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

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918 Part D Materials Performance Testing

16.3.3 Detection of the Damage

of Composites on the Mesoscale:

Application Examples

Putting aside the mechanical interpretation of compos-

ite damage in polymers, is it possible to characterize the

damage of these materials based on a single method that

combines the physical and chemical interpretations?

The answer must evidently be no. The problems calls

for nondestructive testing methods, which include ul-

trasonics, x-rays, thermography, and others methods. At

Fig. 16.58 Three cross sections of the specimen after im-

pact

Fig. 16.59 Comparison of radiogra-

phy (right) and tomography (left)of

delamination

the microscopic level, an electron or acoustic micro-

scope is the required nondestructive evaluation tool.

Between the macroscopic and microscopic levels,

there exists a very important middle domain, typical

of composite materials, which is treated at the meso-

scopic level: that of transverse cracking, distribution of

reinforcements and of porosity, that is, the domain of

sequential cells. It is very difﬁcult today to characterize

defects at these different levels other than by radiogra-

phy.

The various applications of x-ray tomography al-

low for the study of composite materials at any level,

macroscopic, mesoscopic or microscopic. This instru-

ment is well adapted to reveal defects on a centimeter

scale, such as a delamination, but it is also capable

of revealing the distribution of the reinforcements on

a millimeter scale, and of detecting ﬁbber cracking as

well as cracking along ﬁbber matrix interfaces on a mi-

crometer scale. In addition, the attenuation of x-rays,

which depends on the physical and chemical nature of

the composites constituents, can be measured by x-rays

scanners.

Changes in the polymeric matrix due to aging, or

even the transformation of amorphous polymers into

crystallites, can be studied by x-ray tomography. This

astonishing tool, long used by the medical ﬁeld, is

also well adapted to the study of composite mater-

ials and polymers due to its ability to detect defects

on a wide range of scales, as well as its ability of

physical and chemical analysis by measuring x-ray at-

tenuation.

X-ray scanning can thus provide information on the

physical and chemical aspects of the material and on

the geometry of ﬂaws and damage on any scale. This

Part D 16.3

Performance Control 16.3 Computerized Tomography – Application to Organic Materials 919

tool provides solutions to many problems faced by the

materials scientist, as demonstrated in the following re-

sults.

Damage in a Carbon Epoxy Composite Plate

Specimen After an Impact

The specimen studied is a rectangular coupon whose di-

mensions are 255 × 172 × 8.5 mm. It is made up of 64

plies of carbon-epoxy ﬁber. The stacking sequence is

(45/90/ −45/0) 8 s.

The specimen was subjected to the impact of an

aluminum ball 12.7 mm in diameter moving at a speed

of 120 m/s. Then it was cycled in compression under

110 MPa to intensify the damage.

In this case, the scanner examination has been com-

plemented by radiography, zinc-iodide-enhanced x-ray,

and by optical micrography. This was done in collab-

oration with the NASA Langley Research Center in

Hampton, Virginia.

Global Examination of the Plain Specimen

A study on the axial slices and on the frontal slices

has been carried out. On the frontal views, we note

a strongly damaged area whose density is higher than

550 H, slightly elliptical shaped, and whose largest

dimension is orientated in the compression direc-

tion (Fig. 16.58).

Modeling of the Damage

The damage of this composite specimen is complex be-

cause of the great number of plies. The envelope of

the damaged area has the shape of a barrel, delamina-

tions being more numerous in the center of the plain

specimen than near the surface.

Synthesis of the Observations

Made on the Impacted Specimen

To facilitate the interpretation of observations made

with medical scanners, damage to the impact speci-

men is examined by radiography at a microscopic scale

and by scanning electronic microscopy at a microscopic

scale.

A comparison between radiography and tomogra-

phy is made (Fig. 16.59). We would like to note that the

scanner provided much better resolution of the outline

of the damaged area. Nevertheless, enhanced x-ray ra-

diography was required to reveal the very complicated

network of microcracks.

To conclude, we emphasize that the resolution of the

images of the microcracks with the medical scanner is

not satisfactory at the moment.

