An Introduc+on to X-ray Reflec+vity

Transcription

1 An Introduc+on to X-ray Reflec+vity Mark Schlossman UIC (presented at the APS Workshop 016) Measure molecular ordering in the direc+on perpendicular to the interface normal to the interface Typical Result (T = 8 C) Intrinsic profile Effec+ve profile Goal: 1) Introduce basic theory of reflec+on ) Simple calcula+onal tools 3) Provide intui+on 4) Inadequate +me to explore many interes+ng ideas in the interac+ons of x-rays with liquid interfaces - see

2 Reflec+on from Flat Surfaces Geometry and Fields z reflected y α i α t α i incident X-rays transmitted 3 lines represent the typical range of angles for water reflec+on Electric fields: s-polariza+on typical for synchrotron X-ray sca_ering from liquid surfaces Wave vector magnitudes given by indices of refrac+on n Specular reflec+on: incident = reflected angle Measurable X-ray angles of incidence are small < 0.05 radians! E i! E r! E t!! ( r)= ˆxE i exp i( k i r! ωt )!! ( r)= ˆxE r exp i( k r r! ωt )!! ( r)= ˆxE t exp i( k t r! ωt )! k k o = n = 1 ζ + iβ Incident Reflected Transmi_ed wave number In vacuum wave length k o = π / λ λ 0.41 Å 1.4 Å (30 kev) (10 kev)

3 X-ray Sca_ering from Electrons Produces the Reflec+on Index of refrac+on n determined by electron density ρ (# electrons per Å 3 )! k k o = n = 1 ζ + iβ real part with ζ = πρr e (1+ f / Z) / k o imaginary part produces X-ray absorp+on as X-rays pass through material β = πρr e ( f / Z) / k o ζ 10 6, β 10 8 at 10 kev for common aqueous and organic solu+ons r e = e mc m classical electron radius sca_ering from free electrons f, f correc+on due to electron binding to atoms depends on X-ray energy anomalous dispersion correc+on to the atomic sca_ering factor Z atomic number Index of refrac+on slightly less than, but nearly equal to 1 We will see that this makes the angle for total reflec+on very small and confines measurable reflec+on to very small angles. Absorp+on: wave intensity passing distance r through material varies as where the absorp+on length l abs = ( k o β) 1 = ( µ l ) 1 linear absorp+on coefficient exp( r / l abs )

4 Boundary Condi+ons determine Reflec+on and Transmission z reflected y α i α t α i incident X-rays transmitted Apply standard boundary condi+ons to the electromagne+c fields at the interface (see, for example, D. J. Griffiths, Introduc+on to Electrodynamics) If the electron density varies only in the z-direc+on: ρ(z) or General laws of reflec+on and refrac+on: n(z) 1) In-plane x- and y-components of the wave vectors are equal k i,x = k r,x = k t,x k i,y = k r,y = k t,y ) Snell s Law: n i cosα i = n t cosα neglec+ng absorp+on ( β 0 t ) 3) z-components of wave vectors k i,z, k r,z, k t,z determine the reflec+on and transmission coefficients

5 Fresnel Reflec+on (for a step-func+on interface) reflec+on coefficient Away from the interface: E(z) exp( ik i,z z)+ r(α i )exp(ik i,z z) when z > 0 E(z) t(α i )exp( ik t,z z) when z < 0 transmission coefficient k r k t z k i z Step-func+on interface (mathema+cally flat and smooth) ρ(z) = ρ i z 0 ρ t z < 0 0 ρ i ρ t ρ(z) At this stage, the boundary condi+ons are simple con+nuous r( α i )= Er E i = k i,z k t.z k i,z + k t,z and t( α i )= Et E i = E(z = 0) k i,z k i,z + k t,z and de(z) dz z=0

