4.2: Hyperfine Hamiltonian

We consider the interaction of a single electron spin \(S\) with a single nuclear spin \(I\) and thus drop the sums and indices \(k\) and \(i\) in \(\hat{\mathcal{H}}_{\mathrm{HFI}}\) in Eq. (2.4). In general, all matrix elements of the hyperfine tensor \(\mathbf{A}\) will be non-zero after the Bleaney transformation to the frame where the electron Zeeman interaction is along the \(z\) axis (see Eq. 3.5). The hyperfine Hamiltonian is then given by

\[\begin{aligned} \hat{\mathcal{H}}_{\mathrm{HFI}}=&\left(\begin{array}{lll} \hat{S}_{x} & \hat{S}_{y} & \hat{S}_{z} \end{array}\right)\left(\begin{array}{ccc} A_{x x} & A_{x y} & A_{x z} \\ A_{y x} & A_{y y} & A_{y z} \\ A_{z x} & A_{z y} & A_{z z} \end{array}\right)\left(\begin{array}{c} \hat{I}_{x} \\ \hat{I}_{y} \\ \hat{I}_{z} \end{array}\right) \\ =& A_{x x} \hat{S}_{x} \hat{I}_{x}+A_{x y} \hat{S}_{x} \hat{I}_{y}+A_{x y} \hat{S}_{x} \hat{I}_{z} \\ &+A_{y x} \hat{S}_{y} \hat{I}_{x}+A_{y y} \hat{S}_{y} \hat{I}_{y}+A_{y z} \hat{S}_{y} \hat{I}_{z} \\ &+A_{z x} \hat{S}_{z} \hat{I}_{x}+A_{z y} \hat{S}_{z} \hat{I}_{y}+A_{z z} \hat{S}_{z} \hat{I}_{z} \end{aligned}\]

Note that the \(z\) axis of the nuclear spin coordinate system is parallel to the magnetic field vector \(\vec{B}_{0}\) whereas the one of the electron spin system is tilted, if \(g\) anisotropy is significant. Hence, the hyperfine tensor is not a tensor in the strict mathematical sense, but rather an interaction matrix.

In Eq. (4.7), the term \(A_{z z} \hat{S}_{z} \hat{I}_{z}\) is secular and must always be kept. Usually, the high-field approximation does hold for the electron spin, so that all terms containing \(\hat{S}_{x}\) or \(\hat{S}_{y}\) operators are non-secular and can be dropped. The truncated hyperfine Hamiltonian thus reads

\[\hat{\mathcal{H}}_{\mathrm{HFI}, \text { trunc }}=A_{z x} \hat{S}_{z} \hat{I}_{x}+A_{z y} \hat{S}_{z} \hat{I}_{y}+A_{z z} \hat{S}_{z} \hat{I}_{z}\]

The first two terms on the right-hand side can be considered as defining an effective transverse coupling that is the sum of a vector with length \(A_{z x}\) along \(x\) and a vector of length \(A_{z y}\) along \(y\). The length of the sum vector is \(B=\sqrt{A_{z x}^{2}+A_{z y}^{2}}\). The truncated hyperfine Hamiltonian simplifies if we take the laboratory frame \(x\) axis for the nuclear spin along the direction of this effective transverse hyperfine coupling. In this frame we have

\[\hat{\mathcal{H}}_{\text {HFI,trunc }}=A \hat{S}_{z} \hat{I}_{z}+B \hat{S}_{z} \hat{I}_{x}\]

where \(A=A_{z z}\) quantifies the secular hyperfine coupling and \(B\) the pseudo-secular hyperfine coupling. The latter coupling must be considered if and only if the hyperfine coupling violates the high-field approximation for the nuclear spin (see Chapter 6).

If \(g\) anisotropy is very small, as is the case for organic radicals, the \(z\) axes of the two spin coordinate systems are parallel. In this situation and for a hyperfine tensor with axial symmetry, \(A\) and \(B\) can be expressed as

\[\begin{aligned} &A=A_{\mathrm{iso}}+T\left(3 \cos ^{2} \theta_{\mathrm{HFI}}-1\right) \\ &B=3 T \sin \theta_{\mathrm{HFI}} \cos \theta_{\mathrm{HFI}} \end{aligned}\]

where \(\theta_{\text {HFI }}\) is the angle between the static magnetic field \(\vec{B}_{0}\) and the symmetry axis of the hyperfine tensor and \(T\) is the anisotropy of the hyperfine coupling. The principal values of the hyperfine tensor are \(A_{x}=A_{y}=A_{\perp}=A_{\text {iso }}-T\) and \(A_{z}=A_{\|}=A_{\text {iso }}+2 T\). The pseudo-secular contribution \(B\) vanishes along the principal axes of the hyperfine tensor, where \(\theta_{\mathrm{HFI}}\) is either \(0^{\circ}\) or \(90^{\circ}\) or for a purely isotropic hyperfine coupling. Hence, the pseudo-secular contribution can also be dropped when considering fast tumbling radicals in the liquid state. We now consider the point-dipole approximation, where the electron spin is well localized on the length scale of the electron-nuclear distance \(r\) and assume that \(T\) arises solely from through-space interactions. This applies to hydrogen, alkali and earth alkali ions. We then find

\[T=\frac{1}{r^{3}} \frac{\mu_{0}}{4 \pi \hbar} g_{e} \mu_{\mathrm{B}} g_{\mathrm{n}} \mu_{\mathrm{n}}\]

For the moment we assume that the pseudo-secular contribution is either negligible or can be considered as a small perturbation. The other case is treated in Chapter 6 . To first order, the contribution of the hyperfine interaction to the energy levels is then given by \(m_{S} m_{I} A\). In the EPR spectrum, each nucleus with spin \(I\) generates \(2 I+1\) electron spin transitions with \(\left|\Delta m_{S}\right|=1\) that can be labeled by the values of \(m_{I}=-I,-I+1, \ldots I\). In the nuclear frequency spectrum, each nucleus exhibits \(2 S+1\) transitions with \(\left|\Delta m_{I}\right|=1\). For nuclear spins \(I>1 / 2\) in the solid state, each transition is further split into \(2 I\) transitions by the nuclear quadrupole interaction. The contribution of the secular hyperfine coupling to the electron transition frequencies is \(m_{I} A\), whereas it is \(m_{S} A\) for nuclear transition frequencies. In both cases, the splitting between adjacent lines of a hyperfine multiplet is given by \(A\).