X-Ray Tomography in Material Science

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X-Ray Tomography

in Material Science


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ERMES Science Publications Paris 2000

HERMES Science Publications

8 quai du MarchC-Neuf

75004 Paris

Serveur web

http://www.hermes science.com

ISBN

2-7462-0115-

All rights reserved. o part of this publication may be reproduced stored in

a

retrieval

system or transmitted in any form or by any means electronic mechanical photocopying

recording

or

otherwise without prior permission in writing from the publisher.

Disclaimer

While every effort has been made to check the accuracy of the information in this book no

responsability is assumed by Author o r Publisher for any damage or injury to o r loss of

property or persons as a matter

of

product liability negligence or otherwise

or

from any

use

of materials techniques methods instructions or ideas contained herein.


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X-Ray

Tomography

in

Material

Science

Jose Baruchel

Jean-Yves Buffiere

Eric

Maire

Paul Merle

Gilles

Peix

•cience


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his page intentionally left blank


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Authors

A N D E R S O N P., Department of Biophysics in Relation to D entistry, St Bartholomew's

and

The

Royal London Scool

of

Medecine

and

Dentistry, Queen Mary

and

Westfield College, Mile End Road, London, E l 4NS, U K

B A B O T D., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex,

France

B A R U C H E L

J.,

European S ynchrotron Ra diation F acility,

B P

220, F-38043 Grenoble,

France

BELLET

D., Laboratoire GP M2,

INPG,

BP 46, 38402

Saint-Martin-d'Heres

B E N O U A L I A.-H., Department of Metallurgy and Materials Engineering, Katholieke

Universiteit Leuven, D e Croylaan 2, B-3001 Heverlee, B elgium

B E R N A R D

D .,

ICMCB, CNRS, 87 avenue du docteur Albert Schweitzer,

33608

Pessac, France

B L A N D I N

J.-J., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46,

F-38402 Sa int-Martin-d'Here

Cedex, France

B O L L E R E., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble,

France

B O U C H E T

S.,

Ecole

des

mines

,

ENSMP,

35 rue St

Honore,

77300

Fontainebleau,

France

B R A C C I N I

M.,

Genie physique

et

mecanique

des

materiaux, ENSPG-UJF,

BP 46,

F-38402

Saint-Martin-d'Here Cedex, France

BUFFIERE

J.-Y., GEMPPM INSA Lyon, 20 avenue Albert Einstein, 69621

Villeurbane

Cedex, France

CLOETENS P ., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble,

France

D A V I S

G .,

Department

of

Biophysics

in

R elation

to

Dentistry,

St

Bartholomew's

and

The Roy al London Scool of Medecine and D entistry, Queen Mary and W estfield

College, Mile

End

Road, London,

E l

4NS,

U K

D E G I S C H E R H.P., Institute

of

Materials Science

and

Testing, Vienna University

of

Technology,

Karlsplatz 13 , A-1040 W ien


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6

X-ray tomography

in

material

science

DERBY

B.,

Manchester Materials Science Centre, UMIST

and the University of

Ma nchester, G rosvenor Street, M anchester, M l 7HS,UK

DUVAUCHELLE

P ., Laboratoire CNDI , INSA, Batiment 303, 69621 Villeurbane

Cedex, France

ELLIOTT

J.,

Department

of

Biophysics

in

Relation

to

Dentistry,

St

Bartholomew 's

and The

Royal London

Scool of

Medecine

and

Dentistry, Queen Mary

and

Westfield College, Mile

E nd

Road, London,

E l

4NS,

U K

FOROUGHI

B .,

Institute

of

Materials Science

and

Testing, Vienna University

of

Technology, Ka rlsplatz 13, A-1040 W ien

FREUD N., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex,

France

FROYEN

L .,

Department

of

Metallurgy

and

M aterials Eng ineering, K atholieke

Universiteit L euven, De Croylaan 2,

B-3001

H everlee, Be lgium

GuiGAY

J .-P., University

of

Antwerp, RUCA Groenenborgerlaan 171, B-2020

Antwerp,

Belgium

HEINTZ J .-M.,

ICMCB,

C NR S , 87 avenue du docteur Albert Schweitzer, 33608

Pessac, France

JOSSEROND

C., Genie physique et

mecanique

des materiaux, ENSPG-UJF, BP 46,

F-38402


JUSTICE I., Department

of

Materials, University

of

O xford, Parks

Rd,

Oxford ,

O X 1

3PH, UK

KAFTANDJIAN

V .,

Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane

Cedex, France

KOTTAR A ., Institute of Materials Science and Testing, Vienna University of

Technology, Karlsplatz

13,

A-1040 W ien

LUDWIG

W., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble,

France

MAIRE E., GEMPPM INSA Lyon, 20 avenue Albert Einstein,

69621

Villeurbane

Cedex, France

MARC

A.,

LETI-CEA/Grenoble, 17 rue des martyrs, 38054 Grenoble Cedex 9,

France

MARTIN C.F., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46 ,

F-38402


PEK

G ., Lab oratoire CN D I, INSA , Batiment 303, 69621 V illeurbane Cedex, France

PEYRIN

F .,

CR EATIS,

INSA-Lyon ,

69621 Villeurbane, France

ROBERT-COUTANT C.,

LETI-CEA/Grenoble,

17 rue des ma rtyrs, 38054 G renoble

Cedex 9,

France

SALVO

L .,

Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46,

F-38402

Saint-Martin-d'Here

Cedex,

France

SAVELLI

S., GEMPPM INSA Lyon , 20 avenue A lbert Einstein, 69621 Villeurba ne

Cedex, France

SCHLENKER M., C N RS, Laboratoire L ouis

N eel,

BP 166, F-38042 Grenoble, France

SUERY M., Genie physique

et

mecanique

des

materiaux, ENSPG-UJF,

BP 46,

F-38402

Saint-Ma rtin-d'Here Cedex, France


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Authors 7

VAN

DYCK D., University of A ntwerp, RU CA G roenenborgerlaan 171,

B-2020

Antwerp, Belgium

V E R R I E R S., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46,

F-38402


VIGNOLES G.-L., LCTS, CNRS-SNECMA-CEA, Universite Bordeaux

1, 3

allee

La

Boetie, F-33600

Pessac, France

WEVERS

M., Department

of

Metallurgy

and

Materials Engineering, Katholieke

Universiteit Leuven,

De

Croylaan

2 ,

B-3001 Heverlee, Belgium


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10/209

Table of contents

Foreword

13

Chapitre

1.

General principles

G . P E I X , P .

D U V A U C H E L L E ,

N .

F R E U D

1 5

1.1. Introd uction 15

1 . 2 . X and gamm a-ray tomography : physical basis 16

1.3.

D ifferent

scales,

different

applications 20

1 . 4 .

Q untitative tomography 23

1.5

C onclusion 26

1 . 6 .

R eferences 26

Chapitre

2. Phase

contrast tomography

P .

C l o E T E N S ,

W .

L U D W I G ,

J.-P .

G U I G A Y ,

J .

B A R U C H E L ,

M . S C H L E N K E R ,

D.

V A N D Y C K

29

2.1.

