Roger Pynn. LECTURE 6: Surface Reflection

Transcription

1 by Roge Pynn LECTURE 6: Suface Reflection

2 Suface Reflection Is Vey Diffeent Fom Most Neuton Scatteing We woed out the neuton coss section by adding scatteing fom diffeent nuclei We ignoed double scatteing pocesses because these ae usually vey wea This appoximation is called the Bon Appoximation Below an angle of incidence called the citical angle, neutons ae pefectly eflected fom a smooth suface This is NOT wea scatteing and the Bon Appoximation is not applicable to this case Specula eflection is used: In neuton guides In multilaye monochomatos and polaies To pobe suface and inteface stuctue in layeed systems

3 This Lectue Why use neuton eflectivity? Refactive index fo neutons Neuton eflection by a smooth suface Neuton penetation depth Effect of suface oughness on specula eflection Reflection fom a suface coveed by a thin film Reflection fom layeed films the Paatt method The inematic appoximation Reflection fom a gaded inteface Compaision of x-ay and neuton eflection Exact detemination of SLD pofiles Science examples Polymes & vesicles on a suface Lipids at the liquid ai inteface Boon self-diffusion Rough sufaces and coelated oughness Gaing incidence diffaction An example of GID: shea aligned wom-lie micelles

4 Why Use Neuton Reflectivity? Neutons ae eflected fom most mateials at gaing angles If the suface is flat and smooth the eflection is specula Pefect eflection below a citical angle Above the citical angle eflectivity is detemined by the vaiation of scatteing length density pependicula to the suface i.e. we can detemine the aveage density pofile nomal to the suface of a film on the suface Images coutesy of M. Tolan & T. Salditt

5 Vaious foms of small (glancing) angle neuton eflection Viewgaph fom M. R. Fitsimmons

6 Refactive Index fo Neutons The nucleus - neuton potential is given by : V ( ) So the aveage potential inside the medium is : V ρ whee ρ is called the nuclea Scatteing Length Density (SLD) - the same one we used fo SANS πh m πh m bδ ( ) fo a single nucleus. ρ volume i b i h The inetic (and total) enegy of neuton in vaccuum is E m Inside the medium the total enegy is h Consevation of enegy gives m h m h m + V + V h m πh + m ρ o 4πρ Since / n efactive index (by definition), and ρ is vey small (~ n -λ ρ / π Since geneally n <, neutons ae extenally eflected fom most mateials. -6 A - ) we get :

7 / Since i.e. Note : Only Neutons With Vey Low Velocities Pependicula to a Suface Ae Reflected The suface cannot change the neuton velocity paallel to the suface so : cosα Neutons obey Snell's The citical value of Fo quat (π / λ)sinα α citical ( o n ) α sin cosα' citical l ( 4πρ α' citical citical o ) sin l ncosα' Law citical (cos α 4πρ.5x α' + sin fo total extenal eflection is 3.λ( A) fo quat A i.e - o.λ ( A) fo nicel n cosα/cosα' α') (cos 4πρ How do we mae a neuton bottle? α α + sin α) 4πρ 4πρ α

8 Reflection of Neutons by a Smooth Suface: Fesnel s Law continuity of ψ & ψ& at a a I I + a + I R a a R R T () a T T n -λ ρ/π components pependicula and paallel to the suface : ai cosα + ar cosα at n cosα () ( ai ar ) sinα at n sinα (3) () & () > Snell's Law : cosα n cosα ( ai ar ) sinα sinα T () & (3) > n ( a + a ) sinα sinα so eflectance is given by I R a R / a I ( I I T ) /( I + T )

9 What do the Amplitudes a R and a T Loo Lie? Fo eflection fom a flat substate, both a R and a T ae complex when < 4πρ I.e. below the citical edge. Fo a I, we find: Real (ed) & imaginay (geen) pats of a R plotted against. The modulus of a R is plotted in blue. The citical edge is at ~.9 A -. Note that the eflected wave is completely out of phase with the incident wave at Real (ed) and imaginay (geen) pats of a T. The modulus of a T is plotted in blue. Note that a T tends to unity at lage values of as one would expect

10 Penetation Depth In the absence of absoption, the penetation depth becomes infinite at lage enough angles Because is imaginay below the citical edge (ecall that 4πρ), the tansmitted wave is evanescent The penetation depth Λ / Im( ) Aound the citical edge, one may tune the penetation depth to pobe diffeent depths in the sample

11 Measued Reflectivity We do not measue the eflectance,, but the eflectivity, R given by: R # of neutons eflected at Q.* # of incident neutons i.e., ust as in diffaction, we lose phase infomation Notice, also, that the measuement aveages the eflectance ove the suface of the sample: i.e. measued eflectivity depends on ρ ( ) dx dyρ( x, y, S ) Measued and Fesnel eflectivities fo wate diffeence is due to suface oughness