6 Fresnel Reflec+on (for a step-func+on interface) z Fresnel reflec+vity (s-polarized X-rays) R F ( α i )= I r I i = Er E i = r α i = k i,z k t.z k i,z + k t,z k r k t k i 1 10 R F (Q z ) R( α i ) I I r I i Reflec+vity frac+on of incident energy that is reflected Intensity average power per unit area (+me-averaged magnitude of the Poyn+ng vector) Reflected intensity measured by detector amer sample Incident intensity measured by detector before sample Fresnel reflec7vity simulated for liquid-gold/ interface Q z (Å -1 ) Wave vector transfer:! Q = Q z ẑ =! k r! k i α i Specular reflec+on k r Q z k r k i α i Q z = 4π λ sinα i Q x = Q y = 0

7 Fresnel Reflec+on (for a step-func+on interface) 1 10 R F (Q z ) Fresnel reflec7vity simulated for liquid-gold/ interface Q z (Å -1 ) R(Q z ) 1 as Q z Q c Q c = 4π λ sinα c 4 πρ t r e (1+ f / Z) Q c 4 π r e Δρ Q c Å 1 or, neglec+ng f with cri+cal Q for total reflec+on liquid- Δρ = ρ t ρ i for the water- interface Rapid decrease of R F with Q z Rewri+ng R F ( Q z )= k i,z k t.z k i,z + k t,z in terms of expressions for k that neglect β leads to R F sin 4 α c / 16sin 4 α i R F Q c Q z 4. Using Q z = ( k! r k! i ) = k z i,z = k o sinα i which is accurate for Q z! > 4Q c (see dashed line above).

8 Below the Cri+cal Angle for Total Reflec+on! k t Let = n = 1 ζ + iβ for liquid- interface, where k! k t = k o cosα i ŷ k t,z ẑ o! k t = ko 1 ζ + iβ = k t,y + k t,z = k o cos α i + k t,z For small ζ and ζ β : k t,z k o ( 1 ζ ) cos α i k t,z k o α 1/ i ζ For small α i : Here, we see that the wave vector k t,z becomes imaginary when α i <α c ζ (or α c Δζ = (ζ t ζ i ) or for a buried, liquid-liquid interface) k t,z k o sin α i ζ 1/ The transmi_ed wave exp ik t,z z exp + k t,z z E(z) t(α i )exp( ik t,z z) does not propagate as a traveling wave into the lower material traveling wave if is real k t,z exponen+al decay if k t,z is imaginary k t,z = Im(k t,z ) k r k i α i < α c Total External Reflec+on: R 1

9 Two physical effects Inverse decay length Below the Cri+cal Angle for Total Reflec+on Total reflec+on, but R slightly less than 1 because of absorp+on of evanescent (exponen+al decaying) wave in lower material. The transmi_ed wave E(z) t(α i )exp + k t,z z with z < R F (Q z ) Λ 1 = k t,z = k o ζ sin α i k o α c α i 0 liquid-gold/ interface Q z (Å -1 ) z k r k i α i < α c Decay length for the field intensity is Λ / ~ 5 to 10 nm at α i = 0.8 α c for water and common organic liquids The evanescent wave is confined to the interface and is useful for grazing-incidence diffrac+on and fluorescence

10 Exercise I Introduc+on to Data Analysis Somware Wei Bu Compute R F for the water- interface without and with absorp+on (µ = 0 and 10)

11 Reflec+on from Stra+fied Media upper bulk phase j = J+1 Mul+ple planar layers at the interface between two bulk materials J layers lower bulk phase j = 0 Model a con+nuous electron density profile ρ(z) (or index of refrac+on n(z) ) with a sequence of thin layers

12 Reflec+on from Stra+fied Media (Parra_ Method) Calculate Fresnel reflec+on and transmission at each internal interface in the stack of layers index of refrac+on n j = 1 ζ j + iβ j wave vector with ζ j = Q c 8k o ρ j ρ j=0 k j,z k o sin α i ζ j + iβ j 1/ At interface between layer j and layer j -1, use the Fresnel expressions: = r j 1, j r j, j 1 = k j,z k j 1.z k j,z + k j 1,z and t j, j 1 = 1+ r j, j 1 = k j,z ( k j,z + k j 1,z )

