Small gap width measurement with a finite X-ray source

S m a l l gap w i d t h m e a s u r e m e n t with a finite X-ray source B. Karp and G. Adam Nuclear Research Center- Negev, POB 9001, Beer-Sheva, Israel Received 30 June 1993; revised 3 January 1994

Theoretical work was carried out in order to assess the feasibility of measuring small gap widths by using X-rays. The work was done using optical geometry, and the effect of the various geometrical parameters (source dimension, geometrical enlargement, gap width, and film position relative to the no-umbra point) on the apparent gap size was investigated. The influence of some other parameters (eg full reflectance of the beam from the gap edges or misalignment of the source and the gap axis) is also discussed. The results, which are presented graphically and analytically, enable the user to plan a measurement in an optimal way and help in analysing the results correctly. Keywords: dimensional measurement, X-rays, gap width

Over the last few years X-ray radiography has been used to obtain quantitative as well as qualitative information about the object under study. Dimensional measurements have been reported in several works u-s] and some of the difficulties involved in these measurements have been outlined especially in the submillimetre range t3'5]. An important case where exact dimensional measurements by X-rays are needed is in the manufacture of fuel elements, where accurate non-destructive measurements of gaps, cracks and displacements are of the utmost importance t3]. In all the works cited there are no specific recommendations concerning the desired experimental setup and the analysis of the resulting radiograms. In this work we perform a parametric analysis (with some simplifying assumptions) for the two-dimensional case of the formation of an image on the film by radiation passage through a gap (Figure 1 describes a typical example). The measurement parameters were divided into two groups. The first is the group of independent parameters (source size, gap width, enlargement, gap thickness and misalignment between the source and the gap axis). The second consists of the dependent parameters which define the result obtained on the film (umbra, penumbra, F W H M , apparent source size in the umbra region Su and the apparent source size in the penumbra region SUg). Obviously, the assignment of the parameters to the groups can be altered, depending on the question at hand. Formulae which connect the dependent parameters to the independent ones were developed. Due to the nature of the problem it was found 0963-8695/94/01/0021-05 © 1994 Butterworth-HeinemannLtd

that different formulae apply to various measurement conditions as described in the following paragraphs.

Assumptions 1. We are dealing with a two-dimensional problem; alternatively, the source, the gap and the film are isotropic and have infinite length in the plane which is perpendicular to the plane of the paper. . The source is a line source of length 2S. It is a monochromatic source and has a constant intensity as a function of position. In reality the source is three-dimensional and emits a wide range of wavelengths tr]. 3. The gap has width 2d between two infinite absorbers of thickness h. 4. Fresnel diffraction was assumed to be negligible for our analysis. Fresnel diffraction contributes to image unsharpness t~'s]. In the case of parallel radiation passage through a gap in a thin wall this effect can be neglected t7] when: 1 >> FFD*2/2d

(1)

where FFD is the film focal point distance, 2 is the radiation wavelength and 2d is the gap width. For X-rays with a wavelength of 1/~, FFD of 1 m and d of 10 -4 m, Equation (1) holds. 5. The contribution of the film's inherent unsharpness was neglected. A linear response of the film was

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Small gap width measurement. B. Karp and G. Adam

The F W H M therefore has the form: FWHM

/

['~

Aligned

T--

~.."~ .....

t

_!_1 7M-1 --

./"

j_! ~ . I . ~r,.-~ ....

~

1 .,-~.

t.. l. tug ,n

\

I

/

J[-density

(7)

= 2d • M

and the height of the trapezium at the Ug region S U 9 as a function of its distance x from the axis of symmetry is given by: S U g oz S + d ( M / ( m -

1 ) ) - x ( 1 / ( m - 1))

(8)

1)

(9)

In case (b) we get: U = -2d,M

FeD

='

. I~

FFD

J l

UO = 2d • M

(1 O)

Su oc 2 d ( M / ( M - 1))

(11)

FNu Figure 1 Schematic drawing of X-ray passage through an "intermediate' gap with absorber thickness h. On the right-hand side of the film plane the density of the exposed film is drawn as a function of the distance from the axis of symmetry

assumed. One can use the actual response as given by the manufacturer if needed. We also assumed that the film is never exposed to saturation. 6. It was assumed that the absorber is completely opaque. It does not emit or scatter radiation (apart from total reflection which is dealt with separately). No radiation passes through it even when the radiation 'sees' only a thin edge at a corner.

