Small-Angle X-ray Scattering(SAXS)SPring-8/JASRI
Naoto Yagi1Wikipedia
Small-angle X-ray scattering (SAXS) is a small-angle scattering (SAS)technique where the elastic scattering of X-rays (wavelength 0.1 - 0.2 nm) bya sample which has inhomogeneities in the nm-range, is recorded at very low
angles (typically 0.1 - 10°). This angular range contains information about
the shape and size of macromolecules, characteristic distances of partiallyordered materials, pore sizes, and other data. SAXS is capable of delivering
structural information of macromolecules between 5 and 25 nm, of repeat
distances in partially ordered systems of up to 150 nm. USAXS (ultra-smallangle X-ray scattering) can resolve even larger dimensions.
SAXS and USAXS belong to a family of X-ray scattering techniques that areused in the characterization of materials. In the case of biologicalmacromolecules such as proteins, the advantage of SAXS overcrystallography is that a crystalline sample is not needed. NMR methods
encounter problems with macromolecules of higher molecular mass (>30000-40000). However, owing to the random orientation of dissolved or
partially ordered molecules, the spatial averaging leads to a loss of
information in SAXS compared to crystallography.
2

“scattering” vs. “diffraction”
scattering
X-ray changes its direction by interaction
with a non-periodic material
diffraction
X-ray changes its direction by interaction
with a periodic material

However, these definitions are not always obeyed.
3Crystal does not diffract X-ray
Bragg reflection --- not a reflection

Scattered X-rays form apattern on the detector
because of interference.

An X-ray changes its direction
because of Thomson scattering
X-ray

Diffraction is not involved.

Young’s slit experiment
Intensity variationis created byinterference.
brightdark

Light changes direction
because of diffraction.

“Diffraction” in a dictionary
Webster's 1913 Dictionary
Diffraction, n. (Opt.)
The deflection and decomposition of light in passing by
the edges of opaque bodies or through narrow slits,causing the appearance of parallel bands or fringes ofprismatic colors, as by the action of a grating of finelines or bars.
Remarked by Grimaldi (1665), and referred by him
to a property of light which he called diffraction.
--Whewell

Crystals do NOT diffract X-rays

Thomson scattering
X-ray is a travelling wave
wavelength λElectric field by an X-ray

𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 𝑒𝑒 −𝑖𝑖(2𝜋𝜋𝜋𝜋𝑡𝑡+𝜙𝜙0 )X-ray with thesame wavelength

an electron vibrates
in an electric fieldsame wavelength= elastic scattering

This is the scattered X-ray
Thomson scattering takes
place in all directions,although the intensity
depends on the direction.

“Interference” is the key phenomenon

At a large distancescattered waves interferewith each other.
“interference” of waves
X-ray
travellingwave

two electrons
an interference pattern is
not always periodic.

Scattering from two electrons
path difference∆= 𝑘𝑘 � 𝑟𝑟⃗ − 𝑘𝑘0 � 𝑟𝑟⃗ = 𝑆𝑆⃗ � 𝑟𝑟⃗

scattering vector 𝑆𝑆⃗ = 𝑘𝑘 − 𝑘𝑘0
phase difference in radianδφ = 2π∆ / λAk0

= 𝐸𝐸0 (𝑒𝑒 −𝑖𝑖(2𝜋𝜋𝜔𝜔𝑡𝑡) +𝑒𝑒 −𝑖𝑖(2𝜋𝜋𝜔𝜔𝑡𝑡+𝛿𝛿𝛿𝛿) )kincident vectork0superposition of waves at the detector
𝐸𝐸 𝑡𝑡

scattered vectorkr

2θB − k0 ⋅ r

 k ⋅rwith N electrons
𝑁𝑁𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 𝑒𝑒 −2𝜋𝜋𝑖𝑖𝜔𝜔𝑡𝑡 � 𝑒𝑒 −𝑖𝑖𝛿𝛿𝜙𝜙𝑗𝑗
𝑗𝑗=1

detectors measure energy
2𝐼𝐼 = 𝐸𝐸 𝑡𝑡

Comparison with Bragg refelction
condition of constructive interference is the same phase
𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 (𝑒𝑒 −𝑖𝑖(2𝜋𝜋𝜔𝜔𝑡𝑡) +𝑒𝑒 −𝑖𝑖(2𝜋𝜋𝜔𝜔𝑡𝑡+𝛿𝛿𝛿𝛿) )phase differenceδφ = 2π∆ / λwhen this is an integer
multiple of 2π, constructive
interference takes place.
This also applied to a slit
diffraction experiment.

