An alternative indicator of annihilated electrons in atoms: Rahm's electronegativity scale
Yu Wu, Xiaoguang Ma, Fang Yuan, Jipeng Sui, Meishan Wang, Chuanlu Yang
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a r An alternative indicator of annihilated electrons in atoms:Rahm’s electronegativity scale
Yu Wu, Xiaoguang Ma, ∗ Fang Yuan, Jipeng Sui, Meishan Wang, and Chuanlu Yang
School of Physics and Optoelectronic Engineering, Ludong University,Yantai, Shandong 264025, People’s Republic of China (Dated: March 30, 2020)
Abstract
This paper presents a new explanation of the width of gamma-ray spectra based on Rahm’selectronegativity scale. This quantitatively proves, for the first time, that positrophilic electronsin the positron-electron annihilation process are exactly the valence electrons. This suggests thereplacement of Full Width at Half Maximum (FWHM) of the gamma-ray spectra with the newlydefined physical quantity Average Doppler Shift (ADS). Both FWHM and ADS of the gamma-ray spectra in light elements agree well with the corresponding Rahm’s electronegativity values,respectively. Further, ADS provides strong evidence in favor of Rahm’s electronegativity scale.It is expected that this will be useful in understanding the mechanism of the positron-electronannihilation process.
PACS numbers: 78.70.Bj, 82.30.Gg, 36.10.DrKeywords: positron-electron annihilation; positrophilic electrons; electronegativity; average Doppler shift NTRODUCTION
The gamma-ray spectra for low-energy positron annihilation in molecules have been stud-ied extensively over the several past decades[1–5]. However, understanding of low-energypositrons and their behavior in molecules is still incomplete compared to more familiar elec-tron problems[6–8]. Most of the experimental measurements exhibit good agreement withthe theoretical gamma-ray spectra of the valence electrons in molecules[7–10]. However,next to no research has been conducted to quantitatively establish the reason why valenceelectrons dominate the positron-electron annihilation process.Moreover, the width FWHM of the gamma-ray spectra is only an experimental spectrumanalysis parameter, which has almost no physical interpretation. In other words, it is justa technical term, and does not correspond to a physical quantity. This makes it difficult tounderstand the mechanism of the positron-electron annihilation process. For instance, thereason behind the empirically observed higher annihilation rate compared to the theoreticalprediction is one of many issues that eludes explanation. Some explanations and suggestionsbased on the comparison between theoretical calculations and experimental measurementshave been given. However, the physical meaning of the width of the gamma-ray spectraremains elusive.Recently, M. Rahm et. al. introduced a new electronegativity scale[11], which providessome hints to explain the dominance of valence electrons in the positron-electron annihilationprocess. Electronegativity assigns a measure to atoms (actually nuclear or positrons) basedon their proclivity to attract electrons in a molecule. It is an important physical quantityin chemical reactions. It can naturally be used to find positrophilic electrons in molecules.The gamma-ray spectra are generally represented by the Doppler shift parameter. TheDoppler shift, i.e., the width (usually using the full width at half maximum, FWHM) ofgamma-ray spectra represents the probability of the existence of positron-electron pairs atcertain energies. The integration of the gamma-ray spectra produces the annihilation rate,i.e., the total number of electrons annihilated with an incoming positron[10]. The maximumtheoretical annihilation rate is taken to be the number of electrons in a molecule in generaltheories[8]. However, as mentioned previously, experimental measurements indicate that theannihilation rate is usually considerably larger than the number of electrons in a molecule[9].Moreover, only the number of valence electrons has been usually considered in translating2he annihilation process.In order to fundamentally understand the mechanism of the positron-electron annihilationprocess, identification of a new physical quantity to represent the gamma-ray spectra and toexplain the positrophilic electrons in the positron-electron annihilation process is necessary.The present study attempts to identify such an alternative physical quantity and explainthe dominance of valence electrons in the positron-electron annihilation process by usingRahm’s electronegativity scale.
