Invariance of U_harm of crystal to rotation

In summary, we need to show that the energy of a crystal only depends on the symmetric strain tensor epsilon_ij. This can be done by considering a pure rotational displacement field u(r) and manipulating the equations to remove all dependence on antisymmetric terms.
  • #1
sam_bell
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Homework Statement



Show that because a pure rotational displacement field u(r) has no effect, that the energy of a crystal only depends on the symmetric strain tensor epsilon_ij.

Homework Equations



As in Ashcroft & Mermin (22.72), the energy of a crystal as a function of displacement field u_i(r) and dynamical matrix D_ij(R) (R = direct lattice vector) is:
U_harm = 1/2 Integral[dr Sum[ijkl, (du_j/dx_i)(du_l/dx_k)E_ijkl ]]
where
E_ijkl = -1/2 Sum[R, R_i D_jl(R) R_k] (ijkl indices go over x,y,z)

We need to show that this only depends on the strain in the symmetric combination
epsilon_ij = 1/2( du_i/dx_j + du_j/dx_i ) (Ashcroft & Mermin 22.77)

The Attempt at a Solution



For simplicity, consider the 2D case. Then u(x,y) = (-ay, ax) for small a is an infinitesimal rotation of the plane. Writing (22.72) for the crystal energy, we get

0 = U_harm = 1/2 Integral[dr (du_y/dx)(du_y/dx) E_xyxy + (du_y/dx)(du_x/dy)E_xyyx + (du_x/dy)(du_y/dx)E_yxxy + (du_x/dy)(du_x/dy)E_yxyx]
= 1/2 Integral[dr a^2(E_xyxy - E_xyyx - Eyxxy + E_yxyx) ] = 0

Or, E_xyxy - E_xyyx - Eyxxy + E_yxyx = 0.

Now, we rewrite du_y/dx = eps_xy + ant_xy = 1/2(du_y/dx+du_x/dy) + 1/2(du_y/x-du_x/dy), and similarly du_x/dy = eps_xy - ant_xy as symmetric and antisymmetric parts. Then we can substitute in (22.72):

U_harm = 1/2 Integral[dr (du_y/dx)(du_y/dx) E_xyxy + (du_y/dx)(du_x/dy)E_xyyx + (du_x/dy)(du_y/dx)E_yxxy + (du_x/dy)(du_x/dy)E_yxyx + (remaining terms)]
= 1/2 Integral[ dr, (eps_xy + ant_xy)(eps_xy + ant_xy)(E_xyxy) + (eps_xy + ant_xy)(eps_xy - ant_xy)(E_xyyx + E_yxxy) + (eps_xy - ant_xy)(eps_xy - ant_xy)(E_yxyx) + (remaining terms)]
= 1/2 Integral[ dr, eps_xy^2 (E_xyxy + E_xyyx + Eyxxy + E_yxyx) + (eps_xy ant_xy)(2 E_xyxy - 2 E_yxyx) + ant_xy^2(E_xyxy - E_xyyx - Eyxxy + E_yxyx) + (remaining terms)]

Now the coefficient of ant_xy^2 vanishes, but I can't see why the coefficient for eps_xy*ant_xy would vanish. The goal is to remove all dependence on antisymmetric terms. I see no reason from what I read in Ashcroft & Mermin that E_xyxy = E_yxyx generally.
 
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  • #2
I figured it out
 

1. What is the concept of invariance of U_harm of crystal to rotation?

The concept of invariance of U_harm (or harmonic potential energy) of a crystal to rotation refers to the idea that the potential energy of a crystal lattice does not change when the crystal is rotated. This means that the forces between atoms in a crystal remain the same regardless of the orientation of the crystal.

2. Why is it important for the U_harm of a crystal to be invariant to rotation?

It is important for the U_harm of a crystal to be invariant to rotation because it allows for the crystal to have the same physical properties in all directions. This is crucial for the structural and mechanical stability of the crystal, as well as its ability to exhibit certain physical phenomena such as piezoelectricity and thermal expansion.

3. How is the invariance of U_harm of a crystal to rotation related to its symmetry?

The invariance of U_harm to rotation is closely related to the concept of symmetry in crystals. This is because a crystal with a high degree of symmetry will have the same potential energy in all orientations, while a crystal with lower symmetry will have different potential energies in different orientations.

4. Are there any exceptions to the invariance of U_harm of a crystal to rotation?

While the invariance of U_harm to rotation is a fundamental principle in crystal physics, there are some exceptions to this rule. In certain cases, the potential energy of a crystal may change when it is rotated due to the presence of defects or impurities in the crystal lattice.

5. How is the invariance of U_harm of a crystal to rotation experimentally determined?

The invariance of U_harm to rotation can be experimentally determined through techniques such as X-ray diffraction and neutron scattering. These methods allow for the measurement of the crystal's atomic positions and lattice parameters, which can then be used to calculate the potential energy of the crystal in different orientations and confirm its invariance to rotation.

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