+ 2S(M-

= 2 S ( M - 1)

FWHM

(12)

S U 9 remains the same as in the previous ease (Equation

(8)). Some insight to these formulae can be gained by plotting U and UO as a function of 2d (Figure 2). It can be seen from this figure that estimating the gap width from the F W H M value of the image leads to a wrong conclusion in the case of 'small' gaps. In this case the estimation should be carried out by analysing the U or U9 values, a procedure which is difficult to carry out and often needs microdensitometric measurements. It is also apparent from the formulae and the figure that to estimate a gap from one radiograph one should know the source dimension and approximate gap width in order to decide which formulae to apply. FWHM,

Gap between t w o absorbers of negligible thickness (h 0) =

The unsharpness U9 obtained when illuminating a sharp edge with a source of dimension 2S is given by[9]: U9 = 2 S ( M -

1)

(2)

M being the geometrical enlargement. A gap consists of two edges with a distance 2d between them. The film density (D) of the image (as a function of the distance x from the axis of symmetry) will have the shape of a trapezium with height Su, a small base of length U and a large base of length U + 2 , UO. The full width at half maximum (FWHM) of this trapezium is given by: FWHM

= U + Ug

Gap between t w o absorbers of thickness h ( > 0 ) The fact that the absorber has a thickness h which is not zero is equivalent to the case of two thin absorbers with a distance h between them. The resulting image will be built from the images produced by the 'two gaps'. This introduces another parameter M' which characterizes the

"Small"gap region

=s -r

"Large"gap region

(3)

Trying to express these quantities in terms of the parameters M, 2d, 2S one finds that there is a need to distinguish between two cases: (a) A 'large' gap (S < d or S > d and F N u > F F D ) (b) A 'small' gap (S > d and F N u < F F D )

2S{M-1)

~ ".~. "~,..

/ 1 "/

..J

where F N u is the focal no-umbra distance (Figure 1).

Ug = 2 S ( M Su oc 2S

22

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1)

../

..

j..U/" ,#

In case (a) we get from geometry the following relations: U = 2 d , M - 2 S ( M - 1)

J

///

.. . f . . J ' " 2S(MI~I)

(4)

2d

Gapwidth

(5)

Figure 2 FWHM, U and Ug as a function of 2dfor a source length of 2S, geometrical enlargement of M and thin absorber thickness

(6)

(h=0)

I n t e r n a t i o n a l 1 9 9 4 V o l u m e 27, N u m b e r 1

Small gap width measurement: B. Karp and G. Adam The formulae are:

image. M' is defined as: (13)

M ' = F F D / ( F O D - h)

For h = 0, Equation (13) yields M = M'. When h is not zero we find that we have to define four different geometrical cases. (a) 'Large' gap (S < d). In this case the situation is identical to that of a large gap with h = 0, and the same formulae apply. (b) 'Intermediate' gap (S > d and F N u > FFD). F N u is given by: F N u = (FOD - h)/(1 - d/s)

(14)

It follows from this equation that F N u is greater than the F F D when (15)

S(M' - 1)/M' < d < S

Inequality (15) defines the region where the formulae for an intermediate gap are valid. Calculating the film density at each point for this case shows that the Ug area is now divided into two areas: Ug~n with density SUo~,, and Ugout with density SUgo,.,. The shape is now two trapezia, a small one above a bigger one. The large base of the larger trapezium is of length U + 2Ug (where UO = Ugi,, + UO.=), the large base of the smaller trapezium is 2d = U + 2Ugi,, and the small base of the small trapezium is U (Figure 1). The relevant relations are: UO = 2S((M' + M ) / 2 - 1) - d ( M ' -

M)(16)

U = 2dM' - 2 S ( M ' - 1) F W H M = d(M' + M ) - S ( M ' -

(17) (18)

M)

SUgom oc S + d ( M / ( M - 1)) - x ( 1 / ( M - 1))

for x > d

(19)

SUg~n oc S + d ( M ' / ( M ' - 1)) - x ( 1 / ( M ' - 1))

forx
(20) (21)

Su oc 2S

(c) 'Small' gap (S > d and F N u < FFD). The formulae for this region are valid when: S(M' - M)/(M' + M) < d < S(M'-

1)/M'

(22)

This case itself is divided into two subgroups. U, Ug and F W H M are common to the subgroups and are defined as: Ug = d(M' + M ) - S ( M ' - M ) U = 2 S ( M ' - 1) - 2dM'

(23) (24)

F W H M = 2S[(M' + M ) / 2 - 1] - d(M' - M )

(25) The difference between the two subgroups is in the way the small trapezium divides the illuminated zone. When U < 2d the Ug area is divided to Ugin and Ugo= with illumination (or density) S U g i , and SUgo= respectively. When U > 2d the Ug area is uniform with density S U g , and the U area is divided into U~ and Uo~t with respective illumination SU~n and SUo,t.

