Radiation damping: the mathematics, and its application in indirect detection of spins

David Rourke

Radiation damping occurs when precessing spins generate a magnetic field, which reacts back on them, and causes them to move back to their lowest energy state, emitting a ``radiation damping burst'' of rf energy. This effect is becoming increasingly important with higher Q coils, and stronger magnetic fields, particularly with experiments involving laser-polarized noble gases.

The dynamics of the spin system under radiation damping can be derived using energy considerations, or by constructing a circuit equation [1,2]. For the simplest case of a single spin species with a precisely known free-precession frequency, the system evolves according to the third order nonlinear Bloch equations

$\displaystyle {\frac{\partial}{\partial t}}$ $\displaystyle \begin{pmatrix}m\\ \bar m\\ m_z\end{pmatrix}$ = $\displaystyle \begin{pmatrix}i\omega_3 m -i\omega m_z\\ -i\omega_3 \bar m +i\bar\omega m_z\\ -\frac{i}{2}\bar \omega m +\frac{i}{2}\omega \bar m \end{pmatrix}$ + $\displaystyle {\frac{1}{m_0 \tau_r}}$ $\displaystyle \begin{pmatrix}- m m_z\\ - \bar m m_z\\ m \bar m\end{pmatrix}$ .

(1)

Here, the system is being described by a magnetization vector (m_x, m_y, m_z). m is the complex transverse magnetization, m = m_x + im_y. The magnetization has magnitude m₀ = |m|. The external applied driving field is (in units of angular frequency) $\omega$ = $\omega_{1}^{}$ + i $\omega_{2}^{}$ . The spin's resonance offset is $\omega_{3}^{}$ (also in units of angular frequency), and $\tau_{r}^{}$ is the radiation damping time constant of the system. All quantities here are, in general, time-varying. The complex conjugate of any quantity z is denoted by $\bar{z}$ .

We have recently shown [3,4] that this nonlinear system is essentially no more complex than the usual linear Bloch equations without radiation damping. Hence we are able to apply techniques usually applied to the linear system, such as the design of rf pulses, or the determination of the effect of stochastic fields.

An area that is being investigated is whether the radiation damped system can act as a sensitive indirect detector of dilute spins. This is analogous to an effect in optics, where Rydberg atoms exhibit a ``superfluorescent burst'', the timing of which is dependent on the strength of the radiation field that they experience [5]. In the same way, the timing of the radiation damping burst for nuclear spins depends on their magnetic environment, including any effects due to other spins being present.

Important generalizations to system (1) that are being investigated include the effect of imprecise knowledge of the free-precession frequency of the spins (i.e., inhomogeneous broadening or T₂ effects), and the effects of T₁ and T₂ relaxation.

Bibliography

1: N. Bloembergen and R. V. Pound.
Radiation damping in magnetic resonance experiments.
Phys. Rev., 95(1):8-12, 1954.
2: Stanley Bloom.
Effects of radiation damping on spin dynamics.
J. Appl. Phys., 28(7):800-805, 1957.
3: David E. Rourke and Matthew P. Augustine.
Exact Linearization of the Radiation-Damped Spin System.
Phys. Rev. Lett., 84(8):1685-1688, 2000.
4: David E. Rourke.
Solutions and Linearization of the Nonlinear Dynamics of Radiation Damping.
Concepts Magn. Reson., 14(2):112-129, 2002.
5: M. Gross, P. Goy, C. Fabre, S. Haroche, and J. M. Raimond.
Maser oscillation and microwave superradiance in small systems of Rydberg atoms.
Phys. Rev. Lett., 43(5):343-346, 1979.

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David Rourke 2002-02-19