Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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268 Part B Chemical and Microstructural Analysis

that allow chromatography to become useful as a pre-

process for more sophisticated analysis.

Electrophoresis utilizes the difference in the drift

mobility of ionic molecules in a stationary phase under

an electric ﬁeld. Electrophoresis using gel as the station-

ary phase is called gel electrophoresis (GE), and that

using aqueous solution in a capillary is called capillary

electrophoresis (CE). CE has advantages over GE be-

cause of reduced problems with Joule heating and the

action of the capillary itself as the pumping system.

Circular Dichroism (CD)

Circular dichroism (CD) arises from the helicity or

chirality of molecules that exhibit optical absorption

due to electronic excitations at ultraviolet to visible

wavelengths. CD spectra of chiral proteins, peptides

and nucleic acids have distinct structures and are sen-

sitive to conformational changes. Figure 5.96 shows

a schematic diagram of experimental setup. Linearly

polarized light can be regarded as a superposition of

left- and right-hand circularly polarized light. Optically

active substances such as chiral molecules transmit

the opposite circularly polarized light with slightly

different absorption coefﬁcients. As a consequence,

the linearly polarized light incident on the sample

becomes elliptically polarized light without changing

the main polarization axis. The degree of elliptic-

ity (CD) depends on the difference in the absorption

coefﬁcients.

In contrast, optical rotation arises from differences

in the index of refraction that cause a difference of

the rotation angle between the opposite components

of circularly polarized light. The optical rotary dis-

persion (ORD) spectrum, the dependence of rotation

angle on wavelength, is related to the CD spectrum

by a Kramers–Kronig transformation. The ORD curve

changes sign at the extremum peak of the CD spectrum,

known as the Cotton effect, as illustrated in Fig. 5.97.

Owing to the fact that CD/ORD curves exhibit fea-

tures that are characteristic to the molecular structures,

CD/ORD measurements provide structural information

about the molecule. In proteins, for example, CD sig-

Wavelength λ

Polarizer Analyzer

Light

source

Mono-

chromator

Sample Optical

detector

Fig. 5.96 Experimental setup of circular dichroism (CD)

and optical rotary dispersion (ORD) measurements

nals are more sensitive to α-helical residues than those

of random coils and β-sheets. Experimental data are ﬁt-

ted to reference model spectra to determine the amounts

(composition) of the constituent secondary structure of

proteins. The reliability of composition analysis is en-

hanced by the use of proteins of known structure as the

basis set. More sophisticated analysis of molecular po-

sitions, conformation and absolute conﬁguration can be

conducted with detailed knowledge of the inﬂuence of

these factors on the Cotton effect.

Samples are usually prepared as a solute in an opti-

cally nonactive solution. CD/ORD measurements allow

quantitative analysis since the signal intensity is propor-

tional to the molecular density in the solution. Structural

analysis, however, is only possible at densities that yield

an optical absorbance of 1–2 at the absorption maxi-

mum, where the Cotton effect is observed, in order to

avoid signal weakening due to excessive absorption.

Although simple CD and ORD are observed only

in optically active molecules, similar dichroic effects

are also induced in achiral substances under a mag-

netic ﬁeld applied parallel to the measuring light beam.

Magnetic circular dichroism (MCD) combined with

photoabsorption measurements is most commonly used

to analyze chromophoric groups in which the magnetic

ﬁeld lifts the degeneracy of optically excited energy

levels differing only in magnetic spin (Sect. 5.3.1). In

biology, MCD is also applicable to metalloproteins, pro-

Elipciticy

Rotation angle

Wavelength μ

max

Fig. 5.97 Schematic spectra of circular dichroism (CD)

and optical rotary dispersion (ORD)

Part B 5.4

Nanoscopic Architecture and Microstructure 5.5 Texture, Phase Distributions, and Finite Structures Analysis 269

tein molecules that contain a metal substance necessary

for a certain reaction to take place.

