Positron-molecule interactions: Resonant attachment, annihilation, and bound states

This article presents an overview of current understanding of the interaction of low-energy positrons with molecules with emphasis on resonances, positron attachment, and annihilation. Measurements of annihilation rates resolved as a function of positron energy reveal the presence of vibrational Feshbach resonances (VFRs) for many polyatomic molecules. These resonances lead to strong enhancement of the annihilation rates. They also provide evidence that positrons bind to many molecular species. A quantitative theory of VFR-mediated attachment to small molecules is presented. It is tested successfully for selected molecules (e.g., methyl halides and methanol) where all modes couple to the positron continuum. Combination and overtone resonances are observed and their role is elucidated. Molecules that do not bind positrons and hence do not exhibit such resonances are discussed. In larger molecules, annihilation rates from VFR far exceed those explicable on the basis of single-mode resonances. These enhancements increase rapidly with the number of vibrational degrees of freedom, approximately as the fourth power of the number of atoms in the molecule. While the details are as yet unclear, intramolecular vibrational energy redistribution (IVR) to states that do not couple directly to the positron continuum appears to be responsible for these enhanced annihilation rates. In connection with IVR, experimental evidence indicates that inelastic positron escape channels are relatively rare. Downshifts of the VFR from the vibrational mode energies, obtained by measuring annihilate rates as a function of incident positron energy, have provided binding energies for 30 species. Their dependence upon molecular parameters and their relationship to positron-atom and positron-molecule binding-energy calculations are discussed. Feshbach resonances and positron binding to molecules are compared with the analogous electron-molecule (negative-ion) cases. The relationship of VFR-mediated annihilation to other phenomena such as Doppler broadening of the gamma-ray annihilation spectra, annihilation of thermalized positrons in gases, and annihilation-induced fragmentation of molecules is discussed. Possible areas for future theoretical and experimental investigation are also discussed.

Figure 1

The normalized annihilation rate $Z_{\mathrm{eff}}\left(\epsilon \right)$ for butane ${\mathrm{C}}_4{\mathrm{H}}_{10}$ (●) as a function of the total incident positron energy $\epsilon$: (a) up to the Ps formation threshold, $E_{\mathrm{th}}=3.8\mathrm{eV}$, and (b) in the region of the molecular vibrations; dotted curve, the infrared absorption spectrum (148) (logarithmic vertical scale, arbitrary units); solid curve, the vibrational mode density (in arbitrary units), with the modes represented by Lorentzians with an arbitrary FWHM of $10\mathrm{meV}$; dashed line, mean energy of the C-H stretch fundamentals.

Figure 2

(Color online) Comparison of the $Z_{\mathrm{eff}}$ values calculated using the SMC method for ${\mathrm{C}}_2{\mathrm{H}}_2$ (squares) and ${\mathrm{C}}_2{\mathrm{H}}_4$ (circles) by 225 (see Sec. 2H1) with the fit using Eq. (29) with a constant vertical offset. Parameters of the fit: ${\mathrm{C}}_2{\mathrm{H}}_2$ (solid curve), $F=0.261$, and $\kappa =0.0041$; ${\mathrm{C}}_2{\mathrm{H}}_4$ (dashed curve), $F=0.230$, and $\kappa =0.0372$.

Figure 3

(Color online) Dependence of the annihilation rate ${\Gamma }^a$ for positron bound states on the parameter $\kappa =\sqrt{2{\epsilon }_b}$: solid circles, recent results for six atoms (23, 24, 26; 167, 171); open circles, earlier results for these atoms (167) and LiH molecule (170); dashed line is the fit ${\Gamma }^a=5.3\kappa$ (in ${10}^9{\mathrm{s}}^{-1}$) which corresponds to $F=0.66\mathrm{a.u.}$

Figure 4

(Color online) Vibrational level densities in alkanes as functions of the positron binding energy: solid curves, calculated from Eq. (44) for room-temperature incident positrons; circles, estimated from experimental $Z_{\mathrm{eff}}$ values at $300\mathrm{K}$ for ${\mathrm{C}}_3{\mathrm{H}}_8$ to ${\mathrm{C}}_9{\mathrm{H}}_{20}$, using Eqs. (36, 43) and assuming ${\Gamma }^e∕\Gamma =1$, and plotted against the experimental binding energies (Table ). From 103.

