Roger Pynn. Basic Introduction to Small Angle Scattering

Transcription

1 by Roge Pynn Basic Intoduction to Small Angle Scatteing

2 We Measue Neutons Scatteed fom a Sample Φ = numbe of incident neutons pe cm pe second σ = total numbe of neutons scatteed pe second / Φ dσ numbe of neutons scatteed pe second into dω = dω Φ dω d σ numbe of neutons scatteed pe second into dω & de = dωde Φ dω de σ measued in bans: 1 ban = 10-4 cm Attenuation = exp(-nσt) N = # of atoms/unit volume t = thickness

3 Scatteing fom Many Atoms Neutons ae scatteed by nuclei The ange of nuclea foces is femtometes much less than the neuton wavelength so the scatteing is point like (ipples on a pond) Enegy of (themal) neuton is too small to change nuclea enegy If the nucleus is fixed, the scatteing is elastic We can add up the (elastic) scatteing fom an assembly of nuclei: dσ i( k0 k ').( Ri R j ) iq. ( Ri R j ) = bib je = bib je dω i, j i, j whee the wavevecto tansfe Q is defined by Q = k ' k b i is called the coheent scatteing length of nucleus i k is the incident neuton wavevecto (π/λ); k is the scatteed wavevecto The calculation assumes the scatteing is weak (called Bon Appoximation) 0

4 The Success of Neuton Scatteing is Rooted in the Neuton s Inteactions with Matte Inteact with nuclei not electons Isotopic sensitivity (especially D and H) Penetates sample containment Sensitive to bulk and buied stuctue x-ay neuton Simple intepetation povides statistical aveages, not single instances Wavelength simila to inte-atomic spacings Enegy simila to themal enegies in matte Nuclea and magnetic inteactions of simila stength

5 Scatteing Tiangle Detecto Incident neutons of wavevecto k Scatteed neutons of wavevecto, k Sample Q k θ θ k Neuton diffaction measues the diffeential scatteing coss section dσ/dω as a function of the scatteing wavevecto (Q) Fo elastic scatteing, k = k so Q = k sin θ = ( π/λ) sin θ The distance pobed in the sample is: d = π / Q (Combining the two equations gives Bagg s Law: λ = d sin θ)

6 Small Angle Neuton Scatteing (SANS) Is Used to Measue Lage Objects (~10 nm to ~1 μm) Small Q => lage d (because d=π/q) Lage d => small θ (because λ = d sin θ) Scatteing at small angles pobes lage length scales Typical scatteing angles fo SANS ae ~ 0.3º to 5º

7 Two iews of the Components of a Typical Reacto-based SANS Diffactomete Note that SANS, like othe diffaction methods, pobes mateial stuctue in the diection of (vecto) Q

8 The NIST 30m SANS Instument Unde Constuction

9 Whee Does SANS Fit As a Stuctual Pobe?

10 Typical SANS Applications Biology Oganization of biomolecula complexes in solution Confomation changes affecting function of poteins, enzymes, potein/dna complexes, membanes etc Mechanisms and pathways fo potein folding and DNA supecoiling Polymes Confomation of polyme molecules in solution and in the bulk Stuctue of micophase sepaated block copolymes Factos affecting miscibility of polyme blends Chemisty Stuctue and inteactions in colloid suspensions, micoemeulsions, sufactant phases etc Mechanisms of molecula self-assembly in solutions

11 Scatteing Length Density Remembe dσ dω = b coh iq. d. e n ( ) nuc What happens if Q is vey small? The phase facto will not change significantly between neighboing atoms We can aveage the nuclea scatteing potential ove length scales ~π/10q This aveage is called the scatteing length density and denoted ρ( ) How do we calculate the SLD? Easiest method: go to By hand: let us calculate the scatteing length density fo quatz SiO Density is.66 gm.cm -3 ; Molecula weight is gm. mole -1 Numbe of molecules pe Å 3 = N = 10-4 (.66/60.08)*N avagado = molecules pe Å 3 SLD=Σb/volume = N(b Si + b O ) = 0.067( ) 10-5 Å - = 4.1 x10-6 Å - A unifom SLD causes scatteing only at Q=0; spatial vaiations in the SLD cause scatteing at finite values of Q

12 SLD Calculation Need to know chemical fomula and density Ente Not elevant fo SLD X-ay values Backgound Detemine best sample thickness Note units of the coss section this is coss section pe unit volume of sample

13 SANS Measues Paticle Shapes and Inte-paticle Coelations oientation paticle x iq p R iq P space P p oientation paticle x iq R R iq space space P P iq space space N N e x d Q F e R R G d N Q F d d e x d e R n R n R d R d e n n d d b d d. 3 P ').( 3 3 ').( ) ( : the paticle fom facto is F(Q) s one at the oigin) and thee' is a paticle at R if paticle coelation function (the pobability that thee the paticle - is whee G ). (. ) ( ) (. ) ( ) ' ( ) ( ' ). ' ( ) ( ' = = Ω = = Ω v v v v ρ ρ σ ρ ρ σ

