Theory of slow-atom collisions

Transcription

1 PHYSICAL REVIEW A VOLUME 54, NUMBER 3 SEPEMBER 1996 heory of slow-atom collisions Bo Gao Department of Physics and Astronomy, University of oledo, oledo, Ohio Received 23 January 1996 A general theory of slow-atom collisions is presented with special emphasis on the effects of nuclear statistics and atomic fine and/or hyperfine structures. Symmetry properties of the collision complex and correlations between the molecular states and the separated-atom states are carefully examined. he frame transformations between various angular momentum coupling schemes are derived, which, in combination with the multichannel quantum defect theory, provides a solid foundation for the computation and the physical interpretation of slow-atom collision processes. he theory reduces to those of Stoof et al. Phys. Rev. B 38, and Zygelman et al. Phys. Rev. A 49, ; 50, in their respective ranges of validity. S PACS numbers: 34.10x, Pj I. INRODUCION he importance and the novelty of cold-atom collisions are well documented. A review of the theoretical aspects of this problem can be found in 1. he experimental aspects are reviewed in 2. he recent discovery of Bose-Einstein condensation in a weakly interacting atomic vapor 3 is a further motivation for a careful examination of this subject. Compared to atomic collisions at room temperatures or above, cold-atom collisions have a number of distinctive features. First, long-range interactions are important 4. Second, hyperfine structures, if there are any, play a significant role in determining the collision dynamics 2,5. his is easily understood. For a typical ground hyperfine splitting of 1 GHz, the atomic kinetic energy becomes comparable to the hyperfine structure at around 0.1 K, which is where the hyperfine effects become important. hird, if the collision happens in a near-resonant laser field, the effects of spontaneous decay and optical pumping also become important 6,1,2. Lastly, cold-atom collisions are sensitive to the potential, i.e., a small error in the potential can lead to a large uncertainty in the cross section 4,7,8. his is, for the most part, a general feature of slow-atom collisions, due mainly to the fact that atoms are much heavier than electrons see Sec. III A. here are many theories of slow-atom collisions But the effects of the atomic hyperfine structure and the nuclear statistics have not been treated in a fully systematic manner. It should be pointed out that the notion of nuclear statistics is more fundamental than the notion of atomic statistics. For example, is a 85 Rb atom in the ground F3 state distinguishable from a 85 Rb atom in the ground F2 state? What about a 85 Rb ion and a 85 Rb atom? hese are important questions related to the quantum statistics of an atomic vapor. We will get back to them in Sec. IV. he point to be made here is that at a more fundamental level, the task is to ensure that the total wave function has proper symmetry under the exchange of nuclei. he theory of Stoof and Verhaar and their collaborators 20,21 has captured almost all the essential physics of cold collisions between alkali-metal atoms in their ground states. However, owing to the effective-atom nature of their theory, one basically has to develop a different theory for atoms with different angular momenta or for the same atoms in different electronic states. he theory presented in this paper is more general. For example, it is capable of treating resonant charge exchange processes such as Rb Rb RbRb, which have not been studied at cold temperatures achievable in atom traps. It is also capable of treating resonance exchange processes such as Rb*Rb RbRb*, provided that the collision proceeds sufficiently fast so that spontaneous emission and laser pumping if the collision happens inside a laser field can be ignored during the collision. Furthermore, as the success of previous works of Mies and Julienne 17,22 and Gribakin and Flambaum 4,8,23 indicates, the effects of long-range interactions can be treated most naturally and effectively using the multichannel quantum defect theory MQD he work of Stoof and Verhaar and their collaborators 20,21 has not taken full advantage of it. By incorporating MQD and the frame transformation technique 29 33, we provide a method for quickly obtaining complete information about hyperfine collisions from existing single channel calculations 34,8,23. he method can be refined systematically, and it leaves the door open for an R-matrix type of treatment of short-range interactions, which may be needed if highly accurate results are desired. Our theory of a slow-atom collision between atoms A and B is presented in Sec. II and is divided into subsections according to the classification of a collision into three different categories. Section II A deals with the case of Z A Z B, which represents a collision between two different atoms, such as Rb Ar. It is of interest in the understanding of optical properties of gas cells and atomic spectroscopy. Section II B deals with the case of a collision between two similar atoms whose nuclei have the same charge (Z A Z B ), but are otherwise distinguishable due to either different spin or different mass. An example of this type of collision is 85 Rb 87 Rb. Section II C deals with the collision between two atoms with identical nuclei, such as 85 Rb 85 Rb and /96/543/202218/$ he American Physical Society

