Ab initio structure determination using nano electron diffraction

Ab Initio Structure Determination Using

Nano Electron Diffraction

Ab Initio Structure Determination Using

Nano Electron Diffraction

PROEFSCHRIFT

Ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. dr. ir. J. T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op

woensdag 26 maart 2008 om 10:00 uur

Door

Qiang XU

Master of Science

Tsinghua University, Beijing

Geboren te Wuhan, Hubei Province, China

Samenstelling promotiecommissie:

voorzitter

Prof. dr. H.W. Zandbergen Promotor Technische Universiteit Delft

Prof. dr. C. J. Gilmore Glasgow University

Prof. dr. J. Schoonman Technische Universiteit Delft

Prof. dr. J.P. Abrahams Leiden Universiteit

Prof. dr.ir. S. Van der Zwaag Technische Universiteit Delft Prof. dr.ir. T.M. Klapwijk Technische Universiteit Delft Dr. J. Jansen Adviseur Technische Universiteit Delft

This thesis is part of the research program Stichting voor Fundamenteel Onderzoek der Materie (FOM), financially supported by Nederlandse Organisatie voor

Wetenschapperlijk Onderzoek (NWO).

I would like to thank Prof. dr. H.W. Zandbergen for his substantial guidance and support during the preparation of this thesis.

Qiang Xu

Ab Inito Structure Determination Using Nano Electron Diffraction/ Qiang Xu, PhD thesis, Delft University of Technology, with summary in Dutch.

Keywords: Electron Microscopy, Electron Diffraction, Structure Determination, Dynamical scattering

ISBN: 978-90-9022879-2 Copywrght © by Qiang Xu All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means without the prior written permission of the copyright owner.

Chapter 1 Introduction……….. 1

Chapter 2 Structure analysis methods in electron crystallography…….…..… 13

Chapter 3 Preparation of TEM samples for hard ceramic powders………….. 33

Chapter 4 Structure and Magnetic Properties of Eu2CaCu2O6……….. 45

Chapter 5 Ab initio structure determination of the novel superconductor Mg10Ir19B16……… 63

Chapter 6 Correlated approach to structure determination of microcrystalline material………. 87 Summary……… 107 Samenvatting………. 109 Acknowledgements………... 111 Curriculum Vitae………... 115 List of publications…...……….……… 117

1.1 Structure and structure analysis

Way of life is strongly dependent on the use of all kinds of materials, including alloys, semiconductors, ceramics, polymers, and biomaterials, each with their own specific properties. Most properties of matter can be understood from its crystallographic ideal structure and various kinds of imperfections such as point defects, dislocations, stacking faults, and so forth. Structure analysis is thus essential for understanding the properties of materials and in turn, gives us the ability to design, synthesis, and search for novel materials with superior properties such as the development of high-temperature superconductors.

During the last century, techniques were developed for structure analysis. Among these, diffraction methods remain the most important methods for studying the average structure of materials. Recent developments in super electron microscopes, which allow sub-angstrom resolution imaging and thus the direct imaging of structures, may change this, but lower costs mean that diffraction methods will continue to play an important role in structure determination. Diffraction, which refers to a phenomenon associated with wave propagation, provides information about atom arrangements, since the intensities of the diffracted wave passing through crystal vary in the different propagation directions (provided its wavelength is comparable to the interatomic distance, around 1Å). In addition to X-rays, (electro-magnetic waves with a wavelength around 1 Å), it was also found that it was possible to obtain diffraction from solid crystal using some particles, including electrons, protons, neutrons, and even molecules, because of the existence of wave-particle duality. For the investigation of solids, however, X-ray, neutron and electron diffraction have been found to have the most practical applications.

1.2 X-ray, neutron and electron diffraction

1, 2

Although all three diffraction methods for structure analysis—X-ray, neutron, and electron diffraction— have much in common, each has specific features that determine

Figure 1.1: Scatters of X-rays, neutrons and electrons. X-rays, neutrons and electron

diffraction are scattered by electron density, nucleus and electrostatic potential respectively.

application fields. The similarities and differences between these three methods are rooted in the interaction mechanism between the given radiation and the specimen: X-rays—electromagnetic waves—are scattered by electron clouds. The inner atomic nuclei with their positive charges are ‘invisible’ to such radiation. Thus experimental X-ray diffraction data gives information about electron distribution in a crystal.

Neutrons are scattered by nuclei, which act as delta-function potentials; the scattering potential is thus sharply peaked. Aside for those contributing to the magnetic moment of an atom, neutrons do not interact with electrons. Therefore, neutron diffraction experiments provide information about the position of the nuclei and the magnetic moments of an atom.

Scattering of electron diffraction is due to the interaction between the incident electron and electrostatic potential of an atom, which is composed of the positive charge of the nuclei and the negative charge of the electrons.

A comparative picture of the distribution of scattering matter in a lattice of atoms for the three types of radiation is presented in Figure 1.1. As shown in Figure 1.1, the peak positions of the three scattering types all correspond to the atom positions (nuclei), such that all methods can be used to determine the coordinates of the centres of gravity

of the atoms within a crystal. While all three permit the geometric theory of diffraction, which was originally developed for X-rays, the physical difference in the interactions between the radiation and the scatter of matter means that each of these three diffractions has specific features. This thesis will focus on the use of electron diffraction.

As shown by pioneers in electron crystallography, electron diffraction has two major advantages when compared to X-ray and neutron diffraction. The first is that electrons, unlike X-rays or neutrons, are charged particles and can therefore be focused by an electromagnetic lens. This allows both images and diffraction patterns to be recorded for structural analysis. The second advantage is that the scattering cross-section of matter for electrons is approximately 103 _{times greater than for X-rays (meaning 10}6

times greater intensities), and 104 _{times greater than for neutrons. This means that}

much smaller objects can be studied as single crystals with electron beams than with the other two structure analysis methods. A third potential advantage of electron diffraction over X-ray and neutron diffraction is the enhanced detectability of light atoms in the presence of heavier atoms. This advantage is also fundamentally associated with the strong interaction between electron and matter and will be explained further in later sections (1.3).