Figure 5

(Color online) Comparison of experimental $Z_{\mathrm{eff}}$ for butane (solid symbols) and octane (open symbols) with the predictions of the statistical model: diamonds, experimental $Z_{\mathrm{eff}}^{\mathrm{th}}$ values for thermal positrons at $300\mathrm{K}$ (cf. Table ); circles, $Z_{\mathrm{eff}}$ measured as a function of positron energy (14); curves, $Z_{\mathrm{eff}}$ calculated from Eq. (45), using ${\epsilon }_b$ fitted to reproduce thermal $Z_{\mathrm{eff}}$.

Figure 6

(Color online) Comparison of the experimental $Z_{\mathrm{eff}}$ for (●) butane, ${\mathrm{C}}_4{\mathrm{H}}_{10}$ and (○) octane, ${\mathrm{C}}_8{\mathrm{H}}_{18}$ (scaled by a factor $1∕50$) (14), with $Z_{\mathrm{eff}}$ from Eq. (49). The theoretical $Z_{\mathrm{eff}}$ is averaged over the positron beam energy distribution. The fit for butane uses ${\epsilon }_b=35\mathrm{meV}$, ${\Gamma }_n^e∕\Gamma =7.2$ for C-H stretch modes and ${\Gamma }_n^e∕\Gamma =1.2$ for the rest; for octane, ${\epsilon }_b=122\mathrm{meV}$, ${\Gamma }_n^e∕\Gamma =84$ for C-H stretch modes and ${\Gamma }_n^e∕\Gamma =14$ for the rest. From 106.

Figure 7

(Color online) Binding energies of positron-atom complexes as a function of their ionization potential: squares, SVM and CI calculations (167; 25, 26); dashed curve, model “alkali atom” (166). For positron binding to metastable states the latter are indicated in round brackets. The two bound states for Ca and Sr are shown in square brackets.

Figure 8

Positron density in $e^{+}\mathrm{Li}\mathrm{H}$ along the molecular axis: dashed curve, HF $({\epsilon }_b=0.16\mathrm{eV})$; solid curve, ECG $({\epsilon }_b=0.94\mathrm{eV})$; Li atom is at $X=-1.51\mathrm{a.u.}$, and H at $X=1.51\mathrm{a.u.}$ From 206.

Figure 9

(Color online) Positron binding energies for alkanes from experiment (crosses) (Table ) and ZRP model calculation (circles). The ZRP model is fit to ${\epsilon }_b=220\mathrm{meV}$ for dodecane. From 106.

Figure 10

Two-dimensional density of the positron wave functions in the ZRP model for the first (top) and second (bottom) bound states in tetradecane $(n=14)$. In these plots, the first carbon atom is at the origin, and the C-C bonds are alternately parallel and at 67° to the $x$ axis. From 106.

Figure 11

Schematic diagram of the apparatus used to study the interaction of trapped, thermal positrons with low-pressure gases. The positrons are contained in a fourth trapping stage surrounded by a cold surface to minimize the effects of impurities present in the vacuum system. From 123. From 106.

Figure 12

(Color online) Measurement of the distribution of total positron energies in the trap-based beam using a vibrational Feshbach annihilation resonance: solid curve, prediction of Eq. (56), normalized arbitrarily, shifted in energy, and fitted to the energy-reversed, normalized, C-H stretch peak in $Z_{\mathrm{eff}}$ for propane (●), with $T_{\perp }=26\mathrm{meV}$ and a parallel energy spread (FWHM) of $27\mathrm{meV}$. From 237.

Figure 13

(Color online) Schematic diagram of the gas cell, shielding, and detection apparatus used for energy-resolved positron annihilation measurements (not to scale). From 237.

