Article pubs.acs.org/JPCC
Second-Order Semiclassical Perturbation Theory for Diffractive Scattering from a Surface Eli Pollak*,† and S. Miret-Artés‡ †
Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel Instituto de Física Fundamental (IFF-CSIC), Department of Atomic, Molecular and Cluster Physics, Serrano 123, 28006 Madrid, Spain
‡
ABSTRACT: A second-order semiclassical perturbation theory is developed and applied to the elastic scattering of an atom from a corrugated surface. Analytical expressions for the diffraction pattern in the momentum space are obtained based on a sine corrugation function and a Morse potential for the interaction of the particle with the surface. The theory is implemented for a model of the in-plane scattering of Ar atoms from a LiF(100) surface. The resulting diffraction intensities are compared with second-order perturbation theory classical distributions and closecoupling results for two incident energies of 300 and 700 meV. The previous first-order perturbation theory predicts a symmetric diffraction pattern about the elastic peak, while the second-order semiclassical perturbation theory accounts correctly for the asymmetry in the diffraction pattern.
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INTRODUCTION The scattering of atoms from surfaces has been actively studied for the past 50 years.1−4 Old and new experimental results are still lacking a proper interpretation, in particular, extracting reliable potential parameters and surface properties such as corrugation amplitudes. When dealing with heavy atoms as probe particles, classical features are very prominent and can shed light on surface properties. Rainbow features5−7 in the final angular distributions are especially important because the angular distance between them is directly related to the surface corrugation. Their energy dependence is related to the physisorption well depth. These observations have also been corroborated in recent experiments on grazing collisions at high incident energies.11 Some reviews of rainbow scattering can be found in refs 8−10. A typical example is the scattering of Ar atoms from a LiF(100) surface. Detailed experimental results have been reported by Kondo et al.12 The angular distributions measured at fixed geometry are asymmetric in two senses. The intensity of the subspecular peak is larger than that of the superspecular peak, and the center of the distributions is shifted toward subspecular scattering angles. Kondo et al. analyzed their experimental results using Tully’s washboard model,13 fitted with energy-dependent corrugation height parameters. This system has been analyzed quite recently by our group14 using a second-order classical perturbation theory with respect to the corrugation height, which is considered to be the small parameter. By including a shallow physisorption well, whose depth was taken to be 88 meV (in reasonable agreement with quantum chemistry estimates15), the energy dependence of the rainbow angles as well as the asymmetry in the measured distribution was qualitatively well accounted for; there is no need to fit parameters for each energy separately. In this paper, only the elastic scattering from a corrugated surface was © XXXX American Chemical Society
considered. The second-order perturbation theory enabled us to obtain an analytic description of the classical elastic in-plane scattering dynamics, valid to second-order in the corrugation height. This was achieved by assuming a single sine function for the corrugation and an interaction potential given by a Morse function. The classical analysis is reasonable under the experimental conditions used by Kondo et al. The energy of their beam has a standard deviation of the order of 20%, implying that the coherence wavelength was substantially shorter than the lattice length, validating a classical description of the dynamics.16 Moreover, the experiments were carried out at room temperature, so that phonon scattering would obliterate any coherence. However, the same system probed at lower surface temperature and with a more collimated incident beam could reveal quantum diffractive scattering.16,17 The central theme of the present paper is to use the same ideas developed in the classical theory but in the context of semiclassical perturbation theory. Miller and Smith formulated a first-order semiclassical perturbation theory.18 Hubbard and Miller19 applied it to atom surface scattering using the same type of interaction potential as discussed in our classical theory. The expression they derived for the diffraction pattern has Bragg peaks whose intensities are determined by integer Bessel functions whose indices are the diffraction orders, and their arguments are proportional to the corrugation height. More recently, we noted that the arguments of the Bessel functions may be directly related to the energy-dependent classical rainbow Special Issue: Steven J. Sibener Festschrift Received: September 19, 2014 Revised: November 16, 2014
A
dx.doi.org/10.1021/jp509500v | J. Phys. Chem. C XXXX, XXX, XXX−XXX
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peaks.20 With suitable averaging, the semiclassical first-order distribution reduces to the classical distribution derived with a first-order perturbation theory. In their treatment, Miller and Hubbard set the prefactor in the semiclassical expression equal to unity. When considered in momentum space, the first-order diffraction spectrum is symmetric about the elastic channel. In angular space, the intensity distribution is asymmetric due to the Jacobian factor of cos θ between the horizontal momentum and final scattering angle distributions. This asymmetry is thus in a sense trivial. To really understand any asymmetry it is necessary to consider the distribution in the momentum space. The classical first-order distribution in momentum space is also symmetric about the elastic zero momentum transfer position; it is thus not surprising that the semiclassical theory displays the same symmetry. However, measured distributions are not symmetric,12 even when considered in momentum space. This motivates the search for a more accurate theory that would explain and characterize the asymmetry. We present a second-order semiclassical perturbation theory, which complements our previous second-order classical treatment. The resulting semiclassical diffraction pattern is no longer symmetric about the elastic channel and thus differs qualitatively from the first-order result. Quantitatively, the differences are not huge, at least as long as the corrugation height is not too large. To implement the theory, we used the same model as before, that is, a Morse potential for the interaction and a sine corrugation function. This choice allows for the derivation of analytic expressions. However, the diffraction peaks are now expressed in terms of sums of products of Anger functions and Bessel functions. The present theory includes the prefactor to first order in the corrugation height. The second order contribution to the prefactor becomes very involved, and the indications from the first order theory are that it does not change much the diffraction pattern. The semiclassical results are compared with both the classical second-order distribution as well as numerically exact solution of the quantum close-coupling equations.21 We show that the second-order classical theory agrees quantitatively with numerically exact classical results. Agreement between the secondorder semiclassical theory and the numerically exact quantum results is reasonable, though not quantitative. The paper is organized as follows. In Section II, we briefly review the second-order classical theory and then apply it to derive the second-order semiclassical theory for the transition amplitudes and the corresponding diffraction patterns in the momentum space. In Section III, the theory is applied to a model for the diffractive scattering of Ar atoms on a LiF(100) surface by assuming a Morse potential model and a sine corrugation function. Some of the technical details are relegated to two appendices. Numerical results are presented in Section IV. We end with a discussion of the results as well as further possible extensions to include also interactions with thermal surfaces.