Fluorescence Resonant Energy Transfer (FRET)

Fluorescence resonant energy transfer (FRET) occurs

when the distance between a ﬂuorescent donor dye

molecule and a photoabsorbing acceptor dye molecule

is close enough for the electronic excited energy of

the donor molecule to be transferred to the acceptor

molecule nonradiatively. Since the efﬁciency of FRET

depends on the inverse sixth power of the intermolecu-

lar distance, it is useful for determining the proximity of

the two molecules within 1–10 nm, which is a distance

range of particular importance in biological macro-

molecules. For FRET to take place, the ﬂuorescence

spectrum of the donor and the absorption spectrum of

the acceptor must overlap, as shown in Fig. 5.98. Based

on this technique, the approach of two molecules can

be detected by quenching the donor ﬂuorescence, and

Acceptor

absorption

Donor

fluorescence

Fig. 5.98 Spectral overlap necessary for ﬂuorescence res-

onant energy transfer (FRET)

one can study the conformation change of a single

molecule by detecting FRET under a laser scanning

confocal microscope (LSCM) (Sect. 5.1.2)oratotal

internal reﬂection ﬂuorescence microscope (TIRFM)

(Sect. 5.1.2)[5.78].

5.5 Texture, Phase Distributions, and Finite Structures Analysis

This section deals with materials that are inhomoge-

neous on a relatively large scale with respect to their

composition, structure, and physical properties. Poly-

crystals are said to have a texture when they have

a nonrandom distribution of grain orientations. The

phase distribution or element distributions may play

an important role in the macroscopic properties of the

material. Another issue addressed is solid structures

with ﬁnite three-dimensional size that may have their

own functions, such as biological cells, nanoparticles,

etc. In the last subsection, we describe some of the

recent progress in stereology, an emerging ﬁeld of three-

dimensional analysis of materials.

5.5.1 Texture Analysis

Textures evolve by various mechanisms. Plastic de-

formation of crystals by glide motion of dislocations

proceeds on preferential crystallographic slip planes,

usually low-index planes with the largest spacing. If

polycrystalline wires are drawn through a die, the wires

have a texture (wire texture) such that the slip planes

tend to align parallel to the wire axis, because otherwise

each grain would be subject to further deformation. The

texture of polycrystalline materials is of technological

importance for various physical properties, mechanical

properties such as strength and elastic constants, mag-

netic permeability, ﬂux pinning in superconductors of

the second kind, and so on.

Texture Analysis by X-Ray Diffraction

The best established technique for assessing textures is

x-ray diffraction. In powder diffraction, each diffrac-

tion ring consists of ﬁne spots which correspond to

crystallites in different orientations. If the crystallites

are randomly oriented, the spots or the diffraction in-

tensity is uniformly distributed around the circle. If

textures are present, however, one observes an inhomo-

geneous distribution of diffraction spots or diffraction

arcs, as shown in Fig. 5.99. In modern experiments, the

data are collected by using a diffractometer equipped

with a four-circle goniometer (Fig. 5.37) that scans all

orientations by a combinatorial rotation around the Eu-

lerian ω–φ–χ axes. For thick or large-grained samples,

neutron diffraction may offer an alternative to x-ray

diffraction, if a facility is accessible. The merit of

diffraction techniques is the ease with which good sta-

tistical data can be acquired.

The texture of materials is expressed by several

different representations. The pole ﬁgure (Fig. 5.100)

presents the orientational distribution of a speciﬁc pole

(crystallographic (0001) axis in Fig. 5.100a) with re-

spect to the sample orientation, which is displayed on

a stereographic projection with the surface normal at

Part B 5.5

270 Part B Chemical and Microstructural Analysis

Fig. 5.99 X-ray diffraction pattern from a mechanically

drawn Al wire. Exposure time 30 s using x-ray source Mo

(50 kV, 40 mA) recorded on an imaging plate placed

10 cm from the sample (Courtesy of Rigaku Co)

Normal

direction

Rolling

direction

Transverse

direction

(0001)

Rolling direction

Normal

direction

Transverse

direction

(0001)

Fig. 5.100a,b A presentation of the texture (a) by the

stereographic pole ﬁgure in

(b)