Damage in Rubber Submitted

to High Hydrostatic Pressure

It is well known that triaxiality of stresses including

a high hydrostatic pressure has a large effect on the me-

chanical behavior of rubbers and particularly on fatigue

mechanisms.

In a pancake specimen loaded in tension the hy-

drostatic pressure is about 2.5 × the shear modulus of

rubber for an elongation of 20% in NR. From observa-

tions by x-ray tomography, it seems that the ﬁrst cavities

are formed in the center of the specimen at 20% of

elongation for a fracture at 380% (Fig. 16.60).

It must be pointed out that no other NDT method is

available to observe cavitation in rubber.

16.3.4 Observation of Elastomers

at the Nanoscale

The damage caused by fatigue and cracking in rubber

or in composites with elastomer matrices seems to be

considerably inﬂuenced by the transformation from an

15 %

20 %

28 %

60 %

140 %

200 %

290 %

320 %

Fig. 16.60 Cavitation growth due to high hydrostatic pressure (NR)

Part D 16.3

920 Part D Materials Performance Testing

amorphous to a crystalline phase, when this latter phase

exists. This hard-to-detect transformation is generally

studied by x-ray diffraction, a method by which we can-

not detect the gradients of transformation within the

rubber nor make local observations.

In contrast, x-ray tomography allows for the obser-

vation of the crystalline transformation of rubber and

its localization owing to the images generated by scan-

ner. The attenuation of the intensity of the x-ray beam,

similar to x-ray diffraction, reveals the existence of

crystallites.

In order to validate this hypothesis, test samples

of natural, crystallisable rubber NR and of synthetic,

noncrystallizable rubber (SBR) were studied at differ-

ent temperatures, knowing that low temperatures induce

a partial crystallization of NR.

DT mean value (pixel)

TD mean value (pixel)

Temperature (K)

3500 100 150 200 250 300

350

1180

1170

1175

1100

1200

1160

1150

1140

1130

1190

1120

50 125 200 275

NR_1

NR_2

SBR1

SBR2

Amorphous zone

Semicristal zone

1225

1212

1200

1187

1175

1162

1150

1137

1125

1112

1100

1087

1075

1062

1237

1050

Maximum crystallization

temperature

Transition zone

Glassy zone

Rubbery zone

Glassy zone

Fig. 16.61a,b Evolution of the TD depending on the temperature in

(a) and SBR (b)

The crystallization of rubber (NR) may be ob-

served on the samples, which were cooled down to

−200

◦

C. This microscopic phenomenon can clearly

be evaluated by means of x-rays CT on the meso-

scopic scale, which is another beneﬁt of medical

scanning. To show this effect, some of the tests were

carried out on the axisymmetric hourglass-shaped spec-

imens of NR and compared with SBR, which does

not show this phenomenon as it is always amor-

phous. Hourglass-shaped specimens are very suitable

for this test due to their large outer surface [16.32, 33].

So, the variation of mean values of the TD becomes

complex; the TD value is maximal at the level of

227 K (−46

◦

C) (Fig. 16.61), it exceeds 1180, whereas

at lower temperatures (83 K, −190

◦

C) the value is

1135.

This variation in TD displayed a crystallite state be-

tween 300 K (27

◦

C) and 227 K (−46

◦

C) above which

the amorphous state again becomes predominant be-

tween 227 K (−46

◦

C) and the transition temperature

) of rubber. When there is no crystallization, varia-

tion in density certainly modiﬁes the attenuation (TD).

In the case of SBR, which is always amorphous,

only the density varies with temperature, related to the

dilatation. An increasing in the temperature causes natu-

rally the increasing of the volume and the decreasing of

the density and ﬁnally the decreasing of the TD (pixel).

Figure 16.61b shows the evolution of the TD curve as

a function of the temperature for the SBR specimens. It

decreases from 1185 to 1120 in the temperature range

from 175 (−98

◦

C) to 300 K (27

◦

C). However, it shows

a transition zone at the level of 200 K (−73

◦

C), corre-

sponding to the transition temperature, T

, between the

glassy and rubbery phase. This zone is found between

1155 and 1160, corresponding to this glassy transfor-

mation.

