New effective interaction for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi mathvariant="italic">pg</mml:mi><mml:mrow><mml:mn>9</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>-shell nuclei

We present a new effective interaction for shell-model calculations in the model space consisting of the single-particle orbits $1{p}_{3/2}$, $0{f}_{5/2}$, $1{p}_{1/2}$, and $0{g}_{9/2}$. Starting with a realistic interaction based on the Bonn-C potential, 133 two-body matrix elements and four single-particle energies are modified empirically so as to fit $400$ experimental energy data out of $69$ nuclei with mass numbers $A=63~96$. The systematics of binding energies, electromagnetic moments and transitions, and low-lying energy levels are described. The soft $Z=28$ closed core is observed, in contrast to the stable $N=50$ shell closure. The new interaction is applied to systematic studies of three different chains of nuclei, Ge isotopes around $N=40$, $N=Z$ nuclei with $A=64~70$, and $N=49$ odd-odd nuclei, focusing especially on the role of the ${g}_{9/2}$ orbit. The irregular behavior of the ${0}_{2}^{+}$ state in Ge isotopes is understood as a result of detailed balance between the $N=40$ single-particle energy gap and the collective effects. The development of the band structure in $N=Z$ nuclei is interpreted in terms of successive excitations of nucleons into the ${g}_{9/2}$ orbit. The triaxial/$\ensuremath{\gamma}$-soft structure in $^{64}\mathrm{Ge}$ and the prolate/oblate shape coexistence in $^{68}\mathrm{Se}$ are predicted, showing a good correspondence with the experimental data. The isomeric states in $^{66}\mathrm{As}$ and $^{70}\mathrm{Br}$ are obtained with the structure of an aligned proton-neutron pair in the ${g}_{9/2}$ orbit. Low-lying energy levels in $N=49$ odd-odd nuclei can be classified as proton-neutron pair multiplets, implying that the obtained single-particle structure in this neutron-rich region appears to be appropriate. These results demonstrate that, in spite of the modest model space, the new interaction turns out to describe rather well properties related to the ${g}_{9/2}$ orbit in various cases, including moderately deformed nuclei.