Annihilation On Atoms And Molecules

Electronics Repair Manuals

Schematic Diagrams and Service Manuals

Get Instant Access

The annihilation rate is a sensitive measure of short-range correlations between the positron and the bound electrons. We follow the convention of expressing the annihilation rate r in terms of the parameter Zeg, which is the annihilation rate relative to that for positrons in a gas of uncorrelated electrons (i.e., the Dirac annihilation rate).

where r0 is the classical electron radius, c is the speed of light, and nm is the number density of atoms or molecules. For large molecules, it is well established that Zeff can gready exceed the total number of electrons Z in the molecule [5, 8, 12-14]. Consequently, Z^should be viewed as a normalized annihilation rate - it bears no relation to the charge on the nucleus or the number of electrons in the molecule. While the physical process responsible for these high annihilation rates is not fully understood, these large rates have been viewed as evidence for the existence of (long-lived) positron-molecule complexes [5, 6, 12, 13, 16].

The experimental arrangement for positron annihilation studies in the positron accumulator is shown in Fig. 1 [12-15]. Annihilation rates are measured after the positrons are trapped and cooled to room temperature by measuring the number of positrons, Np, remaining as a function of time in the presence of a test gas. Typically the number of positrons remaining is measured by dumping the positron plasma on a metal plate and measuring the annihilation gamma rays. The rate is then given by r= d[ln(Np)]/dt, where t is time. These experiments can be done either in the presence or absence of the buffer gas used for positron trapping. In order to reduce the density of impurity molecules in the system (base pressure < 1 x 10 torr), a cryosurface was placed in situ in the vacuum chamber as necessary. It was cooled with either liquid nitrogen (to 77K) or with an ethanol-water mixture (to ~ 266 K), depending on the atomic or molecular species studied. The measured annihilation rates are found to be a linear function of the test gas pressure (i.e., proportional to nm), and the slope yields Zeff. The linearity of the slope provides evidence that annihilation is due to isolated two-body interactions between the positrons and the test molecule.

Shown in Fig. 2 are data for a wide range of chemical species [12-15]. Note the very large differences in rates observed for only modest changes in chemical structure. Values of Zeff ~ 104 had been measured previously in the high-pressure experiments [5, 6, 8]. The development of the positron trap enabled the extension of these studies to even larger molecular species including those that are liquids and solids at room temperature. The extremely broad range of

Figure 1. Schematic diagram of the apparatus used to study positron annihilation. The positrons are confined in a Penning trap by potentials applied to the electrodes and a magnetic field, B. Annihilation rates are measured by storing the positrons for various times in the presence of a test gas, then measuring the number of positrons remaining by dumping them on a plate and measuring the gamma ray signal. The Doppler linewidth of the gamma rays is measured using a Ge detector placed in close proximity to the trapped positrons.

observed values of provides evidence of qualitative changes in the nature of the positron molecule interaction for relatively modest changes in chemical species. While the smaller values (e.g., ZeS ~ Z) can be explained in terms of a simple collision model, larger values appear to require a different physical picture, such as the formation of positron-molecule resonances. Murphy et al., pointed out that Ztff for atoms and single-bonded molecules obeys a universal scaling as a function of (£,■ - EPs)'\ which is shown in Fig. 3 [13], To date, there has been no satisfactory explanation of this empirical relationship, beyond the speculation that large annihilation rates might be thought of in terms of positron-molecule complexes in which a positronium atom is attached to the corresponding positive ion [13].

