CN113792471B - Multi-particle multi-collision micro-discharge threshold Monte Carlo calculation method - Google Patents
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Abstract
A method for calculating a multi-particle multi-collision micro-discharge threshold Monte Carlo belongs to the technical field of space microwaves. The invention provides a Monte Carlo method for efficiently and accurately calculating micro-discharge threshold values of a satellite-borne microwave component, which carries out randomization treatment on the emergent energy, angles and phases of a plurality of initial electrons, and the physical process of the occurrence of a micro-discharge effect is more objectively and accurately represented by the multi-particle-multi-collision process according to the number of actual secondary electrons generated by collision electrons and the corresponding energy participating in the micro-discharge process after the collision moment. The calculation result shows that the calculation accuracy of the proposed method is obviously improved compared with the existing Monte Carlo method by taking the commercial particle simulation software result as a reference, and the calculation efficiency of the proposed method is obviously improved compared with that of the particle simulation method.
Description
Technical Field
The invention relates to a multi-particle multi-collision micro-discharge threshold Monte Carlo calculation method, and belongs to the technical field of space microwaves.
Background
The microdischarge effect, also known as secondary electron multiplication, refers to a component at 1 x 10 -3 Pa or lower, a resonance discharge phenomenon that occurs in the case of being subjected to a large power. The high-power microwave components in the spacecraft load such as output multiplexers, filters, switch matrixes, antenna feeds and the like are extremely easy to generate micro-discharge effects, and once the micro-discharge effects occur, serious consequences are caused: noise level rises and output power drops; the standing wave ratio of the microwave transmission system is increased, the reflected power is increased, and the channel is blocked; microwave component meterSurface damage, reduced load life; the load of the spacecraft is permanently invalid, so that micro discharge is one of effects which must be overcome in the development process of the high-power microwave component.
In the design stage, the micro-discharge risk assessment of the satellite-borne microwave component by a micro-discharge sensitive area or an analysis method is an effective means for reducing repeated design and avoiding long-period ground tests. At present, theoretical analysis methods including classical theory and statistical theory, and numerical simulation methods including Monte Carlo (MC) and Particle-in-Cell (PIC) have been developed in terms of microdischarge analysis methods. The theoretical method needs an analytic solution of the electron motion trail, so that the method is difficult to apply to micro-discharge threshold calculation of the microwave component with a complex structure. The PIC method simulates the movement of particles under an electromagnetic field and ignores and assumes few collision situations of the particles and the surfaces of the components, considers complete movement parameters such as the speed, the direction, the phase and the like of an electron emergent surface, considers interaction between electrons and the electromagnetic field and can realize micro-discharge threshold simulation of complex microwave component structures such as an impedance converter, a ridge waveguide filter, a coaxial cavity filter and the like. However, the particle simulation method needs to update the particle state and the field distribution on the grid nodes in the split grid, so that the calculation hardware cost is high and the time is long. The MC method has advantages over the PIC method in terms of time overhead because the interaction between electrons and electromagnetic fields is not considered, however, the three kinds of monte carlo microdischarge threshold calculation methods of single particle-multiple collision, multiple particle-single collision, pseudo multiple particle-multiple collision, which are proposed at present, still need to be improved in calculation accuracy due to the problems of the algorithm itself.
