Chapter 588: Nonlinear Neutron Transport Equation Published in Advance
"????"
Inside the secret room.
Hearing these words from Hua Yun's mouth.
A confused expression suddenly appeared on Lu Guangda’s fair and round face, known as ‘Doctor Doll’:
o.O?
What?
Are the neutron transport equations nonlinear?
How is this possible?
To know.
The essence of the phenomenon of the neutron transport equation is the derivation of slowing down + diffusion.
The slowing down process can be described in terms of energy reduction.
The process of diffusion introduces flow density - both concepts have been mentioned before.
The diffusion process is the free diffusion of large-scale thermal neutrons in the reactor, participating in the fission reaction and maintaining the operation of the nuclear reactor.
This is the core and most critical in nuclear fission, and it is also a relatively complex research object.
But at the end of the day.
The so-called diffusion process is still an evolution of neutron distribution as the nuclear reaction proceeds.
at the same time.
The concept of neutron flux density has been defined above, that is, flow density.
The change in neutron density is obviously divided into three parts:
First, a source comes to produce neutrons.
Second, neutrons are absorbed and consumed for fission.
Eventually, the neutron leaked out of the system.
Here the source can be recorded as S(r,t), and the leakage is represented by a divergence J(r,t), where J(r,t) is the flow density of neutrons leaving the system.
The nuclear reaction rate is as above R=Σa.
If n is used to represent the neutron density, a continuity equation appears:
n(r,t)t=S(r,t)Σa(r,t)J(r,t)
At the same time, the flow of neutrons into and out of the system is driven by distribution, which is determined by the gradient.
J(r,t)=D(r,t).
Where D=λs/3 is the coefficient, called the diffusion coefficient.
It’s not hard to see from here.
The neutron transport equation is obviously a linear partial differential equation. Wait!
Think of this.
Lu Guangda suddenly realized something and suddenly looked at Hua Yun, the leader of the second group:
"Lao Hua, you mean that the neutron transport equation actually has a situation similar to the nonlinear Schrödinger equation?"
Hua Yun nodded vigorously:
"That's right."
Speaking of Schrödinger's name, everyone must be familiar with it - the descendant of Xue Rengui in the marketing slogan is a well-known cat abuser.
Among the many deeds of this big man, the Schrödinger equation is obviously a focus.
It is a basic equation in quantum mechanics proposed by Schrödinger himself and is also a basic assumption of quantum mechanics.
In the future life where Xu Yun traveled through time.
Many people regard it as the most important equation in modern physics, if not one.
At the same time.
It is also a very complex linear partial differential equation.
For any atom - as long as the force field experienced by the electron can be represented by a centered force field, its Schrödinger equation can separate the variables.
So in almost every situation.
The Schrödinger equation is a standard linear equation.
But there is one very special situation.
That is, when the potential field depends on the wave function, the derived Schrödinger equation is nonlinear.
This situation generally occurs in plasma or optics in application fields, and is considered an extremely rare situation.
But now according to what Hua Yun said.
This seems to be the case if the neutron transport equation changes in a specific region.
Is it really possible?
Think of this.
Lu Guangda picked up the documents Hua Yun brought over and read them carefully.
The top document is a report on the Petrov Capillary Reactor. This is one of the only dozen pieces of non-cold explosion nuclear reactor center data that the Rabbits have.
However, this report was not sent back to China by the rabbits through special channels, but a commendation given by the bear:
three years ago.
When Wang Ganchang was a researcher at the Joint Institute for Nuclear Research in Dubna, he found an example of the generation of anti-sigma negative hyperons from 40,000 pairs of negatives. This was also the first time in human history that the anti-sigma hyperon was discovered. particle.
Negative hyperons were one of the key areas that Mao Xiong and Hai Duan were competing for at the time. Wang Ganchang's discovery gave Mao Xiong a very useful chess piece in the field of theoretical physics.
Therefore, Mao Xiong presented this picture to Mr. Wang Ganchang as a reward.
Of course.
According to some situations decrypted by later generations, this reward should be a choice made by the rabbits on their own initiative after internal discussions.
In addition, Mao Xiong also gave Mr. Wang Ganchang an invitation at that time:
As long as he changes his nationality, he can stay in Moscow forever.
However, Mr. Wang finally refused the invitation and returned to his motherland without hesitation. (This is something I only found out when I checked the information for this report, so I didn’t mention it when I introduced Mr. Wang. In those days, you could see the shadow of these seniors in everything)
alright.
His eyes returned to reality.
However, the data carrier on this document is not the black and white image that many people think, but a special tool from the early days of the scientific community:
paper tape.