Introduction

29

2.2. X-ray phase modulation

30

2.3. Phase sensitive imaging methods

32

2.4. Direct imaging

38

2.5. Q uantitative imag ing 38

2.6. Conclusion

42

2.7. References

43

Chapitre 3. Microtomography at a third generation

syncrotron

radiation

facility

J. B A R U C H E L , E. B O L L E R , P. C L O E T E N S , W. L U D W I G , F. P E Y R I N 45

3.1. Introduction 45

3.2. Syncrotron radiation

and

m icrotomography

46


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10 X-ray tomography

in

material science

3.3. Improvement

in the

signal

to

noise ratio

in the 3D

images

49

3.4. Imp rovem ent in the spatial resolution 50

3.5. Q uantitative measurement (absorption case) 51

3.6. Present state

of

"local" tomography

53

3.7. Sample env ironmen t in microtomography 54

3.8. Phase Imaging 55

3.9. O ther new approaches in microtomography 56

3.10. Conclusion 57

3.11. R eferences 57

Chapitre

4. Introduction to reconstruction

methods

C.

RO BERT-COUTA N T,

A.

MA RC

61


4.2. Description

of

projection m easurements

62

4.3. B ackprojection

65

4.4. P rojection-slice theorem

66

4.5. Fourier reconstruction methods 67

4.6. Filtering in Fourier methods 69

4.7. ART-type methods 70

4.8. Conclusion

74

4.9. References

74

Chapitre

5.

Study

of

materials

in the

semi-solid

state

S . V E R R E E R , M . B R A C C I N I , C . J O S S E R O N D , L . S A L V O , M . S U E R Y , W . L U D W I G ,

P. C L O E T E N S , J. B A R U C H E L 77


5.2. Experimental device and procedure 79

5.3. Results on Al-Si alloys 80

5.4. Results on

Al-Cu

alloys 85

5.5. Conclusion

and

perspectives

86

5.6. References ,...

87

Chapitre

6. Characterisation of

void

and

reinforcement distributions

by edge contrast

I .

J U S T I C E ,

B .

D E R B Y ,

G .

D A V I S ,

P .

A N D E R S O N ,

J .

E L L I O T T

8 9

6.1. Intro du ction 89

6.2. Dual energy X-ray microtomography 90

6.3. Experimental materials 92

6.4. Results and discussion 94

6.5. Conclusions 100

6.6. References 101


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Table of

contents

1 1

Chapitre 7. Characterisation of MMCp and cast A luminium alloys

J . - Y . B U F F I E R E , S . S A V E L L I , E . M A I R E 1 0 3

7.1. Introduction

103

7.2. Experimental methods 104

7.3. Results and discussion 107

7.4. Conclusion 112

7.5. References

113

Chapitre 8. X-ray tomography of Aluminium foams and Ti/SiC composites

E. M A I R E , J .- Y . B U F F I E R E 115

8.1. General introduction

115

8.2. A luminium foams

116

8.3. Titanium composites

121

8.4. G eneral conclusion 124

8.5. References 125

Chapitre 9. Simulation tool for X-ray imaging techniques

P. D U V A U C H E L L E , N. F R E U D , V. K A F T A N D J I A N , G. P E I X , D. B A B O T 127


9.2. Background

128

9.3. S imu lation possibilities

129

9.4. Simulation

examples

in tomography 13 2

9.5. C onclusions and future directions 135

9.6. References 136

Chapitre 10. Micro focus computed tomogrgraphy of Aluminium foams

A.-H.

BENAOULI, L. FROYEN, M. WEVERS 139


10.2. Production

process

of

A luminium foams

140

10.3. Mechanics

of

foams

142

10.4. N on-destructive investigation of A lum iniu m foams 144

10.5. Conclusion

151

10.6. References 152

Chapitre

11. 3D

observation

of

grain boundary penetration

in

Al alloys

W .

L U D W I G ,

S.

BOU C HE T,

D .

BELLET, J.-Y. BUF FIERE

.*. 15 5

11.1.

Introduction

155

11.2. Experimental set-up

15 7

11.3. Result

158

11.4.

C onclusions

160

11.5.

References

163


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12 X-ray tomography

in

material science

Chapitre 12. Determination of local mass density

distribution

H . P . D E S I S C H E R ,

A . K O T T A R , B .

F O R O U G H I

1 6 5

12.1. Introduction

165

12.2. Material 166

12.3. X-ray rad iography

166

12.4. R esult

168

12.5. Application of the mean local density distribution 17 2


Chapitre 13. Modelling porous materials evolution

D .

B E R N A R D ,

G .-L.

V I G N O L E S ,

J.-M.

H E I N T Z

1 7 7


13.2. E volution of sand stone reservoir rocks by pressure solution

179

13.3. C-C 185

13.4. Ceramics sintering

187

13.5. Conclusions a nd forthcoming w orks 190


Chapitre 14.

Study

of damage during

superplastic

deformation

C . - F . M A R T I N ,

J.-J.

B L A N D I N ,

L .

S A L V O ,

C .

J O S S E R O N D ,

P . C L O E T E N S ,

E.

B O L L E R

193

14.1. Introduction to damage in superplas ticity 193

14.2. U sual techniqu es

of

characterisation

197

14.3. E xperimen tal procedure

198

14.4. X-ray microtomography results

199

14.5. Quantification of the

coalescence

process 200

14.6. Conclusions 203



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Foreword

This book collects

the

texts

of the

lectures given during

the

Workshop

on the

application ofX Ray tomography in material science which wa s organised by the

G roupe d'Etudes

de

M etallurgie P hysique

et de

Physique

des

Materiaux (GEMPPM)

in V illeu rba nne on O ctober 28-29 1999. R esearchers from several European

universities, research centres

and

companies attended

the lectures

which were given

by experts in both materials science and X-ray tomography. The workshop was

subsidised

by the

INSA Lyon,

the

MMC

Assess

european network

and the

Region Rhone

Alpes and we

w ould like

to

acknow ledge their support.

The

scope

of this European workshop was to provide material scientists with a

detailed presentation of X-Ray tomography techniques, including the latest

developments, and to present recent applications of these techniques in the field of

structural materials.

The interest of material scientists in X ray tomography arises

from

two facts:

1)

most structural materials are opaque, and 2) it is of very crucial importance to

observe what occurs in the bulk of materials when they are subjected to a

mechanical loading.

The

apparent contradiction between

these

tw o

facts

ha s

been

overcome by recent progress in X Ray tomography which has allowed 3D non

destructive images of structural materials, with a resolution around 1 micron, to be

achieved. Synchrotron radiation sources are necessary to record these very high

resolution images. Moreover,

the

phase contrast images, easily obtainable with

X

ray sources emitting photons with a high

spatial

coherence, even permits the

visualisation of features with weak attenuation differences. This technique is

especially well adapted for studying metal matrix composites which are among the

most promising structural materials and for which damage development under

stress

is of

crucial importance.

Within

this framework, the workshop was divided into two parts. The

first

one

included

a

global description

of the

technique itself,

an

introduction

to the


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14

X-ray tomography in ma terial science

reconstruction algorithms, and an overview of the new possibilities offered by

synchrotron X ray sources with an emphasis on the phase contrast images. The

second part was devoted to the presentation of some examples of the application of

X - R a y

tomography

to

investigating micro-heterogeneous structural materials.

The

use

of synchrotron and laboratory X-Ray sources w as illustrated.

The workshop was a stimulating event which has given scientists with various

backgrounds the opportunity to discuss and exchange ideas and experiences. We do

hope that this book will bring useful information

to

material scientists looking

for

new

characterisation methods in their research fields.

The

organisers,

Jose Baruchel

Staff Scientist, Group Leader ESRF

Jean-Yves

Buffiere

Maitre de conferences INSA Lyon

Eric Maire

Charge de recherches INSA Lyon

Paul

Merle

Professeur INSA Lyon

Gilles Peix

Maitre de

conferences INSA Lyon


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Chapitre

1

General principles

Among the different methods allowing to obtain, in a non-invasive way, the

image

of a

slice

of

matter within

a

bulky object, X-ray transmission tomography

is

widely used

in

both

the

medical

and the

industrial fields.

