ARTICLE IN PRESS

0921-4526/$ - se

doi:10.1016/j.ph

�CorrespondiE-mail addre

Physica B 385–386 (2006) 917–920

www.elsevier.com/locate/physb

Optimising visibility for the neutron radiography of titanium and nickelgas turbine components

John Thorntona,�, Benedicta D. Arhatarib, Andrew G. Peelec, Keith A. Nugentb

aAir Vehicles Division, DSTO, 506 Lorimer St, Fishermens Bend, AustraliabSchool of Physics, The University of Melbourne, Victoria 3010, Australia

cDepartment of Physics, LaTrobe University, Victoria 3086, Australia

Abstract

An analytical expression for the visibility of radiographic images with phase and absorption contrast has been derived. It was used to

theoretically optimise the visibility of cracks in titanium and nickel for the case of neutron radiography. The image-to-object distance,

object-to-detector distance and the crack depths were varied in the optimisation. The different neutron refractive indices of the titanium

and nickel produced qualitative differences in the predicted visibility as a function of the spatial frequencies of the object. Significant

visibility from phase contrast was predicted for 50mm deep cracks in nickel and 200mm deep cracks in titanium.

Crown Copyright r 2006 Published by Elsevier B.V. All rights reserved.

Keywords: Neutrons; Phase contrast; Gas turbine engine materials

Contrast in radiographic images is produced by differ-ences in absorption (imaginary part of the refractive index)and phase (real part of the refractive index) [1–3]. Withcontact images only absorption contributes to contrast. Asthe distance of the film, or position sensitive detector(PSD), from the object is increased the phase contributionto contrast increases. The absorption contribution de-creases with film distance because of parallax and the non-zero source size i.e., the absorption image becomes blurred.Analytical expressions of the visibility of radiographicimages as functions of film distance, source size (s), and thedifferences in absorption and phase have been developed[4,5]. For the contrast due to phase to become apparent theincident radiation must be spatially coherent; radiationfrom pinholes or micro-focused sources [2,6] meet thisrequirement. Such sources emit spherical wavefronts. Thewavefronts are distorted as they pass through the object togive a distorted transmitted wavefront. For neutrons manymaterials like nickel (that have a refractive index less thanunity) this causes the wavefront passing through these

e front matter Crown Copyright r 2006 Published by Elsevie

ysb.2006.05.249

ng author. Tel.: +613 96267083; fax: +61 3 96267311.

ss: john.thornton@dsto.defence.gov.au (J. Thornton).

materials to advance in phase relative to that passingthrough empty space (air). However, any cracks in thesematerials will then produce slight dips or ‘phase shadows’in the advanced portion of the wavefront (Fig. 1, Niwavefront).The depth of the dips will be DnT where Dn is difference

of the refractive index of the crack and the material and T

is the depth of the crack. These slight dips mean thatdiffraction (Fresnel) and refraction will cause some of theneutrons to be scattered through very small angles and thusproduce contrast at large object to detector distances. Incontrast, neutron wavefronts passing through materialssuch as titanium (that have a refractive index slightlygreater than unity) are retarded and a crack thereforeproduces a slight peak in the transmitted wavefront (Fig. 1,Ti wavefront). For both titanium and nickel the intensity ofthe transmitted wavefronts will peak from the lowerabsorption of the cracks. The peak in intensity will alsoproduce diffraction.In Fig. 1 no attempt was made to show the overall

curvature of the wavefront, particularly as the curvaturewas small over the small distances shown. An additionaladvantage of the spherical geometry is the geometric

r B.V. All rights reserved.

ARTICLE IN PRESS

T

W

G

Z

Plate

Ti wavefront

ΔN T titanium

ΔN T nickel

Ni wavefront

Fig. 1. A schematic diagram of a crack in a metal plate. The crack has

length T, interfacial separation G, and projected width W. The radiation

source is below the plate and propagation is in the z direction. The

spherical nature of the wavefronts is not represented.

Table 1

Typical crack parameters as shown in Fig. 1

Material T (mm) G (mm) W (mm)

Ni 0.05 2 20

Ti 0.2 2 20

J. Thornton et al. / Physica B 385–386 (2006) 917–920918

magnification it provides [7]. The magnification is given bythe standard magnification equation

M ¼Z1þ Z2

Z1, (1)

where Z1 is the source to object distance, and Z2 is theobject to detector distance.

