Measurements of micro-focus spot size (1-10 μm) for X-ray. tubes and Kumakhov lenses.

Transcription

1 Measurements of micro-focus spot size (1-10 μm) for X-ray tubes and Kumakhov lenses. V.Ya. Shovkun 1 Institute for Roentgen Optics, 1Volokolamskii pr.10, Moscow, Russia. ABSTRACT X-ray lenses and X-ray tubes that are produced at our Institute should be certified in accordance with any test method or standard procedure in the energy range from 6 kev to 50 kev. Early it was demonstrated that five different measurement techniques (the pinhole, the slit, the edge, the scanning and the grid- wire methods) can be exploited for experimental estimation of focal spot sizes. It had been shown that the wire and edge method were rather good for estimation of focal spot sizes in the range from 10 μm to 50 μm. Now we are developing the wire and edge method for focal spot size measurements in the size range between 1 μm up to 10 μm for X-ray tube operating up to 50 kv high voltage and power W. The aim of developing the facility for X-ray tube and X-ray lens focal spot size measurements at our Institute is providing our X-ray tube and lens- designers and engineers with metrology and arrangements. Some experiments and numerical simulation have been performed to estimate accuracy of the above methods. The arrangement consists of an X-ray tube and X-ray lens under investigation, set of W-wires with Au-coating or mercury liquid drops as test-objects, X-ray CCD- cameras, and optical rails with rotation and translation stages. Agreement between the wire and edge methods of measurements and simulated data is rather good in the considered focal spot sizes range with an accuracy ±20%. Keywords: X-ray lens, X-ray tube, focal spot size, mercury liquid test-object. 1 shov@iroptic.ru, chovkoun@mail.ru. Tel.:

2 1. INTRODUCTION Though, there are several distinguishing features between X-ray fields generated by an X-ray lens and an X- ray tube, we are exploiting the same methods of focal spot size measurements in both cases. These distinguishing features are as follows: 1) in lenses, there are both divergent and convergent regions of X-rays, 2) in lenses, there is chromatic dependence of the X-ray focal spot size on the focus distance, 3) the width of angular distribution of focused X-rays in lenses can be rather small (from radians and up) and rather irregular (from 1% and up). There are several recommendations and standards for measuring focal spot sizes of X-ray tubes [1-11]. All of these standards are simple to perform [7-11] but demand special laboratory equipment [1-6]. The Russian standard [1] allows to be used for radiographic measuring focal spot size smaller than 100 μm by means of grid-mesh method. This standard declares the accuracy of evaluation of the focal spot dimensions within ±35% range in the confidence interval. But this procedure can give a poor accuracy under estimate focal spot sizes in the range from 1 μm to 10 μm if the used W-wire has a diameter smaller than 10 μm. The British and European standard procedures [2, 5] describe radiographic method for measuring effective focal spot size above 5 μm. Unfortunately, in these standard procedures, the accuracy of evaluation of the focal spot dimensions is not specified. The other standard procedures [3, 4, 6] describe radiographic and radiometric methods for measuring effective focal spot size above 100 μm, and so, can not be used for our purposes. All above method do not allow a real-time determination of focal spot size (i. e. on-line measurements). For X-ray lenses there is not any dedicated standard procedure of measuring focal spot sizes, and one commonly used method is knife-edge scanning [12]. In this case the knife with sharp edge is scanned across the output X-ray beam of the lens near focal plane. The X-ray intensity deviation is recorded by a detector as a function knife edge position. Then the size of output beam can be obtained from the derivative of the recorded intensity profile curve. Having performed several such measurements for different distances from the tip of the lens it is possible to determine the true size of focal spot. The disadvantages of knife-edge method are: 1) it needs precision moving translation stages and precision alignment of all apparatus parts, 2) considerable time for conducting the experiment is required, typically, a quarter of an hour, and 3) it demands long-time vibration stability all parts of the apparatus. 2

3 The aim of the present study was the further development radiometric method of a real-time determination of focal spot sizes in lenses and tubes [13] in the sizes range from 1 μm to 10 μm with an accuracy of 20-30%. 2. METHODOLOGY 2.1. Commonly used methods for X-ray tubes The simplest and more used method of a nominal feature definition of focal spot size is the measurement of the intensity profile of sharp edges of a ball or wire that are performed with heavy metals (Pt, Au, Pb, Ta, W and so on). The physical situation is shown in Fig.1. Incident photons fall on the W-wire or the ball with a diameter d. X-rays entering the wire are absorbed and scattered. If the absorption predominates over the scattering, the image of the wire will be essentially shadow picture of the wire. Another part of X-rays from the X-ray source travels by air and then reaches the detector. The scattered and secondary radiation from the wire, air and construction materials contributes to the recorded wire image and thus reduces image contrast of shadow picture of the wire. Hence, this scattered and secondary radiation is unwanted. R0 X-ray source f d R1 I(y) Intensity, arb. units Detector A) Test-object plane B) U g IMAX E C A B D U g F1 Position, arb. units. F2 B0 y Fig.1. A) Geometric illustration for focal spot size measurements. B) Schematic drawing of the measurements of the wire image intensity profile, I(y). The measured intensity curve (y) has three remarkable regions: 1) (y) = B 0 is the geometrical shadow or background field; 2) B 0 < (y) < (I max +B 0 ) is the penumbra region; 3) (y) = (I max +B 0 ) is the brightest part of the W-wire image field. The source of X-rays used in the radiographic imaging process is not a point source, but is rather distributed over some spatial domain, f. The combination of the X-ray source intensity distribution and the source to object distance R 0, and the object to detector distance R 1 causes a blurring of the X-ray pattern because of penumbra effects. All information about dimensions of a focal spot can be obtained within the transient region of the geometric un-sharpness Ug,, with Ug = (G-1) f. Here G is the projective magnification, and G = (R 0 +R 1 )/R 0. 3

