I'm getting a contradiction as follows and am not sure how to resolve it:
leads to the conclusion that the higher order minimas of single slit diffraction will be given by $b\sin\theta = n(\lambda)$ where $n = 1,2,3,\dots$ (this is given in textbooks)
leads to the conclusion that the higher order minimas of single slit diffraction should be given by $b\sin\theta = (2n+1)\lambda$ where $n = 0,1,2,3,\ldots$(i.e. only for odd multiples of wavelength.
I need to understand why logic 2 is wrong.
logic 2: if we divide the slit into two equal halves, and assume that light from top half destructively interferes with light from the bottom half, then path difference between corresponding pairs of points will be $\lambda/2,$ for the given angle theta at which first dark fringe occurs. Next dark fringe occurs at angle theta2 which is bigger and thereby increases the path difference between the same corresponding sets of rays by lambda so that total path difference is now (3/2)lambda resulting in dark fringe. So this condition can be set up as $(d/2)\sin\theta = (2n+1)(\lambda/2)$ which gives: $d\sin\theta = (2n+1)(\lambda)$ But this is supposed to be wrong, how?
Logic 1 says we just keep dividing the slit into even pairs like 2, 4, 6, 8, giving rise to higher order dark fringes as integral multiples of lambda including both odd and even multiples.
So am confused!