Quasielastic Neutron Scattering (QENS)

Joachim WuttkeForschungszentrum JülichJülich Centre for Neutron Science at MLZ GarchingScientific Computing Grouphttp://apps.jcns.fz-juelich.de

SISN data analysis school, June 2016

MLZ is a cooperation between

outline

What is QENS?What makes QENS analysis special and difficult?What software do we need?

2

ressources

http://apps.jcns.fz-juelich.deposters and talks > slides of this talkpublications > Quasielastic Scattering, spring school 2012Frida > download, installation instructions, tutorial

3

TOF spectracoenzyme Q10

Smuda

etal:J

ChemPhys

2008,measured

onTO

FTOF

(FRMII)

4

quasielastic scatteringcoenzyme Q10

Smuda

etal:J

ChemPhys

2008,measured

onTO

FTOF

(FRMII)

5

what does ‘quasielastic’ mean ?

Def 1|ℏω| ≪ Ei

6

decoupling scattering ←→ sample physics

∂2σ

∂ω∂Ω=

kfki

Nσ4π S(q, ω)

7

what does ‘quasielastic’ mean ?

Def 1|ℏω| ≪ Ei

Def 2broadened elastic peak

8

what does ‘quasielastic’ mean ?

Def 1|ℏω| ≪ Ei

Def 2broadened elastic peak

8

methyl group dynamics in a molecular solidtetramethylpyrazine : picric acid

Sawka-Dobrowolska

etal:J.M

ol.Struct.

2010

9

methyl group dynamics in a molecular solidtetramethylpyrazine : picric acid

−5 −2.5 0 2.5 5

hω (µeV)

10

100

1000

S(q

,ω)

(µe

V−

1)

80 K

50 K

Sawka-Dobrowolska

etal:J.M

ol.Struct.

2010,measured

onSPH

ERES(JCN

SatFRM

II)

10

methyl group dynamics in a molecular solidtetramethylpyrazine : picric acid

−5 −2.5 0 2.5 5

hω (µeV)

10

100

1000

S(q

,ω)

(µe

V−

1)

80 K

50 K

35 K

Sawka-Dobrowolska

etal:J.M

ol.Struct.

2010,measured

onSPH

ERES(JCN

SatFRM

II)

10

methyl group dynamics in a molecular solidtetramethylpyrazine : picric acid

−5 −2.5 0 2.5 5

hω (µeV)

10

100

1000

S(q

,ω)

(µe

V−

1)

80 K

50 K

35 K

30 K

Sawka-Dobrowolska

etal:J.M

ol.Struct.

2010,measured

onSPH

ERES(JCN

SatFRM

II)

10

methyl group dynamics in a molecular solidtetramethylpyrazine : picric acid

−5 −2.5 0 2.5 5

hω (µeV)

10

100

1000

S(q

,ω)

(µe

V−

1)

80 K

50 K

35 K

30 K

4 K

Sawka-Dobrowolska

etal:J.M

ol.Struct.

2010,measured

onSPH

ERES(JCN

SatFRM

II)

10

rotational tunnelingmethyl group CH3

ψA = ψ1 + ψ2 + ψ3

ψE = ψ1 + e±i2π/3ψ2 + e∓i2π/3ψ3

Press,Single-particlerotation

inm

olecularcrystals(1981)

11

what does ‘quasielastic’ mean ?

Def 1|ℏω| ≪ Ei

Def 2broadened elastic peak

Def 3slow modes

Def 4measured on TOF/BS/NSE spectrometer

Def 5presented at QENS conference

12

what does ‘quasielastic’ mean ?

Def 1|ℏω| ≪ Ei

Def 2broadened elastic peak

Def 3slow modes

Def 4measured on TOF/BS/NSE spectrometer

Def 5presented at QENS conference

12

what does ‘quasielastic’ mean ?

Def 1|ℏω| ≪ Ei

Def 2broadened elastic peak

Def 3slow modes

Def 4measured on TOF/BS/NSE spectrometer

Def 5presented at QENS conference

12

what does ‘quasielastic’ mean ?

