Small-Angle Scattering:

Principles and Practice

Yimin Mao1,2

1. NCNR, National Institute of Standard and Technology, Gaithersburg, MD.

2. Dept. of Materials Science and Engineering, University of Maryland, College Park, MD.

The 28th CHRNS School on Methods and Applications of Small Angle Neutron Scattering

and Neutron Reflectivity

Oct. 17-21, 2022, NCNR

Yimin Mao UMD/NIST SAS Principles and Practice Oct. 17-21, 2022
An Overview of Different Scattering Phenomena

Elastic scattering experiment measures the intensity change as a function of angle

Incident Beam

Beam Stop

Detector

ω0

ω0

ω0 ± ∆ω

−ω1 ω ω1

0

Y. Mao UMD/NIST SAS Principles & Practice (1 / 36) Oct. 17-21, 2022
Primary beam

Scattered beam

2θ

Y. Mao UMD/NIST SAS Principles & Practice (2 / 36) Oct. 17-21, 2022

Scattering

Geometry
θ

θ

O

eˆ0 2π/λ

eˆs

2π/λ

q

=

2

λ

π

(eˆ

s

−

eˆ

0)

q ≡ |q| =

4π

λ

sin θ

Y. Mao UMD/NIST SAS Principles & Practice (3 / 36) Oct. 17-21, 2022
Scattering Fundamentals: Interference

O

A P

B

2π

λ

eˆ0

2π

λ

eˆs

2θ

x1

Primary beam

Primary beam

Scattered wave 1

Scattered wave 2

Detection Plane

Y. Mao UMD/NIST SAS Principles & Practice (4 / 36) Oct. 17-21, 2022

❦ eˆ0, eˆs: unit vector of incident and scattered wave

❦ Path difference:

δ = |AP| + |BP| = eˆ0 · x1 − eˆs · x1

❦ Phase difference: φ = 2π

λ

δ = −q · x1

❦ Amplitude of the scattered waves (2):

F(q) = F0 exp(i qx1)
Scattered Intensity: NParticles

Y. Mao UMD/NIST SAS Principles & Practice (5 / 36) Oct. 17-21, 2022

❦ Superposed amplitude of waves scattered byN particles

F(q) =

N

∑

i=1

bi exp(−iq · x) = Zρ(x) exp(−iq · x) dx

* bi: Scattering length (atomic scattering factor, fi, in X-ray scattering).

* ρ(x): Scattering length density (SLD, electron density in X-ray scattering).

❦ Scattered intensity due toN atoms

IN(q) = |F(q)|

2 ≡ F(q)F∗

(q)

=

Z

ρ(x) exp(−iq · x) dx

Z

ρ(x

′

) exp(iq · x

′

) dx

′

=

Z

P(r) exp(−iq · r) dr, where

P(r) =

Z

ρ(x

′ + r)ρ(x′

) dx

′
Form Factor & Structure Factor

Intra- & Inter-Particle Interference

I(q) ∝ |F(q)|

2 × S(q)

Y. Mao UMD/NIST SAS Principles & Practice (6 / 36) Oct. 17-21, 2022
A Roadmap Toward Scattering Function: A Refresher

Y. Mao UMD/NIST SAS Principles & Practice (7 / 36) Oct. 17-21, 2022
Form Factor & Structure Factor: Dilute Solution

S(q) = 1; I(q) ∝ |F(q)|

2

❦ Total scattering is a summation of scattering from each individual particle.

Dilute hard sphere suspension The scattering pattern

Y. Mao UMD/NIST SAS Principles & Practice (8 / 36) Oct. 17-21, 2022
Form Factor & Structure Factor: Concentrated Solution

S(q) 6= 1; I(q) ∝ S(q)|F(q)|

2

❦ Need to consider interference of the scattered waves from different particles.

Concentrated hard sphere suspension The scattering pattern

Y. Mao UMD/NIST SAS Principles & Practice (9 / 36) Oct. 17-21, 2022
Structure Factor: An Example of Hard-Shpere Model1−3

1. J. K. Percus, G. J. Yevick; Phys. Rev.,110(1958) : 1-13. 2. M. S. Wertheim; Phys. Rev. Lett.,10(1963) :

321-323. 3. E. Thiele; J. Chem. Phys., 39(1963) : 474-479.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 5 10 15 20