Disclosure of Invention
The invention solves the technical problems that: overcomes the defects of the prior art, provides a multi-particle multi-collision micro-discharge threshold Monte Carlo calculation method,
the technical scheme of the invention is as follows: a method for calculating a multi-particle multi-collision micro-discharge threshold Monte Carlo comprises the following steps:
s1, according to probability distribution which accords with initial electron energy, angle and phase, N initial values are randomly generatedThe electron, N initial electron states are: position r i Energy E i Angle (theta) i ,Ф i ) Phase t i ；
S2, calculating an electromagnetic field inside the microwave component according to the input frequency f and the input power W;
s3, updating the state of electrons entering an electromagnetic field, wherein the electronic state is as follows: position r in Energy E in Angle (theta) in ,Ф in ) Phase t in The same state for both initial electrons;
s4, calculating an electronic motion trail in a time step delta t;
s5, judging whether the position of the electron in the time step delta t reaches the boundary of the part, and repeating S4-S5 if the position does not reach the boundary; otherwise, the state of electron collision at the arrival boundary is recorded: position r p Energy E p Angle (theta) p ,Ф p ) Phase t p ；
S6, adopting electron energy E at collision time p And angle (theta) p ,Ф p ) From the surface material properties of the microwave component, a secondary electron emission coefficient (delta) is calculated ts 、δ e 、δ r ) Spectral distribution (f) of secondary electrons 1,e 、f 1,r 、f n,ts ) Probability P of generating n electrons n And calculate the energy E of the outgoing electrons si Angle (theta) si ,Ф si )；i＝0,1,…,n；
S7, judging whether a set cut-off condition is reached, if so, judging whether micro discharge occurs, and ending; if not, return to S3.
Further, the probability of generating n electrons is that
Wherein,respectively is notProbability of generating intrinsic secondary electrons, probability of generating 1 intrinsic secondary electron, probability of generating n intrinsic secondary electrons.
Further, the energy E of the outgoing electrons is calculated s The method of (1) comprises the following steps:
generating random number RAND within 0-1 1 Comparing RAND 1 And probability P of generating n electrons n Judging the number n of electrons actually generated;
if n=0, no secondary electrons are generated;
if n=1, according to X (1) =f 1,e +f 1,r +f 1,ts Calculating the energy E of the outgoing electrons s1 Wherein f 1,e To generate an energy spectrum of 1 elastically scattered electron, f 1,r To generate an energy spectrum of 1 reflected electron, f 1,ts To generate an energy spectrum of 1 intrinsic secondary electron;
if n.gtoreq.2, according to X (n) =f n,ts Calculating the energy E of the outgoing electrons s1 、E s2 …E sn The method comprises the steps of carrying out a first treatment on the surface of the Wherein f n,ts To generate an energy spectrum of n intrinsic secondary electrons.
Further, the exit angle (θ si ,Ф si ) The method comprises the following steps of:
sinθ si is [ -1,1]Random number, phi of samples between si Is [0,2 pi ]]Random numbers sampled in between.
Further, the cut-off condition is that the maximum number of cycles is reached, or the time is calculated, or the number of electrons is counted, or other conditions are set manually.
Further, the method for judging whether the micro discharge occurs comprises the following steps:
if the cycle number color_num is used as the method for judging the micro-discharge threshold value, delta is calculated equl ＝(δ 1 +δ 2 +…+δ collsion_num ) Color_num; wherein delta 1 、δ 2 、…、δ collsion_num Respectively the secondary electron emission coefficient average value of the ith collision; if delta equl >1, then the calculation condition issuesGenerating micro-discharge; if delta equl <1, no microdischarge occurs under the calculation conditions; if delta equl =1, which is the microdischarge threshold voltage;
if the change trend of the electron number evolving along with time is used as a method for judging the micro-discharge threshold value, the electron number is reduced along with time, and micro-discharge does not occur under the calculation condition; if the number of electrons increases with time, micro-discharge occurs under the calculation condition; if the number of electrons is constant over time, the voltage is the microdischarge threshold.
Further, the secondary electron emission coefficient δ=δ ts +δ e +δ r ，δ ts Is the emission coefficient of intrinsic secondary electrons, delta e Is the emissivity coefficient delta of elastically scattered electrons r Is the emissivity of the reflected electrons.
Further, the method for solving the motion trail of the electron in the electromagnetic field comprises the following steps:
in rectangular coordinate system, the electron track is composed ofDetermining;
in the cylindrical coordinate system, the track of electrons is composed ofDetermining;
wherein (x, y, z) is the x, y, z direction position of the electron in the electromagnetic field in the rectangular coordinate system,the radial, angular and axial positions of electrons in an electromagnetic field in a cylindrical coordinate system are represented by adding a point to the position of the electrons, the speed in the direction is represented by adding two points to the position of the electrons, the acceleration in the direction is represented by adding two points to the position of the electrons, the E, B is respectively an electric field and a magnetic field in the direction, q is the unit electric quantity of the electrons, and m is the mass of the electrons.