H=
px2 + pz2
+ V̅ (z) + V̅ ′(z)h(x) +
2M
1 V̅ ″(z)h2(x) 2 (2.1)
where the interaction potential is split into three terms: (i) V̅ (z) gives the zeroth order term (for analytic purposes it will be taken as a Morse potential), and (ii) the dependence on the corrugation height function h(x) is obtained from an expansion of the potential V̅ (z + h(x)) up to second order in the corrugation h(x), which is considered to be small. The corrugation function h(x) is assumed to be periodic with period l, and the perturbation theory is established with respect to the corrugation amplitude. The particle is initiated at the time −t0 with initial vertical (negative) momentum pzi and (positive) horizontal momentum pxi. To zeroth order in the corrugation, at t = 0 the particle impacts the surface. We are then interested in the final momenta of the particle at the time +t0, which is taken to be sufficiently large to ensure that the scattering event is over. In the analytical formalism developed below, we then take the limit t0 → ∞. The exact equations of motion for the horizontal and vertical coordinates are Mxẗ = −V̅ ′(zt )h′(xt ) − V̅ ″(zt )h′(xt )h(xt ) Mzẗ = −V̅ ′(zt ) − V̅ ″(zt )h(xt ) −
1 V̅ ″′(zt )h2(xt ) 2
(2.2)
(2.3)
The coordinates and momenta are then expanded according to ∞
rt =
∞
∑ rt ,j , j=0
pr = t
∑ pr ,j ; j=0
t
r = x, z (2.4)
where the j subscript denotes the corresponding power in the corrugation strength. This leads to the following equations of motion for the zeroth, first-order, and second-order contributions
Mxẗ ,0 = 0
(2.5)
Mzẗ ,0 = −V̅ ′(zt ,0)
(2.6)
Mxẗ ,1 = −V̅ ′(zt ,0)h′(xt ,0)
(2.7)
Mzẗ ,1 = −V̅ ″(zt ,0)[h(xt ) + zt ,1]
(2.8)
Mxẗ ,2 = −V̅ ″(zt ,0)h′(xt ,0)zt ,1 − V̅ ′(zt ,0)h″(xt ,0)xt ,1 − V̅ ″(zt ,0)h(xt ,0)h′(xt ,0)
(2.9)
The corresponding second-order equation for the vertical coordinate is not written explicitly because it will not be needed. These equations then imply the zeroth order solutions for the horizontal direction: px −t0 (t + t0) xt ,0 = x−t0 + (2.10) M
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SECOND-ORDER SEMICLASSICAL THEORY Classical Perturbation Theory. The dynamics is modeled in terms of two degrees of freedom, a vertical coordinate z (with conjugate momentum pz) describing the distance of the atom from the surface, and a horizontal coordinate x (with conjugate momentum px) for motion parallel to the surface. The Hamiltonian for an atom of mass M colliding with a corrugated surface is written as
px = px t ,0
−t0
(2.11)
and we assume that the solution for the zeroth order vertical equation of motion (eq 2.6) is also known. The first-order horizontal coordinate and momentum are then readily seen to be B
dx.doi.org/10.1021/jp509500v | J. Phys. Chem. C XXXX, XXX, XXX−XXX
The Journal of Physical Chemistry C t
px ,1 = − t
≡−
∫−t
dt ′ V̅ ′(zt ′ ,0)h′(xt ′ ,0) = − 0
∂ ∂x −t0
Article t
∫−t
dt ′ V̅ ′(zt ′ ,0)h(xt ′ ,0)
Wk =
0
∂A1(x −t0 , pxi , pzi ) ∂x −t0 t′
∫−t dt′ ∫−t dt″ V̅ ′(zt″,0)h′(xt″,0) 0
0
−t0
pz
t
t
−
t ,0
pz
(2.19)
∫−t dt′ [xtpẋ 0
t
+ zt pż ]
(2.21)
t
We also note the general relationship derived by Miller23 between the action and the final horizontal momentum, which is that for times t sufficiently long, such that the collision is over and the particle is again in the asymptotic (free particle) region ∂ϕt ∂x−t0
∂px t = −xt̅ ∂x−t0
(2.22)
In the asymptotic region, the final momenta are constant so that the time derivative of xt̅ vanishes, and it is therefore a constant. It then follows from eqs 2.12 and 2.16 that after the collision is over
(2.15)
and one may readily see by inspection that eq 2.14 is a solution of eq 2.8. With these results, one may also solve the secondorder horizontal equation of motion 2.9
∂ϕt ∂x−t0
=
∂ϕt ∂x ̅
= xt̅
∂ 2[A1(x−t0 , pxi , pzi ) + A 2 (x−t0 , pxi , pzi )] ∂x ̅ 2 (2.23)
t
px ,2 = − t
∫−t dt′ [V̅ ″(zt′,0)h′(xt′,0)zt′,1
where we introduced the notation
0
x ̅ ≡ xt̅ ,0
+ V̅ ′(zt ′ ,0)h″(xt ′ ,0)xt ′ ,1 + V̅ ″(zt ′ ,0)h(xt ′ ,0)h′(xt ′ ,0)] ≡−
∂x−t0
∂x ̅ =1 ∂x−t0
(2.16)
and here, too, we have defined the (second-order) action function A2(x−t0,pxi,pzi) in anticipation of the semiclassical theory. Semiclassical Perturbation Theory. The exact quantum final momentum distribution for an initial state characterized by the initial momenta pxi, pzi and ending with the final momenta pxf, pzf with amplitude Wk for a transition to the kth Bragg channel is
