Virtual and bound states in low-energy positron scattering by atoms and molecules via modified effective range theory

Modified effective range theory is applied as a tool to determine bound and virtual state energies in low-energy positron elastic scattering by atoms and molecules. This is achieved by the S-matrix continuation into the complex momentum plane, allowing to identify poles related to shallow energy states. The influence of the long-range polarization potential (∼r −4) on the bound and virtual-state pole positions is analyzed for noble gases and nonpolar molecules such as H2, N2, and CH4. The quantitative relations between the S-matrix poles and the s-wave scattering length accounting for dipole polarization are introduced.


Introduction
The association of huge scattering cross-sections with bound or virtual states in the case of low-energy collisions is a standard topic in elementary scattering theory [1][2][3].This effect occurs when a slow particle is scattered in a field with a shallow (close to zero) energy level (ε).Such a discrete level may weakly bind (when negative) or almost bind (when positive) a particle with a target, enhancing the strength of the interaction.For instance, the concept of virtual state (i.e., positive 'quasi-discrete' energy level) has been used to explain the unexpectedly high rise of the integral elastic cross section as the scattering energy decreases to zero in electron collisions with CO 2 [4], SF 6 [5] and other nonpolar molecules [6,7].The shallow energy level is also vital in low-energy positron scattering by many atoms and molecules [8,9].The effect is manifested by unusually high probabilities for elastic collision and direct annihilation when the energy of the positron is close to zero.
Unlike the bound state, the energy of virtual level in positron scattering is hardly ever calculated directly, and its existence is usually postulated when a considerable negative s-wave scattering length (a 0 ) is found (e.g.see [8,[30][31][32]).The energy of the virtual level is then estimated as ma 2 , where m is the mass of the positron.Since near-zero energy ab initio calculations are rare, the scattering length is usually estimated through lowenergy extrapolation of theoretical scattering phase-shift or experimental elastic cross-section using the modified effective range theory (MERT) [33].Alternatively, as theoretical studies indicate, an energy-resolved positron annihilation can also be used to determine a 0 (e.g.see [27,[30][31][32] and their supplemental materials).However, the latter approach may give substantially different values than MERT estimates.
The existence of bound and virtual states in low-energy collisions can also be proven by examining the analytical properties of the S-matrix.Mathematically, both states are associated with the poles of the S-matrix in the complex momentum plane (k) [1][2][3].The pole laying precisely on the positive imaginary momentum axis reflects the bound state.In contrast, the virtual state corresponds to the pole on the negative imaginary axis just below the origin.However, such a simple picture is valid only for the finite-range potentials, which disappear beyond a certain distance from the target.On the other hand, analytical studies for pure long-range interactions show that the S-matrix can have an infinite number of virtual-state poles.Moreover, the S-matrix for infinite range potentials can be plagued with redundant poles and zeros, which do not correspond to any physical state, see [34,35] and references therein.
The interaction of charged particles with neutral targets comprises both long-and short-range potentials.In particular, the polarisation potential (∼r −4 ) is a dominant long-range component in low-energy collisions with atoms and nonpolar molecules due to the dipole induced by the charge of the incoming projectile.Hence, the contribution of this component must be addressed when estimating the poles' positions related to bound and virtual states.Recently, we showed [36] that the virtual states in electron-molecule collisions can be revealed and studied within the frame of MERT using the properties of Mathieu functions, that is, the exact analytical solutions for the Schrödinger equation with pure polarisation potential.We followed the conclusions of Khrebtukov [37], who noted that Mathieu functions can be exploited for the S-matrix continuation into the complex momentum plane.Ward and Macek [38] also used this property to analyze virtual states in positronium-proton collisions.The influence of the short-range part of the interaction potential on the pole positions has been taken into account semi-empirically using the effective range approximation applied exclusively to this component (as described in [39,40]).
This work shows that the modified effective-range theory extended to the complex momentum plane can reveal both bound and virtual states in positron scattering from atoms and nonpolar molecules.Moreover, we demonstrate that MERT can be a valuable tool for estimating the values of binding and virtual states' energies from the position of S-matrix poles.The proposed method may benefit theories that calculate low-energy positron scattering phase shifts (or elastic cross-sections) but do not estimate the binding energy (e.g.see [41]).Finally, we study the relation between the scattering length and the poles' positions in the presence of long-range polarisation.Recently, we concluded [36] that in the presence of infinitive interactions, the scattering length can not be treated as the reciprocal of the pole's position along the imaginary axis of the complex momentum plane as it is commonly assumed.In the present paper, we analyze this issue more quantitatively.In particular, we discuss the validity of the shallow-state energy estimation based only on the value of the scattering length.Moreover, when addressing the effect of dipole polarisability in determining energy levels, we show that the discrepancy between scattering lengths in positron collisions estimated by MERT and from the positron annihilation data (as can be found in [27,[30][31][32]) disappears.
The paper is organized as follows.Section 2 briefly describes the basic virtual and bound states theory for finite-range potentials.We also provide principles of the modified effective range theory, allowing us to calculate the S-matrix poles in the presence of the long-range polarisation potential.Section 3 includes MERT analysis of bound states in positron scattering on highly polarizable atoms such as Be, Zn, Cu, Cd, and Mg.Section 4 is devoted to the virtual states in positron scattering on noble gases and simple molecules (H 2 , N 2 and CH 4 ).Section 5 discusses the relation between the scattering length and the S-matrix poles' positions.The main conclusions are summarized in section 6.