16.3.5 Application Assessment of CT

with a Medical Scanner

The application examples show quite obviously that the

medical scanner is adapted to nondestructive evaluation

of composite materials from a technical point of view

and if we do not take into account the price of the de-

vice. The advantages of the method have been presented

with its excellent results. They confer on the medical

scanner a double role, which may form the subject of

two developments.

1. A nondestructive testing instrument with an indus-

trial purpose,

Part D 16.3

Performance Control 16.4 Computerized Tomography – Application to Inorganic Materials 921

2. An instrument comparable to the electronic micro-

scope with a more scientiﬁc purpose and able to give

information at a microscopic scale (several hun-

dreds of microns).

In the ﬁrst case, the resolution power of the medical

scanner is good enough for a device of classical NDT.

We have found only one major inconvenience, namely

the lack of reliability of the scanner to make precise

measurements of the dimensions of the defects within

more than 25%.

In the second case, the resolution of the scan-

ner is not good enough to study the transverse cracks

in the matrix. We can estimate the spatial resolu-

tion of the scanner at some hundreds of micrometers,

whereas we would need 10 μm. We are dealing here

with an important need which needs more investi-

gation.

Nevertheless, we should point out that the reso-

lution in density of the scanner is of the order of

−3

, which makes it possible to resolve all the de-

fects contained by composites whether they are cracks,

delaminations, ﬁber bundles, or porosities. The varia-

tion of the Hounsﬁeld density must be related at the

volumetric density but also to the modiﬁcation of the

chemical microstructure of the materials. From this

point of view, the scanner enables studies of the crys-

tallization of polymeric matrix of rubbers, for example.

This means that x-ray tomography is a multiscale NDT

method.

16.4 Computerized Tomography – Application to Inorganic Materials

The study of volume properties as well as of dimen-

sional features with computed tomography requires

an optimized selection of source–detector combina-

tion depending on the material composition (energy-

dependent linear attenuation coefﬁcient μ), the size

of the samples and the maximum thickness of ma-

terial that has to be irradiated, d. Additionally the

manipulator system and the mounting need to suf-

ﬁce the required accuracy. The maximum SNR of

a CT measurement of a homogenous sample is given

for μd

∼

2, corresponding to a transmission of about

11% [16.40]. This value is valid only for line detec-

tors with a sufﬁcient detector collimator and shielding

between single detector elements. Due to some lim-

itations, especially for ﬂat detectors, the conditions

for optimum image quality differ from this theoretical

value.

For organic materials with low atomic numbers

and low density, medical scanners can be applied

in many cases. As an example ﬁber-reinforced heli-

copter blades have been investigated for many years

by computed tomography, using commercial medical

scanners [16.41]. Inorganic materials require radiation

sources with higher energy and appropriate detector

systems. The different kinds of computed tomography

equipment developed at Federal Institute for Materials

Research and Testing (BAM) represents the state of

the art on this ﬁeld and are described in the following

knowing well that commercial solutions are in general

more efﬁcient with regard to saving of time and user

guidance.

16.4.1 High-Energy CT

This ﬁeld covers roughly the energy range given by

420 kV x-ray tubes over some radionuclide sources up

to electron linear accelerators with a maximum en-

ergy of 12 MeV. Most of the high-energy scanners used

have line detectors which can be shielded against the

scattered radiation. Starting with a translation/rotation

scanning principle and multidetector systems, today

line detectors are common with several hundred de-

tector elements and scanning times in the range of

some minutes and lower. At BAM a scanner for a 380-

kV x-ray tube and a

Co radionuclide source was

described as early as 1985 [16.42]. This universal scan-

ner was extended for measurements with a 12-MeV

electron linear accelerator (LINAC, Raytech 4000) in

combination with a multi-detector system with step-

motor-controlled collimator slits [16.43]. The actual

research covers the study of some effects of high-energy

cone beam CT with a LINAC and

Co using an a-

Si ﬂat-panel detector (Perkin Elmer, 16-bit ADC, 256×

256 pixel, pixel size is 0.8mm

). Due to the focal spot

size of high-energy radiation sources, which is on the

order of about 1.5 mm, the spatial resolution is limited

to a few tenths of a mm.

