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`4.Techniques for Surface Structure DeterminationLong-range order - size and symmetry of surface unit cell Short-range order - atomic positions within unit cell4.1 Low Energy Electron Diffraction (LEED)Most suitable for long-range order determination (does not work on disordered or amorphous materials) Can provide short range information using detailed theoretical analysis 4.1.1 General Diffraction Ideas5000 eV photon (x-ray): E= = hc  6.63x10 -34 Js × 3.00x10 8 ms-1 5x103 V (= JC -1 ) ×1.60x10 -19 C= 2.479x10 -10 m = 2.5 Å 20 eV electron: = h p de Broglie wavelengthp = m  v = 2m(KE) = 2.42x10-24 kg  ms -1 = 6.63x10-34 J  s 2.42x10-24 kg  m s-1 (J = kg s-2  m2 )= 2.74x10-10 m = 2.7 Å Electrons of KE 20-200 eV can be diffracted by lattice of atomic dimensionsCEM 924 6.1 Spring 2001Low energy electrons are strongly back-scattered by the electrons of the surface atoms (unlike x-ray diffraction) - electrons of 20-200 eV penetrate only ~10-50 Å into surface  pathlength=d·sin d n  n = 2dsin  h   = 2dsin   2meV Bragg EquationFeatures to note: (1) Sin is proportional to becomes smaller (2) Sin is proportional to1 - diffraction angle gets bigger as d d 1 - diffraction angle becomes eVbigger as electron KE becomes smaller (3) Diffraction has same probability with n=1 and n=-1 diffraction pattern is symmetricCEM 9246.2Spring 2001· Possible to explain surface diffraction in terms of scattering from rows of atoms but quickly becomes cumbersomeSurface FCC(110) a Diffraction Patterna'b a&lt;bb'a'&gt;b'Can build a general theory based on reciprocal relationship of diffraction angle and electron wavelength 4.1.2 Ewald Sphere Construction in 3-DDefine wavenumber as reciprocal of wavelength r 2 k= r It can be shown that k is really a measure of momentum = h p r 2 p  2 p k= = =  h hDuring scattering (diffraction): - both energy and the magnitude of the electron momentum are conserved - direction of momentum is changedCEM 9246.3Spring 2001r So how do we determine scattering angle knowing k is conserved? Draw reciprocal lattice (k-space lattice):Real Space d Reciprocal Space 2/dReciprocal lattice is a scaled version of real lattice but lattice points are spaced proportional to k not d.Reciprocal Space 2/d Sphere of constant k Ewald sphere2 k(=2/ ) _ (200) k' (101)_ (301) lg (000)h(1)Draw a scaled version of incident beam. The point of the beam should touch one lattice point. The direction of the arrow corresponds to the real-space angle with respect to the bulk lattice vectors6.4 Spring 2001CEM 924(2) (3) (4)This point is labeled as the origin hkl = (000). r A sphere of constant k is drawn with center at the start of the incident beam. Diffraction will occur for any reciprocal space point that crosses this circle such that r r r k' = k + g r where g is the scattering vector. r Draw a scattered reciprocal k-space vector, k', for the outgoing wavevector from the center of the Ewald sphere (circle) to any point where the sphere and reciprocal lattice points intersect(5)Pythagoras' theorem (in 3-D) gives  2 r2  g =  h2 + k 2 + l2     d r   2 g =  h2 + k 2 + l2     d and since r r 2 k k = and sin = r  g/2 sin = 2  2  2  h + k 2 + l2     2d  2d =  h2 + k 2 + l2     CEM 9246.5Spring 2001This is essentially the Bragg equation (n = 2d sin) where (h2+k2+l2)1/2 is the diffraction order Remember that the above drawing is a 2-D &quot;slice&quot; through true 3-D reciprocal lattice 4.1.3 Ewald Sphere Construction in 2-D For surface scattering, we are only concerned with periodicity in 2-D of surface, not into bulk. Can remove one dimension from reciprocal lattice - points become &quot;rods&quot;&quot;Side view&quot; (10) Real space  k' k 2 g k  (00) hk' 2&quot;Top view&quot;hin-plane diffraction k k' g kCEM 924 6.6 Spring 2001In fact, will get diffraction for many beams including forward and backscattered and a specular beam (00) &quot;reflected&quot; at the incidence angle.