Nuclear measurement energy spectrum result full width at half maximum analysis method and system
In nuclear fusion research, the Gaussian peak superposition of multiple single-energy hydrogen nuclei is used to calculate the relationship coefficients between different heights and the half-height and full width of the original Gaussian peak in the neutron scattering energy spectrum results, which solves the problem of difficulty in extracting the half-height and full width, and realizes effective analysis of the energy spectrum results.
Patent Information
- Application Number
- CN202510155400.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-12
- Publication Date
- 2025-05-13
AI Technical Summary
In nuclear fusion research, it is difficult to directly extract the full width of half height (FWHM) information in the energy spectrum results measured by the neutron scattering method, because the scattering of single-energy neutrons and hydrogen nuclei causes the energy distribution of the hydrogen nuclei to be rectangular, hindering the direct extraction of half height and full width.
By using the Gaussian peak superposition of multiple single-energy hydrogen nuclei on the rising or falling edge of the energy spectrum, the relationship coefficient k of different height h% and the original Gaussian peak half-height full width FWHM is calculated, thereby calculating the half-height full width FWHM_.
The analysis of half-height full width from the energy spectrum measured by the neutron scattering method is achieved, and the problem of difficulty in extracting half-height full width in traditional methods is solved.
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Figure CN119986764A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to the field of fusion research, and in particular to an analysis method and system for the full width at half maximum of a nuclear measurement energy spectrum result. Background Art
[0002] In the field of nuclear fusion research, neutron spectroscopy is a key technology used to obtain important information about plasma. Traditionally, the characteristics of spectral measurement results are characterized by the full width at half maximum (FWHM) of the peak. Since neutrons are not charged, data cannot be obtained through direct measurement, so indirect measurement methods are needed.
[0003] A common indirect measurement method is to use the scattering process of neutrons and hydrogen nuclei, but in most cases, no peak can be obtained in this process, making it difficult to extract the full width at half maximum (FWHM) information from the energy spectrum measurement results. This is because: in the scattering of monoenergetic neutrons and hydrogen nuclei, the energy distribution of hydrogen nuclei is theoretically rectangular, and this distribution characteristic prevents us from directly extracting the full width at half maximum (FWHM) information from the measurement results.
[0004] In view of this, this application is hereby filed. Summary of the invention
[0005] The present invention aims to provide a method and system for analyzing the half-width of the energy spectrum results of nuclear measurement. In order to analyze the half-width from the energy spectrum results measured by the neutron scattering method, the present invention is based on the rising edge or falling edge of the energy spectrum, through the superposition of Gaussian peaks of multiple monoenergetic hydrogen nuclei, and calculates the relationship coefficient k between the different heights h% of the rising edge and the half-width FWHM of the original Gaussian peak at the first point or the relationship coefficient k between the different heights h% of the falling edge and the half-width FWHM of the original Gaussian peak at the second point, and then calculates the half-width FWHM_. The present invention can analyze the half-width from the energy spectrum results measured by the neutron scattering method.
[0006] The present invention is achieved through the following technical solutions:
[0007] In a first aspect, the present invention provides a method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result, the method comprising:
[0008] Acquire a nuclear measurement energy spectrum result, and identify an edge of the nuclear measurement energy spectrum result; the edge includes a rising edge or a falling edge;
[0009] Subtracting the background from the edge of the energy spectrum result of the identified nuclear measurement to obtain edge data after the background is subtracted;
[0010] In the edge data after background deduction, the full width at half maximum is calculated according to the first position and the second position corresponding to the first edge height and the second edge height respectively and the relationship coefficient;
[0011] Among them, the first edge height and the second edge height are any edge heights in the edge data after background subtraction, and the first edge height is not equal to the second edge height.
[0012] Furthermore, the calculation formula for the full width at half maximum is:
[0013] FWHM_ = |(x1 - x2) / (k1 - k2)|
[0014] In the formula, FWHM_ is the full width at half maximum; x1 is the first position corresponding to the first edge height h1%; x2 is the second position corresponding to the second edge height h2%; k1 is the relationship coefficient between the first edge height h1% and the full width at half maximum EWHM of the original Gaussian peak; k2 is the relationship coefficient between the second edge height h2% and the full width at half maximum FWHM of the original Gaussian peak; || represents the absolute value.