Despite these potential advantages, and the fact that electron crystallography has a long history of application to structure analysis, the strong interaction between electrons and matter causes that the results of such determinations have not been so well accepted by structure chemists as those from X-rays and neutron diffraction analysis. The reason is that due to this strong interaction and to the very small wavelength of fast electrons, electron diffraction from even a thin crystal is a process of multiple scattering (also called dynamical scattering), rather than the primary scattering (also called kinematical scattering) that occurs with X-ray and neutron diffraction. Some early crystallographers argued that multiple scattering complicates the relationship between electron diffraction and projected potential, making ab initio structure determination using electron diffraction data intractable. Thus, in the early stage (see, for example, Rigamonti, 1936 3_{; Karpov, 1941} 4_{; Lobachev, 1954} 5_{; Vainstein, 1971} 6_{) most}

structure analysis work using electron diffraction was limited to solving the structure of molecular organic crystal structures, such as polymethylene compounds, copper salts of amino acid and so forth, in which dynamical effects could be ignored. This approach is still used for the structure determination of proteins (Gilmore, 1996 7_{). The}

similar way to X-ray and neutrons diffractions. Other research was carried out into resolving the structure of inorganic materials using electron diffraction, and especially combined with electron microscope images (see, for example, Heidenreich, 1964 8_;

Spence, 1980 9_{; Humphreys and Bithell, 1992} 10_{; Fan, 1993} 11_{; Eades, 1994} 12_{, and}

Gilmore, 2003 13_{) within a weak-phase-object approximation, which holds if one}

assumes kinematical diffraction. This assumption is far from adequate, however, especially for those crystals containing heavy atoms. These kinds of structures thus limited to providing possible structure models. Moreover, this approximation does not take advantage of the positive features associated with the dynamical scattering effect of electron diffraction, such as enhanced detection efficiency for light atoms.

With the development of dynamical scattering theory and the huge increase in computing speed, it is now possible to use quantitative electron diffraction to determine

ab initio complex inorganic structures with accuracies comparable to those achieved by

single crystal X-ray diffraction. Our group successfully determined some inorganic structures using this approach (for example, Zandbergen, 1997 14_{, 1998} 15_{, and 2002} 16_).

This thesis will mainly focus on structure analysis using dynamical electron diffraction on other more complicated inorganic structures.

1.3 Dynamical scattering theory

The theory of dynamical electron diffraction was developed by a number of authors over a large part of the twentieth century. Methods for calculating electron diffraction can generally be classified into two groups: the Bloch wave method, and the multislice method. The Bloch wave method starts with the Schrödinger equation, and Fourier expands the crystal potential and the electron wave function with components that match the underlying periodicity of the crystal lattice (implying that the specimen has to be a perfect crystal). Multislice methods approach the dynamical scattering problem by dividing the specimen into a series of slices along the electron beam direction. Each slice is thin enough to allow the kinematical scattering approximation to be used. The latter method has the main advantage of being much less time-consuming than the Bloch wave method (the time scales of the multislice method and the Bloch wave method are Nlog2N and N3 respectively, where N refers to the number of diffraction

beams in the calculation). Another advantage of using the multislice method is that the sample does not have to be a perfect crystal. Using the real space method proposed by

D. van Dyck allows imperfections to be easily taken into account in the multislice calculation. The refinement software used is based on the multislice method.

1.3.1 Multislice methods

17

The multi-slice algorithm, developed by Cowley and Moodie, is shown schematically in Figure 1.2.

In the simulation of the exit wave, the specimen is divided into slices of a thickness ∆z that is small enough to neglect the probability of multiple scattering. The structure information of each slice is expressed by a (two-dimensional) projected potential VP_,

projected on a plane perpendicular to the direction of the incident electrons. If absorption is neglected, the interaction of the incident electron wave with the projected potential is merely a phase shift given by the transmission function qn(x,y) and a new

electron wave can be written as

Figure 1.2: Multislice method for treating dynamical diffraction for a thick

specimen. (a) original thick specimen; (b) the specimen divided into thin slices.; (c) each slice is treated as a transmission step (solid) line using kinematical theory followed by a propagation step (vacuum between solid lines). 17

'( , )

( , )

( , )

(1 1)

,

( , ) exp(

( , ))

(1 2)

n n n p n n

x y

x y q x y

where

q x y

i V x y

ϕ

ϕ

σ

=

⋅

−

=

−

This new wave will then propagate through vacuum to the next (projected) slice. The propagation function pn(x,y) used to describe this transfer can be written in the small

angle approximation as 1 2 2

( , )

'( , )

( , ),

(1 3)

( , ) exp(

)

(1 4)

n n n n n

x y

x y

p x y

x

y

where

p x y

i k

z

ϕ

ϕ

π

+

=

⊗

−

+

=

−

Δ

This transmission-propagation cycle will be repeated until the assumed thickness of the specimen is reached. The wave function at the back exit plane then can be written as

0 1 1 2 2

( , ) ((((

x y

q

)

p

)

q

)

p

)...)

q

_n

),

(1 5)

ϕ

=

ϕ

⋅

⊗

⋅

⊗

⋅

−

To save computing time, the convolution is generally calculated as a multiplication in Fourier space, using

1

₍

₎

_{(1 6)}

f

_{⊗ =}

g

FT

−

F G

_⋅

₋

where F and G are the Fourier transforms of the function f and g and FT-1 denotes an inverse Fourier transform.

1.3.2 A simple intuitive theory: electron channeling

18

Multislice methods are valuable for numerical calculation, but they do not provide much physical insight in the diffraction process. For this purpose, a simpler intuitive theory, van Dyck’s electron-channeling theory, can be used. This theory is important because it provides an intuitive interpretation of the dynamical effect.

Van Dyck’s theory expounds that in an exact zone axis orientation of a perfect crystal, without taking higher order Laue zones reflection into account, the dynamical wave function represents the interaction between the incident electrons and the crystal potential consisting of a two-dimensional assembly of potential wells corresponding with the different projected columns (the z-dependence of the crystal potential is removed in the first order approximation, due to the fact that the incident electrons move so fast that they do not feel the potential difference along z, the electron incident

direction). The solution of (1-4) can be expanded in eigenfunctions of the Hamiltonian (similar to the Bloch wave approach)

( , )

( ) exp(

n

),

(1 7)

n n

E z

R z

C

R

i

E

ψ

φ

π

λ

=

∑

−

−

where

H

_φ

_n

( )

R

=

E

_{n n}

_φ

( )

R

(1 8)

−

with

( ),

(1 9)

2

H

eU R

m

= −

=

Δ −

−

and 2 2

2

k

E

m

=

=

is the incident electron energy. λ is the electron wavelength.