In the

latter case, defect

detection, dimensional inspection as well as local characterization are possible.

Non

destructive testing,

process

tomography

and

reverse

engineering

are

thus

feasible. A

w ide range

of

sizes

can be 1 mm

small inspected, starting from

a

sample,

up to a whole rocket motor (several meters in d iameter). The present paper describes

the physical basis and give examples of some industrial applications. The main

reconstruction

a rtifacts are described.

1.1. Introduction

Tomography

is

referred

to as the

quantitative description

of a

slice

of

matter

within a bulky object. Several methods are available, delivering specific images,

depending on the

selected

physical

excitation:

- ultrasonics,

- magnetic field (in the case of nuclear magnetic resonance imaging),

- X and gamma-rays (y rays) ,

-

electric

field (in the

case

of

electrical imped ance

or

capacitance tomograp hy).

In the

field

of

indus trial non-de structive testing

(NOT), as

w ell

as in the field of

materials characterization, X -ray or

y-ray

tomography is mostly used today.

Tom ography is a relatively "new" technique. The very first images w ere obtained

in 1957 by Bartolomew and Casagrande [BAR 57]: they characterized the d ensity of


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16 X-ray tomography in material

science

particles

of a

fluidized bed, inside

a

steel-walled riser.

The

first medical images

were performed by H ounsfield in 1972, and most indu strial applications w ere

developed much later, in the 1980's. This slow development can be explained by the

huge amount of data to handle, and thus by the need for high speed and high

memory computers. Industrial benefits of what is called computed tomography (CT)

today are numerous. This is due to the wide range of potential applications, starting

from the small sample, 1mm in size, dedicated to the characterization of advanced

composite materials, and displayed in three dimensions with a one micrometer (urn)

voxel size, up to the single

slice

image, across a 1 meter diameter riser, with a five

centimeter pixel size.

1.2.

X and gamma-ray

tomography:

physical basis

1.2.1.

D ifferent

acquisition set ups

The simplest set-up consists in detecting the photons which are transmitted

through the investigated object (Fig. 1.1): transmission tomography delivers a map

of u, the linear attenua tion coefficient, q ua ntity wh ich is in turn a

function

of p (the

density) and Z (the atomic number).

Figure

1.1. X-ray transmission tomography

The clear separation between p and Z implies to perform either bi-energy

tomography or scattered photons tomography [ZHU 95, DUV 98] (Fig. 1.2). This

last technique is based on the clear differentiation between Compton and Rayleigh

scattered photons. The ratio between those tw o measured quanti t ies is purely

proportional

to Z and is not

affected

by the

density.

The

third possibility

is to

detect photons emitted

by the

investigated object itself.

Such is the

case

when gamma-ray sources are distributed inside a nuclear waste

container, for instance. Emission tomography is thus performed [THI 99] (Fig. 1.3).

A n alternative

is

encountered when

the

distributed source

is a

positon emitter:

the


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General principles 17

local positon annihilation delivers pairs of 0.51 M eV annihilation photons which are

detected outsid e. This is the PET tech niqu e, used in the medical field.

Figure

1.2.

Scattered photons tomography Figure

1.3.

Emission tomography

1.2.2. X ray transmission tomography

The present paper

will

be focused on transmission tomography, which is widely

used in both industrial and medical fields. It is based on the application of equation

[1], known

as the

Beer-Lambert law,

or

attenuation law. Figure

1.4

describes

the

basic experimental set-up

for

transmission tomography inside

a

single

slice.

N

l =

A r

0

e x p [ - v ( x , y i ) d x ] [ 1 ]

path

Measuring the num ber N

0

of photons emitted by the source and the nu mb er N, of

photons transmitted throughout a single line across the sample allows to calculate

the integral of ja along the considered path:

[2]

N

l path

The term ( ( x , y )

represents

the value of the linear attenuation coefficient at the

point (x,y). Repeating such a measurement along a sufficient number of straight

lines

within

the same slice delivers the Radon transform of the object. Radon

demonstrated in 1917 the po ssibility to

find

an inverse to that transform and thus to

reconstruct the n( x, y) map of the slice [KA K 87].


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18 X-ray

tomography

in

material

science

As industrial tomography makes

frequently

use of an X-ray generator, we wil l

focus our discussion on that kind of experimental set-up. Nevertheless, some

comments

wil l be

made

on

gamma-ray tomography.

Figure 1.4. Physical basis of transmission tomography inside a slice

f

1.2.3. The linear

attenuation

co efficient

Transmission tomography delivers

a map of

(x,y),

the

linear attenuation

coefficient, which is correlated to i) the photon energy E, ii) the density p and iii

the atomic number Z of the investigated material . Figure 1.5 displays the

dependance between those quantit ies for carbon (Z=6) and iron (Z=26). It must be

noticed that the quantity displayed on Fig. 1.5 is in fact the ratio /p, the mass

attenuation coefficient.

Figure 1.5. Value of the

m ass attenu ation

coefficient for

carbon

and iron

Tw o

main domains appear

in

Fig. 1.5. Below

200

keV,

the

photoelectric

effect

dominates

and

jj/p

is

sharply dependant

on E and Z.

Equat ion

[3] is often

used

to

describe this behaviour [ATT 68]:


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General principles

19

[ ]

where K is a consta nt. Such an equ ation implies that, for any given pho ton energy ,

is

proportional to p and to Z

4

. Performing images in the photoelectric domain

implies

two

main characteristics:

- a comparison of p between two areas of the object (or between two objects)

can

be

achieved only

in the

case when

Z is

constant (same atomic element

or

same

composition),

- a

change

in p

between

two

areas

can be

cancelled

by a

change

in Z in the

opposite direction.

It thus appears that a clear separation between Z and p can not be obtained, in

the photoelectric domain, unless two tomographic images are performed, using two

different

energies.

Within the Compton domain, above 200 keV, u can be considered as weakly

dependant on Z and on photon energy. Tomography thus delivers an information

which is

nearly proportional

to p.

However ,

due to the

higher photons energy,

and

hence

to the

lower value

of u, the

contrast within

the

object image

is

lower,

as can

be

derived from Beer-Lamb ert law.

1.2.4.

D ifferent

experimen tal set ups

In

th e field of industrial tomography, three

different

configurat ions are mainly

encountered. They are displayed on figu re 1.6.

Figure

1.6. D i ff e r e n t expe rimental set-ups

in the field

of industrial tomography:

a) first

generation scanner, b fan-beam scanner, c)

cone-beam scanner


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20 X-ray tomography in material science

Figure 1.6.a corresponds

to the

simplest experimental set-up.

A

single sensitive

element

is

used

and a

rather long scanning time

is

needed,

as the

acquisi t ion

of a

single linear "projection" needs a set of elementary translations. Successive

projections are then acquired, corresponding to

different

value of the angle of

rotation.

A

half turn

is

sufficient

to

reconstruct

the

image

of a

slice. Figure 1.6.b

implies the use of a linear array. Acquisition is shorter, as a w hole linear projection

is acquired at a time. A complete turn is needed since the beam diverges. Figure

1.6.c makes

the best use of the

X-ray cone-beam;

one

turn

of the

object

is

needed.

The Feldkamp algorithm [PEL 84] allows the direct 3D reconstruction of the w hole

object.

1.3. D ifferent

scales,

different applications

1.3.1. Industrial tomogra phy

The main application in the field of X-ray tomography is Non-Destructive

Testing (NOT) of manufactured components, i.e. detection of internal defects.

