Muon-Pair Separation Measurements and Comparison with Transverse-Momentum Models

In principle, there exists a second invariant in the fourth-order derivatives of ~. However, in a system with translational invariance this term is identical with that given in (1). See K. G. Wilson and J. Kogut, Phys. Bep. 12C, 75 (1974); S.-k. Ma, Bev. Mod. Phys. 45, 589 (1973); M. E. Fisher, Bev. Mod. Phys. 46, 597 (1974); and A. Aharony, in "Phase Transitions and Critical Phenomena, " edited by C. Domb and M. S. Green (Academic, New York, to be published), Uol. 6, for excellent general introductions to the renormalization group and its application to critical phenomena. This point divides the ~ line into two segments (see Fig. 1). The Lifshitz condition restricts the representations to which the order parameter may belong on only one of these segments. See E. M. Lifshitz, J. Phys. (Moscow) 6, 61 (1942); L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, New York, 1968), 2nd ed. , Chap. XIV; I. E. Dzyaloshinski, Zh. Eksp. Teor. Fiz. 46, 1420 (1964) (Sov. Phys. JETP 19, 960 (1964)]; S. Goshen, D. Mukamel, and S. Shtrikman, lnt. J. Magn. 6, 221 (1974). E. K. Riedel and E. J. %'egner, Phys. Bev. Lett. 29, 349 (1972); M. J. Stephen and J. L. McCauley, Jr., Phys. Lett. 44A, 89 (1973); T. S. Chang, G. F. Tuthill, and H. E. Stanley, Phys .Bev B 9, 4882 (1974). ~We use the term helicoidal to include a variety of periodic structures, such as screw and cone spirals and sinusoids. Note that all the structures we are considering are not restricted by the system Hamiltonian to be either rightor left-handed. They thus differ from Dzyaloshinski-type spirals fL. L. Liu, Phys. Bev. Lett. 31, 459 (1973)]. For a discussion of this basic difference in a somewhat different context, see P. G. de Gennes, Mol. Cryst. Liq. Cryst 7,. 325 (1969). A. Yoshimori, J. Phys. Soc. Jpn. 14, 807 (1959). J. Uillain, J. Chem. Phys. Solids 11, 303 (1959). T. Kaplan, Phys. Rev. 116, 888 (1959), and 124, 329 (1961). B. Bausch, Z. Phys. 254, 81 (1972); D. J. Amit, J. Phys. C: Solid State Phys. 7, 3369 (1974). The system anisotropy will, in general, affect other terms in (2) [and also in (5)1. This will be discussed elsewhere. The classical correlation-function exponents for a I,ifshitz point are ~„=, ~g, =-, ', g)4=pe, =o. Alternatively, Eq. (4) can be obtained by examining the divergence of the lowest-order diagrams contributing to the four-point function. To first order in &q the critical exponent for a uniaxial Ising-type Lifshitz point (i.e., d=3, m=n =1) are 0g4~ 0, v&4 = v/2/2 -0.31, &j =p) —0.25, l'( = 1.25, &g =5 0. M. E. Fisher and D. B. Nelson, Phys. Bev. Lett. 32, 1350 (1974). D. E. Cox, lEEE Trans. Magn. 8, 161 (1972) . G. H. Lander, M. H. Mueller, and J. F. Beddy, Phys. Bev. B 6, 1880 (1972).