Further, after the electron collides with the boundary, updating the state that the electron enters the electromagnetic field as follows: electronic position r in Updated to the position, energy E, of electron impact in Update to the energy and angle (θ) of the outgoing electrons in step 6 in ,Ф in ) Updating to the angle, phase t, of the emergent electrons in the step 6 in Updated to the time at which the electron hit the boundary.
A computer readable storage medium storing a computer program which when executed by a processor implements the steps of the multi-particle multi-collision microdischarge threshold monte carlo calculation method.
Compared with the prior art, the invention has the advantages that:
(1) The invention overcomes the problem that no matter what the secondary electron emission coefficient of the collision electrons is in the calculation process, only one electron participates in the next collision, and according to the actual secondary electron emission process, the actual number of secondary electrons can be emitted, the number of the emitted electrons accords with certain probability distribution, but the statistical value of the number of electrons generated by the collision of a plurality of electrons in the same state is equal to the secondary electron emission coefficient, thereby improving the accuracy of the calculation result;
(2) According to the method, the random state of a plurality of initial electrons is taken as an initial state, and the electrons with all energies in the secondary electron emission process after the electrons collide with the surface of the component are contained in the calculation, so that the accuracy of a calculation result is remarkably improved;
(3) In the process of propelling the electronic motion trail in the electromagnetic field, the particle state in the subdivision grid and the field distribution on the grid nodes do not need to be updated according to the time step by a particle simulation method, so that the problems of high calculation hardware cost, long time consumption and the like are avoided. The method comprises the steps of carrying out a first treatment on the surface of the
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FIG. 1 is a schematic flow chart of the method of the present invention;
FIG. 2 is a comparison result of the micro-discharge threshold calculation according to the embodiment of the present invention and other calculation methods.
Detailed Description
In order to better understand the technical solutions described above, the following detailed description of the technical solutions of the present application is provided through the accompanying drawings and specific embodiments, and it should be understood that the specific features of the embodiments and embodiments of the present application are detailed descriptions of the technical solutions of the present application, and not limit the technical solutions of the present application, and the technical features of the embodiments and embodiments of the present application may be combined with each other without conflict.
The following describes in further detail a method for calculating a multi-particle multi-collision micro-discharge threshold value monte carlo according to the embodiments of the present application with reference to the accompanying drawings of the specification, and a specific implementation manner may include (as shown in fig. 1-2):
(1) According to probability distribution which accords with initial electron energy, angle and phase, N initial electrons are randomly generated, and N initial electron states are as follows: position r i Energy E i Angle (theta) i ,Ф i ) Phase t i ；
(2) Calculating the electromagnetic field inside the microwave component according to the input frequency f and the input power W;
(3) Updating the state of electrons entering an electromagnetic field, wherein the electronic state is as follows: position r in Energy E in Angle (theta) in ,Ф in ) Phase t in The same state for both initial electrons;
(4) Calculating an electronic motion trail in a time step delta t by adopting a fourth-order Dragon-Kutta method;
(5) Judging whether the position of the electron in the time step delta t reaches the boundary of the component, if not, advancing continuously according to the step (4), and if the position reaches the boundary, recording the collision state of the electron at the position of the boundary: position r p Energy E p Angle (theta) p ,Ф p ) Phase t p ；
(6) Using electron energy E at the moment of collision p And angle (theta) p ,Ф p ) Based on the surface material characteristics of the microwave component, the secondary electron emission coefficient (delta) is calculated based on the Furman model ts 、δ e 、δ r ) Spectral distribution (f) of secondary electrons 1,e 、f 1,r 、f n,ts ) Probability of generating n electrons
(6a) Generating random number RAND within 0-1 1 Comparison of RND 1 And P n Judging the number n of electrons actually generated;
(6b) If n=0, no secondary electrons are generated;
(7) If n=1, according to X (1) =f 1,e +f 1,r +f 1,ts Calculating the energy E of the outgoing electrons s1 Wherein f 1,e To generate an energy spectrum of 1 elastically scattered electron, f 1,r To generate an energy spectrum of 1 reflected electron, f 1,ts To generate an energy spectrum of 1 intrinsic secondary electron; the method comprises the steps of carrying out a first treatment on the surface of the
If n.gtoreq.2, according to X (n) =f n,ts Calculating the energy E of the outgoing electrons s1 、E s2 …E sn ；
(8) Judging whether the set cut-off condition is reached, if so, judging whether micro discharge occurs, and if not, returning to the step (3).
