An Introduction to Neutron and X-Ray Scattering: SANS ... · by. Roger Pynn. Indiana University and...

51
by Roger Pynn Indiana University and Oak Ridge National Lab An Introduction to Neutron and X-Ray Scattering: SANS, SAXS & Reflectometry With thanks to various friends for allowing me to use their materials: especially Jen Als-Nielsen, Mike Fitzsimmons & Sunhil Sinha

Transcript of An Introduction to Neutron and X-Ray Scattering: SANS ... · by. Roger Pynn. Indiana University and...

  • by

    Roger PynnIndiana University and Oak Ridge National Lab

    An Introduction to Neutron and X-Ray Scattering:SANS, SAXS & Reflectometry

    With thanks to various friends for allowing me to use their materials: especially Jen Als-Nielsen, Mike Fitzsimmons & Sunhil Sinha

  • Small Angle Scattering (SANS or SAXS) Is Used to Measure Large Objects (~10 nm to ~1 µm)

    θλθ

    θπθππλ

    θ

    /2d smallfor or 2as sin2 law sBragg' rewritecan weso

    /sin4)sin2//(2/2 thatand

    scattering elasticfor sin2

    : thatRecall

    00

    ≈==

    ===

    ==

    π/QdθdλQQk

    kk'-kQ

    i.e. small Q => large length scales

    Scattering at small angles probeslarge length scales

    Typical scattering angles for SANS are ~ 0.3º to 5º

  • Typical SANS/SAXS Applications

    • Biology– Organization of biomolecular complexes in solution– Conformation changes affecting function of proteins, enzymes, protein/DNA

    complexes, membranes etc– Mechanisms and pathways for protein folding and DNA supercoiling

    • Polymers– Conformation of polymer molecules in solution and in the bulk– Structure of microphase separated block copolymers– Factors affecting miscibility of polymer blends

    • Chemistry– Structure and interactions in colloid suspensions, microemeulsions,

    surfactant phases etc– Mechanisms of molecular self-assembly in solutions

  • Biological Applications of SANS

    • Studying Biological Macromolecules in Solution– Proteins– Nucleic Acids– Protein-nucleic acid complexes– Multi-subunit protein complexes – Membranes and membrane components– Protein-lipid complexes

    • One of the issues with studying bio-molecules is that most contain H which gives a large, constant background of incoherent scattering. To avoid this:– Use D2O instead of water as a fluid for suspension – May need to deuterate some molecular components

  • The Concept of SANS & SAXS

    • Measure scattering at small angles & azimuthally average for isotropic samples

  • SANS in Practice

    1. Need a way to determine neutron wavelength– Velocity selector or TOF

    2. To measure the scattering angle accurately, we need to define the incident beam trajectory very well

    3. Increase scattering by increasing lateral size of sample– (2) & (3) => the instrument has to be long (40m max for D11 at ILL)

  • The NIST 30m SANS Instrument Under Construction

  • Where Does SANS Fit As a Structural Probe?

    • SANS resolves structures on length scales of 1 – 1000 nm

    • Neutrons can be used with bulk samples (1-2 mm thick)

    • SANS is sensitive to light elements such as H, C & N

    • SANS is sensitive to isotopes such as H and D

  • Instrumental Resolution for SANS/SAXS

    dominates. resolutionh wavelengtsample, thefrom distance and sizedetector by theset , of lueslargest va at the that Note

    neutrons. Å 15 using instrument SANS m 40 ILL at the m 5about of maximum a achieves This

    . /~Q/2~ isobject observablelargest The /)/2(~/4~)Q/(~Q

    and dominates resolutionangular , of valueAt this/5.1~2 :size beamdirect by the determined is of aluesmallest v The

    h/L.12/5 h, of sample and sourceat apertures

    equal and L of distancesdetector -sample & sample-source equalFor

    0025.0 so small, is and 5% ~ )/( SANS,For

    sin.cos sin4

    rms

    rmsminminrmsrms

    min

    2

    22

    2

    2

    22

    2

    22

    2

    θ

    µλδπ

    λπλπδθθδθδθ

    θθδθ

    θδθδθλδλ

    θδθθ

    λδλδθ

    λπ

    hLLh

    Lh

    QQ

    QQQ

    rms

    rms

    =

    +=

    +=⇒=

    PresenterPresentation NotesTHIS IS A BACK UP SLIDEDelta-theta/theta-min ~ 1 so delta_Q_rms ~ Q_min ~ (2.pi/lambda) h/L

