How to calculate an atomic bomb's critical mass

Step-by-step guide on how to derive and solve the neutron diffusion equation to determine the critical mass of an atomic bomb, following the steps of Rudolf Peierls and Otto Fritsch. It involves solving a partial differential equation but every step is presented. In the end a general formula for a nuclear critical mass is found.

Relevant links:

Errata:
• at 10:10 the Laplacian of f should say: ∇²f= (1/r²)*d/dr(r² df/dr) (not 1/r² in the derivative bracket), this is a typo and not an error in the calculation; thanks to @fatassmonkey37 for pointing this out
• at 15:58 the subscripts are reversed, it should say: λ_f = 16.89 cm, λ_t = 3.596 cm; thanks to @lucacastronuovo9789 for pointing this out

Credits:

Chapters:
00:00 Neutron diffusion equation
05:41 Solution
15:24 Critical mass
20:13 Advanced solution

[jakubstawarczyk]

Amazing, as a theoretical physicist myself, I'm looking forward to see more mathematical videos about this topic 😁

Edit: makes me wanna sit down with numerical methods and analyze more reallistic example of shrinking and expanding core (true R(t))

[hypercomms]

Absolutely fascinating. I have always been interested how this was calculated ever since I did a school assignment on the Manhattan project when I was 14 in 1974.

[verdienthusiast]

Man, I am a high schooler and I am trying my best to learn Physics, a subject that I love with all of my heart, I am self studing calculus to, one day, contribute a little to this wonderful field, the things that I find more interesting are the possibilities that the Island of stability can give us, and I want to dedicate my life to that.
Take my money (earned by surveys so no problem for me) and buy a cup of something

[MrFib]

This is the first time that I am excited for a homework assignment and I can't believe it came from YouTube, I really want to try to do this! Brilliant video and thanks for educating us on this fascinating topic!

[kirkhamandy]

17:15 _This will very quickly start blowing up_ LOL, I've often heard of exponentials "blowing up" but here, never a more true definition 😂

[footbballa]

This channel deserves more views and subs, one of the best Physics YT channels out there keep it up!

[KiwiExpressCream]

Not going to lie, the maths is a bit beyond me, but I love how you worked out the critical masses. All of your videos have been amazing and I always look forward to the next one.

[stevebeal]

I graduated in physics in 1971 in Bristol, UK. When we were shown the neutron diffusion equation, our lecturer said "you all have all of the mathematical tools that you need to solve this equation, but I'm not allowed to show you how to do it, for reasons of national security." How times have changed!

[Hexnano]

Thank you for this video, I got stuck while working on building my atomic bomb and this was very helpful.

[pawelbjanowski]

A coffee/beer from a nuclear colleague from Poland, with kind regards!⚛

[tomashernandez]

Man, this theory was taught to me by a professor the last semester, amazing the application and the smooth way this works to explain such a beauty phenomenon

[CarlosMartinez-epmi]

A little mistake, min 10:17. On radial term you have inside the brackets (1/r^2 df/dt) but it is ( r^2 df/dt ). After you use the function f=u/r all come back to be good.

[omargaber]

Actually, I am amazed by your work. I don’t know what to say. I thought a lot about the effort you made to make this video. It must have been hard. I don’t think anyone can thank you as you deserve. In fact, I am traveling to Russia, where there are no means of support or Communication with the outside world, even YouTube ads are prohibited, so I turned on a VPN so that ads would appear on your videos and I could watch them. Indeed, I watched 14 ads and visited 11 advertiser’s websites. This took some time, but it is nothing compared to the time that you spent making this video, and perhaps this is worth a few cents to the channel, but I feel that it is more moral support than material support. Thank you from the bottom of my heart, you brilliant mind.❤

[danieleambrosini]

Awesome! Make a lot of videos on physics because you are really talented! :) Love from Italy <3

[Isaac-mthx]

This is an amazing video! Made me even more excited to learn DPEs next semester!

[fablearchitect]

Can you please do a video on deriving the transfer function for a nuclear reactor in terms of control rods vs output thermal power?

[Amsterdampardoc]

Thank you I was trying to find this for a long long time

[latefoolstalk]

Wow, so well explained. Such advanced physics explained step by step. I hope more people checks your wonderful channel

[Asterism_Desmos]

You know it’s a good day when you get a notification for Dr. Jorge S. Diaz 🔥

[spencerwenzel]

I really appreciate that in every formula you describe what each variable represents and how it relates to the physical interpretation. Even if I don't completely understand the math, the implications are not completely lost and I am still learning. Thanks for making these.