Introduction to Neutron (and X-ray) 

Scattering Techniques 

Jeffrey Lynn

NIST Center for Neutron Research

University of Texas—Dallas, Arlington

April 12, 2021
Outline

• Scattering Basics

– Cross sections, form factors, x-rays vs. neutrons

• Powder Diffraction (crystal and magnetic diffraction)

– Profile refinement, subtraction technique, polarized neutrons

• Single Crystal Diffraction (structure and magnetic)

• Small Angle Neutron Scattering (SANS)

– Nanoparticles, Vortex lattice, ferromagnetic superconductor, 

skyrmions, domains, ... polymers, biological systems,

• Reflectometry (thin films and multilayers)

– Structural and Magnetic Depth Profile

• Inelastic Scattering

– Phonons, Magnons, Crystal Field Levels, Spin Liquids, 

• Reference Materials
Neutron (and X-ray) scattering

Neutron scattering experiments measure the flux of neutrons 

scattered by a sample into a detector as a function of the change

in neutron wave vector (Q) and energy (hω).

The expressions for the scattered neutron flux Φ depend on the 

positions and motions of atomic nuclei or unpaired electron spins.

Momentum Energy

hQ = hki - hkf E = Ei - Ef

hkn = h(2π/λn)

Φ provides information about 

all of these quantities! Φ = f{ri

(t), rj(t), Si(t), Sj(t)}

Φ(Q,hω) = neutrons

sec–cm2

φ

E = h2kn

2/2m
Reciprocal (Scattering) Space 

Periodic array of atoms

in Real Space

d

2π/d

Real space ↔ Reciprocal (Fourier) Space
Other Probes

Ephoton (keV) = 2.0k =12.4 / λ

2 2 Eelectron (eV) = 3.8k =150 / λ

1 8.06 ( / ) 1 meV cm E hc − =

1meV 11.6 K (k T) =

B

1 / 17.3 ( / ) meV μ B = T E μ B

1meV = 0.2418 THz (E / h)

2 2 Eneutron (meV) = 2.0719k = 81.7968/ λ

λ = 1 Å: En=82 meV; Ep=12,400,000 meV; Ee=150,000 meV

300 K  25 meV
Neutron and X-ray Scattering

• Both techniques collect data as functions of the energy and the momentum transferred 

from the system to the neutron or photon beam. The resulting five-dimensional data 

sets serve as powerful probes of materials. Elastic scattering elucidates the crystal 

structure, magnetic configuration, direction of the spins, symmetry of the magnetic 

state, spatial distribution of the magnetization density, and dependence of the order(s) 

parameter on thermodynamic fields such as temperature, pressure, magnetic and 

electric fields. Inelastic scattering determines the energies of the fundamental 

excitations which can be used to elucidate the nature, strength, and range of the 

interactions. 

• Both techniques can measure crystal and magnetic structures and their dynamics.

• Neutron advantages:

– Magnetic and structural scattering are comparable in strength; Elastic scattering yields quantitative 

information; energy resolution is orders-of-magnitude better than x-rays; simplicity of sample 

environment; low T accessible. Theory has solid theoretical basis.

• X-ray advantages:

– High Flux  small samples; individual domains, topography; pump probe capability; resonant x-ray 

scattering element specific; magnetic resonant x-ray scattering; RIXS
Neutrons and X-rays are Complementary

Nucleus looks like a point particle  b is just a constant independent 

of scattering angle.