constructive interference occurs when
the path difference is a multiple ofwavelength
This also applied to Bragg reflection

Relation to the scattering angle
definition of q2𝜋𝜋𝑞𝑞⃗ =𝑆𝑆⃗𝜆𝜆4𝜋𝜋 sin 𝜃𝜃𝑞𝑞 = 𝑞𝑞⃗ =𝜆𝜆k1k0

Ascattered vector

path difference𝛿𝛿𝛿𝛿 = 2𝜋𝜋∆/𝜆𝜆 = 2𝜋𝜋𝑆𝑆⃗ � 𝑟𝑟/𝜆𝜆⃗ = 𝑞𝑞⃗ � 𝑟𝑟⃗

incident vector𝑟𝑟⃗

Q (Momentum transfer)

𝑘𝑘1
Bwith N electronsθ𝑘𝑘0

𝑆𝑆⃗ = 𝑘𝑘1 − 𝑘𝑘0

When B is the origin,𝑟𝑟⃗ is the coordinate of the electron

𝑁𝑁𝐸𝐸 𝑡𝑡, 𝑞𝑞⃗ = 𝐸𝐸0 𝑒𝑒 −2𝜋𝜋𝑖𝑖𝜔𝜔𝑡𝑡 � 𝑒𝑒 −2𝜋𝜋𝑖𝑖𝜔𝜔𝑡𝑡𝑁𝑁

𝑗𝑗=1= 𝐸𝐸0 𝑒𝑒 −2𝜋𝜋𝑖𝑖𝜔𝜔𝑡𝑡 � 𝑒𝑒 −𝑞𝑞�𝑟𝑟𝑗𝑗
𝑗𝑗=1

Atomic scattering factordistribution of electrons
differs in each element

Assumption:• influence by nucleus is ignored• nucleus is too heavy to scatter X-rays

Scattering from a molecule
With N electrons
𝑁𝑁𝐸𝐸 𝑞𝑞⃗ = 𝐸𝐸0 � 𝑒𝑒 −𝑖𝑖𝑞𝑞�𝑟𝑟⃗𝑗𝑗With N atoms

𝑗𝑗=1

𝑁𝑁F 𝑞𝑞⃗ = � 𝑓𝑓𝑗𝑗 𝑒𝑒 −𝑖𝑖𝑞𝑞�𝑟𝑟⃗𝑗𝑗
𝑗𝑗=1

fj atomic scattering factor
rj coordinate of the atom

Scattering and interference due to a crystal
In an atomic crystal
𝑁𝑁𝐸𝐸 𝑞𝑞⃗ = � 𝑓𝑓𝑗𝑗 𝑒𝑒 −𝑖𝑖𝑞𝑞�𝑟𝑟⃗𝑗𝑗
𝑗𝑗=1

Fourier transform∞𝐹𝐹 𝜔𝜔 = � 𝑓𝑓 𝑡𝑡 𝑒𝑒 −𝑖𝑖𝜔𝜔𝑡𝑡 𝑑𝑑𝑑𝑑−∞

In a crystal, rj is a periodic. Thus, at aparticular q, q·rj is always a multiple of2π, causing constructive interference.
This is Bragg reflection.

Scattering and interference from a non-crystalline
material (with a random orientation)
electron density distribution

𝐼𝐼 𝑞𝑞⃗ = 𝐹𝐹 2 𝑞𝑞⃗ = � 𝜌𝜌 𝑟𝑟⃗ 𝑒𝑒 −𝑖𝑖𝑞𝑞�𝑟𝑟⃗

This is Fourier transform. Thus,
reverse Fourier transform ispossible, but the phase is notavailable.

2

average over all directions
𝐼𝐼 𝑞𝑞⃗ = 𝐼𝐼 𝑞𝑞⃗

Ω𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖 𝑞𝑞⃗ � 𝑟𝑟⃗

Ω =sin 𝑞𝑞𝑞𝑞
𝑞𝑞𝑞𝑞in a centrosymmetric object, the density ρ is a function ofradius r, ρ(r).
𝐹𝐹 𝑞𝑞 = � 4𝜋𝜋𝑟𝑟 2 𝑑𝑑𝑑𝑑 𝜌𝜌 𝑟𝑟 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑞𝑞⃗ � 𝑟𝑟⃗
𝑗𝑗

all electrons in a shellwith radius r2
Ω = 4𝜋𝜋 � 𝜌𝜌 𝑟𝑟 𝑟𝑟

sin 𝑞𝑞𝑞𝑞
𝑑𝑑𝑑𝑑𝑞𝑞𝑞𝑞

In the case of a solid spherea homogeneous sphericalparticle (radius R)“form factor”

