Wikipedia:Reference desk/Archives/Science/2017 December 30

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December 30

Feynman Lectures. Exercises. Exercise 16-5 JPG

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The Berkeley "bevatron" was designed to accelerate protons to sufficiently high energy to produce proton-anti-proton pairs by the reactio

p + p --> p + p + (p + p)

The so-called threshold energy of this reaction corresponds to the situation when the four particles on the right move along together as a single particle of rest mass M = 4 m_p. If the target proton is at rest before the collision, what kinetic energy must be bombarding proton have at threshold?

—  R. B. Leighton , Feynman Lectures on Physics. Exercises

${\begin{cases}m_{\text{pv1}}+m_{\text{p0}}=4m_{\text{pv2}}\\m_{\text{pv1}}v_{1}=4m_{\text{pv2}}v_{2}\\m_{\text{pv1}}={\tfrac {m_{\text{p0}}}{\sqrt {1-v_{1}^{2}/c^{2}}}}\\m_{\text{pv2}}={\tfrac {m_{\text{p0}}}{\sqrt {1-v_{2}^{2}/c^{2}}}}\end{cases}}$

${\begin{cases}v_{1}={\tfrac {4{\sqrt {3}}}{7}}c\\v_{2}={\tfrac {\sqrt {3}}{2}}c\\m_{\text{pv1}}=7m_{\text{p0}}\\m_{\text{pv2}}=2m_{\text{p0}}\end{cases}}$

So I got $0.5m_{\text{pv1}}v_{1}^{2}={\tfrac {24}{7}}m_{\text{p0}}c^{2}$

But Feynman answer is $6m_{\text{p0}}c^{2}$.

Why?

Username160611000000 (talk) 12:55, 30 December 2017 (UTC)

It seems Feynman has used a formula $T=(m_{\text{pv1}}-m_{\text{p0}})c^{2}$. Username160611000000 (talk) 15:45, 30 December 2017 (UTC)

Suppose that the full 4D momentum of the 4 resulting particles is $u$. Since at the threshold it behaves like a single particle with mass $4m_{p}$ we have $u^{2}=16m_{p}^{2}c^{2}$. On the other hand the 4D momentum of initial proton at rest is $u_{1}=(m_{p}c,0)$ and that of the moving initial proton $u_{2}=(m_{p}c+K/c,\mathbf {p} )$ where $K$ is its kinetic energy and $\mathbf {p}$ its 3D momentum. So, we have $(u_{1}+u_{2})^{2}=2m_{p}^{2}c^{2}+2m_{p}c(m_{p}c+K/c)=u^{2}=16m_{p}^{2}c^{2}$. Quite obvious that $K=6m_{p}c^{2}$. Ruslik_Zero 20:57, 30 December 2017 (UTC)

@Ruslik0: That's a great derivation, except ... aren't you saying that the four resulting particles are at rest? If they're at rest, they should be in the frame where the two colliding protons have equal and opposite velocities. Wnt (talk) 17:05, 31 December 2017 (UTC)

Yes, it is true that they are at rest in the reference frame of their center of mass (at the threshold). In the laboratory reference frame they have exactly the same 4d momentums. You can easily understand why: in the center of mass frame two initial protons have the same energy and momentum but move in opposite directions towards each other. So, their total 3d momentum is zero. If we are at threshold of the reaction, all kinetic energy of both moving protons is spent to create a proton-antiproton pair. Therefore in the final state all four particles are at rest in the center of mass frame. When we transform the reference frame back to the laboratory frame, 4d momentums of the final four particles remain equal to each other. So, if $u'$ is the 4d momentum of one of those four particles, then their total momentum is $u=4u'$ and $u^{2}=16u'^{2}=16m_{p}^{2}c^{2}$. Ruslik_Zero 17:34, 31 December 2017 (UTC)

Momentum and energy is conserved, not momentum and inertial mass. So your first equation is the one that is wrong. Dragons flight (talk) 22:30, 30 December 2017 (UTC)

I feel like an idiot when I try to do this, but hopefully it is, by definition, a method any idiot can follow. Looking at the system in the final frame of the four proton masses, we need two proton masses of mass + two proton masses of energy to begin with, so each proton has a relativistic mass twice that of its rest mass. Actually that just lets us directly set gamma = 2 = 1/sqrt(1-v^2/c^2). So v = sqrt(3)/2 c. Now using the velocity-addition formula I get that if one of the initial moving protons is set at v=0, the other (with twice the speed) is now at sqrt(3) c / (1 + 3/4) = 4 sqrt(3)/7 c. Taking that gamma formula, that gets 1/sqrt(49) i.e. gamma=7. One mc^2 of that is actual rest mass and 6mc^2 is kinetic energy. Double checking with the energy-momentum relation and saying that E^2 = (m c^2)^2 + (p c)^2, we first use p = gamma m v with the two sets of gamma and v values to get 2 * m * sqrt(3)/2 c = sqrt(3) m c and 7 * m * 4 sqrt(3)/7 c = 4 sqrt(3) m c. Then we put that into the formula and we have E = 2 m c^2 and E = 7 m c^2 respectively. So we do indeed have 4 m c^2 on two protons with equal speed in the final frame and 7 m c^2 on one proton moving + 1 m c^2 on one proton at rest in the lab frame. In the lab frame we know that all the protons have 2 m c^2 total energy after the collision x 4 so this energy is properly conserved. And the 4 sqrt(3) m c value for p on one proton in the lab frame allows all the protons coming out to end up moving with momentum sqrt(3) m c.

