# How can I calculate the total rotation a detuned pulse will apply a nuclear spin?

I'm trying to model the effect a radiofrequency pulse will have on a nuclear spin at different detunings. The pulse has a sech lineshape, a pulse area (time integral of the pulse envelope) of $$\frac{π}{2}$$ and a width of 5MHz centred at 123MHz. This can (I believe) be represented by the equation

$$P(\omega)=\frac{1}{4}\operatorname{sech}\left(\frac{\omega-123}{2}\right)$$

Then I want to calculate the overlap between this pulse and a range of nuclear spins at different detunings from the pulse. The nuclear spins can be represented by a rectangular function in frequency space with a linewidth of 10kHz:

$$R(\omega)= \begin{cases} R_0 & \text{if } \omega_R-\Delta\leq\omega\leq\omega_R+\Delta\\ 0 & \text{otherwise} \end{cases}$$

where $$\omega_R$$ is the resonant frequency of the pulse and $$2\Delta$$ is the linewidth of the pulse.

I believe the overlap can be represented by the integral

$$\int_{-\infty}^{\infty}|R(\omega)P(\omega)|^2d\omega$$

but am unsure on a couple of details. I am unclear on whether the value of $$R_0$$ can be arbitrarily chosen to be $$1$$, as the only details I have about the nuclei are their frequency and linewidth and I am unclear whether the absolute square I have used is in the correct place in the equation or whether it should be outside the integral (or not included at all).

• Is $S(\omega)$ same as $P(\omega)$? If not then what is $S(\omega)$ – Jitendra Oct 14 '18 at 16:56
• Yes it is, I have edited. – JJH Oct 14 '18 at 17:00

Overlap will be $$\int_{-\infty}^{\infty}R(\omega)P(\omega)d\omega = \int_{\omega_R-\Delta}^{\omega_R+\Delta}\frac{R_0}{4}\operatorname{sech}\left(\frac{\omega-123}{2}\right)d\omega = \frac{R_0}{2}{tan}^{-1}(sinh(\Delta))$$