For En < 0 the states are bound to the columns. (1-7) can then be rewritten as

( , )

( )[1

n

)

( )[exp(

n

) 1

n

],

(1 10)

n n n n

E z

E z

E z

R z

C

R

i

C

R

i

i

E

E

E

ψ

φ

π

φ

π

π

λ

λ

λ

=

∑

−

+

∑

−

− +

−

The coefficients Cn are determined from the boundary condition

( )

( ,0)

(1 11)

n n

C

_φ

R

=

_ψ

R

−

∑

In case of plane wave incidence one thus has

( ) 1

(1 12)

n n

C

_φ

R

=

−

∑

And from (1-8) and (1-9)

( )

( ,0)

1

( )

(1 13)

n n n

C

_φ

R E

=

H

_ψ

R

= ⋅ = −

H

eU R

−

∑

Thus (1-10) becomes

( )

( , ) 1

( )[exp(

n

) 1

n

],

(1 14)

n n

E z

E z

eU R z

R z

i

C

R

i

i

E

E

E

ψ

π

φ

π

π

λ

λ

λ

= +

+

∑

−

− +

−

The first two terms yield the well-known weak phase object approximation. In the third term only these states will appear in the summation for which

(1 15)

n

E

E

z

λ

≥

−

In the case that the object is very thin; no states obey (1-15). The third term thus

Figure 1. 4:

Each atom in an atom column is considered as a thin lens. By passing

successive lenses, the electron wave is focused at periodic distances. When the crystal thickness is equal to the extinction distance, the exit wave is equal to the entrance wave.

approximates to zero and the weak phase object approximation is valid (kinematical theory). For a thicker object, only bound states with very deep energy levels will be exited. These low states are localized near the column cores. Moreover, a two-dimensional projected column potential has very few low energy states, and when the overlap between adjacent columns is small only the radial symmetric states will be excited. In practice, only one state appears for most types of atom columns, which can be compared with the 1s state of an atom.

In the case of an isolated column of type i, taking the origin in the centre of the column, we have

( )

( , ) 1

i

( )[exp(

i

) 1

i

],

(1 16)

i i i

eU R z

E z

E z

R z

i

C

R

i

i

E

E

E

ψ

π

φ

π

π

λ

λ

λ

= +

+

−

− +

−

Since the states

φ

_i are localized at the atom cores, the wave function for the total crystal can be expressed as a superposition of the individual column functions

( )

( , ) 1

(

)[exp(

i

) 1

i

],

(1 17)

i i i i

E z

E z

eU R z

R z

i

C

R R

i

i

E

E

E

ψ

π

φ

π

π

λ

λ

λ

= +

+

∑

−

−

− +

−

with

( , )

_i

(

_i

)

(1 18)

i

U R t

=

∑

U R R

−

−

The interpretation of (1-17) is simple. Each column i acts as a channel in which the wave function oscillates periodically with depth. The periodicity is related to the ‘weight’ of the column, thus, proportional to the atomic number of the atoms in the columns and inversely proportional to their distance along the column. The physical reason for this channeling theory is that due to the positive electrostatic potential of the atoms, a column acts as a guide for the electron within which the electron can scatter dynamically without leaving the column (see Figures 1.3 and 1.4 ). The importance of these results lies in two key aspects. The first is that for thicknesses greater than the usual kinematical approximation, the wavefunction at the exit face still retains a one-to-one relation with the configuration of columns even in the presence of dynamical scattering. Hence this description is very useful for the interpretation of high-resolution images and for providing a possible starting model for structure analysis (the application is described in Chapters 4 and 5). The second is that the ‘effective’ scattering power of each column (at the exit face) varies with the sample thickness, such that the light atom column may scatter electrons more at certain thicknesses than the heavy atom column. Hence the detection efficiency of light atoms will be enhanced

for the electron diffraction from these sample thicknesses, which is obviously one way in which dynamical scattering benefits structure analysis using electron diffraction.

1.4 Summary and outline

The presence of strong interaction between incident electrons and the crystal potential of matter means that there is an inherent need to take the dynamical effect in electron diffraction into account. Compared with conventional X-rays diffraction, this dynamical effect provides improved sensitivity of electron diffraction to light atoms. To understand and utilise this sensitivity in structure analysis, one needs a proper incorporation of dynamical scattering theory.

This thesis uses election crystallography to address the structure determination of complex inorganic materials (composed of both heavy and light atoms). Due to the inherent dynamical effect of electron diffraction, the routine of structure analysis using electron diffraction is quite different from that using X-rays diffraction, and structure refinement from electron diffraction data is still in progress. This thesis will focus on how the routines can be optimized, so as to allow for a quick and accurate structure analysis using electron diffraction. To aid the reader, Chapter 2 introduces the methods used in the later structure determination, such as high-resolution electron microscopy (HREM), nano diffraction, convergent beam electron diffraction (CBED), and the first- principles calculations), together with a general description of the theoretical background. Chapter 3 describes an improved transmission electron microscope (TEM) specimen preparation method for multi-phase polycrystalline powders, the sample type investigated in the thesis. Emphasis is given to this sample preparation method’s ability to provide very thin TEM specimens of all relevant orientations for hard ceramics. In Chapter 4, electron crystallography is applied to determine the structure of Eu2CaCu2O6 and (Eu0.5Ca0.5)2CaCu2O6. It is shown that these contain a novel CuO3

plane, aside from the common CuO2 plane present in high-temperature

superconducting cuprites (HTSC). The results highlight electron crystallography’s ability to determine a completely new structure with no analogous structure model. Oxygen positions can also be determined with almost the same accuracy as for heavy atoms. The first-principles calculation can be combined with electron crystallography to overcome the fact that a refinement is trapped in a local minimum. In Chapter 5, the structure of another new superconducting compound Mg10Ir19B16, with a larger unit

with other structure analysis methods. Due to the complexity of the structure, — including the large unit cell, lack of centro-symmetry, the presence of heavy (Ir) and light atoms (B) —, an optimized structure analysis procedure is used, which allows an accurate structure determination of Mg10Ir19B16. A combination of possible structure

analysis methods is emphasized. The last chapter offers a summary of a general routine for structure determination, according to our experience. This routine aims to achieve a quick and accurate structure analysis from a reaction mixture of searching a new phase of interested properties in the early stage. Correlative Structure Analysis (a combination of possible structure analysis methods) is proposed and discussed.

References

1_{B.K.Vainshtein, Structure Analysis by Electron Diffraction, Oxford,}

Pergamon,1964

2 _{D.L.Dorset, Structure Electron Crystallography, New York, Plenum Press, 1995} 3 _{R.Rigamonti, Gazz.Chim.Ital. 66, 174, 1936}

4 _{V.L.Karpov, Zh. Fiz. Khim. 15, 577, 1941}

5 _{A.N.Lobachev, Trud. Inst. Krist. SSSR 10, 71, 1954}

6 _{B.K.Vainshtein, I.A.D’yakon, and A.V.Alov, Sov.Phys. Crystallogr. 3, 452, 1971} 7 _{C.J.Gilmore, W.V.Nicholson, and D.L.Dorset, Acta Cryst. A52, 937-946. 1996} 8 _{R.D.Heidenreich, Fundamentals of Transmission Electron Microscopy, New York,}

