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please solve a,b,c
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answerhappygod
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please solve a,b,c
please solve a,b,c
In 1946, Felix Bloch proposed the following set of equations to describe NMR relaxation: dM Mx/T2 My/T₂ = M x H - (3) dt (M₂-Mo)/T₁ where M = (Mx, My, M₂) is the total magnetization of nuclear spins in the system, H = (hx, hy, Ho) is the magnetic field, y is the gyromagnetic ratio, T₁ and T₂ are the longitudinal and the transverse relaxation times respectively, and Mo is the equilibrium magnetization in magnetic field Ho. Firstly, let us consider the case, where hx = hy = 0, H₁ ‡0 and the system is initially in equilibrium, i. e. M(0) = (0, 0, Mo). a.) (Free induction decay.) Obtain M(t) after a fast (π/2), radio-frequency pulse rotating the nuclear spins by angle π/2 around the y-axis. b.) (T₁ measurement.) Obtain M(t) after a fast (π), pulse. c.) Now we consider the case when he = ho coswt, hy = ho sin wt, Ho ‡ 0 and take the limit when yho 1/√T₁T₂ < yHo. Obtain in the leading approximation the steady-state solution M(t). From this solution obtain the energy absorption per unit time using the following relation: dE dH = -M dt dt
In 1946, Felix Bloch proposed the following set of equations to describe NMR relaxation: dM Mx/T2 My/T₂ = M x H - (3) dt (M₂-Mo)/T₁ where M = (Mx, My, M₂) is the total magnetization of nuclear spins in the system, H = (hx, hy, Ho) is the magnetic field, y is the gyromagnetic ratio, T₁ and T₂ are the longitudinal and the transverse relaxation times respectively, and Mo is the equilibrium magnetization in magnetic field Ho. Firstly, let us consider the case, where hx = hy = 0, H₁ ‡0 and the system is initially in equilibrium, i. e. M(0) = (0, 0, Mo). a.) (Free induction decay.) Obtain M(t) after a fast (π/2), radio-frequency pulse rotating the nuclear spins by angle π/2 around the y-axis. b.) (T₁ measurement.) Obtain M(t) after a fast (π), pulse. c.) Now we consider the case when he = ho coswt, hy = ho sin wt, Ho ‡ 0 and take the limit when yho 1/√T₁T₂ < yHo. Obtain in the leading approximation the steady-state solution M(t). From this solution obtain the energy absorption per unit time using the following relation: dE dH = -M dt dt