Numerical analysis method for determining application range of cathode wall surface emission mechanism

Abstract

The invention discloses a numerical analysis method for determining the application range of a cathode wall surface emission mechanism. The numerical analysis method specifically comprises the following steps: acquiring an M-G formula calculation result and electron emission current densities under three emission mechanisms of hot electrons, field electrons and thermal field electrons according to a work function, a field enhancement factor, a cathode wall surface temperature and a surface electric field intensity of a determined cathode material; the calculation result of the M-G formula is used as a reference solution, the ratio of the reference solution to the electron emission current density under the three emission mechanisms is calculated, the ratio is defined as a deviation factor, and the corresponding parameter interval when the deviation factor is 0.75-1.25 is the application range of the corresponding emission mechanism. The invention aims to provide a numerical analysis method for determining the application range of a cathode wall surface emission mechanism, and solves the problems that an emission mechanism mathematical model in the prior art selects a proper model according to experience of experimenters to calculate electron emission current density; and the problems of gas discharge numerical simulation errors or inconsistency of simulation results are solved.

Description

Technical Field

[0001] This invention belongs to the field of low-temperature plasma physics technology and relates to a numerical analysis method for determining the applicable scope of cathode wall emission mechanisms. Background Technology

[0002] The widespread application of plasma has greatly promoted the development of modern science and technology, such as integrated circuit manufacturing, magnetic confinement fusion, plasma welding and cutting, plasma-assisted combustion, material modification, and biomedicine. Generating plasma by artificially breaking down gas is a common method used in industry and laboratories. The physical process of plasma interacting with electrode surfaces is crucial, being a key factor determining the microscopic mechanisms of particles and their industrial applications.

[0003] In plasma generation systems, metal electrodes are typically used. The role of the cathode in the electron injection process requires particular attention. The metal cathode contains a large number of electrons, which normally remain inside the cathode and do not escape. Electrons are only released when they acquire energy greater than the work function of the cathode material at the surface, or when the electrons weaken the potential barrier of the cathode surface through tunneling. The former is called thermionic emission, and the latter is called field-induced electron emission. When both emission mechanisms exist simultaneously, it is called thermal field-induced electron emission. In summary, cathode wall emission mechanisms include thermionic emission, field-induced electron emission, and thermal field-induced electron emission. Each emission mechanism has a corresponding mathematical model describing it. In numerical simulations of gas discharge, the selection of the mathematical model for the emission mechanism significantly affects the accuracy of the numerical simulation results.

[0004] In current numerical simulations of gas discharge, the mathematical models for emission mechanisms are selected based on the experience of experimenters to calculate electron emission current density, without a unified standard. This can lead to errors or inconsistencies in the numerical simulation results. Selecting the appropriate range of mathematical models for the three emission mechanisms—thermal electrons, field-induced electrons, and thermal fields—is the theoretical basis for accurately selecting suitable electron emission models in the numerical simulation of gas discharge. Therefore, this is a technical challenge that urgently needs to be solved. Summary of the Invention

[0005] The purpose of this invention is to provide a numerical analysis method for determining the applicable scope of the cathode wall emission mechanism. This method solves the problem in the prior art where the mathematical model of the emission mechanism is selected based on the experience of the experimenters to calculate the electron emission current density, which leads to errors in the numerical simulation of gas discharge or inconsistencies in the simulation results.

[0006] The technical solution adopted in this invention is a numerical analysis method for determining the applicable scope of the cathode wall emission mechanism, which is implemented according to the following steps: Step 1: Obtain the MG formula calculation results for the work function, field enhancement factor, cathode wall temperature, and surface electric field intensity of the determined cathode material; Step 2: Given the work function, field enhancement factor, cathode wall temperature, and surface electric field strength of the determined cathode material, obtain the electron emission current density under the three emission mechanisms of hot electrons, field-induced electrons, and hot field electrons. Step 3: Using the calculation result of the MG formula obtained in Step 1 as the reference solution, calculate the ratio of the reference solution to the electron emission current density under the three emission mechanisms respectively, and define the ratio as the deviation factor. The parameter range corresponding to the deviation factor being between 0.75 and 1.25 is the applicable range of the corresponding emission mechanism.

