Quantum ricochets: surface capture, release and energy loss of fast ions hitting a polar surface at grazing incidence

Here we propose a novel mechanism for both capture and escape based on quantum mechanical diffraction of the ion, involving no thermal effects, and capable of taking place on a perfect, atomically smooth surface. Of course, diffraction is omnipresent in slow atom scattering, e.g. in HAS (helium atom scattering), as well as in electron scattering at all energies, such as LEED (low energy electron diffraction), RHEED (reflection high energy electron diffraction), or TEM (transmission electron microscopy), but diffraction appears to have been overlooked in the context of surface channeling of energetic ions, although the concept has been developed in recent works for fast neutral atoms [13]. Perhaps, as pointed out by Rousseau et al [12], the minuscule de Broglie wavelength (≪⃒1 pm) of a keV ion or atom has hindered the realization that at grazing incidence diffraction can lead to selective adsorption of even a heavy, energetic particle such as Ne⁺.

We shall shortly present the proper kinematic argument, but it might be useful to start with figure 3, which explains the elementary mechanism of Bragg reflection of a plane wave by an atomic array of period d, at oblique incidence Φ.

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Diffraction at emerging angle Ψ occurs when the difference of path lengths between two adjacent reflected rays is an integer multiple n of the wavelength:

$$\begin{eqnarray}&&d({\rm cos} \;\psi \;{\rm cos} \;\Phi )=n\lambda .\end{eqnarray} \tag{ 1 }$$

At grazing incidence this condition is written

$$\begin{eqnarray}&&{{\Phi }^{2}}-{{\psi }^{2}}=2n\lambda /d.\end{eqnarray} \tag{ 2 }$$

Specular reflection (Ψ = Φ) corresponds to n = 0. Selective beam adsorption will occur for Ψ = 0, i.e. at incidence Φ such that

$$\begin{eqnarray}&&{{\Phi }^{2}}=2n\lambda /d.\end{eqnarray} \tag{ 3 }$$

This shows that, even for λ/d = 10⁻⁴, adsorption can take place at incidences of the order of 1°.

Applying this idea to the present problem, diffraction of Ne⁺ can deflect the particle in and out of a bound state E_B of the ion in its image potential V_im(z) (see figure 4) by exchange of a parallel crystal momentum ħG with the periodic surface lattice. In the capture process the ion is slightly accelerated in its forward, parallel motion by acquiring the extra momentum ħG and the increase of parallel kinetic energy is subtracted from the energy of the perpendicular motion. The vertical momentum need not be conserved at the surface of the target, which, by recoiling, acts as a momentum sink.

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To work out the kinematics of the process, one must take account of the increase of perpendicular energy just prior to impact due to the ion acceleration in its average electronic (adiabatic) image potential |V_im|. From energy conservation we can write

$$\begin{eqnarray}&&\frac{{{\hbar }^{2}}k_{\bot }^{2}}{2M}+\frac{{{\hbar }^{2}}{{k}_{\shortparallel }}^{2}}{2M}+\left| {{V}_{im}} \right|={{E}_{B}}+\frac{{{\hbar }^{2}}}{2M}{{\left( {{{\vec{k}}}_{\shortparallel }}+\vec{G} \right)}^{2}}.\end{eqnarray} \tag{ 4 }$$

The final perpendicular energy E_B after diffraction is given by

$$\begin{eqnarray}&&{{E}_{B}}=\left| {{V}_{im}} \right|+\frac{{{\hbar }^{2}}}{2M}\left[ k_{\bot }^{2}-(2{{{\vec{k}}}_{\shortparallel }}.\vec{G}+{{G}^{2}}) \right]\approx E\Phi _{im}^{2}-\hbar vG\end{eqnarray} \tag{ 5 }$$

where v is the ion velocity parallel to the surface and where

$$\begin{eqnarray}&&{{\Phi }_{im}}^{2}={{\Phi }^{2}}+\left| {{V}_{im}} \right|/E.\end{eqnarray} \tag{ 6 }$$

Φ_im is the incidence angle at the height z where diffraction takes place, most likely at the average distance of closest approach z₀ where the surface potential corrugation is maximum. The G ² term in (5) can be neglected since G ≪⃒ k_||. The average image potential can be modeled by

$$\begin{eqnarray}&&V(z)=\left\{ \begin{array}{lllllllllllllll} -\frac{{{e}^{2}}\epsilon }{4z}\quad z\geqslant {{z}_{0}} \\ \quad \infty \quad z<{{z}_{0}} \\ \end{array} \right.\end{eqnarray} \tag{ 7 }$$

where  = (_e − 1)/(_e + 1) ≈ 1/3 is the electronic response function of the substrate (_e = 2 for LiF) and z is the distance from the surface plane. For capture to occur one must have V₀ < E_B < 0, where V₀ = −e²/4z₀ = −0.63 eV is the bottom of the image potential of the ion at the distance of closest approach to the surface estimated as z₀ = 1.8 Å in [1] (figure 4). This gives

$$\begin{eqnarray}&&\frac{{{V}_{0}}}{E}+\frac{2{{G}_{\shortparallel }}}{{{k}_{\shortparallel }}}<\Phi _{im}^{2}<\frac{2{{G}_{\shortparallel }}}{{{k}_{\shortparallel }}}\end{eqnarray} \tag{ 8 }$$