12 When Does a Rough Suface Scatte Diffusely? Rayleigh citeion γ γ path diffeence: Δ h sinγ γ γ h phase diffeence: Δφ (4πh/λ) sinγ bounday between ough and smooth: Δφ π/ that is h < λ/(8sinγ) fo a smooth suface whee g 4 π h sin γ / λ Q h

13 Suface Roughness Suface oughness causes diffuse (non-specula) scatteing and so educes the magnitude of the specula eflectivity θ θ t x The way in which the specula eflection is damped depends on the length scale of the oughness in the suface as well as on the magnitude and distibution of oughness Note that oughness intoduces a SLD pofile aveaged ove the sample suface spaling sea model -- specula fom many facets each piece of suface scattes indepedently -- Nevot Coce model R R F e σ I t

14 Fesnel s Law fo a Thin Film ( - )/( + ) is Fesnel s law Evaluate with ρ4. -6 A - gives the ed cuve with citical wavevecto given by (4πρ) / If we add a thin laye on top of the substate we get intefeence finges & the eflectance is given by: Log(.*) + + e e i i and we measue the eflectivity R.* t t Film thicness t substate If the film has a highe scatteing length density than the substate we get the geen cuve (if the film scatteing is weae than the substance, the geen cuve is below the ed one) The finge spacing at lage is ~ π/t (a 5 A film was used fo the figue)

15 One can also thin about Neuton Reflection fom a Suface as a -d Poblem V() π ρ() h /m n V() substate -4π ρ() Whee V() is the potential seen by the neuton & ρ() is the scatteing length density Film Vacuum

16 Multiple Layes Paatt Iteation (954) The same method of matching wavefunctions and deivatives at intefaces can be used to obtain an expession fo the eflectivity of multiple layes i i i e X e X e T R X,,,,, ,,,,, whee i ) Then R( substate & unit amplitude incident wave) nothing coming bac fom inside (i.e. and Stat iteation with X T X R N N + + α Image fom M. Tolan

17 Dealing with Complex Density Pofiles Any SLD depth pofile can be chopped into slices The Paatt fomalism allows the eflectivity to be calculated A thicness esolution of Å is adequate this coesponds to a value of Q whee the eflectivity has dopped below what neutons can nomally measue Computationally intensive!! Image fom M. Tolan

18 Kinematic (Bon) Appoximation We defined the scatteing coss section in tems of an incident plane wave & a wealy scatteed spheical wave (called the Bon Appoximation) This pictue is not coect fo suface eflection, except at lage values of Q Fo lage Q, one may use the definition of the scatteing coss section to calculate R fo a flat suface (in the Bon Appoximation) as follows: R L x numbe of σ L sinα because numbe of y x Fom the definition of dσ ρ dω d neutons eflected by a sample of neutons incident on sample ( ΦL L x L cosα d ' e sinα y a so iq.( v ') dσ dω dω d x 4π Q It is easy to show that this is the same as the Fesnel fom at lage Q sinα sinα dα. coss section we get fo a smooth substate : ρ L L x x L y y L δ ( Q x x ) δ ( Q sie L y L ) y so x L y sinα) dσ dxd y dω sinα R 6π ρ / Q 4

19 of ρ gives : R Reflection by a Gaded Inteface Repeating the bottom line of the pevious viewgaph but eeping the - dependence 6π Q ρ( ) e equality follows afte integating by pats. iq d 6π Q 4 dρ( ) e d iq d whee the second If we eplace the pefacto by the Fesnel eflectivity R fo a smooth inteface, as well as the coect fom at lage Q This can be solved analytically fo seveal convenient foms of dρ/d such as/cosh R R F dρ( ) e d, we get the ight answe ().This appoximate equation illustates an impotant point : eflectivity data cannot be inveted uniquely to obtain ρ(), because we geneally lac impotant phase infomation. This means that models efined to fit eflectivity data must have good physical ustification. iq d F

20 Compaison of Neuton and X-Ray Reflectivity Neutons often povide bette contast and don t damage samples X-ays povide bette Q esolution and highe Q values Viewgaph coutesy of M. Tolan

21 The Goal of Reflectivity Measuements Is to Infe a Density Pofile Pependicula to a Flat Inteface In geneal the esults ae not unique, but independent nowledge of the system often maes them vey eliable Fequently, laye models ae used to fit the data Advantages of neutons include: Contast vaiation (using H and D, fo example) Low absoption pobe buied intefaces, solid/liquid intefaces etc Non-destuctive Sensitive to magnetism Thicness length scale 5 Å Issues include Geneally no unique solution fo the SLD pofile (use pio nowledge) Lage samples (~ cm ) with good scatteing contast ae needed

22 Analying Reflectivity Data We want to find ρ() given a measuement of R(Q ) This invese poblem is not geneally solvable Two methods ae used:. Modelling Paameteie ρ() and use the Paatt method to calculate R(Q ) Refine the paametes of ρ() BUT thee is a family of ρ() that poduce diffeent (Q ) but exactly the same R(Q ): many moe ρ() that poduce simila (Q ). This non-uniqueness can often be satisfactoily ovecome by using additional infomation about the sample (e.g. nown ode of layes). Multiple measuements on the same sample Use two diffeent bacings o fontings fo the unnown layes Allows (Q ) to be calculated R(Q ) can be inveted to give ρ() unless ρ() has bound states (unusual)