Levels:

0.7

1.0

1.4

2.0

2.8

4.0

4.7

max

: 4.7

Stereographic

projection

{111}

Fig. 5.101 An experimental pole ﬁgure of an aluminum

sheet rolled and annealed. After [5.79]

the origin, a speciﬁc direction along the sample surface

(e.g., the rolling direction in rolled metals) at the north

pole, and the transverse direction in the horizontal direc-

tion. Figure 5.101 shows an experimental pole ﬁgure of

an aluminium sheet thermally annealed after mechani-

cal rolling [5.79, p. 102]. The four crystallographically

equivalent (111) poles are distributed in some dominant

directions. Also widely used is the inverse pole ﬁgure

that presents the orientation distribution of the specimen

coordinate system with respect to the crystal coordinate

system, the latter being reduced to a standard stereo-

graphic triangle. Whichever the case, however, the pro-

jection of the three-dimensional orientation distribution

onto a two-dimensional ﬁgure causes a loss of infor-

mation. The orientation distribution function (ODF)is

a three-dimensional presentation thus devised for a full

description of texture. The ODF can only be obtained by

calculations from a data set of several pole ﬁgures. For

details, see the textbook by Randle and Engler [5.79].

Texture Analysis by SEM Electron Channeling

Pattern (ECP)

The disadvantage of diffraction techniques for the anal-

ysis of texture is the lack of direct information about the

grain size. Local crystal orientations may be determined

by microscopic techniques that also reveal the dimen-

sion of each grain directly [5.80]. TEM is a routine

method to conduct such measurements, but the samples

are limited to thin ﬁlms. The electron channeling pat-

tern (ECP), which can be obtained by scanning electron

microscopy (SEM), is a more convenient approach for

studies of texture using bulk samples.

Part B 5.5

Nanoscopic Architecture and Microstructure 5.5 Texture, Phase Distributions, and Finite Structures Analysis 271

SEM-ECP corresponds to the Kikuchi lines

(Sect. 5.1.2) or bands observed in TEM. Figure 5.102a il-

lustrates how the Kikuchi lines are formed. As men-

tionedinSect.5.1.2, the Bloch wave with its largest

amplitude on the atomic column (Fig. 5.102b) interacts

more strongly with the atoms and is more likely to be

scattered inelastically, losing coherency with the direct

wave. As shown in Fig. 5.102a, some of the electrons

thus inelastically scattered in various directions will be

reﬂected by lattice planes satisfying the Bragg condi-

tion and, if the specimen is moderately thick, propagate

along the channel direction with an anomalously small

absorption coefﬁcient. The electrons are projected onto

the back focal plane of the objective lens as a pair

of traces of the diffraction cones. Line pairs (Kikuchi

lines) or bands formed in this way indicate the orien-

tation of the crystal. In a similar way to the Kikuchi

lines in TEM, if the electron beam in SEM is incident

to a crystalline sample in an orientation along a chan-

neling direction (Fig. 5.102b), the electrons penetrate

deeply into the sample and have less chance to be back

scattered and detected by a back-scattering electron de-

tector. If one rocks the beam direction as in Fig. 5.102b

and synchronously displays a two-dimensional map of

the intensity of back-scattered electrons in the SEM

screen as a function of the beam orientation, one ob-

tains patterns as shown in Fig. 5.103a,b. Since the

SEM-ECP pattern reﬂects the crystalline orientation

of the sample position at which the beam is rocked,

beam scanning with the beam direction ﬁxed gives

a) b)

c) d)

100μm

Fig. 5.103a–d Electron channeling patterns (a),(b) and

images

(c),(d) at different tilt angles. After [5.17]

a) b)

Back-scattered

electron detector

Channeling

Reduction of

back-scattered electrons

Bulk

sample

Thin

sample

Beam

rocking

Back

focal

plane

Kikuchi

lines

Fig. 5.102a,b Formation of Kikuchi lines in TEM (a) and electron

channeling pattern in SEM (b)

ECP images, as shown in Fig. 5.103c,d. From such

crystallographic orientation contrasts, one can deter-

mine the orientation distribution and the size of each

grain.