16.4.2 High-Resolution CT

Using magniﬁcation techniques a much higher spa-

tial resolution can be reached. The condition hereby

is that the focal spot size can be lowered to the mi-

Part D 16.4

922 Part D Materials Performance Testing

crometer range, which is possible with microfocus x-ray

tubes. The usable energy range of commercial x-ray

tubes extends up to 225 kV. For many objects it would

be desirable to extend the energy range of microfocus

x-ray tubes. As an example most cellular metals are

made from aluminum and thus have a high penetration

depth for x-rays. However, they are often used to pro-

duce objects of irregular shapes (with an outer cover

of a different material or the foam skin itself). This

produces artefacts due to the high-attenuation parts. In

some applications foams of high-attenuation materials

(up to iron) are used. In these cases the choice of an

x-ray energy as high as possible reduces the artefacts

from beam hardening and exponential edge-gradient

effects.

Figure 16.62a gives a graph that shows the gain

in attenuation possible with an extended range of kV.

For CT measurements the maximum absorption to free

beam ratio is about 20. From the graph it is seen that this

means an extension of measurable material in object

thickness from 20 to 30 mm of steel. At the end of 2002

a 320-kV microfocus x-tray tube (MX-5 tube, build by

YXLON, Halfdangsgade 8, 2300 S Copenhagen) was

integrated into a CT system (Fig. 16.62a). The bipolar

tube is build up from a standard −200 kV tube. The

original probe head is changed to the bipolar part which

has a build in anode with an additional high voltage sup-

plyofupto+120 kV. The distance between x-ray spot

to tube outside is enlarged to 25 mm because the target

position is no longer at ground voltage level. This means

that the tube in this modiﬁcation has a lower maximal

magniﬁcation, but this corresponds to the bigger object

Attenuation

100 kV

160 kV

200 kV

320 kV

380 kV

Thickness (mm)

3001020

–1

–2

–3

a) b)

Fig. 16.62 (a) Attenuation curve for iron as function of x-ray energy (b) 320-kV micro-focus tube

size. For example with 320 kV and 0.1 mA the optimum

spatial resolution is about 35 μm.

In practical operation the second part is ﬁxed to

+120 kV while the variation in energy is done with the

ﬁrst. A range of 130 kV to 320 kV results. To the ﬁrst

part a standard microfocus target can be attached and

the tube can thus be used as normal 200 kV microfocus

tube with high magniﬁcation. As detector an a-Si ﬂat-

panel detector (Perkin Elmer, 16-bit ADC, 1024 × 1024

pixel, pixel size is 0.4mm

) is used [16.44].

16.4.3 Synchrotron CT

With transmission target x-ray tubes the focal spot size

can be lowered to 1–5 μm, corresponding to a spatial

resolution of less than 10 μm. Due to the limited inten-

sity given by electron scattering and the heat generated,

which can melt the target, measuring times are increas-

ing to some hours and more. Synchrotron radiation

sources offer photon ﬂux densities, which are several or-

ders of magnitude higher compared to laboratory x-ray

tubes. The additional advantage is the fact that the high

photon ﬂux density can be reached with monochromatic

radiation giving some advantages over tomography,

such as the absence of beam-hardening artefacts.

To extend the usable energy range at the Berlin Elec-

tron Storage Ring Company for Synchrotron Radiation

(BESSY) a 7-T wavelength shifter with a critical en-

ergy of 13.5 keV has been installed in the storage ring

to operate the ﬁrst hard x-ray beam line at BESSY,

called BAMline. The main optical components of the

beam line are a double crystal monochromator (DCM)

Part D 16.4

Performance Control 16.4 Computerized Tomography – Application to Inorganic Materials 923

and, for the ﬁrst time for imaging, a double multilayer

monochromator (DMM, 320 double layers of W and

Si, with thicknesses of 1.2and1.6 nm, respectively).

The latter is preferred for use in tomographic facilities,

giving a factor of 100 higher photon ﬂux due to the

increased bandwidth of 2% compared with 0.01% of

the DCM [16.45–47]. The detector system consists of

a 2048× 2048 photometric camera system together with

a scintillator screen of GdOS. Depending on the lens

combination used nominal voxel sizes of 1.5–7.2 μm

can be used.