where $\hbar {{G}_{||}}=2\pi n/a$, with a = 4.02 Å the LiF lattice constant and n any positive integer. At sufficiently small Φ_im, equation (5) may however be fulfilled only for the smallest ${{G}_{||}}$, i.e., n = 1, which is the case considered here. Condition (8) actually conforms to the didactic calculation in equation (3) for n = 1. Other angular thresholds may occur for n > 1, but the matrix elements for diffraction by the surface corrugated potential are likely to decrease rapidly with increasing n (calculation of the periodic surface potential and of the dynamics of ion scattering by this potential will be considered in a future publication). Using numerical values appropriate to the experiment of figure 1, i.e. 1 keV Ne⁺ at 0.95° incidence [2], one obtains ħvG = 0.9755 eV and a shallow bound state energy E_B = −0.07 eV is found. This argument shows that the diffraction mechanism is allowed and can produce the capture and the following bouncing motion for this incidence angle. The spectrum of bound states of the heavy particle Ne⁺ being quasi-continuous (see the appendix), the capture is non-resonant. The quantized length L_B = L_osc is then the fixed parallel length travelled by the ion while it executes one full vertical oscillation between the classical turning points z₀ and z₁ of the image potential at the selected, constant energy E_B (figure 4). After executing a number of ricochets (figure 2), the captured ion can be released at the next surface encounter in the initial specular direction by the reverse elastic diffraction process. The probability of escape at each impact is the same as the capture probability, by virtue of time reversal symmetry.

In the absence of thermal effects, the quantization of L_osc is a consequence of the quantization of the crystal momentum ħG , which selects the bound state energy E_B in (5) in a precise manner.

The value of the time of flight T_osc and bounce length L_osc can be obtained by solving Newton's equation for the (perpendicular) motion of the ion in its image potential (7). This is done in section 6. The oscillation time T_osc and hence the bounce length are found to be

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{L}_{{\mbox{osc}}}}=v{{T}_{{\mbox{osc}}}}=v{{z}_{1}}\sqrt{\frac{2M}{\left| {{E}_{B}} \right|}}\left[ arccos\sqrt{\frac{{{z}_{0}}}{{{z}_{1}}}}+\sqrt{\frac{{{z}_{0}}}{{{z}_{1}}}\left( 1-\frac{{{z}_{0}}}{{{z}_{1}}} \right)} \right] \\ \quad \approx v{{z}_{1}}\sqrt{\frac{2M}{\left| {{E}_{B}} \right|}}\frac{\pi }{2}\quad ({\mbox{for}}\ \;{{z}_{1}}\gg \;\;{{z}_{0}}) \\ \end{array}\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray}&&{{z}_{1}}\equiv \frac{\epsilon {{e}^{2}}}{4\left| {{E}_{B}} \right|}\end{eqnarray} \tag{ 10 }$$

is the upper turning point of the arc trajectory (figure 4). Inserting the values of z₀ and E_B we find z₁ ≈ 16.2 Å, T_osc = 6.08 ps and L_osc ≈ 6080 Å, about 1500 lattice parameters. It will be shown in sections 5 and 6 that this result allows quite accurate prediction of the observed constant separation of about 48 eV between the echo peaks in figure 1. The mechanism of diffraction-assisted capture and release is therefore consistent with these basic experimental data.

The diffraction mechanism proposed here can be experimentally tested in a number of ways. Most convincing would be to see ricocheting persist at low temperature ⁶. Another test would be to check for the existence of a discrete sequence of critical incidence angles Φ_c( G ) for a given incident energy at which the normal kinetic energy falls at the vacuum level for that particular G -vector. Since above each of these angles there should be no skipping observed for the corresponding G -vector, a sequence of spikes should occur in the final skipping intensity as a function of Φ_im, provided there is enough sensitivity and resolution. A third test would consist in changing G while keeping Φ constant by changing the azimuthal orientation of the single crystal surface, which should modify E_B and the observed energy loss at each ricochet.

Before we apply the semi-classical theory to the present situation, we first analyze the experimental data of figure 1 using only general physical arguments. For the sake of maximum simplicity let us assume here that the surface FK wake field [19] can be represented by one single, effective oscillator of frequency ω. In reality, at grazing incidence, the relevant FK modes making up the wake field form a 2D continuum of wave vector Q satisfying the surfing condition ω −  Q  ·  v  = 0. This condition expresses the fact that only the modes of phase velocity equal to the ion velocity are resonantly excited. However it will be found in section 6 that the physical arguments which we are about to develop apply independently to each of the Q -modes.

Referring to figure 5, let us then call α = ia (a positive) the amplitude of the Glauber state into which this effective FK oscillator is driven by the incoming Ne⁺ ion during its initial approach to the surface (for grazing incidence, all amplitudes are pure imaginary; see (31) and (34)). By time reversal symmetry the amplitude for the specularly reflected path is −α* = α (see equation (14)), giving a total (additive) amplitude of 2α and hence an energy loss for the Ne⁺ parent peak

$$\begin{eqnarray}&&{{I}_{0}}=4{{\left| \alpha \right|}^{2}}\hbar \;\omega =4{{a}^{2}}\hbar \;\omega =29\;{\mbox{eV}}{\mbox{.}}\end{eqnarray} \tag{ 17 }$$

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Notice that here the excitation amplitudes created by the incoming and reflected paths add up coherently and therefore the associated energy loss is four times (not twice) the energy loss of either the incident or the reflected paths alone. This constructive interference effect is a property of the phonon coherent state, which also holds in the classical mechanics of the forced oscillator: at resonance, doubling the applied external force quadruples the energy transfer.

Similarly, call β = ib (b positive) the coherent amplitude of the Glauber state into which the FK field mode is further driven by the Ne⁺ ion in its ascending path in the capture bound state (figure 5). From the additivity property of the coherent amplitudes, the total amplitude for the first capture impact is then α + β and the corresponding mean energy loss is (a + b)² ℏω.