[0015] Furthermore, the relationship coefficient is calculated based on the superposition of Gaussian peaks of multiple monoenergetic particles. The calculation formula for the relationship coefficient k is:
[0016] g(a + k * FWHM) ≈ g max (x) * h%
[0017]
[0018] In the formula, FWHM is the full width at half maximum of the original Gaussian peak; g(x) is the total density function after the superposition of Gaussian peaks of multiple monoenergetic particles; a is the first position, b is the second position, the centers of the Gaussian peaks are evenly distributed between a and b, where a < b; σ is the standard deviation of the Gaussian peak; μ is the center of the Gaussian peak; erf() is the error function; g max (x) is the highest value of g(x); h% is the edge height.
[0019] Furthermore, the value ranges of k1 and k2 are (-0.699, -0.544, -0.499, -0.44, -0.357, 0, 0.357, 0.44, 0.499, 0.544, 0.699);
[0020] The corresponding value ranges of h1 and h2 are (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5).
[0021] Furthermore, the method further includes:
[0022] Performing smoothing processing on the edge data after background subtraction.
[0023] Furthermore, this method is applicable to the energy spectrum results of neutron measurement using hydrogen nuclei, helium nuclei or carbon nuclei.
[0024] Furthermore, this method is applicable to the energy spectrum results of electrons generated by the Compton effect of gamma rays.
[0025] In a second aspect, the present invention further provides an analysis system for the full width at half maximum (FWHM) of nuclear measurement energy spectrum results, which system includes:
[0026] An acquisition unit, configured to acquire nuclear measurement energy spectrum results;
[0027] An edge recognition unit, configured to recognize the edges of the nuclear measurement energy spectrum results; the edges include rising edges or falling edges;
[0028] A background subtraction unit, configured to subtract the background from the recognized edges of the nuclear measurement energy spectrum results to obtain edge data after background subtraction;
[0029] An FWHM calculation unit, configured to calculate the FWHM in the edge data after background subtraction according to the first position and the second position corresponding to the first edge height and the second edge height respectively and the relationship coefficient; wherein, the first edge height and the second edge height are any edge heights in the edge data after background subtraction, and the first edge height is not equal to the second edge height.
[0030] Furthermore, the calculation formula for the FWHM is:
[0031] FWHM_ = |(x1 - x2) / (k1 - k2)|
[0032] In the formula, FWHM_ is the FWHM; x1 is the first position corresponding to the first edge height h1%; x2 is the second position corresponding to the second edge height h2%; k1 is the relationship coefficient between the first edge height h1% and the FWHM of the original Gaussian peak EWHM; k2 is the relationship coefficient between the second edge height h2% and the FWHM of the original Gaussian peak FWHM; || represents the absolute value.
[0033] Furthermore, the relationship coefficient is calculated based on the superposition of Gaussian peaks of multiple monoenergetic particles, and the calculation formula for the relationship coefficient k is:
[0034] g(a + k*FWHM) ≈ g max (x)*h%
[0035]
[0036] In the formula, FWHM is the FWHM of the original Gaussian peak; g(x) is the total density function after the superposition of Gaussian peaks of multiple monoenergetic particles; a is the first position, b is the second position, the centers of the Gaussian peaks are evenly distributed between a and b, where a < b; σ is the standard deviation of the Gaussian peak; μ is the center of the Gaussian peak; erf() is the error function; g max(x) is the maximum value of g(x); h% is the edge height.
[0037] Furthermore, the value range of k1 and k2 is (-0.699,-0.544,-0.499,-0.44,-0.357,0,0.357,0.44,0.499,0.544,0.699);
[0038] The value range of h1 and h2 in one-to-one correspondence is (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5).
[0039] Furthermore, the system also includes: a smoothing processing unit, which is used to smooth the edge data after deducting the background.
[0040] Compared with the prior art, the present invention has the following advantages and beneficial effects:
[0041] The present invention provides a method and system for analyzing the half-width at half maximum of a nuclear measurement energy spectrum result. In order to analyze the half-width at half maximum from the energy spectrum result measured by a neutron scattering method, the present invention is based on the rising edge or falling edge of the energy spectrum, through the superposition of Gaussian peaks of multiple monoenergetic hydrogen nuclei, and calculates the relationship coefficient k between different heights h% of the rising edge and the half-width at half maximum of the original Gaussian peak at the first point, or the relationship coefficient k between different heights h% of the falling edge and the half-width at half maximum of the original Gaussian peak at the second point, and then calculates the half-width at half maximum FWHM_. The present invention can analyze the half-width at half maximum from the energy spectrum result measured by a neutron scattering method. BRIEF DESCRIPTION OF THE DRAWINGS
[0042] The drawings described herein are used to provide a further understanding of the embodiments of the present invention, constitute a part of this application, and do not constitute a limitation of the embodiments of the present invention. In the drawings:
[0043] Figure 1 The present invention is a method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result. Figure 1 ;
[0044] Figure 2 The present invention is a method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result. Figure 2 ;
[0045] Figure 3 The present invention is a nuclear measurement spectrum result half-maximum full width analysis system structure frame Figure 1 ;
[0046] Figure 4 The present invention is a nuclear measurement spectrum result half-maximum full width analysis system structure frame Figure 2 . DETAILED DESCRIPTION
[0047] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with embodiments and drawings. The exemplary embodiments of the present invention and their description are only used to explain the present invention and are not intended to limit the present invention.
