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Relativity at the LHC

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  1. Oct 21, 2013 #1

    bobie

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    1. The problem statement, all variables and given/known data
    This is not really homework, I am trying to understand relativity and I suppose my question is too simple to post in other forums.
    I took the following data from the Cern site when they reached 7TeV:
    2. Relevant data
    Ek = 7 Tev (1.6926*1027 Hz)
    B = 8.33 T
    RLch = 4243 00 cm
    Ep(m) = 0.938 GeV = 2,2685 * 1023 Hz (= 1,67 *10-24 g)

    β= 0.999 999 991 c : I'm not sure how many 9 should be there. Can you show me how to find it?

    Relativistic mass: can we find the mass increase only if we know v/c?
    [tex]M+ = \frac{1} {\sqrt (1-β^2)}[/tex]

    Has this curve got a name? is it similar to sec α?
    Could someone plot the two curves together in a graph, please?

    Can I find the exact value of the mass increase just dividing Ek by Ep:
    M+ = 7000/0.938 = 7461 rest masses ?
     
    Last edited: Oct 21, 2013
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  3. Oct 21, 2013 #2

    vanhees71

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    In modern high-energy physics the mass is always taken as the rest mass of the particle. In our case we deal with protons which have a restmass of [itex]m=938 \; \mathrm{MeV}/c^2[/itex]. The energy is related to momentum via
    [tex]E=c \sqrt{m^2 c^2+p^2}.[/tex]
    Note that in the runs so far the protons in each beam had an energy of [itex]E=3.5 \; \mathrm{TeV}[/itex]. The total center-of-mass energy was [itex]\sqrt{s}=7 \; \mathrm{TeV}.[/itex] This will be doubled in the future after the just running upgrade to [itex]\sqrt{s}=14 \; \mathrm{TeV}[/itex], i.e., to an energy of [itex]E=3.5 \; \mathrm{TeV}[/itex] for the protons in each beam.

    From the given energy you can easily calculate [itex]p[/itex] and the velocity then is given by
    [tex]v=c^2 p/E.[/tex]
    I guess now you can figure everything out yourself :-).
     
    Last edited: Oct 21, 2013
  4. Oct 21, 2013 #3

    bobie

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    Thanks, vanhees,

    - why is Ep divided by c2?
    - my value for Ek is wrong? are the other data correct? R should be correct
    - isn't there a Lorenz formula for v, like the one for M+?
    - I suppose I need a lot of precision to get 0.999 999 991; before I try to solve it by myself, can you tell me at least if that value is correct?
    - do you have the possibility of showing me that curve and how it is related to sec α?

    - is M+ = 7461 correct?, I get same result with (1-√1-0.9999999912)-1: 0.000134-1 =7461

    - if I use the formula mv/B =r => m = R*B/ c, what is B ? 82,200 G?
    M+ = 424,300* 82,200/2.99*1010 = 1.162582 it is how many masses?

    Thanks
     
    Last edited: Oct 21, 2013
  5. Oct 21, 2013 #4

    bobie

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    Is that a typo , should it be 7 ?
     
  6. Oct 22, 2013 #5

    bobie

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    I checked with the data but each proton seems to have 7 TeV:
    [tex]E = \frac{m_0c^2}{\sqrt{1-(v/c)^2}}[/tex]
    [tex]\frac{0.938 GeV}{\sqrt{1-(0.999 999 991)^2}}[/tex]
    = 0.9383 GeV / 0.000134 = 7000 GeV

    Where do I go wrong?
     
  7. Oct 22, 2013 #6

    vanhees71

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    Ok, then obviously your velocity refers to the design energy of [itex]\sqrt{s}=14 \mathrm{TeV}[/itex], i.e., for [itex]E=7 \mathrm{TeV}[/itex] for the protons in one of the beams.
     
  8. Oct 22, 2013 #7

    BruceW

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    yeah, it's the right answer for 7TeV per beam. But last I heard, the LHC was running two 4TeV beams (not 7TeV).
     
  9. Oct 23, 2013 #8

    bobie

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    I took these data before the reached 8TeV, and probably the experiment regarded only one beam.

    Can you help me now check the radius?, I think the formula is
    R = mv/ qB
    (1.67*10-27 Kg * 7461) * 3*108 / 1.6*10-19 * 8.22
    I get 2840 m instead of 4243, where do I go wrong?
     
    Last edited: Oct 23, 2013
  10. Oct 24, 2013 #9

    BruceW

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    that is the non-relativistic equation. There is only a small change to turn it into the relativistic version. instead of using the rest mass, you need to use the total (relativistic) energy of the particle. (which will approximately be equal to its relativistic momentum at those speeds). you can derive this relativistic equation if you think about the four-force and relate the spatial part of the four-force to the Lorentz force. Remember that the Lorentz force is defined as:
    [tex]\frac{d\vec{p}}{dt}[/tex]
    while the four-force is usually defined as:
    [tex]\frac{dP^\mu}{d\tau}[/tex]
    so there is going to be a factor of gamma there (which I missed the first time that I tried to do this derivation). So be careful not to forget about this.
     