Def 1|ℏω| ≪ Ei

Def 2broadened elastic peak

Def 3slow modes

Def 4measured on TOF/BS/NSE spectrometer

Def 5presented at QENS conference

12

quasielastic light scattering

Quasielastic light scattering

dynamic light scatteringphoton-correlation spectroscopy

figurefrom

wikipedia

13

confused terminology

Quasielastic light scattering

photon-correlation spectroscopy |E|4 µs …s

Raman-Brillouin scattering

grating spectrometer, interferometer |E|2 GHz …THz

Neutron scattering

quasielastic ≃ high-resolution |ψ|2 GHz …THz

14

scattering kinematicsgeneric case

2ϑ

kf (ω>0)

kf (ω=0)

kf (ω<0)

ki

q (ki,ϑ,ω)

15

scattering kinematics

−15 −10 −5 0 5

Ei − Ef (meV)

0

1

2

3

4q (

A−

1)

Ei = 4 meV0°

60°

120°

180°

16

scattering kinematicsquasielastic case |ℏω| ≪ Ei

ϑ

kf (ω≈0)

ki

q = 2 ki sin ϑ

17

resolution broadening

−10 −5 0 5 10

ω

10−4

10−3

10−2

10−1

S (

ω)

ΩS :

0.025

0.1

0.4

1.6

scattering function:

S(ω) = 1π

ΩSω2+Ω2

S

resolution:

R(ω) = 1√2πΩR

exp(− ω2

2Ω2R

)convolution:

SR(ω) =∫

dω′ R(ω−ω′)S(ω′)

18

resolution broadening

−10 −5 0 5 10

ω / ΩR

10−4

10−3

10−2

10−1

S (

ω)

(ΩR

−1)

ΩS / ΩR :

0.025

0.1

0.4

1.6

scattering function:

S(ω) = 1π

ΩSω2+Ω2

S

resolution:

R(ω) = 1√2πΩR

exp(− ω2

2Ω2R

)

convolution:

SR(ω) =∫

dω′ R(ω−ω′)S(ω′)

18

resolution broadening

−10 −5 0 5 10

ω / ΩR

10−4

10−3

10−2

10−1

S (

ω)

(ΩR

−1)

ΩS / ΩR :

0.025

0.1

0.4

1.6

scattering function:

S(ω) = 1π

ΩSω2+Ω2

S

resolution:

R(ω) = 1√2πΩR

exp(− ω2

2Ω2R

)convolution:

SR(ω) =∫

dω′ R(ω−ω′)S(ω′)

18

resolution histogram ⊗ sharply peaked theory

−2 0 2

hω (µeV)

0.1

1

10

100

1000

S(q

,ω)

(µe

V−

1)

(R⊗ T)(ω) =∑

ω′ R(ω − ω′)T(ω′)

must be replaced by

(R⊗ T)(ω) =∑

ω′ R(ω − ω′) [P(ω′+∆ω/2)− P(ω′−∆ω/2]

Wuttke,Algorithm

s(2012)

19

-10 -5 0 5 10

hω (µeV)

0.001

0.01

S (

q,ω

) (

µeV

-1)

1.0 ± 0.15 A-1

300 K274 K250 K240 K230 K217 K200 K100 K

0 0.5 1

hω (µeV)

0.1

1

S (

q,ω

) /

S (

q,0

)

3 4 5 6

1000 K / T

10-11

10-10

10 -9

10 -8

10 -7

10 -6

10 -5

⟨τ⟩ (s

)

CPC improved fit

CPC standard fit

lysozyme

myoglobin

myoglobin (NMR)

myoglobin (diel)

TL

(∆ω)-1

Fourier deconvolutionrelaxation in a molecular glass former

SR(ω) =∫

dω′ R(ω−ω′)S(ω′) =⇒ I(t) = IR(t)/R(t)

IN13IN6IN5

o-terphenyl 1.2A-1

293 K

298 K

306 K

312 K

320 K

327 K

1 10 1000

0.2

0.4

0.6

0.8

t (psec)

I (q

,t)

Wuttke

etal:Z.Phys.

B(1993),m

easuredatthe

ILL

22

S(ω) vs I(t)

0

ω

0

S (

ω)

t

0

1

I (t

)

23

S(ω) vs I(t)

10−13 s−1 0 10−13 s−1

ω

0

S (

ω)

ps

log t

0

1

f

I (t

)

S(ω) = fδ(ω) + (1− f)Sphonons(ω)

f =

Debye-Waller factor (coherent scattering)Lamb-Mössbauer factor (incoherent scattering)

24

S(ω) vs I(t)localized motion

10−13 s−1 0 10−13 s−1

ω

0

S (

ω)

ps

log t

0

1

f

f A

I (t

)

S(ω) = f [Aδ(ω) + (1− A)Squasiel.(ω)] + (1− f)Sphonons(ω)

A = elastic incoherent structure factor (EISF)

25

elastic incoherent structure factor2-site jump model, jump length 1.2 Å

0 2.5 5 7.5 10 12.5

q (A−1)