Turning now to the second impact, it is essential to observe that, because the time of flight T_osc of one bounce lasts about 6 ps (see section 3), by the time the Ne⁺ ion returns to the surface for the second impact the FK mode excited at the first one will have decayed back to its ground state. This is because the phonon damping constant in ω + iγ is typically γ ≈ 0.05ω, i.e. the phonon lifetime of the order of 1.2 ps is short compared to T_osc. We can therefore assume that, at each new approach to the surface, the trapped Ne⁺ meets the phonon field anew in its ground state and that the amplitudes of the events occurring at the successive impacts do not add up coherently. In other words, at each impact the coherent amplitudes add up coherently, while from one impact to the next the energy losses rather than their amplitudes are additive. Now, by the property of time reversal symmetry, the additional mean energy loss of Ne⁺ released into the specular direction at the second impact is identical to the loss at the initial capture impact. Hence the first echo peak, as indicated in figure 5, should occur at

$$\begin{eqnarray}&&{{I}_{1}}=2{{\left| \alpha +\beta \right|}^{2}}\hbar \;\omega =2{{\left( a+b \right)}^{2}}\hbar \;\omega =76\;{\mbox{eV}}{\mbox{.}}\end{eqnarray} \tag{ 18 }$$

Solving (17) and (18) for a and b, one finds

$$\begin{eqnarray}&&a={{\left( {{I}_{0}}/4\hbar \omega \right)}^{1/2}}=2.692{{\left( {\mbox{eV}}/\hbar \omega \right)}^{1/2}}\end{eqnarray} \tag{ 19 }$$

and

$$\begin{eqnarray} &&b = \left( {1/2} \right)\left[ { - \sqrt {{I_0}} + \sqrt {2{I_1}} } \right]/{\left( {\hbar \omega } \right)^{1/2}} = 3.472{\left( {{\rm{eV}}/\hbar \omega } \right)^{1/2}}.\end{eqnarray} \tag{ 20 }$$

Next, referring to figure 5, we see that, while travelling in the capture state, the Ne⁺ creates a FK amplitude of 2β and hence suffers an additional mean energy loss ΔI = 4b² ℏω to the FK field every time it ricochets from the surface. Using (20) for b one finds

$$\begin{eqnarray}&&\Delta I = 4{b^2}\hbar \:\omega = {\left[ {\sqrt {2{I_1}} - \sqrt {{I_0}} } \right]^2} = 48.2\:{\rm{eV}}{\rm{.}}\end{eqnarray} \tag{ 21 }$$

The spectacular numerical fact here is that, by using the observed separation I₁−I₀ = 47 eV in (18) between the first two peaks, the model automatically generates a numerical value of ΔI which is very close to the observed periodic repetition throughout the entire spectrum. Our reckoning of the quantum mechanical additivity of coherent amplitudes at each individual impact and of the lack of such additivity between the successive ricochets is therefore vindicated. Indeed, the model provides an explanation for the unique periodicity in the echo spectrum of figure 1 where the mth echo peak occurs at the mean energy loss

$$\begin{eqnarray}&&{{I}_{m}}={{I}_{1}}+4{{b}^{2}}\left( m-1 \right)\;\hbar \omega ={{I}_{0}}+m\Delta I.\end{eqnarray} \tag{ 22 }$$

For future reference we note that the observed ratio between the periodic loss ΔI and the parent peak loss I₀ is

$$\begin{eqnarray}&&\Delta I/{{I}_{0}}={{\left( \beta /\alpha \right)}^{2}}={{\left( b/a \right)}^{2}}\approx 47/29=1.62.\end{eqnarray} \tag{ 23 }$$

The ratio ΔI/I₀ is what determines the relative scale of the mean periodic losses observed in figure 1. The physical meaning of this number (fortuitously close to the golden ratio!) is simply that the interaction time of the ion when impacting the surface in the trapped state is longer than the interaction time of the first specular impact (see [20] for a quantitative estimation of the interaction time in specular reflection). Section 6 will show that the model Hamiltonian developed in [1] does lead to the empirical values of the individual excitation amplitudes α and β in (19) and (20). Therefore, both the relative energy scale (β/α)² in (23) and the absolute energy scale |α|² in (17) of the Ne⁺ experimental spectrum are accounted for by the present theory.

The heuristic evaluation developed above is further validated by the following consideration of the relative peak positions I_n and J_n of the Ne⁺ and Ne⁰ spectra shown in figure 1. It was shown in [1] that the average energy loss of the Ne⁰ parent peak J₀ has several components:

$$\begin{eqnarray}&&{{J}_{0}}={{I}_{0}}/4+\Delta E=17.75\;{\mbox{eV}}{\mbox{.}}\end{eqnarray} \tag{ 24 }$$

Here I₀/4 = a² ℏω = 7.25 eV is the energy loss of the Ne⁺ on the incident portion of its path (see figure 5), i.e., as emphasized above, one-quarter of the parent peak loss in (17). ΔE = ΔE_hole + ΔE_Auger = 10.5 eV, where ΔE_hole = 6.5 eV is the screening energy of the fixed 'hole' F⁰ [1] and ΔE_Auger = 4 eV is the Auger energy defect [1, 2] (i.e. the difference between the energy necessary to create a LiF exciton at 12 eV above the valence band and the 8 eV energy liberated from a valence band electron falling into the Ne⁰ ground state). Hence the parent peak separation in figure 1 amounts to

$$\begin{eqnarray}&&{{I}_{0}}-{{J}_{0}}=29-17.75=11.25\;{\mbox{eV}},\end{eqnarray} \tag{ 25 }$$

which is close to the observed separation of 11.5 eV. This is not surprising given that in [1] ΔE_hole = 6.5 eV was adjusted to obtain the shift of 12 eV discussed there for interpreting the experiment of Borisov et al [11].

