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In this work, we probe the weak gravitational lensing by a static sphericallysymmetric black hole in view of $f(R)$ gravity in the background of thenon-plasma medium (vacuum). We provide a discussion on a light ray in a staticblack hole solution in $f(R)$ gravity. To adore this purpose, we find theGaussian optical curvature in weak gravitational lensing by utilizing theoptical geometry of this black hole solution. Furthermore, we find thedeflection angle up to the leading order by employing the Gauss-Bonnet theorem.We present the graphical analysis of the deflection angle with respect to thevarious parameters that govern the black hole. Further, we calculate theHawking temperature for this black hole via a topological method and compare itwith a standard method of deriving the Hawking temperature. We also analyze theSchr\"odinger-like Regge-Wheeler equation and derive a bound on the greybodyfactor for a static black hole in the framework of $f(R)$ gravity andgraphically inquire that bound converges to 1. We also investigate thesilhouette or shadow generated by this static $f(R)$ black hole. Moreover, weconstrain the non-negative real constant and cosmological constant from theobserved angular diameters of M87* and Sgr A* released by the EHT. We thenprobe how cosmological constant, non-negative real constant and mass affectedthe radius of shadow. Finally, we demonstrate that, in the eikonal limit, thereal part of scalar field quasinormal mode frequency can be determined from theshadow radius.

Weak Deflection Angle, Hawking Radiation, Greybody Bound and Shadow Cast for Static Black Hole in the Framework of $f(R)$ Gravity