Wiley-Inter-science. 1964

9 _{J.C.H.Spence, Experimental High-resolution Electron Microscopy, Oxford,}

Clarendon. 1980

10 _{C.J.Humphreys, and E.G.Bithell, Electron Diffraction Techniques (Vol.1) 1992} 11 _{H.F.Fan, Modern Crystallography. (Proceedings of the Seventh Chinese}

International Summer School of Physics. Beijing International Workshop. 1993

12 _{J.A.Eades, Acta Cryst. A50, 292, 1994}

13 _{C.J.Gilmore, Crystallography Reviews 9, 17-32}

14 _{H.W.Zandbergen, S.J.Andersen, J.Jansen, Science 29, 1221, 1997}

15 _{J.Jansen, D.Tang, H.W.Zandbergen, and H.Schenk, Acta Cryst. A54, 91 1998} 16 _{H.W.Zandbergen et al, Nature 372, 759 2002}

17 _{E.J.Kirkland, Advanced Computing in Electron Microscopy, New York, Plenum}

Press, 1998

methods in electron

crystallography

Abstract

In this chapter, we discuss the range of electron microscopy methods needed to obtain the necessary crystallographic information, including unit cell parameters, space groups, and basis positions.

Unit cell parameters can be routinely determined using tilt series, by which a sample is tilted along a reciprocal lattice vector (which is generally, but not necessarily, short) to record a series of diffraction patterns. From this series, the reciprocal lattice parameters can be reconstructed based on reflections in the diffraction patterns and the corresponding tilt angles between the various diffraction patterns (Vainshtein, 1964 1; Dorset, 1995 2).

A sample’s space group can be unambiguously determined using the convergent beam electron diffraction (CBED) method, by which a conical electron beam with a convergence larger than 10-3 rad is utilized to form the diffraction pattern. Due to the convergence, diffraction disks are produced instead of the usual diffraction spots. The contrast variation in the disks reflects the crystal’s symmetry of a crystal, thereby allowing one to derive its point and space groups. The procedure for determining a space group using CBED has been thoroughly discussed by Buxton, 1971 3 and Tanaka, 1994 4. In this chapter, we will thus focus on how to determine atomic basis positions using electron microscopy.

2.1 Transmission electron microscopy

5

When a beam of high-energy electrons passes through a thin foil of material, it is scattered in part elastically and in part in-elastically by the crystal potential of the solid. The resulting electron wave at the back exit-face (exit-wave) of the crystal contains information about crystal potential. The exit wave can be focused with suitable electron

lenses, to form an image or a diffraction pattern. This thesis mainly considers elastically scattered electrons, which maintain a phase relationship (coherency) during the interaction with the material and during the image or diffraction pattern formation. In the case of an inelastic event, the incident electrons interact with the sample’s phonons, plasmons and core electrons. Other signals are produced, such as characteristic X-rays, secondary electrons, back scattering electrons, or the in-elastically scattered electrons themselves (in an energy-loss spectrum), and thus can be collected to provide other useful information of the material, including its chemical composition, electronic structure, and so forth. The recording of these signals exited by the incident electrons and of in-elastically scattered electrons is generally known as ‘analytical microscopy’, and will not be discussed in this chapter.

A schematic representation of a transmission electron microscope (TEM) is shown in Figure 2.1. The microscope, which operates in a high vacuum, is an electron optic lens

Figure 2.1: An electron microscope. Both diffraction (left) and image (right) can be

system that is used to form high-resolution electron images or diffraction patterns. The imperfection of the imaging can be described with a point-spread function or contrast transfer function, through which the wave function at the exit face is transformed into the wave function in the image plane (see Figure 2.2). Mathematically, the microscope’s point-spread function or contrast transfer function can be written as

)

1

.

2

(

)

(

)

(

)

(

R

_ex

R

h

R

i

=

ψ

⊗

ψ

)

2

.

2

(

)

(

)

(

)

(

g

_ex

g

H

g

i

=

ψ

⋅

ψ

Here, R is a real space vector; g is a reciprocal vector, the spatial frequency for a particular direction. Ψi , Ψex are the wave functions at the image plane and the exit

plane respectively. H(R), H(g) are the microscope’s point-spread function and contrast transfer function. In fact, the point-spread function is exactly equivalent to the contrast transfer function, in the sense that H(g) is the Fourier transform of H(R). Correspondingly Ψex(g) is the Fourier transform of Ψex(R) and Ψi(g) is the Fourier

transform of Ψi(R).

The contrast transfer function

H

(

g

)

can be more easily attributed to three main factors： apertures (the aperture function A(g)), attenuation of the wave (the envelope function E(g)), and aberration of the lens (the aberration function B(g)). We can write

)

(g

H

as the product of these three terms

)

3

.

2

(

)

(

)

(

)

(

)

(

g

A

g

E

g

B

g

H

=

Due to its geometrical construction, the electron microscope has its own physical apertures, and it allows for the insertion of an aperture in the back focal plane to cut off all information with a g greater than a certain value. This effect is described using aperture function A(g). The envelope function has a damping effect related to the properties of the microscope itself (mainly due to the spatial and temporal coherency of the electron beam). It seems to impose a virtual aperture in the diffraction plane (back focal plane), thus imposing a new resolution limit on the microscope, which is known as the ‘information limit’. The last term, B(g), know as the aberration function, is usually expressed as

)

4

.

2

(

))

(

exp(

)

(

g

i

g

B

=

−

_χ

where the term

_χ

(g

)

can be written as

2 1 3 4

( ) ... (2.5)

2 s

g f g C g

χ

= Δ

π λ

+

π λ

+

and mainly determined by defocus Δf and the spherical aberration of objective lens Cs.

Figure 2.3 shows the typical plots of A(g,) E(g) and B(g) versus g. According to equations (2.1)-(2.5),

)

6

.

2

(

))

(

exp(

)

(

)

(

)

(

)

(

g

_ex

g

A

g

E

g

i

g

i

ψ

χ

ψ

=

⋅

−

)

7

.

2

(

)

(

)

(

)

(

)

(

)

(

2 2 2 2

g

E

g

A

g

g

g

I

_i

=

_ψ

_i

=

_ψ

_ex

Figure 2.3:

An illustration of A(g), E(g),B(g) in phase, for a microscope under the conditions: HT 200 kV, defocus spread 8nm, Cs 0.7 mm, defocus 17 nm, beam half

convergence 0.71 mrad, specimen drift 0.05 nm. The aperture function A(g)( grey frame) cuts off the high frequency information, depending on the size of the apperture. The total envelope function E(g) (red dash-dot-dot line in (b)) is close to a top-hat function, contributed by the spatial part (purple dashed line in (a)) and temporal part (orange dotted line in (a)). The aberration function B(g) (blue solid line in (a)) fluctuates, especially in the high frequency region, which makes HREM difficult to interpret.

)

8

.