16.4.4 Dimensional Control

of Engine Components

A current research topic is the improvement of dimen-

sional control with CT.AsanexampleFig.16.63 shows

a cross section of an aluminum test sample investigated

with the 320-kV equipment. This sample was measured

additionally with the LINAC CT. Standard operation

for a comparison is the conversion of the voxel data in

a point cloud and the analysis of ﬁles in the stereolitho-

graphic data format (STL).

The comparison performed using the STL data

shows principal deviations between the computer-aided

design (CAD) model and real sample, which are due to

the limited energy of the 320-kV equipment and espe-

cially the smoothed edges in the LINAC measurement

due to the limited detector pixel size.

Fault Detection

in Telecommunication Equipment

The transmission characteristics of glass-ﬁber cables

can be changed after installation or during lifetime.

Fig. 16.63 Test sample of aluminum (∅ about 100 mm). The image (left) shows a cross section of the measurement

with the 320-kV equipment (320-kV x-ray: 220 kV, 100 μA; preﬁlter: 1.5 mm Sn; projections: 900/360

◦

; exposure time:

2.28 s per projection; voxel: (0.12 mm

); matrix: 1023× 1023 ×605). The image (middle) shows the deviation against the

CAD data and the image (right) the deviation against the measurement with LINAC (10.5 MeV, 20 Gy; preﬁlter: 90 mm

Fe; projections: 720/360

◦

; exposure time: 0.8 s per projection; voxel: (0.65 mm

); matrix 255×255×201)

To localize such ﬂaws optical time-delay reﬂection

(OTDR) methods are frequently used. The location of

ﬂaws can be determined with an accuracy of a few

mm. An analysis of the type and size of ﬂaws to-

gether with geometry control of the complex-shaped

cable can be performed with CT. Figure 16.64 shows

a vertical slice of a section (length 48 mm) of the ca-

ble together with two horizontal slices. The bunches

of four glass ﬁbers are embedded in a protective cov-

ering. One of the seven coverings contains no ﬁbers.

The diameter of each glass ﬁber is 125 μm. The

outer diameter of the cable is 16 mm. Some protec-

tive coverings show ﬂaws (Fig. 16.64 bottom right)

and an incomplete embedding of ﬁber bunches. To

avoid the inﬂuence of sample preparation the investi-

gated sample volume was part of a 15-m-long cable

segment.

Flaw Extraction in Cu Samples

As a ﬁrst example results are shown for ﬂaw extraction

in electron-beam-welded Cu samples (size about 40 ×

100 × 300 mm) using high-energy 3-D CT (Fig. 16.65).

For studies of probability of detection (POD) a 100%

inspection of volumetric ﬂaws are performed. Due to

the limited size of the ﬂat-panel detector the sample

was measured in three positions and the overlapping

data sets are joined after the image reconstruction. The

reconstructed images show some artefacts given by

scattered radiation, beam hardening and inherent detec-

tor effects. Therefore a ﬁlter was used to reduce these

artefacts. Figure 16.65 shows as result a cross section

trough the sample.

The ﬂaws are evaluated twofold; by a local threshold

operation using some image-processing modules (e.g.

Part D 16.4

924 Part D Materials Performance Testing

an STL converter module) developed for the image-

processing system AVS and the Volume Graphics (VG

Studio Max) image-processing tool. The geometry of

the envelope and the voids are lead together and con-

verted to STL format for comparison with ultrasonic

results and for theoretical simulation of the irradia-

tion process. The volume of all ﬂaws of the sample is

1100 voxels, determined with the AVS system, which is

in good agreement with the result from the VG system,

which gives a volume of a 1072voxel. The total volume

is 4 304 500 voxels.