2 54 HEORY OF SLOW-AOM COLLISIONS Rb 85 Rb. It is our hope that by putting these different cases together in a concise fashion, their similarities and differences will become more transparent. Various angular momentum coupling schemes are discussed in Sec. II D and Appendix A. Frame transformations and the MQD are discussed in Sec. II E. II. HEORY We consider a collision between two atoms A and B in a pair of LS coupled manifolds ( 1 L 1 S 1 ) and ( 2 L 2 S 2 ). he atoms are identified by their respective nuclei. If the two nuclei are identical, this identification is facilitated by labeling one of them A and the other one B. he Hamiltonian describing the two colliding atoms can be written as H 2 2 R 2 H BO H f H hf, 1 where is the reduced mass of the two atoms. H f represents the spin-orbital interactions, and H hf represents the hyperfine interactions. Equation 1 serves to define the adiabatic Born- Oppenheimer Hamiltonian H BO to be used in this paper 41. For a collision between two atoms having fine and/or hyperfine structures, a fragmentation channel 42 having a total angular momentum and a projection on a space-fixed axis M is specified by a set of quantum numbers such as 1 L 1 S 1 J 1 I 1 F 1 A 2 L 2 S 2 J 2 I 2 F 2 B Fl, where F results from the coupling of F 1 and F 2 ; l is the relative orbital angular momentum of the centers of mass of the two atoms. his set of quantum numbers specifies an angular momentum coupling scheme which diagonalizes both the spin-orbital and the hyperfine interactions at large interatomic separations. We will identify this coupling scheme as J 1 F 1 J 2 F 2 F,orFF coupling for short. here are other possible representations of channels corresponding to different angular momentum coupling schemes. One of them is specified by quantum numbers such as 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B LlLSKI, where LLl, and KLS. hus L represents the total orbital angular momentum, and K represents the total angular momentum excluding nuclear spin. his coupling scheme will be identified by LLSKI, or LS coupling for short. Another channel representation is specified by a set of quantum numbers such as 1 L 1 S 1 J 1 I 1 A 2 L 2 S 2 J 2 I 2 B JlKI, where JJ 1 J 2, KJl, and KI. his coupling scheme will be identified by J 1 J 2 JKI, or JJ coupling for short. he total wave function for the collision complex having definite and M can be expanded in terms of adiabatic channel functions as P b b P RG b P R/R, 2 where the summation is over one set of channels. I will use indices a and b to refer to any set of channels, i and j to refer to the fragmentation channels. Indices and will be reserved for condensation channels 43. he channel functions a P (R) contain both the electronic wave function and the angular part of the relative motion of the centers of mass. hey have different symmetry properties for three classes of collisions mentioned earlier, and will be discussed in detail in Secs. II A II C. Substituting Eq. 2 into the Schrödinger equation, H P E P, and making use of the orthogonality properties of the channel functions, we obtain, upon ignoring both the radial and the angular nonadiabatic coupling terms 41, a set of closecoupling equations: 2 d 2 2 dr 2 l al a 1 2 2R 2 EG P a R b in which BO f RV ab RV hf ab RG P b R0, V ab V BO ab R P a H BO P b, f V ab R P a H f P b, V hf ab R P a H hf P b. 3a 3b 3c 3d We look for solutions which satisfy the physical boundary conditions i P R P j f j ji g j K ji jo E/R, where the sum over j is over the open fragmentation channels only. Functions f j and g j are determined by the asymptotic behavior of V BO ij (R). For a collision between two neutral atoms, they are given explicitly by f k j l j R 1 2 1/2 k j k j Rj lj k j R, 5a g k j l j R 1 2 1/2 k j k j Ry lj k j R, 5b where j l and y l are the spherical Bessel functions 44, and we have chosen the normalization constants such that the wave function defined by Eq. 4 is normalized per unit energy. he collision cross sections and other physical observables can be extracted from K () ji (E) in a pretty standard 4

