Title: Group theoretic approach to rf-coil design

A group-theoretic method is presented for finding closed-form solutions for the normal mode frequencies for coils possessing at least a modicum of symmetry. The coil symmetry allows one to block-diagonalize Kirchhoff's laws. One may then find the normal modes and their frequencies for each block independently. For many coils of interest in MR applications, the frequencies can therefore be expressed in closed form. Such solutions allow one to consider design questions about mode degeneracy of particular relevance to the parallel imaging community. Also, recent work has shown the value of having such solutions in developing coil 'simulators.'
To demonstrate the utility of this method, an eight-rung three-ring birdcage was modeled and constructed. This coil has been successfully used as an eight channel SENSE coil. A theoretical approach based on group theory was used to calculate in closed-form the seventeen normal mode frequencies of the coil. The symmetry of the system is a key to finding analytical expressions for the normal mode frequencies. Good agreement was found between the frequency measurements and calculations. The method thus represents an effective approach to symmetric systems with larger degrees of freedom.