  • The Fermi Pseudo-Potential for Neutrons

    )(2)( so )(2

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    rbm

    rVrderVmkVkdmkYd

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    kmY

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    mkE

    Y

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    kVkW

    Wdd

    d

    rkki

    k

    k

    k

    Y

    rkkikk

    kkk

    kkk

    δπππ

    πσ

    πρ

    πρ

    π

    ρ

    ρπρπ

    σ

    =

    =Ω

    Ω=

    ===Φ

    Ω==

    Ω=Ω=

    =⇒=

    =

    ==

    ΩΦ=

    ∫∑

    Ω→

    Ω→

    Fermi pseudopotential

    PresenterPresentation NotesThe argument here is that we can use Fermi’s golden rule to get an expression for the scattering cross section in terms of the interaction potential between neutron and nucleus. Given our previous expression in terms of b, this allows us to identify the pseudo potential (basically a potential that gives the same phase shift as the actual potential)Fermi’s Golden rule: The transition rate out of a state is: sum over final states of 2*pi/hbar times the square of a matrix element that links the final and initial states times the density of final states per unit energy range. This gives d(sigma)/d(omega) = (m/2*pi*hbar^2)^2 times the square of the FT of V(r)

  • Use V(r) to Calculate the Refractive Index for Neutrons

    materials.most from reflected externally are neutrons 1,ngenerally Since2/1

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    4or 2222

    givesenergy of onConservati

    2 isenergy total themedium theInside

    2

    is vaccuumin neutron ofenergy total)(and kinetic The

    tryreflectome & SANSfor used - (SLD) Density LengthScatteringnuclear thecalled is

    1 where2 :is medium theinside potential average theSo

    nucleus. singlea for )(2)( :by given is potential neutron-nucleus The

    2

    2-6-0

    220

    2222220

    2

    22

    20

    2

    2

    2

    <=

    ==

    =−+=+=

    +

    =

    ==

    =

    πρλ

    ρ

    πρρπ

    ρ

    ρρπ

    δπ

    -nnk/k

    kkmm

    kVmk

    mk

    Vmk

    mkE

    bvolumem

    V

    rbm

    rV

    ii

  • Why do we Care about the Refractive Index?

    • When the wavevector transfer Q is small, the phase factors in the cross section do not vary much from nucleus to nucleus & we can use a continuum approximation

    • We can use all of the apparatus of optics to calculate effects such as:– External reflection from single surfaces (for example from guide surfaces)– External reflection from multilayer stacks (including supermirrors)– Focusing by (normally) concave lenses or Fresnel lenses– The phase change of the neutron wave through a material for applications

    such as interferometry or phase radiography– Fresnel edge enhancement in radiography

  • Refractive Index for x-rays

    { }

    aaaaa

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    za

    a

    rk

    krk

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    rifffk

    rn

    ee

    e

    kkrf

    in

    σρµσπ

    µπρ

    βπρ

    πρβπρ

    µ

    ρ

    µβπρδ

    βδ

    µ

    =

    −=

    −=

    −=

    −=++−≡

    ==

    +−=

    because 4222

    " so

    "2 with"')0(21

    : writealso can we theNote

    isit medium theinside ; is medium theoutside waveThe

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    2 and )0(2 where

    1

    00

    2

    0

    2

    200

    20

    20

    0

    PresenterPresentation NotesSigma_a is the absorption cross section

  • Scattering Length Density

    • Remember

    • What happens if Q is very small? – The phase factor will not change significantly between neighboring atoms– We can average the nuclear scattering potential over length scales ~2�/10Q – This average is called the scattering length density and denoted