[0007] Furthermore, step 1 specifically involves: Step 1.1: Give an expression for the MG formula that accurately describes the electron emission phenomenon under the influence of a wide range of cathode surface temperature and surface electric field intensity; Step 1.2: Dimensionlessize the MG formula to obtain a dimensionless expression; Step 1.3: Given the initial parameter ranges for four parameters of the cathode material: work function, field enhancement factor, cathode wall temperature, and surface electric field intensity; Step 1.4: Discretize the initial parameter intervals, and use the left boundary value of each interval as a reference to approximately solve the first and second integrals of the dimensionless MG formula, thereby obtaining the calculation results of the MG formula.

[0008] Furthermore, the expression for the MG formula, which accurately describes the electron emission phenomenon under the influence of a wide range of cathode surface temperature and surface electric field intensity in step 1.1, is as follows:

[0009] in, J MG The cathode electron emission current density is calculated using the MG formula, where e is the electron charge. The work function of the cathode material. T C The cathode wall temperature, E C The electric field strength on the cathode surface. β As the field enhancement factor, ɛ The component representing electron energy perpendicular to the surface of the metal cathode. N The Fermi-Dirac distribution, representing the free electrons inside a metal, D The tunneling probability of free electrons inside the metal is expressed as:

[0010]

[0011] In the formula, This is the critical electron energy. m e For electronic quality, k B Boltzmann's constant, a These are parameters related to the electric field strength on the cathode surface. For feature parameters, To and The relevant function expression is specifically described as follows:

[0012]

[0013]

[0014]

[0015] In the formula, e 0 is the vacuum permittivity, and π is pi. h is Planck's constant. and These are the first and second kind of complete elliptic integrals, respectively. x and x ' is the independent variable, and the expression is:

[0016]

[0017] The MG formula is obtained by reorganizing as follows: .

[0018] Furthermore, the dimensionless expression in step 1.2 is as follows: Define the dimensionless terms:

[0019]

[0020]

[0021]

[0022]

[0023] in, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression; but The dimensionless expression for MG's formula is ultimately: .

[0024] Furthermore, the initial parameter range in step 1.3 is: the work function of the cathode material. Cathode wall temperature T C Cathode surface electric field strength E C and field enhancement factor β The initial intervals are as follows: = 1~5eV; T C = 300~6000 K; E C = 10 5 ~10 10 V / m; β =1~55.

[0025] Furthermore, step 1.4 specifically includes: Discretize the initial parameter intervals, and using the left boundary value of each interval as a reference, approximately solve the first and second integrals of the dimensionless expression of the MG formula, thereby obtaining the calculation result of the MG formula, specifically: Work function of five cathode materials Perform calculations. = 1eV, 2eV, 3eV, 4eV and 5eV; Work function for each cathode material Three field enhancement factors were selected for calculation. β= 1, 10, and 55; Given the work function and field enhancement factor of the cathode material, the initial range of the cathode wall temperature is divided into 57 parameter intervals, with the value at the left boundary of each interval being [value missing]. T Ci =200+100 i In the formula iThis represents the number of temperature ranges at the cathode wall. i =1,2,3……,57; The initial range of the electric field intensity on the cathode surface is divided into 100 parameter intervals, with the value at the left boundary of each interval being [value missing]. E Cj =10 4.95+0.05j In the formula j This represents the number of temperature ranges at the cathode wall. j =1,2,3……,100; For a given work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field strength, by setting the upper limit of the second integral to infinity to 100 and the lower limit of the first integral to infinity to -100, the integral result of the dimensionless expression of the MG formula can be calculated. Then, based on the dimensionless expression of the MG formula and the relationship between the MG formula, we have:

[0026] Obtain the calculation results of the MG formula. J MG .

