Structural characterization of amorphous materials using … · Structural characterization of amorphous materials using x-ray scattering ... A gentle introduction to the theory of

Structural characterization ofamorphous materials using x-ray scattering

Todd C. HufnagelDepartment of Materials Science and Engineering

Johns Hopkins University, Baltimore, Maryland

[email protected]

Funding for work on scattering from metallic glasses provided by NSF-DMR, DOE-BES, ARO, ARLScattering data presented collected at SSRL (7-2, 10-2), APS (1-ID), NSLS (X-14A)

Outline

1.How is scattering from amorphous materials different from diffraction from crystalline materials?

2.A gentle introduction to the theory of scattering from amorphous materials, and the structural information you get.

3.The path to enlightenment: From raw data to real-space structure

4.Experimental techniques and considerations

5.Resonant scattering

6.Resources

T. C. Hufnagel, Johns Hopkins University

!q " 2!

"

!s # s$

"

|!q| = 4! sin "

#= 2!

dhkl

X-ray diffraction (crystalline materials)

.hkl/ planes 2!

EqOsOsı

Scattering vector:

Bragg’s Law:

Incident x-rays Diffracted x-rays


q (A!1)

Composite consisting of ~4 vol. % Ta particles in Zr-based metallic glass matrix

Example: Mixed amorphous + crystalline structure

Ta

Ta

Ta Ta

Ta Ta

Ta

Metallic glass matrix


Crystalline AmorphousScattering is strong

(intense peaks)Scattering is weak

Scattering concentrated into a few sharp diffraction peaks

Scattering spread throughout reciprocal space (all values of q)

Data easilyinterpreted using simple equations

Detailed analysisrequired to obtain

real-space information


Amorphous scattering theory I

An D fn exp.!i Eq " Ern/

Scattered amplitude from a single atom:

Scattered intensity from n atoms (arbitrarily arranged):

D X

n

fn exp.!i Eq " Ern/

! X

m

f !m exp.i Eq " Erm/

!

DX

m

X

n

f !mfn exp.i Eq ! Ernm/

Ieu.Eq/ D X

n

An

! X

m

A!m

!

fn D Atomic scattering factor

Ern D Position of nth atom

Ernm D Vector joining nth and mth atoms


Amorphous scattering theory II

Assume that the material is isotropic (so we can averageover all orientations of rnm) to obtain the Debye scattering equation:

Ieu.q/ DX

m

X

n

f !mfn

sin qrnm

qrnm

This applies to any assemblage of identical scattering units(e.g. molecules) so long as they scatter independently andare randomly oriented.


Example: Ehrenfest relation

P. Ehrenfest Proc. Amsterdam Acad. 17, 1184 (1915)A. Guinier X-Ray Diffraction (Dover, 1994), pp. 72-74.

Amorphous case is too hard; instead, consider scattering from diatomic gas:

d

If the molecules scatterindependently, then: I(q) ! "

m"n

fm fnsinqrmn

qrmn= 2 f 2

!1+

sinqdqd

"

If the atoms are point scatters, then f is independent of q and the scattering has a maximum at

qmax =1.23(2!)

d!

!


Amorphous scattering theory III

Define a new quantity, the structure factor, S(q):

S.q/ ! Ieu.q/

N hf .q/i2

S(q) is related to the real-space structure, the pair distribution function ρ(r), by a Fourier transform:

!.r/ ! !ı D 1

8"3

Z 1

0

4"q2 .S.q/ ! 1/sin qr

qrdq

Note the limits on the integral!

N D number of atoms

!ı D average atomic density


S(q)

!(r

)(A

!3)

q (A!1) r (A)

(a) (b)

RD

F,4!

r2 "(r

)(A

!1)

r (A)

Data in reciprocal space and real spaceReciprocal space

“Structure factor”Real space

“Pair distribution function”

S(q) = I (q)

N ! f (q)"2

!(r) ! !" = 18"3

! #

04"q2 (S(q) ! 1)

sin qrqr

dq

Real space“Radial distribution function”

RDF = 4!r2"(r)

g(r) = !(r)

!!T. C. Hufnagel, Johns Hopkins University

Physical interpretation of real-space functionsS(q) = I (q)

N ! f (q)"2 !(r) ! !" = 18"3

! #

04"q2 (S(q) ! 1)

sin qrqr

dq

RDF = 4!r2"(r)

g(r) = !(r)

!!

