Rainbow scattering at shadowing and blocking critical angles

355

Surface Science 234 (1990) 355-360 North-Holland

RAINBOW

SCATTERING

Richard S. DALEY, Department

of Chemistry

AT SHADOWING

David FARRELLY and Biochemistry

AND BLOCKING

CRITICAL

ANGLES

and R. Stanley WILLIAMS

and Solid State Science Center,

Received 21 November 1989; accepted for publication

UCLA,

Los Angeles,

CA 90024-1569,

USA

11 April 1990

We show that blocking and shadowing effects for energetic ions scattering from more than one atom are special cases of rainbow scattering. Even at keV energies and above, the cross section at the critical angle for scattering must be evaluated by quantum or semi-classical means to avoid the singularity in the classically calculated cross sections. Experimentally observed scattered distributions are averaged over a large range of atomic orientations because of lattice vibrations, so supernumerary rainbows are not observed. Practical considerations for numeric evaluation of the cross section are discussed.

Recently, we reported on a new method that includes all quasi-single, -double, and -triple scattering events for calculating the three-dimensional cross section for ions that scatter sequentially and classically from two atoms [l]. In this approach, the projectile forward scatters from the first atom of the two-atom pair (shadowing atom) and then backscatters from the second atom (target atom). When the total scattering angle is large, the projectile can forward scatter from the shadowing atom again on its outward trajectory (triple scattering). The incident and final ion trajectories are confined to the plane which contains the interatomic axis of the two-atom pair. In addition to the calculation of a two-atom scattering cross section, the formalism can be used to calculate shadowing and blocking critical angles [2] as a function of the total scattering angle as well as providing a two-atom method for the simulation of impact-collision ion scattering spectroscopy (ICISS) data [2,3]. For comparison to ICISS data, the cross-section calculations are performed for a fixed laboratory scattering angle 8, as a function of (Y, the angle between the incident ion trajectory and the interatomic axis of the atom pair. The coordinate system is chosen such that s1 is measured with respect to the first atom in the scattering sequence (fig. 1). The classical two-atom cross section of ref. 0039-6028/90/$03.50

[l] is written in the form

u$3,,a) =

&-

L

a%,,, -’ -ae, asl a+in ’

I

I

(1)

where the second term in the Jacobian accounts for the breaking of axial symmetry when two

Fig. 1. Illustration of the trajectory for an ion that scatters sequentially from two atoms into a total scattering angle of or; where s, (sa) is the impact parameter with respect to scattering into an angle 0, (8,) from the first (second) atom, a is the rotation of the internuclear axis with respect to the incident ion beam direction, and R is the interatomic distance.

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

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et al. / Rainbow scattering at shadowing and blocking critical angles

scattering centers are present. There is a minimum in the LY(S,) function which defines the shadowing (or blocking) critical angle (Ye, and numerical calculations show (figs. 2 and 3) that the classical cross section diverges at the critical angle because

aa

-

=0 as,@ L

when

%

24

=O. 1 %

This is exactly the condition for rainbow scattering; the total scattering angle B,(s,) is stationary with respect to variations in the impact parameter [4] at the shadowing and blocking critical angles. The true cross section does not actually diverge at CX,because of quantum mechanical uncertainty in the deflection function. Near the rainbow angle, the classical formula for the cross section must therefore be replaced by a quantum-mechanical version. In practical applications, determination of the actual value for the cross section at (Y, can be important to avoid the singularity of eq. (1) in numeric computations of the a, versus cxcurves. If the classical cross section is computed at a value of si too near the critical value, the calculated intensity of scattered ions will be much too high. In order to make detailed comparisons between experimental data and calculated intensities, the divergence in the classical cross section must be handled properly. In this paper, we compare cross sections calculated with the classical two-atom scattering formalism and the semiclassical JWKB [5] method, show over what range of (Y the classical cross section is not valid, and then discuss the practical evaluation of the cross sections in the neighborhood of the critical angle. A complete discussion of the classical cross section is contained in ref. [l]. For every value of s1 a corresponding value of (Y is determined such that the sum of the angular deflections experienced by the projectile as it scatters from the two-atom pair is equal to eL (fig. 1). A set of LY versus si curves in the region of the shadowing critical angle for 5 keV He+ scattering from two Ag atoms separated by 2.89 A is shown in fig. 2 for various total scattering angles. The individual ion-atom interactions were calculated using the Thomas-Fermi-Moliere potential, with the Firsov