Among other issues there

are i)

"reverse engineering", whose purpose

is the

geometrical inspection of a component, in such a way to assist the design, ii) local

characterization

of

materials (density measurements,

for

instance)

and

Hi)

process

tomog raphy , able to deliver some kind of control on a con tinuou s ma nu factu ring

process. A s industrial applications involve a broad range of sizes and a great variety

of materials to be inspected, the corresponding devices may be very different.

1.3.1.1. Different photon sources

Inspection of small components can be performed using a standard industrial X -

ra y

tube

(160

kV for instance). Much attention must be paid to the stability of both

the

high-voltage

and the anode current, because the consecutive projections must be

acquired within constant conditions. A focus size within the range 1 to 3 mm is

acceptable. Inspecting heavier components may require a 450kV tube, or even a

linear

accelerator.

Tw o different high capacity scanners were constructed by the

french Atomic Energy Commission

(CEA-LETI,

Grenoble).

A 420 kV

X-ray

generator

in the

first case

and an 8 MeV

linear

accelerator in the

other case allow

the complete inspection of a whole (empty) rocket motor, up to 2.3 meters in

diametre,

of a

nuclear waste container

or of a

w hole

ca r

engine.

Gamma-ray sources

can be

used,

in

spite

of the

very

low

emitted photon flux.

The Elf Research Centre (Solaize-France) uses a cesium 137 source with an activity

up

to 18 GBq

(gigabecquerels).

The

high monochromatic energy (662 keV)

delivered

by the

source allows

to map the

density

of

solid particles inside

a fluidized


22/209

General principles 21

bed, through the steel wall of the riser (0.85 meter in diametre). A single source and

a single

detector (Nal)

are used, thus constituting a first generation tomograph, as

shown in Fig. 1.6a. The scan lasts 3 hours [BER 95]. The University of Bergen and

the

Norsk-Hydro Company built

a

static device using

a set of

five

americium

241

sources (energy: 60 keV) distributed around a pipe [JOH 96]. A linear array

comprising

17

semiconductor detectors

is set

opposite

to

each source, allowing

a

near real-time

(0.1

second) imaging

of the

slice.

The

purpose

is to

visualize

the

liquid components (oil, water) apart from gas within a pipe. This application is an

example of process tomography, i.e. fast imaging dedicated to the control of a

manufacturing process.

1.3.1.2.

Different

families

of

detectors

Four ma in families

of

detectors

can be

found:

1. gas ionisation detectors were used in the early medical scanners. They are still

in use today in some industrial applications. Their main characteristic is their high

dynamic

range. Filled with gas having a high atomic number, they can be used even

with high energies. Linear arrays are av ailable.

2.

image intensifiers (I.I.) are used in

"desktop"

scanners for industrial NDT of

small components. Their low d yn am ic range and the inherent distortion of the image

need some care. Significative 3D images can nevertheless be obtained.

3.

scintillation detectors, composed

of a

fluorescent material (e.g. gadolinium

oxysulphide

Gd

2

O

2

S,

or caesium iodide Csl) are nowadays widely used. Those

detectors are of two kinds: i) the fluorescent material is directly coupled to an array

of photodiodes [KAF

96] or of

photomultipliers

(in

some cases

the

coupling

is

realized using tapered optic fibers), ii) the fluorescent material is spread on a screen,

which is optically coupled to a CCD camera v ia a lens [CEN

99].

4. arrays of semiconductors (e.g. CdTe or Z nCdTe), w hich allow a direct photo n

detection

are

promising. H igh energy applications

are

possible.

1.3.2.

Microtomography

Considering advanced materials characterization, the need of 3D images with a

very

h igh resolution

(a few

um ) obtained through

a non

invas ive method

is

growing.

Figures 1.7 and 1.8 show two specific examples of 3D tomographic images

performed with tw o different scanners, conceived and built in our laboratory [KAF

96]

[CEN

99].

Such

3D

images

are

then used

by the

researchers

for the

modelisation

of

the mechanical properties of materials, within

finite elements models

computations. For such applications micro-focus X-ray tubes, with a focus size in

the range

5 - 1 0

micrometers, are used. A very low focus size allows to set the

investigated object directly at the window of the tube. A geometrical magnification


23/209

22 X-ray tomogra phy in material science

can thus be obtained. Figure 1.9 shows that th e

magnification

can be easily

modified.

A

limit

exists to the

magnification:

the geometrical unsharpness

[HAL

92]

m us t

be

kept

lower than p, the

size

of the sensitive element of the

detector

(sampling

step). In

practice, this upper

boundary to the magnification G

g

can be

computed

according

to equation [4],

where

represents the size of the focus:

[ ]

Figure

1.7.

3D

rendered view

of a

tomographic image of a composite material

with 400 yon glass balls inside an organic

matrix.

(Herve

Lebail; Laboratory

G EMP P M) .

The voxe l size is set to 42 jjm

Figure 1.8.

3D rendered view of a

tomographic image of an aluminium foam

(density

0.06)

(Eric

M aire; Laboratory

G EMP P M) .

The voxel size is 150 pm. The

size of

the

sample

is 3cm

Figure 1.9. According

to the

location

of the

inve stigated

object

between

the

focus

and the

screen,

d i f f e r e n t

geome trical magnifications

are

attained


24/209

General principles

23

Designing and b uilding such a kind of scanner implies some care in at least three

domains:

- the low photon

flux

delivered by the micro-focus X-ray tube results in long

exposure times; the camera must therefore deliver a very low noise,

- the choice of the photon energy is important: low energy photons deliver

images w ith an higher contrast, but also with an higher relative noise,

- the accuracy of the mechanical setting mus t be better than the expected image

resolution.

Today, the most powerful tool involves the use of synchrotron radiation. The

European Synchrotron Radiation Facility (ESRF-Grenoble) delivers

a

huge X-ray

flux

and thus allows very short exposure times. A complete scan can be acquired

within a few minutes, with a spatial resolution down to 1 um. O n beam-line ID 19,

the source is located far

from

the working hutch (145 meters), thus delivering

photons

with

a

high spatial coherence. This property

of the

X-ray

flux

generates

diffraction features which underline

the

edges within

the

sample,

and

thus

highlighting sharp defects. Such

a

phenomenon,

the

so-called "phase contrast" [CLO

97], allows very small defects

to be

detected.

As the

beam

is

non-diverging,

the

resolution is set by the detector itself. Transparent luminescent screens are used,

with a 5 jam sensitive layer of an yttrium-aluminium (YAG) or lutetium-aluminium

(LUAG) garnet , epitaxially grown on a YAG monocrystal , 170 jim in thickness;

they allows

a

high resolution

(1

fim)

and a 4% to 8% efficiency for 14 keV

photons .

1.4.

Quantitative

tomography

As mentionned

earlier,

tomography offers many possibilities. If the goal is just

defect detection,

the

selected resolution m us t therefore

be

adjusted

to the

size

of the

details to be observed. Much attention must also be paid to the noise of the camera

or, more precisely, to its dynamic range [CEN 99]. When the inspection's issue is

the determination of the

accurate size

of some internal feature, or the

local

characterization of ma terials (dens ity measu rement for instance), then an increased

attention must

be

paid

to the reconstruction

artifacts. They create artificial patterns

inside

the reconstructed

slice

(streak artifacts), or they locally modify the

pixels

values (cupping effect),

and

hence

the

quantitative result [ISO

99]

[SCH 90].

In the

following lines, we will describe the main physical mechanisms leading to

erroneous reconstructions, as well as the shape of the corresponding artifact in the

reconstructed image.