∑
xt̅ ≃ x ̅ + x1̅ (x ̅ )
x1̅ (x ̅ ) ≡ xt ,1 +
δ[pzf − pz (pzi )]δ(pxf − pxi − αk)|Wk|2 t0
pxi pzi
zt ,1 +
⎛ p2 ⎞ zt ,0⎜⎜1 + xi2 ⎟⎟ pzi pzi ⎠ ⎝
px ,1 t
(2.27)
This implies that to second order in the corrugation the partial derivative of the action function with respect to x̅ is
(2.17)
2π ℏ k l
(2.26)
with
where in the argument of the second Delta function we have used the notation αk =
(2.25)
Expanding the coordinate xt̅ to first order in the corrugation we have that
∞ k =−∞
(2.24)
and noted that
∂A 2 (x−t0 , pxi , pzi )
P(pxf , pzf ; pxi , pzi ) =
(2.20)
t
ϕt = −
(zt ,1 + h(xt ,0))
t ,0
∂x−t0
The action function is
(2.14)
pz ,1 = −
1/2
zt
∫
MV̅ ′(zt ,0)
dx −t 0
where the coordinate xt̅ is defined to be px xt̅ = xt − t zt p
(2.13)
where the (first-order) action function A1(x−t0,pxi,pzi) is defined in anticipation of the semiclassical theory, as described below. The solution of the first-order equation of motion for the vertical position is more involved. In principle, eq 2.8 is that of a parametrically driven, forced oscillator, and so not easily solved. It is this stumbling block that we believe has prevented extension of the perturbation theory beyond the first-order solution. In recent work we have shown14,22 how this equation may be solved using the conservation of energy to reduce the equation to a first order in time differential equation, which is then readily solved. The resulting solutions are px px ,1 ⎞ t 1 ⎛ −t0 t ′ ⎟ dt ′ 2 ⎜V̅ ′(zt ′ ,0)h(xt ′ ,0) + zt ,1 = −pz t ,0 M ⎠ −t 0 pz ⎝ t ′ ,0 px px ,1
∫0
∂xt̅ 0
l
⎛i ⎞ exp⎜ [ϕt + xt̅ 0(px − pxf )]⎟ t0 ⎝ℏ 0 ⎠
(2.12)
t
1 xt ,1 = − M
1 l
∂ϕt
≃ x̅ ∂x ̅
∂ 2[A1(x ̅ , pxi , pzi ) + A 2 (x ̅ , pxi , pzi )]
∂x ̅ 2 ∂ 2A1(x ̅ , pxi , pzi ) + x1̅ (x ̅ ) ∂x ̅ 2
(2.18)
to express the Bragg condition and k is referred to as the Bragg index. The final angular distribution is obtained by integrating the momentum probability over all final momenta subject to the condition that θf = tan−1(pxf/pzf). A semiclassical expression for the transition amplitude Wk has been derived by Miller, Smith, and Hubbard18,19
(2.28)
This equation is readily solved for the action, and one finds that to second order in the corrugation the phase appearing in the semiclassical expression for the transition amplitude (eq 2.19) is C
dx.doi.org/10.1021/jp509500v | J. Phys. Chem. C XXXX, XXX, XXX−XXX
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⎛ px ⎞ ϕt + ⎜⎜xt0 − t z t0⎟⎟(px − pxf ) ≃ ϕt ,0 + x ̅ (pxi − pxf ) 0 pz ⎠ t0 ⎝ t −
∫
x̅
dx ̅ ′ x1̅ (x ̅ ′)
∂px
t0
px ,1 = −M[cos(χ )Fc(t ) − sin(χ )Fs(t )] pz ,0 t
zt ,1 = −
(x ′) ,1 ̅
∂x ̅ ′
M
(2.29)
2πh lM
∫−∞ dt′ V̅ ′(z0,t′) cos(ωxt′)
Fs(t ) =
2πh lM
∫−∞ dt′ V̅ ′(z0,t′) sin(ωxt′)
Gc(t ) =
hM pz2 ,0
∫−∞ dt′
hM pz2 ,0
∫−∞ dt′
Gs(t ) =
(3.7)
t
t
dV̅ ′(z 0, t ′) dt ′
t
dV̅ ′(z 0, t ′) dt ′
(3.8)
cos(ωxt ′) (3.9)
sin(ωxt ′)
t
(3.10)
and
2πpxi
ωx ≡
(3.11)
Ml
At long time, this implies the central relation between the second-order final horizontal momentum and the impact parameter px
t0 ,2
= 2Pc + Ps sin(2χ )
(3.12)
with
SECOND-ORDER SEMICLASSICAL THEORY FOR A SINE CORRUGATION FUNCTION Sine Corrugation Function. In this section, we will specify the corrugation function to be a single sine term
⎡ ⎤ pz2 ,0 1 t ⎢ dt Fc(t )Fs(t ) − 2 Gc(t )Gs(t )⎥ ≡ pxi Kcc ⎢ ⎥ 2 −t 0 M ⎣ ⎦
πM Pc = l
∫
t0
(3.13) Ps = −
(3.1)
It is useful to define a dimensionless horizontal coordinate variable
−
πM l
πh2 l
⎛ p2 [G (t )G (− t ) + G2(t )] ⎞ s s c z ,0 + Fc(t )Fc(− t ) + Fs2(t )⎟ dt ⎜ t 2 ⎜ ⎟ −t0 M ⎝ ⎠ t0 pxi dt V̅ ″(z 0, t ) cos(2ωxt ) ≡ (K sc + Vsc,2) (3.14) −t0 2
∫
t0
∫
where we note the relationship of these results to the constants Kcc, Ksc, and Vsc,2 defined in ref 14. From these results, one finds that the action functions A1 and A2 (eqs 2.12 and 2.16) can be expressed as
(3.2)
so that χ is the (reduced) value of the horizontal coordinate when according to the zeroth order dynamics the particle hits the zeroth order vertical turning point (t = 0). As shown in ref 14 for this specific choice of the corrugation function, one readily finds the following perturbation theory solutions for the coordinates and momenta t