    • How do we calculate the SLD?– Easiest method: go to www.ncnr.nist.gov/resources/sldcalc.html– By hand: let us calculate the scattering length density for quartz – SiO2– Density is 2.66 gm.cm-3; Molecular weight is 60.08 gm. mole-1– Number of molecules per Å3 = N = 10-24(2.66/60.08)*Navagadro = 0.0267 molecules per Å3– SLD=Σb/volume = N(bSi + 2bO) = 0.0267(4.15 + 11.6) 10-5 Å-2 = 4.21 x10-6 Å-2

    • A uniform SLD causes scattering only at Q=0; variations in the SLD cause scattering at finite values of Q

    2.2 )(. rnerdb

    dd

    nucrQi

    coh

    ∫ −=Ωσ

    )(r

    ρ

    PresenterPresentation NotesDensity/MW = # of moles per cm^3; multiply by Avagadro’s number gives number of molecules per cm^3.

  • SLD Calculation

    • www.ncnr.nist.gov/resources/sldcalc.html• Need to know chemical formula

    and density

    Not relevant for SLD

    X-ray values

    Background

    Determine best sample thickness

    Note units of the cross section – this is cross section per unit volume of sample

    Enter

  • SANS Measures Particle Shapes and Inter-particle Correlations

    norientatioparticle

    xQi

    p

    RQiP

    spacePp

    norientatioparticle

    xQiRRQi

    space spacePP

    rrQi

    space spaceNN

    exdV

    QF

    eRGRdNVQFdd

    exdeRnRnRdRd

    ernrnrdrdbdd

    2

    .32

    2

    2P

    .32220

    2

    .30

    )'.(33

    )'.(332

    .1)(

    :factor form particle theis )QF( and origin) at the one s there'if Rat particlea is

    rey that theprobabilit (the function ncorrelatio particle-particle theis G where

    ).(.)()(

    .)()'()('

    ).'()('

    ∫∫ ∫

    ∫ ∫

    =

    −=Ω

    −=

    =Ω

    ρρσ

    ρρ

    σ

    These expressions are the same as those for nuclear scattering except for the additionof a form factor that arises because the scattering is no longer from point-like particles

    PresenterPresentation NotesNote that these expressions look exactly the same as the ones for scattering from an assembly of nuclei, except that we now have a form factor involved because the scatterer (the particle) is no longer point like (I.e. it’s dimensions are not
  • Scattering from Independent Particles

    2)(QF

    2

    .220

    2.

    1)()(

    particles identicalFor

    )(11)( sample of eunit volumper intensity Scattered

    −=

    =Ω

    ==

    particle

    rQi

    ppp

    rQi

    rdeV

    VVNQI

    rderVd

    dV

    QI

    ρρ

    ρσ

    contrast factorparticle form factor

    weightmolecular particle thefind way toa provides )()0( so

    number s Avagadro'isN anddensity mass particle theis whereweight molecular particle and / ionconcentrat Particle

    )()0( that Note

    20

    A

    220

    ρρρ

    ρ

    ρ

    ρρ

    −=

    ==

    −=

    pA

    w

    Apwp

    pp

    NcMI

    NVMVNVc

    VVNI

    PresenterPresentation NotesAverage is over particle orientations

  • Scattering for Spherical Particles

    particles. ofnumber thetimes volumeparticle theof square the toalproportion is )]r( )rG( when [i.e. particles

    spherical eduncorrelat ofassembly an from scattering total the0,Q as Thus,

    0Qat )(3)(

    cossin3)(

    :Q of magnitude on the dependsonly F(Q) R, radius of sphere aFor

    shape. particle by the determined is )(factor form particle The

    010

    30

    2

    .2

    δ→→

    =→≡

    −=

    = ∫

    VQRjQRV

    QRQRQRQRVQF

    erdQF

    sphere

    V

    rQi

    2 4 6 8 10

    0.2

    0.4

    0.6

    0.8

    1

    3j1(x)/x

    xQ and (a) axismajor the

    between angle theis where)cossin (

    :by R replace particles ellipticalFor

    2/12222

    ϑ

    ϑϑ baR +→

  • Radius of Gyration Is the Particle “Size” Usually Deduced From SANS Measurements

    result. general thisof case speciala is spherea offactor form for the expression that theifiedeasily ver isIt /3.r is Q of ueslowest val at thedata theof slope The

    .Q vty)ln(Intensi plottingby or region)Guinier called-so the(in Qlow at data SANS to

    fita from obtainedusually isIt ./ is gyration of radius theis r where

    ...6

    1....sin

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    21

    ....).(21.)(

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    2g

    2

    332g

    60

    22

    03

    32

    0

    0

    22

    0

    3230

    .