for U < 2d SUgou t oC S q- d [ M / ( M - 1)] - x [ 1 / ( M - 1)]

(26) SUgin oc. S + d [ M ' / ( M - 1)] - x [ 1 / ( M ' - 1)3

(27) Su oc 2 d [ M ' / ( M ' - 1)]

(28)

U + 2Ug~n = 2d

(29)

whereas for U > 2d S U g oc_ S + d [ M / ( M - 1)] - x [ 1 / ( M - 1)]

(30)

SUout OC d [ M / ( M - 1) + M ' ( M ' - 1)] - x[M/(M-

1 ) - M ' / ( M ' - 1)]

Sui, oc. 2dCM'/(M' - 1)]

(31) (32)

Ui. = 2d

(33)

(d) 'Very small' gap (S >> d) is a situation where the following condition holds: e < d < S(M' - M ) / ( M ' + M)

(34)

where e is a small number for which assumption 4 is no longer valid. In this case the source is so large compared to the gap that the edges of the source are never seen by the film. As a result, the effective source size is given by: 2S = 2dl-(M' + M ) / ( M ' -

M)-]

(35)

Inserting this source size into Equation (23) we get that Ug is zero, ie the unsharpness region disappears. We get in this case only the secondary (small) trapezium which is produced by the division of the U region into two parts. The first part consists of a centre area where the density is really uniform. Its dimension is 2d and the density is Suin. The second part consists of the areas at the edge which, in a sense, become the unsharpness zone because the density there decreases linearly to zero at the edge. The dimension of this area is (U - 2d)/2 on each side, and the density is Suout. U, Ug and F W H M , as a function of 2d, are plotted in Figure 3 for all the regions. This is, in a sense, a generalization of Figure 2 for the case M < M' (see expression (13)), apart from the case of a 'very small' gap which does not exist when h = 0 (M = M').

Looking at Figure 3 and the different formulae, a few points are worth mentioning: (1) It can be seen from the formulae that when h is assumed to be zero, the accuracy of the results depends on the ratio M ' / M . When this ratio is near 1, the assumption is valid. (2) The slopes of the trapezia are independent of the gap width 2d and are a function only of M and M' which are known independently from the system geometry and the film exposure. This fact can be used when

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Small gap width measurement: B. Karp and G. Adam

/

"Small. . . . gap region

Intermediate" gap

reg

/

/"

1 >> a/FOD

"Large" gap

.,

/FWHM

/"/"

region

:

SI2M-I-~,) -

\ 2S(M-1)

\/

//

,,/'\,\ 2S(M~,~M)

/

---.og

"---

..

2S(~)

2S

(36)

an assumption which is usually true. In this case, when h = 0 the value o f - a - has no influence on the image, so that we will deal only with the ease of non-zero h. In that case we will deal with two setups: (a) A 'large' gap (S < d). We have here three different regions: the first when - a - is small enough so that the orthogonal projection of the source on the gap plane is fully inside the gap [a < (d - S)]. The second is when - a - is such that the projection of the source is partially beyond the gap (d - S) < a < (d + S). The third region is when - a - is large enough so that the whole projection of the source is outside the gap but not too far out so that part or all the source will still be visible from the film (S + d ) < a < [d(2FOD/h - 1) - S].

/ i..1

2S(M,Z~M-1)-

between the centres. The nature of the problem is such that each case discussed above is now divided into three subcases, according to the value o f - a - . We will only deal with a few examples (other cases can be analysed in a similar way). We assume that:

2d

Gap width

F i g u r e 3 FWHM, U and Ug as a function of 2dfor a source length of 2S, geometrical enlargements of M and M' with absorber thickness h > 0

analysing the density distribution of the film. By plotting the trapezium's slope more accurately, U and Ug can be determined with a higher precision. (3) The different formulae developed for Su in different regions and different exposure geometry enables one to determine ratios of exposure values for the different cases. Thus, from one density measurement, conditions for future exposures can be calculated for optimal results. (4) As metals can have an index of refraction smaller than 1 for X-rays tal, total reflection of the X-rays from the edges of the absorber can occur for grazing angles of a few arc minutes. This effect can cause rays which, rather than being absorbed by the edges, hit the film and change the density distribution calculated above. This effect is important mainly in the case of smooth surfaces [l°]. One can estimate the contribution of this effect by calculating the index of refraction for the specific material of the absorber using the formula in Reference 8 and assuming monochromatic X-rays and a completely smooth surface with the aid of simple geometry. A more detailed examination of this problem can be found in References 10 and 11.

