Neutron scattering presentation series

(1) Basic concepts and neutron diffraction

Xin Li Department of Chemistry Louisiana State University June 1st, 2015 (2012 ESS Technical Design Report) Length scale Time scale Type Technique Reciprocal space Real space (r/ Energy space Time space (τ/ (Q/Å-1) nm) (Δω/µeV) ps) Ultra-Small Angle Neutron Scaering 5×10-6 ~ 0.005 100 ~ 105 (USANS)

Small Angle Neutron Stac 0.001 ~ 0.5 1 ~ 500 Scaering (SANS) N/A scaering Neutron Diffracon 0.1 ~ 20 0.05 ~ 5 Neutron 0.001 ~ 0.5 1 ~ 500 Reflectometry Neutron Spin Echo 0.01 ~ 0.5 1 ~ 50 0.01 ~ 100 10 ~ 105 (NSE) Dynamic Quasi-Elasc Neutron 0.1 ~ 10 0.05 ~ 5 1 ~ 100 0.1 ~ 103 scaering Scaering (QENS) Inelasc Neutron 0.1 ~ 10 0.05 ~ 5 10 ~ 105 0.01 ~ 100 Scaering (INS) Measurement λ=0.1 ~ 10 Å Source Sample size Incident energy me

Reactor Neutron min ~ hour cm, mL meV Spallaon source

Synchrotron µs ~ ms X-ray mm, µL keV In-house min ~ hour Advantages and Disadvantages of Scaering Techniques

Advantages: 1. Dynamical and structural informaon in several orders 2. Ensemble sampling 3. Non-destrucve penetraon 4. Contrast variaon available 5. Sensive to magnec fields (neutron)

Disadvantages: 1. Inverse problem 2. Ensemble sampling 3. Radiaon resistance (X-ray) 4. Sample amount 5. Beamme accessibility (neutron)

Outline

Basic concepts: 1. Scaering cross secon 2. Scaering length and scaering length density 3. Coherent and incoherent scaering 4. Reciprocal space 5. Spaal and me correlaon funcons

Neutron diffracon: 1. Single crystal diffracon 2. Powder diffracon 3. Rietveld refinement method 4. Pair distribuon funcon (PDF) method Cross Secon – Scaering Ability

dΩ Number of incident neutrons: I Number of scaered neutrons: Θ φ Number density of scaerers in the sample: N [L-3] dφ dθ Beam size: A [L2] area A Sample thickness: Δx [L] θ Solid angle: Intensity I polar axis Ω Scaering probability: density N Δx ​Θ/� ∝​��∆�/� =�∆�

azimuthal axis ​Θ/� =�∆�� ​1/� ​�Θ/�Ω =�∆�​��/�� =�∆��(�) Cross Secon and Scaering Length

​Θ/� =�∆�� ​1/� ​�Θ/�Ω =�∆�​��/�� =�∆��(�)

σ [L2] (microscopic cross secon): describes the scaering ability of the material.

For neutrons scaered by the nuclei: �(�)=​��/�Ω =​�↑2 b [L]: constant, scaering length

�=∫Ω↑ ▒�(�)�Ω =​4��↑2

Units: σ: 1 barn = 10-24 cm2 = 10-28 m2 b: 1 fm = 10-15 m = 10-5 Å Cross Secon and Scaering Length (cont’d)

1. X-ray sensive to heavy atoms (high electron density) 2. Neutrons sensive to light nuclei 3. Hydrogen: negave neutron scaering length (isotope substuon) 4. Chlorine and sulfur in the solvent strongly scaer X-ray 5. Boron: neutron absorpon Cross Secon and Scaering Length (cont’d)

Example 1: scaering by 1mm thick water

Mass density: 0.99997 g/cm3 Cross secon: H: 82.02 barn, O: 4.232 barn ​�↓� =1−​Θ/� =1−​�↓​�↓2 � Δ�​�↓​�↓2 � =1−​�↓​�↓2 � Δ�(2​�↓� +​�↓� )=1−​0.99997/18.01528 ×6.0221413×​10↑23 ×0.1×(2×82.02+4.232)×​10↑−24 =�.���� Scaering Length Density

�=​1/� ∑�↑▒​�↓� Unit: 10-10 cm-2=10-6 Å-2

Contrast comes from the scaering length density. (Pynn IUB lecture 2006) Scaering Length Density (cont’d)

Example 2: scaering length density of water

Mass density: 0.99997 g/cm3 Scaering length: H: -3.7423 fm, O: 5.805 fm ​�↓​�↓2 � =​���� �������/��������� weight ​�↓� ∑�↑▒​�↓� =​���� �������/��������� weight ​�↓� (2​�↓� +​�↓� )=​ 0.99997/18.01528 ×6.0221413×​10↑23 ×​1/​10↑24 ×(−2×3.7423+5.805)×​10↑−5 =−�.����×​��↑−�

(Å-2) Scaering Length Density (cont’d)

materials SLD (10-6 Å-2)

H2O -0.56

D2O 6.39 h-styrene 1.413 d-styrene 6.5 h-cyclohexane -0.24 d-cyclohexane 6.01

SiO2 4.186

(Pynn IUB lecture 2006) Scaering Length Density Profile

Fitzsimmons lecture 2005) (S.-H. Chen Macromolecules 1998) Coherent and Incoherent Scaering

The neutron scaering length depends on the nuclear isotope, spin relave to the neutron, and nuclear eigenstate.