4 The grid-mesh method. The Russian standard [1] This standard describes the method for measuring the effective X-ray tube focal spot size smaller than 100 μm. The test object shall consist of accurately machined wire meshes of tungsten or materials with atomic numbers Z more than 42. The grid spacing has an accuracy of ±5%, and the wire diameter accuracy is factoryinstalled. The projective magnification G should be taken more then x50, if f < 10 μm, and G should be determined with an accuracy better than ±10%. Diameter of the wires d shall be at least three times more than the focal spot size f. The micro densitometer shall permit contrast measurements of optical density with an accuracy of ΔD=±0.02 (i.e. ±2% for X-ray films with the value of film gradient K=2.3). Evaluation of actual focal spot dimensions is performed according to the equation: f^= (F 2 - F 1 )/(G-1) (1), where the projective magnification G can be determined from the equation G=IS/AS. The value of IS is the image of the grid spacing of the wire mesh and AS is the actual grid spacing of wire mesh. Other values are defined as follows. F 1 is the full width of the W-wire image defined at the level 0.5 I max and F 2 is the full width of the W-wire image defined at the level 0.9 I max (typically, this value is taken from the manufacturers specification). The accuracy of evaluation of the focal spot dimensions is established within ±35% range in the confidence interval. In this method we do not need to define distances R 0 and R The British standard procedure [2] The British standard describes a method for measuring effective focal spot size within the range of 5 μm to 300 μm of X-ray tubes operating at 50kV to 200kV. The test object shall consist of an accurately machined, high precision spherical ball of tungsten carbide, having a diameter of 5 mm. The projective magnification G should be taken more then x20 and G can be obtained by measurement of the diameter of the image on the radiograph, knowing the true diameter of the ball. The actual focal spot dimension can be obtained using the equation f= ED/(G-1), where ED is defined by means of the plot optical density versus the distance as follows (see Fig. 1B). Measure the height I max between the maximum and minimum densities close to the edge trace and draw horizontal lines at 0.25 I max and 0.75 I max to cut the density trace at A and B. The straight line, AB, is then projected to cut the maximum and minimum density lines at C and D. The horizontal distance between C and D, that is ED, is a measure of the focal spot diameter, f. The parameter G is defined as G=F 1 /d. In this case, we do not need to determine distances R 0 and R 1. When the magnification G is evaluated via the 4

5 equation G= (R 0 + R 1 )/ R 0, we need to measure distances R 0 and R 1. Note, that the accuracy of a ball and the roughness of the ball surface are not specified. The accuracy of evaluation of the focal spot dimensions is not declared in the British standard The European standard [5] The European (also approved as the British) standard specifies a method for the measurement of focal spot dimensions within the range of 5 microns to 300 microns of X-ray systems up to and including 225kV tube voltage, by means of radiographs of sharp edges. This method is based on indirect measurement of the focal spot size by measuring the geometric un-sharpness. For this purpose sharp edges are imaged either on a film or by means of a radioscopic device using a relatively high geometric magnification. The test object shall be either a cross wire or a ball consisting of highly absorbing material (e.g., tungsten, tungsten alloy or platinum), having a diameter between 0.9mm and 1.1mm, which has an accuracy of ±0.01 mm. The distances R 0 and R 1 shall enable projective magnification G between x20 and x50. The minimal distance R 0 shall be at least five times the wire or ball diameter, d. Evaluation of actual focal spot dimensions is performed according to the equation f=(f 2 - F 1 )/G, where magnification G is defined as G=F 1 /d. Here, F 1 is the full width of the W- wire image defined at the level 0.5 I max, and F 2 is the full width of the test object image defined at the level 0.9 I max. The parameter G can also be determined from the equation G = (R 0 +R 1 )/R 0. In the last case we have to define distances R 0 and R 1. Note, that the roughness of the wire (ball) surface is not specified. The accuracy of evaluation of the focal spot dimensions is not declared in the European standard also Agreement on focal spot sizes for X-ray tubes Different standards use different levels of the measured intensity profile to evaluate the focal spot size, and so, these values differ one from the other too. If the focal spot intensity distribution of emitted X-rays follows the Gauss law, then it becomes possible to express values f^rus, f^gb and f^eu via the common value ζ: f^rus = 2.56 ζ [1], f^gb = 2.7 ζ [2], and f^eu = 3.3 ζ [3], where ζ represents the Gauss intensity distribution parameter (standard deviation), and FWHM=2.36 ζ. The focal spot size can be also determined by using the first derivative of the intensity profile- I'(y), which is the focal spot intensity distribution of emitted x-rays (for y Є {Ug}). Here focal spot size can be defined either as the full width at half maximum of the curve I'(y): FWHM=2.36 ζ, or as the full width at tenth maximum of the curve I'(y): f* GB = f* EU = FWTM = 4.3 ζ [3]. 5