13 Reflec+on from Stra+fied Media (Parra_ Method) Electric field in a layer is the sum of the net field traveling downward and the net field traveling upward E j (z) = E j (z) E j + (z) = A j exp( ik j,z z) B j exp(ik j,z z) Fields undergo two processes as they pass through a layer (1) Propaga+on from top of layer j (z = z j ) to bo_om of layer j (z = z j 1 ) E j (z j 1 ) = P z j 1 =z j d j E j (z j ) = exp(ik j,zd j ) 0 j 0 exp( ik j,z d j ) E j (z j ) E j + (z j ) phase shim k j,z d j () Interface reflec+on and transmission at z j r j 1, j E j 1 (z j 1 ) = I j 1, j E j (z j 1 ) = 1+ r j 1, j r j 1, j 1 E j (z j 1 ) E j + (z j 1 ) Fresnel reflec+on

14 Reflec+on from Stra+fied Media (Parra_ Method) Each layer in the stack is accounted for by an Interface reflec+on/transmission matrix and a Propaga+on matrix upper bulk phase j = J+1 I j 1, j P j E 0 (z 0 ) E 0 + (z 0 ) E J+1 (z J+1 ) = I 0,1 P 1 I 1, P!P J I J,J+1 E + J+1 (z J+1 ) = M 11 M 1 E J+1 (z J+1 ) M 1 M E + J+1 (z J+1 ) lower bulk phase j = 0 SinceE + 0 (z 0 ) = 0 (no upward ray in the lower phase), 0 = M 1 E J+1 (z J+1 )+ M E + J+1 (z J+1 ) Reflec+vity coefficient r = E J+1 + (z J+1 ) E J+1 (z J+1 ) = M 1 M Reflec+vity R = M 1 M

15 Exercise II Wei s Data Analysis Somware uses the Parra_ Method to calculate reflec+vity (also uses a different sign conven+on for z, z > 0 going into the material) z Calculate R and R/R F ρ(z) ) Try Log and linear scales ρ(z) ) Subs+tute ρ (= 0.333) (1.5, 1, 0.75) for ρ(z) 3) Vary layer thickness ρ(z) bulk water bulk water

16 The Parra_ Method and the Master Formula Parra_ Method is exact for flat, stra+fied interfaces omen used to fit reflec+vity data (1) Complex calcula+ons may not be intui+ve () Cannot be extended to in-plane (x-y) varia+ons in electron density that produce sca_ering for Q xy 0 Approximate approaches are useful in both areas Master Formula derived from the Parra_ Method Recall k j,z As we saw earlier, when k o sin α i ζ j + iβ j α i α c sinα i k j,z k o sinα i ζ j iβ j Q z Q c absorp+on can be neglected and ζ j α c, j r j 1, j = k j 1,z k j,z k j 1,z + k j,z α c, j α c, j 1 4α i 1

17 Master Formula derived from the Parra_ Method r j 1, j = k j 1,z k j,z α c, j α c, j 1 1 k j 1,z + k j,z 4α i α c, j Use = 4πρ j r e / k o to write in terms of r j, j+1 ρ(z) r j 1, j 1 ρ Q c 4Q z ( ρ j ρ j 1 ) 1 ρ Q c 4Q z ( z j z j 1 ) ρ(z) z z=z j 1 R(Q z ) = M 1 M 1 α i α c Q c Q z 4 1! (see Pershan & Schlossman book for details) + ρ ρ(z) dz z exp[ iq z z] Master Formula R(Q z ) R F (Q z ) Φ eff (Q z ) Effec+ve surface structure factor Φ eff (Q z ) = 1 ρ [ ] exp [ +iq zz] + ρ(z) dz z exp[ iq z z] Note: let exp iq z z become when z is posi+ve in the lower phase