S(qR)

qR

η = 0.1

η = 0.2

η = 0.3

η = 0.4

η = 0.5

Y. Mao UMD/NIST SAS Principles & Practice (10 / 36) Oct. 17-21, 2022
X-Ray, Light, and Neutron

❦ X-ray & visible light are electromagnetic waves

E = E0 exp(i ωt) E: EM field; ω: angular frequency

❦ X-ray interacts with electrons, light with electron cloud

p =

e

2/m

ω2

0 − ω2

· E

p: induced-dipole moment

e: elementary charge;e=1.6 × 10−19 C

m: electron mass; m=9.1 × 10−31 kg

❦ Neutrons are particles(mass=1.68 × 10−27 kg)

λ = h/p

λ: (particle) wavelength; h: Planck’s constant

p: (particle) momentum

A B

C D

E

Schematics: propagation of a EM wave (A-B) and neutron beam (C-E).

Y. Mao UMD/NIST SAS Principles & Practice (11 / 36) Oct. 17-21, 2022
X-ray & Neutron as Complementary Probes

* Illustration, not in absolute scale.

Y. Mao UMD/NIST SAS Principles & Practice (12 / 36) Oct. 17-21, 2022

H D C O Al Si Fe

X-ray scattering cross section

Neutron scattering cross section
Scattering Length Density

Y. Mao UMD/NIST SAS Principles & Practice (13 / 36) Oct. 17-21, 2022

https://www.ncnr.nist.gov/resources/activation/

ρ =

∑

N

i=1

bi

Vm

❧ ρ: scattering length density

❧ bi: coherent scattering length of the i-th atom

❧ Vm: volume of the molecule
Contrast: Why Structures Can be Probed at All

❦ Structure:spatial variation of

some ’property’

❦ To examine a structure, there must be a

sufficient contrast between the objectives

and the surroundings.

❦ Interpret∆ρ = ρp − ρs;

what does ρ refers to?

❧ NS: scattering length density

❧ XS: electron density

❧ LS: polarizability ’Density’ difference between the two phasess

makes the bi-continous structure ’visible’.

Y. Mao UMD/NIST SAS Principles & Practice (14 / 36) Oct. 17-21, 2022
A General Definition of Structure: The Essence of ρ(x)

❦ Variation of some material property overspace.

...

Y. Mao UMD/NIST SAS Principles & Practice (15 / 36) Oct. 17-21, 2022
Scattering Function of Particulate System: General Features

0.01

1

100

10000

1e+06

1e+08

1e+10

1e+12

1e+14

0.01 0.1 1

Intensity (a.u.)

q (nm −1)

10 nm

30 nm

100 nm

Y. Mao UMD/NIST SAS Principles & Practice (16 / 36) Oct. 17-21, 2022

F(q) = 3

(qR)

3

[sin(qR) − qRcos(qR)]

I(q) = [F(q)]2
Influence of Polydispersity

0.001

0.01

0.1

1

10

100

1000

10000

100000

1e+06

1e+07

1e+08

0.01 0.1 1

Intensity (a.u.)

q (nm −1)

∼ q

−4

R = 30 nm

R ̄ = 30 nm, polydisperse

Y. Mao UMD/NIST SAS Principles & Practice (17 / 36) Oct. 17-21, 2022

Ipoly(q) = Z ∞

0

D(R)Imono(q,R) dR
Guinier Approximation

I(q) = ∑

i

∑

i

fifj

sin( qrij )

qrij

sin(qr)

qr

≈ 1 −

q

2

r

2

6

+

q

4

r

4

120 + O(qr)

I(q) ≈ ∑

i

∑

j

f

2 − ∑

i

∑

j

f

2

q

2

r

2

ij

6

∑

i

∑

j

r

2

ij = 2N

2R2

g I(q) ≈ Nf 2



1−

q

2R2

g

3



;

e

−q

2

r

2/3 ≈ 1 −

q

2

r

2

3

+

q

4

r

4

18 + O(qr)

I(q) ∝ exp(−q

2R2

g/3)

when qR <∼ 1

Y. Mao UMD/NIST SAS Principles & Practice (18 / 36) Oct. 17-21, 2022

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

0.01 0.1

Intensity (a.u.)

q (nm −1)

Guinier Region

qRg <∼ 1

exp(−1

3

R

2

g

q

2

)

Sphere, R = 50 nm

Rg = 38.7 nm

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 0.0002 0.0004 0.0006 0.0008 0.001

ln[I(q)] (a.u.)

q

2

( nm−2)

y = −kx + b

k = R

2

g/3 = 510.4

Rg = 39.1 (nm)

qRg <∼ 1
Radius of Gyration

▲

❘

❛

❜

❝

R

2

g =

3

5

R

2

1

2

R

2 +

1

12L

2

1

2

R

2 +

1

12d

2

1

12(a

2 + b2 + L2

)