Theory
The s-wave components (l = 0) of the scattering amplitude f l and the S-matrix S l (=2ikf l + 1) can be represented by the following formulas when the slow particle is scattered in the finite-range field with shallow bound or virtual state [1][2][3]: Here, l is the angular momentum quantum number.The components of f l and S l with l > 0 can be neglected since the contribution of higher partial waves is negligible compared to s-wave at sufficiently low energies.The quantity κ is a wavenumber corresponding to the energy of bound or virtual state: |ε| = κ 2 /2 (in atomic units).For a bound state, κ is positive, while for a virtual state is negative.Notice that for k = iκ, the scattering amplitude and the S-matrix have poles on the positive (bound state) or negative (virtual state) imaginary axis of the complex momentum plane.
The energy-dependence of the elastic scattering cross-section is described by while annihilation probability expressed by the effective number of electrons participating in annihilation [42], is given by where F is the proportionality constant.It is clear from equations (2) and (3) that for k → 0, both σ el and Z eff can be very large if κ is relatively small.In the case of k = 0, the elastic cross-section is expressed by the scattering length as a 4 el 0 2 s p = .Hence, the following relation holds: The scattering length a 0 distinguishes a virtual state (a 0 large and negative) from a weakly-bound state (a 0 significant and positive).Moreover, according to the model, a 0 alone is sufficient to estimate both states' energies.
The formulas in equations (1) to (4) originate from the theoretical approach, which does not consider the background scattering, i.e., the scattering on the potential unrelated directly to the resonant interaction of particle wave function with a shallow energy level.In particular, this model does not include long-range forces.
The impact of infinite-range interactions on the S-matrix components can be included through the scattering phase shifts of partial waves (η l ): In particular, the phase shifts affected by the interaction of the charged particle with induced dipole can be expressed by exact analytical solutions of the Schrödinger equation with pure r −4 potential (in atomic units) [33,43,44]: Here m l (k, α) and ν l (k, α) denote the energy-dependent parameters which can be determined numerically from properties of the Mathieu functions for a target (atom/molecule) characterized by the static dipole polarisability α.See the numerical procedures described in [39,40].The contribution of the unknown finite-range potential to the phase shift in equation ( 6) is considered in the boundary conditions imposed on the Schrödinger equation.O'Malley et al [33] showed that energy-dependent parameter B l (k), related to the short-range part of the interaction, has the following general form: where a l is the scattering length of l th partial wave and R a = * .Here where Φ l (r, k) is the exact radial wavefunction of the system, while Ψ l (r, k) is the Mathieu function (the linear combination of both independent solutions of the Schrödinger equation for polarisation potential, for more details see [33]).O'Malley et al [33] proposed to approximate the ρ l (k) by the value at zero-energy, anticipating that it changes slowly with k at low energies: where R l = ρ l (0).Equation ( 9) is similar to the effective range expansion of the scattering phase-shift in the absence of the long-range potentials used to describe neutron-proton collisions [45,46].Hence, in analogy to the original effective-range theory, we can call R l as the 'effective-range,' though the physical meaning of this parameter is rather different.We have already shown in a series of papers that the approximation in equation ( 9) is valid over a broad range of energies (even up to the ionization threshold for atomic targets), e.g.see [39,40,47].Importantly, Mathieu functions used to derive equations (6) -(9) are valid not only for real and positive k but in the sector −π < Arg k <π [33].Hence, equation (6) can be used for the S-matrix continuation (equation ( 5)) into the complex momentum plane.We checked that the S-matrix calculated with the present approach (equations (5)-9) fulfills the unitary condition ( ) ( ) for any complex value of k [2,3].This implies that η l is real when k is real.
Note that the Mathieu coefficients m l (k, α) and ν l (k, α) describe the contribution of the long-range polarisation potential (∼r −4 ), while the effective range expansion parameters a l and R l describe the 'strength' of the short-range potential.Since the system's exact wavefunction is unknown, parameters a l and R l can be determined numerically by comparing the model with experimental low-energy cross-sections or calculations based on advanced quantum mechanical models.
The current approach exclusively applies the effective range approximation to the short-range part of the interaction potential.O'Malley [33] expanded coefficients m 0 (k) and ν 0 (k) around zero-energy, and keeping only the lowest order terms, the more familiar form of the MERT was derived: The latter is frequently used to extrapolate scattering data (such as theoretical phase shifts or experimental cross-sections) towards zero energy to determine the s − wave scattering length (e.g.see [27,[30][31][32]48]).However, the energy range of the applicability (kR * = 1) of equation (10) is much narrower than equation (6).Hence, expansion (10) can not be used to calculate S-matrix poles precisely.Therefore, our current approach is entirely based on equation 6, which was semi-empirically proved to be valid also for kR * > 1. (see [39,40,47]).