18 Example, then Exercise III Consider the profile in the previous exercise ρ(z) ρ = 0 z < z < z > 30 1 z 30 ρ(z) / ρ bulk water z d ρ(z) ρ dz d ρ(z) ρ dz = 1.5δ (z) 0.5δ (z 0)+ 0.5δ (z 30) R(Q z ) R F (Q z ) 1 ρ + ρ(z) dz z exp [ +iq z z] R(Q z ) R F (Q z ) exp ( i 0Q z)+ 0.5exp( i 30Q z )

19 Example con+nued Now for some algebra ρ(z) / ρ D 1 A B C bulk water D R(Q z ) R F (Q z ) exp ( i 0Q z)+ 0.5exp( i 30Q z ) A Bexp( i D 1 Q z )+ C exp( i D Q z ) Recall = A Bexp( i D 1 Q z )+ C exp i D Q z A Bexp( i D 1 Q z )+ C exp i D Q z exp(ix)+ exp( ix) = [ cos x + i sinx]+ [ cos( x)+ i sin( x) ]= cos x R(Q z ) R F (Q z ) A + B + C ABcos(D 1 Q z )+ AC cos(d Q z ) BC cos ( D D 1 )Q z This illustrates that each internal interface generates a reflec+on and that reflec+ons from each pair of internal interfaces interfere with each other ( cos(dq z ) ).

20 Exercise III Use the Master formula to calculate R/R F for the profile shown bulk water Plot R/R F or sketch it by hand, then compare it to the predic+on of the data analysis program. ρ(z) / ρ 0 0

21 Exercise III Solu+on Use the Master formula to calculate R/R F for the profile shown. Plot R/R F or sketch it by hand, then compare it to the predic+on of the data analysis program ρ(z) / ρ 0 0 bulk water d ρ(z) ρ dz = 1.5δ (z) 0.5δ (z 0) 1 d ρ(z) ρ dz R(Q z ) R F (Q z ) 1 ρ + ρ(z) dz z exp [ +iq z z] R(Q z ) R F (Q z ) exp ( i 0Q z)

22 Exercise III Solu+on ρ(z) / ρ D A B bulk water R(Q z ) R F (Q z ) exp ( i 0Q z) 0 0 A Bexp i DQ z where I used = A Bexp( i DQ z ) = A + B AB exp( i DQ z )+ exp i DQ z A Bexp i DQ z exp(ix)+ exp( ix) = [ cos x + i sinx]+ [ cos( x)+ i sin( x) ]= cos x = A + B ABcos DQ z R(Q z ) R F (Q z ) cos 0Q z R/R F 36/ π/d π/d Q z (Å -1 ) If you compare this curve to the calcula+on from the data_analysis program, you will find that the curve differs from the Parra_ calcula+on primarily at small Qz. The master formula does not exhibit a cri+cal Q z and is quan+ta+vely accurate only for Q z! > 4Q c

23 Average Electron Density ρ(z) What is Averaged? Master Formula R(Q z ) R F (Q z ) 1 ρ + ρ(z) dz z exp [ +iq z z] Parra_ method deriva+on of Master formula it appears as if ρ(z) is the average from one layer to the next, but the more complete Born approxima+on deriva+on (not shown here) shows that it is an average over the x-y plane What depends upon x-y? ρ(z) xy SoH MaJer 10, 7353 (014) Structural inhomogenei+es within the plane (such as domains) Chemical inhomogenei+es within the interfacial plane Thermal fluctua+ons (capillary waves) out of plane Each requires special considera+on to understand the X-ray sca_ering

24 Average Electron Density ρ(z) What is Averaged? Capillary wave (thermal) fluctua+ons of liquid interfaces Local density profile 1 d! ξ A ξ ρ! r xy, z r xy ρ r! xy r! xy, z = ρ z h r! xy averages over granular molecular nature of the surface provides an intrinsic width or intrinsic profile of the interface ξ bulk correla+on length ( ~ 1 molecular diameters) A ξ ξ and ρ z h ( r! xy ) is the intrinsic profile Macroscopic average over Gaussian fluctua+ons σ cap ρ(z) = dh( r! xy ) ρ z h( r! xy ) = δ h r! ( xy ) = h r! ( xy ) h r! xy 1 πσ cap h(! r xy ) Ilan Benjamin water/nitrobenzene exp δ h r! xy σ cap = h! ( rxy ) h ( r! xy ) σ cap = h! rxy if h ( r! xy ) = 0