Specifically, in the scheme provided by the embodiment of the application, the method includes the following steps: parallel plates with a 1mm gap and surfaces made of Furman-Ag material
(1) Generating n=1000 initial electrons, 1000 initial electron energies E 0 Satisfy distributionσ=3 eV, μ=5 eV; the initial electron angle satisfies the distribution f (θ 0 )-sin2θ 0 The method comprises the steps of carrying out a first treatment on the surface of the Initial electronic azimuth +.>At [0,2 pi ]]Randomly distributed, initial electron phase t 0 At [0, T wave ]Randomly distributed, the emergent electron velocity is
(2) Calculating parallel flat electromagnetic field with x=1mm gap, and setting the frequency as followsf=1.06 GHz, 2.14GHz, 3.71GHz, 5.33GHz, 7.97GHz, the internal electromagnetic field of which is determined by the formula e= [ -Vsin (ωt- βz), 0]
(3) State of electron entering electromagnetic fieldInitial time->
(4) The propulsion of the motion trail of the electrons in the electromagnetic field is solved by a fourth-order Dragon lattice-Kutta method,
setting a time step Δt=t wave Solving the electron trajectory in the dt range according to the above, the first time [ v ] x v y v z ]An initial electronic initial velocity for the solution of step (1);
(5) Determining whether the x-direction trajectory of the electron satisfies x every time the electron trajectory is solved within the Δt time step>1mm or x<0mm, if not, go to step (4), [ v ] x v y v z ]Setting the last solving result in the step (4);
(6) The electron collides with the surface of the device at the moment, and the collision position x p Time of collision t p The number of collisions colour_num=1 (once per collision subsequently, colour_num+1), and the collision velocity is obtained by solving in step (4) [ v ] px v py v pz ]Incident electron energy relative to the surface of the materialAngle of incidence of electrons->According toThe Furman model calculates the number of emitted electrons and secondary electron emission coefficient delta 1 And energy, assuming 2 electrons are emitted, the corresponding energy is [ E ] s1 ,E s2 ]Angle->
(7) Judging color_num<10000, if not, returning to the step (3), and the electron enters an electromagnetic field stateRespectively->And->
The method for judging whether the micro discharge occurs comprises the following steps:
if in the number of cyclescollsion_num is used as a method for judging the micro-discharge threshold, and when color_num is used>10000 time, delta is calculated equl ＝(δ 1 +δ 2 +…+δ collsion_num ) Color_num; wherein delta 1 、δ 2 、…、δ collsion_num Respectively the secondary electron emission coefficient average value of the ith collision; if delta equl >1, micro discharge occurs under the calculation condition; if delta equl <1, no microdischarge occurs under the calculation conditions; if delta equl =1, which is the microdischarge threshold voltage.
Further, secondary electron emission coefficient δ=δ ts +δ e +δ r ，δ ts Is the emission coefficient of intrinsic secondary electrons, delta e Is the emissivity coefficient delta of elastically scattered electrons r Is the emissivity of the reflected electrons.
If the change trend of the electron number evolving along with time is used as a method for judging the micro-discharge threshold value, the electron number is reduced along with time, and micro-discharge does not occur under the calculation condition; if the number of electrons increases with time, micro-discharge occurs under the calculation condition; if the number of electrons is constant over time, the voltage is the microdischarge threshold.