Reading tapes was a magical skill in the 1960s and 1970s, and there were a large number of top masters at home and abroad. Unfortunately, it is now almost lost.
In the process of looking at the paper tape, scientists will use numerical simulation images in their minds to analyze the calculation data recorded on the paper tape.
For example, the Manhattan Project back then.
Seaberg and Lawrence were experts at reading tapes and played a key role in the development of the atomic bomb across the sea.
Then Lu Guangda carefully picked up the paper tape and read it carefully:
"Collision record number 45242, fission secondary neutrons are isotropically approximated."
"The energy range in which high-order neutrons dominate is from 0.12 to 0.16, and the energy level of a single-energy strong neutron source is 14MeV"
"V1 is 2738 centimeters per microsecond, and the upper level energy zone is 42MeV"
The results were watched.
Lu Guangda's pupils suddenly shrank:
"Huh? This is."
I only see this moment.
A set of numbers are clearly recorded on a piece of paper:
8.27^14g/cm.
The parameters corresponding to this set of numbers are clearly written
Neutron density inside the device!
Then Lu Guangda stared at this set of numbers and remained silent.
As everyone knows.
The reason why the neutron transport equation can be regarded as a linear equation is essentially that the neutron density in the system is usually much smaller than the atomic nucleus density - here the small refers to the difference in magnitude, which is the so-called [much smaller than ] degree.
For example, the earth and watermelon, and people and ants.
This is one of the basic physical assumptions made when deriving the neutron transport equation, and is the basis for all subsequent inferences.
Under this assumption.
We can only consider the collision between neutrons and medium nuclei, and ignore the collision between neutrons, and finally obtain a linear neutron transport equation.
But if the neutron density is so high that it is close to the nuclear density or the two are equivalent.
This assumption naturally fails.
And generally.
The magnitude of the density of an atomic nucleus is usually.
10.14^14g/cm!
Although this number is not completely consistent with the neutron density on the paper tape, there is no difference in magnitude between the two:
For example, there are two people, A and B. A has assets of 1 million and B has assets of 800,000.
You can say that A is richer than B, but the gap between the two is not big. Maybe B will catch up with A in a few months.
In other words.
In this situation.
The neutron transport equation can no longer be regarded as a linear equation.
Then Lu Guangda looked at several other sets of data.
It was eventually discovered that the neutron density would indeed increase dramatically in some special circumstances, approaching or even reaching the level of the density of the atomic nucleus.
These data include a large number of institutions in three countries, China and the United States, and there is no possibility of occasional errors.
That is to say
The theory devised by Norris Bradbury was indeed wrong.
See this situation.
Lu Guangda's mood suddenly became a little dazed.
He was not sentimental about the project team spending a lot of time on the wrong path, but rather surprised.
The plan designed across the sea can actually go wrong?
But soon.
Another question popped up in Lu Guangda's mind:
The authority on the other side of the sea is also a human being. If all technologies are not implemented yet, why can't they make mistakes?
Don’t say anything else.
If they were so invincible that everything was correct, would they have been beaten so badly by us on the peninsula?
There are also Wang Ganchang and Zhao Zhongyao in the base, as well as Lao Yang and Lu Guangda who are still on the other side of the sea. Haven't they corrected a lot of wrong theories on the other side of the sea in the past? ——It’s just that the altitude is not as amazing as nuclear weapons.
Think of this.
Lu Guangda couldn't help but take a deep breath, his eyes were no longer misty, but were gradually replaced by a look of determination.
Then he pondered for a moment, looked up at Hua Yun, and said:
"Lao Hua, thank you for your hard work this time. Obviously, your verification is correct."
"Here I would like to apologize to you and Comrade Ruiping. Because we did not discover the problems in the model, the comrades in the second and third groups were criticized and pressured for no reason."
"As the leader of the project team, this is my dereliction of duty. I will take the initiative to review this matter at the next summary meeting."
Say it.
Lu Guangda turned to Xu Yun who was aside, and the expression on his face softened a lot:
"Comrade Han Li, I also want to express my gratitude to you - not only thank you for finding out the problem, but more importantly, for making me understand a truth."
"That is, although the atomic bomb technology has been realized across the sea, it is still far from being fully understood. There are still many errors and omissions that even an authority like Norris Bradbury cannot discover."
"This is undoubtedly good news. It means that although we are temporarily lagging behind, we have not yet been pulled back to a point where we are beyond our reach!"
"Maybe one day we can surpass them."
While Lu Guangda was speaking, he waved his right hand in the air a few times, which looked extremely powerful.
Hear this.
Xu Yun quickly waved his hand and shook his head quickly:
"Director Lu, you are serious. I just did some trivial work. I really don't deserve the credit."
