A method for determining radiation margin of weak areas of spacecraft
Through ray tracing method and curve conversion technology, the accuracy of radiation margin estimation in the weak areas of the spacecraft is solved, and a more accurate radiation protection design for spacecraft sensitive components is achieved, and the service life of the equipment is extended.
Patent Information
- Application Number
- CN202210768541.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-06-30
- Publication Date
- 2025-04-15
- Estimated Expiration
- 2042-06-30
AI Technical Summary
The prior art is difficult to effectively analyze the radiation residual of the weak radiation protection parts in the three-dimensional complex structure of the spacecraft, resulting in the radiation dose estimation being too optimistic and cannot effectively guide the optimized design of sensitive components.
The three-dimensional geometric structure of the spacecraft was analyzed by ray tracing method, and the depth-three-dimensional angle proportion curve and the dose-three-dimensional angle proportion curve were calculated, and it was converted into a dose multiple-three-dimensional angle proportion curve and a dose multiple-depth curve. Combined with the three-dimensional angle proportion of the weak area and the upper limit of the shielding depth, the final total dose multiple was calculated to determine the radiation balance.
This method can more accurately analyze the radiation margin of the spacecraft's weak areas, provide multi-standard and multi-dimensional design references, enhance radiation protection for weak parts, and extend the service life of sensitive components.
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Figure CN115186464B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of satellite space environment analysis, and in particular to a method for determining the radiation margin of a weak area of a spacecraft. Background Art
[0002] During the on-orbit mission cycle of a spacecraft, there are complex and diverse radiation sources. During the entire on-orbit mission cycle, the radiation sources can be considered isotropic, but at the instantaneous level, the distribution of the radiation sources in the spatial direction is uncertain. For the three-dimensional complex structure of the spacecraft, the weak parts of the shielding thickness are the shortcomings of radiation protection, which determines whether sensitive components can work normally according to the established performance. Estimating the radiation dose margin according to the average overall shielding situation is prone to insufficient margin, which will cause the radiation dose in the weak area to exceed the limit and cause the device to fail. Therefore, it is very necessary to conduct radiation margin analysis on sensitive components according to the weak parts of radiation protection.
[0003] In the prior art, the estimation of polar radiation dose margin is usually based on ray tracing method, which can provide the distribution of radiation shielding depth along the spatial direction, and then calculate the total dose by weighted accumulation in spatial orientation, so as to estimate the radiation margin. However, this estimation is based on the equal weighted accumulation of the entire spatial direction according to the principle of isotropy, and loses the information about the weak direction. Although it can provide an average radiation margin estimation, it is difficult to reflect the special hazards of the weak parts of radiation protection. For the highly variable space environment, this estimation method is too optimistic and cannot provide effective guidance for the optimal design of sensitive components. Summary of the invention
[0004] The problem solved by the present invention is how to provide a method for performing radiation redundancy analysis on sensitive components according to weak points of radiation protection, so as to provide a multi-standard and multi-dimensional design reference for spacecraft.
[0005] In order to solve at least one aspect of the above problems, the present invention provides a method for determining the radiation margin of a weak area of a spacecraft, comprising the following steps:
[0006] Step S1, performing ray tracing calculation on the three-dimensional geometric structure of the spacecraft to obtain shielding depth data, as well as the dose value and spatial angle size in each spatial angle region;
[0007] Step S2, calculating the depth-solid angle ratio curve, the dose ratio-solid angle ratio curve and the total dose;
[0008] Step S3, converting the calculation result in step S2 into a dose multiple-solid angle ratio curve and a dose multiple-depth curve;
[0009] Step S4, taking the solid angle ratio and the upper limit of the shielding depth of the weak area as criteria, substituting the criteria into the dose multiple-solid angle ratio curve and the dose multiple-depth curve, respectively, to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively, to obtain the final total dose multiple by taking the larger value of the two or taking the average value of the two, and multiplying the total dose multiple by the total dose to obtain the radiation margin determined according to the weak area.