∫−t dt′ Fc(t′) + sin(χ ) ∫−t dt′Fs(t′) 0
t
Fc(t ) =
t
■
2π x0 l
t
and we used the notations
The first-order contribution to the prefactor necessitates a solution for the time dependence of the first-order contribution to the vertical coordinate. This solution was not available previously; it was considered to be small and so ignored. The second-order contribution to the prefactor necessitates the solution for the second-order contribution to the vertical coordinate. Although, in principle, this solution may be obtained by using the second-order contribution to energy conservation and thus deriving a first order in time equation of motion for zt,2, this does become quite involved in practice, so that we limit ourselves in this paper only to the first-order contribution to the prefactor. In summary, we have derived explicit expressions for the semiclassical transition amplitude, with the phase expanded to second order in the corrugation. This solution is rather general, in the sense that the interaction potential V̅ (z) and the corrugation function h(x) have not yet been specifically defined. This will be carried out in the next Section.
⎛ 2π ⎞ h(x−t0) = h sin⎜ x−t0⎟ ⎝ l ⎠
zt ,1 − pz ,0 [sin(χ )Gc(t ) + cos(χ )Gs(t )]
t ,0
(3.6)
(2.30)
xt ,1 = −cos(χ )
0
(3.5)
pz
t
p ∂zt ,1 p2 ⎞ ∂p ∂xt ,1 zt ,0 ⎛ ∂(xt̅ ) ⎜1 + xi ⎟ xt ,1 =1+ + xi + ∂x−t0 ∂x−t0 pzi ∂x−t0 pzi ⎜⎝ pzi2 ⎟⎠ ∂x−t0
t
t
∫−t dt′ Gc(t′) + cos(χ ) ∫−t dt′ 0
MV ′(zt ,0)
pz ,1 = −
The zeroth order action is independent of the horizontal coordinate so it just gives a phase that disappears when taking the absolute value of the matrix element and is not important. One notes that this expression, to first order in the corrugation, is identical to the first-order theory derived by Hubbard and Miller.19 It is also of interest to consider the prefactor. To first order in the corrugation one finds that
χ=
t
[sin(χ )
Gs(t ′)]
− [A1(x ̅ , pxi , pzi ) + A 2 (x ̅ , pxi , pzi )]
(3.4)
t
0
A1(x−t0 , pxi , pzi ) =
Ml sin(χ )Fc(t0) 2π
(3.15)
A 2 (x−t0 , pxi , pzi ) =
lPs lP cos(2χ ) − c χ π 4π
(3.16)
One then also finds that at long times
(3.3) D
dx.doi.org/10.1021/jp509500v | J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Article
Inclusion of the first-order correction to the prefactor does not change the symmetry property of the first-order solution, that is, that |Wk,1|2 = |W−k,1|2. This is due to the fact that in the firstorder theory the Bessel functions contributing to the diffraction pattern are of integer order. In the second-order theory, the order is a real number so that the symmetry condition is no longer correct, in general. Analytic Theory for a Morse Potential Model. To complete the theory, one needs to obtain explicit expressions for the parameters Ax, Rc, Pc, and X1. These may be obtained analytically if one specifies the interaction potential to have the Morse form
p ⎤ ⎡ dt ′⎢Fc(t ′) − xi Gs(t ′)⎥ ⎦ ⎣ −t 0 M
∫
t
p 2 ⎞⎤ zt ,0 ⎛ ⎜1 + xi ⎟⎥ ≡ −cos(χ )X1 pzi ⎜⎝ pzi2 ⎟⎠⎥⎦
(3.17)
so that
∫
x̅
∂px
dx ̅ ′x1̅ (x ̅ ′)
t0 ,1
(x ̅ ′)
∂x ̅ ′
=
M Fc(t0)X1 cos(2χ ) 4
(3.18)
Putting this all together, using the notation
VM̅ (z) = V0[(exp( −αz) − 1)2 − 1]
Ml Fc(t0) Ax = 2π
Rc =
which is defined through the physisorption well depth V0 and the stiffness parameter α. The trajectory for the Morse potential at an incident energy Ez (or negative incident momentum pzi) is known analytically14
(3.19)
lPs MFc(t0)X1 + 4π 4
(3.20)
and the Bragg condition
exp(αzt ,0) = −
pxf = pxi + αk
(3.21)
1 |Wk| = 2π +
∫0
2π
Ω2 =
⎛ ⎛ ⎡ ⎤ lP ⎞ π dχ ⎢1 + X1 sin(χ )⎥ exp⎜− iχ ⎜k − c ⎟ ⎣ ⎦ ⎝ πℏ ⎠ l ⎝
⎞ i [− A x sin(χ ) − R c cos(2χ )]⎟ ℏ ⎠
2α 2Ez M
(3.28)
and cos Φ = −
⎡ V (E + V ) ⎤ lim (αzt ,0 − Ωt ) = ln⎢ 0 z 2 0 ⎥ t →∞ ⎢⎣ ⎥⎦ 4Ez
⎛ Ax ⎞ ⎛ R ⎞⎡ ⎜ ⎟ ( −i)m Jm ⎜ c ⎟⎢Jν ( k − 2 m ) ⎝ ⎠ ⎝ ⎠ ℏ ℏ ⎣ m =−∞ ∞
∑
⎛ Ax ⎞ ⎛ A x ⎞⎞⎤ πX1 ⎛ ⎜ ⎟ − J ⎜ ⎟⎟⎥ + ⎜Jν ν(k + 1 − 2m) ⎝ 2il ⎝ (k −1−2m) ⎝ ℏ ⎠ ℏ ⎠⎠⎦
V0 Ez + V0
(3.29)
This implies that at asymptotic times
The integration over the impact parameter can be carried out analytically (see Appendix A), and one finds the central result of this paper
(3.30)
As shown in Appendix B, using the notation 2
Ω̅ = (3.23)
lPc πℏ
Fc(∞) =
(3.24)
