Marder M.P. Condensed Matter Physics

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Further

Features

of Scattering Experiments 63

3.4.3 Neutrons

Neutrons scatter only off nuclei, and apart from interesting corrections due to rela-

tive spins of incoming neutron and nucleus that make it possible to probe magnetic

structures, the interaction is completely isotropie. The scattering form factor f(r)

for neutrons is a constant a for any given element called the

scattering

length. This

simplifying fact is the great advantage of

neutrons.

Their main disadvantage is sim-

ply the great expense required to generate beams of useful intensities. A neutron

wavelength of 2 Â corresponds to an energy of 0.02 eV, which corresponds ap-

proximately to room temperature, giving neutrons of this energy the name thermal

neutrons. Figure 3.14 shows neutron powder data used to uncover antiferromag-

netic ordering in MnO; the structure is depicted in Figure 24.6.

80K

)

293K

- i -

(111)

(311) /

■ 1

(222)

(400) (331)

(111)

(200)

(511) (440)

(220)

(622)

(311)

0 20 40 60 80

Scattering angle

2Θ

(degrees)

Figure 3.14. Powder pattern for neutron scattering from MnO. Above an antiferromagnetic

transition temperature, the scattering pattern is that of a crystal in the NaCl structure, with

a lattice parameter of 4.42 Â. Below the transition temperature, an entirely new set of

scattering peaks appears, indicating the antiferromagnetic ordering depicted in Figure 24.6.

[Source: Shull et al. (1951), p. 337.]

Elastic scattering, in which the outgoing neutron has the same energy as the

incoming neutron, is important for deciphering complicated structures, but inelastic

scattering, where the neutron loses or gains energy during its passage through the

material, is equally important and will be discussed in Section 13.4, as a means of

studying excited states of lattices.

3.4.4 Electrons

Electrons interact more strongly with matter than either X-rays or neutrons. This

fact might at first seem to present an advantage, but the scattering is so strong

that electrons cannot escape through samples without suffering multiple collisions

unless the samples are less than approximately 100 Â thick and unless the electrons

travel at fairly high energies, on the order of 100 keV. Electrons therefore are not

64 Chapter 3. Scattering and Structures

employed to deduce the structures of bulk crystals, but they do find use in the study

of solid surfaces (Section 4.3), and microscopically thin samples.

Incoming electrons interact with the complete electrostatic potential of

the

mat-

ter into which they travel, including both electrons and nuclei, and in the Born

approximation this leads to a scattering amplitude f

,(q)

= — / dr V(7)e""

See Landau and Lifshitz (1977) p. 513, or (3.46)

J Schiff (1968), p. 324.

which is related to the scattering function for X-rays by

ine

(q) = —~ \Z — n(q)] Z is the atomic number; n is the electron den- (3.47)

2/j Jt? sin Θ

d '

s re

ate<

the X-ray scattering form

factor by Eq. (3.44).

as shown in Problem 6.

3.4.5 Deciphering Complex Structures

Teasing crystalline structures out of arrays of spots on film is relatively easy when

the structure in question is composed of only a few elements and the unit cell is

small. Faced, however, with a protein crystal, with a unit cell containing tens or

hundreds of thousands of atoms, the problem of reconstructing atomic positions

becomes a daunting one. Ideally, there would be some purely automatic procedure

that operates upon the scattering data and reconstructs the scattering potential. In

the fields of light and electron microscopy this procedure exists; it is known as a

lens,

which should be understood as a device for recombining scattered waves in a

fashion so as to reveal the structure from which they came. An important feature

of recombining waves by focusing them is that all information about the relative

phase of waves scattered from different parts of the structure is preserved. Math-

ematically, this information is needed for structure to be deduced from scattering

data, and in X-ray and neutron experiments it is almost always lost (see Hauptman

(1989)).