Adjacent elements, heavy + light elements, isotope substitution
Neutron Cross Sections
Neutrons and X-rays are Complementary
Magnetic X-ray Cross Sections
Reciprocal Space for Powder

TiAu

glass/fluid/nano/amorphous
The scattered neutron flux Φ(Q,hω) is proportional to the space (r) 

and time (t) Fourier transform of the probability G(r,t) of finding one or 

two atoms (spins) separated by a particular distance (angle) at a 

particular time.

e G r t d rdt i Q r

 t   

( ) 3

2

( , ) ∫∫ ⋅ − ∝

∂Ω∂

∂ Φ ∝ ω

ω

σ

Time space

ω

2π/T

Ε-space

d

Real space

Q

2π/d

Q-space

~1/ξ ~1/Γ

T

Correlations

time

Φ Φ

ξ Γ
Neutron Scattering Techniques

• Crystallography—powder, single crystal

Atomic positions, site occupancies, lattice parameters, bond distances, meansquare vibrations as a function of T, H, P, E, ...

• Magnetism

Magnetic structure, order parameter, spin directions, spin density distribution

Phase Transitions and Critical Phenomena (Scaling, Universality)

• Small Angle Neutron Scattering (SANS)

Ferromagnetic Correlations, Vortex Structures, Domain Structures, Grain 

boundaries, twin boundaries, defect structures, nanoparticles, skyrmions, ...

• Thin Film Reflectometry

Density profiles, Magnetic structures, Magnetization profiles, Surface and 

Interface properties (flatness, roughness)

Diffraction
Inelastic Scattering

Lattice Dynamics

Phonon Dispersion, Density of States

Interatomic Force constants

Mean-square vibrations

Diffusion

Spin Dynamics

Magnon Dispersion, Exchange interactions

Magnetic Anisotropy

Magnetic Fluctuation Behavior

Crystal Field Levels

Magnetic-Structural Coupling
May 2017

NIST Center for Neutron Research
Oak Ridge National Laboratory

High Flux Isotope 

Reactor 

Spallation Neutron 

Source
Materials that are both 

Magnetic and 

Superconducting
Magnetic Impurities

Cause Spin depairing

M. B. Maple, Appl. Phys. 9,179 (1976) 

Magnetic Concentration

T

C

Temperature

Field

Magnetic Fields and 

Superconductivity are Antagonists
• Concentrated Magnetic Systems (Exceptions to the Rule!)

C-15 Cubic Laves phase (Ce-Ho)Ru2 (’60’s-’70’s)

• Magnetic Sublattice—Long Range Order

Chevrel Phase DyMo6S8 (’70’s)

• Ferromagnets—Competition & Coexistence

– Chevrel Phase HoMo6(S-Se)8, ErRh4B4 (’70’s – 80’s)

• High TC cuprates—Cu spin order & fluctuations

Cuprates RBa2Cu3O7 [123], R2CuO4 [214] (’80’s→...)

• Borocarbides

– HoNi2B2C, ErNi2B2C (’90’s→...)

• New Ferromagnetic Superconductors 

– Ruthenates RuSr2GdCu2O8, RuSr2(Eu-Ce)2Cu2O10; ZrZn2, UGe2 (2000’s →...)

• Sodium cobaltates (Magnetic, thermoelectric, and Superconducting) 2000’s→...

– NaxCoO2 (+ H20) [just add water for superconductivity !]

• Iron-based superconductors (2008→...)

– R(O1-xFx)FeAs; Sr1-xKxFe2As2; LiFeAs; Fe(Se1-xTex)

– 1:1:1:1 1:2:2 1:1:1 1:1

Magnetic Superconductor History

• Pure Superconductors (1911→ ...) (k ;- k ) • X Magnetic Impurities X

⇐
Ferromagnetic Superconductors

(Ce-Ho)Ru2

HoMo6S8 , HoMo6Se8 , ErRh4B4

ErNi2B2C

RuSr2GdCu2O8 & RuSr2Eu2-xCexCu2O10

(Li-Fe)OHFeSe

UGe2 ZrZn2, UCoGe, URhGe, C/S
Magnetic Structures

Ferromagnet

Antiferromagnet

Spin Density Wave
Magnetic Structures

Ferromagnet

Antiferromagnet

Spin Density Wave
Chevrel Phase Superconductors

[(HoS8)Mo6 Magnetic Lattice Isolated]

