ON THE QUESTION OF BETATRON STEREOGRAPHY

G. D. BRASLAVSKII, V. I. GORBUNOV

In a number of cases in nondestructive testing of materials and products it is necessary to have information not only about the presence of a defect, but also to know its location and dimensions.

Ordinary planar radiography cannot answer these questions, since it is two-dimensional, and one has to resort to stereoradiography, i.e., to obtaining one or two radiographs taken from two points of transillumination.

A stereoradiogram is usually obtained by moving the radiation source—the betatron—or the product under examination in such a way that the inspected region of the product falls, in both cases, within the field of the gamma-radiation beam. Naturally, the inspection time in this case increases considerably: first, because of the need for two exposures and, second, because of the time required to carry out the preparatory operations—moving the betatron or the product, installing the film, adjusting the betatron, etc.

The use of a two-beam betatron substantially reduces the time of stereoradiographic inspection, making it possible in some cases to bring it down to a time approximately equal to that of ordinary radiography (a small difference follows from the inequality of the volumes being inspected).

Let us consider possible variants of stereoradiographic inspection with the aid of a two-beam betatron (Fig. 1).

Variant “a”. A stereo pair is obtained by two successive exposures on two films, placed successively at the surface of the product.

Since transillumination is carried out on two films and the transillumination exposures (operation of both betatron beams) are separated in time, the radiographic sensitivity is calculated by the usual formula:

\[ \frac{\Delta x}{x} = 2{,}3 \cdot \frac{\Delta D B}{\gamma_D \cdot \mu x}, \tag{1} \]

where \(\Delta D\) is the minimum perceptible change in blackening density;
\(\Delta X\) is the corresponding change in the thickness of the product;
\(\gamma_D\) is the gradient of the characteristic curve of the film at density \(D\);

\(\mu\) is the total attenuation coefficient;

\(B\) is the buildup factor \(\left(B=\dfrac{I_{\mathrm{pr}}+I_{\mathrm{r}}}{I_{\mathrm{pr}}}\right)\)

(\(I_{\mathrm{pr}}\) and \(I_{\mathrm{r}}\) are, respectively, the intensities of the direct and scattered radiation at the film).

The time required for stereoradiographic inspection consists of the time of two exposures plus the time for changing the film and switching on the betatron

Fig. 1. Variants of stereoradiographic inspection using a two-beam betatron.

and, consequently, is somewhat reduced, since the need to move the betatron or the article is eliminated.

The depth of occurrence of the defect from the surface of the article facing the film is readily determined from the formula:

\[ x_y=\frac{(d'_2-d'_1)F}{a+(d'_2-d'_1)}\simeq \frac{d'_2-d'_1}{a}F, \tag{2} \]

where \(a\) is the distance between the targets;

\(F\) is the distance from the targets to the surface of the article facing the film (when the film is at the article, the distance to the film);

\(d'_1\) and \(d'_2\) are the distances between the images of the defect and the reference mark on the films.

Variant “b”. The film is placed at the surface of the article, and stereographic inspection is carried out with one exposure while the two betatron beams operate simultaneously on one film. This variant is used when stereoradiography is performed for the purpose of measuring the dimensions of a defect and the depth of its occurrence (without viewing in a stereoscope).

Since radiography is performed from two points (targets \(M_1\) and \(M_2\)), each defect will have two images on the film, and, depending on the size and position of the defect relative to the film, these images will be superimposed on one another or will be separated by some distance. For such an inspection mode, it is necessary to introduce into the formula for radiographic sensitivity a coefficient \(K\), characterizing the degree of overlap of the images. Then

\[ \left(\frac{\Delta x}{x}\right)_{\mathrm{I+II}} = 2.3\,\frac{\Delta D\cdot B}{\gamma_D \mu x}\cdot K, \tag{3} \]

where \(\gamma_D\) is the film gradient when two beams operate on one film,

and the index \((\mathrm{I}+\mathrm{II})\) of radiographic sensitivity indicates the operation of two beams.

When two beams are operating, for defects whose images are separated, the radiographic sensitivity is equal, from the image produced by beam I, to:

\[ \left(\frac{\Delta x}{x}\right)_{\mathrm{I}+\mathrm{II}}^{\mathrm{I}} = 2.3\,\frac{\Delta D}{\gamma D^{\mu}x}\cdot \frac{(I_{\mathrm{pr}}+I_{\mathrm p})_{\mathrm I}+(I_{\mathrm{pr}}+I_{\mathrm p})_{\mathrm{II}}} {I_{\mathrm{pr.I}}}, \tag{4} \]

from the image produced by beam II:

\[ \left(\frac{\Delta x}{x}\right)_{\mathrm{I}+\mathrm{II}}^{\mathrm{II}} = 2.3\,\frac{\Delta D}{\gamma D^{\mu}x}\cdot \frac{(I_{\mathrm{pr}}+I_{\mathrm p})_{\mathrm I}+(I_{\mathrm{pr}}+I_{\mathrm p})_{\mathrm{II}}} {I_{\mathrm{pr.II}}}. \tag{5} \]