"If everyone is as polite as you and always elevates trivial matters to the level of the state, then I, as a consultant, will not dare to speak out easily in the future."
Xu Yun's words were partly polite, but mostly his true thoughts.
After all
The concept of nonlinear neutron transport equation is originally the result of the 596 project team.
Based on information decrypted by later generations.
throughout the development process of nuclear weapons.
The rabbits discovered a total of 11 mistakes across the sea and the furry bear. Among them, there were two most important mistakes that could shake the foundation of nuclear engineering.
The first is the error discovered by Mr. Zhou Guangzhao regarding the energy distribution and angular distribution of secondary neutrons.
Mentioned earlier.
At that time, in the early days of atomic bomb research, Mao Xiong experts provided some technical data related to the atomic bomb.
But later, when researchers used "nine calculations," a simulation method for solving equations, they found that the indicators of secondary neutron energy distribution and angle distribution were inconsistent with those provided by Mao Xiong.
In the end, Mr. Zhou Guangzhao started from the energy utilization rate and used the "maximum work principle" to prove the correctness of the "nine calculations" results and the impossibility of the Mao Xiong data.
It was later revealed based on the documents declassified by Mao Xiong.
This mistake was really not intentional, but a major mistake made by Sakharov, the first person to have nuclear weapons across the sea.
Except for Mr. Zhou Guangzhao.
The second major error corrected by the rabbits was the nonlinear neutron transport equation.
This error was corrected relatively late, and the discoverer was Academician Du Xiangwan, who is still alive today.
Academician Du Xiangwan has so far won one special prize for national scientific and technological progress, one first prize, two second prizes, and more than ten provincial first and second prizes. He is also a typical wall-hanger.
At that time, the rabbits had already begun to study the nuclear fusion of hydrogen bombs.
As a result, Academician Du Xiangwan's team found that in actual engineering, in some places where the fusion reaction is very violent, the neutron density may be greater than the nuclear density.
This situation was later extended to the field of nuclear fission, that is, the atomic bomb, which brought a comprehensive innovation to the field of neutron transportation of nuclear weapons.
That's right.
This was during the development of the hydrogen bomb - by which time the Rabbits' first atomic bomb had exploded.
The rabbits on that atomic bomb used another approximate perturbation method, which did not involve the nonlinear neutron transport equation.
How to say it
From the perspective of future generations.
It's a little better than the original linear neutron transport equation, but there are limits to how good it can be.
If the linear neutron transport equation is a small electric donkey that can travel 10 kilometers, then the approximate perturbation method used by the atomic bomb can only travel 15 kilometers.
As for the conditions for adaptation of the nonlinear neutron transport equation, it is.
One hundred thousand kilometers! ——This is because there are no higher-level nuclear weapons today.
Wait until the 1980s.
In order to have a place in IUPAP, the highest organization in the international physics community, the International Union of Pure and Applied Physics.
The rabbits reluctantly published this technology in "Computational Physics", doi is 10.19596/j.cnki.1001-246x.1984.02.010.
This technology bought the rabbits the seat of vice president of IUPAP, which was held by Mr. Zhou Guangzhao.
By the way.
This seat is not a face-saving project, but a very important node in the modern history of rabbit physics.
Give an example.
Later, more than 90% of the first batch of non-Batumi treaty-imported instruments from domestic universities followed the IUPAP route.
As for that paper, it was still cited even in 2018. It can be said to be an article with extremely far-reaching influence in the domestic physics community.
It is said. It is just said.
It is said that the current hydrogen bomb technology across the sea later adopted this idea - after all, before controllable nuclear fusion, nuclear fusion thermonuclear weapons will definitely not be able to escape the neutron transport equation.
That's why.
Xu Yun still only played the role of a porter this time. It must be a bit hard work, after all, he has been exposed.
But if you want to say how much credit he has contributed, then he really cannot bear it.
True.
As a time traveler, it is impossible not to be a porter or copycat, and no one can avoid this.
But if you still feel complacent and accept it calmly after the move, calling yourself "showing off", then this is another matter. (Yesterday, I saw a comment on a pirated book review website, saying that the protagonist is too cowardly. Even if the other party is Yu Min or Qian Wushi, the protagonist should still pretend to be cool. It’s really weird)
all in all.
At this point, the remaining questions are simple.
Lu Guangda looked around and then said:
"Okay, comrades, now that we have found the problem, we should solve it next."
Say it.
Lu Guangda walked to the small blackboard nearby, picked up the chalk and wrote:
"There are many methods for solving nonlinear equations, but the most commonly used one is the perturbation method, which is to convert the nonlinear equations into a linear system of equations."