Figure from Y. Waseda, Anomalous X-Ray Scattering for Materials Characterization (Springer, 2002)T. C. Hufnagel, Johns Hopkins University

Data reduction 1: Raw data

Zr57Ti5Cu18Ni8Al10 bulk metallic glassTransmission geometry (2.8 mm diameter cylinder)80.72 keV x-rays (APS beamline 1-ID)MAR image plate300 mm camera length

60000

50000

40000

30000

20000

10000

0

Inte

nsi

ty (

counts

)

1614121086420

Scattering vector magnitude, q (Å-1

)


Data reduction 1I: Apply corrections

50000

40000

30000

20000

10000

0

Inte

nsi

ty (

counts

)

1614121086420


)

Corrected Raw data



50000

40000

30000

20000

10000

0

Inte

nsi

ty (

cou

nts

)

1614121086420


)

1.Detector dead time

14x103

12

10

8

6

4

2

0Mea

sure

d in

tens

ity (c

ount

s pe

r se

cond

)

40x103 3020100True intensity (arbitrary units)

3 µs shaping time 10 µs shaping time



50000

40000

30000

20000

10000

0

Inte

nsi

ty (

cou

nts

)

1614121086420


)

1.Detector dead time2.Background (substrate/container)3.Absorption4.Polarization5.Multiple scattering

O1O2

y

x

hr



50000

40000

30000

20000

10000

0

Inte

nsi

ty (

cou

nts

)

1614121086420


)

1.Detector dead time2.Background (substrate/container)3.Absorption4.Polarization5.Multiple scattering6.Inelastic processes (Compton, fluorescence)


2500

2000

1500

1000

500

0

Inte

nsi

ty (

ele

ctro

n u

nits

)

1614121086420


)

Coherent independent scattering Scaled intensity

Data reduction III: Normalize to e- units

Icoh.q/ D hf .q/f !.q/i


4

3

2

1

0

S(q

)

1614121086420


)

Data reduction IV: Normalization, S(q)Ieu.q/ D Ielastic.q/ C Icoh.q/ C Iinc.q/ D ˛Iobs.q/

S.q/ D Ieu.q/

hf .q/ihf !.q/i D ˛Iobs.q/ ! Icoh.q/ ! Iinc.q/

hf .q/ihf !.q/i

Measured (arb. units)Calculated/tabulated

Calculated/tabulated

Normalizationconstant


Data reduction IV: Normalization, S(q)4

3

2

1

0

S(q

)

1614121086420


)

Oscillating about S(q)=1 is equivalent to oscillating about hf .q/f !.q/i

S.q/ D Ieu.q/


hf .q/ihf !.q/i

Warren’s “large angle” method: Ieu(q) should oscillate about the independent coherent scattering at large q.

B. E. Warren, X-Ray Diffraction (Dover, 1990)T. C. Hufnagel, Johns Hopkins University

Data reduction IV: Normalization, S(q)4

3

2

1

0

S(q

)

1614121086420


)

S.q/ D Ieu.q/


hf .q/ihf !.q/i

Norman/Krogh-Moe (“integral”) method: Makes use of the fact that ρ(r)→ at low r to obtain:

˛ DR 1

0 q2Icoh.q/ dq ! 2!2"ıhf .q/ihf !.q/iR 1

0 q2Iobs.q/ dq:

N. Norman, Acta Crystallogr. 10, 370 (1957)J. Krogh-Moe, Acta Crystallogr. 9, 951 (1956)T. C. Hufnagel, Johns Hopkins University

Data reduction V: Real-space functions

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Pa

ir d

istr

ibu

tion

fu

nct

ion

, !

(r)

(Å-3

)

20151050

Distance from central atom, r (Å)

!(r) ! !" = 18"3

! #

04"q2 (S(q) ! 1)

sin qrqr

dq

Upper limit set by experiment

Positions of peaks→Coordination shell distances


Data reduction V: Real-space functions

50

40

30

20

10

0

Ra

dia

l dis

trib

utio

n f

un

ctio

n,

4!r2"

(r)(

ato

ms

Å-1

)

1086420

Distance from central atom, r (Å)

RDF = 4!r2"(r)

Area under peak = coordination number

4!r2"ıInspect the low r region...should be zero!(Large oscillations are a sign of trouble withnormalization, corrections, and/or damping.)


Damping and the high q limit4

3

2

1

0

S(q

)

1614121086420Scattering vector magnitude, q (Å

-1)

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

Dam

pin

g fu

nctio

n

20

15

10

5

0

RD

F (

ato

ms

Å-3

)

4.03.53.02.52.0


)


More on damping and Fourier transform

q (A!1)

q[S

(q)!

1 ]

RD

F,4p

ir2 !