It is important to note that the total thickness of theplate is largely irrelevant to the contrast as it is thedifferences in phase and absorption over the crack lengththat determine the degree of contrast. The excess thicknessdoes decrease the overall intensity but with neutrons thereduction is usually negligible.

The analytical expressions for visibility [4,5] shouldenable the contrast to be optimised for real samples. In thispaper, they are applied to the imaging of cracks found inaircraft gas turbine engine components using thermalneutrons (0.4 nm). One of the components was composedof a titanium alloy (Ti 8Al–1Mo–1V) the other a nickelsuper alloy. It was anticipated that the differences in nickeland titanium’s refractive indices would lead to interestingdifferences in the images of cracks in the two materials.Phase contrast radiography promises to improve sensitivityto short cracks and cracks obliquely aligned to the incidentradiation. Crack sizing may also be possible. In rotatinggas turbine components cracks of p0.1mm can be asignificant hazard.

Typical values of crack widths (G and W in Fig. 1) areshown in Table 1. The Ti crack depth was increased from0.05 to 0.2mm before pronounced visibility (40.2) wasobtained.

It is the projected width, W, that is important in thisstudy as it defines the dip (or peak) in the transmittedwavefront. The interfacial separation G was found to benegligible and was, therefore, ignored in the followingarguments. To visualise 20 mm wide cracks it was antici-pated that the source size should be smaller. However,previous neutron experiments [6] had shown that even witha 0.2mm pinhole several hours exposure were required.

Thus, the largest possible pinhole that would not obscurethe image was selected—FWHM 20 mm.The visibility of features in an image is usually defined as

the ratio of the difference in the maximum and minimumintensity divided by their sum. For a sample with one-dimensional harmonic phase and attenuation variation ithas been shown that the visibility is given by [4,5]

VTIEðx;MÞ ¼ m0 exp �1

2

M � 1

M

� �2

x2 !

�M � 1

M

j0x2

NFexp �

1

2

M � 1

M

� �2

x2 !

, ð2Þ

where m0 is the difference in absorption between the crackand the neighbouring material, x is 2p times the ratio ofsource size and the spatial wavelength of interest in theobject, j0 is the phase shift, NF is the Fresnel number of thesource.The following equations define these terms m0 x, j0, NF

[8]:

m0 ¼ rsT=2, (3)

x ¼ s 2p=lobject, (4)

j0 ¼ l rbT=2, (5)

NF ¼ 2p s2=lZ1, (6)

where s is the one-sigma width of the source that isassumed to have a Gaussian intensity distribution, lobject isthe wavelength of the object’s modulation, l is thewavelength of the neutrons, r is the number density ofatoms, b is the bound neutron coherent scattering length, sis the neutron cross-section [9].In Eqs. (4) and (6) T is twice the amplitude of the one-

dimensional harmonic variation in thickness. If the one-dimensional harmonic variation corresponded to an arrayof cracks then T is again the length of the cracks in the z-direction.The first term in Eq. (2) gives the absorption contribu-

tion to the visibility or contrast. As the detector distance(Z2) is increased the exponential rapidly decays as the non-zero source size blurs the image. The second term in Eq. (2)gives the phase contribution to the contrast. It alsocontains an exponential that decays as the detector distanceincrease. However, this decay is initially countered by thegrowth of the preceding factor which can be expressed asZ2 r b T 2p (l/lobject)

2. Thus, the phase contribution to the

ARTICLE IN PRESSJ. Thornton et al. / Physica B 385–386 (2006) 917–920 919

contrast increases as the detector distance is increased untilit is also reduced by the decaying exponential term.

Eq. (2) was used to produce Figs. 2 and 3 using the datain Table 2 [9,10]. The projected width of the cracks was20 mm, which corresponds to a full spatial period of 40 mmfor use in Eq. (2). Fig. 2 shows the predicted visibility of thetypical cracks as a function of magnification. Fig. 3 shows

Fig. 2. The predicted visibility of a 40mm period variation in titanium and

nickel plates for Z1 ¼ 4 cm and source FWHM ¼ 20 mm.

Fig. 3. The predicted visibility as a function of x for plates of titanium and

nickel with Z1 ¼ 4 cm, Z2 ¼ 16 cm, and source FWHM ¼ 20 mm.

x ¼ 1.33 corresponds to features of spatial wavelength 40mm.