THE PHYSICAL PARAMETERS OF GAMMA-RAY SPECTRA
During the annihilation process, the momentum ~p of an emitted photon is equal to themomentum of the parent positron-electron pair, i.e., ~p = ~k − + ~k + . Therefore, the momentumdistribution of photons can be obtained via Fourier transform from the positron-electron pairwavefunction[10] A ( ~p ) = Z Φ − ( ~r )Φ + ( ~r ) e − i~p · ~r d~r, (1)where Φ − ( ~r ) and Φ + ( ~r ) denote the wavefunctions for electrons and positrons in the realspace, respectively.The momentum of the electron-positron pair is rotationally averaged in gas or liquidexperiments. Hence, in order to compare our results with these experimental measurements,the theoretical momentum distribution must be spherically averaged. The radial distributionfunction in the momentum space can be defined by D ( p ) = Z π dθ Z π dφP sin θ | A ( ~p ) | , (2)where P, θ, φ denote spherical coordinates. Hence, the theoretical spherically averaged mo-mentum distribution is given σ ( p ) = D ( p )4 πp . (3)which gives the mean probability to encounter the electron-positron pair on the surface withthe momentum | P | .Following this, the gamma-ray spectra during the annihilation process is Doppler shiftedin energy due to the longitudinal momentum component of the positron-electron pair[8].Hence, integration must be performed over the plane perpendicular to p in order to obtain3he total probability density corresponding to the momentum p = 2 ǫ/c . Then, the gamma-ray spectra for the positron-electron pair are given byΩ( ǫ ) = 1 c Z ∞ ǫ/c σ ( p ) pdp. (4)The Doppler shift from the center ( mc = 511 keV) is given by ǫ . For low-energy positrons,the gamma-ray spectra are dominated by the bound electrons in molecules. The Dopplershift ǫ is directly related to the bonding energy ǫ n of the annihilated bound electrons ǫ ∝√ ǫ n [10]. In other words, the Doppler shift ( ǫ ) is determined by the bonding energyof the annihilated bound electrons. If we integrate the gamma-ray spectra, the annihilationrate is given by Z eff = Z Ω( ǫ ) dǫ. (5)Then, the average binding energy (Doppler shift, actually) of the annihilation electrons willbe given by ¯ ǫ = R Ω( ǫ ) ǫdǫ R Ω( ǫ ) dǫ . (6)The gamma-ray spectra carry information about the electron momentum distribution inthe bound state orbitals[1]. In the present study, ¯ ǫ is named the Average Doppler Shift(ADS). The Doppler shift of the gamma-ray spectra is proportional to the square root of theabsolute value of the binding energy of the annihilated bound electrons[10]. Hence, ADS ofgamma-ray spectra represents the average bonding energy of the positron-electron pairs.ADS is a more general measure than FWHM of gamma-ray spectra in terms of rep-resenting the gamma-ray spectra, because FWHM is merely an analysis parameter of thespectrum shape with only one peak. Further, FWHM is merely a special point while ADS isobtained by averaging over all the points of the gamma-ray spectra. A general gamma-rayspectrum comprises more than one peak and shoulder. Hence, FWHM is not applicablein these complicated situations. However, ADS is determined by taking the entire data ofgamma-ray spectra into account, and not merely one value at half maximum. Hence, ADSis capable of representing all characteristics of gamma-ray spectra.Moreover, for a low-energy positron annihilation process, the ADS of valence electronsin a molecule also has the identical definition and agrees well with Rahm’s electronegativityscale[11]. Therefore, in order to understand the mechanism of the positron-electron anni-hilation process, the experimental analysis parameters, such as FWHM, should be replaced4 e Ne Ar Kr Xe Rn1234567 Expt FWHM Valence FWHM Rahm’s electronegativity
Names of atoms
FWHM (keV)
Rahm’s electronegativity (eV e -1 ) FIG. 1: The experimental and theoretical FWHM of gamma-ray spectra of noble gas atoms com-pared with Rahm’s electronegativity. by the physical quantity ADS. Eq.(6) remains the same and is consistent with Rahm’selectronegativity scale[11] ¯ χ = P Ni =1 n i ǫ i P Ni =1 n i . (7)with the summation running over the entire set of valence electrons in a molecule. Eq.(6) andEq.(7) both present very natural definitions that measure the ability to attract positrons.Comparing them with ADS of gamma-ray spectra and Rahm’s electronegativity value, onecan quantitatively prove the dominance of the valence electrons during the annihilationprocess. 5 Expt FWHM Valence FWHM
FWHM (keV)
Rahm’s electronegativity (eV e -1 ) NeArKrXeRn Xe Kr Ar Ney=0.19x-0.28y=0.11x+0.26 FIG. 2: The experimental and theoretical FWHM of gamma-ray spectra of noble gas atoms asfunctions of their electronegativities.