Bound states
Since no direct experimental data prove bound states in low-energy positron-atom collisions, we determine a 0 and R 0 parameters by fitting MERT (equation 6 combined with equation 9) to theoretical s-wave scattering phase shifts.We used the results of model-potential calculations for Be and Mg by Swann and Gribakin [27], the box variational method for Zn and Cu by Mitory et al [21], and the configuration interaction calculations for Cd by Bromley and Mitroy [23].The results of MERT fit for Be, Zn, Cd, and Mg are shown in the upper panel of figure 1.We also acknowledge the calculations by Bromley et al [49], Zubiaga et al [50], Poveda et al [51] and Savage et al [52].All of them are in excellent agreement with each other, showing a rise of the s-wave phase-shift towards π as the energy goes to zero.
The effective range parameters are given in table 1.In our calculations, we used dipole polarisabilities reported by the authors of fitted data.MERT-derived scattering lengths are comparable with other theoretical predictions.As expected, the values of a 0 are large and positive since they support weakly bound states for the positron.
The presence of the bound states is confirmed by the poles of the S-matrix (equation ( 5) + equation ( 6)) occurring on the positive imaginary axis of the complex momentum plane, as shown by the lower panel of figure 1.The poles' positions along the axis (k = iκ) and the corresponding binding energies (|ε| = κ 2 /2) are given in table 2. Obtained energies correlate excellently with the best currently available theoretical values.This agreement confirms that the calculated singularities are not false (redundant) poles but reflect the presence of real bound states.Therefore, MERT can be a valuable tool for estimating binding energies from theoretical and experimental scattering data.Notice also that the S-matrix poles shown in the lower panel of figure 1 are accompanied by the zeros close to the real axis in the second quadrant of the k -plane (particularly visible for Mg).These zeros correspond to poles on the third quadrant (not shown in figure 1), reflecting the S-matrix symmetry property: the poles on the upper half-plane correspond to zeros on the lower half-plane and vice versa.
In the last column of table 2, we give κ estimated by Swann and Gribakin [27] from positron annihilation data.They showed that low-energy Z eff (k) for Be and Mg, calculated using their model, can be fitted by equation (3) with κ values given in the table.Both values are close to the pole positions derived in the present work.The following section shows that it is not a coincidence since similar convergence (even better) is obtained for virtual states.Note also that presently determined a 0 (table 1) and κ (table 2) are not related through equation 4.This is due to the presence of non-negligible background (i.e., non-resonant) scattering, e.g.see discussion in [57].The vital component of the latter is the long-range polarisation when a neutral target scatters a charged particle.The relation between a 0 and κ accounting for the dipole polarisability is discussed in detail in section 5.