2

(

)

(

)

(

)

(

)

(

R

R

2

R

*

R

I

i

=

ψ

i

=

ψ

i

ψ

i

If we use weak phase object approximation such that

( )

R

1

i

U

(

R

)

(

1

.

14

)

ex

σ

ψ

=

+

then (2.8) can be simplified by ignoring terms in

σ

2 as

)

9

.

2

(

)

(

)

(

))

(

sin(

)

(

2

1

)

(

R

U

R

R

A

R

E

R

I

_i

=

+

σ

⊗

χ

⊗

⊗

Thus

I

_i

(

g

)

=

_δ

(

g

)

+

2

_σ

U

(

g

)

sin(

_χ

(

g

))

A

(

g

)

E

(

g

)

(

2

.

10

)

Note the resolution of an image is mainly determined by the cut-off g value of the envelope function (the information limit), which is around 1.0 Å for a Philips 300UT. Not all information in the image is easily interpretable in real space, however, since the image is also modulated bysin(_χ(g)). For an objective lens without a Cs corrector, the

high frequency information in an image is strongly delocalized bysin(_χ(g)), resulting in a limited interpretable resolution. For this reason, the concept of point resolution is introduced. Point resolution gsch is mainly governed by Cs and λ (gsch=0.66Cs1/4λ3/4)

and describes the maximum direct interpretable resolution of TEM images. For normal TEM, the point resolution is around 1.5-2Å (1.7 Å for a Philips 300UT), which is much lower than the information limit.

Electron diffraction intensities are not hampered by the lens aberrations. The intensities of electron diffraction can be written as:

)

11

.

2

(

)

(

)

(

)

(

g

g

*

g

I

_d

=

_ψ

_ex

⋅

_ψ

_ex

Electron diffraction therefore provides higher-resolution experimental data sets for structure analysis (easily reach 0.7 Å for electron accelerated by 300 kV). Although high-resolution imaging can only provide much lower resolution structural information, it is nevertheless valuable for structure analysis due to its direct interpretation.

It should be noted that in the latest super microscope equipped with a Cs corrector, the

image resolution reaches 0.8 Å, comparable with that of diffraction patterns. This might be taken to suggest that we are about to enter an ear of ab initio structure determination using direct imaging, and that the electron diffraction method will shortly become out of date. The high signal-noise-ratio and low cost of the electron diffraction method will continue to be utilized in crystal structure analysis, however, especially for small precipitates embedded in a matrix and protein nano-crystal.

2.2 Quantitative electron diffraction (MSLS)

6

As shown in Figure 2.4, experimental diffraction intensities recorded on image plates can be quantitatively obtained using the data reduction program GREED from the Elstru package. In this process, the background is estimated from the intensity on the edge of the box and then subtracted for the area within the box. The background

contains mainly inelastic scattering electrons that have interacted with phonons and core electrons. After subtraction of the background, the intensities within the box are those elastic scattering and part of inelastic scattering related to Plasmon excitations. Jansen et al. showed that the inelastic scattering intensities would not change structure refinement results (atom positions). Thus, after background subtraction, the intensity within each box can be summed and taken as the experimental intensity of the corresponding diffracted beam suited for structure determination, implying that it closely resemble the intensity for pure elastic diffraction.

Note that the experimental diffraction used in later MSLS refinement is neither convergent beam electron diffraction (CBED) nor selected area diffraction (SAD), but is rather nano parallel beam electron diffraction (NPBED). Figure 2.5 compares these three types of diffraction. SAD cannot be used, since with this method, the recorded diffraction comes from a large sample area (~ 1μm), resulting in large variations of misorientation and thickness, that can not currently be quantitatively simulated. The CBED pattern is recorded from a small area (~10nm) with a large C2 aperture (~70μm),

meaning that it can be simulated. CBED is disk pattern, however, with each disk

containing many details, and would therefore be time-consuming to calculate. Alternatively, the diffraction (nano-PBED) is recorded using a spot size (10 nm or less) and the smallest C2 aperture (10μm). The latter limits the beam’s convergence, giving a

spots diffraction pattern rather than the disks pattern recorded using CBED.

The electron intensities for a chosen structure model can be calculated using the multi-slice method. The calculated diffraction intensities Igcal can then be compared to the

observed diffraction intensities Igobs. The mismatch is expressed in an R factor defined

by

)

12

.

2

(

}

{

}

{

)

(

₂ 2

∑

∑

−

=

g obs g g calc g g obs g g

I

w

I

I

w

I

R

Figure 2.5: Comparison of SAD, CBED and NPED. Compared with normal SAED,

CBED and NPED are recorded from a local area (generaly around 10nm). Thus both can be quantitively calculated for structure ananlysis. NPED’s advantage over CBED is that by using the smallest C2 aperture, a spots diffraction pattern can be obtained rather than a disks pattern. The CBED disks pattern is very detailed and simulation requires a longer calculation time.

or

(

2

.

13

)

}

{

)

(

2

∑

∑

−

=

g obs g g calc g g obs g g

I

w

I

I

w

I

R

where wg is the weights for different reflections (usually 1 for all reflections), which

can be used to change the contribution of certain groups of diffraction beams in the refinement of the minimization of the R-value.

Figure 2.6 shows schematically how the Multi-slice Least-Squares (MSLS) software package combines the multi-slice algorithm with a least squares refinement to obtain accurate crystal structure parameters.

To minimize the R-value in equation 2.10 or 2.11, one needs to solve a set of equations

)

14

.

2

(

0

i

P

R

i

∀

=

∂

∂

Figure 2.6: The scheme for MSLS refinement. The MSLS software consists of two

main parts. One is the multi-slice method for calculating the diffraction based on the initial structure model. The other is a least squares algorithm for calculating the modification of the structure model based on the difference between experimental and calculated diffraction intensities.

where Pi are the parameters to be refined, such as an atomic coordination, the sample

thickness or the crystal misorientation. As equation 2.14 is not a linear set of equations, it cannot be solved directly. Therefore, MSLS’s least-squares algorithm linearises equation 2.12 by using a Taylor expansion of (Pi, current-Pi, true) (parameter shift) and

keeping only the first terms. This is an algorithm commonly used in crystallographic structure refinement and therefore requires initial values for the parameters Pi. A

parameter shift

s

is calculated using the least squares algorithm using the linear set of equations given as

)

15

.

2

(

ν

=

s

M

where M is a

n

_p

×

n

_p(

n

_pis the number of parameters) matrix with elements

)

16

.

2

(

)

)(

(

i calc g j calc g g ij

p

I

p

I

w

M

∂

∂

∂

∂

=

∑

The vector

ν

in equation 2.15 is given by

)

17

.

