Sufficiency conditions for the validity of the j z-conserving coupled states approximation

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Abstract
The jz‐conserving coupled states (jzCCS) approximation is examined by beginning with the exact body frame Lippmann–Schwinger (BFLS) equation derived recently by Kouri, Heil, and Shimoni. It is shown how one may derive the jzCCS equation in such a way as to enable sufficiency conditions to be obtained for the validity of the jzCCS approximation. In addition, the method of derivation leads to a more general jzCCS approximation that includes states of definite parity. The ordinary jzCCS amplitude density equations result when one assumes that the even and odd parity amplitude densities are equal. The behavior of the jzCCs method for fixed total angular momentum J and varying energy E and for fixed E and varying J is discussed. In addition, the method enables us to show clearly, for fixed E and J, the roles played by the various regions of internuclear scattering distance. The resulting sufficiency conditions may be stated as follows: (a) When the turning point Rt is larger than the impact parameter X0, the jzCCS method will be accurate. (b) When the turning point is smaller than the impact parameter but the difference is not too large, then the accuracy deteriorates as X0−Rt increases. The jzCCS method depends on X0−Rt through the constants Z(i)J(jλ‖j0λ0), i=1,2, defined by Σλ′∫Rtx0dR R2 J̃λλ′Jj(kjR) ζ+j(jλ′‖j0λ0‖R) and Σλ′∫xRtdR R2 H̃λλ′Jj(kjR) ζ+J(jλ′‖j0λ0‖R), and if these constants are sufficiently small, then the jzCCS method will be accurate. Conversely, if Z(i)J(jλ‖j0λ0) are large, or the potential is sufficently long ranged [even though the Z(i)J(jλ‖j0λ0) are small] so as to build up significant λ transitions, then the jzCCS method may break down. However, breakdown is initiated in the short range (Rt⩽R⩽X0) region. (c) The jzCCS method works least well for transitions j→j′ where both j, j′ are large since then, the impact parameter X0 (approximately equal to (J+J+1/2)/kj) is very large and X0−Rt can be quite large. (d) Transitions are poorly treated by the jzCCS close to their threshold because X0≃ (J+j+1/2)/kj will be very large. We emphasize that our criterion (a) is a sufficient condition only. Also, even though condiitons (b) – (d) apply good results may still be obtained owing to fortuitous cancellation of errors.