5.5.2 Microanalysis of Elements and Phases

Dark-Field TEM Imaging

Dark-ﬁeld (DF) TEM images (Sect. 5.1.2)formedby

a diffracted beam of a speciﬁc diffraction vector provide

the most standard technique for microscopic analysis of

coexisting multiphases. Figure 5.104 shows a bright-

ﬁeld image of a metallic compound of Nb

and

the corresponding DF images obtained for two differ-

ent diffraction vectors. The lateral resolution is similar

to that of BF-TEM images as long as the sample is

thin enough for the overlap of different phases to be

avoided.

0.5 μm

b) c)

Fig. 5.104 (a) Bright-ﬁeld and (b),(c) dark-ﬁeld images of multi-

phase Nb

(courtesy of M. Ichihara)

Part B 5.5

272 Part B Chemical and Microstructural Analysis

Electron Probe Microanalysis (EPMA)

Another common method to investigate element dis-

tribution in solid materials is scanning electron mi-

croscopy operated in the x-ray ﬂuorescence mode,

usually referred to as an electron probe microanalyzer

(EPMA). On the impingement of electrons in the energy

range of 1–40 keV, the samples emit x-rays at wave-

lengths characteristic of the elements contained. The

emitted x-ray is usually detected with an energy dis-

persive x-ray (EDX) solid-state detector placed above

the specimen (Fig. 5.105). Scanning the beam gives us

a map of the element distribution with a spatial reso-

lution of a few nm to 100 μm. The technique becomes

more sensitive as the mass number increases.

Similar measurements could be conducted using

TEM. In ordinary SAD, however, it is difﬁcult to re-

duce the size of a selected area to smaller than several

tens of nm, whereas in CBED the beam size can be re-

duced down to a subnanometer scale. The best choice

is to use a focused beam in STEM that allows one to

conduct CBED analysis as well as mapping element

distribution over the scanned area. For heavy elements,

STEM-EDX has advantages over the following tech-

niques, though there are technical problems with the

energy resolution (≈ 150 eV) and the time for data

acquisition.

Electron impingements induce the emission of

Auger electrons whose energies are characteristic of

speciﬁc elements (Fig. 5.67). Since the energy of Auger

electrons is so small that the escape distance is as short

as ≈ 1nm (Fig.5.59), Auger electron spectroscopy

(AES) is overly sensitive to the surface.

Energy Loss Analysis

Energy ﬁltering TEM (EFTEM)isaTEM or STEM

equipped with an energy ﬁlter passing only electrons

of a speciﬁc energy that are used to construct an im-

age of the spatial distribution of the corresponding

elements. The energy ﬁlter may be placed inside the op-

tical column of the TEM (in-column type) or in front

of the camera chamber (post-column type). EFTEM al-

Fig. 5.106 STEM-EELS image of Gd (white) atoms in a carbon

(gray) nanotube. Scale bar =3nm.After[5.81]

Focused electron beam

Beam scanning

–ΔE

Energy

dispersive

x-ray

detector

Fluorescent x-ray

Sample

Electron energy

loss spectrometer

Fig. 5.105 Element analysis by an electron probe

lows spectrum imaging of the element distribution with

an energy resolution of ≈ 30 eV and a spatial resolu-

tionofdownto≈ 1 nm at 200 kV and ≈ 0.5nm at

300 kV.

In contrast to EFTEM, the spatial resolution in

STEM-EELS, if equipped with a ﬁeld emission gun,

is much higher owing to the fact that STEM generally

does not need an objective lens to image the aberration

that governs the spatial resolution of TEM micrographs.

A disadvantage of STEM-EELS is the long acquisi-

tion time due to the scanning nature of the microscopy,

but the use of a parallel EELS (PEELS) detector and

the great assistance of a computer have advanced the

progress of STEM-EELS as a tool able to identify even

a single atom in small molecules [5.81], as demon-

strated in Fig. 5.106. STEM-EELS has advantages over

STEM-EDX for light elements.