0

0.25

0.5

0.75

1

EIS

F

IN10, IN16, HFBS, SPHERES

IN13

Bée:Q

uasielasticneutron

scattering(1988,outofprint)

26

S(ω) vs I(t)localized motion

10−13 s−1 0 10−13 s−1

ω

0

S (

ω)

ps

log t

0

1

f

f A

I (t

)

S(ω) = f [Aδ(ω) + (1− A)Squasiel.(ω)] + (1− f)Sphonons(ω)

A = elastic incoherent structure factor (EISF)

27

S(ω) vs I(t)long-ranged motion

10−13 s−1 0 10−13 s−1

ω

0

S (

ω)

ps

log t

0

1

f

f A

I (t

)

S(ω) = fSquasiel.(ω) + (1− f)Sphonons(ω)

28

basic functions for Squasiel.(ω)

ballistic short-time limit:S(ω) ∼ exp(−τ2ω2/2) I(t) = 1− t2/2τ2 + . . .

diffusion, rotational diffusion, jump models:

S(ω) ∼(1 + τ2ω2)−1 I(t) = exp(−t/τ)

complex relaxation: e.g.S(ω) ∼ Re (1 + i(ωτ)α)−γ I(t) = exp(−(t/τ)β)

29

localized motion: rotation in a molecular solid(CH3NH3)5Bi2Br11

−10 0

hω (µeV)

0.1

1

10

100

S(q

,ω)

(ve

rtic

ally

sh

ifte

d) (CH3NH3)5Bi2Br

65 K

75 K

85 K

100 K 10 12.5 15 17.5

1000 K / T

0.1

1

10

τ (

ns)

Tc

weight 2

weight 3

Piechaetal,publication

overdue,measured

onSPH

ERES(JCN

SatFRM

II)

30

Ni diffusion in Ni:Zr melt

Ni36Zr64

Q = 0.9 A−1

T = 1650 K

T = 1290 K

0 1 2

hω (meV)

0.01

0.1

1

S(q

,ω)

(m

eV

−1)

Ni36Zr64

T = 1650 K

T = 1345 K

T = 1210 K

1 2 3

Q2 (A−2)

0

0.2

0.4

0.6

Γ Q (m

eV

)

Lorentzian fits ⇒ width Γ ⇒ diffusion coefficents D = Γ/q2

Holland-M

oritzetal:

PhysRev

B(2009),m

easuredon

TOFTO

F(FRM

II)

31

H motion in n-alkanes

CnH2n+2

NMR

QENS

100 1000

molecular mass

10−10

10−9

10−8

D

(m2/s

)

C32H66

1 10 100 1000

resolution time (ps)

0

1

2

3

D (1

0−

9 m

2/s

)

small-q (NMR): center-of-mass diffusionlarge-q (QENS): intra-chain (Rouse) motion

Smuda

etal:J

ChemPhys;Unruh

etal:ibid

(2008),measured

onTO

FTOF

(FRMII)

32

two steps of data analysis

neutron counts N(j,i)

data reduction

scattering law S(q,ω)

data analysis, fitting, interpretation

results

33

two steps of data analysisbut: reduced data still contain resolution and multiple scatering

neutron counts N(j,i)

data reduction

scattering law S(q,ω) ⊗ R(ω)

data analysis, fitting, interpretation

results

34

data analysis is circular

neutron counts N(j,i)

data reduction

scattering law Sexp(q,ω) ⊗ R(ω)

inspection fit

fit model Stheo(q,ω) ⊗ R(ω)

fit parameters

35

shall we fit raw data?

neutron counts N(j,i)

data reduction

scattering law Sexp(q,ω) ⊗ R(ω)

inspection

fit

fit model Stheo(q,ω) ⊗ R(ω) ⊗ D(q,ω)

fit parameters

36

rank of datarank 3:S(q, ω;T) inelastic temperature scanS(q, ω; t) other inelastic time scan

rank 2:S(q, ω) regular scanS(ω; t) q-averaged/selected time scanS(q, 0; t) elastic time scan

rank 1:S(ω) q-averaged/selected spectrumS(q, 0) elastic intensity

37

inelastic T scan at SPHERES

Mg(NH3)6Cl2

|E| < 0.25µeV

|E| > 1.2 µeV

inelast x 100

elast

0 50 100 150 200 250

T (K)

0

200

400

600

800

counts

(s

−1)

Q.Lietal,m

easuredon

SPHERES,unpublished

38

Real-time kineticsdecomposition of sodium alanate

3 NaAlH4 → Na3AlH6 + 2 Al + 3 H2

Na3AlH6 → 3 NaH + Al + 1.5 H2

3 NaH → 3 Na + 1.5 H2

-5 0 5

hω (µeV)