✷❘

❘ ❂ Pr✐

R

2

g =

3

5

R

5

1 − R

5

0

R

3

1 − R

3

0

1

2

R

4

1 − R

4

0

R

2

1 − R

2

0

+

1

12L

2 2R

2

1

2

Nb2

Y. Mao UMD/NIST SAS Principles & Practice (19 / 36) Oct. 17-21, 2022

R

2

g =

R

V

ρ(r)r

2 dv

R

V

ρ(r) dv

; R

2

g =

1

2N2 ∑

i

∑

j

r

2

ij
Guinier Regions at

Multiple Length Scales

Y. Mao UMD/NIST SAS Principles & Practice (20 / 36) Oct. 17-21, 2022

✵✳✵✶

✵✳✶

✶

✶✵

✶✵✵

✶✵✵✵

✶✵✵✵✵

✵✳✵✵✶ ✵✳✵✶ ✵✳✶ ✶

■♥

t❡

♥

s✐t2

✭❛

✉

✮

q ✁✂♠ ✄☎✆

✝✞✟✺

✝✞✟✻

✝✞✟✼

✝✞✟✽

✝✞✟✾

✝✝✟✞

✞✟✞ ✞✟✺ ✝✟✞ ✝✟✺ ✷✟✞ ✷✟✺ ✸✟✞ ✸✟✺ ✹✟✞

❧✠❬✡☛☞✌❪✍✎

✏✑

✏✒

✓

✔ ✕ ✖✗✘✙ ✚ ✛✜✘✔ ✢

✣ ❂ ✤❦1 ✰ ❜

✥ ✦ ❘✧

❣✴★ ✦ ✩✪✫✫✬✯

❘❣ ✦ ✱✫✬✫ ✲✿❀❁

✓❃❄ ❅❆ ✖

❇❈❉

❇❈❊

❋❈●

❋❈❍

❋❈❏

❋❈❇

❋❈❋

●❈●●❑ ●❈●❍ ●❈●❍❑ ●❈●❏ ●❈●❏❑

▲▼

❚

◆❖P◗❙

◗❯❱❲

❳❨

❳❩

❭

❫

❴ ❵❝❞❫ ❢

❤ ❥ ♦♣r ✈ ✇

3 4 56

789 4 910❶❷

57 4 ❸❶❹ ❺❻❼❽

❭❾❿ ➀➁ ➂ ➃ ➄ ➅➆➆ ➇➈

➉

➊

➋

➌

➍

➎
I(q) = F[P(r)]

hI(q)iΩ ≡ I(q) = ρ

2

Z ∞

0

V(r)

sin(qr)

qr

4πr

2 dr

where

ρ

2 V(r) = hP(r)iω

Define

γ0(r) = V(r)/V(0) ≡ V(r)/V

p(r) = 4πr

2Vγ0(r)

Particles and their phantom identicals shifted by r, at the same r = |r|

Y. Mao UMD/NIST SAS Principles & Practice (21 / 36) Oct. 17-21, 2022

r

Correlation function & Distance Distribution Function
I(q) = Z ∞

0

ρ

2Vγ0(r)sinc(qr)4πr2 dr =

Z ∞

0

p(r)sinc(qr) dr

p(r) ∝

Z ∞

0

I(q)qr sin(qr) dq

▲

❘

✷❘

r ✭ ♥♠ ✮

✵

✵✳✺

✶

✶✳✺

✷

✷✳✺

✸

✵ ✶✵ ✷✵ ✸✵ ✹✵ ✺✵ ✻✵

♣✁✂

❘ ❂ ✄☎ ✆✝

❘ ❂ ✞ ✆✝✱ ▲ ❂ ✟☎ ✆✝

❞ ❂ ✞ ✆✝✱ ❘ ❂ ✄☎ ✆✝

✠❘ ❂ ✟☎ ✆✝

Y. Mao UMD/NIST SAS Principles & Practice (22 / 36) Oct. 17-21, 2022

❆ r ❇

p(r): Distance Distribution Function
Obtaining p(r): Indirect Fourier Transform

p(r) =

N

∑

ν=1

cνφν(r); I(q) =

N

∑

ν=1

cνΦν(q); where Φν(q) = F[φν(r)]