Tsuper = 1.8 K 5.6 K 8.6 K 

Tferro = 0.7 K 0.5 K 0.9 K

Treentrant= 0.7 K < 0 K 0.9 K

HoMo6S8, HoMo6Se8, ErRh4B4
Small Angle Neutron Scattering
HoMo6Se8

~100 Å 

(10 nm)

periodicity 

TS=5.6 K

TM=0.5K

J. W. Lynn, J. A. Gotaas, R. W. Erwin, R. A. Ferrell, J. K. 

Bhattacharjee, R. N. Shelton and P. Klavins, 

Phys. Rev. Lett. 52, 133 (1984)
HoMo6Se8

Center for Neutron Research
Iron-based High TC Superconductors
Iron-based superconductors under 

Investigation at the NCNR

• FeSe, Fe1+x(Se-Te), KxFe2-ySe2 (1:1) 

• LiFeAs (1:1:1)

• LaO1-xFxFeAs LaOFeAs (1:1:1:1)

• CeO1-xFxFe(As,P) CeOFeAs

• NdO1-xFxFeAs Nd...

• PrO1-xFxFeAs Pr...

• BaFe2As2, SrFe2As2, CaFe2As2 (1:2:2)

• CaFe2As2, Under Pressure; doping 

– https://www.nist.gov/people/jeffrey-w-lynn
7Li1-xFexODFeSe

Neutron Investigation of the Magnetic Scattering in an Iron-based 

Ferromagnetic Superconductor

Jeffrey W. Lynn1, Xiuquan Zhou2, Christopher K. H. Borg2

, Shanta R. Saha3, 

Johnpierre Paglione3, and Efrain E. Rodriguez2

Phys. Rev. B 92, 060510(R) (2015).

The Preparation and Phase Diagram of Superconducting 

(7Li1-xFexOD)FeSe

Xiuquan Zhou2, Christopher K. H. Borg2, Jeffrey W. Lynn1

, Shanta R. Saha3, 

Johnpierre Paglione3, and Efrain E. Rodriguez2

, 

J. Mater. Chem. C 4, 3934 (2016).

Long range magnetic order in Mn-doped (LiOH)FeSe, Brandon Wilfong, 

Xiuquan Zhou, Huafei Zhang, Navneeth Babra, Craig M. Brown, Jeffery W. Lynn, 

Simon A. J. Kimber, Keith M. Taddei, Johnpierre Paglione, and Efrain E. 

Rodriguez, Phys. Rev. Materials 4, 034803 (2020).

1NIST Center for Neutron Research, Gaithersburg, MD (USA)

2Department of Chemistry and Biochemistry, University of Maryland, College Park, MD (USA)

3Department of Physics, University of Maryland, College Park, MD (USA)
Coexistence of 3d-ferromagnetism and superconductivity in [(Li1-xFex)OH](Fe1-yLiy)Se, 

Ursula Pachmayr, Fabian Nitsche, Hubertus Luetkens, Sirko Kamusella, Felix Bruckner, 

Rajib Sarkar, Hans-Hennig Klauss, and Dirk Johrendt, Angew. Chem. Int. Ed. 54, 293 (2015)

(Li-Fe)OHFeSe Ferromagnetic Superconductor

TS=43 K

TF=10 K
FeSe Superconducting Layer

Magnetic Layer

Incommensurate Long wavelength 

Ordered state?

Spontaneous Vortex Lattice?

Crystal Structure
Synthesis and Structure
Neutron and X-ray Diffraction
Magnetization for Tc = 18 K 

(polycrystalline sample)
Ferromagnetic Magnetization

2 K
No Applied Magnetic Field
B = 0.4 T
Singular angular magnetoresistance in a magnetic nodal 

semimetal

by T. Suzuki, L. Savary, J.-P. Liu, J. W. Lynn, L. Balents, and J. G. Checkelsky

Science

Volume 365(6451):377-381

July 26, 2019

Copyright © 2019, American Association for the Advancement of Science
Fig. 1 Singular angular magnetoresistance (SAMR).