0.01

0.1

1

S(q

,ω)

(µe

V-1

)

NaAlH4

177°C 0.. 5 h

15..20 h

30..35 h

45..50 h

64..69 h

Léon&

Wuttke,J

PhysCondensed

Matt(2011),m

easuredon

SPHERES

(JCNS

atFRMII)

39

real-time kineticsdecomposition of sodium alanate

-5 0 5

hω (µeV)

0.01

0.1

1

S(q

,ω)

(µe

V-1

)

NaAlH4

177°C 0.. 5 h

15..20 h

30..35 h

45..50 h

64..69 h

0 25 50 75 100

t (h)

0

0.2

0.4

0.6

0.8

am

plit

udes

0 25 50 75 100

t (h)

0

0.2

0.4

0.6

0.8

am

plit

udes

NaAlH4 → Na3AlH6 → NaH

total

elastic

Lorentzian (Na3AlH6)

d[A]/dt = −k00[A]− k01[A][B]2

d[B]/dt = −d[A]/dt− d[C]/dtd[C]/dt = k10[B]4/3 + k11[B]4/3[C]2/3

Léon&

Wuttke,J

PhysCondensed

Matt(2011),m

easuredon

SPHERES

(JCNS

atFRMII)

40

rank reduction cascadescattering law Sexp(ω; q,T) ⊗ R(ω)

inspection fits for individual q,T

fit model Stheo(ω) ⊗ R(ω)

parameters Pexp(q;T)

inspection fits for individual T

fit model Ptheo(q)

parameters pexp(T)

inspection fit

fit model ptheo(T)

global parameters

41

feed outcome back into modelscattering law Sexp(ω; q,T) ⊗ R(ω)

fit model Stheo(ω) ⊗ R(ω) global fit Stheo(ω,q,T) ⊗ R(ω)

fit

parameters Pexp(q;T)

fit model Ptheo(q) derive constraints

parameters pexp(T)

fit model ptheo(T)

global parameters

42

software: Unix vs workbench principle

file plot

tool

file plot

tool

file plot

tool

file plot

tool

file

file

workspaceworkspaceworkspaceworkspaceworkspace

transform

fit

file plot

43

why workbench?

file

workspaceworkspaceworkspaceworkspaceworkspace

transform

fit

file plot

advantages:saves disk I/Opermanent storage only when requiredcan be GUIfiedefficient CLIsupport out of one hand

disadvantages:less concurrence for best toolsscripting less easy, less standard

perspective:Jupyter notebook

44

QENS data analysis software

1990s:INX, SQW data reduction for time-of-flight / backscatteringIDA → Frida command-line workbench for data analysis

2000s:LAMP GUI workbench for data reduction and analysisDAVE, DANSE

2010s:Mantid GUI workbench for spallation data reduction

2020s ???

45

Frida

a Swiss Army Knife forfast reliable interactive data analysis

History:1990−2001 Ida → Frida1 in Fortran772001− Frida2 in C++

46

Frida

a Swiss Army Knife forfast reliable interactive data analysis

Usage:used by a few groupstaught to users of SPHERES

47

Frida

a Swiss Army Knife forfast reliable interactive data analysis

Status:legacy one-man projectnot an official project of MLZ Scientific Computing Group

48

Frida

a Swiss Army Knife forfast reliable interactive data analysis

Hasopen-source licencedownload page, CMakeversion controlfrequent releasessome tutorialssome tests

Hasn’tGUIuser manualfull test coverage

49

Frida

a Swiss Army Knife forfast reliable interactive data analysis

a collection of algorithmsoperating on data files in RAM (»workspaces«)controlled by a concise (cryptic) command-line interfacewith dedicated fit models for QENSgenerating human editable PostScript graphics

50

−10 0 10

E (ueV)

0.1

1

10

S(E

,q)

(u

eV

−1

)

gly5 mfj # gly5 is merger of: − gly255 glycerol measured on SPHERES by J.Wuttke reduced data set for Frida tutorial fs gly255.y08 # Fri Aug 10 16:16:30 2012 mpaf 3 mr j==8 − gly275 ===fs gly275.y08 # Fri Aug 10 16:16:31 2012 − gly295 ===fs gly295.y08 # Fri Aug 10 16:16:31 2012 − gly305 ===fs gly305.y08 # Fri Aug 10 16:16:31 2012

