  We compute the crossover exponent associated with the spin-3 operator in
three-dimensional O(N) models. A six-loop field-theoretical calculation in the
fixed-dimension approach gives $\phi_3 = 0.601(10)$ for the experimentally
relevant case N=2 (XY model). The corresponding exponent $\beta_3 = 1.413(10)$
is compared with the experimental estimates obtained in materials undergoing a
normal-incommensurate structural transition and in liquid crystals at the
smectic-A--hexatic-B phase transition, finding good agreement.