[0010] Preferably, in step S2, calculating the depth-solid angle ratio curve includes: sorting the shielding depth data in each spatial area according to the shielding depth, and then traversing it, taking the proportion of the accumulated spatial angle in the entire space as the X-axis data, and taking the shielding depth value as the Y-axis data, thereby obtaining the depth-solid angle ratio curve.
[0011] Preferably, when calculating the depth-solid angle ratio curve, firstly, in the program, a data structure including the shielding depth, the total dose in the area, and the solid angle weight is constructed, the shielding depth data in each spatial area is sorted according to the shielding depth, and then it is traversed, and the ratio of the accumulated spatial angle to the entire space is used as the X-axis data, and the shielding depth value is used as the Y-axis data, so as to obtain the depth-solid angle ratio curve.
[0012] Preferably, in step S2, calculating the dose ratio-solid angle ratio curve includes: sorting the shielding depth data in each spatial region according to the shielding depth, and then traversing it, accumulating the dose items in all spatial regions to obtain the total dose, and then traversing again, accumulating the spatial angle ratio, using it as the X-axis, calculating the ratio of the dose in each space to the total dose, accumulating the dose ratio, and using it as the Y-axis, thereby obtaining the dose ratio-solid angle ratio curve.
[0013] Preferably, when calculating the depth-solid angle share curve, first, in the program, a data structure including the shielding depth, the total dose in the area, and the solid angle weight is constructed, the shielding depth data in each spatial area is sorted according to the shielding depth, and then it is traversed, the dose items in all spatial areas are accumulated to obtain the total dose, and then it is traversed again, the spatial angle share is accumulated and used as the X-axis, the share of the dose in each space to the total dose is calculated, the dose share is accumulated and used as the Y-axis, thereby obtaining the dose share-solid angle share curve.
[0014] Preferably, in step S3, the dose multiple-solid angle ratio curve conversion includes:
[0015] According to the dose ratio-solid angle ratio curve in step S2, the value of the dose ratio divided by the solid angle ratio is calculated, and it is used as the new Y-axis and the solid angle ratio as the X-axis to obtain the dose multiple-solid angle ratio curve.
[0016] Preferably, the dose multiple-depth curve conversion includes:
[0017] According to the corresponding relationship between the depth and the solid angle ratio in the depth-solid angle ratio curve obtained in step S2, a one-to-one search is performed on the X-axis in the dose multiple-solid angle ratio curve, and the depth is used as the new X-axis to obtain the dose multiple-depth curve.
[0018] Preferably, in step S4, when the maximum value of the two is chosen to obtain the final total dose multiple, the solid angle ratio and the upper limit of the shielding depth of the weak area identified as the weak area are used as criteria, and the dose multiple-solid angle ratio curve and the dose multiple-depth curve are consulted to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively. The two are compared, and the larger value is taken as the final dose multiple. Multiplying it by the total dose can obtain the radiation margin determined according to the weak area.
[0019] Preferably, in step S4, when the method of taking the average of the two is selected to obtain the final total dose multiple, the solid angle ratio and the upper limit of the shielding depth of the identified weak area are used as criteria, and the dose multiple-solid angle ratio curve and the dose multiple-depth curve are consulted to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively. The two are compared, and the average of the two is calculated as the final dose multiple, which is multiplied by the total dose to obtain the radiation margin determined according to the weak area.