Virtual states
As in the previous section, we decided to determine a 0 and R 0 parameters for positron elastic scattering by noble gases and small molecules H 2 , N 2 , and CH 4 from the theoretical s-wave phase shifts.Although these parameters can be determined by fitting MERT to experimental cross-sections, discrepancies between experimental data from different laboratories prevent the current analysis from being consistent.As a reference, we used the manybody calculations by Green et al [30] for atomic targets and Rawlins et al [32] for molecules since the same theoretical approach is used in both works.The results of MERT fit are shown in figure 2, while the effective range parameters are reported in table 3.
MERT-derived scattering lengths are comparable with other theoretical predictions.All values of a 0 are negative, suggesting no bound states for the positron.Indeed, the S-matrix continuation into the complex momentum plane does not reveal any poles on the positive imaginary axis.On the other hand, without difficulty, we found poles in the third quadrant of the plane; see figure 3. Except for Xe, all poles are displaced from the negative imaginary k axis, so they do not represent true virtual states as defined for finite-range potentials.This result is consistent with our recent findings for electron scattering from molecules [36].This shows that a polarisation potential (r −4 ) affects the poles' positions (related to virtual states) similarly to a long-range dipole potential (r −2 ) in polar molecules as reported by Herzenberg and Saha [58].Ward and Macek found a similar pole shift [38] due to the dipole polarisability when analyzing virtual states in positronium-proton collisions.Since the poles are off the imaginary axis, one would expect a pair of poles located symmetrically relative to this axis.However, since the long-range potentials, such r −4 , cause a branch point at k = 0 [59], the mirror image of the pole is located on a different Riemann sheet.This sheet can be found by the proper S-matrix rotation around the origin [60].
The pole moves towards the plane's origin as the scattering length becomes more negative; see figure 3. A more prominent scattering length reflects a more significant role of the short-range interactions compared to long-range polarisation potential.For very large and negative a 0 , the pole reaches the imaginary axis just below the origin (a case of Xe), as predicted by the textbook theory.We can define the distance of the pole from the plane's origin as , where κ r and κ i are the pole coordinates along the real and imaginary axis, respectively.We found, see table 4, that |κ| almost perfectly correlates with κ determined in other works by fitting equations (3) to theoretical low-energy positron annihilation data (see [27,[30][31][32]).Since theoretical Z eff (k) follows equation (3), the increase in the cross-section for positron direct annihilation with decreasing interaction energy is primarily due to the positron wave function being 'in resonance' with the shallow energy level (for all considered targets).This suggest that the energies of the states related to the singularities in the S-matrix shown in figure 3 can be estimated as ε = |κ| 2 /2.The following energy values were found in the present analysis: 2.842 eV for He, 1.564 eV for Ne, 186 meV for Ar, 61 meV for Kr, 14 meV for Xe, 38 meV for CH 4 , 174 meV for N 2 , and 426 meV for H 2 .

S-matrix poles and scattering length
The simple relation between the s − wave scattering length (a 0 ) and the S-matrix pole's location (κ) expressed by the equation (4) holds only when resonant interaction between positron wavefunction and shallow energy level is dominant over other components of the interaction potential.To address the effect of long-range polarisation tail (r −4 ) of interaction potential on the relation between a 0 and κ we analyzed the evolution of S-matrix (S 0 ) poles on the complex and dimensionless plane kR * for different scattering lengths.The typical evolution is shown in figure 4 (calculated for R 0 ≈ 0).
Only two poles were identified near the origin of the complex momentum plane: on the positive imaginary axis (bound states) and in the third quadrant of the plane (virtual states).Both poles coexist independently of each other for both positive and negative scattering lengths.When a 0 < 0, the bound state pole is far from the origin.As the scattering length increases from negative to positive, this pole moves continuously along the axis towards the origin.The pole is located at Im (k)R * ≈ 3.5 (a.u.) for a 0 ≈ 0 (not shown in figure 4).On the Table 1.Positron scattering parameters determined from the present MERT analysis of s-wave scattering phase-shifts for Be, Zn, Cu, Cd, and Mg.Here α is the dipole polarisability (a.u.), R 0 is the 'effective' range (a.u.), a 0 is the scattering length (a.u.).The results of other calculations of a 0 are also presented.[26], 423 [27], 425 [13], 439 [15], 463 [20] 0.167 [27] Figure 2. MERT fits to the s-wave scattering phase-shifts calculated by Green et al [30] (squares), and Rawlins et al [32] (circles) for positron elastic scattering by noble gases and H 2 , N 2 and CH 4 .