Non-Crystalline Diffraction(NCD)• NCD includes all interactions between
non-periodic materials and X-rays.
• Diffraction and scattering are the same
phenomenon in principle, diffraction being aspecial case of scattering• Since a single-crystal diffraction is a
special case, it is not included in NCD.
17

NCD in material science• Most materials around us are non-crystalline.
• Why crystalline materials are so important?
-- Proteins: atomic structure cannot be
obtained without crystallization-- Crystalline materials have uniquecharacteristics• Metallic materials are not single crystals.
• Non-crystalline materials are not simple instructure (form factor vs. structure factor)-- hierarchical structure18Form and structure factors
Bragg diffraction
form factor × structure factor“speckle pattern”
form factor (with fixedorientation)

by Prof. Nakasako

by Prof. Nakasako

Form and structure factors

20

Hierarchy• At different size scales, there are differentstructures.
• Small angle scattering: large structure• In the case of protein solution scattering
Small-angle: shape of molecules
Medium-angle: domain structure
Wide-angle : secondary structure
21

Protein solution scattering• All particles have the same structure• All particles are random in orientation• There is no interaction between theparticles.
• This is an ideal and special case of smallangle scattering.
22

SAXS from protein solution is an averageby Prof. Nakasako

Apo-Ferritin, Mw. 480 kDaq (nm-1)

0.0

0.5

1.0

1.5

23

SPring-8, BL45 HP

Meaning of the scattering curve24Raw data25

saxs8Circular-averaging: obtain a
one-dimensional profile
from a two-dimensional
scattering pattern.

Calibration of q is donewith the known spacing ofAgBehenate (58.38Å)26

Circular-averaged data
2 d sin θ =λ1/d = 2 sin θ / λq = 4π sin θ / λ (=h)
q=2π/d----- q = 6 / d----- d = 6 / q
S=1/dθ

ors = 4π sin θ / λ2θ
27

Scattering from protein• Subtract scattering from the solvent (buffer)
IProtein = ISample – IBuffer• Absorption by protein is so small that it canbe neglected.
Scattering from protein and solvent must
be measured under the same conditions(same exposure time, same sample cell).
28

Subtraction of solvent scattering

protein + solvent
protein only

solvent

29

Scattering from protein
All analysis will be made on
this profile.
molecular shape

It is recommended tomeasure this with different
protein concentrations.

domain structure
secondary structure30Software:
ATSAS
Downloadable
World-standard?
With English manual

http://www.emblhamburg.de/ExternalInfo/Research/Sax/downl
oad.html

31

PRIMUS

Abscissa iss(=q) in Å-132

If the protein is a sphere
Scattering from a sphere with radius R

If sphere, radius can be obtained from the peak
position.

In the data in the previous slide,the peak is at q=0.07 Å-1 andthus R is 83Å.

33

Calmodulin

http://scattering.tripod.com/
34

Radius of gyration : Rg
• 𝑅𝑅𝑔𝑔2 =∫ 𝜌𝜌(𝑟𝑟)�𝑟𝑟 2 �𝑑𝑑𝑑𝑑∫ 𝜌𝜌(𝑟𝑟)�𝑑𝑑𝑑𝑑

• 𝐼𝐼 𝑞𝑞 = 𝑁𝑁𝑝𝑝 𝑛𝑛𝑒𝑒2 exp −2𝑅𝑅𝑔𝑔

3𝑞𝑞 2

approximation

i.e. near the origin, the intensity distribution is Gaussian• ln(I(q)) vs. q2

Guinier plot
35

Radius of gyration
Rg depends on the shape.

36

Rg “Guinier”
𝐼𝐼 𝑞𝑞= 𝑁𝑁𝑝𝑝 𝑛𝑛𝑒𝑒2 exp

𝑅𝑅𝑔𝑔2 2− 𝑞𝑞337

The region where the Guinier
approximation is valid
Rgqmax < 1.3

region to be fitted

38

Radius of gyration of a sphere•

𝑅𝑅𝑔𝑔2 =∫ 𝜌𝜌(𝑟𝑟)�𝑟𝑟 2 �𝑑𝑑𝑑𝑑∫ 𝜌𝜌(𝑟𝑟)�𝑑𝑑𝑑𝑑

Rg= R√3/5spheroid Rg=√(2a2+b2)/5

If sphere, radius R can be calculated from Rg• Rg =59.7Å gives R=77Å
close to 83Å obtained from the peak position→This protein can be approximated as a sphere with aradius of about 80Å.
39

Dependence on protein
concentration
Ideally, infinite dilution

80
70

Rg (A)

60

y = -0.628x + 67.05

50
40
30
20
10002

4

68

10

12

Protein conc. (mg/mL)40

I(0) scattering at the origin
I(0)/c is proportional
to the molecularweight.