    22

    0

    0

    ∫∫

    ∫∫∫

    =

    +−=

    +−=

    +−+≈=

    VVg

    rQg

    V

    V

    VV

    rQi

    V

    rdrdRr

    eVrQ

    Vrd

    rdr

    d

    dQV

    rdrQrdrQiVerdQF

    r

    g

    π

    π

    θθ

    θθθ

    PresenterPresentation NotesThe radius of gyration formula is easily verified for a sphere from the form given on the previous VG. The way I have written the eqn with integral of cos squared times sin over intergral of sin comes about because I need the cos squared for the dot product of Q and r. The sin theta d(theta) piece is just the usual volume element in spherical coordinates. I divide by the integral of sin theta d(theta) because that’s the theta dependence that I would normally have in the integral of r squared over the bodyNeed to be careful when we fit SANS data at low Q that we really are in the Guinier region

  • Determining Particle Size From Dilute Suspensions

    • Particle size is usually deduced from dilute suspensions in which inter-particle correlations are absent

    • In practice, instrumental resolution (finite beam coherence) will smear out minima in the form factor

    • This effect can be accounted for if the spheres are mono-disperse • For poly-disperse particles, maximum entropy techniques have been used

    successfully to obtain the distribution of particles sizes

  • Correlations Can Be Measured in Concentrated Systems

    • A series of experiments in the late 1980’s by Hayter et al and Chen et al produced accurate measurements of S(Q) for colloidal and micellar systems

    • To a large extent these data could be fit by S(Q) calculated from the mean spherical model using a Yukawa potential to yield surface charge and screening length

  • Contrast & Contrast Matching

    Both tubes contain borosilicate beads + pyrex fibers + solvent. (A) solvent refractive index matched to pyrex;. (B) solvent index different from both beadsand fibers – scattering from fibers dominates

    O

    O

    Wate

    rD2O

    H2O

    * Chart courtesy of Rex Hjelm

  • Contrast Variation

    CD2

    Deuterated Lipid Head Group

    CONTRAST∆ρ

    Lipid Head Group

    CH2

  • Isotopic Contrast for Neutrons

    PresenterPresentation NotesNeutron scattering lengths of isotopes can be very differentPotential for changing local scattering length density by isotopic labeling: 1H -> 2D to change r of a component or portion of the structure is an important application of this.Applications in neutron optics.Neutron scattering is due largely to the nuclear potential. Consequently, different isotopes have different scattering lengths. The difference between hydrogen and deuterium is particularly important in this regard, as this provides a ready means of labeling molecules containing light elements. Other elements, such as tungsten and nickel, share this attribute. The later is important for the development of neutron optical devices, such as neutron reflecting mirrors.

    Hydrogen Isotope

    Scattering Length

    b (fm)

    1H

    -3.7409 (11)

    2D

    6.674 (6)

    3T

    4.792 (27)

    Nickel

    Isotope

    Scattering Lengths b (fm)

    58Ni

    15.0 (5)

    60Ni

    2.8 (1)

    61Ni

    7.60 (6)

    62Ni

    -8.7 (2)

    64Ni

    -0.38 (7)

  • Using Contrast Variation to Study Compound Particles

    Viewgraph from Charles Glinka (NIST)

    Examples include nucleosomes(protein/DNA) and ribosomes(poteins/RNA)

    12

    122121

    22

    22

    21

    21

    2

    2.

    21.

