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By measuring second occurring times of factors of an infinite word $x$,Bugeaud and Kim introduced a new quantity ${\rm rep}(x)$ called the exponent ofrepetition of $x$. It was proved by Bugeaud and Kim that $1 \leq {\rm rep}(x)\leq r_{\max} = \sqrt{10} - 3/2$ if $x$ is a Sturmian words. In this paper, wedetermine the value $r_1$ such that there is no Sturmian word $x$ satisfying$r_1 < {\rm rep}(x) < r_{\max}$ and $r_1$ is an accumulation point of the setof ${\rm rep}(x)$ when $x$ runs over the Sturmian words.

A gap of the exponents of repetitions of Sturmian words