6 2.2. The wire-edge method [13] We are developing a method for measuring effective focal spot size within the range 1 μm up to 10 μm of X- ray lenses and X-ray tubes operating at 10kV to 50kV. The method is based on indirect measurement of the focal spot size by measuring the geometric un-sharpness. For this purpose the image of sharp edges of the test object (W, Ta, Au, Pt- wires or plates, and mercury liquid drops) are recorded either by means of a digital CCD-camera or with mechanical scanning of a collimated detector with pinhole (or narrow slit) at the entrance. The wires have diameters between 30 μm and 300 μm which were verified with an accuracy of ±1μm. The evaluation of the actual focal spot dimensions is performed according to the equation: f^ = (F 2 - F 1 )/ (F 1 /d-1) (2), where d is the W-wire diameter; F 2 is the full width of the intensity profile of the W-wire image defined at the level 0.9 I max and F 1 is the full width of the intensity profile of the W-wire image defined at the level 0.5 I max As it follows from the equation (2) this method does not require measurements of distances R 0 and R 1 (see Fig.1). Besides an estimate f^ which is determined from the equation (2), we can also determine an estimate f*, which is defined as the full width at half maximum of the first derivative I'(y): f* = FWHM Theory and experiment of the method proposed With a good approximation, measured profile (y) can be represented as sum of the true intensity profile of the W-wire image- I(y), the correction term of the recorded image- C(y), and background- B(y): (y) = I(y) + C(y) + B(y) (3) The true intensity profile I(y) can be obtained if terms (y), B(y) and C(y) are known. The profile I(y) can be described as convolution of functions: W(E), F(r'', E), exp [-µ(e) L(y, r', r'')], and ε (E). Here W(E) is the X- ray source spectrum, and E is quanta energy. The function F(r'', E) is the Gaussian intensity distribution of emitted X-rays at the r''-point of the X-ray source. The term exp [-µ(e) L(y, r', r'')] is the energy-flux intensity transmitted by the wire object, where L(y, r', r'') is the X-ray path through the W-wire. Values of y, r' and r'' are coordinates of points in the image plane, the object plane, and the X-ray source plane, respectively. The term µ(e) is the X-ray linear attenuation coefficient. The term ε(e) is the efficiency of the X-ray detector. The convolution in the equation (3) is performed with respect to variables E, r' and r''. 6

7 In practice, the shape of the intensity distribution, I(y) is complicated by the background radiation and interactions of primary X-rays with the W-wire. These interactions are: coherent and incoherent scattering, and fluorescence radiation. Here we should also take into account specific interactions of incident X- rays with borders of the wire: reflection, refraction, and diffraction. It can be shown [13] that the contribution of the above interactions to the measured intensity profile (y) can be reduced to values not greater than one percent of I(y) in the penumbra region (y Є {Ug}), and so we can neglect the above interactions. Besides, we can express the influence of all above interactions on the measured intensity (y) as the correction function C(y) for our model (3): C(y) k(y) I(y), where k(y) is the correction factor, and k(y)<< 1 for y Є {Ug}. On the contrary, the scattered and secondary radiation that originates from interactions of primary X-rays with air, construction materials, and collimators can be rather high. This component of background radiation is denoted as B(y) in the equation (3). The background B(y) should be decreased to the minimal value, as much as possible in experiments. The shape of the focal spot intensity distribution of emitted X-rays is usually complicated and varies with many parameters such as tube current, tube voltage, and so on. In our model (3), we assume, that the shape of the X-ray tube focal spot is the round and the intensity distribution of emitted X-rays obeys the Gauss law with the full width at half maximum FWHM = 2.36 ζ, where ζ represents the Gauss intensity distribution parameter (standard deviation). In order to check validity of equations (2) and (3) for obtaining estimates of f^, we performed several numerical model experiments. First, we simulated intensity profiles I TH (y) for different values of R 0, R 1, d, and f in accordance with our absorption model of the W-wire image. In these calculations we have used published data of an X-ray Mo-tube operating at 30kV [14, 15], and tabulated data for X-ray absorption coefficients in tungsten [16]. Then we used the calculated profile I TH (y) to find values of F 1 and F 2 graphically. Finally, we evaluated the estimate f^ in accordance with formula (2) and found the difference between values of f and f^ as the measure of the method s error (available accuracy). The simulations show that the method s error is ranging from 0.1% to 20% in the wide range of changing values of R 0, R 1, d, and f that are of interest (note, that for larger d and smaller f the method s error is smaller). So, we can consider that the expected accuracy of our approach of evaluating f^ (based on equations (2) and (3)) is about of ±10% or better. Comparison of the measured intensity profile with theoretical intensity profile is demonstrated in Fig.2, where the measured focal spot size is 10.8 ±2 μm and the theoretical intensity profile was calculated with the assumption of f = 11 7

8 Intensity, ADU. μm. The calculated intensity profile and the measured profile are normalized to the same values of I max, R 0, R 1, and d. Note, that the measured profile (y) is given in uncorrected form and without subtraction of the background in order to draw attention of the readers to existing differences between (y) and I(y). For convenience, horizontal axis is the distance on the detector plane in units of pixels (2 μm), while vertical axis is ADU units (Analog to Digital Unit as it is specified for CCD-cameras) o-o- Experiment Theory Distance, pixels. B Fig.2. Comparison of the measured intensity profile (-0-0-) with the theoretical intensity profile ( ). As it can be seen from Fig.2, the measured and theoretical profile are close to each other. But profiles differ in the vicinity of the dark and light zone where off-focus radiation and scattering appear. Hence, we should await that estimates f* and f^ can be differ one from the other Model for X-ray lenses Geometric illustration of the focal spot size measurements for an X-ray lens is shown in Fig. 3. The divergent part of X-rays emerging from the lens can be considered as an X-ray source with unknown size-f. The testobject (a W-wire) is situated behind the lens focus (F). After measuring the intensity profile I(y), it is possible to find values of F 1 and F 2. Here F 1 is the full width of the intensity profile of the W-wire image defined at the level 0.5 I max and F 2 is the full width of the intensity profile of the W-wire image defined at the level 0.9 I max (see Fig.1). After that, the estimate f^ can be derived from the equation (2), and an estimate f*can be determined as the full width at half maximum of the curve I'(y): f* = FWHM. 8

