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Problem

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problem description

# Problem setup: Consider a simple cubic crystal with a static periodic strain field of long wavelength described by a displacement vector u→=ε→sin(Q→⋅r→)\vec{u}=\vec{\varepsilon}\sin(\vec{Q}\cdot\vec{r}), where u→(r→)\vec{u}(\vec{r}) is the vector displacement from the perfect-crystal position for the atom at lattice site r→\vec{r}, and Q→\vec{Q} denotes the wave vector of the displacement wave (not a scattering wave vector). You may assume that the lattice-wave period is a large integer (MM) multiple of the lattice spacing aa and that all displacements are much smaller than the lattice spacing. # Main problem: Calculate the structure factor to first order in ε\varepsilon for diffraction peaks at reciprocal space vector (nx,ny,xz)(n_x, n_y, x_z) for ε→∥[100]\vec{\varepsilon} \parallel [100], Q→∥[100]\vec{Q} \parallel [100]. Write down the criteria for the structural factor to be nonvanishing (besides nx=Mn_x = M) in the lowest-possible-order Brillouin Zone and the corresponding structural factor.
Plain-text mathematical notation (without MathML)
# Problem setup:

Consider a simple cubic crystal with a static periodic strain field of long wavelength described by a displacement vector (u)→=(ε)→sin((Q)→⋅(r)→), where (u)→((r)→) is the vector displacement from the perfect-crystal position for the atom at lattice site (r)→, and (Q)→ denotes the wave vector of the displacement wave (not a scattering wave vector).

You may assume that the lattice-wave period is a large integer (M) multiple of the lattice spacing a and that all displacements are much smaller than the lattice spacing.



# Main problem:

Calculate the structure factor to first order in ε for diffraction peaks at reciprocal space vector (n_(x),n_(y),x_(z)) for (ε)→∥[100], (Q)→∥[100]. Write down the criteria for the structural factor to be nonvanishing (besides n_(x)=M) in the lowest-possible-order Brillouin Zone and the corresponding structural factor.
Original LaTeX notation
# Problem setup:

Consider a simple cubic crystal with a static periodic strain field of long wavelength described by a displacement vector $\vec{u}=\vec{\varepsilon}\sin(\vec{Q}\cdot\vec{r})$, where $\vec{u}(\vec{r})$ is the vector displacement from the perfect-crystal position for the atom at lattice site $\vec{r}$, and $\vec{Q}$ denotes the wave vector of the displacement wave (not a scattering wave vector).

You may assume that the lattice-wave period is a large integer ($M$) multiple of the lattice spacing $a$ and that all displacements are much smaller than the lattice spacing.



# Main problem:

Calculate the structure factor to first order in $\varepsilon$ for diffraction peaks at reciprocal space vector $(n_x, n_y, x_z)$ for $\vec{\varepsilon} \parallel [100]$, $\vec{Q} \parallel [100]$. Write down the criteria for the structural factor to be nonvanishing (besides $n_x = M$) in the lowest-possible-order Brillouin Zone and the corresponding structural factor.

code template

Code

import sympy as sp

M = sp.symbols('M')
epsilon = sp.symbols('epsilon')
a = sp.symbols('a')

def answer(M, epsilon, a):
    r"""
    Return the expressions of the $n_x$ criteria for structural factor to be nonvanishing
    and the corresponding structure factors in Sympy format.

    Inputs
    ----------
    M         : sympy.Symbol
        Large integer relating the strain wavelength to the lattice spacing.
    epsilon   : sympy.Symbol
        Amplitude of the static periodic strain.
    a         : sympy.Symbol
        Lattice spacing of the simple-cubic crystal.

    Outputs
    ----------
    allowed : set[(sympy.Expr, sympy.Expr)], Set of nonvanishing $n_x$ criteria with corresponding structure factor, {(nx, S)}
        nx: sympy.Expr, $n_x$ component of the reciprocal space vector in the lowest possible order Brillouin Zone
          for which the structure factor is nonvanishing (besides $n_x = M$).
        S: sympy.Expr, corresponding structure factor to the first order in $\varepsilon$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    allowed = ...
    # ---------------------------------------------------------------

    return allowed

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Source and history

upstream_id / Original problem ID
Challenge_59_main
task_id
86c86e08-b464-5459-b5b9-23141c16de50
task_key
train--Challenge~5f59~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import