APPLICATION AND DISCUSSION
The recently developed Rahm’s electronegativity scale[11] provides significant hints toexplain the physical meaning of the gamma-ray spectra. In most of the previous studies,the characteristics of gamma-ray spectra have always been represented by the width FWHMof the profiles[1–4]. However, FWHM represents only one point in the gamma-ray spectrawhich lacks a direct physical interpretation. ADS takes into account all the points in gamma-ray spectra and incorporates the entirety of the information in itself. In the benchmark ofthe experimental studies of gamma-ray spectra[9], the C. M. Surko group presented recentaccurate measurements of the gamma-ray spectra for several molecules. In order to explaintheir measurements, one-Gaussian function was fitted to their measurements, leading to6
Li Na K Rb Cs Be Mg Ca Sr Ba B Al Ga In C Si Ge Sn N P As Sb O S Se Te F Cl Br I He Ne Ar Kr Xe
FWHM Rahm’s electronegativity
Names of atoms
FWHM (keV)
A A A A A A A A
Groups of atoms
Rahm’s electronegativity (eV e -1 ) FIG. 3: The Comparison between FWHM of gamma-ray spectra and electronegativity of all the36 atoms. the loss of certain structures, such as the shoulders of the profiles. They also mentionedthat their presentation of experimental data was merely analytical and quantitative, ac-knowledging that the physical meanings of the fitting parameters remain unknown.[9] Theirmeasurements were precise enough to study the line shape of the spectra and not just thewidth[1]. Moreover, most experimental gamma-ray spectra exhibit some shoulders, and arenot exactly smooth to fit. The presence of more than one shoulder in a gamma-ray spec-trum necessitates multiple Gaussian functions to fit the spectrum approximately[9]. Hence,a reasonable physical interpretation of the gamma-ray spectra seems necessary.As depicted in Fig.1 of reference[3] and Fig.2 of reference[4], the experimentally obtainedwidths of gamma-ray spectra for several molecules have been compared with the corre-sponding theoretical predictions. The comparison reveals that the two values agree well for7
FWHM (keV)
Rahm’s electronegativity (eV e -1 )y=0.18x-0.1H He FIG. 4: FWHM of gamma-ray spectra of 36 atoms as functions of their electronegativities. valence electrons. This induces the conclusion that valence electrons in molecules dominatethe annihilation process. However, the reason behind this is unclear. A physical meaning ofthe width of gamma-ray spectra is also elusive. The benchmark measurements are preciseenough to study the line shape of the spectra, and not just their widths[9]. Hence, a newmethod to translate the spectra needs to be developed. As depicted in Fig.1, in additionto the comparison of the experimental and theoretical width, we also compared the widthwith Rahm’s electronegativity. Electronegativity assigns a measure to the propensity ofatoms in molecules to attract electrons[11]. In the present work, the exact definition of theelectronegativity of an atom is alternatively taken to be its ability to attract positrons.We find that the electronegativity of a noble gas atom exhibits the same variation rulewith respect to the widths of the gamma-ray spectra. According to Rahm’s electronegativityscale, electronegativity is defined to be the average valence electron binding energy in atoms8 i Na K Rb Cs Be Mg Ca Sr Ba B Al Ga In C Si Ge Sn N P As Sb O S Se Te F Cl Br I Ne Ar Kr Xe
FWHM Rahm’s electronegativity
Names of atoms
FWHM (keV)
A A A A A A A A
Groups of atoms