More significant is the position of the first Ne⁰ echo peak J₁ given by (see figure 5)

$$\begin{eqnarray}&&{{J}_{1}}={{I}_{1}}/2+{{b}^{2}}\hbar \omega +\Delta E=60.5\;{\mbox{eV}}{\mbox{.}}\end{eqnarray} \tag{ 26 }$$

The term I₁/2 = (a + b)²ℏω = 38 eV is the energy loss of Ne⁺ upon entering the bound state at the first capture impact (figure 5); from the fact that processes at successive impact do not interfere, this term is just one-half of the I₁ peak loss in (18). The term b²ℏω = ΔI/4 = 12 eV is the additional energy loss of the Ne⁺ ion on its descending path towards the second impact just prior to neutralization, again one-quarter of the total 48 eV loss in (21). Hence the separation of the two first order echo peaks in figure 1 is

$$\begin{eqnarray}&&{{I}_{1}}-{{J}_{1}}=76-60.5=15.5\;eV.\end{eqnarray} \tag{ 27 }$$

This is clearly different from the parent peak separation and spectacularly close to the observed echo separation of 15 eV. Moreover, since all higher order pairs of echo peaks (I_n , J_n ) derive from the first order pair (I₁ , J₁ ) by adding ΔI = 4b² ℏω = 48 eV to both terms (see figure 5), the separation ΔI_n  = I_n  − J_n will remain constant at about 15 eV throughout the spectrum, as indeed shown in figure 1. Therefore, the relative unequal shift (I_n  − J_n )/(I₀ − J₀) ≈ 1.3 is also accounted for by our model.

In conclusion, both crucial ratios ΔI/I₀ and (I_n  − J_n )/(I₀ − J₀), which together determine the relative energy loss scale of the experiment in figure 1, are correctly reproduced by our toy model, in which (i) the FK phonon field responds quantum mechanically to every particle impact (where coherent amplitudes add up) and (ii) there is independence between the inelastic events occurring at successive impacts (where square amplitudes or energy loss add up). Notice that the interpretation of the spectra developed above implies no special assumption on the nature of the capture–release mechanism and demands only time reversal symmetry.

In view of obtaining explicit theoretical expressions for the amplitudes α and β we now need to specify the analytical form of the various trajectories. Referring to figure 6(a), the exact Ne⁺ incident or outgoing trajectory can be deduced starting from the equation for energy conservation in the perpendicular motion

$$\begin{eqnarray}&&{{\left( \frac{dz}{dt} \right)}^{2}}=v_{\bot 0}^{2}+{\mkern 1mu} \frac{2}{M}\frac{{{e}^{2}}\epsilon }{4z(t)}\quad \to \quad {{v}_{\bot 0}}\;dt=\sqrt{\frac{z}{{{z}_{e}}+z}}dz\end{eqnarray} \tag{ 28 }$$

where v_⊥0 = vΦ is the incident perpendicular velocity and where we have introduced a length z_e

$$\begin{eqnarray}&&{{z}_{e}}=\frac{{{e}^{2}}\epsilon }{4{{E}_{\bot 0}}}\;=\frac{{{e}^{2}}\epsilon }{4E{{\Phi }^{2}}}=0.39\;{\AA}\end{eqnarray} \tag{ 29 }$$

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$$\begin{eqnarray}&&{{t}_{sp}}(z)=\frac{{{z}_{e}}}{v\Phi }\left[ \sqrt{\frac{z}{{{z}_{e}}}(1+\frac{z}{{{z}_{e}}})}-\sqrt{\frac{{{z}_{0}}}{{{z}_{e}}}(1+\frac{{{z}_{0}}}{{{z}_{e}}})}-\;{\rm ln} \;\frac{\sqrt{\frac{z}{{{z}_{e}}}}+\sqrt{1+\frac{z}{{{z}_{e}}}}}{\sqrt{\frac{{{z}_{0}}}{{{z}_{e}}}}+\sqrt{1+\frac{{{z}_{0}}}{{{z}_{e}}}}} \right].\end{eqnarray} \tag{ 30 }$$

From this, we obtain the coherent amplitude of the FK phonon of wave vector Q excited during the specular path

$$\begin{eqnarray}\begin{array}{} {{I}_{sp}}(Q) & = & \frac{i}{\hbar }{{C}_{Q}}{{e}^{-Q{{z}_{im}}}}\left[ \mathop \int \limits_{-\infty }^{0} dt{{e}^{-Qz(t)-i\Omega t}}+\mathop \int \limits_{0}^{\infty } dt{{e}^{-Qz(t)-i\Omega t}} \right] \\ {} & = & \frac{2i}{\hbar }{{C}_{Q}}{{e}^{-Q{{z}_{im}}}}\frac{1}{{{v}_{\bot 0}}}\left[ \mathop \int \limits_{{{z}_{0}}}^{\infty } dz\sqrt{\frac{z}{{{z}_{e}}+z}}{{e}^{-Qz}}\;{\rm cos} \;\Omega \;{{t}_{sp}}(z) \right] \\ \end{array}\end{eqnarray} \tag{ 31 }$$

where Ω = ω −  Q  ·  v is the Doppler-shifted frequency and where the integral over t has been converted into an integral over z using (28). As stated above, the total amplitude is purely imaginary, like its two equal incident and outgoing components. I_sp( Q ) can be calculated numerically by inserting the explicit z dependence of t_sp from (30) into the z integral (31). The total strength in (13) would then be calculated via a further 2D integral over Q . However, below, we shall have recourse to a physical approximation to the integral (31), which allows us to dispense with such cumbersome multiple integration and which makes the result much more transparent both physically and algebraically (see [20] for an alternative, accurate method of evaluation).