0.20

0.50

0.80

1.10

S,/X Fig. 2. Values function of s, Ag-Ag atomic The minimum

of a near the shadowing critical angle as a and 8, for scattering of 5 keV He+ from a pair with an interatomic separation of 2.89 A. in the curve for each value of ~9, is defined as the critical angle a,.

screening length reduced by 0.70 [4]. There are two main features to note about this plot for each 8,: (i) the curve possesses a minimum value of LX, the critical angle, that occurs at the critical impact parameter sic and, (ii) for (Y> (Y, there are two different values of si that give a particular value of CL As shown, the critical angle decreases with decreasing 0,_, but only slowly for large 8, [l]. The basic criteria for observing rainbows in surface scattering have been discussed by Horn and Kleyn [5], and applications to lo-100 eV Na and K atoms scattering from Ag(ll1) have been discussed [6]. The origin of the singularity of eq. (1) at (Y, can be seen graphically by referring to fig. 3. Here constant contours of the total laboratory deflection function 8, are plotted as a function of both si and (Yfor the same system as fig. 2. For small values of (Y, the total deflection function monotonically decreases with increasing S, because there is no interaction of the projectile with the second atom of the pair; it is hidden in the shadow cone of the first atom. However, for each (Y> 6 O, both a minimum (s, = 0.2 A) and maximum (si = 0.5 A) are observed in the total deflection function. As cx is increased, the second atom of the pair interacts with the projectile, forcing the projectile to turn away from its original trajectory

351

R.S. Daley et al. / Rainbow scattering at shadowing and blocking critical angles

where 0, is the rainbow angle, L is the angular momentum, L=s,JzmE,

(3)

and

where k is the semi-classical

wave vector, and

k=m/h,

Fig. 3. Constant contours of the classical deflection function 8, for the system of fig. 2 as a function of both S, and a. The in ~9,. The local maximum in 6’,_ contour interval is loo defines a shadowing rainbow angle, and the minimum corresponds to a blocking rainbow.

and scatter into a larger deflection angle. The minimum in the total deflection function corresponds to a forward scattering, i.e., blocking, critical angle. After passing through the minimum, the 8, versus sr curve rises abruptly as the projectile backscatters from the second target atom of the pair. The maximum in 8, defines the shadowing critical angle. Although the scattering potentials normally employed at keV energies are strictly repulsive [l], eq. (1) yields two rainbows in the two-atom deflection function because the total interaction is not spherically symmetric. These may thus be properly called the blocking and shadowing (or forward and backward) rainbow angles. For a (Y, there are two values of sr near the critical impact parameter where the deflection function coincides with the detector angle. To calculate the cross sections correctly in the region of the critical angle, the semi-classical (JWKB) method for approximating scattering intensities near a rainbow angle is employed [7-91. Near the rainbow angle, the deflection function is expanded in the form

e=e*+Q(L-LJ2P+

. ..)

(2)

(5)

To establish where eq. (1) is expected to fail, i.e., for what values of sr eq. (1) becomes unstable and a quantum-mechanical expression for the cross section is required, consider the relationship between the quantum-mechanical uncertainties in the angular momentum and the total deflection angle. From eq. (2), an uncertainty in the angular momentum AL is implied, since two classical trajectories have the same angular deflection Af3 from e,, AL=A[(B-8,)/Q]“‘=

(AHz2,‘Q)1’2.