- Beam hardening

As an X-ray tube delivers a polychromatic spectrum, differential attenuation of

photons within the investigated object leads to the rapid attenuation of the lowest


25/209

24

X-ray tomography

in material science

energy photons,

and

hence

to the gradual

increase

of the

mean energy along

the

path. The reconstruction algorithm uses, for the reconstruction of any single point,

experimental data corresponding

to

individua l rays impinging

the

point

of

interest,

but

coming from

different

orientations. The corresponding information therefore

corresponds

to

different attenuations,

and

hence different energies,

and

different

values of ji. Two kinds of artifacts are generated by beam-hardening: i) cupping

effect and ii) streaks. Cupping

effect

corresponds to measured values of \ J L which are

corrupted, thus preventing the measurement of the "true" density. As the measured

values, inside

an

homogeneous sample,

are

lower

a t the

center than

a t the

edges,

the

name

of

cupping effect

is

generally used

to

describe this artifact. Projections

can be

corrected

by

acquiring

an image of a

step-wedge, made

of the

same material,

in

such

a way to correlate the mesured attenuation to the true material thickness. Streaks

artifact correspond

to

abnormal values along lines which correspond, inside

the

object, to high attenuation. Beam hardening artifacts can be avoided when using

some filter, i.e. a metallic foil, directly set at the window of the X-ray tube and

intended to

pre-harden

the

spectrum [KAF 96]. Figure 1.10 displays

an

example

of

streaks inside the tomographic image of a set of six samples surrounded by air (Fig

l . lO .a ) ; the streaks are suppressed by the use of a copper filter, 0.1 mm in thickness

(Fig. l . lO .b).

Figure

1.10.

T he reconstructed slice (l.lO.a) is corrupted by streaks due to beam-

hardening (l.lO.a). Filtration

with a

foil of copper, (0.1

mm)

nearly suppresses

the

streaks

(l.lO.b). T he high voltage used for

both

images is 100 kV

Beam hardening

is

also avoided when using

a

monochromatic y-ray source.

But

it

mus t be kept in mind that y-ray sources deliver a very low photon flux (typically

one hundredth of the flux delivered by a tube). Tomography using synchrotron


26/209

General

principles 25

radiation does not generates artifacts because a monochromator is always used,

thanks to the huge X-ray f lux.

-

Detector saturation

To

obtain

a

reconstruction which

is

free

of

defect,

the

signal delivered

by

every

cell of the detector must be strictly proportional to the photon f lux. Thus high values

(approaching the upper limit of the digitization range) as well as low values

(approaching the noise level) of the

flux

must be avoided. Streaks artifacts, similar

to those obtained in the case of beam-hardening, are generated along lines which

correspond to high attenuation.

-

Aliasing

High (spatial) frequencies are encountered in the signal corresponding to every

projection.

They are due to the steep edges which are eventually present in the

object. As the detector samples the signal (all along the projection) with a non-zero

step,

high frequencies corrupt the data, within the Fourier domain. Streaks are

generated [KAK 87].

On

figure 1.11, aliasing

is

visible

at the

corners

of the

objects.

-

Scattered photons

Photons

scattered

by the

sample

or by its

environment deliver

a

wrong

information

which leads to cupping effect. Collimation can improve th e

reconstructed image.

Figure 1.11. A liasing at the corners Figure 1.12. Ring

artifacts

-111 corrected detector

The signal delivered by every sensitive cell of the detector must be linearly

spread between the offset level (corresponding to the absence of photons) and the

gain level (corresponding

to the

non-attenuated

f lux). A bad

correction

of one

cell

will generate, in the reconstructed image a "ring artifact", i.e. the image of a ring,


27/209

26

X-ray tomography

in

material science

centered on the pixel corresponding to the location of the rotation axis. On figure

1.12

a great numb er of concentric rings are visible.

-

Spatial distortion

of

the

detector

Distortions of the projections, due for instance to the camera (e.g. distortions due

to the

lens) deliver artifacts which

can be

corrected

by

software.

- Centering error

The reconstruction requires the knowledge of the location of the projection of

the

center

of rotation within the

detector.

Distortions are generated when the

reference

to the

centre

is

erroneous.

1.5.

Conclusions

X and y-ray tomography allow a great number of potential applications. The

measured quantity is in fact the linear attenuation coefficient \i , and not directly the

density. A careful choice of the pho tons energy a nd the selection of a detector w ith a

high

dynamic range allows

to

lessen

the

noise

to a

reasonable level.

Coefficient \ L

can be

estimated

with

an

accuracy slightly

better

than

1% .

1.6. References

[ATT 68]

Anrx

F.H.R., ROESCH W.C. , Radiation Dosimetry, Academic Press, 1968.

[BAR 57] BARTHOLOMEW R.N. , CASAGRANDE, R.M., "Measuring solids

concentration

in

fluidized

systems

by

gamma-ray absorption", Industrial

and

Engineering Chemistry, vol . 49, n. 3, p. 428-43 1, 1957.

[ B E R

95]

BERNARD

J .R. ,

Frontiers

in

Industrial Process Tomography,

Engineering

Foundation,

Ed. DM

SCOTT&

RA

WILLIAMS, New-York ,

p.

197, 1995.

[CEN 99] CENDRE, E. et

al.,

"Conception of a high resolution X-ray com puted

tomography device; Application to damage initiation imaging inside materials",

Proceedings

of the 1st

World Congress

on

Industrial Process Tom ography,

Umist Univ. (U.K .) , p. 362-369, 1999.

[CLO 97 ] CLOETENS P ., PATEYRON-SALOME M., BUFFIERE J.-Y.,

PEK

G.,

BARUCHEL

J., PEYRIN F., SCHLENKER M ., "Observation of microstructure and damage in

materials by phase sensitive radiography and tomography", J. Appl. Phys., vol .

81,

n. 9, p. 5878-5886, 1997.

[ D U V

98]

DUVAUCHELLE

P .,

Tomographie

par

diffusion

Rayleigh

et

Compton avec

un rayonnement synchrotron: Application a la

pathologic

cerebrale, these de

doctoral, univ ersite de G renoble 1, 1998.

[PEL 84 ] FELDKAMP L.A. ,

DAVIS

L.C.,

KRESS

J.W., "Practical cone-beam

algorithm",

J.

Opt. Soc., vol. 1, n. 6, p. 612-619, 1984.


28/209

General principles

27

[HAL 92] HALMSHAW R ., "The effect of focal spot size in industrial

radiography",flrif/s/i Journal

of

NOT, vol. 34, n. 8, p. 389-394, 1992.

[HAR 99 ] HARTEVELD W.K. et al. "A fast active differencial capacitance

transducer

fo r

electrical capacitance tomography", Proceedings

of the 1st

World

Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 571-574,

1999.

[ISO

99] iso/TC 135/SC 5 , ISO document "NDT Radiation methods- Computed

tomography", Part I: Principles; Part II: Examination Practices, 1999.

[JOH 96]

J O HA N S E N

G.A, FR0YSTEIN T.,

HJERTAKER

B.T., OLSEN O., "A dual

sensor flow imaging tomographic system", M e as.

Sci.

Techn., vol. 7, n. 3, p.

297-307, 1996.

[KAF 96] KAFTANDJIAN V., P E D C G ., BABOT D ., PEYRIN F., "High resolution X-ray

computed tomography using

a

solid-state linear

detector",

Journal of X-ray

Science

and

Technology,

vol. 6, p.

94-106,

1996.

[KAK

87 ] KAK A.C., SLANEY M., Principles of Computerized Tomographic Imaging,

IEEE

Press,

1987.

[PIN 99] PlNHEIRO P.A.T. et al., "Developments of 3-D Reconstruction Algorithms

for ERT", Proceedings of the 1st World Congress on Industrial Process

Tomography, Umist Univ. (U.K.), p. 563-570,

1999.