14 Scatteing fom Independent Paticles Scatteed intensity pe unit volume of Fo identical paticles sample = I( Q) = 1 dσ = dω 1 ρ( ) e v v iq. v d I( Q) = N ( ρ p ρ ) 0 p 1 p e paticle v v iq. v d contast facto paticle fom facto F(Q) N Note that I(0) = ( ρ p ρ0) Paticle concentation c = N whee ρ is so I(0) = the cm ρn paticle mass density and w A ( ρ p ρ ) 0 p p / and povides a paticle molecula weight N A is Avagado's numbe way to find the M = ρ paticle molecula weight w p N A

15 Scatteing fo Spheical Paticles The paticle fom facto F( Q) Fo a sphee of adius R, F(Q) only depends on the magnitude of Q : sin QR QR cosqr 3 0 Fsphee( Q) = 3 0 ( ) at Q 0 3 ( ) j1 QR 0 = QR QR Thus, as Q 0, the total scatteing fom an assembly of uncoelated spheical v paticles[i.e. when G() δ ()]is popotional to the squae of the paticle volume times the numbe of paticles. = de iq. 1 is detemined by the paticle shape. 3j 1 (x)/x x

16 If whee Radius of Gyation Is the Paticle Size Usually Deduced Fom SANS Measuements we measue fom the centoid of the paticle and expand the exponential in the definition of the fom facto at small Q : iq. 3 1 F( Q) = de 0 + iq. d (. ) Q d = g 0 Q 1 cos is the adius of gyation is to SANS data at low Q π 0 π 0 θ sinθ. dθ sinθ. dθ +... (in the so - called Guinie egion) o by plotting ln(intensity) v Q The slope of the data at the lowest values of Q is / d +... = 3 d 0 g = It is usually obtained fom a fit It is easily veified that the expession fo the fom facto of a sphee is a special case of this geneal esult. R 3 d 3 / g 0 Q 1 6 d 3. g e g Q 6.

17 Incoheent Backgound and Absoption In addition to coheent (Q-dependent) scatteing, neutons may be scatteed incoheently Incoheent scatteing is not diectionally (Q) dependent In SANS (o eflectomety) measuements it is a unifom backgound Incoheent scatteing aises fom two souces: Spin incoheent scatteing (the neuton-nucleus state can be singlet o tiplet and these have diffeent scatteing lengths) Isotopic incoheent scatteing Look up incoheent scatteing lengths (included in NIST SLD calculato see next G) Neutons may also be absobed by some nuclei

18 Calculating Fom Factos Note: T(1 mm H O) = 0.5; T(1 mm D O) = 0.9 dσ/dω (H O) = 1 cm -1 ; dσ/dω (D O) = 0.06 cm -1 No backgound H O backgound

19 Contast & Contast Matching O D O Wate HO O * Chat coutesy of Rex Hjelm Both tubes contain boosilicate beads + pyex fibes + solvent. (A) solvent efactive index matched to pyex;. (B) solvent index diffeent fom both beads and fibes scatteing fom fibes dominates

20 Contast aiation CD Deuteated Lipid Head Goup Deuteated Potein Wate Potein Deuteated RNA RNA DNA CONTRAST Δρ Lipid Head Goup CH

21 Isotopic Contast fo Neutons Hydogen Isotope Scatteing Length b (fm) 1 H (11) D (6) 3 T 4.79 (7) Nickel Isotope Scatteing Lengths b (fm) 58 Ni 15.0 (5) 60 Ni.8 (1) 61 Ni 7.60 (6) 6 Ni -8.7 () 64 Ni (7)

22 Using Contast aiation to Study Compound Paticles Examples include nucleosomes (potein/dna) and ibosomes (poteins/rna) iewgaph fom Chales Glinka (NIST)

23 What can we Lean fom SANS? Zeo Q intecept - gives paticle volume if concentation is known ln(i) Guinie egion (slope = - g /3 gives paticle size ) Dimensionality of paticle (slope = -1 fo ods, - fo sheets, -D f fo a mass factal) Pood egion - gives suface aea and suface factal dimension {slope = -(6-D s )} ln(q)

24 Sample Requiements fo SANS Monodispese paticles, non-inteacting to measue shape Concentation: 1-5 mg/ml olume: μl pe sample Data collection time: hs pe sample Typical biology expeiment: -4 days Deuteated solvent is highly desiable Multiple concentations ae usually necessay. Specific deuteation may be necessay. Multiple solvents of diffeent deuteation ae highly desiable contast vaiation.

25 Refeences iewgaphs descibing the NIST 30-m SANS instument SANS data can be simulated fo vaious paticle shapes using the pogams available at: To choose instument paametes fo a SANS expeiment at NIST go to: A vey good desciption of SANS expeiments can be found at:

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