Rahm’s electronegativity (eV e -1 ) FIG. 5: The Comparison between FWHM of gamma-ray spectra and electronegativities of 34 atomswithout helium and hydrogen. in Eq.(7). On the one hand, good agreement of electronegativity with the experimentalmeasurement implies the dominance of valence electrons during the annihilation process.On the other hand, the width of gamma-ray spectra might be related to electronegativity,or the average valence electron binding energy in atoms.As depicted in Fig.2, both the theoretical and the experimental widths are observed to belinearly related to the electronegativity of the corresponding atoms. This strong linear cor-relation of electronegativity with the widths of the gamma-ray spectra hints at the presenceof positrophilic electrons in molecules. It is known that Rahm’s electronegativity scale isdefined by averaging the ground state valence electron bonding energies. The agreement ofthe experimental widths of gamma-ray spectra with Rahm’s electronegativity verifies thatthe valence electrons dominate the annihilation process. The present calculated width for9
FWHM (keV)
Rahm’s electronegativity (eV e -1 )y=0.2x-0.24 FIG. 6: FWHM of gamma-ray spectra of 34 atoms as functions of their electronegativities excepthelium and hydrogen. the valence electrons has been depicted in Fig.2 and it also agrees well with Rahm’s elec-tronegativity. This probably indicates that the width of gamma-ray spectra shares a strongrelationship with electronegativity. Further, whether or not the width represents the averagevalence electron binding energy is also a relevant question. Only helium exhibits behaviordifferent from that of other noble gas atoms both with respect to width and electronegativity.This is explored later in the present work.In order to identify the physical meaning of the width of a gamma-ray spectrum and therelationship it shares with electronegativity, we studied the gamma-ray spectra of 36 atomscorresponding to 1-56 elements of group A in the periodic table, as depicted in Fig.3 andFig.4. The valence electrons of these atoms have been defined by Rahm[11]. Heavy andgroup B atoms were not considered due to the difficulty of obtaining accurate wavefunctions10
Li Na K Rb Cs Be Mg Ca Sr Ba B Al Ga In C Si Ge Sn N P As Sb O S Se Te F Cl Br I He Ne Ar Kr Xe
ADS Rahm’s electronegativity
Names of atoms
ADS
A A A A A A A A
Groups of atoms
Rahm’s electronegativity (eV e -1 ) FIG. 7: The Comparison between ADS of gamma-ray spectra and electronegativity of 36 atoms. for them. In Fig.3, the widths of the gamma-ray spectra of 36 light atoms have beencompared with their corresponding Rahm’s electronegativities. Within the same group, thevariation in the width of gamma-ray spectra is observed to be completely consistent withthe variation in electronegativity of these atoms. As the widths and the electronegativitiesare obtained from the same valence electrons, the notion that the width of the gamma-ray spectra is related with Rahm’s electronegativity of atoms is further strengthened. Thewidth of gamma-ray spectra and Rahm’s electronegativity are observed to share an obviouslinear correlation, as depicted in Fig.4. The electronegativity of all A group elements werebasically linearly related with their corresponding widths, except for helium. Thus, Rahm’selectronegativity can be considered to be an alternative indicator of the annihilated electronsin molecules.Fig.5 compares the electronegativities and widths of gamma spectra for 34 atoms except11
ADS
Electronegativity (eV e -1 )y=0.08x-0.03 FIG. 8: ADS of gamma-ray spectra of 34 atoms as functions of their electronegativities excepthelium and hydrogen. hydrogen and helium. The linear relationship only exhibited slight differences over the36 atoms. Based on the aforementioned analysis, it can be concluded that the widths ofgamma-ray spectra produced by the annihilation of positrons in atoms are closely related tothe electronegativities of the corresponding atoms. This relationship can be used to obtaina physical interpretation of the width of a gamma-ray spectrum. Further obvious evidencecan be found in Fig.6. The width of gamma-ray spectra might represent the ability ofvalence electrons to attract a positron. Hence, electronegativity was observed to be aboutfive times of the FWHM of the gamma-ray spectra. As mentioned previously, FWHM isan ideal parameter used to characterize the shape of the spectrum, and lacks any physicalmeaning. Eq.