·Table of Contents ·Methods and Instrumentation | Calculation of Protective Shields Made of Lead Glass for X-Rays N.D.T. Industrial EquipmentJ. Reyes-Romero ,Universidad Central de Venezuela, Facultad de Ingenieria, Departamento de Física Aplicada, Los Chaguaramos, Caracas 1041, Venezuela. Email : reyesj@camelot.rect.ucv.ve. A-M Ghiordanescu, Universitatea din Bucuresti., Facultatea de Fizica, Platforma Magurele, Bucuresti, Romania. B. Hurtado, Universidad Experimental Politecnica " Luis Caballero Mejias", Departamento de Ciencias Basicas, La Yaguara, Caracas, Venezuela. Contact |
(1) |
(2) |
(3) |
(4) |
In the other hand, the spectrum of x-radiation emitted is considered continuous and therefore the maximum frequency or the minimum wavelength of the radiation emitted is given by the following equations [6],
(5) | (6) |
The passage of a type of spectrum in function with the frequency of a spectrum in function with the wavelength is given thought the equation [7],
(7) | (8) |
For the calculation we selected the type of empirical spectrum [2],
(9) |
in which a = 2,5.10^{15} for n < n_{0} y a = 0 for n > n_{0} , for v measured in Hz and C_{t} is a constant. Substituting in (9) and doing obtained,
(10) |
in which the spectrum is taken by E_{min} < E < E_{0} ; the energies are measured in KeV and therefore the value of E_{0} in KeV is equal to the value of V_{0} in KV. The area below the curve (10) in the interval, [E_{min},E_{0}] , is given by the expression,
(11) |
in which V_{0} is measured in KV and E_{min} in KeV.
The expression for the absorbed dose rate "d" to the distance r from the focus for the installation of x-rays, supposing valid the spectrum (10) and also taking into consideration that there is no absorbent in the trajectory of the radiation, we obtain by the definition of the absorbed dose D, as the energy transferred by the radiation, in mass units, the irradiated substance
(12) |
being J the total energy of radiation per time unit and per area unit, m(E) the absorbance coefficient of the x-radiation when E is the energy of the quanta and r the density of the absorbent substance. Taking into consideration the focus of a punctual source that emits uniformly in all directions, for the quanta that have the energy included between E and E+dE , we have the energy where
(13) | (14) |
P_{x} is the x-radiation potential given by the equation (1), is the spectrum given by the equation (10) and A is the area below the curve given by the equation (11) . Substituting (13) and (14) in (12) we obtain
(15) |
If we measured then [d] = W/Kg. To measured [d] = R/h we multiply the equation (15) by 4, 1.10^{5},
(16) |
Taking into consideration the trajectory of the x-radiation, present as absorbents. A film of air or oil that cools down the anode of a thickness , a window of Be of the x-ray tube of a thickness X_{Be}, the air between the Be window and the point located at a distance r from the focus to the point where we calculate the dose supposed to have a determined thickness(r is the radius of the tube) Xair,, a glass window with lead Pb of a thickness X_{ST}. In this case, the expression below the integral is multiplied by S_{2}S_{3}S_{4}S_{5} which represent the probabilities that the x-radiation would not be absorbed, respectively by the Berilium window , the film of water (oil), the layer of air , the lead window. This is presented under the form of
Under these circumstances we have the following expression for the dose rate "d" in (R/h)
(17) |
For the calculation of the absorbance coefficients that are part of the equation 17, we will use for an element with Z < 50 the empirical expression proposed in [4]
(18) | (19) |
Where Z is the number of the order of the element, A the mass number, l the wavelength in is measured in . The equation (18) is valid for the wavelength where E_{K} is the binding energy of the electron of the K level of the element with a number of order Z. For l_{k} < l < l_{L1},
(20) |
And for l_{L1} < l < l_{M1},
(21) |
Where E_{L1} and E_{M1} are the binding energies in the levels L_{1} and M_{1} respectively. For the light elements that are part of the composition of the window, the equations (18)-(21) are not enough for energies greater than 40 KeV. Therefore using the values for the absorbance coefficients given in [6], fitting m/r as a function of the type
(22) |
Being
(23) |
In the case where the material is made of K elements, the absorbance coefficient is given by
(24) |
being , the number of atoms of the lot "i" of the molecule, A_{i} the atomic mass of the element and M the molecular mass. The lead glass (st), having in its composition S_{i}O_{2}, PbO, Na_{2}O, K_{2}O,CeO_{2} and with concentrations C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, respectively, the absorbance coefficient was calculated using the following expression.