O. Glatter; Small Angle X-Ray Scattering, Academic Press, 1982, Chapter 4.

Y. Mao UMD/NIST SAS Principles & Practice (23 / 36) Oct. 17-21, 2022
Computing 1D Scattering Curve: Rod

0.0001

0.01

1

100

10000

1e+06

0.01 0.1 1

Intensity (a.u.)

q (nm −1)

∼ q

−1

R = 8 nm, L = 80 nm

R = 8 nm, L = 160 nm

R = 8 nm, L = 400 nm

Y. Mao UMD/NIST SAS Principles & Practice (24 / 36) Oct. 17-21, 2022

▲

❘

F(q, φ) = 2πR J1(qR sin φ)

q sin φ

· L sinc 

L q cos φ

2



I(q) = Z π

0

| F(q, φ)|

2

sin φ dφ

(1)
Computing 1D Scattering Curve: Disk

0.0001

0.01

1

100

10000

1e+06

0.01 0.1 1

Intensity (a.u.)

q (nm −1)

∼ q

−2

L = 8 nm, R = 60 nm

L = 8 nm, R = 120 nm

L = 8 nm, R = 400 nm

Y. Mao UMD/NIST SAS Principles & Practice (25 / 36) Oct. 17-21, 2022

F(q) = 2πR J1(qRsin φ)

q sin φ

· L · sinc(qL cos φ/2)

I(q) = Z π/2

0

[F(s)]2sin φ dφ
Debye Formula (Summation): Treating a Group of Particles

I(q) ≡




|F(q)|

2



Ω

=



∑

i

∑

j

FiFj exp(iq · r)



= ∑

i

∑

j

FiFj

sin qr

qr

=

N

∑

i=1

Ii(q) + 2 · ∑ ∑

i6=j

FiFj

sin qr

qr

Proteins Clusters Polymer Chains

Lysozyme (1DPX) Icosahedron (55 particle) Gaussian Chain

Y. Mao UMD/NIST SAS Principles & Practice (26 / 36) Oct. 17-21, 2022
I(q) = 1

N

N

∑

i=0

N

∑

j=0



sin(qRij)

qRij 



sin(qr)

qr 

=

Z ∞

0

sin(qr)

qr

W(r,R2) · 4πr

2 dr

= exp[−q

2R2/6]

W(r,R2) =  3

2πR2

3/2

exp −

3

2

r

2

R2



I(q) = 1

N

N

∑

i=0

N

∑

j=0

exp − q

2

(|i − j|)b

2/6

=

2

N2

Z N

0

(N − n) exp(−q

2nb2/6) dn

= 2x

−2

(e

−x + x − 1) with x = q2R2/6

Y. Mao UMD/NIST SAS Principles & Practice (27 / 36) Oct. 17-21, 2022

0.01

0.1

1

0.01 0.1

Intensity (a.u.)

q (A ̊ −1)

∼ q

−2

R = 80 A ̊

R = 40 A ̊

❘ ❂ Pr✐

Debye Function: Scattering from a Gaussian Chain
Spherical Harmonic Expansion

Y. Mao UMD/NIST SAS Principles & Practice (28 / 36) Oct. 17-21, 2022

N

∑

i=0

N

∑

j=0

sin(qRij)

qRij
❦ Density (scalar field, real space) expansion

ρ(r, θ, ξ) ≈

L

∑

l=0

l

∑

m=−l

ρlm(r)Ylm(θ, ξ)

ρlm =

Z 2π

0

Z π

0

ρ(r, θ, ξ)Y

∗

lm(θ, ξ) sin θ dθ dξ

❦ Amplitude expansion

A(q, φ, ψ) =

L

∑

l=0

l

∑

m=−l

Alm(q)Ylm(φ, ψ)

Alm(q) = i

l

r

2

π

Z ∞

0

jl(qr)ρlm(r)r

2 dr

ρlm(q) = (−i)

l

r

2

π

Z ∞

0

jl(qr)Alm(r)q

2 dq

❦ Intensity expansion

I(q) =

L

∑

l=0

Il(q) ∝

L

∑

l=0

l

∑

m=−l

|Alm(q)|

2

; compare

I(q) ∝ ∑

i

∑

j

sin(qRij)

qRij

Y. Mao UMD/NIST SAS Principles & Practice (29 / 36) Oct. 17-21, 2022

a a

a a | Y

0

0

|

a

a | Y

1

1

| | Y

0

1

|

| Y

2

2

| | Y

1

2

| | Y

0

2

|

Spherical Harmonic Expansion
Spherical Harmonic Expansion: An Example Using CRYSOL

ATSAS by EMBL

❦ Monodisperse system;

❦ Atomic coordinates (finite number) are known. (Then, why bother?)

