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I want to know how a radiofrequency pulse of known flip angle $\beta$ changes the population difference of a set of isolated spin 1/2 nuclei.

I know that the equilibrium magnetization is related to the population difference by:

$M_0= \frac{1}{2}\gamma\hbar\Delta{n_{eq}}$

I would like to know how to calculate the x, y, z components of the magnetization after the pulse, and then calculate the population difference from these values.

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immediately after the pulse:

$M_z = M_0cos(\beta )$

$M_x = M_0sin(\beta )$

$M_y = 0$

so for example after a 90 degree pulse, all the magnetization has been rotated into the x-y plane, and after a 180 pulse, the magnetization is in the -z direction.

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  • $\begingroup$ Thanks. How can I use this to calculate population differences ? $\endgroup$
    – J. LS
    Commented Apr 8, 2015 at 22:14
  • $\begingroup$ @J.LS $M_z= \frac{1}{2}\gamma\hbar\Delta{n_{eq}}$, so no population difference after 90 degree pulse, and inverted population after 180 degree pulse $\endgroup$
    – DavePhD
    Commented Apr 9, 2015 at 13:47
  • $\begingroup$ If there is no population difference after a 90 degree pulse, why does spin-spin relaxation occur ? $\endgroup$
    – J. LS
    Commented Apr 10, 2015 at 9:52
  • $\begingroup$ @J.LS the applied magnetic field is in the z direction. The axes about which the protons are spinning are tilted, either up or down relative to the z direction. The axes are precessing about the z axis, and are free to have any x and y value (not quantized with respect to these directions). After the pulse, the x and y components of the spins are not randomly oriented in the x-y plane, but instead are preferentially oriented in the +x direction. (continues below) $\endgroup$
    – DavePhD
    Commented Apr 10, 2015 at 12:39
  • $\begingroup$ (continued from above) Due to local magnetic field inhomgeneity, the spins do not all precess at the same rate, and become out of phase with respect to each other, eventually becoming randomly oriented with respect to x and y. $\endgroup$
    – DavePhD
    Commented Apr 10, 2015 at 12:39

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