Micro-Raman Scattering

Raman scattering measurements (Sects. 5.1.2, 5.2.3,

and 5.3.1) can be coupled with optical microscopes to

investigate the local distribution of particular phases

with a spatial resolution of ≈1 μm. The diffraction limit

of OM is now overcome by the use of scanning near-

ﬁeld optical microscopy (SNOM) (Sect. 5.1.2), in which

the enhancement of the optical ﬁeld by surface plas-

mon resonance at the metallic tip allows the detection of

Raman spectra for even a single molecule [5.82] under

good conditions.

Part B 5.5

Nanoscopic Architecture and Microstructure 5.5 Texture, Phase Distributions, and Finite Structures Analysis 273

Scanning Nano-Indenter

Materials often encountered may be in the form of

thin ﬁlms or nanoscale omplexes of multiple phases.

One may need to conduct mechanical tests for very

thin ﬁlms grown on a substrate. The difference in ma-

terial phase may reﬂect the mechanical strength. The

scanning nano-indenter is a modern version of the

micro-hardness tester with which the local mechanical

strength of crystallites smaller or thinner than ≈1 μm

can be measured with a lateral resolution of several hun-

dreds nanometres from the indent size in situ observed

by a microscope similar to an AFM and STM. The ex-

tremely tiny indenter can be driven into and retracted

from the sample surface at a controlled rate, so that one

can also deduce the local elastic modulus.

5.5.3 Diffraction Analysis of Fine Structures

Tiny objects such as precipitates, crystallites, and ﬁne

particles smaller than ≈ 100 nm can be investigated by

not only microscopic but also diffraction and spectro-

scopic techniques.

Small-Angle Scattering Analysis of Tiny Phases

The Guinier–Preston (GP) zone, an extremely tiny pre-

cipitate with a thickness of only one atomic layer

and lateral size of several atomic distances, is directly

observable by HRTEM at present; it was originally

discovered in x-ray diffraction as a source of diffuse

scattering. In Guinier’s approximation, in which a par-

ticle is a Gaussian sphere

(

)

=ρ

exp



−



(5.37)

with a representing the particle size, the diffraction in-

tensity or the Fourier transform of (5.37) will be

I(K) =ρ

π exp



−



. (5.38)

The magnitude of (5.38) is small except for small |K|

values or small scattering angles, which means that the

primary beam is diffusely scattered within a small an-

gle. In terms of the radius of gyration R

, the root mean

square of the mass-weighted distances of all subvol-

umes in a particle from the center of mass,

I(K) ∝exp



−R



. (5.39)

Thus, from the slope of the so-called Guinier plot,

log I(K) versus K

, we can evaluate a measure of the

particle radii R

experimentally.

Line-Broadening Analysis

of Crystallite Size and Stress

The broadening of x-ray beams occurs not only in the

primary beam but also in diffracted beams when the

crystallites or domains are small [5.83]. This arises from

the fact that the diffraction intensity depends on the in-

terference function, which reﬂects the external shape of

the crystal (Sect. 5.1.1).

More generally, the line broadening is induced by

causes other than the ﬁnite crystal size. The main causes

are instrumental errors and the strain distribution. The

instrumental broadening due to spatial, angular and

wavelength distribution of the incident x-ray beam is

experimentally evaluated by using a standard sample,

usually a large and perfect single crystal such as Si. The

line broadening due to lattice strain is in most cases due

to lattice defects such as dislocations.

