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One-Dimensional chain

  1. Sep 24, 2013 #1
    1. The problem statement, all variables and given/known data

    The atoms having in each case, the mass m and are located at positions with x i i ∈ Z. The rest position of the ith atom is i · a where a is the lattice constant of the crystal.
    The interaction between the atoms can be modeled in a simple approximation, as a spring force of the spring constant D between adjacent atoms. The i-th atom causes therefore the (i - 1) th power of the atom has a size Fi→i−1 = D (xi − xi−1 − a).

    (1) Set the equation of motion for the position of the ith atom in the crystal lattice and show that the equations of motion of the atoms are solved by standing waves of the form
    xi (t) x = sin (a i k) sin (ω t) + a i. Look for the solution ω as a function of D, m, a and k, and determine the maximum value of ω as a function of the parameters of the crystal. Also sketch the history of ω as a function of k.


    2. Relevant equations

    |k | = 2 π/λ
    ω = 2 π f


    3. The attempt at a solution

    Equation of motion for the position of the i-th atom:
    Mx ̈i = C(xi+1 + xi-1 2xi)


    I don´t know how i should go on.
     

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  3. Sep 25, 2013 #2
    isn´t there anyone who could help me??
    it is quite important for me to solve this problem.
    i just can´t go on, i have no idea
     
  4. Sep 25, 2013 #3

    haruspex

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    You are more likely to get help if you take the trouble to use LaTex to lay out your equations properly.
    I assume your equation is supposed to read ##m \ddot x = D(x_{i+1}+x_{i-1}-2 x_i)##. Plug in the solution form you are given and see what you get.
     
  5. Sep 25, 2013 #4
    Do you mean this is the right right equation of motion?

    But how should i solve this equation with standing waves?

    How should i go on?
     
  6. Sep 25, 2013 #5

    haruspex

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    You are given a form of solution and asked to show that it is a solution of the equation (if the parameters are set appropriately). So substitute the given form for xi into the differential equation and deduce the values of the parameters.
     
  7. Sep 26, 2013 #6
    xi = Xo*e^i(kia-ωt)

    xi-1 = xi*e^(-ika)
    xi+1 = xi*e^(ika)

    -Mω^2*xi =C(e^(ika) +e^(-ika) - 2)xi

    ω^2 =2*C/M*(1-coska) = 4C/M*sin^2 * ka/2

    ω = 2*√C/M * sin(ka/2)


    is this right?
    what have i done wrong?
    what should i do know?
     
  8. Sep 28, 2013 #7
    sorry, but can nobody say me if this is right?
    i have no idea how i should go on...


    thank for all your great help :) :)
     
  9. Sep 29, 2013 #8

    haruspex

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    Sorry, but I had no net access for a few days. Your failure to use LaTex makes it very hard to follow what you are doing. Sometimes i should be a subscript (perhaps), other times it stands for root of minus 1. I'm not even sure what the harmonic solution suggested in the OP is saying. Is that really ai in two places? If you repost in LaTex I'll take another look.
     
  10. Sep 29, 2013 #9
    xi = x0 * ei*(k*i*a-ω*t)

    xi-1 = xi*e(-ika)
    xi+1 = xi*e(ika)

    -Mω2xi=C(e(ika)+e(-ika)- 2)xi

    ω2 =2*C/M*(1-coska) = 4C/M*sin2 * ka/2

    ω = 2*√C/M * sin(ka/2)

    I hope you can help me now
     
  11. Sep 30, 2013 #10

    haruspex

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    Well, it's still a bit confusing because you appear to be using i for two roles. How about using j for one of them?
    But I don't understand how you have converted the given form of solution, a product of two sine functions, into the exponential form above. The original is purely real, whereas your form has an imaginary component.
    Or maybe the i in your form is always referring to the index i, as in the original, and not sometimes meaning the square root of minus 1? In that case I don't understand how you changed sine into exp at all.
     
  12. Sep 30, 2013 #11
    I hope you can help me now
     
  13. Sep 30, 2013 #12

    haruspex

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    I fail to see what these exponential functions have to do with answering the given question. You are given a form of solution involving a product of two sine functions. No exponentials. Why not just substitute that form into the differential equation and demonstrate that it is a solution (when the parameters are set appropriately)?
    If the exponential form you introduced is right then there's something about it that puzzles me. In the exponent you appear to have a quadratic in i. Is that correct or does one of the two i's represent the square root of -1? I keep asking this question in different ways and you have still not answered it.
     
  14. Oct 1, 2013 #13
    you think I should stop right here.?
    no the i`s stand for i`s.

    can you please show me what i should do
     
  15. Oct 1, 2013 #14

    haruspex

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    We don't seem to be making much progress. I have the feeling that you are quoting some textbook theory (the source of these equations with exponentials) but the xi terms in them do not correspond to those in the OP, so those equations are not relevant.
    Is this possibly the case?
     
  16. Oct 2, 2013 #15
    Okay, this equations are not the solution for this task.
    But how should i go on? I have no other idea at the moment.
    Can you recommend an information site where I can read up about this theme?
    Or can you give me a tip where I should start to find the right solution.
     
  17. Oct 2, 2013 #16

    haruspex

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    I have suggested another approach three times - posts #3, #5, and #12. Do you understand what I suggested?
     
  18. Oct 2, 2013 #17
    I have got the equation of movement :
    mx = D(xi+1 + xi-1 - 2xi)

    and the equation for standing waves:
    xi(t) = x * sin (a * i * k) * sin (ω * t) + a * i

    I am not sure if I understand you right.

    Maybe:
    x = (D(xi+1 + xi-1 - 2xi) / m

    and this in (2)
    xi(t) = (D(xi+1 + xi-1 - 2xi) / m * sin (a * i * k) * sin (ω * t) + a * i

    I think that this is not the right way.
    Please help me
     
  19. Oct 3, 2013 #18

    haruspex

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    Need to get those equations right first:
    ##m \ddot x_i = D(x_{i+1} + x_{i-1}-2x_i)##
    ##x_i = sin(a i k) sin(\omega t) + a i##
    OK?
    Try again with those.
     
  20. Oct 3, 2013 #19
    These are the two given equations and now i should solve this problem:

    Look for the solution ω as a function of D, m, a and k, and determine the maximum value of ω as a function of the parameters of the crystal. Also sketch the history of ω as a function of k.

    And I really don´t know what I should do with these two equations to get a solution.
    Should I equate the formulas, or transform to a variable?
    I do not know.
     
  21. Oct 3, 2013 #20

    haruspex

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    Use the second formula to substitute for all the xi, xi-1 and xi+1 references in the first.
     
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