Similarly, the exact form of the arc trajectory of a ricochet (figure 6(b)) can be deduced from the equation of perpendicular motion

$$\begin{eqnarray}&&{{\left( \frac{dz}{dt} \right)}^{2}}=\frac{2}{M}\frac{{{e}^{2}}\epsilon }{4{{z}_{1}}}\left( \frac{{{z}_{1}}}{z}-1 \right)\quad \to \quad dt=dz\sqrt{\frac{M}{2\left| {{E}_{B}} \right|}}\sqrt{\frac{z}{{{z}_{1}}-z}}.\end{eqnarray} \tag{ 32 }$$

By integration (z₀  ≦̸ z ≦̸ z₁ )

$$\begin{eqnarray}&&{{t}_{b}}(z)={{z}_{1}}\sqrt{\frac{M}{2\left| {{E}_{B}} \right|}}\left[ {\mbox{arc}}\;{\rm cos} \sqrt{\frac{{{z}_{0}}}{{{z}_{1}}}}+\sqrt{\frac{{{z}_{0}}}{{{z}_{1}}}(1-\frac{{{z}_{0}}}{{{z}_{1}}})}-{\mbox{arc}}\;{\rm cos} \sqrt{\frac{z}{{{z}_{1}}}}-\sqrt{\frac{z}{{{z}_{1}}}(1-\frac{z}{{{z}_{1}}})} \right].\end{eqnarray} \tag{ 33 }$$

This exact trajectory is plotted in figure 6(b). For z = z₁, (33) gives half the oscillating time T_osc (9) of section 3. The FK amplitude is now given by the pure imaginary function of Q

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{I}_{b}}(Q)=\frac{i}{\hbar }{{C}_{Q}}\left[ \mathop \int \limits_{-T/2}^{0} dt{{e}^{-Q\,z(t)-i\Omega \,t}}+\mathop \int \limits_{0}^{T/2} dt{{e}^{-Q\,z(t)-i\Omega \,t}} \right] \\ =\frac{2i}{\hbar }{{C}_{Q}}\sqrt{\frac{M}{2\left| {{E}_{B}} \right|}}\left[ \mathop \int \limits_{{{z}_{0}}}^{{{z}_{1}}} dz\sqrt{\frac{z}{{{z}_{1}}-z}}{{e}^{-Q\,z}}cos\;\Omega \;{{t}_{b}}(z) \right]. \\ \end{array}\end{eqnarray} \tag{ 34 }$$

Again, the z integral can be calculated numerically by inserting (33) into (34), but we shall dispense with this below. Later, we shall need the takeoff (or landing) angle φ of the ricochet (figures 5 and 6):

$$\begin{eqnarray}&&\varphi ({{z}_{0}})=\frac{1}{v}\frac{dz}{dt}=\sqrt{\frac{\left| {{E}_{B}} \right|}{E}\ \;\frac{{{z}_{1}}-{{z}_{0}}}{{{z}_{0}}}}=0.02366\ \;rad.\end{eqnarray} \tag{ 35 }$$

Rather than perform the integrals in (31) and (34) by blind numerical force, we shall obtain upper and lower bounds for each of them, which will then allow us to proceed analytically up to the end.

Referring to figure 6(a), we note that the exact, image-curved trajectory is bounded by two fictitious straight trajectories (broken lines labelled Φ and Φ_im). The lower one, of (experimental) incidence angle Φ, is the specular path in the absence of image attraction, while the upper one, of incidence angle Φ_im, is tangent to the real image trajectory at the impact point z₀. If we substitute one or the other of these two approximations for the real path z(t) into (31), we either overestimate or underestimate the interaction time [20] with the FK phonons and we therefore expect to generate an upper bound I_sp(Φ) or a lower bound I_sp(Φ_im), respectively. Using straight trajectories makes the integral in (31) elementary. The result is (as in [1])

$$\begin{eqnarray}&&{{I}_{sp}}(\Psi )=-{{C}_{Q}}\ \;\frac{2i}{\hbar }\frac{Q{{v}_{\bot }}}{{{\Omega }^{2}}+{{Q}^{2}}v_{\bot }^{2}}\quad \to \quad \mathop{{\rm lim} }\limits_{{{v}_{\bot }}={{0}_{+}}}{{\left| {{I}_{sp}} \right|}^{2}}=C_{Q}^{2}\ \;\frac{4\pi }{{{\hbar }^{2}}}\frac{1}{Qv\Psi }\delta (\Omega )\end{eqnarray} \tag{ 36 }$$

where Ψ is either Φ or Φ_im.

