Tutorial on Using Excel Solver to Analyze Spin-Lattice Relaxation Time Data

Tutorial on Using Excel Solver to Analyze Spin-Lattice Relaxation Time Data oIn the measurement of the Spin-Lattice Relaxation time T , a 180 pulse is followed 1oafter a delay time of t with a 90 pulse, and at each delay time the magnetization M is measured by measuring the voltage level of the signal that is detected. zAccording to the theory, the magnetization M is given by z −tT/ 1⎡ ⎤ M=M 1 −2 e, (0.1) z0 ⎣ ⎦ owhere M is the initial magnetization following the 180 pulse, t is the delay time, and 0T is the spin-lattice relaxation time. There are 2 unknowns in this equation, M 1 0and T . Frequently with exponentially decaying quantities, the equation can be 1manipulated and the logarithm of both sides of the equation taken so that the natural exponential is removed. This usually results in a linear equation of dependence with the time t and a direct relationship between the slope of the linear graph and the decay rate constant, 1/T . However, in this case, this is not possible 1so that a non-linear least squares fitting procedure is needed. Excel provides a method of doing this called “Solver.” In this application the measurements results in a N data pairs, t and V , where V is i i ithe voltage measurement representative of the magnetization M at each time t . z iSince the voltage V is proportional to the magnetization M , Equation (0.1) zbecomes −tT/ 1⎡ ⎤ V=V 1 −2e (0.2) z0 ⎣ ⎦ and this is the equation we wish to fit to the data to ...