x=\displaystyle \frac{p{R}_{0}}{\hslash }\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{\gamma }=\displaystyle \frac{cp}{2},\end{eqnarray} \tag{ 6 }$$

where ${\rm{\Delta }}{E}_{\gamma }$ denotes the shift of the photon energy from 511 keV, $c$ is the speed of light and $p$ denotes the momentum of p-Ps. For the comparison with the experimental results, the Doppler profile needs to be convoluted with our experimental resolution function with a full width at half maximum of 1.35 keV. Then, the S parameter can be calculated as the ratio of the area in the central window to the total area of the convoluted profile. The width of the central window is set to 2.0 keV, as in Sato et al. ³⁰⁾

Figure 2 displays the plot of ${S}_{{\rm{p}}-{\rm{Ps}}}$ versus $R$ for polymers and silica glass. The line shows the tendency from the convoluted Doppler profile based on Eqs. (4)–(6) for the thermalized p-Ps trapped in a free volume. The experimental ${S}_{{\rm{p}}-{\rm{Ps}}}$ for C-and O-group polymers are situated not far from the tendency expected for thermalized p-Ps. Meanwhile, ${S}_{{\rm{p}}-{\rm{Ps}}}$ for F-group polymers and silica glass considerably deviate downward. This implies that Ps possesses appreciable excess energy in substances that contain fluorine or silicon, heavier elements than carbon, oxygen or hydrogen. We roughly estimate the total energy of p-Ps in such substances as the zero-point energy of Ps trapped in a smaller hole. Using the convoluted Doppler profile based on Eqs. (4)–(6), the hole radius in accordance with the S parameters for F-group polymers and silica glass in Table I is deduced. This hole radius is smaller than that evaluated from the o-Ps lifetime based on the Tao-Eldrup model, providing a larger zero-point energy, Eq. (3). We regard this larger zero-point energy as the total energy of p-Ps to evaluate the excess energy of p-Ps as the difference between this zero-point energy and that corresponds to the hole radius $R$ from the o-Ps lifetime (Table II).

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Table II. Excess energy of p-Ps, ${E}_{{\rm{ex}}},$ for fluorinated F-group polymers and silica glass, roughly estimated from the deviation of ${S}_{{\rm{p}}-{\rm{Ps}}}$ from the tendency for thermalized p-Ps trapped in a free volume in Fig. 2. For more details, see text.

Substance	Structure	${E}_{{\rm{ex}}}$/eV	$R$/nm
PTFE	${\left(\mbox{--}{{\rm{CF}}}_{2} \mbox{-} {{\rm{CF}}}_{2}\mbox{--}\right)}_{{\rm{n}}}$	∼2.4	0.42
ETFE	${\left(\mbox{--}{{\rm{CF}}}_{2} \mbox{-} {{\rm{CF}}}_{2}\mbox{--}\right)}_{{\rm{m}}}{\left(\mbox{--}{{\rm{CH}}}_{2} \mbox{-} {{\rm{CH}}}_{2}\mbox{--}\right)}_{{\rm{n}}}$	∼2.0	0.36
PVF	${\left(\mbox{--}{{\rm{CH}}}_{2} \mbox{-} {\rm{CHF}}\mbox{--}\right)}_{{\rm{n}}}$	∼1.6	0.30
Silica glass	SiO2 network	∼1.4	0.25

From Table II, we see that the estimated p-Ps excess energy depends on the composition of the substance. For example, the excess energy of p-Ps in PTFE is about 2.4 eV and much higher than that in PVF, about 1.6 eV. In fully fluorinated PTFE molecules all the hydrogen atoms in polyethylene are substituted by fluorine atoms, whereas only one fourth of hydrogen is substituted by fluorine in PVF.

According to Nagashima et al., ²⁵⁾ the average energy $\overline{{\rm{\Delta }}E}$ that Ps emitted from the solid surface loses per collision with the solid surface is expressed as follows:

$$\begin{eqnarray}&&-\left(\overline{{\rm{\Delta }}E}\right)=\displaystyle \frac{2{m}_{{\rm{Ps}}}}{\overline{{M}_{{\rm{S}}}}}\left(\displaystyle \frac{{m}_{{\rm{Ps}}}{v}^{2}}{2}-\displaystyle \frac{3}{2}{k}_{{\rm{B}}}T\right),\end{eqnarray} \tag{ 7 }$$

where $v$ denotes the Ps velocity, ${k}_{{\rm{B}}}$ denotes the Boltzmann constant, $T$ denotes the temperature, and $\overline{{M}_{{\rm{S}}}}$ denotes the effective mass of the surface atoms, with which Ps collides. Strictly speaking, p-Ps in a free volume should be quantum mechanically treated. However, as no simple quantum mechanical relationship has been reported, we rely on Eq. (7) based on the classical picture of Ps interaction in the following discussion. Equation. (7) predicts that it is difficult for Ps to lose energy in a substance comprising heavy elements, thereby making $\overline{{M}_{{\rm{S}}}}$ larger. This qualitatively agrees with the result in Table II for fluorinated polymers in the sense that the more the polymer is fluorinated the larger the excess energy.

Notably, the p-Ps excess energy of silica glass containing enormous silicon atoms heavier than fluorine is about 1.4 eV, which is smaller than that in the fluorinated polymers. In addition, the excess energy for F-group polymers and silica glass increases in the order of increasing the free volume radius (Table II). The total amount of Ps energy loss depends not only on the average energy loss per collision but also on the number of collisions Ps undergoes inside the free volume during its lifetime. ²⁵⁾ As the mean free path of Ps trapped inside a spherical hole is proportional to its radius, it is expected that Ps undergoes a larger number of collisions to lose more energy in a smaller free volume. This "mean free path effect" might be somewhat responsible for the lower p-Ps excess energy in silica glass, whose free volume radius (0.25 nm) is the lowest among all the substances studied. However, compared with F-group polymers, appreciably smaller p-Ps excess energies in C- and O-group polymers with relatively small free volume radii shown in Fig. 2 cannot be explained in terms of the "mean free path effect," indicating that elemental composition is of primary significance in the energy dissipation of Ps.

So far, the energy dissipation of Ps has been exclusively discussed for long-lived o-Ps in silica aerogel ^25,28) and porous silica. ^26,33) He et al. ²⁶⁾ reported that Ps slowing down is more feasible in the pores, whose surfaces are covered with –H, than the pores covered with heavier –CH₃ groups. Cassidy et al. ³³⁾ highlighted that a lower limit to the mean kinetic energy of o-Ps emitted from porous silica into a vacuum is equal to the confinement energy of Ps in the nanometer scale pores and loose confinement of Ps is advantageous for the further cooling of Ps. In this study, we have discussed the energy dissipation of short-lived p-Ps in polymers of various compositions and silica glass based on the previously reported AMOC data. ³⁰⁾ A comparison of the ${S}_{{\rm{p}}-{\rm{Ps}}}$ parameter with that expected for thermalized p-Ps trapped in a free volume reveals that p-Ps loses excess energy less effectively in substances containing heavier elements. The somewhat lower excess energy of p-Ps in silica glass than in fluorinated polymers suggests a certain role played by the "mean free path effect" due to its smaller free volume. However, the observed appreciably smaller p-Ps excess energies in C- and O-group polymers with relatively small free volumes indicate that elemental composition is of primary significance. It is concluded that efficient cooling of Ps requires a substance comprising light elements kept at low temperatures that can loosely confine Ps.

Acknowledgments