Nuclear magnetic resonance (NMR) spectroscopy uses the intrinsic magnetic moment that arises from the spin angular momentum of a spin-active nucleus. If the element of interest has a nuclear spin that is not 0, the nucleus may exist in different spin angular momentum states, where the energy of these states can be affected by an external magnetic field. For a spin I = 1 2 {\displaystyle I={\frac {1}{2}}} nucleus, there are two spin states of consideration: spin up and spin down. The presence of an external magnetic field causes the two states to separate in energy, with the relative ordering depending on the gyromagnetic ratio of the nucleus. The sample's bulk magnetization, that is, the sum of the total magnetic moments of all nuclei in the sample, will determine the strength of the NMR signal. In addition, the energy of the applied radio frequency used in NMR must be consistent with the energy difference between the spin states.
Eigenvalues of nuclear spin states The Hamiltonian operator corresponds with the total energy of a quantum system. The Hamiltonian for a single spin 1/2 nucleus in the presence of an applied magnetic field B 0 {\displaystyle B_{0}} aligned along the z axis is
H ^ = − γ B 0 I ^ z {\displaystyle {\hat {H}}=-\gamma B_{0}{\hat {I}}_{z}}
where γ {\displaystyle \gamma } is the gyromagnetic ratio and I ^ z {\displaystyle {\hat {I}}_{z}} is the z-component of the nuclear spin angular momentum. Denoting | α ⟩ {\displaystyle |\alpha \rangle } the spin state with m s = + 1 2 {\displaystyle m_{s}=+{\frac {1}{2}}} and | β ⟩ {\displaystyle |\beta \rangle } the spin state with m s = − 1 2 {\displaystyle m_{s}=-{\frac {1}{2}}} , the eigenvalues of the Hamiltonian are
H ^ | α ⟩ = − 1 2 γ ℏ B 0 | α ⟩ , H ^ | β ⟩ = + 1 2 γ ℏ B 0 | β ⟩ {\displaystyle {\hat {H}}|\alpha \rangle =-{\frac {1}{2}}\gamma \hbar B_{0}|\alpha \rangle ,\quad {\hat {H}}|\beta \rangle =+{\frac {1}{2}}\gamma \hbar B_{0}|\beta \rangle }
We can see that the ordering of energies for the two states depends on the gyromagnetic ratio of the nucleus. For example, for 1 H {\displaystyle ^{1}{\textrm {H}}} nuclei ( γ = 2.675 × 10 8 {\displaystyle \gamma =2.675\times 10^{8}} s -1 T -1 ) the | α ⟩ {\displaystyle |\alpha \rangle } states are lower in energy than | β ⟩ {\displaystyle |\beta \rangle } , whereas the situation is reversed for 15 N {\displaystyle ^{15}{\textrm {N}}} ( γ = − 2.711 × 10 7 {\displaystyle \gamma =-2.711\times 10^{7}} s -1 T -1 ).
Two spins without coupling If there are two spin states, then we have to change the Hamiltonian in such a way that it accommodates both the spin states. Ĥtwo spins, no coupling = v0,1Î1Z + v0,2Î2Z v0,1 is the Larmor frequency of first spin and v0,2 is the Larmor frequency of second spin. Similarly Î1Z is the z-component of angular momentum operator of first spin and Î2Z is the z-component of angular momentum operator of first spin. Here in this case coupling is not considered. Here while considering the wave function we have to look into both spin states of both spin 1 and 2. The spin up state is represented by α and spin down is β. The wave functions hence will have four combinations as below. ψα,1 ψα,2 = αα ψα,1 ψβ,2 = αβ ψβ,1 ψα,2 = βα ψβ,1 ψβ,2 = ββ Applying these combinations into the two spin Hamiltonian above will give the eigenvalue which is the energy state. This is tabulated below.
In general, the energy level (eigenvalue) can be written as; Em = m1v0,1 + m2v0,2
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