[0020] The present invention obtains shielding depth data of a three-dimensional structure of a spacecraft, as well as a dose value and a spatial angle size in each spatial angle region by a ray tracing method, calculates a depth-solid angle ratio curve and a dose ratio-solid angle ratio curve, and further converts them into a dose multiple-solid angle ratio curve and a dose multiple-depth curve, and then uses the solid angle ratio and the upper limit of the shielding depth of the weak area identified as a criterion, consults the dose multiple-solid angle ratio and the dose multiple-depth curve, obtains the dose multiple, and then obtains the radiation margin determined according to the weak area by multiplying the dose multiple with the total dose; the present invention makes full use of the shielding depth information obtained by the ray tracing method, calculates the relationship curve between the dose multiple and the solid angle ratio and the depth, and when it is necessary to judge the radiation margin of the weak area, substitutes the solid angle ratio and the upper limit of the shielding depth of the weak area as a criterion into the relationship curve, and obtains its dose multiple, thereby obtaining the radiation margin, and can conduct targeted analysis on the weak areas of the spacecraft that users are concerned about, which is helpful to provide a multi-standard and multi-dimensional reference for the design of the spacecraft, strengthen the radiation protection of the weak parts, and extend the service life of sensitive components. BRIEF DESCRIPTION OF THE DRAWINGS
[0021] Figure 1 A flow chart of a method for determining radiation margin of a spacecraft weak area in an embodiment of the present invention;
[0022] Figure 2 A schematic diagram of a three-dimensional geometric structure of a spacecraft used for calculation in an embodiment of the present invention;
[0023] Figure 3 is a depth-solid angle ratio curve diagram in an embodiment of the present invention;
[0024] Figure 4 It is a dose ratio-solid angle ratio curve diagram in an embodiment of the present invention;
[0025] Figure 5 It is a dose multiple-solid angle ratio curve diagram in an embodiment of the present invention;
[0026] Figure 6 This is a dose multiple-depth curve diagram in an embodiment of the present invention. DETAILED DESCRIPTION
[0027] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, specific embodiments of the present invention are described in detail below.
[0028] It should be noted that, in the absence of conflict, the features in the embodiments of the present invention can be combined with each other. The meanings of the terms "comprising", "including", "containing", and "having" are non-restrictive, that is, other steps and other ingredients that do not affect the results can be added. The above terms cover the terms "consisting of" and "essentially consisting of". Unless otherwise specified, materials, equipment, and reagents are commercially available.
[0029] The embodiment of the present invention provides a method for determining the radiation margin of a weak area of a spacecraft, such as Figure 1 As shown, the following steps are included:
[0030] Step S1, performing ray tracing calculation on the three-dimensional geometric structure of the spacecraft to obtain shielding depth data, as well as the dose value and spatial angle size in each spatial angle region;
[0031] Step S2, calculating the depth-solid angle ratio curve, the dose ratio-solid angle ratio curve and the total dose;
[0032] Step S3, converting the calculation result in step S2 into a dose multiple-solid angle ratio curve and a dose multiple-depth curve;
[0033] Step S4, taking the solid angle ratio and the upper limit of the shielding depth of the weak area as criteria, substituting the criteria into the dose multiple-solid angle ratio curve and the dose multiple-depth curve, respectively, to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively, to obtain the final total dose multiple by taking the larger value of the two or taking the average value of the two, and multiplying the total dose multiple by the total dose to obtain the radiation margin determined according to the weak area.
[0034] The shielding depth data of the three-dimensional structure of the spacecraft, as well as the dose value and spatial angle size in each spatial angle area are obtained through the ray tracing method. The depth-solid angle ratio curve and the dose ratio-solid angle ratio curve are calculated, and further converted into the dose multiple-solid angle ratio curve and the dose multiple-depth curve. The solid angle ratio and the upper limit of the shielding depth of the weak area are used as the judgment criteria, and the dose multiple-solid angle ratio and dose multiple-depth curves are consulted to obtain the dose multiple. Then, by multiplying the dose multiple by the total dose, the radiation margin determined according to the weak area can be obtained.
[0035] Among them, in step S1, using a ray tracing tool implemented based on the Geant4 framework, the number of divisions of the solid angle in the azimuth and polar angles is set, and ray tracing operations are performed on the three-dimensional geometric structure of the spacecraft to obtain shielding depth data, as well as dose values and spatial angle sizes in each spatial angle region.
[0036] In step S2, calculating the depth-solid angle ratio curve includes:
[0037] First, in the implementation program of this method, a data structure containing shielding depth, total dose in the area, and solid angle weight is constructed. The shielding depth data in each spatial area is sorted according to the shielding depth, and then it is traversed. The proportion of the accumulated spatial angle in the entire space is used as the X-axis data, and the shielding depth value is used as the Y-axis data, so as to obtain the depth-solid angle proportion curve.