The total strength of the Poisson distribution of the specular, parent peak S₀ = 4a² = I₀/ℏω is obtained from integrating (36) over Q (see [1] for details of the calculations). The strength is found to be

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{{\mbox{S}}}_{0}}(\Psi )=\frac{2{{e}^{2}}\bar{\epsilon }}{{\mkern 1mu} \hbar {{v}_{\bot }}}{{I}_{\operatorname{int}}}(u)\equiv S\frac{1}{\Psi } \\ S=\frac{2{{e}^{2}}\bar{\epsilon }}{{\mkern 1mu} \hbar v}{{I}_{\operatorname{int}}}(u)\quad \\ \end{array}\end{eqnarray} \tag{ 37 }$$

where the quadrature

$$\begin{eqnarray}&&{{I}_{\operatorname{int}}}(u)=\mathop \int \limits_{1}^{{{x}_{c}}} dx\frac{{{e}^{-ux}}}{x\sqrt{{{x}^{2}}-1}},\quad \quad {{x}_{c}}=\frac{Gv}{\omega {\mkern 1mu} },\quad \quad u=\frac{2\omega \bar{z}}{v}\end{eqnarray} \tag{ 38 }$$

is plotted in figure 7 [1].

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Finally. the true strength S₀ is bracketed by the two bounds

$$\begin{eqnarray}&&S\frac{1}{{{\Phi }_{im}}}\;<\ \;{{{\mbox{S}}}_{0}}=4{{{\mbox{a}}}^{2}}<\ \;{\mbox{S}}\frac{1}{\Phi }.\end{eqnarray} \tag{ 39 }$$

Turning now to the ricochet trajectory of figure 6(b), we similarly approximate the real path by one or the other of the straight, broken paths (labelled φ and φ₁) chosen to encompass the arc. The upper line (takeoff angle φ) will give a lower bound to the FK amplitude I_b(φ), while the lower line (takeoff angle φ₁) will give an upper bound I_b(φ₁). These approximations again make the integral in (34) elementary. The result is, up to negligible terms proportional to exp(−Qz₁),

$$\begin{eqnarray}&&{{I}_{b}}(\psi )=-{{C}_{Q}}\ \;\frac{2i}{\hbar }\frac{Q{{v}_{\bot }}}{{{\Omega }^{2}}+{{Q}^{2}}v_{\bot }^{2}}{\rm cos} \frac{\Omega \;T}{2}\quad \to \quad \mathop{{\rm lim} }\limits_{{{v}_{\bot }}={{0}_{+}}}{{\left| {{I}_{sp}} \right|}^{2}}=C_{Q}^{2}\frac{4\pi }{{{\hbar }^{2}}}\frac{1}{Qv\psi }\delta (\Omega )\quad \end{eqnarray} \tag{ 40 }$$

where ψ is either φ or φ₁. Neglecting the upper limit, exp(−Qz₁) amounts to taking z₁ = ∞, which explains why (40) and (36) have identical forms. The end result for the FK excitation strength S_b = 4b² caused by one single ricochet is therefore bounded by

$$\begin{eqnarray}&&S\frac{1}{\varphi }\;<\ \;{{{\mbox{S}}}_{{\mbox{b}}}}\;=4{{b}^{2}}<\ \;{\mbox{S}}\frac{1}{{{\varphi }_{1}}}\;\quad \end{eqnarray} \tag{ 41 }$$

where S is given in equation (37).

We are now prepared to make the crucial comparison between the theoretical predictions and the observed data of figure 1. Since the predicted strengths in (39) and (41) are bracketed by two incidence or two takeoff angles there must exist two intermediate angles, which we can call Ψ and ψ respectively, for which the strengths (39) and (41) coincide with the exact values. We shall make the reasonable assumption that these intermediate angles can be approximated by the arithmetic means (see figure 6 where the mean paths are shown in red)

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \Psi \;=(\Phi +{{\Phi }_{im}})/2=0.0233 \\ \psi \;=(\varphi +{{\varphi }_{1}})/2=0.0142. \\ \end{array}\end{eqnarray} \tag{ 42 }$$

The numerical values of the various angles (in rad) are as follows: Φ = 0.0166 (experimental incidence 0.95°), Φ_im = √(Φ² + |V_im|/E) = 0.03 (image potential 0.63 eV), φ = 0.023 66 (takeoff angle for E_B = −0.07 eV, see (35)) and φ₁ = 2(z₁ − z₀)/L_osc = 0.0047 (z₁ = 16.2 Å, z₀ = 1.8 Å and L_osc = 6080 nm). The mean angles, indicated in (42), are then Ψ = 0.0233 and ψ  = 0.0142. Notice that, apart from the incident angle Φ, all other angles are fixed by the bounce state energy E_B, itself a consequence of the kinematics of the proposed diffraction mechanism of capture. The ratio between the two strengths S_b/S₀ is given by

$$\begin{eqnarray}&&{{S}_{b}}/{{S}_{0}}={{\left( b/a \right)}^{2}}=\Psi /\psi =1.64.\end{eqnarray} \tag{ 43 }$$

The numerical value obtained here for |E_B| = 0.07 eV is astonishingly close to the empirical scale ratio in (23). This spectacular agreement alone strongly supports our diffraction mechanism of capture and release. The theoretical ratio Ψ/ψ is plotted in figure 8 as a function of binding energy E_B. One observes that in the immediate neighborhood of |E_B| = 0.1 eV the energy loss per ricochet is insensitive to |E_B| in a narrow range. This however ceases to hold for deeper bound states, for which the energy loss rapidly increases with |E_B| (figure 8). For example, at |E_B| = 0.25 eV, the echo period would be 51.3 eV instead of 48, which is way outside the uncertainty on peak position in figure 1.