2

(

)

)(

(

i calc g obs g calc g g i

p

I

I

I

w

v

∂

∂

−

=

∑

Solving this set of equations provide

s

for all the refined parameters. This allows us to calculate new approximate values of Pi. These are mostly (but not necessarily) better

than the input values, as the shifts may be overestimated for some parameters. This can be controlled by setting appropriate damping factors to reduce the shifts. The new Pi

values can be used to repeat the cycle until the shifts are less than the expected standard deviations, or if another criterion is met.

Before using the MSLS software, a starting structure model has to be constructed, which can be obtained from high-resolution electron microscopy (HREM), direct methods (see section 2.4), or other methods.

2.3 High-resolution electron microscopy and exit

wave reconstruction

7

Figure 2.7: Schematic representation of the through focus exit wave reconstruction.

A series of HREM images are recorded with precisely known focus steps. The images are corrected for specimen drift by cross correlation. With these HREM images and the microscope parameters a computer program estimates a first exit wave, which is subjected to the transfer of the microscope, allowing a comparsion between the experimental HREM images and the calculated ones. This, in turn, allows further refinement of the exit wave. For details see Ref. 7.

As mentioned in the last section, successful structure determination using MSLS requests having a good starting model. HREM is one of the methods that can be used to obtain a starting model. Although normal high-resolution imaging has been widely applied to provide structure models, it should be noted that high-resolution images are distorted by lens aberrations (mostly defocus Δf and the spherical aberration of objective lens Cs). The imperfection of electron optics yields a limited resolution (~1.7

Å for CM300UT) for direct interpretation, provided that the specimen is very thin. Information delocalization in the imaging means that in general, correctly interpreting HREM is not an easy process, even under optimized imaging conditions.

A number of approaches have been developed to solve this problem. One is through focus exit wave reconstruction (TF-EWR), which allows the electron wave to be

obtained at the exit of the specimen, by combining information from a series of high-resolution images with different defocus values, as shown in Figure 2.7. Van Dyck and Kirkland et al originally proposed this method, and now a software package is available to extract all of the useful information from a series of HREM images taken at different defocus values with well-defined defocus increments. This results in a reconstructed exit wave function, containing amplitude as well as phase information up to the microscope’s information limit (which is mainly determined by the energy spread of the electrons and the microscope’s mechanical vibration). The reconstructed exit wave is not dependent on electron microscope optics, and provides quite accurate atomic positions once all imaging aberration has been taken into account. The exit wave is still dependent on the specimen’s thickness, however, and the actual scattering potential at those positions is much more difficult to interpret — as can be easily understood from channeling theory (see section 1.3.2).

2.4 Direct methods

Direct methods allow us to obtain an initial structure model from electron diffraction instead of images. Direct methods were widely developed for single crystal X-ray structure analysis. These methods are based on estimation of the unknown phases of the reflections, which are lost during the diffraction recording. The lost phase information prevents a direct structure reconstruction since phase φgof structure factors

Fg (Fg=|Fg|exp(iφg)) and intensities (Ig = |Fg|2) are both required for retrieving the

structure. This is known as the ‘phase problem’

.

Due to the redundancy of intensity information (a large number of reflections), the phase problem is over-determined and is therefore solvable. This over-determination implies the existence of relationships among the sets of |Fg| and their phases. So-called direct methods are those that exploit

these relationships in order to estimate the phase. The use of direct methods has been very successful in X-ray crystallography. The application of direct methods to electron diffraction has been limited, however, because of dynamical scattering and the fact that typically, a smaller number of reflections are recorded. In the case of dynamical diffractions, all reflections are also dependent on sample thickness in a nonlinear way, such that they lack the one-to-one relationship between the diffraction intensity and the magnitude of the structure factor in electron diffraction. This has led some diffraction physicists to question the viability of ab initio structure determination using direct methods in electron crystallography. However, some pioneers have carried out

successful research into the application of direct methods to the problems of ab initio structure determination using electron diffraction on structures, including Fan(1991 8， 1993 9) and Dorset (1992a 10), among others. As pointed out by Dorset, these successes are partially due to the fact that the solved structures are small, and partially due to the high degree of control over the process, (for instance, that of collecting more kinematical electron diffraction data). When L.D. Marks (1998 11，1999 12) applied direct methods to electron dynamical diffraction, a structure model containing O atoms instead of heavy metal atoms was obtained, clearly indicating the dynamical effect of electron diffraction. It has been pointed out that based on channeling theory, the dynamical scattering effect results in changes in the effective scattering power of the atom columns, but that the projected positions of the atom columns remain unchanged under certain conditions. Once the electron wave exits from the back face of the specimen, the relationship between phases and amplitudes of diffractions is fixed (note that here we refer to the electron diffraction phase, rather than the structure factor phase). Thus direct methods should in principle be applicable to electron diffraction in order to infer phase information. In this case, though, the solved ‘model’ corresponds to the exit wave instead of the direct real structure. Therefore the map generated by direct methods is also thickness-dependent, and only reflects the real structure in a complex way. To obtain the correct atomic type, a later dynamical diffraction calculation may be of use. One example is in the structure determination of Ce5Cu19P12 13_{: after obtaining the projected position of the atoms, a refinement assuming that all}

atoms are Cu can first return better atoms positions, a later refinement of occupancies of each atom indicated three atomic types (the higher occupancy suggests the heavier atoms).

Recently Vincent and Midgely proposed a new method called ‘precision’, which allows one to partly overcome the dynamical effect and generate quasi-kinematical electron diffraction data sets (see Figure 2.8) 14_{. They claim that direct methods can be}

straightforwardly applied to those ‘kinematical’ data. Some structures 15,16,17 _{have been}

successfully solved in this way. Although the validity and limitation of this method are still being researched, we believe that this method of creating quasi-kinematical electron diffraction is not the best solution for structure determination, although if it succeeds, it will have made a valuable contribution by providing a simple and direct way to construct an initial structure model. Accurate structure determination

necessitates dynamical electron diffraction refinement, because some of electron diffraction’s advantages over X-ray diffractions are derived from its dynamical scattering characteristics, such as higher sensitivity to light atoms. Once the dynamical effect has been eliminated, such advantages are also lost.

