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Mathematics
Linear and Abstract Algebra
Understanding the Derivation of Reciprocal Lattice Basis from Equations 5 and 6
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[QUOTE="Infrared, post: 6282810, member: 467682"] You can check directly that those formulas for the ##b_i## satisfy the equations that you want. For example, ##b_1\cdot a_1=2\pi\frac{a_1\cdot (a_2\times a_3)}{a_1\cdot (a_2\times a_3)}=2\pi##. Since the vectors ##a_i## are linearly independent, the equations ##b_i\cdot a_j=2\pi\delta_{ij}## can be uniquely solved for the ##b_i##, and hence the above are the unique solutions. If you wanted to figure out these formulas from scratch, you could argue like this: since ##b_1## is orthogonal to both ##a_2## and ##a_3##, you know that ##b_1=c_1 (a_2\times a_3)## for some constant ##c_1##. Then the equation ##a_1\cdot b_1=2\pi## let's you solve for ##c_1##, etc. [/QUOTE]
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Forums
Mathematics
Linear and Abstract Algebra
Understanding the Derivation of Reciprocal Lattice Basis from Equations 5 and 6
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