What is the minimum value of the radius ratio | vedclass

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(A) In an ionic crystal,the stability of the lattice depends on the radius ratio ${r_ + }/{r_ - }$. For an octahedral void (coordination number $6$),w

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$0.414$

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(A) In an ionic crystal,the stability of the lattice depends on the radius ratio ${r_ + }/{r_ - }$. For an octahedral void (coordination number $6$),which is the structure of $NaCl$,the limiting radius ratio is calculated as follows: In an octahedral void,the cation is surrounded by $6$ anions. For the condition where the cation just touches the anions and the anions do not touch each other,the geometry leads to the relation: ${r_ + } + {r_ - } = \sqrt{2} {r_ - }$. Rearranging this,we get ${r_ + } = (\sqrt{2} - 1) {r_ - }$. Thus,${r_ + }/{r_ - } = \sqrt{2} - 1 \approx 1.414 - 1 = 0.414$. Therefore,the minimum radius ratio for an octahedral arrangement is $0.414$.

What is the minimum value of the radius ratio ${r_ + }/{r_ - }$ for $NaCl$ to ensure that anion-anion contact does not occur?

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