[0027] Furthermore, step 2 specifically involves: The formulas for the three emission mechanisms—thermal electrons, field electrons, and thermal field electrons—are given as follows: Given the RD formula describing thermionic emission:

[0028]

[0029] In the formula, J RD The thermionic emission current density, A RD It is the thermionic emission constant; Given the FN formula for field-induced electron emission:

[0030] In the formula, J FN This represents the field-induced electron emission current density; A FN and B FN These are the field-induced electron emission constants, expressed as:

[0031]

[0032] v (z )and t 2 ( z ) is a modified value based on the original FN equation. z These are parameters relating to the electric field and work function at the cathode surface, and their corresponding expressions are:

[0033]

[0034]

[0035] Given the simplified Hantzsche formula for thermal electron emission:

[0036] In the formula, J Hantzsche This represents the electron emission current density in the thermal field. K 1 is an optimization item; k For adaptive coefficients, k =1.45; A Han Represents the thermal field electron emission constant; B Han , C Han and D Han It is a thermal field electron emission correction coefficient related to the work function of the metal cathode material, and its corresponding expression is:

[0037]

[0038]

[0039]

[0040]

[0041] Then, based on the discrete intervals defined in step 1.4, the electron emission current density is calculated under the three emission mechanisms of thermionic electrons, field-induced electrons, and thermal field electrons, given the corresponding determined work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field intensity. J RD , J FN , J Hantzsche .

[0042] Furthermore, step 3 specifically involves: The calculations will be performed under the given work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field intensity. J MG Electron emission current density calculated under the corresponding three emission mechanisms J RD , J FN , J Hantzsche Find the ratio J MG / J RD , J MG / J FN , J MG / J Hantzsche The ratio is defined as the deviation factor. If the deviation factor is between 0.75 and 1.25, it proves that the corresponding parameter range is the applicable range of the corresponding emission mechanism.

[0043] The beneficial effects of this invention are: This invention uses the Murphy and Good formula (MG) as a unified analytical solution encompassing the entire electron emission process. First, the MG formula is dimensionless and then discretely solved. Using the MG formula as a benchmark, a specific metallic material is selected. Under different cathode surface temperatures and electric field intensities, the calculation results of the simplified formulas (RD, FN, and Hantzsche) are compared with the MG formula. This provides the applicable range of the three simplified formulas for the given metallic cathode material. The applicability of this applicable range is then evaluated by comparing it with existing numerical solutions. The verification method provided by this invention can effectively guide the design and simulation of hot cathode and field emission cathode devices, helping researchers quickly and accurately select the most suitable emission model when facing complex operating conditions, thereby avoiding calculation inaccuracies caused by model misuse. Attached Figure Description

[0044] Figure 1 This is a graph showing the variation of the deviation factor corresponding to the thermionic emission mechanism in the numerical analysis method for determining the applicable scope of the cathode wall emission mechanism in this invention; Figure 2 This is a diagram showing the variation of the deviation factor corresponding to the field-induced electron emission mechanism in the numerical analysis method for determining the applicable scope of the cathode wall emission mechanism in this invention; Figure 3 This is a graph showing the variation of the deviation factor corresponding to the thermal field electron emission mechanism in the numerical analysis method for determining the applicable scope of the cathode wall emission mechanism in this invention. Figure 4 This is a graph showing the deviation factor obtained using the method of this invention and existing methods. Detailed Implementation

[0045] The following detailed description is provided in conjunction with specific implementation methods.

