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A simple expression for the Gamow factor is obtained using a second ordercurvature corrected tunneling potential. Our results show that it approximatesaccurately the `exact-WKB' transmission coefficient obtained by numericallyintegrating over the tunneling region to obtain the Gamow factor. The averagedifference in current density using the respective transmission coefficients isabout $1.5 \%$, across a range of work-functions $\phi \in [3-5.5]$eV, Fermienergy ${\cal{E}}_F$ in [5-10]eV, local electric fields $E_l$ in[3-9]eV andradius of curvature $R \geq 5$nm). An easy-to-use correction factor $\lambda_P$is also provided to approximately map the `exact-WKB' current density to the`exact' current density in terms of ${\cal{E}}_F/\phi$. The average error onusing $\lambda_P$ is found to be around $3.5\%$. This is a vast improvementover the average error of $15\%$ when $\lambda_P = 1$. Finally, an analyticalexpression for the curvature-corrected current density is obtained using theGamow factor. It is found to compare well with the `exact-WKB' current densityeven at small values of local electric field and radius of curvature.

Higher order curvature corrections to the field emission current density