[0048] Example 1
[0049] like Figure 1 As shown, the present invention provides a method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result, the method comprising:
[0050] S1: Obtain a nuclear measurement energy spectrum result, and identify an edge of the nuclear measurement energy spectrum result; the edge includes a rising edge or a falling edge;
[0051] S2: subtracting the background from the edge of the energy spectrum result of the identified nucleus measurement to obtain edge data after background subtraction;
[0052] S3: In the edge data after deducting the background, calculate the half-height full width according to the first position and the second position corresponding to the first edge height and the second edge height respectively and the relationship coefficient; wherein the first edge height and the second edge height are any edge heights in the edge data after deducting the background, and the first edge height is not equal to the second edge height.
[0053] As a further implementation, the method further includes, before performing step S3: smoothing the edge data after deducting the background, such as Figure 2 shown.
[0054] This example briefly describes the basis of this analytical method as follows:
[0055] For the measurement of monoenergetic particles, the peak obtained is a Gaussian peak, and its density distribution f(x) is:
[0056]
[0057] Where μ is the center of the Gaussian peak and σ is the standard deviation. The highest position of the Gaussian peak is at x = μ, and the highest value is:
[0058]
[0059] Solve the following equation to find the position of the half-maximum value:
[0060]
[0061] Get the position of the half-maximum value:
[0062]
[0063] Then the original Gaussian peak half-maximum full width FWHM is:
[0064]
[0065] On this basis, the improvements and innovations of the present invention are as follows:
[0066] When measuring the hydrogen nuclei of neutron scattering, it can be considered as the superposition of Gaussian peaks of multiple mono-energetic hydrogen nuclei. Assume that the centers of the Gaussian peaks are uniformly distributed between a and b, where a < b, and the standard deviation of each Gaussian peak is σ. The total density function of these Gaussian peaks is g(x):
[0067]
[0068] In the formula, erf() is the error function;
[0069] If a - b > 6σ, the maximum value of g(x) can be calculated as:
[0070]
[0071] By solving the following equation, the relationship coefficient k between the different heights h% of the falling edge and the full width at half maximum FWHM of the original Gaussian peak at point b can be calculated.
[0072] g(b + k * FWHM) ≈ g max (x) * h% (8)
[0073] In the formula, FWHM is the full width at half maximum of the original Gaussian peak; g(x) is the total density function after the superposition of Gaussian peaks of multiple mono-energetic particles; b is the second position, and the centers of the Gaussian peaks are uniformly distributed between a and b, where a < b; σ is the standard deviation of the Gaussian peak; μ is the center of the Gaussian peak; g max (x) is the maximum value of g(x); h% is the edge height.
[0074] Substitute Equation (5) and Equation (7) into Equation (8). When h takes the values of 95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5 respectively, the k values obtained are -0.699, -0.544, -0.499, -0.44, -0.357, 0, 0.357, 0.44, 0.499, 0.544, 0.699 respectively.
[0075] Similarly, the relationship coefficient k between the different heights h% of the rising edge and the full width at half maximum FWHM of the original Gaussian peak at point a can be calculated.
[0076] g(a + k * FWHM) ≈ g max (x) * h% (9)
[0077] Wherein, FWHM is the full width at half maximum of the original Gaussian peak; g(x) is the total density function after the superposition of Gaussian peaks of multiple monoenergetic particles; a is the first position, and the centers of the Gaussian peaks are uniformly distributed between a and b, where a < b; σ is the standard deviation of the Gaussian peak; μ is the center of the Gaussian peak; g max (x) is the maximum value of g(x); h% is the edge height.
[0078] When h takes values of 95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5 respectively, the k values are 0.699, 0.544, 0.499, 0.44, 0.357, 0, -0.357, -0.44, -0.499, -0.544, -0.699 respectively.