[0046] Example 1 The numerical analysis method for determining the applicable scope of the cathode wall emission mechanism in this invention is implemented according to the following steps: Step 1: Obtain the MG formula calculation results for the work function, field enhancement factor, cathode wall temperature, and surface electric field intensity of the determined cathode material; Step 2: Given the work function, field enhancement factor, cathode wall temperature, and surface electric field strength of the determined cathode material, obtain the electron emission current density under the three emission mechanisms of hot electrons, field-induced electrons, and hot field electrons. Step 3: Using the calculation result of the MG formula obtained in Step 1 as the reference solution, calculate the ratio of the reference solution to the electron emission current density under the three emission mechanisms respectively, and define the ratio as the deviation factor. The parameter range corresponding to the deviation factor being between 0.75 and 1.25 is the applicable range of the corresponding emission mechanism.

[0047] Example 2 Based on Example 1, step 1 specifically includes: Step 1.1: Give an expression for the MG formula that accurately describes the electron emission phenomenon under the influence of a wide range of cathode surface temperature and surface electric field intensity; Step 1.2: Dimensionlessize the MG formula to obtain a dimensionless expression; Step 1.3: Given the initial parameter ranges for four parameters of the cathode material: work function, field enhancement factor, cathode wall temperature, and surface electric field intensity; Step 1.4: Discretize the initial parameter intervals, and use the left boundary value of each interval as a reference to approximately solve the first and second integrals of the dimensionless MG formula, thereby obtaining the calculation results of the MG formula.

[0048] Example 3 Based on Example 2, the expression for the MG formula that accurately describes the electron emission phenomenon under the influence of a wide range of cathode surface temperature and surface electric field intensity in step 1.1 is as follows:

[0049] in, J MG The cathode electron emission current density is calculated using the MG formula, where e is the electron charge. The work function of the cathode material. T C The cathode wall temperature, E CThe electric field strength on the cathode surface. β As the field enhancement factor, ɛ The component representing electron energy perpendicular to the surface of the metal cathode. N The Fermi-Dirac distribution, representing the free electrons inside a metal, D The tunneling probability of free electrons inside the metal is expressed as:

[0050]

[0051] In the formula, This is the critical electron energy. m e For electronic quality, k B Boltzmann's constant, a These are parameters related to the electric field strength on the cathode surface. For feature parameters, To and The relevant function expression is specifically described as follows:

[0052]

[0053]

[0054]

[0055] In the formula, e 0 is the vacuum permittivity, and π is pi. h is Planck's constant. and These are the first and second kind of complete elliptic integrals, respectively. x and x ' is the independent variable, and the expression is:

[0056]

[0057] The MG formula is obtained by reorganizing as follows: .

[0058] The dimensionless expression in step 1.2 is as follows: Define the dimensionless terms:

[0059]

[0060]

[0061]

[0062]

[0063] in, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression; but The dimensionless expression for MG's formula is ultimately: .

[0064] The initial parameter range in step 1.3 is: the work function of the cathode material. Cathode wall temperature T C Cathode surface electric field strength E C and field enhancement factor β The initial intervals are as follows: = 1~5eV; T C = 300~6000 K; E C = 10 5 ~10 10 V / m; β =1~55.

[0065] Step 1.4 specifically involves: Discretize the initial parameter intervals, and using the left boundary value of each interval as a reference, approximately solve the first and second integrals of the dimensionless expression of the MG formula, thereby obtaining the calculation result of the MG formula, specifically: Work function of five cathode materials Perform calculations. = 1eV, 2eV, 3eV, 4eV and 5eV; Work function for each cathode material Three field enhancement factors were selected for calculation. β= 1, 10, and 55; Given the work function and field enhancement factor of the cathode material, the initial range of the cathode wall temperature is divided into 57 parameter intervals, with the value at the left boundary of each interval being [value missing]. T Ci =200+100 i In the formula i This represents the number of temperature ranges at the cathode wall. i =1,2,3……,57; The initial range of the electric field intensity on the cathode surface is divided into 100 parameter intervals, with the value at the left boundary of each interval being [value missing]. E Cj =10 4.95+0.05j In the formula j This represents the number of temperature ranges at the cathode wall. j =1,2,3……,100; For a given work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field strength, by setting the upper limit of the second integral to infinity to 100 and the lower limit of the first integral to infinity to -100, the integral result of the dimensionless expression of the MG formula can be calculated. Then, based on the dimensionless expression of the MG formula and the relationship between the MG formula, we have:

[0066] Obtain the calculation results of the MG formula. J MG .