Automated procedures for analyzing X-ray data are nevertheless possible, to

a point. Suppose one carries out a rotating crystal experiment and measures I(q),

as in Eq. (3.45). Performing an inverse Fourier transform in 3 on these data gives

immediately the

Patterson

function, after Patterson (1934), depicted in Figure 3.15,

which is proportional to

n is the number density per volume of electrons. Con-

dr'n(~r*\n(~r — 7

)

stant

mum

ners

sucn as

factors of 2π are irrelevant for /τ ΛΟΛ

\ / V -'the ensuing arguments, and are being dropped. ^ ' '

How much of the information originally present in phases has irredeemably been

lost? As Figure 3.15 makes clear, not so much. There is a peak in the Patterson

function at all locations R\

—

R2,

where R\ and R2 are peaks of the true density,

and the height of each peak is given by the product of the amplitudes of the two

original density peaks that produce it. So, while one does not know where all the

atoms are from Figure 3.15(B), one can read off a net of vector distances between

them. Matters are simplified further if

the

structure contains a heavy

atom:

an atom

Further

Features

of Scattering Experiments

Figure 3.15. (A) A contour map of electron density of an imaginary structure with three

atoms per unit cell, a large heavy atom at the origin, and two lighter atoms displaced

from it. (B) The Patterson function resulting from this density. Notice that the Patterson

function is centrosymmetric and that there is a peak in the density at a distance from the

origin corresponding to the distance between the two light atoms.

whose scattering is stronger than any other. Say this atom is at the origin. Then the

largest peaks of

the

Patterson function will be at the origin and at ±/?,, where

are

the locations of all the other atoms. So, if a structure has heavy atoms, the greatest

remaining uncertainty is whether peaks of ±/?, correspond to atoms at

/?,·,

—/?,·,

or at both locations. One of the techniques essential to unraveling the structures of

large proteins has been the artificial chemical insertion of heavy atoms at various

points in the proteins as replacement for lighter atoms, to provide guides to the

structure.

Even the stubborn tendency of X-ray data to make all structures centrosymmet-

ric can be overcome. When incoming X-rays have an energy near to that at which

one of the atoms in the sample has an absorption resonance, a bit more phase in-

formation becomes available in the region of anomalous

dispersion;

this fact is the

subject of Problem 5.

3.4.6 Accuracy of Structure Determinations

Catalogs of crystal structures must always be employed with the possibility in mind

that published results are inaccurate or incorrect. Abrahams et al. (1967) describe

an effort to determine how big these errors might be, by taking a carefully pre-

pared crystal to multiple research groups and asking them to measure Bragg peak

intensities independently. Although each group claimed accuracy in determining

structures of around 1%, discrepancies between different groups were typically on

the order of 5%-6%, and measurements of

the

intensity of the (111) reflection var-

ied by 50%. In revisiting the results of this experiment, Mackenzie and Maslen

(1968) call the results "grossly discordant." An example of an incorrect structure

determination is provided by the high-cristobalite structure of

S1O2,

on page 318 of

Chapter 3. Scattering and Structures

Wyckoff (1963); the correct structure is discussed in Liu et al. (1993). How many

other examples like this latter one may exist is difficult to say.

3.5 Correlation Functions

3.5.1 Why Bragg Peaks Survive Atomic Motions

Sommerfeld objected in 1912 to the search for X-ray scattering on the grounds

that atoms constantly move large distances from their ideal positions, and the ex-

periment could not work. The theory presented so far assumes atoms sit perfectly

still at ideal locations, yet it compares well with experiments performed at room

temperature. How can this be?

This is a special case of an even more primitive question, which concerns the

nature of crystals. What really defines them? It is not enough to say that all their

atoms sit in a perfect lattice, since thermal motions and occasional impurities do

not destroy the crystal's essential nature.

The essential nature of a crystal is that positions of atoms are correlated at long

distances. This idea can be made precise in any monatomic system by defining the

two-particle or Van Hove (1954) correlation function

(7\,

; t). This function

gives the probability that if some particle is at position r\ at time t\, a particle is to

be found at position r

at time t\+t. Formally,

«2(n,

~h\

0 = (Σ

(

-Ri{h)W

-Ri'{t\ +t)) ) . SSe °

ngmal

(3.49)

\ / // I condition / ji /' is

' imposed on the sum.

The brackets denote either a thermal average or a time average over t\, as according

to principles of statistical mechanics the two averages should be the same. Vari-

ables R[(t) track the locations of the particles.

From the correlation function, define the dynamic structure factor, a dimen-

sionless measure of scattering, by

S(q, ή Ξ = — y^ /

'HRi('i)-Ri>(.ti+t))\ This is proportional to the time (3.50)

yV/iitnm N ^^ \ ' average or thermal average of

1,1'

Eq. (3.11).