Calibration of NDT Methods

The CT imaging method combined with the geomet-

rical correctness of the dimensions of the investigated

i240 k250

k550

5mm0616 m

0 50 100 150 200 250

Fig. 16.64 Vertical and two horizontal cross sections of

a glass-ﬁber cable

Fig. 16.65 Electron beam welding of a section (size 40 × 100 × 300 mm) of a Cu canister. The image (left)showsacross

section containing the ﬂaws, ﬂaws after segmentation (middle) projected into a plane and the ﬂaws after conversion to the

STL data format (right). Source: LINAC 10.5 MeV, 15 Gy; preﬁlter: 90 mm Fe; projections: 720/360

◦

; exposure time:

0.2 s per projection; voxel: (0.653 mm

); matrix: 255×255×532

samples are the main advantages to the use of CT for

the calibration of other NDT methods, such as ultra-

sonic technique (UT) or eddy-current techniques. As an

example Fig. 16.66 shows the investigation of ﬂaws of

coated turbine blades. With optical methods only the

ﬂaws in the coating layer can be visualized but not in

the matrix material. With high-resolution CT the crack

conﬁguration was analyzed. The result was used for

calibration of other NDT techniques (the eddy-current

method).

Pore Detection and Pore-Size Calculation

in Al Foam

Depending on the type of foam there are two ways to

measure the size of the pores inside.

•

If the pores are closed, a threshold operation is ﬁrst

used to obtain the binary area of all pores. Start-

ing on one edge, an algorithm then searches in 3-D

for the ﬁrst marked voxel and colors all adjacent

as belonging to the same pore until no more con-

nected voxels are found. The searching algorithm

then goes on to the next pore. Figure 16.67 (left im-

age) shows the result; the image is a slice through

the searched foam, the different colors of each pore

are used to detect pores, which were not separated.

The size distribution is given in Fig. 16.67 (right im-

age). While the volume is searched for pores, the

volume and the center of gravity are also stored, and

further parameters can be calculated. From these

parameters a simpliﬁed model of the foam can be

generated and calculation of real foams with ﬁnite-

element method (FEM) programs becomes possible.

The center image of Fig. 16.67 shows the same

foam, representing the pores as spheres with the

same diameter.

•

If the pores are open, the inverted nonmetal part has

to be 3-D-eroded until single areas result for all the

Part D 16.4

Performance Control 16.4 Computerized Tomography – Application to Inorganic Materials 925

Fig. 16.66 Turbine blade with protective layer (layer thickness of about 400 μm). The CT cross sections show a crack

only in the protective layer (right image) and a crack in the matrix material

Number of spheres

Radius (mm)

1.120.910.70.490.28 1.330.07

140

120

100

160

Fig. 16.67 Pore detection (left). Spheres with a diameter as calculated from the volume of found pores (right) at position

of pore centers. The pore size distribution is shown in the diagram on the right

Fig. 16.68 A vertical slice through the foam before and after three compression states (from left to right). From the

different 3-D image data sets the same slice was extracted showing the internal deformation of the foam

pores. The pore radius can then be calculated from

the number of voxels counted per pore. Knowing

the number of erode steps, the pore radius has to

be enlarged by this value.

Internal Deformation

The failure mechanisms of strength-tested foams were

studied on samples before compression and after com-

pression. 3-D CT images of the samples were used

Part D 16.4

926 Part D Materials Performance Testing

100 mm

Fig. 16.69 Digital radiography of the antique bronze statue Idolino (left image). A cross section is shown on the right,

representing the inner structure as well as the technique to ﬁt together the separate cast parts of the statue

Fig. 16.70 Enlarged detail (ﬁvefold magniﬁcation), marked

in Fig. 16.69 by the white square

to ﬁnd out where foam deformation has started, for

which a program for image comparison in 3-D was de-

veloped. First the size of the region to be compared

with the region of the sample after the compression

test has to be deﬁned. This region has to be big-

ger than the mean pore size. The region is moved in

different directions in 3-D until it ﬁts best with the

initial sample, and the shift of the parts of the strength-

tested sample is written to an array. In this way, for

all small regions of the probe we obtain shifts with re-

spect to the initial sample in three directions: Δx, Δy

and Δz.

The displacement after the compression test is

shown (unit mm) using different grey levels, which are

painted over the original foam structure, in order to

show the displacement of compressed foam compared

to original foam (Fig. 16.68).