The microscopic nature of positron interactions with atoms and molecules can be studied by measuring the Doppler-broadening of the 511-keV annihilation gamma-ray line [30, 31]. The Doppler linewidth is determined by the momentum test gas in

positrons

Figure 1. Schematic diagram of the apparatus used to study positron annihilation. The positrons are confined in a Penning trap by potentials applied to the electrodes and a magnetic field, B. Annihilation rates are measured by storing the positrons for various times in the presence of a test gas, then measuring the number of positrons remaining by dumping them on a plate and measuring the gamma ray signal. The Doppler linewidth of the gamma rays is measured using a Ge detector placed in close proximity to the trapped positrons.

pump test gas in clectrodcs gamma ray detector positrons pump

Figure 2. (a) Experimental values of Zen/Z plotted against Z, illustrating the fact that this quantity varies by orders of magnitude for modest changes in chemical species: (•) noble gases, (V) simple molecules, (O) alkanes, (A) perfluorinated alkanes, (□) perchlorinated alkanes, (0) perbrominated and periodated alkanes, (■) alkenes, (▲) oxygen-containing hydrocarbons, (O) ring hydrocarbons, (▼) substituted benzenes, and (♦) large organic molecules.

Figure 2. (a) Experimental values of Zen/Z plotted against Z, illustrating the fact that this quantity varies by orders of magnitude for modest changes in chemical species: (•) noble gases, (V) simple molecules, (O) alkanes, (A) perfluorinated alkanes, (□) perchlorinated alkanes, (0) perbrominated and periodated alkanes, (■) alkenes, (▲) oxygen-containing hydrocarbons, (O) ring hydrocarbons, (▼) substituted benzenes, and (♦) large organic molecules.

distribution of the electrons (i.e., determined by the electron quantum states) participating in the annihilation process. Shown in Fig. 4 is the gamma ray spectrum for positron annihilation on helium atoms. Also shown is a theoretical calculation by Van Reeth and Humberston [37]. Theory and experiment are in excellent agreement for the shape of the linewidth. There is also good agreement between theory and experiment for the annihilation linewidths of other noble gases [31]. These comparisons were made for annihilation on valence electrons. A careful search was also made in noble gas atoms for evidence of inner-shell annihilation, which would produce a broad, low-amplitude wing on the annihilation line. Annihilation was observed on the next inner shell, but only at the few percent level, and then only in larger atoms, Kr (1.3 %) and Xe (2.4 %) [31]. The fact that these percentages are low is consistent with the highly repulsive (core) potential that the positron experiences once it begins to penetrate the valence electrons.

The gamma-ray linewidth provides relatively direct information about the specific electronic states participating in the annihilation process. A systematic study was done in alkane molecules in which the linewidth was measured as a function of the fraction of C - C and C - H bonds in the molecule, and the results

Atoms And Molecules Poster Values
Figure 3. Values of Z^g for noble gas atoms and single-bonded molecules as a function of (Ej - Ep,)1, where Ej is the ionization energy and Ep, is the positronium formation energy. (See Ref. [14] for details.)

506 508 510 512 514 516 y-ray energy (keV)

Figure 4. Annihilation gamma-ray spectrum from He: (O) experiment, (-) theoretical calculation including detector response, and (■■■) a Gaussian fit. (From Ref. [37])

506 508 510 512 514 516 y-ray energy (keV)

Figure 4. Annihilation gamma-ray spectrum from He: (O) experiment, (-) theoretical calculation including detector response, and (■■■) a Gaussian fit. (From Ref. [37])

are shown in Fig. 5 (a) [15]. When compared with calculations for the linewidths of the C-C and C-H bonds [38], these data are consistent with annihilation occurring with roughly equal probability on any of the valence electrons. This is only an approximate statement; and since the calculations of Ref. [38] are now more than three decades old, further theoretical study of the momentum distribution expected for electrons in valence orbitals in hydrocarbons would be helpful. A similar study of Doppler linewidths was done in hydrocarbons in which the H atoms were systematically substituted with fluorines [15]. In this case, the measured linewidths can be accurately fit by a linear combination of the linewidths measured for pure fluorocarbons and pure hydrocarbons. The results of this analysis are shown in Fig. 5 (b). This analysis also implies that annihilation occurs with approximately equal probability on any valence electron, which in this case includes the valence electrons in the fluorine atoms in addition to those in the C - H and C - C bonds.