Table 2

Absorption and phase parameters for nickel and titanium (l ¼ 0.4 nm)

Material Phase shift (j0) Absorption difference

(m0)

Titanium �7.835 3.47� 10�3

Nickel 9.38 1.02� 10�3

Fig. 4. The predicted visibility as a function of x for plates of titanium and

nickel with Z1 ¼ 1mm, Z2 ¼ 1 cm, and source FWHM ¼ 20mm. x ¼ 1.33

corresponds to features of spatial wavelength 40mm.

the visibility as a function of x, which is equivalent tospatial frequency for the constant source size (FWHM20 mm).The nickel and titanium plots are similar in form despite

the differences in the sign of the phase shifts in Eq. (3).However, differences are evident in Fig. 4 where smaller Z1and Z2 are used (Z1 ¼ 1mm, Z2 ¼ 1 cm).With nickel the phase term of Eq. (2) subtracts from the

absorption term. For increasing x this leads to a dip beforea rise. In the case of titanium the phase term adds to theabsorption term (j0 is negative) and there is a continuousrise in visibility. The difference becomes apparent when thephase and absorption terms of Eq. (2) are comparable.However, a Z1 of 1mm will give an impractically smallfield of view. The Z1 ¼ 4 cm case will give a reasonablefield of view and is also small enough to allow reasonableexposure times. Previous experiments [6] with 0.2mmpinholes and Z1 ¼ 1.8m needed 4 h exposures, thus0.02mm pinholes with Z1 ¼ 4 cm should require 12minexposures. For a typical detector resolution of 50 mm [6] amagnification of � 5 should allow reasonable resolution ofthe 20 mm wide crack. The ability to detect cracks ofp0.1mm in nickel turbine components makes phase-contrast a useful tool for non-destructive inspection.The displacement, x, of the refracted rays on the detector

from their unrefracted position because of refraction canbe estimated simply. The phase shift numbers in Table 2correspond to shifts in the wavefronts of j0l/2p. This is0.997 nm for the dip in the nickel wavefront and 1.19 nmfor the peak in the titanium wavefront. Assuming atriangular geometry for the dip and peak in the twotransmitted wavefronts (Fig. 1) the displacement, x, wouldbe

x ¼ Z2 ð2j0l=pW Þ. (7)

Thus, at x ¼W/2 or 8.4 cm the nickel wavefront dip willproduce a crude focus, no crude focus is likely from peaks

ARTICLE IN PRESSJ. Thornton et al. / Physica B 385–386 (2006) 917–920920

as the divergent radiation will not intersect. To be visible x

must be greater than the detector resolution. Thus for adetector resolution of 50 mm Z2 must be greater than 42 cmfor the nickel and 50 cm for the titanium.

References

[1] S.W. Wilkins, T.E. Gureyev, D. Gao, A. Pogany, A.W. Stevenson,

Nature 384 (1996) 335.

[2] P. Cloetens, W. Ludwig, J. Baruchel, J-P. Guigay, P. Pernot-

Rejmankova, M. Salome-Pateyon, M. Schlenker, J-Y. Buffiere, E.

Maire, G. Peix, J. Phys. D 32 (1999) A145.

[3] K.A. Nugent, T.E. Gureyev, D.F. Cookson, D. Paganin, Z. Barnea,

Phys. Rev. Lett. 77 (1996) 2961.

[4] B.D. Arhatari, K.A. Nugent, A.G. Peele, J. Thornton, Phase contrast

radiography II: imaging of complex object, Rev. Sci. Instrum.

(accepted for publication).

[5] B.D. Arhatari, A.P. Mancuso, A.G. Peele, K.A. Nugent, Rev. Sci.

Instrum. 75 (2004) 5271.

[6] J. Thornton, P.J. McMahon, B.E. Allman, J.E. Murphy, K.A. Nugent,

D.L. Jacobson, M. Arif, S.A. Werner, NDT&E Int. 36 (2003) 289.

[7] V.E. Cosslett, W.C. Nixon, Proc. Roy. Soc. (London) B 140 (1952)

422.

[8] V.F. Sears, Neutron Optics, Oxford University Press, 1989 64pp.

[9] V.F. Sears, Neutron News 3 (3) (1992) 29–37 Available from: /http://

www.ncnr.nist.gov/resourcesS (5 June 2002).