This paper presents the design and evaluation of the optical zoom system for an LWIR camera. The 12.8operating wavelength range of this system is from $7.7{mu}m$ to $12.8{mu}m$. Through a paraxial design and optimization process, we have obtained the extended four-group inner-focus zoom system with focal lengths of 10 to 100 mm, which consists of the six lenses including four aspheric surfaces and two diffractive surfaces. The diffractive lenses were used to balance the higher-order aberrations, and its diffraction properties were evaluated by scalar diffraction theory. We have calculated the polychromatic integrated diffraction efficiency and the MTF drop generated by background noise. The f-number of the zoom system is F/1.4 at all positions. Fields of view are given by $51.28^{circ}{ imes}38.46^{circ}$ at wide field and $5.50^{circ}{ imes}4.12^{circ}$ at narrow field positions. In conclusion, this design procedure results in a $10{ imes}$ compact zoom lens system useful for an LWIR camera.