Figure 14

(Color online) Annihilation rates $Z_{\mathrm{eff}}$ (●) and IR absorption (solid curves) for methyl halides: (a) $\mathrm{C}{\mathrm{H}}_3\mathrm{F}$, (b) $\mathrm{C}{\mathrm{H}}_3\mathrm{Cl}$, and (c) $\mathrm{C}{\mathrm{H}}_3\mathrm{Br}$ (15). Vertical bars below each plot indicate the vibrational energies from IR and Raman measurements, and selected mode frequencies (“all modes”). From 148.

Figure 15

(Color online) Comparison of experimental and theoretical $Z_{\mathrm{eff}}$ for methyl halides $\mathrm{C}{\mathrm{H}}_{3}X$ (● and solid curves) and $\mathrm{C}{\mathrm{D}}_{3}X$ (○ and dashed curves) for (a) $X=\mathrm{F}$, ${\epsilon }_b=0.3\mathrm{meV}$; (b) $X=\mathrm{Cl}$, ${\epsilon }_b=25\mathrm{meV}$; and (c) $X=\mathrm{Br}$, ${\epsilon }_b=40\mathrm{meV}$ (15; 104; 238). Dotted curves show the contributions of direct annihilation.

Figure 16

(Color online) Comparison of experimental $Z_{\mathrm{eff}}$ (●) for methanol $\left(\mathrm{C}{\mathrm{H}}_3\mathrm{O}\mathrm{H}\right)$ with theory: dotted curve, contribution of direct annihilation $Z_{\mathrm{eff}}^{\left(\mathrm{dir}\right)}$; dashed curve, total $Z_{\mathrm{eff}}$ due to VFR of vibrational fundamentals for ${\epsilon }_b=2\mathrm{meV}$; solid curve, total $Z_{\mathrm{eff}}$ due to resonant annihilation involving the 12 modes and 9 overtones and combinations. See 238 for details.

Figure 17

(Color online) Comparison of the experimental $Z_{\mathrm{eff}}$ (●) with theory for (a) ethylene (${\mathrm{C}}_2{\mathrm{H}}_4$, ${\epsilon }_b=10\mathrm{meV}$) and (b) acetylene (${\mathrm{C}}_2{\mathrm{H}}_2$, ${\epsilon }_b=5\mathrm{meV}$). Dotted curves, direct annihilation $Z_{\mathrm{eff}}^{\left(\mathrm{dir}\right)}$; solid curves, total $Z_{\mathrm{eff}}$ due to all IR active modes. In (a) dashed curve, total $Z_{\mathrm{eff}}$ due to $A_g$, $B_g$, and $B_u$ modes that capture $s$-, $p$-, and $d$-wave positrons; chain curve, same with the addition of 14 IR-active overtones and combinations listed in 81. In (b) long-dashed curve, all modes; chain curve, all modes with ${\Sigma }_{g,u}$, ${\Pi }_{g,u}$, and ${\Delta }_g$ symmetries. The contributions of all overtones and combinations are ad hoc weighted by a factor $1∕n$, where $n$ is the number of vibrational quanta.

Figure 18

(Color online) Experimental $Z_{\mathrm{eff}}$ spectrum (●) and theory for ethane $\left({\mathrm{C}}_2{\mathrm{H}}_6\right)$. The calculations use ${\epsilon }_b=1\mathrm{meV}$ and include: dotted curve, direct $Z_{\mathrm{eff}}$; solid curve, same with VFR of IR-active modes; dashed curve, same with the addition of $A_{1g}$ modes; chain curve, same with the addition of $E_g$ modes.

Figure 19

(Color online) Comparison of experimental $Z_{\mathrm{eff}}$ (●), IR absorption spectrum (solid curve), and theoretical $Z_{\mathrm{eff}}$ (dashed curve) for ethanol $\left({\mathrm{C}}_2{\mathrm{H}}_5\mathrm{O}\mathrm{H}\right)$. The calculation [cf. Eq. (42)] uses ${\epsilon }_b=45\mathrm{meV}$ and the mode frequencies and transition dipole amplitudes from 201.