It is the second-order constant shift (Pc) in the final horizontal momentum (eq 3.12) that imposes the use of the Anger functions. It is this shift that is also responsible for the asymmetry in position of the center of the diffraction pattern. We also note in Appendix A that eq 3.23 will obey unitarity, provided that the corrugation is such that the perturbation theory treatment is valid. For the sake of comparison, it is also useful to write down the first-order expression for the matrix element, including the prefactor, expanded to first order ⎛ A ⎞⎤ ⎛ A ⎞ iπX1 ⎡ ⎛ Ax ⎞ Jk − 1⎜ ⎟ − Jk + 1⎜ x ⎟⎥ |Wk ,1|2 = Jk ⎜ x ⎟ − ⎢ ⎝ ℏ ⎠⎦ ⎝ℏ⎠ 2l ⎣ ⎝ ℏ ⎠
ωx 2π = |tan θi| αl Ω
(3.31)
where θi is the (negative) angle of incidence, one finds the following analytical results
where Jν(x) is the Anger function of order ν.24−26 We also used the notation ν(k) ≡ k −
(3.27)
2
(3.22)
|Wk|2 =
cos Φ [cosh(Ωt ) + cos Φ] sin 2 Φ
with
leads to the central result for the square of the absolute value of the transition amplitude (where we included the first-order contribution to the prefactor) 2
(3.26)
4π 2hpzi Ω̅ cosh(ΦΩ̅ ) sinh(π Ω̅ ) Ml
Ax = 2πhpzi
Pc =
Ω̅ cosh(ΦΩ̅ ) sinh(π Ω̅ )
(3.32)
(3.33)
M2Fc2(∞) tan 2|θi| 4pxi ⎛ cos2|θ | + Ω̅ Φ tanh(Ω̅ Φ) − π Ω̅ coth(π Ω̅ ) Ω̅ i ⎜ − 2 2 sin |θi| ⎝
2
⎞ tanh(Ω̅ Φ) sin 2Φ⎟ ⎠
(3.25) E
(3.34)
dx.doi.org/10.1021/jp509500v | J. Phys. Chem. C XXXX, XXX, XXX−XXX
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⎧ ⎛ p2 ⎞ ⎡ −cos Φ ⎤ ⎪ M Fc(∞)⎨⎜⎜1 + xi2 ⎟⎟ ln⎢ ⎥ ⎪ αpzi pzi ⎠ ⎣ 2sin 2 Φ ⎦ ⎩⎝ +
⎫ ⎪ 2 ⎬ + Φ [1 cos ] 2 ⎪ pzi ⎭
For 300 (700) meV, this constant shift is −4.178 au (−4.418 au). In Figure 2, the final horizontal momentum distribution is plotted as a function of the Bragg index k for the two incident energies. Blue bars with asterisks show the results for the second-order theory, and green bars with full circles are the first-order semiclassical results. In both cases, we included the first-order correction for the prefactor. For 300 meV, the prefactor makes a negligible contribution because the parameter X1 = −0.089 au is very small. At 700 meV, X1 = −0.912 au, and this leads to a contribution of at most a 20% change in the intensity of a given peak. As already mentioned, the prefactor correction is not very big in any case, justifying and limiting its computation only to the first order. (See also its small impact on the normalization, as given in Appendix A.) As expected already from the classical results, important differences are observed. Because of energy conservation, diffraction peaks only up to k = 14 (very near to a final angle of 90°) are observed for the lower energy, and k = 21 for the higher energy. Specular diffraction corresponds to k = 0. There are two central differences between the first- and second-order distributions. As already shown in the previous section and readily observed in the two panels, the first-order distributions are symmetric about k = 0. The second-order distributions are shifted downward by k ≃ −5 at both energies. Neither of them has any symmetry property. Finally, in Figure 3 we compare the second-order diffraction pattern (blue bars with asterisks) with the numerically exact quantum results (red bars with open squares) obtained using a close coupling computation and the second-order classical final momentum distribution (brown continuous line) broadened with a Gaussian whose variance in atomic units is 0.4. Several features are to be noted. First, the classical distribution is symmetric, but about a shifted value of k = −5.03 and k = −5.32 for the low and high energies, respectively. This asymmetry was masked in the angular distributions presented in ref 14 due to the cos θf Jacobian factor. Second, the centers of the semiclassical and quantum distributions are shifted by approximately the same amount. Third, all three distributions have rainbow maxima, which are quite close to each other. The first-order perturbation theory is inferior when considering these last two properties. A fourth observation is that the second-order semiclassical theory accounts rather well for the intensities in the classically forbidden region, that is, outside of the rainbow angles. Finally, the semiclassical results, even at second-order, are not able to quantitatively reproduce the numerically exact quantum intensities in the region between the two rainbow peaks.