    1

    )sin()()(

    )()()(

    )(1 2

    QRQRQFQF

    QFQFQI

    rderdeVNQI

    V V

    rQirQi

    ρρ

    ρρ

    ρρ

    ∆∆

    +∆+∆=

    ∆+∆= ∫ ∫

  • Porod Scattering

    function.n correlatio for the form (Debye) h thisfactor wit form theevaluate toandr smallat V)]A/(2 a[with ..brar-1G(r) expand toisit obtain y toAnother wa

    smooth. is surfaceparticle theprovided shape particleany for Q as holds and law sPorod' is This

    surface. ssphere' theof area theisA where2)(2

    9 Thus

    )resolutionby out smeared be willnsoscillatio (the averageon 2/9 Q as cos9

    cossin9)()(

    cos.sin9

    radius) sphere theis R where1/RQ (i.e. particlespherical afor Q of valueslargeat )(F(Q) ofbehavior theexamine usLet

    2

    44

    22

    2

    22

    224

    2

    3242

    42

    π

    π

    =++=

    ∞→

    =→

    =

    ∞→→

    −=

    −=

    >>

    QA

    QRVF(Q)

    VQRV

    QRQRQRVQR

    QRQRQRQRV(QR)F(Q)

    QR

  • Scattering From Fractal Systems

    • Fractals are systems that are “self-similar” under a change of scale I.e. R -> CR

    • For a mass fractal the number of particles within a sphere of radius R is proportional to RD where D is the fractal dimension

    • Growth in 2D leads to a variety of values of D

    2.dimension of surfacessmooth for form

    Porod the toreduces which )( that provecan one fractal, surface aFor

    sin..12

    )4/.(sin..2)(.)( and

    )4/()( dRR and R distancebetween particles ofnumber )(.4

    Thus

    6

    2

    3.

    3

    12

    sD

    DD

    D

    DRQi

    D

    D

    QconstQS

    Qconstxxdx

    Qc

    RcQRRdRQ

    RGeRdQS

    RcRGdRcRRGdRπR

    ==

    ==

    =∴

    =+=

    ∫ ∫ ππ

    π

    PresenterPresentation NotesNote that we should really be FT’ing G(r) –1. However this won’t matter until we get to a fractal dimension of 3, at which point we need the next term in G(r) as we saw on the last slide. Thus, the limit of the surface fractal gives Porod’s law. The mass fractal gives a slope exponent greater >-3, whereas a surface fractal gives an exponent in the range –4< exponent < -3. Porod’s law is the limiting (steepest slope) for a smooth surface

  • Typical Intensity Plot for SANS From Disordered Systems

    ln(I)

    ln(Q)

    Guinier region (slope = -rg2/3 gives particle “size”)

    Mass fractal dimension (slope = -D)

    Porod region - gives surface area andsurface fractal dimension {slope = -(6-Ds)}

    Zero Q intercept - gives particle volume if concentration is known

  • Surface Reflection Is Very Different From Most Neutron and X-ray Scattering

    • We worked out the cross section by adding scattering from different scattering centers– We ignored double scattering processes because these are usually very weak

    • This approximation is called the Born Approximation

    • Below an angle of incidence called the critical angle, neutrons and x-rays are perfectly reflected from a smooth surface– This is NOT weak scattering and the Born Approximation is not applicable to

    this case

    • Specular reflection is used:– In neutron guides and x-ray mirrors– In multilayer monochromators and neutron polarizers– To probe surface and interface structure in layered systems

  • Various forms of small (glancing) angle neutron reflection

    Viewgraph from M. R. Fitzsimmons

  • Only Neutrons With Very Low Velocities Perpendicular to a Surface Are Reflected

    α

    α’nickelfor )(1.0)( :Notequartzfor )(02.0)(

    sin)/2(

    A 1005.2 quartzFor

    4 is reflection external for total of valuecritical The

    4or 4sin'sin i.e.4)sin(cos)'sin'(cos 4 Since

    LawsSnell'obey Neutrons'/coscosn i.e 'cos'coscos

    :so surface the toparallel velocity neutron thechangecannot surface The/

    critical

    critical

    critical

    1-3

    00

    20

    2220

    22

    2220

    22220

    2

    00

    0

    0

    0

    AA

    k

    xk

    kk

    kkkkkkkk

    nkkk

    nkk

    o

    o

    critical

    critical

    zz

    zz

    z

    z

    λα

    λα

    αλπ

    πρ

    πρπραα

    πρααααπρ

    ααααα

    ⇒=

    =

    =

    −=−=

    −+=+−=

    ===

    =

    How do we make a neutron bottle?

    k0

    k

  • magneticpart+

    magneticpart+

    DispersionAbsorptionMinus!!