[0067] Example 4 Based on Example 3, step 2 specifically includes: The formulas for the three emission mechanisms—thermal electrons, field electrons, and thermal field electrons—are given as follows: Given the RD formula describing thermionic emission:

[0068]

[0069] In the formula, J RD The thermionic emission current density, A RD It is the thermionic emission constant; Given the FN formula for field-induced electron emission:

[0070] In the formula, JFN This represents the field-induced electron emission current density; A FN and B FN These are the field-induced electron emission constants, expressed as:

[0071]

[0072] v ( z )and t 2 ( z ) is a modified value based on the original FN equation. z These are parameters relating to the electric field and work function at the cathode surface, and their corresponding expressions are:

[0073]

[0074]

[0075] Given the simplified Hantzsche formula for thermal electron emission:

[0076] In the formula, J Hantzsche This represents the electron emission current density in the thermal field. K 1 is an optimization item; k For adaptive coefficients, k =1.45; A Han Represents the thermal field electron emission constant; B Han , C Han and D Han It is a thermal field electron emission correction coefficient related to the work function of the metal cathode material, and its corresponding expression is:

[0077]

[0078]

[0079]

[0080]

[0081] Then, based on the discrete intervals defined in step 1.4, the electron emission current density is calculated under the three emission mechanisms of thermionic electrons, field-induced electrons, and thermal field electrons, given the corresponding determined work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field intensity. J RD , J FN , J Hantzsche .

[0082] Example 5 Based on Example 4, step 3 specifically includes: The calculations will be performed under the given work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field intensity. J MG Electron emission current density calculated under the corresponding three emission mechanisms J RD , J FN , J Hantzsche Find the ratio J MG / J RD , J MG / J FN , J MG / J Hantzsche The ratio is defined as the deviation factor. If the deviation factor is between 0.75 and 1.25, it proves that the corresponding parameter range is the applicable range of the corresponding emission mechanism.

[0083] Example 6 Based on Example 5, electronic mass ( m e ), electron charge (e), vacuum permittivity ( e 0), Boltzmann constant ( k B Planck constant ( h The specific expressions for π and π are: m e =9.11×10 -31 kg e =1.6022×10 -19 C e 0 = 8.8542 × 10 -12 J / K kB =1.3807×10 -23 J / K h =6.626×10 -34 J s π = 3.141593.

[0084] Example 7 Based on Example 6, initial parameters and cathode work function are set. Φ= 5 eV, field enhancement factor β= 10. Cathode wall temperature T C = 300~6000 K, electric field strength at the cathode surface E C = 10 5 ~10 10 V / m, the calculation results of the MG formula and the calculation results of RD, FN and Hantzsche simplified formula are obtained according to steps 1 and 2 respectively. Then, the result is obtained when J MG / J RD , J MG / J FN , J MG / J Hantzsche The ratio, the result is as follows Figure 1-3 As shown, the results indicate that, under given initial parameters, the applicable range for thermionic emission is approximately [missing information]. T C =2200~6000 K, βE C =10 5 ~10 6 V / m; the applicable range of field electron emission varies depending on the specific circumstances. T C , βE C Different ranges, such as T C =300 K βE C = (150~185) × 10 7 V / m, T C =1000K βE C = (225~250) × 10 7V / m; the applicable range of thermal electron emission is approximately T C =1500~6000 K hours βE C =10 5 ~10 6 V / m, βE C =10 7 V / m T C =2900~6000 K.