pxi2
32π 3h2 cosh(2ΦΩ̅ ) sinh(2π Ω̅ ) l2 2 2 sinh(2ΦΩ̅ ) 4π h α V0 tan Φ cot θi − sinh(2π Ω̅ ) l Ez
(3.35)
Vsc,2 = −
(3.36)
and the rather lengthy expression for Ps is given in eq B.19 in Appendix B.
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NUMERICAL APPLICATION To test the second-order theory and obtain some feeling for the results, we study a model that is applicable for the in-plane scattering of an Ar atom from a LiF(100) surface, a system that has been investigated in detail, both experimentally12 as well as theoretically.14,15 The parameters used previously to fit the experimental results are h = 0.25 au, l = 4 A, αl = 3, and V0 = 88 meV. Here we report results for two incident energies, 300 and 700 meV, and an angle of incidence θi = −45°. To understand the extent of validity of the classical secondorder perturbation theory, we first compare the classical results for the final horizontal momentum with a numerical simulation. In Figure 1, we show the second-order classical final horizontal
Figure 1. Quality of classical perturbation theory. The second-order classical final horizontal momentum (red dashed line) is compared with the first order (blue dashed-dotted line) and numerically exact classical trajectory (solid squares) results as a function of the reduced impact parameter (eq 3.2), which changes from 0 to 2π. The left panel shows the results for an incidence energy of 300 meV (incident horizontal momentum pxi = 28.43 au), and the right panel shows the results for 700 meV (pxi = 43.43 au). The angle of incidence is −45°. Momenta are given in atomic units.
■
DISCUSSION AND CONCLUSIONS The central result of this paper is a second-order perturbation theory solution for the semiclassical amplitude Wk for a transition to the kth Bragg channel in an elastic collision of a particle with a periodically corrugated surface. This generalizes our previous classical second-order perturbation theory to the realm of semiclassics. As in the classical case, the second-order term induces an asymmetry in the diffraction pattern, shifting the center of the distribution in the subspecular direction. In contrast, the first-order semiclassical theory predicts a diffraction pattern that is symmetric with respect to reflection of the Bragg index. The asymmetry becomes especially evident when plotting the distribution as a function of the final horizontal momentum, rather than the final angle. The Jacobian
momentum (red dashed line), the first order (blue dasheddotted line), and the numerically exact classical trajectory result (solution of eqs 2.2 and 2.3, solid squares) as a function of the reduced impact parameter (eq 3.2), which changes from 0 to 2π, for the two incident energies. The first-order results are only in qualitative agreement with the numerical results. The second-order results are in quantitative agreement, the central difference between the first-order and second-order results is the constant shift coming from the second-order perturbation theory, toward lower values of the final horizontal momentum. F
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Figure 2. Comparison of the first (green bars with full circles) and second-order (blue bars with asterisks) final horizontal momentum distributions. The left panel is for an incidence energy of 300 meV, and the right panel is for 700 meV. Negative (positive) values of the Bragg index imply a decrease (increase) in the final horizontal momentum relative to its initial value. There is a noticeable difference between the first- and second-order distributions, implying that first-order perturbation theory is insufficient.
Figure 3. Comparison of different estimates for the final horizontal momentum distribution (in terms of the Bragg index k). The left (right) panel shows the distributions for an incident energy of 300 (700) meV. Blue bars with asterisks and red bars with open squares are the second-order semiclassical results and those obtained from a close-coupling calculation, respectively. For comparison, we have also plotted the classical distribution (brown continuous line), broadened with a Gaussian whose variance in atomic units is 0.4.