Figure 20

(Color online) Measured $Z_{\mathrm{eff}}$ spectra for $\mathrm{C}{\mathrm{H}}_4$ (●) and $\mathrm{C}{\mathrm{H}}_3\mathrm{F}$ (○) and IR absorption spectrum of methane (solid curve).

Figure 21

(Color online) Experimental $Z_{\mathrm{eff}}$ spectrum (○) for water, and a fit based on the direct annihilation model: solid curve, $Z_{\mathrm{eff}}$ from Eq. (29) with ${\kappa }^2∕2=0.3\mathrm{meV}$ plus a constant offset of 20.

Figure 22

$Z_{\mathrm{eff}}$ spectra of hydrogenated butane (solid circles) (14) and nonane (open circles) (241) compared with (the appropriately scaled) fully deuterated analogs (dashed and chain curves) (14). Mode energies for the deuterated species have been scaled by the appropriate reduced mass factor to match those in the hydrogenated compounds, while the positron binding energy was assumed to be independent of deuteration. The magnitudes of $Z_{\mathrm{eff}}$ have been scaled by the appropriate $g$ factor at each incident positron energy.

Figure 23

(Color online) $Z_{\mathrm{eff}}$ spectra as a function of incident positron energy for a variety of alkane molecules, showing the systematic shift of the spectra to lower positron energy as molecular size is increased. Values of $Z_{\mathrm{eff}}$ are absolute except for ${\mathrm{C}}_{14}{\mathrm{H}}_{30}$, which is in arbitrary units due to difficulties measuring its vapor pressure.

Figure 24

(Color online) Normalized and energy-shifted $Z_{\mathrm{eff}}∕g$ spectra for alkanes with $n=3–8$ carbons. For comparison, room-temperature $Z_{\mathrm{eff}}∕g$ data for alkanes are also shown (circles with plus sign) at energies ${\epsilon }_b+{\epsilon }_T$, where ${\epsilon }_T=\frac{3}{2}{k}_{B}T=37.5\mathrm{meV}$. Each room-temperature datum is labeled by the number of carbons $n$ in the molecule. See text for details.

Figure 25

(Color online) $Z_{\mathrm{eff}}$ spectra for (○) dodecane, (diamonds) tetradecane, and (squares) hexadecane. The vertical arrows indicate the positions of the C-H stretch mode VFR peaks for the second bound state (i.e., the first positronically excited state) in each molecule. The spectra for tetradecane and hexadecane have been normalized arbitrarily since their vapor pressures were too low to measure reliably. The large peaks at lower energy are the C-H stretch mode VFR for the first bound states (the positronic ground states).

Figure 26

(Color online) $Z_{\mathrm{eff}}$ at the C-H stretch peak vs binding energy ${\epsilon }_b$ for alkanes, ${\mathrm{C}}_n{\mathrm{H}}_{2n+2}$ (◻), with the number of carbons $n$ indicated; rings (hexagons); halomethanes (◇); ethylene (●); methanol (+); 1-chlorohexane (△); fluoroalkanes (△); and deuterated species (▽).

Figure 27

(Color online) $Z_{\mathrm{eff}}$ at the C-H stretch peak, normalized by the factor $g=\sqrt{{\epsilon }_b∕\epsilon }$, vs the number of atoms in the molecule. The symbols are as in Fig. 26. The solid line is the fit given by Eq. (59). The error bars on 1-fluorohexane indicate the unusually large uncertainty for this datum.

Figure 28

(Color online) The $Z_{\mathrm{eff}}$ spectra for (○) propane and (●) 1-fluoropropane (241) and (△) 2,2-difluoropropane (92). The vertical dot-dashed lines indicate the expected energies of C-F stretch resonances and escape thresholds in each molecule, based on C-F stretch mode energies from 67 and 110. See text for details.

Figure 29

Energy-resolved measurements of $Z_{\mathrm{eff}}$ for pentane $\left({\mathrm{C}}_5{\mathrm{H}}_{12}\right)$ at $300\mathrm{K}$ (○) and $153\mathrm{K}$ (●) using a cold cell (240).