9 R0 R1 F d X-RAY SOURCE X-RAY LENS W-WIRE IMAGE PLANE Fig. 3. Geometric relationships of focal spot size measurements for an X-ray lens. The test-object (the W- wire) is situated behind the lens focus F. After measuring the intensity profile I(y) it is possible to find the estimate f in accordance with the equation (2) Arrangement The schematic drawing of the experimental arrangement is shown in Fig.4. The arrangement consists of an X- ray tube to be characterized (or an X-ray tube and an X-ray lens to be characterized), pre-collimators, set of W-wires, X-ray CCD- cameras and a collimated semiconductor X-ray detector with data acquisition systems, hardware and software. This setup also includes optical rails, translation and rotation stages for installation of the X-ray tube and X-ray lenses, and X-ray detectors. The X-ray unit was mounted on the goniometer, the test objects and the CCD-camera were installed on the stages, which could be moved along the rail and can be securely locked in any position. A simple radiation protection sheets were used for safety. The CCD cameras from the Photonic Science were used as on line X-ray detectors for the intensity profile measurements. The two-dimensional images of W-wires were stored, and the treatment was performed according to our approach of the focal spot size estimation. One of the mainly used CCD-cameras has the spatial resolution about ~ 12 μm (1040x1380 pixels with 2 μm effective pixel size) and the dynamic range of 1000 (12bit). The thickness of the entrance scintillator layer of oxy-sulphide gadolinium is optimized for working in the X-ray energy range from 6 kev to 30 kev. The inherent instrumental background of the CCDcamera is about B CCD 67 ADU and it usually includes both dark current noise and intentional offset of the electronics [17]. This CCD-camera with data acquisition facilities permits contrast measurements of 0.5%. Note, that data acquisition and data processing system for standard radiographic and radiometric procedures 9

10 shall permit contrast measurements of 1% - 2% and shall be stable to at least ±1% over the measuring time [1, 3, 4]. Cu-target W-wire Collimators CCD- camera Magnetic lens C1 C2 R0 R1 Fig. 4. Measuring facility for radiometric determination of the focal spot size. Different test-objects were used in experiments. Among them were tungsten wires, tungsten wires with gold coating, tantalum plates with round polished edges, and mercury liquid drop test-objects. Tungsten wires were polished by electro chemically processing. The surface roughness of some tungsten wires was rather great and we had to apply wires with gold coating and mercury liquid drop test-objects to reduce unwanted scattering. Note, that typical measured values of surface roughness of an accurately polished tantalum plates (that were used in our experiments) were in the range from 1000Å to 500Å. Exploiting of such test objects in experiments give rather smoothed curves I(y) and I'(y). On the contrary, the use of some W-wires without Aucoating gives additional spikes and peaks for curves I(y) and I'(y). The surfaces of these tungsten wires have many original machined marks (surface scratches and abrasions) after wire drawing; typical values of surfaces roughness range from 0.3 μm to 10 μm, which are seen by a microscope. Although, the roughness of the wire (ball) surface is not specified by the considered standards [1, 2, 5], nevertheless for the standard scanning method of X-ray focal spot size measurements [3], the roughness Ra of the slit surface is specified. The last method is used for measurements of focal spot sizes above 100 μm, and the roughness Ra shall be better than 2 μm. The advantages of Hg-drop test-object are that 1) a surface roughness can reach very small values of about 1Å for pure mercury surface [18], and so, such test-object can be considered as an ideal one, and 2) mercury has a rather high absorption coefficient for X-rays in the range from 6 kev to 50 kev [17]. Note that the 10

11 absorption coefficients for tungsten and mercury are close to each other. For example, for Mo-Kα X-rays (17.5 kev) values of X-ray linear absorption coefficient for W and Hg is 0.18 μm -1 and 0.16 μm -1, respectively. For the proposed method, it came out, that the value of the surface roughness of about 0.3μm or smaller is sufficient to avoid the influence of the surface roughness on the focal spot size determination. But this announcement and the determination of true minimum values of Ra demand additional experimental investigation. The Hg - drop test-object with a diameter of drop 1.5mm was installed in an evacuated trough with thin plastic films ( 0.02mm) as entrance and exit windows for X-rays. Besides, a thin W-wire with known diameter is placed in the trough in such way, that it is possible to define the apex of the Hg-drop with an accuracy 0.1mm. The wires (edges) were installed parallel (or perpendicular) to the CCD rows (or columns) with an accuracy of about radian. The crossed wires (edges) were arranged on the goniometer, which was capable of accurately aligning and positioning wire test-objects in the central X-ray beam (it is defined here as an X-ray beam arising from the focal spot in a direction parallel to the electron trajectory). In our experiments for X-ray tube focal spot size measurements we have used the «pencil-beam» geometry. In this case, the incident X-ray beam is formed by the tantalum collimator (C1) with an inner diameter of 0.2mm. The other collimator (C2) served as additional collimator, and in some experiments collimators were omitted. The collimator (C1) can be mounted behind the X-ray source at a distance ranging from 0.5mm up to 20mm. The test objects can be installed at distances R 0 from 0.6mm to 200mm behind the X-ray source. The distance between the test objects and detector R 1 may be changed from 5mm up to 300mm. The image magnification factor G, which is defined as G = (R 0 +R 1 )/R 0, can be changed from G 1 up to G 500. In the experiments we usually used G While conducting these measurements, considerable care was taken to decreasing vibration. The effects of the objects motion (industrial vibration and noise) is accomplished by not synchronously moving of all parts of the apparatus (the image recording system, the test object, and the X-ray tube or lens). Hence, the image sharpness of wire borders is degraded and the observed size of focal spot will be increased. It came out, that 11