[SCH

90],

SCHNEBERK

D.J.,

AZ EVEDO

S.G.,

MARTZ H.E., SKEATE

M.F., "Sources

of

error in industrial tomographic reconstruction", Materials Evaluation,

vol.

48, p.

609-617,

1990.

[THI

99]

THIERRY

R. et

al., "Simultaneous Compensation

fo r

Attenuation, Scatter

and Detector Response for 2D-Emission Tomography on Nuclear Waste wi thin

Reduced Data",

Proceedings of the 1st

World

Congress on Industrial Process

Tomography, Umist Univ. (U.K.), p. 542-551,

1999.

[ZHU 95] ZH U P. ,

PEIX

G., BABOT

D.,

MULLER J., "In-line density measurement

system using X-ray Compton scattering", ND T & E International,

vol.

28, n.

1,

p. 3-7, 1995.


29/209



30/209

Chapitre 2

Phase contrast tomography

Hard X-ray radiography

and

tomography

are

common techniques

fo r

medical

and industrial imaging. They normally rely on absorption contrast. However, the

refractive index for X-rays is slightly

different from

unity and an X-ray beam is

modulated in its optical phase after passing through a sample. The coherence of third

generation synchrotron radiation beams makes

a

simple form

of

phase-contrast

imaging, based on simple propagation, possible. Phase imaging can be used either in

a

quali tat ive way, mainly

useful for

edge-detection,

or in a

quanti tat ive way,

involving numerical retrieval of the phase

from

images

recorded

at different

distances from

the

sample.

2.1.

Introduction

The

phase

of an X-ra y b eam tran sm itted by an object is shifted due to the inter-

action with the electrons in the ma terial. Im ag ing usin g phase contra st as opposed to

attenuation contras t is a powerful method for the inv es t igat ion of light ma terials but

also to dis tingu ish, in ab sorbing samples, phases w ith very similar X-ray attenua tion

but

different

electron dens ities. Phase contrast im ag ing

w as

pioneered

in the

early

seventies by

A n d o

and

Hosoya [AND 72] ,

w ho

obtained images

of

bone tissues

and

of

a

slice

of

granite us ing

a

Bonse-H art type interferometer [BO N 65]. This technique

developed into a qua ntitativ e three-dim ensional ima ging technique. Because of the

limited

qu al i ty

of

available lenses, elaborate forms

of

phase contrast ima ging such

as

Zernike phase-contrast [ZER

35] or

off-axis

holography [LEI

62] are

presently ruled

out for hard X-rays. Three methods of phase sensitive imaging exist: the interfero-

metric techniqu e [MO M

95 , BEC

97],

the

Schlieren technique [FOR

80, ING 95] and

the propagation technique [SNI

95, CLO

96].

They are compared in

section

3. The

main advantages of the method used in this work, the propagation technique, are the

extreme s imp licity of the set-up a nd the better spatial resolu tion.


31/209

30

X-ray tomography

in

material science

This techniq ue w as mostly used up to now in the so-called 'edge-detection regime'

to image directly th e discontinuities in refractive index in the object. It is how-

ever possible

to fully

exploit

th e

quantitative information entangled

in the

Fresnel

diffraction

patterns towards h igh resolution qu antita tive phase tomography. The

'holo-

tomographic' reconstruction is performed in two steps: first the optical phase of the

w ave exiting the sample is retrieved n um erica lly from images recorded at different dis-

tances from the sample. The refractive ind ex dis tribu tion is then reconstructed from a

large num be r of phase m aps us ing a classical tomographic a lgorithm . Res ults of q ua n-

titative phase tomography

on

sam ples

of

interest

to

materials science

are

discussed.

2.2. X-ray

phase modulation

The interaction of a wave w ith matter affects its am plitu de and phase. This can

formally be described by the complex refractive index n of the medium. Because its

value is

nearly unity,

it is

usually w ritten

for

X-rays

as

n = l - < J +

i/?

[1]

A

plane monochromatic wave prop aga ting along

th e

z -axi s

in

vacuum

is of the

form

exp(i^

L

z)

with A the X-ray wavelength. In a material with refractive index

n

this

becomes exp(m^

L

z). The refractive index decrement

6

results in a phase va riation

compared

to propagation in vacuum. The imaginary part J determines the attenuation

of

the wav e. The X -ray intens ity is the squared m odu lus of the wave and the absorption

index

(3

is simply proportional to the linear absorption

coefficient

p.

= f > [2]

The absorption index has a complex energy and composition dependence. It va ries

abruptly near the characteristic edges of the elements. The

refractive

index decrement

6

on the other hand is prima rily due to Thomson scattering and has a much sim pler

dependency on the energy and the m aterial characteristics . S is es sentially proportional

to the electron density in the material. Generally, it can be expressed as

where

the sum

extends over

a ll

atoms

p,

with atomic num ber

Z

p

,

in the

vo lume

V,

r

c

= 2.8 fm is the classical electron radius, and f'

p

is the real part of the wav elength-

dependent dispersion correction, s ignificant near absorption edges,

to the

atomic scat-

tering factor. If the composition of the material is known in

terms

of

mass

fractions

q

p

, the follow ing equ iva lent expressions can be used


32/209

Phase contrast tomography 31

[5]

with NA

Av ogadro 's num ber and A

p

the mass number.

6

P

and p

p

are respectively the

refractive index decrement

and mass

density

of the

pure species.

If the

dispersion correction

f

p

can be

neglected,

6 is

proportional

to the

electron

density

p

e

,

i.e.

S

=

r

c

A

2

p

e

/ ( 2 7 r ) . The

ratios

Z

P

/A

P

appearing

in

Equ at ion

4 are

simi-

la r for ma ny atomic species « 1/2), and 6 is thus to a good approx imation determined

by

the

mass density

p

of the

material [GUI

94 ]

[6]

Both 6 and ft are small, typically 10~

5

- 10~

6

and 10~

8

- 10~

9

respectively for l ight

materials, ind icatin g the power of phase sensitive ima ging compared to the absorp-

tion. Figure

1

shows

the

ratio

S /ft, a

figure

of

merit

fo r

phase effects compared

to

attenuation effects,

as a function of the X-ray energy E fo r

a luminium.

The energy

range includes

soft

X-rays

and

hard X-rays.

In the

soft X-ray range, more precisely

in

th e 'water w indow '

where

soft

X -ray microscopes u sua lly operate,

a

gain exists

b ut

it

is relatively mod est. O n the other hand in the hard X-ray range (energies ab ove 6

keV) this ratio increases with energy

to

huge values

(up to 1000).

Practically,

if one

selects

for exam ple an X -ray energy of 25 keV to be able to cross a thick alu m iniu m

sample,

a

hole

in

this metal should hav e

a

diameter

of at

least

20

/zm

to

produce

1 %

Figure 2.1.

Ratio S / / 3

of the

refractive ondex decrement

and the

absorption index

as a

function

of the

X-ray energy

for the

element aluminiu. This

is a

figure

of

merit

for phase

e f f e c t s compared to attenu ation e f f e c t s


33/209


absorption contrast. Using

the

effect

on the

phase,

the

minimum detectable hole

is

reduced

to

about

0.05 y u m .

X-rays

are

adapted

fo r

imaging

of

thick samples thanks

to their low absorption at high energies. If it is possible to visualise the phase of the

transmitted w ave, the sensitivity and spatial resolution remain good.

For

inhomogeneous samples

the

wave

at the

exit

of the

sample

will be

modulated

in

both phase and attenuation. Propagation inside the sample itself can

usually

be

neglected

and it is

possible

to

project

the

object onto

a

single plane perpendicular

to

the propagation direction. The transmission function T(x, y) gives the ratio of the

transmitted

a nd the

incident amplitude s.