[0038] Calculation of dose fraction-solid angle fraction curve and total dose includes:
[0039] First, in the implementation program of this method, a data structure containing shielding depth, total dose in the area, and solid angle weight is constructed. The shielding depth data in each spatial area is sorted according to the shielding depth, and then it is traversed, and the dose items in all spatial areas are accumulated to obtain the total dose. Then, it is traversed again, and the spatial angle proportion is accumulated and used as the X-axis. The proportion of the dose in each space to the total dose is calculated, and the dose proportion is accumulated and used as the Y-axis to obtain the dose proportion-solid angle proportion curve.
[0040] In step S3, the dose multiple-solid angle ratio curve conversion includes:
[0041] According to the dose ratio-solid angle ratio curve in step S2, the value of the dose ratio divided by the solid angle ratio is calculated, and it is used as the new Y-axis and the solid angle ratio as the X-axis to obtain the dose multiple-solid angle ratio curve.
[0042] Dose multiple-depth curve conversion includes:
[0043] According to the corresponding relationship between the depth and the solid angle ratio in the depth-solid angle ratio curve obtained in step S2, a one-to-one search is performed on the X-axis in the dose multiple-solid angle ratio curve, and the depth is used as the new X-axis to obtain the dose multiple-depth curve.
[0044] In step S4, when the maximum value of the two is chosen to obtain the final total dose multiple, the solid angle ratio and the upper limit of the shielding depth of the weak area are used as criteria, and the dose multiple-solid angle ratio curve and the dose multiple-depth curve are consulted to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively. The two are compared, and the larger value is taken as the final dose multiple. Multiplying it by the total dose can obtain the radiation margin determined according to the weak area.
[0045] When the method of taking the average of the two is chosen to obtain the final total dose multiple, the solid angle ratio and the upper limit of the shielding depth of the identified weak area are used as the judgment criteria, and the dose multiple-solid angle ratio curve and the dose multiple-depth curve are consulted to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively. The two are compared, and the average of the two is calculated as the final dose multiple. Multiplying by the total dose, the radiation margin determined according to the weak area can be obtained.
[0046] The present invention will be further described below in conjunction with specific examples. It should be understood that these examples are only used to illustrate the present invention and are not intended to limit the scope of the present invention. The experimental methods in the following examples without specifying specific conditions are usually based on the conditions recommended by the manufacturer.
[0047] An embodiment of the present invention provides a method for determining radiation margin of a weak area of a spacecraft, comprising:
[0048] 1.1、For Figure 2 The simplified three-dimensional geometric structure of the spacecraft shown in FIG. 1 (hereinafter referred to as Geometry 1) is calculated. Based on the ray tracing tool of Geant4, the polar angle is divided into 30 parts and the azimuth angle is divided into 60 parts. Ray tracing operation is performed on it to obtain the shielding depth data, as well as the dose value and spatial angle size in each spatial angle region. In Geometry 1, taking detection point 1 as an example, the obtained data format is shown in Table 1;
[0049] Table 1 Data of detection point 1 obtained by ray tracing operation
[0050] Shielding depth (g / ^cm) Dose in the area (rad) Solid Angle Weight 8.2667E+00 3.6903E-02 5.5556E-04 8.2675E+00 3.6899E-02 5.5556E-04 8.2348E+00 3.7051E-02 5.5556E-04 8.3319E+00 3.6601E-02 5.5556E-04 8.3055E+00 3.6723E-02 5.5556E-04 8.2472E+00 3.6994E-02 5.5556E-04 8.5471E+00 3.5623E-02 5.5556E-04 8.7238E+00 3.4839E-02 5.5556E-04 1.0379E+01 2.8601E-02 5.5556E-04 1.1909E+01 2.4147E-02 5.5556E-04
[0051] 1.2. First, construct a data structure containing shielding depth, total dose in the area, and solid angle weight in the program, and then calculate the depth-solid angle ratio curve and the dose ratio-solid angle ratio curve;
[0052] Among them, calculating the depth-solid angle ratio curve includes:
[0053] The shielding depth data in each spatial region is sorted according to the shielding depth, and then traversed, and the proportion of the accumulated spatial angle in the entire space is used as the X-axis data, and the shielding depth value is used as the Y-axis data, so as to obtain a depth-solid angle proportion curve;
[0054] Calculation of dose fraction-solid angle fraction curve and total dose includes:
[0055] The shielding depth data in each spatial region are sorted according to the shielding depth, and then traversed, and the dose items in all spatial regions are accumulated to obtain the total dose, and then traversed again, and the spatial angle proportion is accumulated, which is used as the X-axis, and the proportion of the dose in each space to the total dose is calculated, and the dose proportion is accumulated, which is used as the Y-axis, so as to obtain the dose proportion-solid angle proportion curve;
[0056] in, Figure 3 The depth-solid angle ratio curve calculated for the detection point 1 in geometry 1 using the data obtained in step 1.1, Figure 4 The dose ratio-solid angle ratio curve calculated for the detection point 1 in geometry 1 by the data obtained in step 1.1;
[0057] 1.3. According to the results calculated in step 1.2, the dose multiple-solid angle ratio curve and the dose multiple-depth curve are converted;
[0058] Among them, the dose multiple-solid angle ratio curve conversion includes:
[0059] According to the dose ratio-solid angle ratio curve in step S2, the value of the dose ratio divided by the solid angle ratio is calculated, and it is used as the new Y-axis and the solid angle ratio as the X-axis to obtain the dose multiple-solid angle ratio curve.