= l Σ / ^1^2 ^'-

?2)

(<5(n -Rifa))S(r2-Ri>(ti +0)) (3-51)

= — I df\ dr

{r\,

\ ήε^'^"

Just insert the definition (3.49). (3.52)

= —η

{3, t) V is the system volume. If the condition / φ /' had been added to (3.53)

N Eq. (3.49), there would be an additive factor of 1.

where

(q, t) = ^ J

drdr>

Ψ,

r; t)e^'. (3.54)

Return to Eq. (3.11). A traditional scattering experiment requires collecting

radiation on a photographic plate over times on the order of seconds. Atomic

Correlation Functions 67

oscillations occur on a time scale of picoseconds. Scattering experiments thus

measure the long-time average of radiation intensity exiting the sample. That is,

these experiments do not really measure the quantity in Eq. (3.11), they measure

its time average, which is the static

structure

factor,

S{q)=S(q,0). (3.55)

The necessary condition for scattering peaks can now be expressed in a more

general way. As Ψ in Eq. (3.54) becomes larger and larger, the static correlation

function must continue to exhibit periodic maxima centered on lattice sites, and the

amplitudes of these maxima should not diminish when the r' becomes large com-

pared to atomic spacings. A specific example comparing solids and liquids appears

in Figure 5.9. This is the type of correlation called long-range positional order.

Putting matters another way, it does not matter if atoms are in thermal motion, just

so long as every atom vibrates around a set of mean locations that constitute a per-

fect crystal. The effect of vibrations is to decrease the amplitude of Bragg peaks,

not their existence or location in frequency space. Explicit expressions for the way

that quantum mechanical and thermal fluctuations reduce Bragg peak amplitude

appear in Eqs. (13.124) and (13.129).

Placing atoms in lattices is a special case of the more general idea of long-

range order or long-range correlations. Long-range order concerns the average

behavior of pairs of particles over time, not their location at any one instant. Long-

range order is the fundamental property that separates solids from liquids. More

generally, the ideas of correlation and ordering explain qualitative phases of matter

including superconducting and superfluid states.

3.5.2 Extended X-Ray Absorption Fine Structure (EXAFS)

Figure 3.16. Extended X-ray absorption fine structure

is observed when a scattered wave impinges on near

neighbors, and returns to interfere with the first scat-

tering source.

Extended X-ray

absorption

une

structure

(EXAFS) is an X-ray scattering tech-

nique that homes in on small-scale details of the correlation function «2 in the

neighborhood of specifically chosen target atoms. To employ the technique, one

illuminates a sample with X-rays whose frequency is chosen to lie near the absorp-

tion resonance of some particular atom. For incoming radiation whose energy £

lies above the onset of absorption at E

, the receiving atom emits an electron of

energy £

—

and wave vector hk

—

\j2m{Z

—

), which has been excited out of

a core state. Traveling out of the atom that produced it, this electron immediately

68 Chapter 3. Scattering and Structures

begins to scatter off neighboring atoms; the scattered wave returns to the original

atom and interferes with the electron emission process, as in Figure 3.16. If

this

re-

turning wave reduces the electron amplitude at the original atom, X-rays will have

increased difficulty ejecting electrons to begin with, and there should be a dip in

the X-ray absorption coefficient a(E). Without developing a detailed theory of this

rather complicated interference process, it is possible to guess the form the result

must have. If X-rays eject electrons from atoms at the origin, and a neighboring

atom at location Rj scatters them back, then because the total path length from

atom to scatterer and back is 2Rj, interference between the original and scattered

wave leads to absorption in the form

(

\ Sum over all neighboring atoms. The

+ \e~

f/Rj]

]

)

squarecomesfr

mthefaathatthe

(3.56)

' ' i ' L J i ji i / wave is scattered twice. / is a

j I scattering form factor.

.-. irs -, n\ —Jr/Ιτ /„. \ Keeping only the leading term ,_ ^_

drn

(0,r, 0)e

2r/

COs{2kr), depending on k. (3.57)

where

is the mean free path of electrons in the solid. In fact the absorption

coef-

ficient is observed experimentally to have small oscillations for energies as much

as 1000 eV above the absorption edge. According to Eq. (3.57), the Fourier trans-

form of these oscillations is proportional to the correlation function of neighboring

atoms times the decay factor

exp[—2r//r].