We will not deal with cases where - a - is still larger. The formulae for these cases are derived directly from geometry and will not be given here. By differentiating the formulae for F W H M and U with respect to - a - , one obtains the dependence of these values on a . It turns out that all the derivatives are proportional to M' - M so that the greater the ratio M ' / M the greater is the influence o f - a - on the resulting image. When M = M' the value o f - a - h a s no influence on the image. Figure 4 describes these values as a function o f - a - . (b) A 'small' gap (S > d). We have here the same regions as above and Figure 5 shows the qualitative dependence of the values of F W H M and U on the value o f - a - for this case. Looking at Figures 4 and 5 one sees that the value of - a - can have opposite influence on F W H M and U according to the relative sizes of a, d and S. One has also to keep in mind that in the case of asymmetry the total reflection effect mentioned above might be of special importance, depending on the specific geometry.

-M

FWHM

21(M'-M)

Effects of asymmetry Until now it has been assumed that the centre of the source was aligned with the centre of the gap. In practice, these two centres will never be aligned and we will now discuss the influence of a misalignment of length - a -

24

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d-S

S÷d

d(2F- -D-1}-S

a

F i g u r e 4 Qualitative dependence of FWHM and U on the misalignment - a - for a 'large' gap (S < d)

Small gap width measurement." B. Karp and G. Adam

M'-M M'-M

i

FWHM I(M'.M) 1, O

S-d

S+d

d(2-~D--1}-S

a

Figure 5 Qualitative dependence of FWHM and U on the misalignment - a - for a 'small' gap (S > d)

Summary We have seen that in order to estimate correctly the width of a gap between two absorbers a detailed understanding of the geometry involved is necessary. Just measuring the width of the radiographic image and calculating the actual dimension by using the geometrical enlargement can lead to erroneous results. This understanding is also needed for good planning of an experimental setup for gap measurement. The available equipment and its characteristic (eg source size) together with the expected gap width should be taken into consideration when planning the geometry of the measurement. The geometrical enlargement should be chosen in such a way that the whole range of the expected gap widths will fall into one category of gap specification (small, intermediate, large etc.). Drawing of a graph of the type of Figure 3 for the actual parameters of the system will help in planning the measurement and in analysing the resulting radiograph. Assessing the form of the expected image (one trapezium or two and their slopes) from the

relevant formulae will help in deciding in advance if the parameters needed will be measurable from the radiographs with the available equipment. An assessment of the importance of the total reflection effect and the effect of misalignment (a) should be carried out and steps should be taken to minimize them if possible (minimizing - a - by accurate positioning or changing the gap into a 'large' one by suitable geometry and source size). It should, however, be borne in mind that before using the results of this work, one should verify that the assumptions on which this analysis was based are valid for the problem at hand. Taking all these steps might help to obtain a more accurate measurement of gap dimensions.

References 1 Segal, E., Notea, A. and Segal, Y. 'Dimensional information through industrial computerized tomography' Mater Eval 40 (1982) pp 1268-1272 2 Cufforth,D.C. 'Dimensioningreactor fuelspecimensfrom thermal neutron radiographs' Nucl Technol 18 (1973) pp 67-70 3 Notea, A., Segal, Y. and Triehtcr, F. 'Gap's dimensions in fuel elements from neutron radiography' Proc 6th Int Conf on NDE in Nuclear Ind (1983) p 433 4 Segal, Y. and Triehter, F. 'Gap width measurements in fuel elements' N D T Intern 22 4 (1989) pp 222-228 5 Segal, Y. and Trichter, F. 'Limitationsin gap width measurements by X-ray radiography' N D T Intern 21 I (1988)pp 11-16 6 Madsen,J.U. 'Focal spot size measurementsfor microfocusX-ray sets' N D T Intern 22 5 (1989)pp 292-296 7 Yavorsky,B. and Detlaf, A. Handbook o f Physics, MIR, Moscow, 3rd edition (1950) 8 Feynman, R.P, Leighton, R.B. and Sands, M. The Feynman Lectures on Physics Vol I, Addison-Wesley,Reading,MA (1963) 9 Kh'kpatrick, P. and Pattee, H.H. 'X-ray microscopy' in X-Ray, HNBKder Physik, ed S. Flngge,Vol. XXX, Springer-Verlag,Berlin (1957) 10 Hogrefe, H. and Knnz, C. 'Soft X-ray from rough surfaces: experimental and theoretical analysis' Appl Opt 26 14 (1987) pp 2851-2859 11 Hogrefe, H., Haelbich, R. and Kunz, C. 'Specular and diffuse reflection of soft X-rays from mirrors' Nucl Inst Meth Phys Res A246 (1986) pp 198-202

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