[0060] Dose multiple-depth curve conversion includes:
[0061] According to the corresponding relationship between the depth and the solid angle ratio in the depth-solid angle ratio curve obtained in step S2, a one-to-one search is performed on the X-axis in the dose multiple-solid angle ratio curve, and the depth is used as the new X-axis to obtain the dose multiple-depth curve;
[0062] in, Figure 5 is the dose multiple-solid angle ratio curve obtained by converting the detection point 1 in geometry 1; Figure 6 is the dose multiple-depth curve converted for detection point 1 in geometry 1;
[0063] 1.4. Taking the solid angle ratio of the weak area of 20% and the upper limit of the shielding depth of 30mmAl as the criterion, the criterion is substituted into the dose multiple-solid angle ratio curve and the dose multiple-depth curve respectively, and the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth of the detection point 1, the detection point 2 and the detection point 3 in the geometry 1 are respectively obtained, and the final total dose multiple is obtained by taking the larger value of the two or taking the average value of the two, and the total dose multiple is multiplied by the total dose to obtain the radiation margin determined according to the weak area;
[0064] Table 2 Calculation results of radiation margin for different detection points
[0065] Detection point 1 Detection point 2 Detection point 3 Dose multiple (according to upper limit of shielding depth) 3.320 1.419 1.314 Dose multiple (based on solid angle ratio) 2.430 1.308 1.471 The dose calculated by taking the larger value of the two (rad) 7.527e+1 7.400e+1 1.039e+2 The dose calculated by taking the average of the two (rad) 6.518e+1 7.110e+1 9.832e+1
[0066] As shown in Table 2, the shielding depth upper limit of 30mmAl and the solid angle ratio of 20% are used as the criteria, and are substituted into the dose multiple-solid angle ratio curve and the dose multiple-depth curve respectively, and then the radiation margin calculation results are obtained by taking the larger value and the average value of the two.
[0067] Although the disclosure is disclosed as above, the protection scope of the disclosure is not limited thereto. Those skilled in the art may make various changes and modifications without departing from the spirit and scope of the disclosure, and these changes and modifications will fall within the protection scope of the present invention.
Claims
1. A method for determining the radiation margin of a spacecraft weak area, characterized in that: The following steps are involved: Step S1, performing ray tracing calculation on the three-dimensional geometric structure of the spacecraft to obtain shielding depth data, as well as the dose value and spatial angle size in each spatial angle region; Step S2, calculating the depth-solid angle ratio curve, the dose ratio-solid angle ratio curve and the total dose; Step S3, converting the calculation result in step S2 into a dose multiple-solid angle ratio curve and a dose multiple-depth curve; Step S4, taking the solid angle ratio and the upper limit of the shielding depth of the weak area as criteria, substituting the criteria into the dose multiple-solid angle ratio curve and the dose multiple-depth curve, respectively, to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively, to obtain the final total dose multiple by taking the larger value of the two or taking the average value of the two, and multiplying the total dose multiple by the total dose to obtain the radiation margin determined according to the weak area.