For a single nucleus of a species, ​�↓� =⟨�⟩+​��↓� where ⟨​��↓� ⟩=0 For the correlaon between two nuclei, ​�↓� ​�↓� =​⟨�⟩↑2 +(​��↓� +​��↓� )⟨�⟩+​��↓� ​��↓�

Average over the whole group of nuclei, ⟨​��↓� +​��↓� ⟩=0

⟨​��↓� ​��↓� ⟩=█0&(�≠�)@⟨​(​��↓� )↑2 ⟩=⟨​�↑2 ⟩−​⟨�⟩↑2 &(�=�) Coherent and Incoherent Scaering (cont’d)

For the correlaon between two nuclei, ​�↓� ​�↓� =​⟨�⟩↑2 +​��↓� ​��↓�

Therefore, the correlaon between all nuclei, ​��/�Ω =∑�,�=1↑�▒​�↓� ​�↓� ​�↑−�​� ∙(​​� ↓� −​​� ↓� ) =​⟨�⟩↑� ∑�,�=�↑�▒​�↓� ​�↓� ​�↑−�​� ∙(​​� ↓� −​​� ↓� ) +�(⟨​�↑� ⟩−​⟨�⟩↑� )

Coherent scaering Incoherent scaering

Correlaon between Individual scaering relave spaal posions contribuon Coherent and Incoherent Scaering (cont’d) ​��/�Ω =​⟨�⟩↑� ∑�,�=�↑�▒​�↓� ​�↓� ​�↑−�​� ∙(​​� ↓� −​​� ↓� ) +�(⟨​�↑� ⟩−​⟨�⟩↑� )

Coherent scaering Incoherent scaering

Correlaon between relave spaal posions Individual scaering contribuon Coherent and Incoherent Scaering (cont’d)

​⟨�⟩↑2 Coherent scaering cross secon

​​⟨�⟩↑2 −⟨�⟩↑2 Incoherent scaering cross secon

Nuclide bcoh (fm) σcoh σinc (barn) (barn) H -3.472 1.8 80.2 D 6.674 5.6 2 C 6.65 5.55 0.001 O 5.805 4.2 0.0008 V -0.443 0.02 5 Reciprocal Space – Spaal Frequency Reciprocal Space – Spaal Frequency (cont’d)

Time space shape: �(�) Frequency space shape: �(�)

ℱ[�(�)]=�(�) ​ℱ↑−1 [�(�)]=�(�)

��=2�

Fourier transform: ℱ[�(�)]=∫−∞↑+∞▒�(�)​�↑−��� �� =�(�)

ℱ[�(​� )]=∫�↑ ▒�(​� )​�↑−�​� ∙​� ​�↑3 ​� =�(​� ) Reciprocal Space – Spaal Frequency (cont’d)

Time space shape: �(�) Frequency space spectrum: �(�)

ℱ[�(�)]=�(�) ​ℱ↑−1 [�(�)]=�(�)

��=2�

Real space distribuon: �(​� ) Reciprocal space spectrum: �(​� )

ℱ[�(​� )]=�(​� ) ​ℱ↑−1 [�(​� )]=�(​� )

��=2� Reciprocal Space – Spaal Frequency (cont’d) Reciprocal Space – Spaal Frequency (cont’d) Correlaon Funcons

Neutron/X-ray/light scaering measures different mathemacal transforms (Fourier, Abel) of two-point correlaon funcons (Debye, van Hove) in different spaces (r/Q, t/ω) and different me or length scales (λ, 2θ). Correlaon Funcons (cont’d)

Interference between two scaered waves: ​Ψ↓� ​Ψ↓�↑∗ =​�↓� ​�↓� ​�↑−�​� ∙(​​� ↓� −​​� ↓� )

Sum over all scaerers: ​��/�Ω =∑�,�=1↑▒​Ψ↓� ​Ψ↓�↑∗ =∑�,�=1↑�▒​�↓� ​�↓� ​�↑−�​� ∙(​​� ↓� −​​� ↓� ) Correlaon Funcons (cont’d)

​��/�Ω =∑�,�=1↑▒​Ψ↓� ​Ψ↓�↑∗ =∑�,�=1↑�▒​�↓� ​�↓� ​�↑−�​� ∙(​​� ↓� −​​� ↓� )

Debye correlaon funcon (structure) van Hove pair correlaon funcon (dynamic) �(​� ,�)=​1/� ∑�,�=1↑�▒�(​� −​​� ↓� (�)+​​� ↓� (0)) �(​� )=∫�↑ ▒�(​� ′)�(​​� ↑′ +​� )​�↑3 ​� ′