0 254.107 1.41697 1 274.146 1.41697 2 293.905 1.41697 3 304.177 1.41697

fit_gly5 cc p0*pconv(kwwp(t,p1,p2)) # z from gly5 p0*pconv(kwwp(t,p1,p2)) data file: 14, conv file: 13, weighing: with reciprocal variance (data and curve) j z0 z1 p0 p1 p2 oc chi^2 1−R^2

0 254.107 1.41697 24.1952 292.671 0.414796 1 1.80418 0.134212 1 274.146 1.41697 23.3787 22.3248 0.484547 1 3.21818 0.352675 2 293.905 1.41697 21.9556 1.25831 0.551861 1 1.21577 0.599272 3 304.177 1.41697 20.9664 0.521076 0.586763 1 1.34236 1.75922

plot −> /home/jwu/pub/V/17/1704−Frida/gly_join.ps

8 minutes manual editing

−10 0 10

hω (µeV)

0.1

1

10

S(q

,ω)

(µe

V−

1)

304 K

294 K

274 K

254 K

180 K

51

PS file → original data points

1 [ 254.107 1.41697 ] zValues1 pstyle % (E (ueV) -> S(E,q) (ueV -1))0.36000 1.58792 0.21947 ti % -13.92 wx 0.04215903 wy0.44000 1.46066 0.23179 t % -13.68 wx 0.03785536 wy0.52000 1.81438 0.19977 t % -13.44 wx 0.051063 wy[...]

52

fl g*msr! 8mpaf! 31:4 mfj1:4 fdel1 cc p0*pconv(kwwp(t,p1,p2))cv 02 op2 .6cx 2cfcu 2cfg20 p1:2 a :gp graphic_file_name

−10 0 10

E (ueV)

0.1

1

10

S(E

,q)

(ueV

−1)

gly180

glycerol

measured on SPHERES by J.Wuttke

reduced data set for Frida tutorial

fs gly180.y08 # Fri Aug 10 16:16:30 2012

mpaf 3

8 1.41697

gly255

....

53

2 oi .6582*p1/p2*gamma(1/p2)ecy <tau>(ns)ox! 1000/xcc p0*exp(p1*t)cwlcfga3,4 pgp graphic_file_name

3.4 3.6 3.8

1000/T (1/(K))

1

10

100

<ta

u>

(n

s)

fit_gly5

cc p0* pconv(kwwp(t,p1 ,p2)) # z from gly5

oi .6582*p1/p2*gamma(1/p2)

ecy <tau>(ns) # old: 0.658200*p1/p2*gamma(1/p2)()

ox 1000/x

0

fit_fit_gly5

cc p0*exp(p1*t) # z from fit_gly5

p0*exp(p1*t)

data file: 5, weighing: logarithmic

j p0 p1 oc chi^2 1−R^2

0 9.1608e−16 10.367 5 0.203248 8.32178e−07

plot −> /home/jwu/pub/V/17/1704−Frida/gly−tau.ps

54

y is shorthand for y[,,] is shorthand for y[k,j,i]

Command oy f(y) is executed asfor k in file_selection:

for j in [0,nj) spectra in filefor i in [0,ni) points in spectrum

y_out[k_out,j,i] := f(y[k,j,i])

This allows foroy y/y[0] normalize to file 0oy y/y[,0] normalize to spec 0 of current fileoy y/y[,,0] normalize to point 0 of current specoy y/y[k-4,0,j] normalize spec j to point j of spec 0 of file k-4

55

−10 0 10

hω (µeV)

0.1

1

10

S(q

,ω)

(µe

V−

1)

304 K

294 K

274 K

254 K

180 K

3.4 3.6 3.8

1000/T (1/(K))

1

10

100

<ta

u>

(n

s)

extract fit parameter

reduce rank

xω

y

Sq(ω;T)

zT

xT

y

τq(T)

56

x

y

z0

z1

z2

x ← z0

y

z0 ← z1

z1 ← z2

Rank-reducing operations:p0 fit parameter 0ni number of points in spectrumsum(y)

∑y[, , i]

avge(y)∑

y[, , i] / niintegral(x,y)

∫dx y(x) per midpoint rule

valmax(y) maxi y[, , i]idxmin(y) index i for which y[, , i] is minimalcog(x,y) center of gravity in x weighed with ywidth(x,y) standard variation in x weighed with ycorr(x,y) correlation coefficient of x and yfirstwith(expr) first i for which expr is true

57

Other functionality:command line as pocket calculatorfunction plotter handling singularities and frame crossingsfunction integration2D color plotsimport/export from/to various tabular formats

58