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Last and importantly, let us verify that the present theory correctly predicts the absolute energy loss scale fixed by the parent peak position at 29 eV (such scale agreement apparently could not be achieved in the simulations of Villette et al [2] based on thermal capture). Using the value of Ψ = 0.0233 in (42) and of the integral I_int = 0.61 indicated in figure 7 (corresponding to u = 0.54, i.e. the parameters  = 0.35, z₀ = 1.8 Å and z_im = 1.0 Å), the theoretical expression of equation (37) gives S₀ = 401 and hence the predicted Ne⁺ parent peak position S₀ħω = 28 eV, which is quite close to the observed value of 29 eV. As announced in section 4, the empirical values of both the coherent amplitudes α and β deduced from the experimental spectrum are therefore correctly given by our energy loss theory.

Successful predictions of other characteristics of the experimental ricochet spectrum will be made below and in section 7, which will further support the validity of the model.

One experimental observation in the paper by Villette et al [2] is the occurrence of a group of neutral particles Ne⁰ in the primary reflected flux from 0.7 to 1.3° grazing exit angles (see figure 2 in [2]). Such low angles are smaller than the main Ne⁰ reflected beam angle expected around 1.8°. This latter value is the normal behavior, pointed out in [2], given that (i) the Ne⁺ impacts the surface at an image-augmented incidence of Φ_im = 1.72° and (ii) the image interaction is turned off for outgoing Ne⁰, which therefore emerge at 1.72° [2]. We propose that the apparent low exit angle anomaly is a possible manifestation of the diffraction of Ne⁺ occurring at the original impact, as illustrated in figure 9, where the low-angle neutral flux is designated Ne⁰DN.

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The diffracted neutral beam Ne⁰DN emerges at the low takeoff angle φ at the first impact and at each of the successive ricochets at L_B, 2L_B etc Naturally all the exit directions are expected to be somewhat dispersed around the discrete values considered here if 'angular straggling' on surface phonons and defects does take place [2].

Consider now a skipping trajectory making m bounces of the same length L_B (figure 2) before escaping in the specular direction. The average energy loss after the mth bounce is now given by (22), i.e. by the Poissonian strength

$$\begin{eqnarray}&&{{S}_{m}}={{S}_{0}}(1+m{{\Phi }_{r}}),{{\Phi }_{r}}=\frac{\Psi }{\Psi }\approx 1.62.\end{eqnarray} \tag{ 44 }$$

Note that with the average energy losses S_m ħω all being several tens of eV, all Poissonian strengths S_m in (44) are very large (several hundred). Now we can replace a (normalized) Poisson distribution of large strength s ≫⃒ 1 by a single (normalized) Gaussian centered at sħω, using

$$\begin{eqnarray}&&{{e}^{-s}}\mathop \sum \limits_{n} \frac{{{s}^{n}}}{n!}\delta (E-n\hbar \omega )\cong \frac{1}{\hbar \omega \sqrt{2\pi s}}{{e}^{-{{(E-s\hbar \omega )}^{2}}/2s{{(\hbar \omega )}^{2}}}}.\end{eqnarray} \tag{ 45 }$$

Substituting for s the various average strengths (44) gives the successive Gaussian peak positions at E_m  = S_m ħω.

Knowing the peak positions, we can now write a compact formula for the complete Ne⁺ spectrum I(E) explicitly, taking account of the cumulative losses along the ion path:

$$\begin{eqnarray}&&I(E)={{e}^{-{{(E-{{S}_{0}}\hbar \omega )}^{2}}/2{{\Delta }_{par}}{{(\hbar \omega )}^{2}}}}+P\mathop \sum \limits_{m=1}^{N} \frac{{{p}^{m-1}}}{\sqrt{1+m{{\Phi }_{r}}}}{{e}^{-{{(E-{{S}_{m}}\hbar \omega )}^{2}}/2{{\Delta }_{m}}{{(\hbar \omega )}^{2}}}}.\end{eqnarray} \tag{ 46 }$$

The first term represents the Ne⁺ parent peak I₀ and the m > 0 terms are the echo peaks I_m . P gives the overall intensity of the Ne⁺ echo phenomenon relative to that of the parent peak. Each echo involves both the capture and the release scattering, so P depends, among other factors (such as escaping neutralization), on the square of the Ne⁺ probability of diffraction by the corrugated surface potential. The parameter p in (46) is the relative probability for the next bouncing to occur without diffraction or neutralization. The calculation from first principles of P, p, the neutralization probability p_n and likely interference effects between them is a complex quantum mechanical problem of combined scattering and charge transfer in the periodic image and Madelung surface potential. Clearly this problem lies outside the scope of the present work. However, ignoring interferences, P, p and p_n can be evaluated from figure 1: P = 0.080 is given by the observed ratio P/√(1 +  Φ_r) = Ι₁/I₀ = 1/20 of the first echo to the parent peak heights, p = 0.25 is obtained from the ratio I₂ /I₁  = 1/6 of the second echo to the first and p_n ≈ 1/3 is the ratio between the Ne⁺ and Ne⁰ parent peak heights. The non-uniform shifts I₀ – J₀ ≈ 11.5 eV and I₁ – J₁ ≈ 15 eV between the Ne⁺ and Ne⁰ spectra (figure 1) have been explained in section 5, so we need not write down a separate spectrum for the Ne⁰ particles. The Δ_m in (46) are Gaussian width parameters that we now explain.