Thus EXAFS gives quantitative descrip-

tions of the neighborhoods of atoms of specific types, even in liquid or amorphous

environments.

3.5.3 Dynamic Light Scattering

Under the general heading of scattering experiments, there is a wide variety of

pos-

sibilities. Dynamic light

scattering

presents an interesting contrast with the X-ray,

electron, neutron scattering experiments discussed so far. Like these techniques,

dynamic light scattering involves sending a beam of radiation into a sample and

measuring the outgoing radiation in order to draw conclusions about particle loca-

tions.

However, almost everything else is different. Dynamic light scattering uses

visible light, so it is most suitable for studying particles or structures whose size is

comparable to visible wavelengths, on the order of

/im.

Instead of Bragg peaks, the

focus of attention moves to speckles of light emerging from the sample, and their

fluctuations in time. The fluctuations in light intensity provide information about

the motions of particles in the path of the light beam. Instead of resulting from

coherent scattering from many different particles at the same time, dynamic light

scattering results from the changes of intensity due to changes in particle locations

over time. In terms of correlation functions, conventional X-ray scattering probes

the correlations of many particle locations with one another at one time, while dy-

namic light scattering probes the correlations of particle locations with their own

histories.

The study of dynamic light scattering begins with the observation that when

electric fields scatter weakly off particles located atR\... R^, then the the result-

Correlation Functions

ing field

E°

<i-

i(')

1=1

Taking the form factor

for polarization.

The index

a is

(3.58)

Thus

-2

Σ(£*(0)Ε

(ί)>

— y y

^^(e-^-^'W-^'iO))

drdr

-^n

(r,

?'; i)

= n

{q,t).

Κ\

Exactly

as in

Eq. (3.54).

(3.59)

Unfortunately, despite this pleasing definition,

ri2(q,

cannot

measured

directly, because experiments measure intensities at a given time, not electric fields.

However,

taking

chance

on one new

assumption, temporal fluctuations

fields can

temporal fluctuations

intensity.

The

assumption

is the

Siegert

relation.

Consider

\Eo\

y (Ε*

(0)Ε

(0)Ε*

(ήΕ

(ή

i\I(t)I(0)

(3.60)

αη

This quantity can be measured directly in experiment by keeping a record over time

of intensity.

Its can be

Eq. (3.59)

one assumes that

the

electric field

components

(0)

and

(ί)

are random variables, normally distributed about their

mean values, and the deviations from the mean independent

one another.

this

assumption holds then one can employ

theorem from statistics discussed by Tri-

antafyllopoulos (2003), analogous

Wick's theorem (Section 13.4.3

Doniach

and Sondheimer (1974), pp. 52-62) saying that the expectation value

products

of random variables

given

summing

up all

possible products

expectation

values

pairs

the operators.

In the case

Eq. (3.60), this means that

</(0/(o))//£

i-^ £(Ε*

(0)Ε

(0)Ε;(ήΕ

(ή)

\Eof

a~f

{Ε*(0)Ε

{0)){Ε*(ήΕ

(ή) Terms such as £*

(0)£;(0

|£θ|

t? +(Ε*

(0)Ε

(ή)(Ε*

(ήΕ

(0))\ factorsof

/|(£*(0)£

(ί))

vanish after averaging over

]

_|-

ß\fi2{q

t) | .

See Gittings

and

Durian (2006)

for an

inves-

tigation

this approximation.

(3.61)

(3.62)

(3.63)

(3.64)

(3.65)

70 Chapter 3. Scattering and Structures

Figure 3.17. Measurement of (I(t)I(O)) for a dilute solution of coiled DNA. The scattering

angle is 60° so q can be determined from Eq. (3.8). The solid line is an exponential

function; it fits the data with a correlation coefficient of

0.996.

Thus the data have the form

predicted by Eqs. (3.67) and (3.65) with β κ. 0.8. [Data of H.L. Swinney and J. Newman

from Berne and Pecora (2000), p. 61.]

Here β is a constant that depends first of all upon the polarization state of the

light. For linearly polarized light, one expects β = 1 and for circularly polarized

light β = 1/2. In addition, depending upon the size of the photo-detector, the

light speckles being measured may be more or less completely spatially correlated,

and the degree of correlation affects the success of the approximation leading to

Eq. (3.65). The constant β is in practice used during experiments as an adjustable

constant to compensate for the approximation.

