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

Abstract

The j_z‐conserving coupled states (j_zCCS) 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 j_zCCS equation in such a way as to enable sufficiency conditions to be obtained for the validity of the j_zCCS approximation. In addition, the method of derivation leads to a more general j_zCCS approximation that includes states of definite parity. The ordinary j_zCCS amplitude density equations result when one assumes that the even and odd parity amplitude densities are equal. The behavior of the j_zCCs 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 R_t is larger than the impact parameter X₀, the j_zCCS 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−R_t increases. The j_zCCS method depends on X₀−R_t through the constants Z⁽ⁱ⁾_J(jλ‖j₀λ₀), i=1,2, defined by Σ_λ′∫_Rt^x₀dR R² J̃^λλ′_Jj(k_jR) ζ⁺_j(jλ′‖j₀λ₀‖R) and Σ_λ′∫^x_RtdR R² H̃^λλ′_Jj(k_jR) ζ⁺_J(jλ′‖j₀λ₀‖R), and if these constants are sufficiently small, then the j_zCCS 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 j_zCCS method may break down. However, breakdown is initiated in the short range (R_t⩽R⩽X₀) region. (c) The j_zCCS 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)/k_j) is very large and X₀−R_t can be quite large. (d) Transitions are poorly treated by the j_zCCS close to their threshold because X₀≃ (J+j+1/2)/k_j 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.