(6) associates real physical meaning to the real quantity ADS. The prospectiverelationship between ADS and electronegativity is also explored.12n Fig.7, we present the ADS of gamma-ray spectra for 36 atoms. As is apparent, ADSexhibits identical variations as electronegativity corresponding to the group A elementsconsidered. From group IA to VIIIA, both electronegativity of atoms and ADS of the cor-responding atomic gamma-ray spectra were observed to increase slowly. The first elementsin each group exhibited maximum electronegativity and ADS, except for helium in groupVIIIA in which neon exhibited the maximum. On the other hand, the maximum in eachgroup was observed to increase from IA to VIIIA, except for hydrogen in group IA. Theelectronegativity and ADS, and even FWHM as presented in Fig.3, of hydrogen were largerthan the largest values in groups II and III.This implies that ADS is capable of representing the gamma-ray spectra comparably tothe FWHM parameters. Fig.8 depicts a strong linear relationship between ADS electroneg-ativity. Meanwhile, the similarity in behavior with electronegativity demonstrates the ADScan represent the ability of an atom to attract a positron. ADS encodes the ability to annihi-late an electron with positron. As ADS arises from the integration of the entire gamma-rayspectrum, it is more general than FWHM. Furthermore, based on Eq.(6) ADS representsthe average Doppler shifts of the gamma-ray spectra, and thus inherently possesses obviousphysical meaning.Like the widths of the gamma-ray spectra, ADS can be taken to be an indicator ofpositrophilic electrons, as illustrated in Fig.8. A strong linear relationship was also foundbetween ADS and electronegativity. In other words, just like electronegativity, the averageDoppler shift of a gamma-ray spectrum represents the ability of the corresponding atom toattract the incoming positrons. The ratio between ADS and FWHM is approximately 0.43,as given in Fig.9. The ADS of gamma-ray spectra can certainly be used to describe thegamma-ray spectra of positron-electron annihilation processes.
CONCLUSOIN
In the present study, the physical quantity Average Doppler Shift (ADS) was introducedto represent the characteristics of gamma-ray spectra instead of the analysis parameterFWHM. The ADS of the gamma-ray spectra exhibit similar physical meaning as the averageenergy of the positron-electron pairs. The ADS of the gamma-ray spectra for the elementsof group A below the atomic number 56 were observed to agree well with the corresponding13
FWHM (keV)
ADS y=2.31x-0.15
FIG. 9: ADS of gamma-ray spectra of 34 atoms as functions of their FWHMs, except helium andhydrogen.
Rahm’s electronegativity values. Using ADS of gamma-ray spectra, the positrophilic elec-trons in the positron-electron annihilation process were, for the first time, quantitativelyproven to be the valence electrons. Based on Rahm’s electronegativity scale, it is concludedthat the present ADS is useful in understanding the mechanism of the positron-electronannihilation process rather than the width (FWHM) of the gamma-ray spectra.
Acknowledgement-
This work is supported by the National Natural Science Foundationof China under grants No. 11674145 and Taishan Scholars Project of Shandong province(Project No. ts2015110055). ∗ Electronic address: [email protected]
1] Swann A R and Gribakin G F 2019
Phys. Rev. Lett. J. Chem. Phys.
Phys. Rev. A Phys. Rev. A Phys. Lett. A
Phys. Rev. A Phys. Lett. A
Rev. Mod. Phys. Phys. Rev. A J.Phys B J. Am. Chem. Soc.342-351.