2.5 Application of chemical information in model

construction

The projected structure model provided by a reconstructed exit wave or generated by direct methods is thickness-dependent due to the dynamical scattering effect. According to channeling theory, peak positions in reconstructed exit waves still reflect those real atom column positions. The intensities of the peaks do not correspond to the atomic weight of the atom columns any more, however, as predicted by kinematical theory. This means that one encounters problems when assigning correct atom types to those indicated positions. This assignment problem can be solved by means of the application of already-available chemical information, such as bond length, bond angle, and the number of nearest neighboring etc. Chemical information is widely used in crystallography. One such application is to check whether a structure model resulting from a structure refinement is reasonable. In fact, chemical information can also assist us with the structure analysis process, as in the case where the unit cell dimension of an unknown oxide structure is determined, and thus the volume of unit cell V is known. For instance, if the volume is ~545 Å3, one can estimate that the number of O atoms in the unit cell should be less than V/18, based on the fact that the volume of O2- should be around ~18Å3. Combined with other information, such as space group, valence, and so forth, one can eliminate certain structure models. Another possible application relates to knowing the interatomic distance: given that from previous chemical experience, the Eu-Eu interatomic distance is known to be not less than 3.3 Å, one can use this as a criterion for checking whether a position fits the atomic basis of the Eu atom in a certain unit cell and space group. Given a basis position P0 in a unit cell with

a known dimension and space group, all other symmetry related positions (P1, P2…Pn)

and then the distance of Pi-Pj (i,j=0,1,2…n) can be easily calculated. The conditions

require that the interatomic distance Pi-Pj is larger than 3.3 Å. If this criterion is fulfilled, it is possible for P0 to be a basis position of the Eu atom; otherwise, P0 is not a

basis position of the Eu atom. Obviously, these possible basis positions fulfilling the requirement of chemical knowledge cannot provide us with a certain basis solution for the Eu atom, but the solution space of Eu is reduced in this way. This may be used to construct a plausible structure model in combination with other structure information, such as that obtained from HREM. For instance, in the case of the structure determination of Eu2CaCu2O6, the assigning problem can be solved using prior

knowledge of the Eu-Eu, Ca-Ca Cu-Cu interatomic distances. In another case of the structure determination of the novel superconductor Mg20Ir38B32, the starting position

of B atoms can be obtained based on known Mg and Ir positions, and prior knowledge of the Mg-B, Ir-B and B-B distances (see Chapters 4 and 5 for more details).

2.6 Quantum mechanical calculations

(first-principles)

18

Given progress in modern computational physics, it is now possible to refine the structure of solids using first-principles calculations, a method that provides a theoretical alternative to experimental diffraction. In comparison with electron and X-ray diffraction refinements, in which the R-value (the mismatch between the experimental and the calculated diffraction intensity) is minimized, a structure relaxation using a first-principles calculation is based on the minimization of the forces between the atoms by relaxation of the atom positions (and if necessary, the unit cell dimensions). One advantage of first-principles calculations is that they are sensitive to light atom positions, since the interatomic forces are primarily set by only the valence electrons. Moreover, apart from the relaxation of atom positions, such calculations also allow for a comparison between the stabilities of postulated models from the calculated formation enthalpy, which may reflect the thermodynamic stability of the system. Since this method provides a completely different criterion for structure determination, it can be used to complement diffraction techniques.

First-principles calculations are based on solving the many-body problem of a collection of heavy, positively-charged nuclei and lighter, negatively-charged electrons. A many-particle Hamiltonian of the quantum mechanics for this system is

2 2 _{2 2} 2 2 2 ^ 0 0 0 1 1 1 _{(2 18)} 2 2 4 8 8 i i R r i i j i i i e i i j i j i j i j i j e Z Z e Z e H M m πε R r πε ≠ r r πε ≠ R R ∇ ∇ = − − − + + − − − −

∑

JJG

∑

JG

∑

∑

∑

= = JJG JG JG JG JJG JJG

where the mass of the nucleus at R is _i M , the mass of electrons at _i r is _i m . The first _e

two terms are kinetic operators for the nuclei and electrons respectively. The last three terms describe the Coulomb interaction between electrons and nuclei, between electrons and other electrons, and between nuclei and other nuclei. Obviously, this

problem cannot be solved exactly. Some approximations at the different levels are required in order to obtain acceptable approximate solutions.

The first is the Born-Oppenheimer approximation, which ‘freeze’ nuclei at fixed positions and assume the electrons to be in an instantaneous equilibrium with them, due to the fact that the nuclei are much heavier and therefore much slower than the electrons. After applying this approximation, the kinetic energy of the nuclei is zero and the first term disappears; the last term reduces to a constant. We are left with the kinetic energy of the electron gas, the potential energy of electron-electron interaction and the potential energy of the electrons in the potential of the nuclei, which can be written formally as H^ e = +T V^_e ^e e− +V^n e− (2 19)−

Thus, by applying this approximation, the structure refinement using the first-principle calculation can be simplified into two-step process. One step is to solve the equilibrium electron states from equation (2-19) from a starting structure model containing all nuclei types and positions (M_i,R ,_i Z ). This is called the electronic step. The other _i

step is to move the nuclei to their equilibrium positions by minimizing the total energy described by the Hamiltonian (2-18). Usually the forces and stress tensors, calculated from solved electron states, are used to determine the search direction for finding equilibrium positions.

It is still not easy to solve the equilibrium electron states from equation (2-19). Several methods exist to reduce equation 2.18 to an approximate but tractable form. For solids, Density Functional Theory (DFT) is currently the most popular and powerful method. According to this method, there is a one-to-one relation between the ground state density_ρ(r) of a many-electron system and the external potential Vn−e

^

. The many-body problem of (2-19) can thus be equivalent to solving the single-particle wave functions (2-20)

^

(2 20)

KS _{i i i}

H φ ε φ= −

And the exact ground state density

_ρ

(

r

)

of an N-electron system is

* 1

( )r N i( ) ( )r i r (2 21)

ρ G =

∑

φ G φ G −

^ ^ ^ ^ ^ 0 ' 2 2 2 _{^ ^} ' ' 0 (2 22) ( ) (2 23) 2 4 i KS H xc n e r xc n e i e H T V V V e r d r V V m _{r r} ρ πε − − = + + + − ∇ = − + + + − −

∑

JG

∫

G G = G G

The third term of the Kohn-Sham Hamiltonian is the exchange-correlation potential, which originates from the many-body interaction problems. The Kohn-Sham scheme described above is exact: apart from the preceding Born-Oppenheimer approximation, no other approximations are made. The exchange-correlation functional, the third term in (2-22), is unknown, however. The evaluation of this term requires another approximation. Some widely used approximations, such as the Local-Density Approximation (LDA), or the Generalised Gradient Approximation (GGA), postulate that the exchange-correlation functional has certain forms related to the ground state electron density

_ρ

(

r

)

, for instance

( ) ( ( )) (2 24)

LDA

xc xc

E =

∫

ρ ε ρrG r d rG H −

Therefore equation (2-23) becomes solvable. Although no law of nature guarantees that these approximations for the form of Exc are true, they appear to be quite accurate in

many realistic cases.