[0079] Therefore, by obtaining the first position x1 and the second position x2 corresponding to the first edge height h1% and the second edge height h2%, the full width at half maximum FWHM_ = |(x1 - x2) / (k1 - k2)| can be calculated, where x1 is the first position corresponding to the first edge height h1%; x2 is the second position corresponding to the second edge height h2%; k1 is the relationship coefficient between the first edge height h1% and the full width at half maximum EWHM of the original Gaussian peak; k2 is the relationship coefficient between the second edge height h2% and the full width at half maximum FWHM of the original Gaussian peak; h1 and h2 are not equal; || represents the absolute value.
[0080] The value ranges of h1 and h2 are (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5), and the corresponding relationship coefficients k1 and k2 are (-0.699, -0.544, -0.499, -0.44, -0.357, 0, 0.357, 0.44, 0.499, 0.544, 0.699). That is:
[0081] The value range of h1 is (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5), and the corresponding relationship coefficient k1 is (-0.699, -0.544, -0.499, -0.44, -0.357, 0, 0.357, 0.44, 0.499, 0.544, 0.699);
[0082] The value range of h2 is (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5), and the corresponding relationship coefficient k2 is (-0.699, -0.544, -0.499, -0.44, -0.357, 0, 0.357, 0.44, 0.499, 0.544, 0.699).
[0083] The present invention can directly analyze the half-maximum full width from the results measured by the scattering method. The method of the present invention is particularly suitable for measuring the energy spectrum results of neutrons using hydrogen nuclei, helium nuclei or carbon nuclei, and is also suitable for the energy spectrum results of electrons generated by the gamma-ray Compton effect.
[0084] Example 2
[0085] like Figure 3 As shown, the difference between this embodiment and embodiment 1 is that this embodiment provides a nuclear measurement energy spectrum result half-maximum full width analysis system, and the system corresponds to the nuclear measurement energy spectrum result half-maximum full width analysis method in embodiment 1 in terms of function; the system includes:
[0086] An acquisition unit, used for acquiring nuclear measurement energy spectrum results;
[0087] An edge recognition unit, used to recognize the edge of the nuclear measurement energy spectrum result; the edge includes a rising edge or a falling edge;
[0088] A background subtraction unit is used to subtract the background from the edge of the identified nuclear measurement spectrum result to obtain edge data after the background is subtracted;
[0089] A half-height full width calculation unit is used to calculate the half-height full width in the edge data after deducting the background, according to the first position and the second position corresponding to the first edge height and the second edge height respectively, and the relationship coefficient; wherein the first edge height and the second edge height are any edge heights in the edge data after deducting the background, and the first edge height is not equal to the second edge height.
[0090] As a further implementation, the calculation formula of the half-height full width is:
[0091] FWHM_=|(x1-x2) / (k1-k2)|
[0092] In the formula, FWHM_ is the full width at half maximum; x1 is the first position corresponding to the first edge height h1%; x2 is the second position corresponding to the second edge height h2%; k1 is the relationship coefficient between the first edge height h1% and the original Gaussian peak half-height full width EWHM; k2 is the relationship coefficient between the second edge height h2% and the original Gaussian peak half-height full width FWHM; || represents the absolute value.
[0093] As a further implementation, the relationship coefficient is calculated based on the superposition of Gaussian peaks of multiple monoenergetic particles, and the calculation formula of the relationship coefficient k is:
[0094] g(a+k*FWHM)≈g max (x)*h%
[0095]
[0096] In the formula, FWHM is the full width at half maximum of the original Gaussian peak; g(x) is the total density function after the superposition of Gaussian peaks of multiple monoenergetic particles; a is the first position, b is the second position, and the centers of the Gaussian peaks are evenly distributed between a and b, where a < b; σ is the standard deviation of the Gaussian peak; μ is the center of the Gaussian peak; erf() is the error function; g max (x) is the highest value of g(x); h% is the edge height.
[0097] As a further implementation, the value ranges of k1 and k2 are (-0.699, -0.544, -0.499, -0.44, -0.357, 0, 0.357, 0.44, 0.499, 0.544, 0.699);
[0098] The corresponding value ranges of h1 and h2 are (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5).
[0099] As a further implementation, as Figure 4 shown, the system further includes: a smoothing processing unit for smoothing the edge data after background subtraction.
[0100] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.
[0101] The present application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or block in the flowchart and / or block diagram can be implemented by computer program instructions, and the combination of processes and / or blocks in the flowchart and / or block diagram can also be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing devices generate a device for realizing the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 one block or multiple blocks.
[0102] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.