    Refractive Index: X-Rays & Neutrons

  • Reflection of Neutrons by a Smooth Surface: Fresnel’s Law

    n = 1-λ2ρ/2π

    TTRRII

    TRI

    kakaka

    aaa

    =+

    =+⇒=

    (1) 0zat & of

    continuityψψ

    )/()(/by given is ereflectanc sosinsin

    sinsin

    )()( (3) & (1)

    coscos :Law sSnell' (2) & (1)(3) sinsin)((2) coscoscos

    :surface the toparallel andlar perpendicu components

    TzIzTzIzIR

    Iz

    Tz

    RI

    RI

    TRI

    TRI

    kkkkaarkkn

    aaaa

    nnkakaa

    nkakaka

    +−==

    =′

    ≈′

    =+−

    =>

    ′==>

    ′−=−−

    ′=+

    αα

    αα

    αααα

    ααα

  • What do the Amplitudes aR and aT Look Like?

    • For reflection from a flat substrate, both aR and aT are complex when k0 < 4�〉I.e. below the critical edge. For aI = 1, we find:

    0.005 0.01 0.015 0.02 0.025

    -1

    -0.5

    0.5

    1

    Real (red) & imaginary (green) parts of aRplotted against k0. The modulus of aR is plotted in blue. The critical edge is at k0 ~ 0.009 A-1 . Note that the reflected wave is completely out of phase with the incident wave at k0 = 0

    Real (red) and imaginary (green) partsof aT. The modulus of aT is plotted inblue. Note that aT tends to unity at large values of k0 as one would expectand that the transmitted intensity peaksat the critical edge.

    0.005 0.01 0.015 0.02 0.025

    -1

    -0.5

    0.5

    1

    1.5

    2

  • Penetration Depth

    • In the absence of absorption, the penetration depth becomes infinite at large enough angles

    • Because kz is imaginary below the critical edge (recall that ), the transmitted wave is evanescent

    • The penetration depth

    • Around the critical edge, one maytune the penetration depth to probedifferent depths in the sample

    πρ4202 −= zz kk

    )Im(/1 k=Λ

  • Measured Reflectivity• We do not measure the reflectance, r, but the reflectivity, R given by:

    R = # of neutrons reflected at Qz = r.r*# of incident neutrons

    i.e., just as in diffraction, we lose phase information

    • Notice, also, that the measurement averagesthe reflectivity over the surface of the sample:i.e. measured reflectivity depends on

    ∫ ∫= ),,(1)( zyxdydxS

    z ρρMeasured and Fresnel reflectivitiesfor water – difference is due to surfaceroughness

  • Fresnel’s Law for a Thin Film• r=(k0z-k1z)/(k1z+k0z) is Fresnel’s law

    • Evaluate with ρ=4.10-6 A-2 gives thered curve with critical wavevectorgiven by k0z = (4�〉 )1/2

    • If we add a thin layer on top of thesubstrate we get interference fringes &the reflectance is given by:

    and we measure the reflectivity R = r.r*

    • If the film has a higher scattering length density than the substrate we get the green curve (if the film scattering is weaker than the substance, the green curve is below the red one)

    • The fringe spacing at large k0z is ~ �/t (a 250 A film was used for the figure)

    0.01 0.02 0.03 0.04 0.05

    -5

    -4

    -3

    -2

    -1

    Log(r.r*)

    k0z

    tki

    tki

    z

    z

    errerrr

    1

    1

    21201

    21201

    1++

    =

    substrate

    01 Film thickness = t2

    PresenterPresentation NotesDon’t forget to say that we measure the reflectivity which is the square of the modulus of the reflectanceNote that we can get the 1/Q**4 law out of the Fresnel form. Multiply top and bottom by (k0+k1) and then replace k0**2-k1**2 by 4 pi ro. The bottom goes to (2 k0)**2 = Q**2 at large k0. This gives r ~ 4 pi ro/Q**2

  • Multiple Layers – Parratt Iteration (1954)

    • The same method of matching wavefunctions and derivatives at interfaces can be used to obtain an expression for the reflectivity of multiple layers

    jjz

    jjzjjz

    zikjjj

    zikjjjzik

    j

    jj

    eXr

    eXre

    TR

    X1,

    1,,

    211,

    211,2

    1 +

    +

    ++

    ++−

    +

    +==

    1,,

    1,,1, where

    +

    ++ +

    −=

    jzjz

    jzjzjj kk

    kkr

    ave)incident w amplitudeunit & substrateinside fromback coming nothing (i.e.