If the beams have the same radiation intensity, then

\[ (I_{\mathrm{pr}}+I_{\mathrm p})_{\mathrm I}=(I_{\mathrm{pr}}+I_{\mathrm p})_{\mathrm{II}}=I_{\mathrm{pr}}+I_{\mathrm p} \]

and (4) and (5) are transformed into:

\[ \left(\frac{\Delta x}{x}\right)_{\mathrm{I}+\mathrm{II}} = \left(\frac{\Delta x}{x}\right)_{\mathrm{I}+\mathrm{II}}^{\mathrm{I}} = \left(\frac{\Delta x}{x}\right)_{\mathrm{I}+\mathrm{II}}^{\mathrm{II}} = \frac{2.3\,\Delta D}{\gamma D^{\mu}x}\cdot \frac{(I_{\mathrm{pr}}+I_{\mathrm p})_2}{I_{\mathrm{pr}}} = \left[ \frac{2.3\,\Delta D B}{\gamma D^{\mu}x} \right]\cdot 2, \tag{6} \]

i.e. \(K=2\).

Under the condition of equality of the beam intensities, for defects whose images on the film coincide (a defect at the film), the radiographic sensitivity is equal to:

\[ \left(\frac{\Delta x}{x}\right)_{\mathrm{I}+\mathrm{II}} = 2.3\,\frac{\Delta D}{\gamma D^{\mu}x}\cdot \frac{2(I_{\mathrm{pr}}+I_{\mathrm p})}{2I_{\mathrm{pr}}} = \left[ 2.3\,\frac{\Delta D B}{\gamma D^{\mu}x} \right]\cdot 1, \tag{7} \]

i.e. \(K=1\).

And for defects whose images overlap one another partially, \(K\) lies in the range from one to two \((1<K<2)\).

It is obvious that the degree of overlap of the defect images, and consequently also the magnitude of \(K\), depend on the extent of the defect in the direction of the targets \((d_{\mathrm d})\), and also on the distance of the defect from the film \((x_{\mathrm d})\). A simple relation is obtained from consideration of similar triangles \((\triangle M_1 d_1 M_2\) and \(\triangle a a_1 b)\) (Fig. 2):

\[ \frac{ab}{M_1M_2}=\frac{x_{\mathrm d}}{F}, \tag{8} \]

where \(ab=d_{\mathrm d}\) is the extent of the defect in the direction of the targets;

\(M_1M_2=a\) is the distance between the targets.

In order for the defect images to overlap, it is necessary that:

\[ \frac{ab}{M_1M_2}>\frac{x_{\mathrm d}}{F} \]

or the overlap condition is:

\[ \frac{x_{\mathrm d}}{d_{\mathrm d}}<\frac{F}{a}. \tag{9} \]

From consideration of the geometry of a two-beam betatron one can obtain the relations:

\[ a=2R\cos\alpha, \tag{10} \]

\[ F=R\cos\alpha\,\ctg(\alpha+\beta)\left(1+\frac{\sin 2\beta}{\sin 2\alpha}\right); \tag{11} \]

\[ A = R \cos \alpha \frac{\sin 2\beta}{\sin 2\alpha}\operatorname{ctg}(\alpha-\beta), \tag{12} \]

where \(R\) is the radius of the target installation;
\(2\alpha\) is the convergence angle of the \(\gamma\)-radiation beams;
\(2\beta\) is the angle cut out by the collimator.

Substituting the values of \(F\) and \(a\) from 10 and 11 into inequality (9), we obtain:

\[ \frac{x_{\mathrm{d}}}{d_{\mathrm{d}}} < \frac{1}{2}\operatorname{ctg}(\alpha-\beta) \left(1+\frac{\sin 2\beta}{\sin 2\alpha}\right). \tag{13} \]

For the selected beam geometry of a two-beam betatron, the quantity on the right-hand side of the inequality is constant and equal to \(C\).

The condition of superposition can then be written as:

\[ \frac{x_{\mathrm{d}}}{d_{\mathrm{d}}} < C . \tag{14} \]

Defects of the “elongated” type, such as cracks, fractures, etc., as well as defects at the surface of the product facing the film, most often satisfy inequality 14; for them \(K\) is approximately equal to unity (\(K \simeq 1\)), and the radiographic sensitivity is not worsened.

Fig. 2. Determination of the condition for superposition of defect images

To compare the radiographic sensitivities of the single-beam \(\left(\dfrac{\Delta x}{x}\right)_0\) and two-beam \(\left(\dfrac{\Delta x}{x}\right)_{\mathrm{I}+\mathrm{II}}\) modes, it is also necessary to take into account that illumination by the second beam, increasing the optical density of blackening, also affects the gradient of the film (in the formula for radiographic sensitivity, \(\gamma_D\) is replaced by \(\gamma'_D\)).

