Magnetic excitations and exchange interactions in the spin-$\frac{1}{2}$ two-leg ladder compound ${\mathrm{La}}_6{\mathrm{Ca}}_8{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$

Using a pulsed neutron source we have been able to observe inelastic signals at high energies from the ladders of ${\mathrm{La}}_6{\mathrm{Ca}}_8{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}.$ These signals constitute the magnon branch $(\sim 30<~E<~\sim 190$ meV) for the lowest triplet excitation. A fit to a theory including a coupling along the leg ${(J}_{leg})$ and rung ${(J}_{\mathrm{rung}})$ alone gives the value $J_{leg}{/J}_{\mathrm{rung}}\sim 2$ that is not consistent with expectations from a geometrical consideration and the exact diagonalization calculation. This inconsistency is removed when our data are fitted with $J_{leg}{=J}_{\mathrm{rung}}=110\mathrm{}$ meV and a four-spin exchange interaction, $J_{\mathrm{ring}}=16.5\mathrm{}$ meV.