12 Newport passive air dampers and rigid construction of all mechanical parts were sufficient to avoid the influence of vibration on the measured focal spot size in the considered size range Image corrections All CCD-images were corrected for: 1) non uniformity of the pixel-to-pixel X-ray sensitivity, and 2) offset variations. For it, gain and offset maps were constructed for image acquisitions of white (X-rays on) and dark (X-rays off) fields. The corrected signal C mn for pixel (m, n) is then given by a linear transformation: C mn =(E mn -b mn )[(F mj -b mj )/(F mn -b mn )] (4) where m is the row index and n is the column index. The term E mn is the experimentally measured pixel signal of a test-object image, the term F mn is the experimentally measured pixel signal without the test-object but at the same geometry. The term b mn is dark field noise and instrumentation pedestal. In the equation (4), the index j is the first pixel number of the high illumination image of the test-object along the region of interest (ROI). This region was a rectangular area with N by M pixels. To reduce an influence of statistical fluctuations of the measured values (E, F and b) on image of test-objects, these values should be averaged along either M rows or N columns. The measured intensity profile I(y) will be expressed as I(k) = C k, where k=m(n), and the other index in the term C mn is the index of summation and it is omitted. Usually we took M (N) in the range from 30 to100 pixels and the length of the ROI was 1040 or 1380 pixels. When liquid mercury was used as a test-object, the pixel trace of the ROI was a column with wide range from 20 to100 pixels and height 1040 pixels. The effectiveness of this simplest correction is illustrated in Fig. 5 where the results are shown before and after correction for the derivative I'(k). Intensity, ADU/PIXEL o o -Raw data Corrected curve Distance, PIXEL. Fig.5. intensity profile (-0-0-). 12

13 To underline the importance of the field correction, we used the CCD-camera with a great value of non uniformity of the pixel-to-pixel X-ray sensitivity: more than ~ 40%. In this experiment the uncorrected profile I'(k) has the inherent additional hump at the right hand side. As it is seen, the correction procedure gives rather smoothed curve. Besides, the full width at half maximum of the corrected histogram (Fig. 5) is two times smaller for corrected profile than it is for uncorrected curve. Usually this correction does not so dramatically improve the shape of curves I'(k), and the full width at half maximum of the corrected curves differs from the uncorrected ones in a value of about 15% - 30% Quantitative measures of focal spot size (on-line determination) Besides conventional graphical methods of focal spot size determination outlined above in section 2.1, there are analytical methods exploiting digital on-line X-ray detectors that enable real-time measurements and determination of the value f. In the course of measurements X-ray sensitive-position detectors are used, such as a CCD-camera or a scanning X-ray detector. In this case, the intensity profile I(k) and the derivative of the intensity profile I'(k) are in a digit form and have own names: I(k) = ESF k and I'(k) = LSF k as the edge spread function (ESF) for an ideal edge test-object and the line spread function (LSF), respectively. Here, k = m (n) is the row (column) index. Then, the estimation method of a size f exploits either the Fourier analysis of the line spread function or the simple least-squares method [19]. In the latter case the size of focal spot can be derived from the well known formula: F RMS = 2.36 ( k 2 LSF k ) 1/2 (5), where the sum k LSF k = 0 and LSF k = A. Here k is the index of summation, k = 1 K and K is the width of the penumbra region Ug, which is measured in pixels (k Є {Ug}, see Fig.1). The term A-is the area under the curve LSF k, k = 1 K, and abbreviation RMS means root mean square. The estimation error of F RMS (the square root of the variance of the estimate F RMS ) is smaller than ΔF RMS 2 p ζ CCD F RMS /A 1/2, (6) where ζ 2 CCD is the maximum observed variance of the measured values I(k) in the light zone. The parameter p can be considered here as a signal to noise ratio (SNR), and amounts typically to p ~

14 Under assumption that the X-ray intensity distribution along any X-direction on the X-ray target plane follows the Gauss law: I(x) = exp (-x 2 /2ζ 2 )/ 2πζ (where ζ is the standard deviation of the Gauss law) we can conclude that: f* = FWHM = F RMS = 2.36 ζ. The advantages of this treatment are that: 1) simplicity and physical evidence of the process of the focal spot size measurement and determination, and 2) for the case of using a CCD-camera with finite spatial resolution, R RMS, the true focal spot size f* is equal to the square root of squared difference of F RMS and R RMS. Here, the value F RMS is considered as the uncorrected focal spot size. Values F RMS and R RMS are considered as normal distributed random variables. Thus, the true size of the measured focal spot can be expressed by the simple expression: f* = ( F 2 RMS R 2 RMS ) 1/2 (7). Illustration of the above formalism is shown in Fig.6 for focal spot size measurements of X-ray tube (by means of the edge method with an Hg-drop, having an ideal surface), where the measured focal spot size is evaluated as f* = 3.0±0.6 μm. Intensity, ADU/PIXEL Raw data o o -The Gauss fit Distance, PIXEL. Fig. 6 The derivative of the intensity profile I'(k) of the mercury liquid drop-edge image fit curve (-0-0-). As it is seen in Fig.6 the Gauss fitted curve is rather close to experimental data points, but there is a pedestal for the fitted curve. So, the estimate f* derived from the Gauss distribution I'(k) will be a biased estimate with respect to the estimate f^, which could be potentially find from the original intensity profile I(k). The relative error δ = (f^- f*)/ f^ is about ~20 % and f^ > f*. Physically this pedestal can be caused by offfocus radiation, scattering and so on. This difference between f^ and f* is due to that the sensitivity of an estimate f* to off-focus radiation and unwanted scattering is lesser than it is for estimate f^ Spatial resolution of the CCD-camera. Geometric distortion correction 14