The calculus program was written in Basic, and can be used in any personal computer. The entry data asked by the program are, zetac= z of the anode, ima= current in mA, ukv= voltage in KV, radio= distance for which we calculate the dose in meters, eneri= minimum energy of the x-radiation in KeV, denerg= integration step, fagua= the thickness of the layer of water on the anode in cm, fBe= thickness of the Be window in cm, abpb= thickness of the window of lead glass in cm, C1= concentration of the S_{i}O_{2} e in %,..., C5= concentration of CeO_{2} in %, rost= density of the lead glass in
. Then we calculate the rand= efficiency of the installation of x-rays, pot x= the potential x and the area below the curve , obtained by (11).
The integration is done by the rectangle method. We calculate for each quantum of energy "energ", the height "intger" of a rectangle of width "denerg", Sumat= value of the obtained integral , dosis= dose in R/h calculated according to (17). The program is running introducing the concentration of the elements that constitute the glass, Ci, its density rost and thickness abpb. We determine the value of the thickness of the glass at a distance "radio", demanded by the norms; an allowed dose of 0.75 m R/h.
Energía x [KeV] | 59 | 80 | 122 |
2,05±0,03 | 1,22±0,03 | 1,46±0,03 | |
1,98±10% | 0,88±10% | 1,42±10% | |
TABLE 1 : |
For the experimental verification of the calculation of the dose, was used an installation of detection with a detector of Na I (TI) and a window of Be connected to a personal computer. In front of the detector , we put a source of ^{57}Co of a known activity. The source emits an x-radiation of 6.4 Kev and y of 14 KeV, 122 KeV and 136 KeV . We fixed the window in such a manner that the equipment could detect the quanta of 122 KeV and 136 KeV ( average of energy is about 128KeV) . Between the source and the detector we put lead glass absorbents of known thicknesses X_{st}.
The dose "d " calculated then has a value, according to the equations (12)...(17).
(26) |
The dose D found experimentally is
(27) |
Where R is the velocity of counting of the installation in pulse/h, "a" is the radius of the detector in m, fab is the probability that the Quantum is not absorbed in the window of the detector and e_{f} is the probability that the quantum g would be register in the photopeak. The values of fab and e_{f} where calculated according to the procedure indicated in[3]. In the equation (27) L is measured in counts/s and E is measured in J. If we measure L in mCi and E in KeV, the other measurements keep in the SI units, then
(28) |
where is the coefficient of mass absorbance of the glass, , X_{st} is the thickness of the lead glass, is the coefficient of mass absorbance of the window, X_{f} is the thickness of the window of the source, is the absorbance coefficient of the air masses , X_{aire} is the thickness between the source and the place where the dose is measured.
V(KeV) | I(mA) | Dose calculate (R/min) |
25 | 20 | 4800 |
50 | 30 | 16700 |
100 | 30 | 36600 |
160 | 19 | 40000 |
TABLE 2 : |
V (KeV) | I (mA) | r(m) | d(cm) |
50 | 20 | 0.20 | 1.40 |
50 | 20 | 0.50 | 1.20 |
50 | 20 | 1.00 | 1.12 |
100 | 30 | 0.20 | 7.00 |
100 | 30 | 0.50 | 6.50 |
100 | 30 | 1.00 | 6.00 |
160 | 19 | 0.20 | 8.00 |
160 | 19 | 0.50 | 6.00 |
160 | 19 | 1.00 | 6.00 |
TABLE 3 : |
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