The equation you mentioned, f = mc^2/h, is not the De Broglie equation but rather an expression derived from it. The actual De Broglie equation relates the wavelength (λ) of a particle to its momentum (p) and is given by λ = h/p, where h is the Planck constant.
The De Broglie equation is valid for particles in any medium, not just in a vacuum. However, it's important to note that the speed of light (c) in the De Broglie equation represents the speed of light in a vacuum. In a medium other than vacuum, the speed of light is different, as it depends on the properties of the medium.
When considering particles in a medium, you need to take into account the refractive index (n) of the medium. The refractive index is defined as the ratio of the speed of light in vacuum to the speed of light in the medium. So, in the De Broglie equation, the speed of light (c) should be replaced by the speed of light in the medium (c/n), where n is the refractive index.
Therefore, the modified De Broglie equation for a particle in a medium becomes λ = h/(p/n), which can be rewritten as λ = hn/(p), where n is the refractive index of the medium. This modified equation accounts for the different speed of light in the medium and allows you to calculate the wavelength of a particle in that specific medium.
To summarize, the De Broglie equation is valid for particles in any medium, and the speed of light in the equation should be replaced by the speed of light in the respective medium, taking into account its refractive index.