Pair distribuon funcon (structure) Self me correlaon funcon (dynamic)

�(​� )=​�/​�↑2 ∑�,�=1↑�▒�(​� −​​� ↓� +​​� ↓� ) ​�↓� (​� ,�)=​1/� ∑�=1↑�▒�(​​� ↓� (0))�(​� −​​� ↓� (�)) Correlaon Funcons (cont’d)

Debye correlaon funcon (structure) van Hove pair correlaon funcon (dynamic) �(​� ,�)=​1/� ∑�,�=1↑�▒�(​� −​​� ↓� (�)+​​� ↓� (0)) �(​� )=∫�↑ ▒�(​� ′)�(​​� ↑′ +​� )​�↑3 ​� ′

�(​� )=ℱ[�(​� )] �(​� ,�)=​ℱ↓� [�(​� ,�)] (SANS,USANS,ND,NR) (NSE) �(​� )=�[�(​� )] �(​� ,�)=​ℱ↓�,� [�(​� ,�)] (SESANS) (INS)

Pair distribuon funcon (structure) Self me correlaon funcon (dynamic) �(​� )=​�/​�↑2 ∑�,�=1↑�▒�(​� −​​� ↓� +​​� ↓� ) ​�↓� (​� ,�)=​1/� ∑�=1↑�▒�(​​� ↓� (0))�(​� −​​� ↓� (�))

�(​� )=ℱ[�(​� )] ​�↓� (​� ,�)=​ℱ↓�,� [​�↓� (​� ,�)] (SANS,USANS) (QENS,incoh) Neutron Diffracon

Scaering triangle ​​� ↓� ​�

θ θ

​​� ↓� ​� : Momentum transfer |​​� ↓� |=|​​� ↓� |=​2�/� ​� =​​� ↓� −​​� ↓�

∴�=|​� |=2|​​� ↓� |����=​4�/� ���� ��=��� Neutron Diffracon (cont’d)

Two axis diffractometer Scaering triangle ​​� ↓� ​�

θ θ

​​� ↓�

Diffracon – Where the atoms are: Clifford Shull, 1994 Nobel Prize (1/2) Notaons

Crystal lace ​� =​�↓1 ​​� ↓1 +​�↓2 ​​� ↓2 +​�↓3 ​​� ↓3

Reciprocal lace ​​� ↓ℎ�� =ℎ​​� ↓1↑∗ +�​​� ↓2↑∗ +�​​� ↓3↑∗ ​​� ↓1↑∗ =​2�/� ​​� ↓2 ×​​� ↓3 Miller indices ℎ,�,� ​​� ↓2↑∗ =​2�/� ​​� ↓3 ×​​� ↓1

(ℎ��): a set of planes perpendicular to ​​� ↓ℎ�� , separated by ​2�/|​​� ↓ℎ�� | ​​� ↓3↑∗ =​2�/� ​​� ↓1 ×​​� ↓2

[ℎ��]: a specific crystallographic direcon �=​​� ↓1 ∙(​​� ↓2 ×​​� ↓3 ) {ℎ��}: a set of symmetry-related lace planes ​​� ↓� ∙​​� ↓�↑∗ =2�​�↓�� ⟨ℎ��⟩: a set of symmetry-equivalent crystallographic direcons Ewald Sphere

Laue’s condion ​� =​​� ↓ℎ��

​� ∙​​� ↓1 =2�ℎ,​� ∙​​� ↓2 =2��, ​� ∙​​� ↓3 =2�� Powder Diffracon

• Phase idenficaon • Crystallinity • Lace parameters • Crystallite size • Orientaon Rietveld Refinement Method

�=​�↓0 ∑ℎ↑▒​�↓ℎ ​�↓ℎ ​�↓ℎ ​�↓ℎ↑2 �(​∆↓ℎ ) +​�↓�

​�↓0 : incident intensity

​�↓ℎ : scale factor for parcular phase

​�↓ℎ : reflecon mulplicity

​�↓ℎ : correcon factors on intensity (texture…)

​�↓ℎ : structure factor for a parcular reflecon ​�↓ℎ�� =∑�↑▒​�↓� ​�↑−�​� ∙​​� ↓� ​�↑−​�↓�

�(​∆↓ℎ ): peak shape funcon (instrument resoluon funcon, crystallite size, strain, defects)

​�↓� : background intensity Example: Polymer Diffracon

• Phase idenficaon <110> • Crystallinity ​�↓�� =​​�↓�� /​�↓�� +�​�↓�� • Lace parameters �=​2�/� • Crystallite size ​�↓ℎ�� =​0.89�/(����−Δ�)���� • Orientaon <200> ​�↓� =​1/2 (3⟨​���↑2 �⟩−1)

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(www.xraydiffrac.com) Example: Polymer Diffracon (cont’d)