As a measure of the line broadening, we conve-

niently employ the integral breadth β, which is deﬁned

as the width of a rectangle having the same area A and

height I

as the line proﬁle of interest, or β ≡ A/I

.For

Gaussian broadening, the relevant breadth β

of sample

origin in our concern is estimated by

=β

−β

, (5.40)

where β

and β

are the experimental and instrumen-

tal breadths, respectively. The separation of the lattice

strain component is achieved by plotting β

versus sin θ

for various diffractions. From the Bragg condition (5.1),

the line broadening in terms of the diffraction angle 2θ

due to a distribution of net-plane spacing Δd is given

Δθ = tan θ

Δd

. (5.41)

When there is another contribution from the ﬁnite crys-

tal, the experimentally measured integral breadth β

generally given by

(

)

=aε tan θ

+β

(5.42)

where ε ≡ Δd/d represents the strain and β

the size

contribution in our concern. Since the integral breadth

is related to the average size of crystallites L by

Scherrer’s formula [5.84, p. 102]

(g) =

L cos θ

, (5.43)

(5.43) is written as

(

)

cos θ

=aε sin θ

. (5.44)

Part B 5.5

274 Part B Chemical and Microstructural Analysis

Willamson–Hall plot

sinθ

(g) cosθ

∝ε

Fig. 5.107 A schematic Williamson–Hall plot for line

broadening analysis

Thus, by plotting β

(g) cos θ

versus sin θ

(the so-

called Williamson–Hall plot), as shown in Fig. 5.107,

we can evaluate the average size L of the crystallites

from the θ

-independent level.

Thus, the analysis of line broadening simultane-

ously enables evaluation of lattice strain ε or the internal

stress in the crystallites. With access to costly facilities

such as synchrotron radiation sources (for x-ray diffrac-

tion) or nuclear reactors (for neutron diffraction), one is

able to conduct experiments with great advantages for

probing the stress state deep into the sample [5.85].

Light Scattering Analysis of Particle Shape

The cloudy or milky appearance of latex and solid

polymer reﬂects a heterogeneous structure with two

coexisting phases [5.29] (colloid particles and liquid

in latex; crystalline and amorphous domains in solid

polymer) (Fig. 5.42). This is due to the high-angle scat-

tering of light over a wide range of wavelength, called

Mie scattering in contrast to the wavelength-dependent

Rayleigh scattering. When the particle size is smaller

than the light wavelength, we enter to a regime of

surface plasmon resonance. The resonance frequency

Table 5.5 Comparison of various stereology methods

Method 3-D reconstruction Resolution Test environment

3-D-SEM Stereogram < 10 nm Vacuum

LSCM+ Stacked tomographs lateral ≈ 1 μm Ambient, liquid, vacuum

nonlinear optical microscopy depth ≈ 1μm

3-DAP-FIM Stacked tomographs < 0.2nm UHV

X-CT Computed tomography ≈ 1 μm Ambient, vacuum

X-PCI Computed tomography afewμm Ambient

3-D-TEM Computed tomography < 10 nm Vacuum

depends on the dielectric function of the particle ma-

terial and the shape of the particle. As the particle is

elongated in one direction, the resonance is shifted to

longer wavelengths, so measuring the resonant wave-

length of the scattering, we can determine how the

particles are distorted from a sphere.

5.5.4 Quantitative Stereology

The great increase of computer power and resources

has made it possible to process the enormous amount

of data necessary for three-dimensional reconstruction

of a sample structure. There are various types of stereo-

logical techniques, as listed in Table 5.5, differing in the

reconstruction method.

Stereogram

Two micrographs are acquired for a sample that is

tilted by ±1−10

◦

in an eucentric way about a tilt-

ing axis at the center of the sample. If one views two

such images placed side by side with an appropriate

separation distance, three-dimensional stereovision of

the object is obtained, as demonstrated in Fig. 5.108.

Fig. 5.108 AsetofTEM stereograms showing disloca-

tions in Si

0.87

0.13

/Si(100)

Part B 5.5

Nanoscopic Architecture and Microstructure 5.5 Texture, Phase Distributions, and Finite Structures Analysis 275

The three-dimensional vision could also be provided

by an anaglyph image, an overlayed image of the two

micrographs that are superimposed with different col-

ors, when viewed with red–blue glasses. Quantitatively,

the surface proﬁle is calculated from the disparity in

the stereo image pairs. The reconstruction of three-

dimensional structure is possible from such stereograms

acquired in TEM and SEM.