Figure 30

(Color online) Energy-resolved $Z_{\mathrm{eff}}$ (●) and infrared absorption (solid curve) for benzene. The $Z_{\mathrm{eff}}$ spectrum has been shifted upward by the binding energy $({\epsilon }_b=150\mathrm{meV})$ for direct comparison. The normalization of the IR absorption (148) is arbitrary. Vertical lines indicate the positions of the vibrational modes.

Figure 31

(Color online) Energy-resolved $Z_{\mathrm{eff}}$ (●) for (a) propane and (b) cyclopropane (15). The solid curves are the normalized IR absorption spectra, and the vertical bars below each plot show the vibrational modes (148). Shown as insets are the molecular structures.

Figure 32

(Color online) Binding energies of positron-atom and positron-molecule complexes as a function of their ionization potential. Calculations: squares, various atoms (167; 26); dashed curve, model alkali atom (166). Experiment: circles, alkanes with $n$ carbons; diamonds, the aromatic molecules, benzene and naphthalene; triangles, methyl halides (see Secs. 4, 5, 6). Adapted from 49.

Figure 33

(Color online) Measured positron binding energies ${\epsilon }_b$ as a function of the dipole polarizability ${\alpha }_d$: large circles, alkane molecules used in the linear fit shown by the solid line; smaller circles, alkane-related molecules; triangles, molecules with permanent dipole moments; squares, aromatics with $\pi$ bonds. Dotted lines, guides to show the linearity of ${\epsilon }_b$ for the different series; diamonds, calculated ${\epsilon }_b$ for the atoms Be, Zn, and Cd (167), shown for comparison. From 49.

Figure 34

(Color online) Binding energy from the fit Eq. (61) (solid line) using the polarizability ${\alpha }_d$, dipole moment $\mu$, and the number of $\pi$ bonds $N_{\pi }$ for aromatic molecules. Symbols as in Fig. 33. From 49.

Figure 35

(Color online) $\gamma$-ray spectrum of hexane $\left({\mathrm{C}}_6{\mathrm{H}}_{14}\right)$: (○) observed spectrum; solid curve, single Gaussian fit; dotted curve, fit using a hydrogenic functional form; dot-dashed curve, fit using a noninteracting hydrogenic form convolved with a Gaussian; dashed curve (indistinguishable from the data), two Gaussian fit. The statistical error bars are comparable to or smaller than the size of the data points. See 126 for details.

Figure 36

Normalized fraction of positrons annihilating on fluorine atoms in partially and fully fluorinated alkanes, plotted against the fraction of valence electrons in the fluorine atoms. Open symbols are methane (circle), ethane (square), propane (triangle), and hexane-based molecules (inverted triangle); filled symbols are six-carbon benzene-based molecules: 1,2-difluorobenzene (square), 1,3-difluorobenzene (triangle), 1,4-difluorobenzene (inverted triangle), and other six-carbon benzene-based molecules (circle). See text and 126 for details.

Figure 37

Time-of-flight mass spectra of ion fragments from 1-dodecene at $1.0\mathrm{eV}$, which is $2.1\mathrm{eV}$ below the Ps formation threshold. Adapted from 232.

Figure 38

Cross sections for fragmentation of 1-dodecene $\left({\mathrm{C}}_{12}{\mathrm{H}}_{24}\right)$ by positron annihilation near and below the Ps-formation threshold at $3.1\mathrm{eV}$. From 232.

Figure 39

Probability $P\left(E\right)$ that the energy deposited into a molecular ion due to positron annihilation with an electron in a valence orbital below the HOMO is greater than a given excess energy for propane, hexane, and decane. For details of the calculation, see 45.

Figure 40

Density dependence of the room-temperature positron annihilation rates for $\mathrm{C}{\mathrm{Cl}}_2{\mathrm{F}}_2$ (◻), ${\mathrm{C}}_2{\mathrm{H}}_6$ (○), and ${\mathrm{C}}_3{\mathrm{H}}_8$ (△). Solid lines are fits of the form Eq. (62). Adapted from 38.