Further, the method for solving the motion trail of the electron in the electromagnetic field comprises the following steps:
in rectangular coordinate system, the electron track is composed ofDetermining;
in the cylindrical coordinate system, the track of electrons is composed ofDetermining;
wherein (x, y, z) is the x, y, z direction position of the electron in the electromagnetic field in the rectangular coordinate system,the radial, angular and axial positions of electrons in an electromagnetic field in a cylindrical coordinate system are represented by adding a point to the position of the electrons, the speed in the direction is represented by adding two points to the position of the electrons, the acceleration in the direction is represented by adding two points to the position of the electrons, the E, B is respectively an electric field and a magnetic field in the direction, q is the unit electric quantity of the electrons, and m is the mass of the electrons.
Optionally, in one possible implementation, after the electron collides with the boundary, updating the state of the electron entering the electromagnetic field is: electronic position r in Updated to the position, energy E, of electron impact in Update to the energy and angle (θ) of the outgoing electrons in step 6 in ,Ф in ) Updating to the angle, phase t, of the emergent electrons in the step 6 in Updated to the time at which the electron hit the boundary.
The calculation results of the traditional MC, single particle-multiple collision (MC 1), multiple particle-single collision (MC 2) and multiple particle-multiple collision (MC 3) based on the Monte Carlo method are from the literature (Linshu, yangjiao, li Yongdong, liu Chunliang, 2014 physical academy 63 147902), the calculation results of CST software are from the literature (Wang Hongan, yonggui, li Jixiao, li Yun, wang Rui, wang Xinbo, cui Mozhao, li Yongdong 2016 physical academy 65 237901), the calculation results of the method are shown in FIG. 2, and the calculation efficiency is shown in Table 1.
Table 1 comparison of micro-discharge threshold calculation efficiency for flat panel transmission lines
The present application provides a computer readable storage medium storing computer instructions that, when run on a computer, cause the computer to perform the method described in fig. 1.
It will be appreciated by those skilled in the art that embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, magnetic disk storage, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
It will be apparent to those skilled in the art that various modifications and variations can be made in the present application without departing from the spirit or scope of the application. Thus, if such modifications and variations of the present application fall within the scope of the claims and the equivalents thereof, the present application is intended to cover such modifications and variations.
What is not described in detail in the present specification is a well known technology to those skilled in the art.
Claims (9)
1. The method for calculating the multi-particle multi-collision micro-discharge threshold Monte Carlo is characterized by comprising the following steps of:
s1, randomly generating N initial electrons according to probability distribution conforming to initial electron energy, angle and phase, wherein the N initial electron states are as follows: position r i Energy E i Angle (theta) i ,Ф i ) Phase t i ；
S2, calculating an electromagnetic field inside the microwave component according to the input frequency f and the input power W;
s3, updating the state of electrons entering an electromagnetic field, wherein the electronic state is as follows: position r in Energy E in Angle (theta) in ,Ф in ) Phase t in The same state for both initial electrons;
s4, calculating an electronic motion trail in a time step delta t;
s5, judging whether the position of the electron in the time step delta t reaches the boundary of the component, if not, repeating S4-S5The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, the state of electron collision at the arrival boundary is recorded: position r p Energy E p Angle (theta) p ,Ф p ) Phase t p ；
S6, adopting electron energy E at collision time p And angle (theta) p ,Ф p ) From the surface material properties of the microwave component, a secondary electron emission coefficient (delta) is calculated ts 、δ e 、δ r ) Spectral distribution (f) of secondary electrons 1,e 、f 1,r 、f n,ts ) Probability P of generating n electrons n And calculate the energy E of the outgoing electrons si Angle (theta) si ,Ф si )；i＝0,1,…,n；
S7, judging whether a set cut-off condition is reached, if so, judging whether micro discharge occurs, and ending; if not, returning to S3;
the method for judging whether micro discharge occurs comprises the following steps:
if the cycle number color_num is used as the method for judging the micro-discharge threshold value, delta is calculated equl ＝(δ 1 +δ 2 +…+δ collsion_num ) Color_num; wherein delta 1 、δ 2 、…、δ collsion_num Respectively the secondary electron emission coefficient average value of the ith collision; if delta equl >1, micro discharge occurs under the calculation condition; if delta equl <1, no microdischarge occurs under the calculation conditions; if delta equl =1, the input voltage is the microdischarge threshold voltage;
if the change trend of the electron number evolving along with time is used as a method for judging the micro-discharge threshold value, the electron number is reduced along with time, and micro-discharge does not occur under the calculation condition; if the number of electrons increases with time, micro-discharge occurs under the calculation condition; if the number of electrons is constant over time, the voltage is the microdischarge threshold.