Finally we discuss the widths of the successive peaks in the spectrum of Ne⁺ (those of Ne⁰ are comparable). The quantum mechanical width of the Gaussian in (45) is proportional to √s, so one expects the mth peak of the Ne⁺ spectrum in (46) to have a width proportional to √S_m . These peak positions and quantum widths are shown as broken red curves in figure 10. Clearly the quantum uncertainty on the excited number of FK phonons falls short of explaining the observed widths. However, it has been shown in [1] that the observed width γ₀ (FWHM) of the parent peak is much larger than the expected quantum width 2(2 ln 2 S₀)^1/2ħω  ≈ 3.47 eV (with S₀ ≈ 416, i.e. I₀ = 29 eV). This arises from a variety of effects such as inhomogeneous broadening, FK phonon resonance width, instrumental energy and angular resolutions [1]. In the work of Villette et al [2], the broad parent peak width is 12 eV (figure 10). If we call Δ_par the broadening of the parent peak one has γ_par = 2[2 ln 2(S₀ + Δ_par)]^1/2ħω  ≈ 2(2 ln 2Δ_par)^1/2ħω ≈ 12 eV, with Δ_par ≈ 5151 ≫⃒ S ₀ . Then the width γ_m of the mth peak reduces to

$$\begin{eqnarray}&&{{\gamma }_{m}}={\mbox{2}}\sqrt{2{\mbox{ln2}}}\sqrt{{{\Delta }_{m}}}\ \;\hbar \omega ={{\gamma }_{par}}\sqrt{(1+m{{\Phi }_{r}})},\quad {{\Delta }_{m}}={{\Delta }_{par}}\left( 1+m{{\Phi }_{r}} \right).\end{eqnarray} \tag{ 47 }$$

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The Δ_m are those appearing in (46). The widths calculated from (47) are then γ_par = 12 eV (measured), γ₁ = 20 eV, γ₂ = 25 eV, γ₃ = 30 eV, etc, in perfect agreement with the increasing widths (FWHMs) shown in figure 10.

The theoretical Ne⁺ energy loss spectrum as calculated from (46) and (47) is shown in figure 10 on the same log scale as the experimental data points. Most satisfactory is that the scale of energy losses based on diffraction capture coincides with the experimental energy scale. The present model fully accounts for the spectrum. Only the calculated fourth echo has a deficit in intensity as compared to the (somewhat diffused) experimental peak. This could be related to the slowing down of the ions which, this far into the skipping motion, have lost about 20% of their kinetic energy to FK phonon excitations. Such slowing down is likely to affect the probability parameters P and p, which were assumed unchanged in the present model.

Here we shall derive the energy levels of the Ne⁺ ion of mass M = 36 000 m_e in the image potential (7) modeled by (see figure 4 where the steep rise of V_im is replaced by a hard wall)

$$\begin{eqnarray}&&V(z)=\left\{ \begin{array}{ccccccccccccccc} -\frac{{{e}^{2}}\epsilon }{4z}=-\frac{{{e}^{2}}\alpha }{z} \\ \ \;\ \;\infty \quad \quad \ \;z<{{z}_{0}} \\ \end{array} \right.\begin{array}{ccccccccccccccc} z\geqslant {{z}_{0}}\quad \quad \quad \quad \quad \quad \\ \end{array}\end{eqnarray} \tag{ A1 }$$

where  = 1/3 and α = 1/12. For a heavy ion with very short de Broglie wavelength it is appropriate to use the WKB approximation [21–23], so the bound state energy levels are given by

$$\begin{eqnarray*}&&\phi \left( E \right)=\left( n+1/2 \right)\pi \end{eqnarray*}$$

where

$$\begin{eqnarray}&&\phi (E)=\mathop \int \limits_{{{z}_{0}}}^{{{z}_{1}}} dz\sqrt{\frac{2M}{{{\hbar }^{2}}}\left| V(z)-E \right|}\end{eqnarray} \tag{ A2 }$$

and the turning points are z₀ and z₁ = 1.133 Å/E_B (eV) = 6.2 Å.

From (A2) we obtain the implicit equation

$$\begin{eqnarray}&&\sqrt{\beta }f\left( \frac{E}{{{V}_{0}}} \right)=22.13\left( n+\frac{1}{2} \right)\sqrt{E\left( eV \right)}\end{eqnarray} \tag{ A3 }$$

where β = M/m, V = V(z) and

$$\begin{eqnarray}&&f(x)={\mbox{arc}}\;{\rm cos} \;\sqrt{x}-\sqrt{x(1-x)}.\end{eqnarray} \tag{ A4 }$$

If we expand f for small E/V₀ = z₀/z₁, we find

$$\begin{eqnarray}&&E(eV)=-\frac{3400}{{{(n+a)}^{2}}}\end{eqnarray} \tag{ A5 }$$

$$\begin{eqnarray}&&a=\frac{1}{2}+\frac{4}{\pi }\sqrt{\frac{\alpha \beta {{z}_{0}}}{2{{a}_{0}}}}\approx {\mkern 1mu} \ \;68\sqrt{{{z}_{0}}}=75.\end{eqnarray} \tag{ A6 }$$

Therefore the lowest bound state n = 0 is at E = −0.61 eV, slightly above V₀ = −0.63 eV. We shall be interested in the higher levels n + a ≫⃒ 1 and their spacing

$$\begin{eqnarray*}&&{{\delta }_{n}}={{E}_{n}}-{{E}_{n+1}}.\end{eqnarray*}$$

At low binding energies

$$\begin{eqnarray}&&\delta \approx \left( 6.8\times {{10}^{3}} \right)/{{\left( n+a \right)}^{3}}\;{\mbox{eV}}\end{eqnarray} \tag{ A7 }$$

This gives the level spacing in the meV range, e.g.

n = 0 12 meV

n = 109 (E = −0.1 eV) 5.2 meV

n = 145 (E = −0.07 eV) 0.6 meV.

This means that for low binding energies the ion levels form a continuum.