[0103] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.
[0104] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.
Claims
1. A method for analyzing the full width at half maximum of nuclear measurement energy spectrum results, characterized in that: The method includes: Acquire a nuclear measurement energy spectrum result, and identify an edge of the nuclear measurement energy spectrum result; the edge includes a rising edge or a falling edge; Subtracting the background from the edge of the energy spectrum result of the identified nuclear measurement to obtain edge data after the background is subtracted; In the edge data after background deduction, the full width at half maximum is calculated according to the first position and the second position corresponding to the first edge height and the second edge height respectively and the relationship coefficient; The first edge height and the second edge height are any edge heights in the edge data after the background is deducted, and the first edge height is not equal to the second edge height.
2. The method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result according to claim 1, characterized in that: The calculation formula of the half-height full width is: FWHM_=|(x1-x2) / (k1-k2)| In the formula, FWHM_ is the full width at half maximum; x1 is the first position corresponding to the first edge height h1%; x2 is the second position corresponding to the second edge height h2%; k1 is the relationship coefficient between the first edge height h1% and the original Gaussian peak half-height full width EWHM; k2 is the relationship coefficient between the second edge height h2% and the original Gaussian peak half-height full width FWHM; || represents the absolute value.
3. The method for analyzing the full width at half maximum of a nuclear measurement spectrum result according to claim 2, characterized in that: The relationship coefficient is calculated based on the superposition of Gaussian peaks of multiple monoenergetic particles. The calculation formula of the relationship coefficient k is: g(a+k*FWHM)≈g max (x)*h% In the formula, FWHM is the full width at half maximum of the original Gaussian peak; g(x) is the total density function after the superposition of Gaussian peaks of multiple monoenergetic particles; a is the first position, b is the second position, and the centers of the Gaussian peaks are uniformly distributed between a and b, where a < b; σ is the standard deviation of the Gaussian peak; μ is the center of the Gaussian peak; erf() is the error function; g max (x) is the highest value of g(x); h% is the edge height.
4. The method for analyzing the full width at half maximum of a nuclear measurement spectrum result according to claim 2, characterized in that: The value range of k1 and k2 is (-0.699,-0.544,-0.499,-0.44,-0.357,0,0.357,0.44,0.499,0.544,0.699); The value range of h1 and h2 in one-to-one correspondence is (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5).
5. The method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result according to claim 1, characterized in that: The method further includes: The edge data after background deduction is smoothed.
6. The method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result according to claim 1, characterized in that: The method is applicable to the energy spectrum results of neutrons measured using hydrogen nuclei, helium nuclei or carbon nuclei.
7. The method for analyzing the full width at half maximum of a nuclear measurement energy spectrum result according to claim 1, characterized in that: The method is applicable to the energy spectrum results of electrons generated by the gamma-ray Compton effect.
8. A system for analyzing the full width at half maximum of nuclear measurement energy spectrum results, characterized in that: The system includes: An acquisition unit, used for acquiring nuclear measurement energy spectrum results; An edge recognition unit, used to recognize the edge of the nuclear measurement energy spectrum result; the edge includes a rising edge or a falling edge; A background subtraction unit is used to subtract the background from the edge of the identified nuclear measurement spectrum result to obtain edge data after the background is subtracted; A half-height full width calculation unit is used to calculate the half-height full width in the edge data after deducting the background, according to the first position and the second position corresponding to the first edge height and the second edge height respectively, and the relationship coefficient; wherein the first edge height and the second edge height are any edge heights in the edge data after deducting the background, and the first edge height is not equal to the second edge height.
9. The nuclear measurement spectrum result full width at half maximum analysis system according to claim 8, characterized in that: The calculation formula of the half-height full width is: FWHM_=|(x1-x2) / (k1-k2)| In the formula, FWHM_ is the full width at half maximum; x1 is the first position corresponding to the first edge height h1%; x2 is the second position corresponding to the second edge height h2%; k1 is the relationship coefficient between the first edge height h1% and the original Gaussian peak half-height full width EWHM; k2 is the relationship coefficient between the second edge height h2% and the original Gaussian peak half-height full width FWHM; || represents the absolute value.
10. The nuclear measurement spectrum result full width at half maximum analysis system according to claim 9, characterized in that: The value range of k1 and k2 is (-0.699,-0.544,-0.499,-0.44,-0.357,0,0.357,0.44,0.499,0.544,0.699); The value range of h1 and h2 in one-to-one correspondence is (95, 90, 88, 85, 80, 50, 20, 15, 12, 10, 5).