Art Objects

Nondestructive methods such as x-ray techniques have

been focused on art objects since the discovery of

x-rays by W. C. Roentgen. In a long-standing col-

laboration with the Antikensammlung Berlin several

large Greek and Roman bronzes were examined, to

evaluate work traces and the interior of these archaeo-

logical samples. In 2000 the interior of the hellenistic

Getty Bronze and the early Augustan Idolino in Flo-

rence were investigated at BAM [16.48, 49]. From the

still being results Fig. 16.69 shows a digital radiogra-

phy (left image) which is used to localize the precise

location of cross sections. The tomogram (right im-

age of Fig. 16.69) gives a view of the interior of the

statue, with parts of the structure ﬁtting the differ-

ent cast parts. A detail of the tomogram shown in

Fig. 16.69 is presented in Fig. 16.70 with a ﬁvefold

enlargement.

Part D 16.4

Performance Control 16.5 Computed Tomography – Application to Composites and Microstructures 927

16.5 Computed Tomography – Application to Composites

and Microstructures

In computed tomography (CT) the interface contrast

of heterogeneous materials can be strongly enforced

by to x-ray refraction effects. This is especially de-

sirable for materials with low absorption or mixed

phases of similar absorption that result in low con-

trast. X-ray refraction [16.50, 51] is an ultrasmall-angle

scattering (USAXS) phenomenon. Refraction contrast

has also been applied for planar refraction topogra-

phy, a scanning technique for improved nondestruc-

tive characterization of high-performance composites,

ceramics and other low-density materials and compo-

nents [16.52].

X-ray refraction occurs when x-rays interact with

interfaces (cracks, pores, particles, phase boundaries),

preferably at low angles of incidence, similarly to the

behavior of visible light in transparent materials, e.g.

lenses or prismatic shapes. X-ray optical effects can be

observed at small scattering angles of between several

seconds and a few minutes of an arc, as the refractive

index n of x-rays is nearly unity (n

∼

1–10

−5

). In other

terms, due to the short x-ray wavelength below 0.1nm,

x-ray light scattering is sensitive to inner surfaces and

interfaces of nanometer dimensions.

16.5.1 Refraction Effect

In analogy to visible optics the interaction of x-rays

with small transparent structures with dimensions above

several nanometers results in coherent scattering gov-

erned by wavelength, structural dimensions and shape,

local phase shift and absorptive attenuation. However,

differently from optical conditions, the refractive index

of x-rays near 1 causes beam deﬂections into the same

small-angle region of several minutes of arc as diffrac-

tion. Thus the resulting interference is due to phase

modulation due to the refractive index and the absorp-

tive and Raleigh diffraction, both of which depend on

the path length through matter.

However if the dimensions of the scattering ob-

jects are much larger than several tens of nanometers,

as is common in classical small-angle scattering, the

interference fringes are no longer observable by clas-

sical small-angle cameras as they are too narrow.

The resulting smeared angular intensity distribution

is then simply described by a continuous decay ac-

cording to the rules for refraction by transparent

media, e.g. applying Snell’s law [16.51]. This purely

geometrical refraction approach is appropriate for

small-angle x-ray (and neutron) scattering effects by

micrometer-sized structures and is applied in the fol-

lowing.

If ε is the real part of the complex index of re-

fraction n, ρ is the electron density and λ is the x-ray

wavelength, then n is

n = 1 −ε, with ε ≈ρλ

and ε

∼

−5

(16.30)

for glass under 8 keV radiation. In contrast to optics

convex lenses cause divergence of x-rays as n < 1.

Figure 16.71 demonstrates the effect of small-angle

scattering by refraction of cylindrical lenses: a bundle

of 15-μm glass ﬁbers (for composites) deﬂects a pin-

hole x-ray beam within several minutes of an arc. In

ﬁbers and spherical particles the deﬂection of x-rays oc-

curs twice, when entering and when leaving the object

(insert Fig. 16.71). The oriented intensity distribution is

collected by an x-ray ﬁlm or a CCD camera while the

straight (primary) beam is removed by a beam stop. The

shape of the intensity distribution of such cylindrical ob-

jects is a universal function independent of materials,

Oriented

small angle

scattering

by refraction

Detector/

film

Sample,

fibers

Collimation

Mo-K

, 20 keV

n =1–ε with ε ≈ρλ ≈ 10

–5

2θ

Refracted

beam I

(2θ)

Fig. 16.71 Effect of oriented small-angle scattering by re-

fraction of glass ﬁbers; n index of refraction, ε real part of

n, λ wavelength

Part D 16.5