4. Someone has provided the following lattice vectors for Li2O2, which is of interest in state-of-the-art lithium-air batteries.



In [1]:

    
a = [1.5915, 2.756559, 0]
b = [1.5915, -2.756559, 0]
c = [0, 0, 7.7258]

a. What are the lattice parameters for Li2O2?

Solution:

a = $|\mathbf{a}|$

b = $|\mathbf{b}|$

c = $|\mathbf{c}|$

$\alpha$ = $cos^{-1}(\frac{\mathbf{b}.\mathbf{c}}{|\mathbf{b}||\mathbf{c}|})$

$\beta$ = $cos^{-1}(\frac{\mathbf{a}.\mathbf{c}}{|\mathbf{a}||\mathbf{c}|})$

$\gamma$ = $cos^{-1}(\frac{\mathbf{a}.\mathbf{b}}{|\mathbf{a}||\mathbf{b}|})$



In [2]:

    
import numpy as np
from math import acos, degrees

a_length = np.linalg.norm(a)
b_length = np.linalg.norm(b)
c_length = np.linalg.norm(c)
alpha_angle = degrees(acos(np.dot(b, c) / (b_length * c_length)))
beta_angle = degrees(acos(np.dot(a, c) / (a_length * c_length)))
gamma_angle = degrees(acos(np.dot(a, b) / (a_length * b_length)))
print "The lattice parameters are {%.4f, %.4f, %.4f, %.4f, %.4f, %.4f}." % (a_length, b_length, c_length, alpha_angle, beta_angle, gamma_angle)









    



The lattice parameters are {3.1830, 3.1830, 7.7258, 90.0000, 90.0000, 120.0000}.

b. What is the crystal system for Li2O2?

Solution: The cell system is hexagonal.