(6)

From the uncertainty principle, AL A8 2 A/2, and thus the classical description can be valid only if A8 > l$Q11’3.

(7)

The actual behavior of the cross section near the rainbow angle can be calculated from a transitional approximation to the semi-classical expression [7,8] us,(e, SJ = k-* ,eQ-*“‘[Ai(z)]*, 2.TLr

where the argument [lo] is z = (er - e)p/3.

of the Airy function

(8) Ai (9)

A fully uniform approximation is not required because of the very small range of sr over which the classical description fails. Although eq. (8) is strictly valid only in center-of-mass coordinates, there is little difference between center-of-mass and laboratory coordinates if the projectile mass is much less than the target mass, and for such a case eq. (8) becomes a good approximation to the semi-classical cross section when laboratory coordinates are used. Additionally, eq. (8) is derived for scattering from a central-force potential. For

358

R.S. Daiey et al. / Rainbow scattering at shadowing and blocking critical angles

two-atom scattering, the effective potential experienced by the projectile is clearly not spherically symmetric, since the deflection function contains both azimuthal and (Ydependences in addition to other geometric factors. However, for two-atom scattering near the rainbow angle, eq. (8) is assumed to parameterize the semi-classical cross section, since the calculated deflection function is parabolic in nature (see fig. 4). In the simulation of ICISS data, the classical cross sections of eq. (1) are calculated for a fixed laboratory scattering angle as a function of (I. Therefore, to apply eq. (8) to the case of two-atom scattering near the rainbow angle, we would like to evaluate the semi-classical cross sections as a function of (Y(s,) for fixed 8,. As shown in fig. 3, each value of (Yhas a corresponding maximum in the deflection function which defines the shadowing rainbow angle. Over a small 1ynear a,(fl,), the rainbow angle 0,( CX)can be expressed a linear function of (Y,

d~(a)=s,(u,)+(a-a,)~l

(10)

LK=LTL

Thus, to calculate the semi-classical cross section as a function of (Y for a constant value of 8,, a new 6, is determined for each value of (Y. The semi-classical expression for the cross section as a function of a is therefore written as

u,,(d,,

a) = kp2 2~i~r~a)

p2/3 [Ai( z)]‘,

01)

LQ

with the argument

to the Airy function

becoming

(12) To establish for what values of cy the classical cross section is expected to fail, i.e., where eq. (11) must be utilized, the deflection function corresponding to (Y= (Y, is calculated in the region of the maximum of tYL,and the 8, versus si curve is fit to the parabola defined by eq. (2). An example of this is shown fig. 4 for the Ag-Ag system of fig. 2. From eq. (6), the range of 8, over which the classical cross section becomes unstable is calculated to be Ae, = 0.8”. Assuming Q is not a strong function of (Y,the corresponding range in (Y where eq. (1) is expected to fail can be found by

149.8 -

149.6-

a ’

149.4c-1

a

149.2149.0 0.450

I” 0.460

”

I, I,,

:

0.470

S,lA Fig. 4. Best fit to the classical deflection function (circles) at a, using the parabolic expansion of eq. (2) (solid line) for 8, = 150 “. The value of p used in the fit is - 9.68 x 10m6.

use of eq. (lo), a-a,=Aa=Ae

aa

-. L ae,

(13)

Referring to fig. 2, for a 0, of 150”, ~LY/M, = 0.01. With a At), of 0.8”, A\cu is approximately 0.01”. To calculate the semi-classical cross sections, e,(a) is calculated from eq. (10) for each 1y within the range defined by eq. (13). From eq. (11) the semi-classical cross section can then be determined for scattering into 8,. The calculated results for both the classical and semi-classical cross sections are shown in fig. 5 over a small range of (Y near the critical angle. The cross sections calculated using the semi-classical approach eliminate the singularity that is classically present at (Y,. The oscillations in the semi-classical cross section for (Y> (Y, are the result of quantum-interference effects between the two trajectories on either side of the rainbow angle that contribute to the scattering. These are not observed experimentally because the period of the oscillations is extremely small compared to the broadening of the distribution caused by the thermal motions of the atoms [l]. There is also a contribution to the total cross section from the semi-classical calculations for (Y< (Y, where none is expected classically. This is quantum-mechanical “leakage” of scattering probability into the dark side of the rainbow angle. The small range of (Yover which eq. (1) breaks down arises from the relative insensitivity of the critical angle with respect to changes in the total