2. The method for determining the radiation margin of a spacecraft weak area according to claim 1, characterized in that: In step S2, calculating the depth-solid angle ratio curve includes: sorting the shielding depth data in each spatial area according to the shielding depth, and then traversing it, taking the ratio of the accumulated spatial angle to the entire space as the X-axis data, and taking the shielding depth value as the Y-axis data, thereby obtaining the depth-solid angle ratio curve.
3. The method for determining the radiation margin of a spacecraft weak area according to claim 2, characterized in that: When calculating the depth-solid angle ratio curve, firstly, in the program, a data structure including shielding depth, total dose in the area, and solid angle weight is constructed, the shielding depth data in each spatial area is sorted according to the shielding depth, and then it is traversed, and the ratio of the accumulated spatial angle to the entire space is used as the X-axis data, and the shielding depth value is used as the Y-axis data, so as to obtain the depth-solid angle ratio curve.
4. The method for determining the radiation margin of a spacecraft weak area according to claim 1, characterized in that: In step S2, calculating the dose ratio-solid angle ratio curve includes: sorting the shielding depth data in each spatial region according to the shielding depth, then traversing it, accumulating the dose items in all spatial regions to obtain the total dose, and then traversing again, accumulating the spatial angle ratio, using it as the X-axis, calculating the ratio of the dose in each space to the total dose, accumulating the dose ratio, and using it as the Y-axis, thereby obtaining the dose ratio-solid angle ratio curve.
5. The method for determining the radiation margin of a spacecraft weak area according to claim 4, characterized in that: When calculating the depth-solid angle ratio curve, firstly, in the program, a data structure including shielding depth, total dose in the area, and solid angle weight is constructed, and the shielding depth data in each spatial area is sorted according to the shielding depth, and then it is traversed, and the dose items in all spatial areas are accumulated to obtain the total dose, and then it is traversed again, and the spatial angle ratio is accumulated and used as the X-axis, and the ratio of the dose in each space to the total dose is calculated, and the dose ratio is accumulated and used as the Y-axis to obtain the dose ratio-solid angle ratio curve.
6. The method for determining the radiation margin of a spacecraft weak area according to claim 1, characterized in that: In step S3, the dose multiple-solid angle ratio curve conversion includes: According to the dose ratio-solid angle ratio curve in step S2, the value of the dose ratio divided by the solid angle ratio is calculated, and it is used as the new Y-axis and the solid angle ratio as the X-axis to obtain the dose multiple-solid angle ratio curve.
7. The method for determining the radiation margin of a spacecraft weak area according to claim 6, characterized in that: Dose multiple-depth curve conversion includes: According to the corresponding relationship between the depth and the solid angle ratio in the depth-solid angle ratio curve obtained in step S2, a one-to-one search is performed on the X-axis in the dose multiple-solid angle ratio curve, and the depth is used as the new X-axis to obtain the dose multiple-depth curve.
8. The method for determining the radiation margin of a spacecraft weak area according to claim 1, characterized in that: In step S4, when the maximum value of the two is chosen to obtain the final total dose multiple, the solid angle ratio and the upper limit of the shielding depth of the weak area are used as criteria, and the dose multiple-solid angle ratio curve and the dose multiple-depth curve are consulted to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively. The two are compared, and the larger value is taken as the final dose multiple. Multiplying it by the total dose can obtain the radiation margin determined according to the weak area.
9. The method for determining the radiation margin of a spacecraft weak area according to claim 1, characterized in that: In step S4, when the method of taking the average of the two is chosen to obtain the final total dose multiple, the solid angle ratio and the upper limit of the shielding depth of the weak area identified as the weak area are used as criteria, and the dose multiple-solid angle ratio curve and the dose multiple-depth curve are consulted to obtain the total dose multiple determined by the solid angle ratio and the total dose multiple determined by the depth, respectively. The two are compared, and the average of the two is calculated as the final dose multiple. Multiplying it by the total dose can obtain the radiation margin determined according to the weak area.
Citation Information
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