If, in the single-beam mode, the exposure \((It)\) is chosen so that \(\gamma_D = \gamma_{D\max}\) is achieved (operation on the linear portion of the characteristic curve), then, for the same irradiation time in the two-beam mode, since the increase in optical density due to the second beam does not lead to an increase in \(\gamma_D\) (\(\gamma_D = \gamma'_D\)), the radiographic sensitivity for defects whose images are separated will be 2 times greater, and for defects whose images on the film are superposed on one another, the radiographic sensitivity remains the same as in the single-beam mode.

In the case of operation in the single-beam mode on the nonlinear part of the characteristic curve \((\gamma_D < \gamma_{D\max})\), illumination by the second beam in the two-beam mode increases the optical density of blackening of the film and thereby \(\gamma_D\) \((\gamma'_D > \gamma_D)\), reducing the difference in radiographic sensitivities.

...capacities of the single-beam and two-beam modes. Thus, if in the single-beam mode one worked at \(\gamma_D = \dfrac{1}{2}\gamma_{D\max}\) and, with illumination by the second beam, \(\gamma'_D\) was brought up to \(\gamma_{D\max}\), then for defects whose images are separated the radiographic sensitivity remains unchanged in the single-beam and two-beam modes, while for defects whose images overlap, the radiographic sensitivity turns out to be even lower because of the transition to the linear part of the characteristic curve.

In stereoradiography on a single film, in order to determine the depth of occurrence of a defect there is no need to set a comparison mark, and (2) is transformed into

\[ x_{\text{d}}=\frac{dF}{a+d}\simeq \frac{dF}{a}, \tag{15} \]

where \(d=d'_2-d'_1\) is the distance between the images of the defect.

The time required for stereoradiographic inspection of an article according to variant “b” is approximately equal to the inspection time in ordinary radiography, since the stereoradiogram is obtained in one exposure with simultaneous operation of both beams of the betatron.

To avoid an excessively high blackening density of the film, the exposure for radiography in the two-beam mode is, as a rule, made shorter than in the single-beam mode.

Variant “c”. The film at a distance \(A\) from the article. Moving the film away from the surface of the article to a distance \(A\), where the gamma-radiation beams diverge, makes it possible, with simultaneous operation of the two beams of a two-beam betatron, to obtain an enlarged stereopair in one exposure.

With this position of the film, owing to a reduction in the amount of scattered radiation reaching the film, the radiographic sensitivity is improved.

The deterioration of image sharpness associated with an increase in geometric unsharpness because of the small size of the betatron focal spot is slight.

The depth of occurrence of a defect from the surface of the article facing the film is determined by the formula:

\[ x_{\text{d}}= \frac{(d_2-d_1)F}{a\left(1+\dfrac{A}{F}\right)+(d_2-d_1)} \simeq \frac{(d_2-d_1)F}{a\left(1+\dfrac{A}{F}\right)}, \tag{16} \]

where \(d_1\) and \(d_2\) are the distances between the images of the defect and the comparison mark on the displaced films.

The exposure duration, as a result of moving the film away from the source, increases in accordance with the inverse-square law:

\[ t_2=t_1\left(\frac{F+A}{F}\right)^2 =t_1\left(1+\frac{A}{F}\right)^2, \tag{17} \]

where \(t_1\) is the exposure time with the film at the article;
\(t_2\) is the exposure time with the film at a distance \(A\) from the article.

Substituting into (17) the values of \(F\) and \(A\) from expressions (11) and (12), we obtain:

\[ t_2=t_1\left(\frac{\sin 2\alpha}{\sin 2\alpha-\sin 2\beta}\right)^2 . \tag{18} \]

Despite the increase in exposure duration in variant “c,” the total time of stereoradiographic inspection when examining...

the inspected object in the stereoscope, depending on the ratio of the angles \(2\beta\) and \(2\alpha\), may be equal to, greater than, or less than the inspection time according to variant “a,” since inspection according to variant “a” requires two exposures.

Thus, for example, for a two-beam betatron B 30/2 at an energy of 30 MeV, \(2\beta=10^\circ\); at \(2\alpha=35^\circ\), \(t_2=2t_1\); at \(2\alpha<35^\circ\), \(t_2>2t_1\); at \(2\alpha>35^\circ\), \(t_2<2t_1\).

Conclusions

From the analysis of the considered variants of stereography with the aid of a two-beam betatron, it follows that:

1. Stereographic inspection can be carried out without moving the betatron or the object.

2. Stereographic images in one exposure can be obtained using the methods considered in variants “b” and “c.”

3. The best radiographic sensitivity is achieved when using the method of variant “c.”

4. The minimum time expenditure for stereoradiographic inspection occurs in variant “b.”

5. Stereo pairs for subsequent viewing in a stereoscope can be obtained when using variants “a” and “c.”

References

1. S. V. Medvedev, L. K. Tatochenko. Industrial Gamma Defectoscopy, Metallurgizdat, Moscow, 1955.