15 The spatial resolution of the CCD-camera was measured by use the tungsten wire mesh with wire diameter 40 μm. The W-wire mesh was arranged at the entrance of the CCD-camera. In these experiments, the X-ray tube with Cu-anode was operated at 25kV high voltage and a power load of around 0.2W. The X-ray source to object distance R 0 and the object to detector distance R 1 were R 0 = 27±2mm and R 1 = 5±1mm. The corrected function LSF k for seven parallel W-wires is shown in Fig.7. The numerical treatment of the data gives the values of spatial non linearity (geometric distortion) and intrinsic CCD-camera resolution along the active field of the camera. The geometric distortion for the presented profiles is about ±5% and may be caused by non uniformity in diameter of the grid wires, non parallel position of wires, and the optics of CCD-camera. Some deviations of the CCD-camera resolution along the active field of the camera are observed within a range of ±10%. The absolute values of the presented resolution R RMS range from 11 μm up to 13 μm. In order to measure focal spot sizes correctly, the resolution R CCD shall be lesser than the geometric unsharpness: q R RMS < (G-1) f (8), where the parameter q ranges from 2 and up. As it is seen from the inequality (8) larger values of CCDcamera resolution require larger magnification G. The maximum value of G is restricted by the finite value of the CCD camera field of view (FOV) and value of wire diameter, d. Under considered values of FOV and d we have G max < 60 (for FOV = 2mm, and d = 30 μm). Let us evaluate the influence of the CCD-camera resolution R RMS on the estimate f*; this influence can be characterized by the parameter q. For values f* 1μm, R RMS = 12μm and G max 60 we find that q max = 5, and the influence of the resolution R CCD on the estimate f* will be 2% only. In the last case, for determination of the true focal spot size we can not consider the CCD-camera resolution at all. For values f* 1μm, R RMS = 12μm and G 25 we find that q = 2, and the influence of the value of the resolution R CCD on the estimate f* will be 15%. And so, it seems reasonable to set range of variation for q from 2 to 5. 15

16 60 Intensity, ADU/PIXEL Distance, PIXEL. Fig.7. The corrected line spread function LSF k for seven parallel W-wires image. In order to reduce the influence of the non uniformity of pixel size and non uniformity of the CCD-camera resolution along the CCD- active field on the accuracy of focal spot size determination one should use the same areas of the CCD-camera (ROI) for all values involved into the treatment process Statistical errors and accuracy We consider two kinds of error sources that define an accuracy of focal spot size determination. They are: 1) the systematic error contributed by geometry configuration and parameters as G = G(R 0, R 1 ), d, spatial resolution and geometrical distortion of the CCD-camera, unwanted scattering, stability of focal spot position and so on, and 2) the random fluctuations of X-ray interactions with the surface of the test-object and surrounding materials, the random fluctuations of the values I(k), the random error caused by uncertainties of the image treatment and so on. We performed error analysis in terms of partial relative measurement errors, for convenience. The total relative error of focal spot size δ f will be approximately equal to the square root of squared sum of partial relative errors: δ G - relative error of magnification factor determination (eq.1), typically it ranges from ±1% to ±10%; δ d - relative error of the wire diameter determination (eq.2), typically it ranges from ±1% to ±3%; δ GD - relative error of the geometric distortion correction, typically it ranges from ±3% to ±5%; δ FC - relative error of the flat-field correction, typically it ranges from ±3% to ±10%; δ F - relative error of the uncorrected focal spot size determination (eqs.5-7), it ranges from ±10% to ±15%; δ R - relative error of the CCD-camera resolution determination (eqs.5-7), typically it is ±10%; 16

17 δ TR - relative error of the data curve treatment methods (smoothing, numerical differentiation, choosing borders of the curve, and so on), typically it ranges from ±5% to ±15%; δ OT - other errors and corrections (the method s error), here it is considered as ±10%. In practice, many of the above errors depend on the quality of the CCD-camera to be used. For the high quality CCD-camera, as a rule, errors (δ GD, δ FC, δ F, and δ R ) can be decreased to minimum values, that are presented here. Consider some numerical examples of relative error calculations and corrections for comparison purposes. First, for the graphical estimation of f either the formula (1) or the formula (2) shall be used. In the case, when the value of G is calculated by means of distances R 0 and R 1, the error δ G will be essentially determined by the uncertainties of the values R 0 and R 1. For example, if the value R 0 is determined with the typical experimental accuracy 0.1mm, then δ G can not be smaller than δ G ~15% for R 0 =0.6mm. On the contrary, from the equation (2) it follows that, the error δ d is due to the uncertainty in the value d only, and it can be rather small because of the wire diameter d can be measured with an accuracy of ±1 μm and thus, δ d can be reduced to δ d ~1 3% for considered wire diameters range from 30 to 100 μm. Hence, in the last case we have a more precision method. Second, let us evaluate the values δ F and δ R numerically. The value δ F can be expressed as follows: δ F 2 p ζ CCD 100%/(M A) 1/2, where ζ 2 CCD is the maximum observed variance of the values E mn in light zone, the term A-is the area under the curve LSF k, M is the wide of the ROI across the edge image of interest, and the parameter p = 3. The same equations are valid for the determination of δ R also. Both δ F and δ R may be made as small as it is allowed by experimental conditions (by choosing A and M large enough). Substitution of conventional numerical values in the last expression gives δ F = 10% (here we take the observed value ζ CCD = 10 ADU for the level of illumination I = 800 ADU, value A 2000 ADU and M 200). Finally, we have that the sum of errors δ F and δ R will be about 15%. The last two components of measurement errors can be decreased to the level of 5% by increasing the exposure time, if all parts of the data acquisition system are stable over the measuring time. Finally, let us consider numerical example of attenuation X-rays by an outer layer of the W-wire. If we use a wire with diameter, d, then the path through the outer layer will be: L (d h) 1/2, where h is the circular arc height (the thickness of outer layer). Substitution of conventional numerical values in the expression for X-ray 17