It can be

compared

to

exp(—

f

n(x,

y, z)dz)

that

gives

the

ratio

of the

transm itted

and the

incident intensities according

to

Lam bert-

Beer's law.

This transm ission

function

corresponds

to the

projection

of the

refractive

index distribution through

T(x,y) = A(x,y)e

i

rt

x

>ri [7]

w ith the amplitude

A(x,y) =e-W*'*) and B(x,y}

y j 0(x,y,z)dz [8]

and the phase m odulation

(p(x, y) = Y

/

[1 - < 5 ( z , y,

z)]dz

= (?

0

-

-̂ /

6(x , y, z}dz .

[9]

( p

0

is the phase modulation that would occur in the absence of the object. In classical

absorption tomography the projection of n is determined for a large number of an-

gular positions of the

sample.

The three-dimensional (3D) distribution of n(x, y, z)

or

eq uivalently

of / 3 ( x , y ,

z}

is

then reconstructed

from the set of

projections using

a

tomographic reconstruction algorithm. Similarly if the phase map (p(x, y) is known

for

a

large enough number

of

angular positions

of the

object,

it is

straightforw ard

to

reconstruct

th e

d istribution

of the

refractive index decrement

8(x, y, z) .

2.3.

Phase

sensitive

imaging

methods

There

are

three methods

of

phase sensitive imaging:

the

interferometric techniq ue

[MOM

95 , EEC

97],

the

Schlieren technique [FOR

80, ING 95] and the

propagation

technique

[SNI 95, CLO

96].

The

co-existence

of the

different

methods shows that

they

all

have their advantages

and

disadvantages with respect

to the

accessible phase-

information,

the

complexity

of the

set-up,

the

requirements

on the

beam

or the

spatial

frequency

range covered.


34/209


33

2.3.1. The interferom etric

technique

Here

contrast

is due to

interference

of the

beam transm itted through

the

object w ith

a reference beam . If the beams are coherent w ith each other, the intensi ty

wil l

be di-

rectly affected by the local phase shift. Bragg-dif f ract ion by p erfect crystal slices cut

out

from a large, almost perfect m ono lithic silicon crystal is used to s plit, dev iate and

recombine the two bea ms. A possible configuration [HAR 75] is shown in Figure 2a.

The recorded inter feren ce pattern cannot be exp loited as it is because the interference

fringes ca nno t be directly linked to a projection of the object and b ecause an intr ins ic

fringe pattern is alw ay s present. The image treatment to

qu anti tat ively

reconstru ct the

phase mo du lat ion introduced

by the

sample

is

howev er rather s traightforw ard. Several

images

for

different

external phase

shifts,

typical ly

8

( inc lud ing

flatfield

images) , mus t

be recorded to reconstruct a single phase-map.

The pos sibility to perform phase tomograp hy a nd to reconstruct the local dis tribu tion

of the

refractive index decrement w ith

a n

X-ray interferometer

w as

demonstrated

by

Momose et al and Beckmann et al in 1995 [MOM 95, BEC

95].

The interest of phase

imaging compared to absorption ima ging w as frequently illustrated [MOM

96].

The

complexity and stability requirements of this technique are however serious draw-

backs.

The

sample must

be

immersed

in a

liquid that matches

the

refractive index

of

the

sample. O therw ise large phase jum ps

a t

air-sample bou nda ries perturb

th e

interfer-

ence fringes and the large deflection in the sam ple reduces the v isib ility of the fringes .

Some

blurring

is

necessarily associated

to the

passage

of the

beam through

the

anal-

yser crystal. This limits the resolution to about 15 /^ m in the best case [BEC 97]. O n

the

other han d

th e

freq uency range covered

is not

limited towards

the low

frequencies

and a spatially homogeneous phase shift can be measured w ith respect to the reference

beam.

2.3.2. Schlieren technique

This differential phase contrast method

is

sensitive

to the

angular deviations

of the

X-ray beam. Phase gra dients present in the object locally deviate the beam by an angle

Forster et al [FOR 80] used a double crystal arrangement similar to the one shown in

Figure

2b. The first

crystal

acts

as a

collimator

in

limiting

the

angular

and

spectral

range.

The

angular deviations introduced

by the

sample change

th e

incidence angle

with

respect

to the

analyser that acts

as an

angular

filter. The

variety

in the

nomen-

clature fo r this approach can be noted: Schlieren-imaging [FOR 80, CLO 96], refrac-

tion

contrast [SOM 91], phase dispersive imaging [ING 95, ING 96], phase contrast

imaging [DAV 95] and diffraction enhanced imaging [CHA 98] are the most com-


35/209


Figure 2.2.

Set-up for phase

sensitive

methods: (a) Interferometric techniqu e, (b) Schlieren

technique and (c) Propagation technique

mon names. The poss ibility to vis ua lise phase gradients (occu ring for exa mple at

edges) was shown by many groups, but no reconstructed

(differential)

phase map was

presented and the method was not extended to 3D im aging through tomographic tech-

niques . Compared to the interferometric technique, th e experimental set-up is sim-

plified

and the

stability requirements

are

less stringent.

To

obtain

a

good sensitivity

to phase gradients, the width of the rocking curve for one of the crystals relative to

th e

other should

be

small, typically 2-10

yurad, and the

angular s tability should

be

about 0.2 yurad [ING 96]. A s the ali gnm ent is less critical, the collima tor and analy ser

crystal do not need to be part of a m onolithic block, and the

space

available for the

sample and its env ironm ent increases. The samples are in general not immersed in a

liquid. The spatial resolution is aga in affected by the passage of the wa ve thro ug h the

analyser crystal. This method

is

less adapted than

th e

previous

one to

covering

th e

low spatial frequen cy range, and very smooth variations of the phase may introduce a

phase gradient that is too small to be detected. This ima ging scheme can be used on a

laboratory X-ray source. Most of the published results w ere obtained und er these con-

ditions, resulting in long exposure times of 15-30 minutes [ING 96] for a radiograph.

This technique corresponds to Schlieren imaging in classical optics [HEC 98].


36/209

Phase contrast tomography 35

2.3.3.

The propag ation technique

The

spatial redistrib ution

of the

photons

due to

deflections

or

more generally Fres-

nel

d iffraction

is considered a nuisa nce in abs orption contact and projection radiogra-

phy and in interferometric and Schlieren phase imaging. It is however also a unique

contrast mecha nism

fo r

phase s ensitive ima ging,

with

advantages

in the

sim plicity

of

the set-up and the achieva ble resolution . In this case there is no distinct reference

beam as in the interferometric techniq ue, and the beam trans mitted throu gh the object

plays this role itself.

The

occurrence

of

contrast

can be

understood

as due to

inter-

ference between parts of the w avefront that have suffered slightly

different

angular

deviations

associated

to different

phase gradients.

The

overlap between parts

of the

wavefront is only possible after propaga tion over a certain distan ce. A s in previous

case,

this is a

differential

phase

ima ging technique .

A

homogeneous phase gradient

cannot be detected because it corresponds to an overall deflection of the beam; de-

tectable contrast requires

th e

second derivative

of the

phase

to be

non-zero . When

the direction of the X-ray beam is tangential to the edge of structures in the sample,

such a perturbation of the wavefront is expected and contrast

wi l l

appear. Po ssib le

internal stru ctures are holes and cracks, inclu sio ns , reinforcin g particles or fibers in a

composite material. Ex perimentally

the

sample

is set in a

(partially) coherent beam

and the trans mitted beam is recorded at a given distance d

with

respect to the sample

[SNI

95 , CL O

96].