3025

y = -0.2349x + 28.426

I(0)/c2015Molecular weight
can be obtained bycomparison with astandard protein.

1050024681012Protein conc. (mg/mL)41

p(r) function• pair distance distribution function• Linear self-convolution of electron density• p(r) is uniquely determined by the protein structure.
• p(r) is obtainable from the SAXS profile.

PRIMUS↓GNOM↓∞

𝑝𝑝 𝑟𝑟 = 4𝜋𝜋 � 𝐼𝐼 𝑞𝑞 𝑞𝑞𝑞𝑞 sin 𝑞𝑞𝑞𝑞 d𝑞𝑞
0Calculate with a givenRmax42

Calmodulinp(r) function

max length

http://scattering.tripod.com/43A hypothesis that can be tested
by a SAXS measurement• Find out a difference in structure.
• Obtain Rg and p(r) from a crystal structure
and compare with the measurement.
• Test a hypothetical structure.

44

Calculate SAXS from atomic coordinates
CRYSOL

Version 2.6 -- 26/01/05-- Program started at 15-Oct-2006 14:17:29—-- No data fitting, parameters entered manually ---------- Real space resolution and grid --------

Solvent density ........................................ : 0.3340
Contrast of the solvation shell ........................ : 3.000e-2
Average atomic radius .................................. : 1.624
Excluded Volume ........................................ : 1.855e+4
Average atomic volume .................................. : 17.92
Radius of gyration from atomic structure
Rg ( Atoms - Excluded volume + Shell ) ................. : 17.28
Rg from the slope of net intensity ..................... : 17.71
Average electron density ............................... : 0.4263

Maximum order of harmonics ............................ : 50
Order of Fibonacci grid ............................... : 18
Total number of directions ............................ : 4182------------ Reciprocal space grid ------------in s = 4*pi*sin(theta)/lambda [1/angstrom]
Maximum scattering angle .............................. : 2.000
Number of angular points .............................. : 201--- Structural parameters (sizes in angstroms) --PDB file name .......................................... : pdb2bni.ent
Number of atoms read .................................. : 1035
Number of discarded waters ............................ : 55
Geometric Center: -5.452 24.939 39.648
Center of the excess electron density: 0.043 0.084 -0.120
Electron Rg : 16.40
Envelope Rg: 16.67
ShapeRg : 16.43
Envelope volume : 0.2084E+05
Shell volume : 0.1176E+05 Envelope surface : 3303.
Shell
Rg : 21.05
Envelope radius : 32.30
Shell width : 3.000
Envelope diameter : 59.99
Molecular Weight: 0.1478E+05 Dry volume: 0.1791E+05
Displaced volume: 0.1855E+05 Average atomic rad.: 1.623
Number of residuals : 128---------------- Output files -----------------Coefficients saved to file 2bni01.flm
CRYSOL data saved to file 2bni01.sav
Intensities saved to file 2bni01.int
Net amplitudes saved to file 2bni01.almC R Y S O L Version 2.6 -- 26/01/05 ---- terminated at15-Oct-2006 14:21:28--

45

Limitation of protein SAXSAtomic structure cannot be obtaineduniquely.
(1)Combine with other information (crystal orNMR structure)(2)Use of physical constraints --- ab initio

46

ab initio modeling
DAMMIN, GASBOR

http://scattering.tripod.com/
47

Optics for SAXS• Basically a two-slit system• A focusing optics is desirable for intensity

beam size
sourcebeam-defining
slits

guard slits
beam stop48

SAXS beamlines at SPring-8
X-ray source optics

Energy (keV)

BL40B2bend magnet

double-Si(111) +bent cylindrical

7 ~ 18BL08B2bend magnet

double-Si(111) +bent cylindrical

7 ~ 18

BL20XUlinear horizontal
undulator

double-Si(111) orSi(511)-(333)

8 ~ 113

BL45XUlinear verticalundulator

double-diamond(111) + doublemirror

13.8 (tunable)

BL40XU

helical horizontal
undulator

double-mirror

8 ~ 16.5 (pinkbeam)

BL03XUlinear horizontal
undulator

double-Si(111) +double-mirror

6 ~ 3549