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Homework Help: Functional Derivative of Long Range Dipole Interaction Energy

  1. Oct 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Dear all, Good day.

    I am currently working on the phase field modeling of ferroelectrics. For this reason, I need to find functional derivative of an expression as presented in attached picture 1.

    Then in picture 2, it shows the final form of equation that I am supposed to arrive at.

    2. Relevant equations

    I am stuck at line P in attached picture 5. Looking at the equation in line P, if I can include only the exponentials in the integral d^3g, then I can change them to 2*Dirac's delta, and the multiplication of dirac's delta with P' leads to P and performing Fourier transform, I can get the form as shown in attached picture 2. However, I doubt if I can do so because n is a function of g. Can anyone please help me on this? I have been breaking my head trying to figure this out for two days.

    3. The attempt at a solution

    My attempt on the question is as attached in picture 3, 4 and 5.

    As shown in attached picture 3, line A, B, C, D is how I started. I bring the unit vector n into the integral of P because it is not a function of r, then I separate exponential into cos real part and sin imaginary part. This is to get rid of the magnitude operator.

    Then in attached picture 4 and 5, from line E to P is how I am trying to find the functional derivative of F dip wrt P1. I am stuck at line P.

    Attached Files:

    • 1.jpg
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    • 2.png
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    • 3.jpg
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    • 4.jpg
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    • 5.jpg
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  2. jcsd
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