R.S. Daley et al. / Rainbow scattering at shadowing and blocking critical angles

12.76

12.78

12.80

12.82

Wdeg Fig. 5. Two-atom classical cross sections calculated by eq. (1) ( eC) and the corresponding semi-classical cross sections (0%) of eq. (11) plotted as a function of (1. The dashed lines are the individual contributions to eC for each branch of the a versus s, curve of Fig. 2. In this case, the cross sections were normalized to each other at a,, the lowest value of a where the classical cross sections are expected to be valid. The singularity in the classical cross sections at a, is removed by use of the semi-classical approach. Note the nonzero semi-classical contribution to the cross section for cx i a,, representing quantum-mechanical “leakage” into the dark side of the rainbow. All cross sections are evaluated for a total laboratory scattering angle of 150 O.

laboratory scattering angle (see fig. 2) because of the high-energy nature of the scattering process. Over this OLrange, the classical values for the cross section are replaced by the semi-classical values if accurate numerical results are required. Experimentally, the observed flux as one passes through the shadowing critical angle is a smooth function of (Y and no quantum interference effects or sharp maxima can be observed. Although the angular acceptance of the detector is comparable to the Al?, over which eq. (1) is expected to fail, variations in the critical angle caused by thermal motions of the two-atom pair are quite large, with the instantaneous value of (Y for a particular sample orientation, typically 50-10° at room temperature. Therefore, for every value of (Ymany different trajectories centered around si = si(c~) contribute to the detected scattered ion yield. In conclusion, true rainbows are shown to occur at the shadowing and blocking critical angles for the scattering of a projectile from a two-atom pair. The calculated classical cross sections possess

359

a singularity at the critical angles because, in the expression for the cross section in eq. (1) the term (ML/as,) vanishes. The JWKB approximation to high-energy scattering can be utilized near the rainbow angle to correctly calculate the scattered ion cross sections. While Oen [ll] suggested that something analogous to rainbow scattering was responsible for the enhancement of scattered ion flux near the edge of a blocking cone in a-particle emission from a single crystal, the cross section formula of eq. (1) allows the first definite identification of rainbows with blocking and shadowing critical angles. Near tic, use of the JWKB cross section removes the singularity, placing an upper limit on the cross section over a small range of (Y near (Y,. The removal of the singularity can be important when absolute values for the cross sections are needed near the critical angle in order to compare calculated to experimental ion scattering intensities. However, in practice, smoothing of the cross sections caused by thermal vibrations of the two-atom pair makes an accurate determination of the critical angle cross section unimportant when a convolution of the cross sections over a range (Y is required for comparison to ICISS data. The cross sections near the critical angle need not be calculated by the semi-classical method but can be approximated quite well by simply replacing them with the classical cross section for the smallest value of LYwhere the classical results are expected to be accurate, i.e., CX,+ 1:Q 1‘/3aa/i36’,. This approximation introduces a less than 1% uncertainty in the calculated cross sections after thermal broadening has been included. Finally, for the case of two-atom scattering, the glory effect is not observed at eL = 180 o because the Jacobian of eq. (1) is proportional to sin 8,. Enhancements in the scattered ion flux for 8, = 180” occur as the shadowing and the blocking critical angles begin to become nearly equal [I]. This effect has recently been observed in experiments performed by Katayama et al. [12].

This work was supported in part by the National Science Foundation. R.S.W. was further supported by the Camille and Henry Dreyfus Foundation.

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et al. / Rainbow scattering at shadowing and blocking critical angles

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