By Philip George Burke

Commencing with a self-contained assessment of atomic collision thought, this monograph offers fresh advancements of R-matrix thought and its functions to a wide-range of atomic molecular and optical strategies. those advancements contain electron and photon collisions with atoms, ions and molecules required within the research of laboratory and astrophysical plasmas, multiphoton methods required within the research of superintense laser interactions with atoms and molecules and positron collisions with atoms and molecules required in antimatter reviews of medical and technologial value. simple mathematical effects and normal and accepted R-matrix laptop courses are summarized within the appendices.

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**Example text**

N . 217) If the trial function u t (r ) depends linearly on the parameters c1 , c2 , . . 217) are a set of n + 1 linear simultaneous equations which can be solved to yield these parameters. 212) to obtain an improved estimate for λ which is correct up to terms of second order in the error in the trial function. 218) where the symbol [λ ] means that this quantity is the variational estimate of λ . The corresponding phase shift, correct up to terms of second order, is then obtained from the variational estimate using the equation [λ ] = tan(δ − τ ) .

157) It follows that when A = 0 then = λ and K (k) = K λ (k). 118) by relating K λ (k) to the R-matrix on the boundary r = a. 116) with replaced everywhere by λ. 11). 158) is an analytic function of k 2 in the neighbourhood of threshold which is a real analytic function when λ is real. 158). 119) in the limit A → 0 so that τ → 0. 4 Effective Range Theory 33 An important feature of scattering by a dipole potential occurs for strong attractive potentials where A < − 14 (2 + 1)2 . 161) where Im λ can be positive or negative.

66) It follows from the asymptotic properties of the Gamma function that the Coulomb S-matrix is analytic in the entire complex k-plane except for poles where +1+iη = −n¯ with n¯ = 0, 1, 2, . . 42) we see that the corresponding values of k are given by kn¯ = −i Z1 Z2 , n¯ + + 1 n¯ = 0, 1, 2, . . 67) Thus for an attractive Coulomb potential (Z 1 Z 2 < 0) the poles of S c (k) lie on the positive imaginary axis of the complex k-plane. 65) that the wave function decays exponentially asymptotically and hence these poles correspond to the familiar bound states with energies En = − 1 Z 12 Z 22 , 2 n2 n = + 1, + 2, .