Chemical potential of mobile magnetic particles in a magnetic field

[Shackleford]

I don't know where the 1/2 comes from in the third image.

If I set mu total(up) = mu total up (0), I don't get the 1/2. I'm missing something very simple here.

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110325_214532.jpg?t=1301107797

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110325_214447.jpg?t=1301107813

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110325_214505.jpg?t=1301107827

 

[Shackleford]

If you equate mu-up total and mu-down total, you get n-up = exp(2mB/tau)*n-down.

Then, solve for B = 0, you get that the concentrations are equal. This allows you to rewrite n(0)= 2n-down, which implies (1/2)n(0) = n-down. Putting this into the original equation yields n-up = (1/2)n(0)*exp(2mB/tau).