3.5.4 Application to Dilute Solutions

For a dilute solution of micron-scale particles in liquid, what should the

correlation function of Eq. (3.49) look like as a function of time? Small parti-

cles undergo Brownian motion which means that they diffuse about, and obey the

diffusion equation (5.2), to be derived in Section 5.2.

Exactly the same equation must apply to the the probability «2(0, r; t) that a

particle at the origin at time t = 0 has moved to r in time

— = DV n

(0, r; t)

(3.66)

Because the particle is originally at the origin, «2(0, r; 0) = δ(τ). Applying the

Fourier transform J dr exp[—q

■ r]

to both sides of Eq. (3.66) gives

dnijq, ή

- 1)q H2 (a, t) The boundary condition is that «2(<? = 0. 0) =

Problems 11

=>n

(q,t)=e-'i

(3.67)

Figure 3.17 shows an example of intensity correlation data for a dilute solution

of coiled DNA. Note that by varying the scattering wave vector q = ko sin

Eq. (3.8) it is possible to obtain many independent measurements of the diffusion

constant D.

Problems

Reciprocal lattice vectors:

(a) Use Eq. (3.24) to verify that the reciprocal lattices of the fee and bec lattices

are as claimed in Table 3.1.

(b) Find reciprocal lattice vectors of smallest magnitude for aluminum, beryl-

lium, and bec iron, and write down their Miller indices.

Hep extinctions:

(a) Show as claimed in Table 3.1 that the reciprocal lattice of a hexagonal lattice

given by Eq. (2.5) is another hexagonal lattice rotated at 30° with respect to

the original one, and find primitive vectors for the reciprocal lattice.

(b) The hep lattice is built upon the hexagonal, with basis given by Eq. (2.6).

Show that the modulation factor induced by the basis is

|ΐ+^|[

("ι+"2)+

"3]|

. (3.68)

cause of an extinction.

Bragg peaks: Bragg's model for reflection of X-rays from a solid was based

on the view that the solid was constructed out of a series of parallel planes. X-

rays bounced off these planes in such a way that angle of incidence

equaled

the angle of reflection, and diffraction peaks occurred when radiation from

successive planes added constructively.

(a) Verify that the values of

ko,

k, and K given in Figure 3.2 are chosen correctly

to produce scattering.

(b) Show that in general the condition for a scattering peak can be written as

2d sin

= IX, (3.69)

where d is the distance between lattice planes,

is the Bragg scattering angle,

λ is the wavelength of incoming light, and / is an integer.

Chapter 3. Scattering and Structures

Laue pattern: Suppose that broad-band X-rays are scattered from a crystal

onto a photographic plate 1.5 inches away, giving the following pattern:

Only the upper half of the four-fold symmetric scattering pattern is shown.

Any scattering peaks at radii falling outside the gray shaded area are not

recorded on the photographic plate.

'""' '

Ilj

30 20 10 0 10 20 30

(a) Use this Mauguin abacus to identify the Bragg angles of each spot. The way

to use the abacus is to copy it (preferably onto transparent paper) and lay it

down so that a tickmark on the left lies on some scattering spot, while the

corresponding tickmark on the right lies on the origin. The Bragg angle is

given by the common value of the two tickmarks.

(b) The lattice has spacing a and is either sc, fee, or bcc. Which one? Assume

the incoming X-rays to have uniform intensity at all wave vectors up to a

cutoff of 32/a. It should be emphasized that the spots are cut off both in real

space by the boundaries of the circle and in reciprocal space by 32/a.

Centrosymmetry:

(a) Suppose that all the ionic form factors // in Eq. (3.10) are real. Show that

the X-ray scattering data will make all crystals appear to be centrosymmetric

whether they are so or not

(FriedeVs

law).

(b) Consider next a crystal that is not centrosymmetric, and with two different

atoms per unit cell. Suppose that the X-ray frequency is chosen near to a

value

ωο

at which one of the atoms has an absorption resonance. Assume that

the frequency dependence of the absorption can be modeled by pretending

that the atom behaves like a mass on a spring with damping. Show that the

amplitude and phase of scattered radiation vary just like the amplitude and

phase of a mass on a spring with damping driven by a periodic force. Show

that the phases of

the

two form factors / for the two atoms should be different,

and that the scattering intensity now contains information from which can be

deduced the fact that the crystal is not centrosymmetric.