18 transmission exp (-µ L), gives the transmission coefficient T 0.17 (here d=100 µm, h=1 µm, µ=0.18 μm -1 for Mo K α -radiation) and T (d=200 µm, h=2 µm, and µ=0.18 μm -1 ). Note, that in our approach of focal spot size determination based on the mathematic model (3) the absorption phenomena is considered, and so, it does not matter what value of the wire diameter is used. But, in order to simplify the absorption correction procedure and to decrease uncertainties of this absorption correction procedure one should use larger wire diameters. For example, the wire diameter shall be more then 300 µm for the checking X-ray tubes operating at high voltage above 30 kv. In order to check the sensitivity of the above evaluating procedure to quantitative changes of X-ray spectrum and scattering we performed two additional experiments. Under conducting experiments we used an X- ray tube with Cu-anode, which was operated at 25 kv high tube voltage (focal spot size was ~ 5μm). In the first experiment we compare estimate f Cu measured in the presence of a cuprum pre filter (30 μm) with the estimate f without filter. Then, in the second experiment, we compare estimate f Be measured in the presence of a pure scattering filter (berillium filter with 800 μm in thickness) with the estimate f without scattering filter. The results show that these estimates are weakly differs from each other by around 4-5%. In this study we neglect a small non-linearity in converting charge to voltage for the CCD-camera. Besides, we did not consider small variations in the pixel width (±3%) because of under point estimates of f it does not so important. 3. RESULTS In this work, the results of the focal spot size measurements are presented as f ± ζ, where f is the mean value, and ζ is the square root of the unbiased variance of estimate f. The result of the focal spot size measurements for X- ray tube is shown in Fig. 8. The measurements were conducted with the especially developed micro focus X- ray tube with Cu-anode, which was operated at 25 kv high tube voltage and power load up to 2W. The source to object distance R 0 and the object to detector distance R 1 were R 0 ~ 0.4см and R 1 ~ 2.6см. The X-ray exposition was 15 sec. The measured focal spot size of the X-ray tube is determined as 2.3±0.6 μm. 18

19 Intensity, ADU/PIXEL Raw data The Gauus fit FWHM Distance, PIXEL. Fig.8. The derivative of the intensity profile of the W ). The measured focal spot size is evaluated as 2.3±0.6 μm. The source to object distance R 0 and the object to detector distance R 1 were R 0 4 mm and R 1 26 mm. The X-ray exposition was 15 sec Illustration of the focal spot size measurements of X-ray lenses is presented in Fig.9 and Fig.10. The X-ray lens has a focal distance F of ~ 3 mm. In Fig. 9 the W-wire image of the focal spot of the X-ray lens is shown and in Fig.10 the intensity profile of the W-wire image is presented. The divergent part of X-rays emerging from the lens was considered as the X-ray source with an unknown size of the focal spot. In this experiment, the X-ray tube (Cu-anode) was operated at 40kV high voltage and current 0.1mA. The source to object distance R 0 and the object to detector distance R 1 were R 0 ~ 0.5см and R 1 ~ 9.2см. The X-ray exposition was 17 sec. After measuring intensity profile I(y), it is possible to find the estimates f^ and f*. The measured focal spot size of the presented X-ray lens is determined as f^ f* = 10±2 μm. Fig.9. The W-wire image of X-ray lens focal spot. The W-wire diameter is 30 μm. The X-ray exposition is 17 sec. The X-ray lens has the focal distance F 3 mm. 19

20 3750 Intensity, ADU F 2 F Distance, microns. Fig.10. The intensity profile of the W-wire image (taken from Fig.9). The measured focal spot size of the lens is evaluated as 10±2 μm. 4. CONCLUSIONS The developing procedure describes a method for measuring effective focal spot size within the range 1 μm up to 10 μm of X-ray lenses and X-ray tubes operating at 10kV up to 50kV. The method is based on indirect measurement of the focal spot size by measuring the geometric un-sharpness. For this purpose the image of sharp edges of the test object (W, Ta, Au, Pt- wires or plates, and Hg-drops) are recorded either by means of a digital CCD-camera or with mechanical scanning of a collimated detector with pinhole (or narrow slit) at the entrance. The projective magnification G shall be larger than or equal q- times the CCD-camera resolution- R CCD divided by the focal spot size- f provided by the manufacturer: (G-1)> q R CCD /f, where the parameter q ranges from 2 to 5. Wires diameter d should be between 30 μm and 300 μm, which have an accuracy of ±3% or better, and the surface roughness should be better than 0.3 μm. Data acquisition and data processing shall permit contrast measurements of 1% and shall be stable to at least ±1% over the measuring time. The mathematical model for the focal spot size measurements for X-ray tubes and lenses by means of the wire and edge methods is presented. Numerical simulation of the focal spot size measurements is performed and it is shown that the expected accuracy of the presented wire and edge methods of focal spot size measurements can be about ±10% or better. Both methods show a good agreement between the measured values of focal spot sizes with an accuracy of ±20%. The accuracy of evaluation of the focal spot dimensions is analyzed. To a first approximation, the most important aspects of method s error of focal spot size determination is considered and it shown that the accuracy is about of 20-30% in the energy range below 30 kev. 20