  11. Oct 24, 2013 #10

    bobie

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    Thanks BruceW, I am just a student: I do not know what factor you are referring to, I thought that the (Lorenz) factor gamma is the formula I mentioned in post #1 as M+ (E/m0c2), and E = 7TeV. I am lost!

    Is it anyway a factor of 1.5? that is the difference I find.
    - Can you tell me at least if R = 4243 m is the correct radius?
    -can you tell me if the curve of M+ (when β varies from 0 to 1) is a secant h/c2, if we consider h= c and c1 = β?

    Thanks for your time.
     
  12. Oct 24, 2013 #11

    BruceW

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    yeah, you say your beam has energy 7TeV, so the total relativistic energy is 7TeV. So your equation for the radius should be: R=Ev/qBc^2 Where E is 7TeV and the c^2 is required to keep to SI units.

    I am not sure what you mean when you talk about this M+ ... is it literally just ##1/\sqrt{1-\beta^2}## ? This is equal to the gamma factor I was talking about. If this is what you meant, then M+ is the relativistic mass, divided by the rest mass. Also, the relativistic mass is equal to the total relativistic energy, divided by c^2.
     
  13. Oct 24, 2013 #12

    vanhees71

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    NO! The mass is the invariant mass. It's confusing to use the old concept of a relativistic mass. That's just energy (modulo a factor [itex]c^2[/itex]). Also when you have [itex]\sqrt{s}=14 \; \mathrm{GeV}[/itex] at a collider with two beams of the same particles (particles with the same invariant mass) this means that a particle in each beam has a total energy of [itex]E=7 \; \mathrm{GeV}[/itex] with the rest energy included!
     
  14. Oct 24, 2013 #13

    bobie

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    M+ stands for relativistic mass, mass increase, M0*γ ( Lorenz factor).
    I think I have already multiplied m by γ , (the mass of the proton M0 by 7461), what else should I do to get the right R?
     
    Last edited: Oct 24, 2013
  15. Oct 24, 2013 #14

    bobie

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    So, what figure should I write instead of [itex]\gamma m[/itex]?
     
    Last edited: Oct 24, 2013
  16. Oct 24, 2013 #15

    BruceW

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    Oh yeah, you did already multiply by gamma. sorry, I missed that. uh, hmm. your answer for R should be correct, as far as I can see.
    The equation ##R=m_0\gamma v/eB## is definitely correct. (and it is in SI units). And this is the equation you used... Are you certain the energy of each beam is 7TeV and the magnetic field is 8.33T ? Using these values, I get roughly the same answer as you do, and since I'm pretty sure the formula is true, I am suspicious of the values.
     
  17. Oct 24, 2013 #16

    bobie

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    OK, so R is definitely around 2800 m, probably I made a typo when I transcribed the data. Thanks.

    Can you tell me what curve is y? I read that it is hyberbolic , is it as a function of β?
    Does it describe the curve of a secant as a function of β?
     
  18. Oct 24, 2013 #17

    BruceW

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    If we use ##\beta=\tanh(\phi)##, then ##\gamma=\cosh(\phi)## So we are creating a new parameterization, with the parameter ##\phi## (called the rapidity). And under this parameterization, the Lorentz factor is a hyperbolic function of the rapidity. I don't know about secant... that's not a hyperbolic function. So I would guess it would not be useful.

    Also, about R - I also read on wikipedia that when they get the beam energy to 7TeV, they will need a magnetic field of 8.3T and they say the radius of the LHC is 4297m ... So I guess maybe your data was correct. But then I don't understand why the formula doesn't work... uh... Maybe we are missing some other piece of information.
     
  19. Oct 24, 2013 #18

    BruceW

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    Ah yeah, here it is: http://www.lhc-closer.es/1/4/2/0 So it says the magnetic field is not the same all the way around the LHC, which is why we get the wrong answer when we assume there is the same magnetic field all the way around.
     
  20. Oct 24, 2013 #19

    bobie

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    Great!
    Is it possible to explain in simple concepts what that means? and what is rapidity?

    If we set h= c, c1 = β , and [itex]\lambda[/itex] the angle opposite c2, then c2 = [tex]\sqrt{1- β^2}[/tex] and
    λ1 = f(β) = h/c2 = sec λ (when β goes from 0 to 1.)
    Is that correct?
    What is the relation of your γ (##cosh(\phi##)) to my γ1?
     
    Last edited: Oct 24, 2013
  21. Oct 24, 2013 #20

    BruceW

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    ah, you've lost me. I don't know what any of h,c,c2,##\lambda##,##\lambda_1## are meant to be.
    In the equations I wrote down, ##\cosh(\phi)## does not mean the cosine of ##h\phi##. I was using ##\cosh## to be the hyperbolic cosine. So you should only need one parameter. Also, I don't understand what you mean when you start talking about 'angles opposite to'. The hyperbolic functions are not really related to triangles... unless they are triangles with imaginary angles maybe.
     
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