(30) (20) (10) (00)(10) SpecularkhNote: Only a few beams diffracted Some beams scattered into surface (invisible) Some beams back-scattered towards source r More beams visible as k increases (smaller , larger KE) 4.1.4 Ewald Sphere Construction for LEED - In LEED, electron beam incident along surface normal   - 2d becomes d in Bragg equation 2d sin =  h2 + k 2      Solving for  by geometry using Ewald sphere construction:CEM 9246.7Spring 2001Reciprocal lattice rods k'  k(01)  (02)(00) 2/dWe have enough information to solve for : For the (01) beam 2 / d r k' r r since k' = k and sin  sin sin =  = sin r  k      = sin -1      d-1  2 /d In fact, for any beam along this azimuth - (01), (02), (03)... - 2/d can be replaced by k(2/d) where k is Miller index of the beam. In general  k 2 /d  = sin -1  r  k    -1  k   = sin    d CEM 924 6.8()  Spring 2001Remember: k is just 2   h + k 2  for constant h in along this azimuth!  4.24.2.1Surface Reciprocal LatticesBulk TerminationLet's define a real space plane lattice (surface net) by lattice vectors a1 and a2:Real Space a2 Lattice Vectors a1 Primitive Unit CellReciprocal lattice (net) defined by reciprocal lattice vectors a'1 and a'2 (surface) or b'1 and b'2 (adsorbate) defined by:  a 1  a' 2 = 0  implies a 1 and a' 2 perpendicular a 2  a' 1 = 0   a 1  a' 1 = 1   implies inverse relation a 1 and a'1 a 2  a' 2 = 1  (Remember dot product a1·a'1 means |a1 x a'1|Cos) When =0° (ie a1 and a'1 parallel)CEM 9246.9Spring 2001a 1  a' 1 = 1 a 1 × a' 1 Cos = 1 a1 = 1 a' 1 Reciprocal relationship(1) FCC(100):Real Space Reciprocal Spacea2 a1 Real Space Neta'2 a'1Reciprocal Space Neta'1  a2 and a'2  a1  (angle between a1 and a'1 is 0°), Cos=1 a 1  a' 1 = 1 a 1 × a' 1 Cos = 1 a1 = 1 a' 1CEM 9246.10Spring 2001(2)FCC(110):Real Space Reciprocal Spacea2 a1 Real Space Net a'1 a'2 Reciprocal Space Neta'1  a2 and a'2  a1  (angle between a1 and a'1 is 0°), Cos=1 a 1  a' 1 = 1 a 1 × a' 1 Cos = 1 a1 = (3) FCC(111):Real Space Reciprocal Space1 a' 1a2 a1 a1a'1 a'2 a'1 Reciprocal Space NetReal Space Neta'1 not  a2 and a'2 not  a1  (angle between a1 and a'1 is 30°), Cos=3/2CEM 924 6.11 Spring 2001a 1  a' 1 = 1 a 1 × a' 1 Cos = 1 a1 × a' 1 3 =1 2 a1 = 2 1 3 a' 1In fact, for qualitative picture, no need to worry about Ewald sphere - diffraction pattern is just a scaled version of reciprocal lattice!LEED pattern obtained from Si(111)7x7 reconstructed surfaceCEM 9246.12Spring 2001Diffraction Patterns of 5 Plane Lattices: Lattice Plane Lattice Diffraction PatternOblique (general)a2 a1a'2 a'1Hexagonala2 a1a'2 a'1Squarea2 a1a'2 a'1a'2Rectangulara2 a1a'1a'2Centered Rectangulara2 a1a'1CEM 9246.13Spring 20014.2.2Adsorbate-Covered SurfacesHow does the presence of adsorbates influence the electron diffraction pattern?Real Space Reciprocal Spacea2 a1a'2 a'1b2 b1b'2 b'1Diffraction Pattern -&gt;CEM 9246.14Spring 20014.2.3Instrumentation for LEEDG1 G2 G3 ScreenSampleE=20~200 eVElectron Gun+5 kVElectron gun produces focussed e- beam 10 nA-10 µA E 20-200 eV Magnetic shield expels residual magnetic fields Sample positioned at &quot;focus&quot; of hemispherical grids Diffracted electrons (elastically scattered) and secondary electrons (inelastically scattered) back-scattered towards LEED optics in field free region Diffracted electrons - spots Secondary electrons - diffuse background After passing through G1 (ground) accelerated towards phosphor screen Negative potentials applied to G2 and G3 to repel secondary electrons Electrons strike phosphor  photons Front-view LEED versus rear-view LEED Image captured on photographic