✶❡✰✵✻

✶❡✰✵✼

✶❡✰✵✽

✶❡✰✵✾

✵✳✶ ✶

■♥

t

♥

s✐t2

✭❛

✁✉

✁✮

q ✂❆✠ ✄☎✆

▼♦✝♦♠✞r ✟ ✷❤✡☛ ☞

❍✞1✌♠✞r✟ ✍✎✡✏ ☞

Y. Mao UMD/NIST SAS Principles & Practice (30 / 36) Oct. 17-21, 2022

https://www.embl-hamburg.de/biosaxs/software.html
Spherical Harmonic Expansion: Approaching Fine Details

❦ Low-order terms are for overall shape; high-order terms are for fine details.

0.1

1

10

100

1000

10000

100000

1e+06

1e+07

0.1 1

Intensity (a.u.)

q (A ̊ −1)

L = 0

L = 2

L = 6

L = 15

L = 0

L = 2

L = 6

L = 15

SAXS curves of egg-white lysozyme. Atomic coordinates are from PDB file 6lyz. Spherical

harmonic expansion algorithm is implemented in CRYSOL; envelops are reconstructed using

MASSHA. Both belongs to EMBL solution-SAXS toolkits ATSAS.

Y. Mao UMD/NIST SAS Principles & Practice (31 / 36) Oct. 17-21, 2022
Planning for Your Experiment

❦ What q-range do you need?

❧ Can we carry out some microscopic study beforehand?

❧ Perhaps a visual inspection of your sample can provide some hints?

❧ Can we do some simple calculation?

❦ Where does the scattering contrast come from?

❧ electron density,scattering length density, polarizability...

❦ What’s the physical form of your sample?

❧ Liquid, film, powder, gel,etc.

❧ Do I need to worry about radiation damage for biological samples?

❦ Is kinetics of interest? How fast is the kinetic process?

❧ Shall we carry out the experiment using an in-house machine or at a

synchrotron beamline?

❦ What would be the scheme of scattering background manipulation?

Y. Mao UMD/NIST SAS Principles & Practice (32 / 36) Oct. 17-21, 2022
Y. Mao UMD/NIST SAS Principles & Practice (33 / 36) Oct. 17-21, 2022

SANS Beamline, NG-7, 30 m, NIST Center for Neutron Research
SDD: A Practical Thinking of the Reciprocal Relation

• tan 2θ = Dp/SDD

↑ SDD 7−→ Observed scattering angle ↓

• q = 4π

λ

sin θ = 2π

d

↓ θ 7−→ d ↑

• To have a full-q range data, different SDDs are used.

Y. Mao UMD/NIST SAS Principles & Practice (34 / 36) Oct. 17-21, 2022

Detector position 2

2θ1

2θ2

SDD2

SDD1

Sample position

Dp

Detector position 1
Detection Range of Different SAS Techniques

100

101

102

103

104

105

106

107

108

0.001 0.01 0.1

Intensity (a.u.)

q (nm −1)

Sphere

Rod

Disk

SLS

USAX(N)S

SAX(N)S

Rg = 150 nm

Sphere:

R = 194 nm

Rod:

R = 20 nm

l = 517 nm

Disk:

R = 212 nm

d = 20 nm

Y. Mao UMD/NIST SAS Principles & Practice (35 / 36) Oct. 17-21, 2022
Background Subtraction 1

I0

I

0

sam

Icor =

I0

I

0

sam

(Isam − Idrk) − (1 − φ)

I0

I

0

buf

(Ibuf − Idrk) − φ

I0

I

0

cel

(Icel − Idrk)

Icor : Background subtracted/final corrected intensity

Isam : Scattering intensity in sample run (empty cell+solute+buffer)

Ibuf : Scattering intensity in buffer run (empty cell+ buffer)

Icel : Scattering intensity in empty cell run (empty cell)

Idrk : Detector response NOT due to scattering event

I

0

sam : Transmitted intensity in sample run (empty cell+solute+buffer)

I

0

buf : Transmitted intensity in buffer run (empty cell +buffer)

I

0

cel : Transmitted intensity in empty cell run (empty cell)

φ : volume fraction of solute

1

Koch, M. H., Vachette, P. & Svergun, D. I. Quart. Rev. Biophys., 36, 147-227 (2003).

Y. Mao UMD/NIST SAS Principles & Practice (36 / 36) Oct. 17-21, 2022