agreement is not quantitative for the individual transition probabilities. The second-order theory presented in this paper can be extended in a few directions. In the analytical development, we assumed that the corrugation consists of a single sine function. One may introduce higher harmonics; this would complicate the analytics a bit but would not pose a fundamental difficulty. We limited ourselves to model in-plane (z, x) scattering, that is, to two degrees of freedom. In reality, the in-plane scattering always occurs in the three dimensions (z, x, y). Even if the interaction in the two surface directions (x, y) is separable, the first-order dynamics introduces a coupling between the two horizontal degrees of freedom. This would complicate the resulting expressions; however, fundamentally, one may derive explicit expressions for the distribution. Is this important? On the fly classical trajectory computations indicate that even for in-plane scattering the third degree of freedom cannot be ignored,15 and one should expect it to introduce significant changes also in the predicted semiclassical distribution. A third direction is to introduce coupling to the surface phonons. We have done this for the first-order theory20 using also a first-order perturbation theory with respect to the coupling strength to the phonons. In particular, for the system considered here, a comparison with the available experimental data was previously carried out10 but keeping first-order
of the transformation from the horizontal momentum to the final angle introduces a trivial cos (θf) asymmetry, which masks the real asymmetry due to the dynamics of the collision. The theory presented gives an explicit expression for the first-order contribution to the prefactor appearing in the semiclassical transition amplitude expression. In principle, one may also obtain results for the second-order contribution; however, this becomes rather lengthy and tedious. For the model case of scattering of an Ar atom on the LiF(100) surface considered in this paper, the numerical indications are that the second-order contribution to the prefactor would change probabilities by at most a few percent, so that it can be safely neglected. Two tests of the second-order theory are presented. First, we compared the classical second-order theory with numerically exact classical trajectories and showed that the classical perturbation theory is quantitative for the parameters studied. Second, we compared the second-order semiclassical diffraction pattern with numerically exact close coupling computations. The qualitative form of the two is the same: both distributions have a center (in k space) that is shifted toward subspecular angles, and both give peaks in the vicinity of the classical rainbow region. (The first-order semiclassical prediction is qualitatively different.) Both predict a non-negligible amount of scattering in the classically disallowed region. However, the G
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perturbation theory in the corrugation amplitude and linear coupling terms to the surface modes for both the vertical and horizontal coordinates. The straightforward generalization of the present theory is to derive a second-order theory with respect to the corrugation and a first-order theory with respect to the coupling to the phonons. Should one go also to second order in the phonon coupling strength? Is this practically of interest? We do not know, but one could use the approach given in this paper for scattering from a thermal uncorrugated surface. This might also shed more light on the difference between classical and semiclassical energy transfer with thermal surfaces. Finally, it is worth mentioning that the LiF surface has been chosen here just for comparative purposes to our previous work. However, it would be interesting to extend this theoretical formalism to many other surfaces such as graphene overlayers, semimetals, or various nanostructures actually being investigated by He atom scattering.27 The only requirement is to have a ratio between the corrugation amplitude and the unit cell length, which is substantially smaller than one.
■
∞
1
∑k =−∞ Wk 2 =
∫ dχ∫0 dχ ′⎡⎣1 + πl X1 sin(χ )⎤⎦⎡⎣1 + πl X1 sin(χ ′)⎤⎦ 2π
((
∞
lPc πℏ
· ∑k =−∞ exp i k + ·exp =
2π
4π 2 0
)(χ′ − χ ))