In summary, all results to date are consistent with most of the annihilation occurring on any of the valence electrons (as opposed to favoring specific sites in the molecule) with a small fraction of the annihilation occurring on the next inner shell when heavier atoms are present. This can be interpreted to mean that the positron has a relatively long de Broglie wavelength in the vicinity of the molecule. Consequently, the positron interacts with roughly equal probability with any of the valence electrons. This picture is in contrast to the case where the positron is localized at a specific molecular site, as would be expected in a tight-binding model. The lack of preference for annihilation on specific valence electrons is consistent with the model developed by Crawford [39] to explain the

fraction of valence e" fraction of valence e" on F

in C-C bonds

Figure 5. (a) Gamma-ray line width for alkanes (•), plotted against the fraction of valence electrons in C-C bonds, and (-) a linear fit to the data, (b) Fraction of annihilation on fluorine atoms for partially fluorinated hydrocarbons (•), plotted against the fraction of valence electrons on these atoms, and (-) the line y = x. These linear relationships provides evidence that positrons annihilate with approximately equal probability on any valence electron. (See Ref. [ 15] for details.)

fraction of valence e" fraction of valence e" on F

in C-C bonds

Figure 5. (a) Gamma-ray line width for alkanes (•), plotted against the fraction of valence electrons in C-C bonds, and (-) a linear fit to the data, (b) Fraction of annihilation on fluorine atoms for partially fluorinated hydrocarbons (•), plotted against the fraction of valence electrons on these atoms, and (-) the line y = x. These linear relationships provides evidence that positrons annihilate with approximately equal probability on any valence electron. (See Ref. [ 15] for details.)

observation of significant molecular fragmentation observed following positron annihilation at energies below the threshold for positronium formation [20, 40]. Crawford predicted that annihilation on any of the valence molecular orbitals occurs with roughly equal probability [39]. Thus if the highest lying molecular orbitals do not dominate the annihilation process, the molecular ion that is produced will frequently be left in an excited electronic state. Then the excess energy in these excited states produces the fragmentation that is observed.

The measurements of annihilation rates shown in Fig. 2 were done with a Maxwellian distribution of positrons having a positron temperature Tp = 300 K (i.e., 0.025 eV). To investigate ZejS(Tp), measurements have also been done with the positrons heated above 300 K by applying radio frequency noise to the confining electrodes [29], The annihilation rate and positron temperature are measured as the positrons cool. The experiments thus far have been limited to Tp < 0.2 eV for hydrocarbon molecules and < 0.8 eV for noble gas atoms; above these temperatures, the heating produces non-Maxwellian positron velocity distributions. In the noble gas studies, the dependence of .Ton Tp is in good agreement with theoretical predictions [29],

Measurements of ZeS(Tp), for CH4 and CH3F are shown in Fig. 6 (a) [15], A study of butane (C4H10) indicates that the temperature dependence of I^n is very similar to that shown for CH4 [15], These data exhibit interesting features, such as an initial slope proportional to Tp'm and a break in slope at higher temperature in CH4 and C4H10, which have recently begun to be considered theoretically [15, 16]. This 'plateau' at higher temperatures may be due to the excitation of molecular vibrations.

Figure 6. Dependence of annihilation rates on positron temperature: (•) methane (CH4), and (O) fluoromethane (CH3F). The annihilation rates are normalized to their room-temperature values. The dotted line (••■) is a power law fit to the lower temperature data with the coefficient of -0.53.

Building upon previous work [6, 12], Gribakin has proposed a comprehensive theoretical model of annihilation in molecules [16]. While many open questions remain, this theory provides a useful framework for considering the annihilation process and its dependence on molecular species. A key assumption is that the positron-hydrocarbon potential is sufficiently attractive to admit bound states. The wide variation in the values of for various species is then explained in terms of positron-molecule resonances. Values of Zeg, much larger than Zbut < 103 are predicted to occur via either low-lying positron-molecule resonances or weakly bound states. However Gribakin concludes that larger values of Ze¡} cannot be explained by such a mechanism. He predicts that values of Ze¡¡ larger than 103 arise from the excitation of vibrationally excited quasi-bound states of the positron-molecule complex, an idea that was proposed previously to explain the values of ZtS observed in large molecules such as alkanes [12]. The vibrational density of states of a molecule increases very rapidly as a function of increasing molecular size. When the positron-molecule potential is attractive, this increased density of states leads to a corresponding increase in Zeff.