    1 and 0 withiterationStart

    111 === ++ TXR NN

    Image from M. Tolan

  • Dealing with Varying Density Profiles

    • Any SLD depth profile can be “chopped” into slices

    • The Parratt formalism allows the reflectivity to be calculated

    • A thickness resolution of 1 Å is adequate – this corresponds to a value of Qz where the reflectivity has dropped below what neutrons can normally measure

    • Computationally intensive!!

    Image from M. Tolan

  • When Does a “Rough” Surface Scatter Diffusely?

    • Rayleigh criterion

    path difference: ⊗r = 2 h sinθ

    phase difference: ⊗∏ = (4�h/ ) sinθ

    boundary between rough and smooth: ⊗∏= �/2

    that is h < /(8sinθ) for a smooth surface

    h

    where g = 4 � h sin θ / = Qz h

  • Surface Roughness

    • Surface roughness causes diffuse (non-specular) scattering and so reduces the magnitude of the specular reflectivity

    k 1 k 2

    k t 1

    z

    x

    θ 1 θ 2 z = 0

    • The way in which the specular reflection is damped depends on the length scale of the roughness in the surface as well as on the magnitude and distribution of roughness

    “sparkling sea”model -- specular from many facets

    each piece of surfacescatters indepedently-- Nevot Croce model

    Note that roughnessintroduces a SLDprofile averaged overthe sample surface

    212 σtzIzkk

    FeRR−=

  • Kinematic (Born) Approximation

    • We defined the scattering cross section in terms of an incident plane wave & a weakly scattered spherical wave (called the Born Approximation)

    • This picture is not correct for surface reflection, except at large values of Qz• For large Qz, one may use the definition of the scattering cross section to

    calculate R for a flat surface (in the Born Approximation) as follows:

    z

    4222

    22)'.(2

    00

    20

    yx

    yx

    Q largeat form Fresnel theas same theis that thisshow easy to isIt

    /16 so )()(4'

    :substratesmooth afor get section we cross a of definition theFrom. sin so cos because

    sinsin1

    sin1

    sin

    )sinLL( sampleon incident neutrons ofnumber LL size of sample aby reflected neutrons ofnumber

    zyxyxz

    rrQi

    xx

    yx

    yxyxyx

    QRQQLLQ

    erdrddd

    dkdkkkk

    dkdkdd

    LLd

    dd

    LLLL

    R

    ρπδδπρρσ

    ααα

    ασ

    ασ

    αασ

    α

    ===Ω

    −==

    Ω=Ω

    Ω==

    Φ==

    ∫ ∫

    ∫∫

    PresenterPresentation NotesMake the point that the Born approx is only valid far from the critical edge where the scattering is weakNote that the d(sigma)/d(omega) integral is done in the following way: one integral over dxdy can be done immediately because of translational invariance in the surface. This gives the L(x)L(y) term. The second integral over dxdy gives the delta functions, each multiplied by 2*pi. Finally, the dz integral is from minus infinity to 0. Doing the integral picks up the factor of one over qz each time. The value of the integral is unity at z=0 and zero at z = minus infinity

  • Reflection by a Graded Interface

    ion.justificat physical good havemust dataty reflectivifit torefinedmodels that means This n.informatio phaseimportant lack generally we

    because (z),obtain ouniquely t inverted becannot dataty reflectivi :pointimportant an sillustrateequation eapproximat This (z).1/cosh as

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    2

    2z

    F

    2

    4

    22

    2

    2

    ρ

    ρ

    ρ

    ρπρπρ

    ∫∫

    =

    ==

    dzedz

    zdRR

    dzedz

    zdQ

    dzezQ

    R

    ziQF

    ziQ

    z

    ziQ

    z

    z

    zz

    PresenterPresentation NotesTry doing for yourself the simple problem of two layers on a substrate using the formula on this slide. This gives 3 delta functions for d(ro)dz: one of strength a at z=0, one of strength b at z=d1 and one of strength c at z=d2. Plugging these into the formula, you immediately find that you can’t tell the difference between the situation mentioned in the last sentence and one in which there is a delta function of strength c at z=d1 and a delta function of strength b at z=d2

  • Electron DensityProfile !