film or video cameraCEM 924 6.15 Spring 2001Rear-view LEED system. Width about 10&quot;, height about 12 &quot;. The grids, G1, G2 and G3, are visible at the top of the picture and the view-port is at the bottom. The magnetic shield has been removed.CEM 9246.16Spring 2001CEM 9246.17Spring 20014.2.4Real Surface Diffraction PatternsIf electron beam diameter is larger than domain size on surface - presence of multiple (rotational) domains increases complexity of diffraction pattern Example 1:Domain A p(3x1)Domain B p(1x3)Domain ADomain A and BCEM 9246.18Spring 2001Example 2:Domain A (2 0 ) 12One DomainAll Domains 120°120°Vicinal surfaces with regularly spaced steps behave like large (1-D!) adsorbate netsLEED Pattern (755) (100)(111)7-fold periodicityPt(755) Pt (S)-[7(111)x(110)]CEM 9246.19Spring 2001Irregularly-spaced steps produce &quot;streaky&quot; or blurred LEED spots or rows of spots Kinked surfaces produce additional spots or rows of spots in different direction to step spots Amorphous, disordered or glassy surfaces produce no LEED pattern - only diffuse background4.3 Dynamical LEEDSo far, only considered position of spots More information in intensity of spots? Kinematic LEED considers incident electron scattered once (top layer of atoms) - works fine for symmetry/size of adsorbate or unit cell Dynamical LEED considers incident electron scattered multiple times (1st, 2nd, 3rd... layers of atoms) - necessary to account for intensity of spotsEssential ingredients of dynamical LEED theory: Calculation of amplitude (A) and phase () due to (a) (b) (c) (d) ion core scattering multiple scattering inelastic events surface vibration (effect of temperature)CEM 9246.20Spring 2001CEM 9246.21Spring 2001CEM 9246.22Spring 20014.3.1Measurement of LEED Spot IntensitiesDigital video camera (CCD) used to measure intensity Computer-controlled potentials &quot;steer&quot; one diffracted beam towards electron multiplier as primary beam energy is changed 4.3.2 Dynamical LEED MethodologyIntensity tells us about interlayer spacing When combined with multiple scattering theory, gives information about &quot;height&quot; of adsorbate atoms and relaxation phenomena Can measure intensity in two ways: (1) vary incidence angle - I() curves (2) vary incidence energy - I(V) curvesCEM 924 6.23 Spring 2001CEM 9246.24Spring 2001Set of I(V) curves with variable :Methodology 1. 2. 3. 4. Initial guess for structure Calculate I(V) or I() curves using dynamical LEED theory Compare theory and experimental data Refine guessed structure until best agreement is obtainedCEM 9246.25Spring 2001Agreement between theory and experimental data given by &quot;goodness of fit&quot; Pendry R-factorCEM 9246.26Spring 2001For very good calculations, Rmin&lt;0.2. Atomic separations to 0.01 Å resolution can be obtained Agreement never perfect because ion core potentials not exact. Unaccepatable values &gt;0.6.CEM 9246.27Spring 20014.4SummaryProvides information about: Symmetry of surface or adsorbate unit cell Size of surface or adsorbate unit cell Steps, domains Temperature dependant phenomena Reconstruction and relaxation (dynamical LEED) Surface sensitive (10-50 Å, 3-10 atomic layers) Relatively simple, inexpensive instrumentation Rapid for simple analysis I(V) or I() curves can be used with theory to measure atomic positions with high accuracy (~0.01 Å) BUT Some electron-induced chemistry Intrusive instrumentation Domains, reconstruction, relaxation, steps, defects complicate simple patterns Intensity measurement requires expensive instrumentation Dynamical LEED calculations not trivial to perform nor perfect No way to go directly from I(V) or I() curves to structureCEM 9246.28Spring 2001`

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