The

experimental set-up shown

in

Figure

2c is

thus essentially

the same

as for

absorption radiography except

for the

increased sample

to

detector

distance. The crystal system upstream of the sample selects a narrow spectral range,

delivering a quasi-monochromatic beam to the sample.

The image contrast changes tremendously with th e sample detector distance d .

The

latter determines

the

defocu sing distance

D

through [BOR

80]

with

/

the source sample distance. In the case of the long ESRF beamline ID 19 (d < C

/ = 145m), th e defocusing distance and the sample-detector distance are practically

equal.

The

absorption radiograph corresponds

of

course

to an

image recorded close

to

the

sample

(D

«

0). The

region

of the

object ma inly contributing

to the

corresponding

point of the image (the first Fresnel zone) has a radius equal to

When it is small compared to the typical transverse dimension a of the features in

th e

sample,

a

separate

fringe

pattern shows

up for

every border

in the

sample,

and the

images

are

cha racteristic

of the

'edge-detection regime'

(rp

< § ;

a).

Three-dimensional

reconstruction of the boundaries inside the volume is feasible with the algorithm fo r

absorption tomography

( c f .

section

4) . At

larger distance

(rp

w

a)

several interfer-

ence fringes show up in the radiographs. These deformed images, corresponding to


37/209


th e

'holographic regime' ,

give little direct inform ation on the sample. How ever, com-

bining

such images

recorded

a t different distances w ith a su itable num erical algorithm

gives access to the phase modulation ( c f . section 5). For the largest dista nces , rarely

accessible with X-rays,

one

reaches

the

Fraunhofer l imit

(rp

a).

Figure

3

shows

as

an exam ple four radiographs of a 0.5 mm thick piece of polys tyren e foam at increa sing

distances D . The beam is monochromatised to 18 keV. As the distance increases, the

contrast and width of the Fresnel fringes both increase. The radiographs are recorded

with a C CD based detector inv olving X-ray / v isible l ight conversion in a transparent

YAG:Ce screen [KOC 98], with

an

effective pixel size

of

0.95 //m.

The most striking advantage of this method is the extreme

simplicity

of the set-

up.

It is

essentially

the

same

as for

absorption radiography.

The

transit ion betw een

abs orption and phase radiog raphy or betw een the

different

regimes of phase ima gin g is

simply o btained by cha ng ing the sam ple detector d istance. The

stab ility

requ irements

on the (few ) elements dow nstream of the monochrom ator, i .e. the samp le and the

detector, are easily met. The m ono chrom ator can be w ell upstream of the sample and

the sample detector distance can often be chosen quite large. The

free

space around

the

sample

can be

used

for all

k inds

of

devices

fo r

in-situ

and

real - t ime ob servat ions .

However the optical elements of the beamline have to be carefully prepared to avoid

spurious phase images.

It

can be shown that for a given defocu sing distance the image is m ost sensit ive

to a specific freque ncy range. The optimum distance to be sensitive to phase features

with spatial frequ ency / is such that

This frequen cy selectivity will intrinsically limit th e access ibil i ty to the low f requency

range , i.e. the smooth v ariation s in the object's phase. The optimu m distance, increas-

ing

as the square of the object size, wil l not be reached for these freq uen cies due to

physical limitations (size of the experimen tal hu tch ) or the coherence co nd itions . The

image is not spoiled by the passage of the modulated wave through a crystal as is the

case in the interferometric and Schlieren techniques. The resolution in the propaga-

tion

technique depends

on the

image processing after recording.

F or

untreated images

recorded

in the

edge-detection regime

the

resolution

is

limited

by the

fringe spacing

to about 2rp. When the

fringes

are disentangled in a holographic recons truction, the

spatial resolution is limited essen tially by the detector. The strin gen t req uire m ents on

th e

beam incident

on the

sample explain

w hy

this

techniq ue emerged only recently

[HAR

94,

SN I

95, CLO 96, NU G 96]

wi th

the

appearance

of

partially co herent X-ray

beams delivered b y third generation sy nchrotron rad iation sources. The geom etrical

resolution in the Fresnel diffraction pattern is equa l to Ds

a

with

s

a

— f the angu-

la r

source

size and s the source size. The condition for observation of the spatial

frequency / is


38/209


37

Figure 2.3.

Phase sensitive radiographs

of a 0.5 mm

thick piece of polystyrene foam, with

the

de tector at various distances dfrom the sample. X-ray energy 1 8 keV . (a) d= 0.03 m, (b) d =

0.2 m, (c) d = 0.5 m and

(d )

d = 0.9 m. The

contrast

and the

width of

the

interference fringes

increase trough the series

The blurring due to source size and detector resolution explains why no interference

fringes are observed with classical laboratory

sources

although the propagation dis-

tances a re also n on-z ero in projection radiography . A lternatively , the interference pat-

tern

involves

the

coherent superposition

of

laterally separated portions

of the

incident

beam. The interfering wav es mu st originate from points that are mutually coherent,

and

thu s latera lly separated

by a

distance sma ller than

the

transverse coherence length

/t

that can be defined as l[ = A/(2s

a

). The incid ent beam mu st be coherent over the

first Fresnel zone fo r o ptimum conditions.

The conditions on the monochromaticity are less stringent. The beam is usually

monochromatised using a monochromator based on perfect silicon crystals. The en-


39/209

38 X-ray tomogra phy in material science

ergy spread AA/A « 10 ~

4

is thus very small. The th ickness of the samples that

can

be

investigated

is not

limited

to the

longi tudinal coherence length

(=

A

2

/ A A )

[CLO 96]. A n increase in energy spread by one or even tw o orders of magn i tude can

still

correspond to quasi-monochromatic conditions. This allows to increase consid-

erably

the flux

us ing

a

m ult i - layer monochromator w ith

a

very high substrate

quali ty.

Using the same propagation principle but w orking w ith the polych rom atic rad iation

delivered by a laboratory X-ray microsource to retain some flux, deflection sensitive

images were obtained [WIL

96 , POG

97]. This

seems

promis ing

for

work

in the

edge

detection regime as the main contrast, a white and black fringe, is unchang ed over a

large spectral range.

2.4. Direct imaging

Most of the tomographic work performed until now using the propagation tech-

nique

is based on the usual algorithm for absorption tomography . This is a workab le

solution especially when the defocusing distance

D

is sma ll and the sample is made

up of different

(metallographic) phases with

different

densities [CLO 97]. This

qual i -

tative approach allows to visua lise, in 3D , density dis continuities , such a s

reinforcing

SiC

particles

in an

alum inium ma trix composite [BUF 99]. D ensity jum ps appear

as

dark / light fringes. A nother adv antage is the poss ibi l i ty to detect and localise features

that are actually sm aller than the pixel size a s the interference fring es produced can be

larger than the feature itself (regime with rp > a). This makes it possible to detect

cracks w ith sub -micron opening. A n exa mp le of a tomogra phic slice of an a luminium-

silicon

alloy obtained using this direct approach based

on a

single distance

is

s hown

on Figure

6b in the

next section. However, with this approach

the

spatial resolution

is

limited by the Fresnel fring e dis tribution and artefacts occur in some cases due to the

ill-suited algorithm. Binarisation of the edge-regime images is extremely tedious and

w as done essentially ma nua lly. I t mu st also be noted that the qua ntitativ e informa tion

on density

and

compos ition

is

lost.

For

these reasons there

is a

need

fo r

more adapted

algorithms preserving resolution and qu antita tive inform ation , at the expense of an

increased data volume and computational effort.

2.5. Quantitative imaging

A new approach, holotomography, has now been im plemented to

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X-Ray Tomography in Material Science