21 The developing method for measuring focal spot sizes has distinguishing features, which are not specified by the considered standards [1-6]: 1) it permits the checking focal spot size in the range from 1 μm and up; 2) focal spot size measurement for X-ray lenses and tubes can be performed by means of the same methods and the same apparatus; 3) it can be used for on-line measurements of focal spot size; the developed mathematical model for the focal spot size measurements with data acquisition and data processing permit determination of the estimate f and variance of estimate f in real-time measurements; the measuring time together with the time of data processing permit determination f during 5sec to 40 sec. 4) it is possible to obtain values of f by using both the edge and wire methods simultaneously; 5) it can be used for an absolute measurement of the focal spot dimensions; 6) it can be used with a new kind of test-object as mercury liquid drops, having almost ideal surface. Further simulations and experiments should be performed to clear up the influence of a wire surface roughness and wire diameters on the accuracy of focal spot size determination in the energy range above 30 kev. ACKNOWLEDGMENTS The author would like to thank to M.A. Kumakhov for his interest and support, V.D. Gelever for the performed X-ray apparatus. The author also would like to acknowledgements A.A Markelov, N.V. Kozlov, M.L. Chibel, and A.N. Garkovenko for manufacturing mechanical components of experimental arrangement. Author is grateful to Elena Seletzkaya for the help in preparing the report. REFERENCES [1] GOST Pribory rentgenovskie. Metody izmereniia razmerov effektivnogo fokusnogo piatna. /in Russian/. X-ray devices. The methods of effective focus spot size measurements. [2] British Standard Institution. BS 6932 Method for measurement of the effective focal spot size of minifocus and micro-focus X-ray tubes used for industrial radiography,

22 [3] The European Standard EN :1999. Non-destructive testing-characteristics of focal spots industrial X-ray systems for use in non-destructive testing, part 1: scanning method. [4] The European Standard EN :1999. Non-destructive testing-characteristics of focal spots industrial X-ray systems for use in non-destructive testing, part 4: edge method. [5] The European Standard EN :1999. Non-destructive testing-characteristics of focal spots industrial X-ray systems for use in non-destructive testing, part 5: measurement if the effective focal spot size of mini and micro focus X-ray tubes. [6] NEMA XR5. Measurement of dimensions and properties of focal spots of diagnostic X-ray tubes. National Electrical Manufactures Association, USA, [7] J.D. Everson, J. E. Gray. Focal-spot Measurement: Comparison of Slit, Pinhole, and Star Resolution Pattern Techniques. Radiology, V.165, pp , [8] A. Cunningham, B.K. Reid, Signal and noise in modulation transfer function determinations using the slit, wire, and edge techniques. Med. Phys., V. 19(4), pp , Jul/Aug [9] H. Kubota, Y. Ozaki, M. Matsumoto, and H. Kanamori. Determination of X-ray tube focal spot position. Med. Phys., V. 20(4), pp , Jul/Aug [10] E.L. Nickoloff, E. Donnelly, L. Eve, J.V. Atherton. Mammographic resolution: influence of local spot intensity distribution and geometry. Med. Phys., V. 17(3), pp , [11] J. Law. The influence of focal spot size on image resolution and test scores in mammography, The British Journal of Radiology, V.66, pp , [12] M.A. Kumakhov. Capillary optics and its use in X-ray analysis. X -ray Spectrometry, V.29, p , [13] V. Ya. Shovkun. Measurements of micro-focus spot size in X-ray tubes and Kumakhov lenses, Proc. of SPIE Vol. 5943, p , [14] J. M. Boone, T.R. Fewell, and R.J. Jennings. Molybdenum, rhodium, and tungsten anode spectral models using interpolating polynomials with application to mammography. Med. Phys., Vol.24 (12), p ,

23 [15] K.P. Ng, C.S. Kwok, and F.H. Tang. Monte Carlo simulation of X-ray spectra in mammography. Phys. Med. Biol., Vol. 45, p , [16] E. Strom and H. Israel. Photon cross sections from to 100 MeV for elements 1 through 100. Los Alamos Scientific Laboratory. New Mexico, [17] Sol M. Gruner, Mark W. Tate, and Eric F. Eikenberry. Charge-coupled device area X-ray detectors. Review of Scientific Instruments, Vol. 73(8), p , [18] H. Kraack, B. M. Ocko, P. S. Pershan, E. Sloutskin, and M. Deutsch. Langmuir films of normal-alkanes on the surface of liquid mercury. J. Chem. Phys., Vol. 119 (19), p.13-22, [19] K. Doi, L.-N. Loo and H.-P. Chan, X-ray tubes focal spot sizes: comprehensive studies of their measurement and effect of measured size in angiography. Radiology, V.144, pp ,