Stacked Tomographs

Laser Scanning Confocal Microscopy (LSCM). The high

depth resolution of ≈ 1 μm in laser scanning confo-

cal microscopy (LSCM) enables reconstruction of 3-D

images from successively acquired optical images of

optically sliced layers by changing the focal position.

The great advantage of LSCM is that there are no spe-

cial requirements for the test environment except that

the sample medium must be optically transparent.

Ordinary LSCM is based on one-photon processes

such as scattering or ﬂuorescence of light. In thick sam-

ples, however, remaining light scattering deteriorates

the spatial resolution. Nonlinear ﬂuorescent microscopy

uses an intense laser light whose photon energy is too

small for a one-photon excitation of a ﬂuorochrome

but is large enough for a two-photon excitation. Due

to the quadratic dependence of the ﬂuorescence in-

tensity on the intensity of excitation laser, the focal

point is effectively reduced so that the spatial resolu-

tion is enhanced whereas the background ﬂuorescence

is removed. Two-photon microscopy has the merit of

avoiding photobleaching or optical damage that would

be caused when excitation lights of higher photon en-

ergy in the ultraviolet range is used for one-photon

excitation. Similar beneﬁts are obtained in second-

harmonic generation microscopy in which the dielectric

response of the sample is nonlinear.

Three-Dimensional Atom Probe (3-DAP) Microanal-

ysis. As described in (Sect. 5.1.2), the ﬁeld ion micro-

scope (FIM)[5.86] is an atom-resolving microscope in

which an imaging gas projects the positions of step-

edge atoms on a phosphor screen without destroying the

surface. If the applied voltage is high enough, the step-

edge atoms themselves evaporate due to the ﬁeld. The

ﬁeld-evaporating atoms (positively charged ions) are

conveyed by the bias ﬁeld to the screen to form a direct

image of the atomic distribution on the tip, now destruc-

tively. As illustrated by Fig. 5.109, this ﬁeld evaporation

can be induced by a pulse bias voltage applied to the

sample that is triggered as the starting signal for time

of ﬂight (TOF) measurements of the evaporated ions.

Sample

tip

Position-

sensitive

detector

Ionized atom

Multichannel plate

(x, y, t)

Pulsed HV

TOF

Stop signalTrigger pulse

Fig. 5.109 Experimental setup of 3-D atom-probe FIM

Nd Cu

20 nm

~10 nm

Fig. 5.110 Elemental mapping of Cu and Nd in Nd

4.5

75.8

18.5

0.2

alloy imaged by 3-D atom-probe FIM

(Courtesy of Dr. K. Hono)

The use of a multichannel plate in front of the phosphor

screen and a gated acquisition of the ion signal allows

one to detect speciﬁc element atoms sorted by the TOF.

Since the ﬁeld evaporation takes place exclusively at

step edges, the evaporation processes are repeated as if

the atomic layers are being peeled back one by one. The

layer-by-layer acquisition of such atom probe signals

are used to reconstruct a three-dimensional map of ele-

ments in the sample tip. Figure 5.110 shows an example

of three-dimensional atom probe (3-DAP) images ob-

tained for a multicomponent alloy. The lateral and depth

resolution of 3-DAP microanalysis is ≈ 0.2nm,thebest

among the stereological methods developed so far.

Computed Tomography (CT)

Computed tomography is based on mathematical trans-

formations that enable reconstruction of the two-

Part B 5.5

276 Part B Chemical and Microstructural Analysis

Beam

source

Object

Detector

f (x,y)

p (r,θ)

Fig. 5.111 Geometry in computed tomography

dimensional distribution function f (x, y) (e.g., of the

absorption coefﬁcient) of an object from a set of

one-dimensional intensity proﬁles p(r,θ) detected by

the detector of a transmitted beam that is emitted

from a source in various directions θ, as illustrated in

Fig. 5.111. The computed tomographs obtained for suc-

cessive cross sections of the object are further stacked

to form a three-dimensional reconstruction of the whole

body.