Atom
Table 3. Positron scattering parameters determined from the MERT analysis of the s-wave scattering phase-shifts calculated by Green et al [30] for noble gases, and Rawlins et al [32] for H contrary, the virtual state pole is far away from the plane's origin when a 0 > 0, and it slowly approaches the real axis at Re (k)R * ≈ −0.5 (a.u.) when a 0 increases.For negative a 0 , the virtual state lies closer to the negative imaginary axis, and for sufficiently large |a 0 |, it reaches this axis.Once on the axis, the virtual state pole moves towards the plane origin as the absolute value of a 0 increases further.
We found that the evolution structure presented in figure 4 is weakly sensitive to the effective range parameter R 0 .However, the exact pole's location depends on the R 0 value in a rather complex manner.Nevertheless, when R 0 is sufficiently small, the term k 2 in the effective range expansion (equation ( 9)) is small and does not affect the poles' positions noticeably.We checked that the following R 0 − free relation describes with 5% accuracy the exact numerical calculations of virtual states' positions (a 0 < 0) when |a 0 | > R * (i.e., for strong short-range interaction) and |R 0 | < R * (i.e., for relatively small effective range correction): where |κ| is a distance (absolute value) of pole's location from the origin of the complex momentum plane.This relation applies to Ar, Kr, Xe, CH 4 , H 2 , and N 2 .The bound states (a 0 > 0) are much more sensitive to R 0 than virtual ones.Nevertheless, the following relation holds within 5% accuracy when Equation ( 12) is applicable for systems with low positron binding energies such as Be, Cu, Zn and Cd.Presently introduced corrections to equation (4) resolve the discrepancies (as can be found in [27,[30][31][32]) between the positron scattering lengths estimated from annihilation data (using equation (3) + (4)) and from effective-range extrapolation of s − wave phase-shift (using equation (10)).Moreover, both equations (11) and (12) show clearly that the dipole polarisability may have an essential impact on the pole's position and, hence, the value of shallow energy levels.

Conclusions
The paper introduces the modified effective range theory (MERT) as a tool to determine the S-matrix poles on the complex momentum plane related to bound and virtual states in low-energy positron scattering by atoms and molecules.The model enables the simultaneous determination of the scattering length (a 0 ) and bound/ virtual state energy (ε).Presently determined positron binding energies to highly polarizable atoms are in excellent agreement with the best currently available theories.Moreover, MERT allows us to study the effect of long-range polarisation potential (i.e., dipole polarisability) on the S-matrix poles' positions (κ).In particular, we confirmed our recent findings [36], showing that the virtual-state poles are displaced off the negative imaginary axis to the left on the complex momentum plane due to an infinite tail in the interaction potential.Comparison of the currently determined virtual states' positions with the same quantity estimated based on positron annihilation cross-sections shows that the virtual state energy can be calculated using the pole's distance (i.e., the absolute value of the pole's location) from the plane's origin: ε = |κ| 2 /2.Unlike the virtual state, the dipole polarizability does not shift the bound state poles from the positive imaginary axis, though it affects their positions along the axis.
We also showed that the s − wave scattering length is not directly proportional to the pole's location reciprocal (as commonly assumed) if the background (non-resonant) elastic scattering is non-negligible.This is usually the case in the presence of infinite range potentials.Hence, the energy of the virtual state can not be estimated directly from the value of the scattering length.We provided simple relations between the scattering length and the pole's position, including the corrections due to the dipole polarisability.Such corrections also explain the discrepancies between the positron scattering lengths determined in two ways: (i) using MERT and (ii) using analysis of energy-resolved annihilation cross-sections.
Finally, we underline that MERT can be a valuable auxiliary tool for numerical approaches that characterize low-energy positron scattering but do not predict scattering lengths and bound/virtual states energies.

Figure 1 .
Figure 1.Upper panel: MERT fits to s -wave phase-shifts for elastic scattering of a positron calculated by Swann and Gribakin [27] for beryllium (Be) and magnesium (Mg), Mitroy et al [21] and Bromley and Mitroy [23] for zinc (Zn) and cadmium (Cd), respectively.For comparison, the calculations of Bromley et al [49] (Be and Mg), Zubiaga et al [50] (Be), Poveda et al [51] (Be) and Savage et al [52] (Mg) are also shown.Lower panel: Poles of |S 0 | matrix on the positive imaginary axis of the complex momentum plane, revealing the presence of the positron-atoms bound states.

Figure 3 .
Figure 3. Poles of |S 0 | matrix on complex momentum plane in positron scattering from noble gases.

Figure 4 .
Figure 4.The typical evolution of S 0 matrix poles (bound and virtual states) on complex momentum plane as a function of the scattering length (a 0 ).
2 , N 2 and CH 4 .The labels as in table1.The results of other calculations of a 0 are also presented.

Table 4 .
The distance (|κ|) of the virtual state S-matrix pole location from the origin of the complex momentum plane for noble gases and H 2 , N 2 , and CH 4 .The second column shows κ estimated in other works using equation (3) fitted to numerical calculations of positron annihilation parameter Z eff (k).