(r)

(A!1

)Dam

pingfunction

r (A)

(a) (b)

R. Lovell, G. R. Mitchell, and A. H. Windle. Acta Crystallogr. Sec. A, 35, 598–603 (1979)T. C. Hufnagel, R. T. Ott, and J. Almer. MRS Symp. Proc. 913, Z18-01 (2006)

The Nyquist theorem puts a limit on the real space resolution:

!r D "=qmax .' 0:2 A for these data/


0.0001

0.001

0.01

0.1

1

10

r(g(r

)-1)

252015105

r (Å)

0% Ta!=3.8±0.2 Å

4% Ta!=5.3±0.2 Å

Extracting MRO information from the RDF

3.0

2.5

2.0

1.5

1.0

0.5

0.0

g(r

)

2520151050r(Å)

(Zr70Cu20Ni10)90-xTaxAl10

0% Ta 4% Ta

Exponential decay “envelope”in terms of a screening length, λ

Write the RDF as:

Oscillations dueto short-range order

ρ(r)4πr2−1 = g(r)−1 =

Ar

exp(−r

λ

)sin

(2πrD

+φ)

Analysis based on A. Bodapati et al. J. Non-Cryst. Sol. (in press)See also C.W. Outhwaite, Stat. Mech. 2, 188 (1975) T. C. Hufnagel, Johns Hopkins University

Experimental geometries and detectors

Evacuatedenclosure

Sample

X-ray beam frommonochromator

Beam slits

Ion chamber(incident beam monitor)

Detector slits

Focusing graphitemonochromator

Position-sensitivex-ray detector

Step scanning

Eq?

Eqjj

Image plate

Specimen

• Excellent energy resolution possible (with the right detector) • Slow scanning (hours) • May be count-rate (dead time) limited

• Excellent signal-to-noise ratio • Very fast “scans” (seconds) • Measure q in multiple orientations simultaneously (anisotropic specimens) • Poor energy resolution • Limited angular range

Area detector


• Need to measure I(q) to the largest q possible:

• The elastic, single-scattering intensity I(q) must be separated from all other sources of intensity. (This usually involves both experimental design and data analysis.)

• High signal-to-noise is essential at all values of q, especially at high q:

Data collection considerations

qmax D 4!

"D 4!E

hcqmax.A!1

/ ! E.keV/

!(r) ! !" = 18"3

! #

04"q2 (S(q) ! 1)

sin qrqr

dq


Anomalous dispersion of x-raysWrite the atomic scattering factor for x-rays as

f (q,E) = f!(q)+ f "(q,E)+ i f ""(q,E)

Energy-dependent anomalous scattering factors

Energy-independentatomic form factor

-8

-6

-4

-2

0

2

4

6

f', f'' (e

lect

rons)

20181614121086Energy (keV)

-12

-10

-8

-6

-4

-2

0

f' (e

lect

rons

)

18100180501800017950179001785017800X-ray energy (eV)

30

25

20

15

10

5

0Scat

terin

g cr

oss-

sect

ion

(1/c

m)

1.00.80.60.40.20.0q (nm-1)

!E = -100 eV

!E = -50 eV

!E = -10 eV

Zr K absorption edgeE=17998 eV

f (q,E) is unique for each element:

f !Zr

f !!Zr

f !!Ni

f !Ni


Energy resolution is criticalPosition-Sensitive Detector Output

energyk (1/Å)

elastic

K RR!

K RR"

intensity(cts/monitor cts)

19.985 keV

Complete spectrum allowsexperimental removal of

inelastic contributions fromCompton and

resonant Raman scattering

Compton scattering nominally eliminated down to ~3 Å below Ge edge and ~5 Å below Mo edge.-1 -1

#E = 394 eV

#E = 121 eV

Mo

Ge

RR

RR

H. Ishii and S. Brennan (unpublished)T. C. Hufnagel, Johns Hopkins University

Partial pair distribution function analysis

Amorphous NiZr studied by combined resonant x-ray scattering and Reverse Monte Carlo simulations.

2

Total structure factor S(q) Partial structure factors Partial distribution functions

J. C. de Lima et al., Phys. Rev. B 67, 094210 (2003)

More complicated alloys are . . . more complicated.


Resonant x-ray scattering on amorphous materials

For a sample with m elements, the total scattered intensity is a weighted sum of m(m+1)/2 partial structure factors:

Only pair correlations that involve atoms of type A contribute to the total scattering

I(q) = n!"

!#

x" f"(q,E) f !#(q,E)S"#(q)

Because f (q, E) only varies rapidly with E near an absorption edge, we can make a differential measurement for an element of interest (A here):

For instance, we can make scattering measurements at two energies below an absorption edge, and take the difference. The resulting real-space information is a distribution function specific to the element of interest.