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24 Peils of fitting Reflectivity Model Model Scatteing length density [Å - ] 5x -6 4x -6 3x -6 x -6 x -6 Model Model Q [Å - ] Depth into sample [Å] Lac of infomation about the phase of the eflected wave means that pofoundly diffeent scatteing length density pofiles can poduce stiingly simila eflectivities. Ambiguities may be esolved with additional infomation and physical intuition. Sample gowes Othe techniques, e.g., TEM, X-ay Neuton data of vey high quality Well-designed expeiments (simulation is a ey tool) D. Sivia et al., J. Appl. Phys. 7, 73 (99).

25 Diect Invesion of Reflectivity Data is Possible* Use diffeent fonting o bacing mateials fo two measuement of the same unnown film E.g. D O and H O bacings fo an unnown film deposited on a quat substate o Si & Al O 3 as substates fo the same unnown sample Allows Re(R) to be obtained fom two simultaneous equations fo R and R Re(R) can be inveted to yield a unique SLD pofile Anothe possibility is to use a magnetic bacing and polaied neutons SiO Si o Al O 3 substate Unnown film HO o DO * Maa et al Biophys Jounal, 79,333 ()

26 Vesicles composed of DMPC molecules fuse ceating almost a pefect lipid bilaye when deposited on the pue, uncoated quat bloc* (blue cuves) When PEI polyme was added only afte quat was coveed by the lipid bilaye, the PEI appeaed to diffuse unde the bilaye (ed cuves) Reflectivity, R*Q Neuton Reflectivities Lipid Bilaye on Quat Lipid Bilaye on Polyme on Quat Q [Å - ] Scatteing Length Density [Å - ] O D O Scatteing Length Density Pofiles Head σ 5.9 Å Hydogenated Tails Head Length, [Å] Polyme σ 4.3 Å Quat Quat * Data coutesy of G. Smith (LANSCE)

27 Polyme-Decoated Lipids at a Liquid-Ai Inteface*.3% PEG lipid in lipids 4.5% PEG lipid in lipids 9.% PEG lipid in lipids 8-6 SLD Pofiles of PEG-Lipids SLD [Å - ] Blac - pue lipid Blue -.3% PEG Geen - 4.5% PEG Red - 9.% PEG Length [Å] mushoom-tobush tansition Inteface boadens as PEG concentation inceases - this is main effect seen with x-ays R*Q Neuton Reflectivities Pue lipid neutons see contast between heads (.6), tails (-.4), D O (6.4) & PEG (.4) R/R F - X-Ray Reflectivities Pue lipid -9 Lipid + 9% PEG Q [Å - ] x-ays see heads (.65), but all else has same electon density within % (-.33) - -3 Lipid + 9% PEG Q *Data coutesy of G. Smith (LANSCE)

28 Q x Diffuse Scatteing If an inteface is ough it will scatte both speculaly and diffusely If (cosθ cosθ) ( θ θ ) Q ( θ θ ) / 4 θ θ Q x.5q. adians 3 nm - diffuse θ θ specula x i.e. the in - plane length scale pobed can be ~micon!! If the oughness of neighboing intefaces is coelated, the diffuse scatteing will appea as constant-q idges extended in Q x Image fom G.Botons & L. Belloni (CEA/SACLAY).

29 Gaing Incidence Diffaction In pincipal, gaing incidence diffaction can be used to pobe lateal (in-plane) stuctue This is difficult with neutons fo seveal easons: Collimation in x-y plane is needed leading to low intensity Had to pevent the beam going in o out though the sample edge and picing up bul ode athe than suface ode A few expeiments have been done New techniques such as neuton spin echo may mae this type of study easie

30 Obsevation of Hexagonal Pacing of Thead-lie Micelles Unde Shea: Scatteing Fom Lateal Inhomogeneities NEUTRON BEAM QUARTZ o SILICON INLET HOLES TEST SECTION TEFLON LIP OUTLET TRENCH RESERVOIRS TEFLON Speculaly eflected beam Quat Single Cystal x 46Å Up to Micons Flow diection Scatteing patten implies hexagonal symmety HO Thead-lie micelle W. A. Hamilton, P. D. Butle, S. M. Bae, G. S. Smith, John B. Hayte, L. J. Magid, and R. Pynn; Phys. Rev. Lett., 7, 9 (994)

31 Planning a Reflectivity Measuement Simulation of eflectivity pofiles using e.g. Paatt is essential Can you see the effect you want to see? What is the best substate? Which mateials should be deuteated? If you sample involves fee liquid suface you will need to use a eflectomete with a vetical scatteing plane Pepaing good (i.e. low suface oughness) samples is ey Bewae of lage islands Laye thicnesses between Å and 5 Å But don t mix extemes of thicness