Spontaneous scalarization of self-gravitating magnetic fields

In this paper, we study the spontaneous scalarization of an extended, self-gravitating system which is static, cylindrically symmetric and possesses electromagnetic fields. We demonstrate that a real massive scalar field condenses on Melvin magnetic universe solution when introducing non-minimal coupling between (a) (b) curvature space-time, respectively. find in both cases, solutions exist finite interval constant with number nodes $k$ exist. For case observe intervals existence are mutually exclusive for different $k$.