X-Ray Computed Tomography (XCT). X-ray computed

tomography (XCT) using an x-ray for the source beam

is common equipment for clinical diagnosis, but can

also be used for materials analysis. In XCT, the beam

is fanned to scan an angle θ rather than scan position r.

The resolution of recent x-ray CT is ≈ 1 μm, compara-

ble with that of OM.

Magnetic Resonance Imaging (MRI). Magnetic reso-

nance imaging (MRI)orNMR-CT is also now common

for clinical diagnosis. MRI is based on the principle that

when the applied magnetic ﬁeld in NMR hasaninten-

tional gradient (i.e., varies with position), the resonance

frequency of an NMR signal becomes dependent on

the position of the nucleus. The positional r-dependent

NMR signal frequency constitutes the proﬁle function

p(r,θ) in which the angle θ is given by the direction of

the magnetic ﬁeld gradient.

Three-Dimensional TEM (3-D-TEM). Although x-ray

crystallography can provide structural information at

atomic resolution, its application depends on whether

we can obtain crystalline arrays of the molecules

with good quality and analyze the diffraction data in

light of relevant phase information [5.87]. Further-

Fig. 5.112 A3-D-TEM image of a selfassembled tri-

block copolymer, poly(styrene-block-isoprene-block-styr-

ene). The box size is 270 nm ×270 nm×210 nm. 120 digital

images were acquired at tilt angles ranging from −60

◦

with 1

◦

steps on a computer-controlled JEOL JEM-

2200FS operated at 200 kV (Courtesy of Dr. H. Jinnai)

more, a serious problem especially in biomolecules

is that crystallization may alter the conformation of

the molecules that could differ signiﬁcantly from that

in which the molecules function in the natural en-

vironment. In single-particle electron microscopy, the

3-D structure of a single macromolecule (MW >

250 000 Da) is reconstructed at a resolution of 1–2.5nm

from a set of TEM images obtained in many differ-

ent sample orientations. In practice, sample damage

due to electron irradiation is reduced by the use of

cryo-microscopes and simultaneous observations of

many sample molecules (usually embedded in ice for

biomolecules) to overcome signal-to-noise problems.

Figure 5.112 shows a graphical rendering of 3-D-TEM

image obtained for a selfassembled polymer mater-

ial. 3-D-TEM is applicable to amorphous solids, but

not suitable for crystalline samples in which the TEM

images are too sensitive to diffraction conditions for

a simple three-dimensional reconstruction to be carried

out.

X-Ray Phase Contrast Imaging (PCI)

The contrast of conventional x-ray transmission images,

except for x-ray topography, is based on the positional

difference of the absorption coefﬁcient [5.88]. For ma-

terials consisting of light elements such as biological

tissues and polymers, conventional x-ray transmission

microscopy, unless staining with heavy elements is

Part B 5.5

Nanoscopic Architecture and Microstructure References 277

possible, is not efﬁcient enough. Phase contrast imag-

ing (PCI), as in optical microscopy (Sect. 5.1.2), is an

approach to overcome this problem also in x-ray re-

gions, though there is a difﬁculty due to the small

size of the phase shift in light elements. However,

in the soft x-ray region, the phase shift can be 10

times larger than in the hard x-ray region, so x-ray

PCI has been introduced. Among the four schemes

so far attempted, interferometric imaging, holography-

like imaging, diffraction enhancement imaging, and

Zernike-type microscopy, the interferometric method

is most sensitive for acquiring phase contrasts. The

method uses a Mach–Zehnder x-ray interferometer cut

out of a large Si single crystal ingot which is mounted

on a multiaxis goniometer set in the beam line of a syn-

chrotron radiation source. X-ray phase shift computed

tomography [5.89] exploiting the ability of phase re-

trieval allows one to reconstruct 3-D images such as

those shown in Fig. 5.113.

500µm

Fig. 5.113 X-ray phase image of a tissue of a rat kidney.

Tubules in the tissue, a part of which was clogged by pro-

tein (T), were depicted. Glomeruli (G) were also revealed

(Courtesy of Drs. A. Momose, J. Wu, and T. Takeda)

References

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Part B 5