There are generally two ways to solve equation (2-23) for solids. One is the pseudopotential method, by which only the states of outer electrons (valence electrons) will be accurately determined. This is because core electrons are taken to be the same as free atoms, due to the fact that the former do not interact with the electrons of other atoms and will not be greatly affected by the formation chemical bonding. Another is the full potential method, whereby the states of all electrons are calculated. Obviously, the pseudopotential method requires less computational power, whereas the full potential method provides more accurate results for the electronic structure. For a structure refinement, however, the fine electronic structure of core electrons is not of interest, so the pseudopotential method proves to be the more convenient choice. In this thesis, all of the first-principles calculations were done using one commercial pseudopotential method code: the Vienna ab-initio simulation package (VASP), using pseudopotentials 19,20,21,22,23_.

References

1 _{B.K.Vainshtein, Structure Analysis by Electron Diffraction, Oxford, Pergamon,1964} 2 _{D.L.Dorset, Structure Electron Crystallography, New York, Plenum Press, 1995} 3 _{B.F.Buxton et al, Philos. Trans. R. Soc. London, 281, 171, 1976}

4 _{M.Tanaka, Acta Cryst. A50, 261}

5 _{D.B.Williams, and C.B.Carter, Transmission Electron Microscopy, New York,}

Plenum Press, 1996

6 _{J.Jansen, D.Tang, H.W.Zandbergen, and H.Schenk, Acta Cryst. A54, 91 1998} 7 _{H.W.Zandbergen, and D. van Dyck, Microscopy Research and Technique 49, 301,}

2000

8 _{H.F.Fan, Direct methods of Solving Crystal Structures, New York, Plenum Press,}

265, 1991

9 _{H.F.Fan, Modern Crystallography. (Proceedings of the Seventh Chinese}

International Summer School of Physics. Beijing International Workshop. 1993)

10 _{D.L.Dorset, Ultramicroscopy 41, 349, 1992}

11 _{W.Sinkler, E.Bengu, and L.D.Marks, Acta Cryst. A54 1998} 12 _{W.Sinkler, and L.D.Marks, Ultramicroscopy 75, 251 1999}

13 _{J.Jansen, D.Tang, H.W.Zandbergen, and H.Schenk, Acta Cryst. A54, 91 1998} 14 _{R.Vincent, and P.A.Midgley, Ultramicroscopy 53 271 1994}

15 _{A.Kverneland, V.Hansen, R.Vincent, and Gjonnes, Ultramicroscopy, 106, 492, 2006} 16 _{C.S.Own, A.K.Subramanian, and L.D.Marks, Micros. and Microanal. 10 96 104}

2004

17 _{C.S.Own, L.D.Marks, and W.Sinkler, Ultramicroscopy 106 114 2006}

18 _{S.Cottenier, Density Functional Theory and the Family of (L)APW-Methods: A} Step-by-Step Introduction 2002

19 _{G.Kresse, and J.Furthmüller, Comput Mater Sci 6: 15. 1996} 20 _{G.Kresse, and J.Furthmüller, Phys Rev B; 54: 11169. 1996} 21 _{G.Kresse, and J.Hafner, J Phys Condens Matter 6:8245. 1994} 22 _{P.E.Blöchl, Phys. Rev. B 50, 17953 1994}

samples for hard ceramic

powders

*

Abstract

It is difficult to prepare a good high-resolution electron microscopy sample using conventional method for polycrystalline ceramic powders containing hard particles or particles with a strong preferential cleavage. In this chapter we show how a Cu matrix in a pressed pellet can be used to overcome this difficulty. To demonstrate its feasibility, the method is applied to Mg10Ir19B16 and Na0.5CoO2. Experimental results

show that TEM samples with crystalline areas thinner than 10nm can be obtained easily.

3.1 Introduction

In the early stages of the search for new compounds, one often has a reaction mixture consisting of multi-phase poly-crystalline powder, of which only one phase will have interesting properties. For instance, the YBa2Cu3O7, Bi2Sr2CaCu2O8+δ, LaNiBN1,2,3

superconducting materials were discovered in reaction mixtures with other superconducting and/or non-superconducting phases. In such cases, TEM provides a quick and reliable tool for obtaining crystallographic and compositional information for interesting phases.

In order to obtain successful experimental TEM results, it is essential to have good samples that contain all the phases and all the relevant crystallographic orientations of each phase in electron transparent areas. This requirement is often not met, however, particularly in the case of hard ceramics, for which only a small number of polycrystalline powders are available for characterization. For such powders, the possible methods for TEM sample preparation can be categorized into three main groups:

Crushing: this is the simplest sample preparation method for powders. The powders are usually crushed in air or in a protective environment, after which a suspension is made (in ethanol or hexane, for instance). Next, a droplet of this suspension is deposited on a carbon-coated, holey film on a Cu or Au grid. Unfortunately, this method may fail to yield electron-transparent samples for at least part of the phases present. Another drawback of the crushing method is that it produces preferential orientations when dripping the crushed material with a preferential cleavage plane (e.g. NaxCoO2) onto

the flat carbon film. As a result, other crystallographic orientations (for example, the orientation 90° away from the preferential orientation) can hardly be reached using normal TEM due to the limited tilt angle (generally about ±30º). A third potential disadvantage of this crushing method is that the overall phase distribution in the suspension might be very different to that in the initial sample.

Cutting using ultramicrotomy: the powders are embedded in an epoxy resin, and the section is cut once the epoxy resin has completely hardened. The hardness of epoxy resin is much lower than that of hard ceramics, however, such that the harder particles can be easily chipped out as a whole from the epoxy matrix, instead of being cut into thin sections. It thus requires considerable effort to identify the optimum conditions (including the operating temperature, matrix materials, and so forth) for using this method to prepare suitable samples of thickness than 10nm for HREM research 4, 5_.

Ion milling: in this method, a pellet of the material is required, which is often obtained by pressing the powders. The rough surfaces of hard powders prevent close-packing of the particles, however, resulting in the presence of numerous voids in the pressed pellets. Generally the fraction of such voids can be 20%~40% 6_{, which, in combination}

with poor adhesion, results in easy collapse during later ion milling. One solution is to fill the voids by adding organic glues to the pellets. However, since the ion-milling rate of organic materials is much faster than that of hard ceramics, the organic part of the pellet will be completely removed before a thin area has been properly created in the ceramic powders, resulting in the disintegration of the thin component. Organic adhesives also add contamination to TEM investigation.

In this chapter, we outline an alternative approach to the preparation of TEM samples using hard ceramic powders, based on the idea of using an easily deformable inorganic material with a low ion-milling rate (for example Cu) to fill the voids between the ceramic grains during the pressing of the pellets. Fine Cu powders (radius ~ 5µm) were chosen for this application. Cu particles can be easily deformed in the process of pressing to fit the rough surfaces of hard ceramics, resulting in good contact for all