[0085] Method verification: Benilov et al. [Benilov MS, Benilova LG. Field to thermo-field tothermionic electron emission: A practical guide to evaluation and electronemission from arc cathodes[J]. Journal of Applied Physics, 2013, 114(6): 1464-256] used the Pad approximate integral to solve the MG formula, compared the thermal field electron emission current density obtained by the Hantzsche simplified formula with the MG formula, and selected the cathode work function. Φ= 4.5 eV, the emission electric field strength at the cathode surface is 10 9 and 10 10 V m -1 Cathode wall temperature T C = 500~6000 K, comparison results are as follows Figure 4 As shown, the dashed line represents the calculation results from the literature, and the solid line represents the calculation results from the method of this invention. The results show that the two curves agree well, proving the correctness of the method of this invention. Compared with the literature, this invention proposes a more universal calculation method that can calculate the emitted electric field intensity of the cathode surface for different cathode work functions and a wider range. It also provides the applicable ranges for the three emission formulas: thermionic electrons, field-induced electrons, and thermal field electrons.

Claims

1. A numerical analysis method for determining the applicable range of cathode wall emission mechanisms, characterized in that, The specific steps are as follows: Step 1: Obtain the MG formula calculation results for the work function, field enhancement factor, cathode wall temperature, and surface electric field intensity of the determined cathode material; Step 2: Given the work function, field enhancement factor, cathode wall temperature, and surface electric field strength of the determined cathode material, obtain the electron emission current density under the three emission mechanisms of hot electrons, field-induced electrons, and hot field electrons. Step 3: Using the calculation result of the MG formula obtained in Step 1 as the reference solution, calculate the ratio of the reference solution to the electron emission current density under the three emission mechanisms respectively, and define the ratio as the deviation factor. The parameter range corresponding to the deviation factor being between 0.75 and 1.25 is the applicable range of the corresponding emission mechanism.

2. The numerical analysis method for determining the applicable range of the cathode wall emission mechanism according to claim 1, characterized in that, Step 1 specifically involves: Step 1.1: Give an expression for the MG formula that accurately describes the electron emission phenomenon under the influence of a wide range of cathode surface temperature and surface electric field intensity; Step 1.2: Dimensionlessize the MG formula to obtain a dimensionless expression; Step 1.3: Given the initial parameter ranges for four parameters of the cathode material: work function, field enhancement factor, cathode wall temperature, and surface electric field intensity; Step 1.4: Discretize the initial parameter intervals, and use the left boundary value of each interval as a reference to approximately solve the first and second integrals of the dimensionless MG formula, thereby obtaining the calculation results of the MG formula.