T. Suzuki et al. Science 2019;365:377-381

Copyright © 2019, American Association for the Advancement of Science

CeAlGe
Fig. 2 Magnetic field dependence of SAMR.

T. Suzuki et al. Science 2019;365:377-381

Copyright © 2019, American Association for the Advancement of Science
Fig. 3 Phase diagram of SAMR.

T. Suzuki et al. Science 2019;365:377-381

Copyright © 2019, American Association for the Advancement of Science
Magnetic Topological Properties in YMn6Sn6

• Competing magnetic states and fluctuation-driven scalar spin chirality in the 

kagome metal YMn6Sn6, Nirmal J. Ghimire, Rebecca L. Dally, L. Poudel, 

D. C. Jones, D. Michel, N. Thapa Magar, M. Bleuel, Michael A. McGuire, J. 

S. Jiang, John F. Mitchell, Jeffrey W. Lynn, and I. I. Mazin, 

Science Advances 6, eabe2680 (2020).

• Chiral properties of the zero-field spiral state and field-induced magnetic 

phases of the itinerant kagome metal YMn6Sn6, Rebecca L. Dally, Jeffrey W. 

Lynn, Nirmal J. Ghimire, Dina Michel, Peter Siegfried, and Igor I. Mazin, 

Phys. Rev. B 103, 094413 (2021).
Fig. 2 Magnetization and Hall effect of YMn6Sn6.

Nirmal J. Ghimire et al. Sci Adv 2020;6:eabe2680

Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association 

for the Advancement of Science. No claim to original U.S. Government Works. Distributed 

under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
Fig. 1 Crystal structure and electrical and magnetic properties of YMn6Sn6.

Nirmal J. Ghimire et al. Sci Adv 2020;6:eabe2680

Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association 

for the Advancement of Science. No claim to original U.S. Government Works. Distributed 

under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
Fig. 3 Single-crystal neutron diffraction of YMn6Sn6.

Nirmal J. Ghimire et al. Sci Adv 2020;6:eabe2680

Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association 

for the Advancement of Science. No claim to original U.S. Government Works. Distributed 

under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
Fig. 4 First-principles calculation and phenomenological model of spin chirality for THE.

Nirmal J. Ghimire et al. Sci Adv 2020;6:eabe2680

Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association 

for the Advancement of Science. No claim to original U.S. Government Works. Distributed 

under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
Summary

 Neutron scattering is a powerful tool investigate crystal and 

magnetic structures, and lattice and magnetic excitations, 

over six orders of magnitude in length scale and energy 

scale.

 Neutron and X-ray scattering are complementary 

experimental tools. 

 National user facilities are available on a proposal basis to 

everyone.
References:

S. W. Lovesey, Theory of neutron scattering from condensed matter, Oxford: 

Clarendon Press - Oxford, 1984.

E. Balcar and S. Lovesey, Theory of Magnetic Neutron and Photon Scattering, 

Oxford: Clarendon Press, 1989.

L. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill and J. van den Brink, 

"Resonant inelastic s-ray scattering studies of elementary excitations," Rev. Mod. Phys., 

vol. 83, p. 705, 2011.

G. E. Bacon, Neutron Diffraction, Third ed., Oxford: Oxford University Press, 1975.

Magnetic Scattering, J. W. Lynn and B. Keimer, in Handbook of Magnetism and 

Magnetic Materials, ed. by Michael Coey and Stuart Parkin, (Springer Nature’s Major 

Reference Work), (in press)

Neutron Nuclear Properties: 

https://www.ncnr.nist.gov/resources/n-lengths/ 

https://www.ncnr.nist.gov/instruments/magik/Periodic.html

Magnetic Form Factors

https://www.ill.eu/sites/ccsl/ffacts/ffachtml.html

JWL website: https://www.nist.gov/people/jeffrey-w-lynn