!!I(q,E)

!E

"

q= xA"

#

!!E

#fA(q,E) f !#(q,E)+ f !A(q,E) f#(q,E)

$SA#(q)


30

25

20

15

10

5

0

RD

F (a

tom

s/Å)

876543210r (Å)


0% Ta 4% Ta

30

25

20

15

10

5

0

Cu D

DF

(ato

ms/

Å)

876543210r (Å)


0% Ta 4% Ta

30

25

20

15

10

5

0

RD

F (a

tom

s/Å)

876543210r (Å)


0% Ta 4% Ta

30

25

20

15

10

5

0

Ta

DD

F (a

tom

s/Å)

876543210r (Å)


4% Ta

30

25

20

15

10

5

0

RD

F (a

tom

s/Å)

876543210r (Å)


0%Ta 4%Ta

30

25

20

15

10

5

0

Zr

DD

F (a

tom

s/Å)

876543210r (Å)


0% Ta 4% Ta

Resonant x-ray scattering from (Zr70Cu20Ni10)90-xTaxAl10

Cu K edgeE=8979 eV

Ta LIII edgeE=9881 eV

Zr K edgeE=17998 eV

Average atom

Cu environment

Average atom

Ta environment

Average atom

Zr environment

Zr-edge data published in T. C. Hufnagel and S. Brennan, Phys. Rev. B 67, 014203 (2003)

0% Ta: CN=13.7 ± 0.54% Ta: CN=13.1 ± 1.1

4% Ta: CN=13.7 ± 1.6


Lessons I’ve learned the hard way

1.Characterize the dead time of your detector at the x-ray energies you use, and over a large range of count rates.

2.Either exclude the Compton scattering experimentally, or count all of it (so you can calculate and subtract it later).

3. If doing a transmission experiment, measure the actual absorption of your specimen (μt product).

4.If possible, measure the fluorescence from the specimen separately from the elastic scattering. (This tells you a lot about potential alignment problems.)

5.Rather than doing a few lengthy scans, do many scans of short duration (to avoid systematic errors).

6.Do the data analysis in near-real time on the first few scans, to identify problems early.


What’s important

1.Signal-to-noise ratio (particularly at high q)

2.Energy resolution on the scattered beam side (critical for resonant scattering)

3.Energy resolution on the incident beam side (resonant scattering only)

4.Minimizing background (anything that is not elastic scattering from your specimen)

What’s not (within reason)Resolution in q space; brightness (divergence)


“X-ray amorphous”

100

80

60

40

20

0

Inte

nsity

(arb

. uni

ts)

5.04.03.02.0Scattering vector magnitude (Å-1)

Laboratory sourceCu Kα radiation (λ=1.54 Å)

100

80

60

40

20

0

Inte

nsity

(arb

. uni

ts)

5.04.03.02.0Scattering vector magnitude (Å-1)

Synchrotron sourceE=17898 eV (λ=0.693 Å)

Is it amorphous? • First peak symmetric • Second peak at ~1.6-1.8 qmax (for a metallic glass) • No other scattering features present • Combine with other techniques (TEM, DSC)

Fundamental problem: All you get (from scattering) is the RDF,which is only sensitive to pair correlations.


0.012

0.010

0.008

0.006

0.004

0.002

0.000

Nor

mal

ized

var

ianc

e

543210Scattering vector magnitude, k (nm-1)

Zr59Ta5Cu18Ni8Al10 Zr57Ti5Cu20Ni8Al10

J. Li, X. Gu, and T. C. Hufnagel. Microscopy and Microanalysis 9, 509 (2003)

T. C. Hufnagel, Nature Mat. 3, 666 (2004)(figure adapted from P. M. Voyles)

Beyond the RDF: Fluctuation microscopy

The variance contains informationabout higher order (3- and 4-body)correlations


L. Fan et al., MRS Symp. Proc. 840, QC6.7 (2005)

Fluctuation x-ray microscopy

7 µm thick layer of277 nm latex spheres

E = 1.83 keV (6.77 Å)

Complement to FEM–great potential for looking at polymers, biologics, self-assembled materials

Future:• Higher spatial resolution• Harder x-rays• Resonant scattering


References and resources• Books

- B. E. Warren, X-Ray Diffraction (Dover, 1990)

- T. Egami and S. Billinge, Underneath the Bragg Peaks: Structural Analysis of Complex Materials (Pergamon, 2003)

• Dissertations/SSRL reports from the Bienenstockgroup (Fuoss, Kortright, Ludwig,Wilson,Ishii...)

• Software

- Matlab routines (http://ssrl.slac.stanford.edu/~bren/files/amorphous/)

- Billinge group software (http://www.pa.msu.edu/ftp/pub/billinge/)


Documents

Structural characterization of amorphous materials using … · Structural characterization of amorphous materials using x-ray scattering ... A gentle introduction to the theory of