3. The numerical analysis method for determining the applicable range of the cathode wall emission mechanism according to claim 2, characterized in that, The expression for the MG formula, which accurately describes the electron emission phenomenon under the influence of a wide range of cathode surface temperature and surface electric field intensity in step 1.1, is as follows: in, J MG The cathode electron emission current density is calculated using the MG formula, where e is the electron charge. The work function of the cathode material. T C The cathode wall temperature, E C The electric field strength on the cathode surface. β As the field enhancement factor, ɛ The component representing electron energy perpendicular to the surface of the metal cathode. N The Fermi-Dirac distribution, representing the free electrons inside a metal, D The tunneling probability of free electrons inside the metal is expressed as: In the formula, This is the critical electron energy. m e For electronic quality, k B Boltzmann's constant, a These are parameters related to the electric field strength on the cathode surface. For feature parameters, To and The relevant function expression is specifically described as follows: In the formula, ε 0 is the vacuum permittivity, and π is pi. h Let be Planck's constant. and These are the first and second kind of complete elliptic integrals, respectively. x and x ' is the independent variable, and the expression is: The MG formula is obtained by reorganizing as follows: 。 4. The numerical analysis method for determining the applicable range of the cathode wall emission mechanism according to claim 3, characterized in that, The dimensionless expression in step 1.2 is specifically as follows: Define the dimensionless terms: in, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression, for Dimensionless expression; but The dimensionless expression for MG's formula is ultimately: 。 5. The numerical analysis method for determining the applicable range of the cathode wall emission mechanism according to claim 4, characterized in that, The initial parameter range in step 1.3 is: the work function of the cathode material. Cathode wall temperature T C Cathode surface electric field strength E C and field enhancement factor β The initial intervals are as follows: = 1~5eV; T C = 300~6000 K； E C = 10 5 ~10 10 V / m; β =1~55。 6. The numerical analysis method for determining the applicable range of the cathode wall emission mechanism according to claim 5, characterized in that, Step 1.4 specifically includes: Discretize the initial parameter intervals, and using the left boundary value of each interval as a reference, approximately solve the first and second integrals of the dimensionless expression of the MG formula, thereby obtaining the calculation result of the MG formula, specifically: Work function of five cathode materials Perform calculations. = 1eV, 2eV, 3eV, 4eV and 5eV; Work function for each cathode material Three field enhancement factors were selected for calculation. β= 1, 10, and 55; Given the work function and field enhancement factor of the cathode material, the initial range of the cathode wall temperature is divided into 57 parameter intervals, with the value at the left boundary of each interval being [value missing]. T Ci =200+100 i In the formula i This represents the number of temperature ranges at the cathode wall. i =1,2,3……,57; The initial range of the electric field intensity on the cathode surface is divided into 100 parameter intervals, with the value at the left boundary of each interval being [value missing]. E Cj =10 4.95+0.05j In the formula j This represents the number of temperature ranges at the cathode wall. j =1,2,3……,100; For a given work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field strength, by setting the upper limit of the second integral to infinity to 100 and the lower limit of the first integral to infinity to -100, the integral result of the dimensionless expression of the MG formula can be calculated. Then, based on the dimensionless expression of the MG formula and the relationship between the MG formula, we have: Obtain the calculation results of the MG formula. J MG .

7. The numerical analysis method for determining the applicable range of the cathode wall emission mechanism according to claim 6, characterized in that, Step 2 specifically involves: The formulas for the three emission mechanisms—thermal electrons, field electrons, and thermal field electrons—are given as follows: Given the RD formula describing thermionic emission: In the formula, J RD The thermionic emission current density, A RD It is the thermionic emission constant; Given the FN formula for field-induced electron emission: In the formula, J FN This represents the field-induced electron emission current density; A FN and B FN These are the field-induced electron emission constants, expressed as: v ( z )and t 2 ( z ) is a correction value based on the original FN equation. z These are parameters relating to the electric field and work function at the cathode surface, and their corresponding expressions are: Given the simplified Hantzsche formula for thermal electron emission: In the formula, J Hantzsche This represents the electron emission current density in the thermal field. K 1 represents an optimization item; k For adaptive coefficients, k = 1.45; A Han Represents the thermal field electron emission constant; B Han , C Han and D Han It is a thermal field electron emission correction coefficient related to the work function of the metal cathode material, and its corresponding expression is: Then, based on the discrete intervals defined in step 1.4, the electron emission current density is calculated under the three emission mechanisms of thermionic electrons, field-induced electrons, and thermal field electrons, given the corresponding determined work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field intensity. J RD , J FN , J Hantzsche .

8. The numerical analysis method for determining the applicable range of the cathode wall emission mechanism according to claim 7, characterized in that, Step 3 specifically involves: The calculations will be performed under the given work function of the cathode material, field enhancement factor, cathode wall temperature, and surface electric field intensity. J MG Electron emission current density calculated under the corresponding three emission mechanisms J RD , J FN , J Hantzsche Find the ratio J MG / J RD , J MG / J FN , J MG / J Hantzsche The ratio is defined as the deviation factor. If the deviation factor is between 0.75 and 1.25, it proves that the corresponding parameter range is the applicable range of the corresponding emission mechanism.

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