I will admit, however, that I am having a very hard time following anyone else's calculation above, and I fear the same may be true here. :( Wnt (talk) 02:42, 1 January 2018 (UTC)

Your derivation is correct but it can be done much simpler as I showed above. Ruslik_Zero 17:13, 1 January 2018 (UTC)

If you can lead me to some background on how to use 4D momentum as you are, I'd welcome it. Our offering on four-momentum seems more mysterious than useful to me at present. I was fiddling about on my own and derived a way to add gamma factors directly: "g1+g2" (by which I really mean, the gamma factor of something with gamma factor g1 viewed from a frame at a velocity in the same direction that would impose gamma factor g2) works out to be g1*g2 + sqrt(g1^2*g2^2-g1^2-g2^2+1). There is probably some way to make that more concise and to make it into a vector result to boot, but it is already an improvement for this problem: knowing that each proton has to have gamma factor = 2 to carry the extra relativistic mass to make a new proton, I can then shift the frame and calculate directly the gamma on the moving proton is 2*2 + sqrt(4*4-4-4+1) = 7, meaning it carries 6 proton masses of kinetic energy plus its rest mass. Wnt (talk) 18:35, 1 January 2018 (UTC)

P.S. when g=g1=g2 this reduces to 2*g^2 - 1, so repeatedly "doubling velocity" should get gamma factors of 2, 7, 97, 18817, 708158977... all apparently numerators of diophantine approximations to the square root of 3,[1] if that means anything. ...Actually, if g(n) is 1,2,7,26,97,362,1351,5042,18817,70226,262087,978122 for n=0,1,2,3,4... then viewing g(n1) from a frame with g(n2) seems to yield an integral gamma value g(n1+n2). Wnt (talk) 20:32, 1 January 2018 (UTC)

PP.S. Actually those diophantine approximations seem relevant -- the velocities for those gamma values are the inverse of the diophantine approximations * sqrt(3) * c: 0, sqrt(3)/2, 4*sqrt(3)/7, 15*sqrt(3)/26, 56*sqrt(3)/97 etc. Each velocity is reached from the preceding by shifting the frame by a velocity of sqrt(3)/2 c. (the momenta are thus 0,1,4,15,56... * m * sqrt(3) * c), appearing here as the numerator, with gamma as the denominator of each speed) Wnt (talk) 03:21, 2 January 2018 (UTC)

Writ, I agree that all these maths can be really hard to follow... and the required gamma factor is 2 as you determined above from considering the center-of-mass frame, a factor that fixes the final kinetic energy of the proton accelerated from rest, so in the lab frame it is $m_{p}c^{2}$ (since their kinetic energies are given by $(m-m_{0})c^{2}$ or $(gamma-1)m_{0}c^{2}$). The total energy of the four protons (which I've assumed behave as a single particle at the threshold as pointed out by Ruslik0) is thus 4*(rest energy + kinetic energy) or $8m_{p}c^{2}$. Given that the energy of the rest masses of the initial two protons are $m_{p}c^{2}$ each, by energy conservation, the additional kinetic energy of the initial bombarding proton is $6m_{p}c^{2}$. This is a simpler derivation, if I'm not mistaken and implied by your double check above, that doesn't mess around with the velocity-addition formula or the four-momentum with this particular problem. -Modocc (talk) 03:34, 2 January 2018 (UTC)

Simply put, bring the 4-momentum you want to eliminate to one side of the equation and square it, as the square of the 4-momentum yields the known mass and thereby indeed eliminates the variables (energy and momentum) you want to get rid of. Count Iblis (talk) 16:08, 2 January 2018 (UTC)

I should mention that I took a sidetrack of this to the Math Desk and eventually learned that a key concept here is rapidity, which is $ln{\sqrt {\frac {1+x}{1-x}}}$, with x = c/v, or $1/2ln{\frac {c+v}{c-v}}$ if you prefer. Unlike velocity, rapidity is additive between frames, so it seems worth learning more about to solve this sort of problem. Wnt (talk) 12:49, 3 January 2018 (UTC)