25 ρ(z) for a Capillary Roughened Simple Interface ρ(z) = dh( r! xy ) ρ z h( r! xy ) 1 πσ cap exp h r! xy σ cap Simple interface fluctua+ng step-func+on interface h(! r xy ) ρ z h( r! xy ) = ρ z < h(! r xy ) 0 z > h( r! xy ) ρ( z) = ρ ρ(z) / ρ 1.0 bulk liquid z 1+ erf σ cap σ cap 0 Bulk liquid Ilan Benjamin water/nitrobenzene The error func+on erf is defined by z erf (z) 1.0 erf (z) = exp( t ) dt π 0 d erf (z) dz [ ]= π exp( z ) 0 z -1.0

26 Apply Master formula to a Simple Interface R(Q z ) R F (Q z ) 1 ρ ρ( z) = ρ exp Q z σ cap R Q z R F Q z 1 ρ + ρ(z) dz z exp [ +iq z z] z 1+ erf σ cap d dz ρ( z) = 1 exp z σ π σ cap ρ(z) / ρ 1.0 d dz ρ z bulk liquid bulk liquid σ cap 0 0 z R/R F falls exponen+ally with (Q z ) R( Q z ) R F ( Q z )exp Q ( z σ cap ) Q c Q z Interfacial capillary roughness has a significant effect on the reflec+vity. 4 exp Q ( z σ cap ) R falls with (Q z ) 4 +mes an exponen+al decay in (Q z )

27 Master Formula and the Capillary Roughened Profile Intrinsic profile (without capillary wave roughness) ρ(z) / ρ D A B bulk water Effec+ve profile (with capillary wave roughness) 0 0 ρ( z) = 1 ρ erf z z erf σ σ 1 ρ d dz ρ z 0 bulk water 0 1 ρ d dz ρ( z) = 1 z 1.5 exp z 0 σ π σ 0.5 exp σ

28 Intrinsic and Effec+ve Profiles ρ(z) / ρ D A B bulk water 0 0 R(Q z ) R F (Q z ) 1.5 exp Q z σ / 0.5exp i 0Q z exp Q z σ / As before, let A = 1.5, B = 0.5, and D = 0 R(Q z ) R F (Q z ) A + B ABcos DQ z exp Q z σ Same as before, without roughness roughness

29 Exercise IV ρ(z) bulk water ρ(z) bulk water Use Data_Analysis program to plot R and R/R F for the profiles shown, but add roughness. Start with a small value (1 Å), then increase to a value typical for the water surface (3 Å), then larger values. Reverse the process. Start with one of your calcula+ons for R/R F and predict the layer thickness and the roughness σ by quan+ta+ve analysis of the R/R F.

30 Exercise V Reverse Profiles (if +me) ρ(z) / ρ A D B bulk liquid ρ(z) / ρ A D bulk liquid B Use Data_Analysis program to plot R and R/R F for the profiles shown, with and without roughness. You should find that these two profiles, known as reverse profiles, lead to the same reflec+vity. Calculate R/R F using the Master formula to be_er understand why they produce the same reflec+vity. Why is it called a reverse profile? Do reverse profiles exist for the two profiles in Exercise IV?

31 What has been lem out (could fill a book) Capillary wave theory provides a way to calculate the capillary (thermal) roughness from the surface or interfacial tension Born approxima+on calculates reflec+vity and off-specular diffuse sca_ering from rough surfaces when the incident and sca_ered angles are larger than the cri+cal angle Distorted wave approxima+on calculates reflec+vity and off-specular diffuse sca_ering from rough surfaces when the incident and sca_ered angles are similar to or smaller than the cri+cal angle Other types of profiles besides a stack of slabs chosen to model different types of interfaces Surface structural and chemical inhomogenei+es how do these affect the reflec+vity?