    Refractive Index: X-Rays

    E = 8 keV λ = 1.54 Å

  • Comparison of Neutron and X-Ray Reflectivity

    Viewgraph courtesy of M. Tolan

    Neutrons often provide better contrast and don’t damage samplesX-rays provide better Q resolution and higher Q values

  • The Goal of Reflectivity Measurements Is to Infer a Density Profile Perpendicular to a Flat Interface

    • In general the results are not unique, but independent knowledge of the system often makes them very reliable– It is possible to get unique results by using different ”backings”

    • Frequently, layer models are used to fit the data• Advantages of neutrons include:

    – Contrast variation (using H and D, for example)– Low absorption – probe buried interfaces, solid/liquid interfaces etc– Non-destructive– Sensitive to magnetism– Thickness length scale 10 – 5000 Å

    • Issues include– Generally no unique solution for the SLD profile (use prior knowledge)– Large samples (> 1 cm2) with good scattering contrast are needed for neutrons

  • What do Specular and Off-specular scattering measure?

    • Specular reflectivity measures variations in scattering density normal to surface (averaged over x,y plane)

    • Off-specular scattering measures (x,y) variations of scattering density, e.g. due to roughness, magnetic domains, etc.

  • Vector Diagram for Q in GISAXS

    Note that the length scales measured parallel to the surface when both ki and kf are in the specular plane are much greater than the length scale measured perpendicular to the surface

  • X-Ray Reflectometers

    LaboratorySetup

    HASYLAB: CEMO

    SynchrotronSetup

  • SynchrotronSetup (APS)

    Reflectivity from Liquids I

  • END

    Small Angle Scattering (SANS or SAXS) Is Used to Measure Large Objects (~10 nm to ~1 mm)Typical SANS/SAXS ApplicationsBiological Applications of SANSThe Concept of SANS & SAXSSANS in PracticeThe NIST 30m SANS Instrument Under ConstructionWhere Does SANS Fit As a Structural Probe?Instrumental Resolution for SANS/SAXSThe Fermi Pseudo-Potential for NeutronsUse V(r) to Calculate the Refractive Index for NeutronsWhy do we Care about the Refractive Index?Refractive Index for x-raysScattering Length DensitySLD CalculationSANS Measures Particle Shapes and Inter-particle CorrelationsScattering from Independent ParticlesScattering for Spherical ParticlesRadius of Gyration Is the Particle “Size” Usually Deduced From SANS MeasurementsDetermining Particle Size From Dilute SuspensionsCorrelations Can Be Measured in Concentrated SystemsContrast & Contrast MatchingContrast VariationIsotopic Contrast for Neutrons Using Contrast Variation to Study Compound ParticlesPorod ScatteringScattering From Fractal SystemsTypical Intensity Plot for SANS From Disordered SystemsSurface Reflection Is Very Different From Most Neutron and X-ray ScatteringVarious forms of small (glancing) angle neutron reflectionOnly Neutrons With Very Low Velocities Perpendicular to a Surface Are ReflectedSlide Number 32Reflection of Neutrons by a Smooth Surface: Fresnel’s LawWhat do the Amplitudes aR and aT Look Like?Penetration DepthMeasured ReflectivityFresnel’s Law for a Thin FilmMultiple Layers – Parratt Iteration (1954)Dealing with Varying Density ProfilesWhen Does a “Rough” Surface Scatter Diffusely?Surface RoughnessKinematic (Born) ApproximationReflection by a Graded InterfaceSlide Number 44Comparison of Neutron and X-Ray ReflectivityThe Goal of Reflectivity Measurements Is to Infer a Density Profile Perpendicular to a Flat InterfaceWhat do Specular and Off-specular scattering measure?Vector Diagram for Q in GISAXSSlide Number 49Slide Number 50END