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rewritten as a sum over g this divergence is concentrated in the g =0 term, which can
be "thrown away". More correctly, it can be explained away as an artifact of
neglecting the 0 term of Eq. (1), which also diverges as N -# - and would cancel
the g = 0 term. Vortices of the same sign require a rigid-body counter-rotation of the
background fluid to contain them, i.e., a rotating reference frame is necessary for
them to appear stationary. In the language of charges (the system here is equivalent
to a collection of line charges of the same sign) a neutralizing background charge of
opposite sign is necessary to prevent divergence of the electrostatic energy density.
iv. Application of Glasser's method for phase modulated sums.[4] Using Jacobi's
transformation from the theory of theta functions, Glasser shows how to reduce sums
of the form .' kx
k k2n
v. Constraint of constant vortex density. The above ingredients give an expression for
E that contains logL1 (or logL2) as a parameter. To fix the density, L1 must scale
properly with the number of vortices per unit cell J. This ensures that the same
energy is obtained for equivalent lattices, i.e., a lattice whose unit cell is doubled,
with no other changes, must have the same energy density as before. Multiply L t
and L2 by a scaling parameter q and set the density to unity to obtain invariance for
equivalent lattices,
1 = qL1 = . (9)
qL1 qL2 sin 0v L2sin
The final result for the energy density is
E = r P sin P- log l2 s J - log H (0,0,m)
P 6 - -I
+ si. r ( I (I 2 z + H (z ,zs,n (10)
_ _ _ _ L 42sn,
where zi.,; - 2,,(ra ro= .L L 2 = 12 and
L 2 2l
H(zt ,m) a 1 - 2e- P 12 + 2m sin 0 cos (zt + 2mcos ml + e' 2.r2+ 2m sin .(11)
E is the relative energy density of lattices containing fixed ratios of vortex species v:'th
fixed strengths. To compare the energies of lattices with different vortex species or species
ratios requires assumptions or physical information about the self-energies of the vortices.
What makes Eqs. (10) and (11) useful for numerical evaluation is the fast convergence of
the function H(zl,z2,m); m 5 is usually sufficient. Explicit periodicity in the y-direction
has been removed for simplification, but is retained in the x-direction. Also, it is
convenient for calculating the partial derivatives of E (used in the conjugate gradient
method) to change the unit cell variables p and 0 to a = 2n sin0/p and z - 2t cos0 /p.
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Capmbell, Laurence J. Vortex lattices in theory and practice, article, January 1, 1988; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc1189559/m1/6/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.