"As for the neutron transport equation, I think the nonlinear neutron transport equation can be solved by turning it into a coupled linear system of equations."
"That is, if you add up the ones with no neutron collision, one collision, two collisions... you can get all the neutrons."
Hear this.
Everyone at the scene nodded.
Perturbation method.
This is indeed a fundamental approach to nonlinear equations.
Xu Yun also mentioned this method when he and Qian Bingqiong mentioned that the world is nonlinear.
introduced earlier.
There are two types of reactions between neutrons and nuclei:
Scattering and absorption.
Scattering is a generalized scattering, that is, neutrons enter and exit the nucleus unchanged, which is referred to as ingress.
This can be divided into two situations:
① Neutrons do not enter the nucleus.
That is to say, the neutron scatters directly with the nucleus. In layman's terms, it bounces away.
This is obviously an elastic scattering, and both energy and momentum are conserved. This kind of scattering is also called potential scattering.
②The neutron is absorbed by the nucleus, but is released again.
This situation is slightly more complicated.
When the energy of the neutron is exactly the energy required for the nucleus to reach an excited state, the neutron is extremely easy to be absorbed:
Knowledge of energy level transitions from quantum mechanics explains why this is, a process called resonant absorption.
The composite nuclei formed then emit neutrons, and are classified as elastic/inelastic scattering according to whether energy is emitted.
The expressions in the two cases are as follows:
Inelastic ZAX+01n[ZA+1X]ZAX+01n
Elastic ZAX+01n[ZA+1X]ZAX+01n+γ.
That’s right!
Smart classmates must have seen it at a glance.
Resonance absorption requires neutron energy, so it has threshold energy characteristics.
In this way, the reaction of neutrons entering and neutrons coming out is the (n, n) reaction.
As for absorption, it is better to understand.
To put it bluntly, neutrons enter but do not exit the nucleus.
There are three types of reactions to this behavior:
1. Radiation capture (n, γ).
The neutrons are absorbed by the nucleus, which releases excess energy in the form of gamma rays and regains a relatively stable state.
Second, nuclear reactions, that is, (n, p), (n, α).
Neutrons are absorbed by the nucleus, and the nucleus releases excess energy in the form of non-neutron particles such as protons and alpha particles to reach a relatively stable state. In particle physics and nuclear physics, due to the quantum tunneling effect, the helium nucleus 24He can be considered is a whole, the so-called alpha particle.
Three is.
Nuclear fission.
That’s right, nuclear fission.
That is, neutrons are absorbed by the nucleus, and the nucleus releases energy by fissioning into multiple daughter nuclei, allowing the daughter nuclei to reach a relatively stable state.
Although such reactions often release neutrons, due to changes in the nucleus, they are still classified as neutron absorption reactions rather than scattering.
But on the other hand.
Not all 235U absorbs neutrons and fission occurs. For example, 92/235U+0/1n[92/236U]92/236U+γ is a radiation capture reaction.
After figuring this out,
All that's left is to have hands.
Extract (n, n), (n, p) and nuclear fission, and then define a concept:
Neutron intensity I.
It represents the number of neutrons passing perpendicularly through unit area per unit time.
This way.
The number of neutrons will change during this process:
It may be bounced back by scattering and unable to pass through the target.
It may also be directly absorbed by the target nucleus.
Then this change is expressed as ΔI=σINΔx, where N is the target core density and Δx is the target core thickness.
It can be seen that σ is a probability, which refers to the average probability of neutrons being scattered or absorbed by the target nucleus.
At this point.
Just add the calculated approximate probabilities together to find the derivation.
Well, look.
About one ten thousandth of the theoretical design of the atomic bomb can be solved so easily. Isn't it very simple?
Ahem
At least it is very simple for Lu Guangda and others.
So it's quick.
The entire project team started calculations in full swing.
"Who can calculate the rationality of the same collision terms at both ends?"
"Director Hua, the speed of neutrons after scattering should not produce ultra-high energy neutrons."
"u(x,t)=z(0)=z(t)=u(xbt,0)=g(xbt)"
"Report, preliminary solution is out!"
"Mmp, is there anyone to go to the toilet with? I'm not allowed to go out alone!!"
that's all.
When the time comes close to 12 o'clock at night.
Lu Guangda wrote a final formula:
∫zJ=uhsΣSφD(r,t)+λs/3=limr→04πDA(rL+1)er/L=SA=S4πD. (I’m not at work late at night for image review, so I’ll just make do with it.)
After finishing writing.
Lu Guangda wiped the sweat from his forehead and breathed a sigh of relief:
"The nonlinear neutron transport equation has finally been calculated." Note:
Believe me, I have written this process as concisely as possible, and there will be no derivation process later
(End of chapter)