( ℏi [−Ax(sin(χ ) − sin(χ′)) − Rc[cos(2χ ) − cos(2χ′)]])
1 2π
⎛
⎞
∫0 dχ ∫0 dχ ′δ⎝⎜χ ′ − χ ⎠⎟⎡⎣1 + πl X1sin(χ )⎤⎦ 2π
=1+
2π
2
2
π X2 2l 2 1
(A.5)
The distribution is normalized to unity up to the second order in the corrugation. Moreover, the magnitude of (π2/(2l2))X21 gives a good indication as to whether higher order terms in the prefactor are important. For the two cases studied here, one finds that for E = 300 and 700 meV, (π2/(2l2))X21 = 0.0007 and 0.072, respectively.
■
APPENDIX B. INTERMEDIATE STEPS FOR THE MORSE POTENTIAL To obtain analytic results for the Morse potential, using the known solution for the classical trajectory as given in eq 3.27, we note that 1 α 2 d(Ωt − X(t )) = 2 3 2 dt pz , t MΩ
(B.1)
0
APPENDIX A. DIFFRACTION PATTERN FOR A SINE CORRUGATION FUNCTION
The transition amplitude Wk can be written in terms of products of Anger and Bessel functions by noting that26 ⎛ ⎞ iA dχ exp⎜ −iνχ − x sin(χ )⎟ = ⎝ ⎠ ℏ π ⎛ ⎞ iA exp( −iνπ ) dχ exp⎜ −iνχ + x sin(χ )⎟ = ⎝ ⎠ −π 2π ℏ ⎛A ⎞ exp( −iπν)Jν ⎜ x ⎟ ⎝ℏ⎠
1 2π
∫0
I1 =
In =
(A.1)
(A.2)
J1 =
=
∂In − 1 1 (n − 1) sin Φ ∂Φ
∫0
∞
⎡ ⎤ 2Ω̅ σ0 sin(Ω̅ t ) dt ⎢ ⎥=− ⎣ cosh(t ) + cos Φ ⎦ sin(Φ)
(B.5)
with
lPc πℏ
∞
(A.3)
σ0 =
we find from eq 3.22 that
∑ (−1)k k=1
m
⎛A ⎞ ⎛ A ⎞⎞⎤ πX1 ⎛ ⎜Jν − 2m − 1 ⎜ x ⎟ − Jν − 2m + 1 ⎜ x ⎟⎟⎥ k ⎝ℏ⎠ ⎝ ℏ ⎠⎠⎦ 2il ⎝ k
sin(k Φ) 2
(Ω̅ + k 2)
(B.6)
Here, too, one notes the recursion relation
⎛A ⎞ ⎛ R ⎞⎡ |Wk| = ∑ ( −i) Jm ⎜ c ⎟⎢Jν − 2m ⎜ x ⎟ k ⎝ℏ⎠ ⎝ ⎠ ℏ ⎣ m =−∞ ∞
+
∞
Ω̅ t ) ∫−∞ dt [cosh(cos( t ) + cosΦ]n
1 2π
one finds after some lengthy algebra the results given in eqs 3.32−3.36. The result for Ps is slightly more involved. First, one notes the following integral25
where Jl(z) denotes the Bessel function of integer order l. Using the notation νk = k −
cos(Ω̅ t ) sinh(ΦΩ̅ ) = cosh(t ) + cos Φ sin Φ sinh(π Ω̅ )
(B.4)
∞
l =−∞
∫−∞ dt
and the recursion relation
where Jν(x) is the Anger function. We then use the identity24 Jl (z)i l exp(ilθ )
∞
1 2π
(B.3)
∫
∑
(B.2)
Then, systematically using the known integral
2π
exp[iz cos θ ] =
cosh(Ωt )[1 + cos 2 Φ] + 2 cos Φ sinh(Ωt )
X (t ) =
Jn =
∫0
∞
dt
∂Jn − 1 sin(Ω̅ t ) 1 n = [cosh(t ) + cosΦ] (n − 1)sin Φ ∂Φ (B.7)
Using the notation
(A.4)
∞
The diffraction intensities are then normalized as seen from the following
σ1 =
∑ (−1)k k=1
H
k cos(k Φ) 2
2
(Ω̅ + k )
=
∂σ0 ∂Φ
(B.8)
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σ2 =
Article
2
∑ (−1)k
2Ω̅ k cos(k Φ)
lim Fc(∞) = Khw
2
(Ω̅ + k 2)2
k=1
Φ→ 0
(B.9)
the identity ∞
∑ (−1)k sin(k Φ) = k=1
lim Fs(∞) = Khw
−sin(Φ /2) 2 cos(Φ/2)
Φ→ 0
(B.10)
∫0
∞
dt ″
+ (B.11)
Ps,G
t0
πM =− l
Ps,F = −
πM l
∫−t
dt
Ω, Φ→ 0
t
dt (Fc(t )Fc( −t ) + Fs2(t ))
0
=
(B.13)
■
(B.14)
Gc(t ) = Gc( −t )
2αV0 tan 2 Φ [1 + cos Φ]
∫0
⎡ π Ω̅ cosh(π Ω̅ ) ⎤ 1 − ⎢ ⎥ 4 sinh(π Ω̅ ) ⎣ sinh(π Ω̅ ) cosh(π Ω̅ ) ⎦
Ω→ 0
2 5πpxi Khw
24
⎡ π Ω̅ cosh(π Ω̅ ) ⎤ 1 − ⎢ ⎥ 4 sinh(π Ω̅ ) ⎣ sinh(π Ω̅ ) cosh(π Ω̅ ) ⎦ 2 πpxi Khw
sinh(π Ω̅ ) → 0
(B.23)
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work has been supported by grants of the Israel Science Foundation, the German-Israel Foundation for Basic Research, the Minerva Foundation, the Einstein center at the Weizmann Institute of Science, and the Ministerio de Economia y Competitividad under Project No. FIS2011-29596−C02-C01. We also acknowledge support from the COST Action MP1006.
(B.16)
and the notation: 2πh lM
(B.21)
*E-mail:
[email protected].
(B.15)
noting that the force upon impact of the zero-th order motion is
Fs0(t ) =
(Ω̅ + k 2)
Corresponding Author
using the symmetry
V̅ ′(z 0,0) = −
k=1
k 2
2 πpxi Khw
lim Ps = lim
pz2 ,0 [Gs(t )Gs( −t ) + Gc2(t )] M2
∑ (−1)k
The hard wall limit is then obtained by letting Ω̅ → 0 so that (B.12)
0
t0
∫−t
1 Vsc,2 2
M
∞
Ω̅
(B.22)
rewriting the expression for Ps (see eq 3.14) as Ps ≡ Ps,G + Ps,F +
|pzi |
(B.20)
⎛ pxi2 ⎞ M2 2 |pzi | π Ω̅ ⎜ lim Ps = ⎜1 + 2 ⎟⎟ Khw π Ω̅ Φ→ 0 M sinh(π Ω̅ ) pzi ⎠ pxi ⎝ |p | ∞ k coth(π Ω̅ ) zi Ω̅ ∑ ( − 1)k 2 M k=1 (Ω̅ + k 2)
dV̅ ′(z 0, t ″)
sin(ωxt ″)X (t ″) dt ″ 2Ω̅ αpzi2 ⎡ 2 =− ⎢Ω̅ sin(Φ)cos Φσ0 + 2σ1 MΩ ⎣ cos Φ[1 − cos(Φ)] ⎤ + ⎥ ⎦ 2
Ω̅ M sinh(π Ω̅ )
In this limit also
and the following intermediate integral 1 Ω
πpzi
t
dt ′ V̅ ′(z 0, t ′) sin(ωxt ′)
(B.17)
■
we find that αp2 2πh 2Ω̅ Fs0(∞) = zi σ1 M lM Ω
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(B.18)
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(B.19)
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