Qualitatively, the basis of Gribakin's model is that, when there are shallow bound states or low-lying resonances, the cross section diverges as a2, where a is the scattering length, and this leads to large values of Zeff. However, for room temperature positrons, the cross section a is limited by the finite wave number, k, of the positron to <T< 4nf¡<?, which in turn limits ZeS to ~ 103. In large molecules in which the positron-molecule potential is attractive, the high density of vibrational states increases greatly the probability of resonance formation, and this results in even larger values of Zeff [12, 15, 16]. The limit occurs when the lifetime of the resonances is comparable to the annihilation time of a positron in the presence of molecular-density electrons, which corresponds to values of Zeff ~ 107 - 108.

The theoretical framework proposed by Gribakin makes a number of predictions, several of which are in qualitative agreement with experiment. The model provides a natural explanation for the qualitative differences in observed for fluorocarbons and hydrocarbons (c.f., Fig. 2) [15, 16]. The positron-fluorine potential is likely to be less attractive than that between the positron and C - H bond electrons. As a result, fluorocarbon molecules are not expected to bind positrons, and hence there will be no resonant enhancement in Ze¡¡. This explains the very large differences in ZeS observed for the two chemical species.

Similarly, the model appears to explain in a natural way the peaks in annihilation rate observed in partially fluorinated hydrocarbons when the molecule contains only one or two fluorine atoms. In this case, Gribakin predicts that the peaks are due to the position of the bound/virtual levels moving to zero energy as a result of changes in the degree of fluorination. This produces a divergence in the scattering length and hence a large value of The data are qualitatively in agreement with Gribakin's predictions. The model predicts that is proportional to the elastic scattering cross section which, as discussed below, could possibly be tested by scattering experiments with a very cold positron beam. The model also predicts that ZeS~ Tpm in the regime whereZe¡¡> 103 and at low values of positron temperature [16], This scaling is observed for both methane and butane. The butane result is consistent with the theoretical prediction while, in the framework of Gribakin's model, the methane scaling appears to be due to a combination of effects [15].

There still remain a number of open questions. One is the observation that deuteration of hydrocarbons produces only relatively small changes in ZeS, even though the C-H vibrational frequencies are changed by ~ 2m [Iwata, 2000], If the large values of ZeS are due to vibrationally excited resonances, then change in the vibrational mode frequencies might be expected to produce changes in the vibrational density of states and hence relatively large changes in Zeff. The experimental results may mean that only low-frequency vibrational modes contribute to the formation of the vibrationally excited resonances.

Another puzzling question is the origin of the empirical scaling of ZeS with (Et - Eps)'1 [i.e., shown in Fig. 3] that is observed for all of the atoms and single-bonded molecules studied. While this scaling fits the data over six orders of magnitude in ZeS to within in a factor < 10, it remains to be seen whether it has any theoretical significance. If there were low-lying electronic excitations of a positron-atom or positron-molecule complex, then a resonance model might be possible without involving molecular vibrations. However, there appears to be no analogous phenomenon involving low-lying electronic excitations in electron-molecule interactions, and so the positron would have to play a fundamental role in such modes. This appears to be unlikely. A more plausible explanation is that the quantity (Et - £/>,)"' is a measure of the attraction of the positron to the atom or molecule, and so increases in this parameter increase Z^f in accord with both the Murphy et al. scaling and Gribakin's model.

The first challenge will be to test the general validity of models for the large values of Beyond this, there also remain a number of trends in with specific chemical species. For example, modest changes in chemical structure can change Z^ by factors of 3 to 10 or more (e.g., differences in ring and chain molecules, for example). Experimental tools such as those discussed in this chapter and the considerable theoretical activity evidenced elsewhere in the volume, may well provide new insights in the not too distant future into the many remaining questions concerning large annihilation rates observed in molecules.

Was this article helpful?

0 0

Post a comment