2. The method for calculating the multi-particle multi-collision micro-discharge threshold value monte carlo according to claim 1, wherein the method comprises the following steps: the probability of generating n electrons is that
Wherein,the probability of not generating intrinsic secondary electrons, the probability of generating 1 intrinsic secondary electron, and the probability of generating n intrinsic secondary electrons are respectively given.
3. The method for calculating the multiple particle multiple collision microdischarge threshold monte carlo according to claim 1, wherein the energy E of the outgoing electrons is calculated s The method of (1) comprises the following steps:
generating random number RAND within 0-1 1 Comparing RAND 1 And probability P of generating n electrons n Judging the number n of electrons actually generated;
if n=0, no secondary electrons are generated;
if n=1, according to X (1) =f 1,e +f 1,r +f 1,ts Calculating the energy E of the outgoing electrons s1 Wherein f 1,e To generate an energy spectrum of 1 elastically scattered electron, f 1,r To generate an energy spectrum of 1 reflected electron, f 1,ts To generate an energy spectrum of 1 intrinsic secondary electron;
if n.gtoreq.2, according to X (n) =f n,ts Calculating the energy E of the emergent electrons under different electron numbers s1 、E s2 …E sn The method comprises the steps of carrying out a first treatment on the surface of the Wherein f n,ts To generate an energy spectrum of n intrinsic secondary electrons.
4. The method of claim 1, wherein the exit angle (θ si ,Ф si ) The method comprises the following steps of:
sinθ si is [ -1,1]Random number, phi of samples between si Is [0,2 pi ]]Random numbers sampled in between.
5. The method for calculating the multi-particle multi-collision micro-discharge threshold value monte carlo according to claim 1, wherein the method comprises the following steps: the cut-off condition is that the maximum circulation times are reached, or the calculation time, or the number of electrons, or other conditions set by people.
6. The method for calculating the multi-particle multi-collision micro-discharge threshold value monte carlo according to claim 1, wherein the method comprises the following steps: the secondary electron emission coefficient delta=delta ts +δ e +δ r ，δ ts Is the emission coefficient of intrinsic secondary electrons, delta e Is the emissivity coefficient delta of elastically scattered electrons r Is the emissivity of the reflected electrons.
7. The method for calculating the multi-particle multi-collision micro-discharge threshold value monte carlo according to claim 1, wherein the method comprises the following steps: the method for solving the motion trail of the electron in the electromagnetic field comprises the following steps:
in rectangular coordinate system, the electron track is composed ofDetermining;
in the cylindrical coordinate system, the track of electrons is composed ofDetermining;
wherein (x, y, z) is the x, y, z direction position of the electron in the electromagnetic field in the rectangular coordinate system,for the radial, angular and axial positions of electrons in an electromagnetic field in a cylindrical coordinate system, adding one point to the sign of the position of the electrons to represent the speed of the original sign representing the direction, adding two points to represent the acceleration of the original sign representing the direction, and E, B is the electric field and the magnetic field with the sign subscripts representing the direction respectivelyQ is the unit electric quantity of electrons, and m is the electron mass.
8. The method for calculating the threshold value of the multi-particle multi-collision microdischarge monte carlo according to claim 1, wherein after the electrons collide with the boundary, updating the state of the electrons entering the electromagnetic field is as follows: electronic position r in Updated to the position, energy E, of electron impact in Update to the energy and angle (θ) of the outgoing electrons in step 6 in ,Ф in ) Updating to the angle, phase t, of the emergent electrons in the step 6 in Updated to the time at which the electron hit the boundary.
9. A computer readable storage medium storing a computer program, which when executed by a processor performs the steps of the method according to any one of claims 1 to 8.
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