3 2024 BO GAO 54 fashion see Appendix B. We note that unless V hf 0, the physical K matrix has to be defined in the fragmentation (FF coupled channels. A. he case of Z A Z B he adiabatic Born-Oppenheimer states and potentials are defined as the eigenfunctions and eigenvalues of H BO at a fixed interatomic separation R: H BO M L Rˆ SM S Rˆ;R E ML SRM L RˆSM S Rˆ;R, where the magnetic quantum numbers M L and M S are both quantized along the molecular axis RR A R B. hese adiabatic states correspond one to one the Wigner-Witmer rule 45,46 to the properly antisymmetrized two-atom states defined by 16 1 L 1 S 1 A 2 L 2 S 2 B LM L SM S N 1N 2! N 1!N 2! A L 1 M L1,L 2 M L2 LM L M L1 M L2 M S1 M S2 S 1 M S1 S 2 M S2 SM S 1 L 1 M L1 S 1 M S1 A 2 L 2 M L2 S 2 M S2 B, where N 1 and N 2 are numbers of electrons corresponding to states 1 L 1 M L1 S 1 M S1 and 2 L 2 M L2 S 2 M S2, respectively. A is an antisymmetrization operator that operates on all electrons. For the sake of simplified notation, the normalization constant and the operator A will be dropped in further references to the two-atom states 47. his one-to-one correspondence implies that there exist quasimolecular states 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R, which are linear combinations of Born-Oppenheimer states, that satisfy 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R R 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ. Here the state 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ is a two-atom state as defined by Eq. 7 except that it is quantized along the molecular axis. hey are related to each other through the relationship 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ M L M S L D ML M L S,,0D M,,0 S M S 1 L 1 S 1 A 2 L 2 S 2 B LM L SM S, where the angles and specify the direction of the molecular axis RR A R B in a space-fixed frame, which by definition always points from the nucleus B to the nucleus A. he symmetry properties of the quasimolecular states defined by Eq. 8 can be derived from their corresponding two-atom states. It can be shown that the two-atom states have the following symmetry properties 48: ˆ e 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ P 1 P 2 1 LM L 1 L 1 S 1 A 2 L 2 S 2 B L M L RˆSM S Rˆ, Pˆ 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ P 1 P 2 1 LSM L M Se im L M S 10 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ, 11 where ˆ e refers to the reflection of electronic coordinates in a plane containing the molecule axis, and Pˆ refers to the total parity operation which inverts both the electronic and the nuclear coordinates. P 1 and P 2 are the parities of LS manifolds ( 1 L 1 S 1 ) and ( 2 L 2 S 2 ), respectively 49. Inarriving at Eq. 11, we have used J D MM,,0e im 1 JM D MM,,0. 12 Since molecular interactions cannot break these symmetries 50, the quasimolecular states defined by Eq. 8, and the corresponding Born-Oppenheimer states from which they are constructed, must also have the same symmetry, i.e., ˆ e 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R P 1 P 2 1 LM L 1 L 1 S 1 A 2 L 2 S 2 B L M L RˆSM S Rˆ;R, Pˆ 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R P 1 P 2 1 LSM L M Se im L M S 1 L 1 S 1 A 2 L 2 S 2 B L M L RˆSM S Rˆ;R. J Specifically, Eq. 13 implies that a quasimolecular state with M L 0 is an eigenstate of ˆ e with an eigenvalue e P 1 P 2 1 L. 15 Other symmetry properties of the quasimolecular states to be discussed later are all derived this way i.e., from their corresponding two-atom states. he quasimolecular state defined by Eq. 8 is generally a linear combination of Born-Oppenheimer states having the same M L, S, and M S, and the symmetry properties as specified by Eqs. 13 and 14. If there is only one such Born-

4 54 HEORY OF SLOW-AOM COLLISIONS 2025 Oppenheimer BO state, then the two have to be equal up to a global phase factor. If there is more than one BO state satisfying the same criteria, further disentanglement can be achieved by comparing additional symmetry properties, if there are any see, e.g., Sec. II B, or by comparing the asymptotic forms of E M L S(R) with the asymptotic potentials calculated using the two-atom states 18. We are now in the position to derive the fragmentation (FF coupled channel functions defined by the desired asymptotic behavior: P 1 L 1 S 1 J 1 I 1 F 1 A 2 L 2 S 2 J 2 I 2 F 2 B Fl R M 1 M 2 M F m l F 1 M 1,F 2 M 2 FM F FM F,lm l 1 L 1 S 1 J 1 I 1 F 1 M 1 A 2 L 2 S 2 J 2 I 2 F 2 M 2 B Y lml,, 16 which diagonalizes both the asymptotic spin-orbital and the asymptotic hyperfine interactions, and in which all magnetic quantum numbers are quantized along a space-fixed axis. he task is to find a proper superposition of the quasimolecular states, which are linear combinations of the adiabatic Born-Oppenheimer states, such that the desired asymptotic behavior is satisfied for all orientations. From Eqs. 7 9, one can show that this proper superposition is given by P J1 1,J L 1 S 1 J 1 I 1 F 1 A 2 L 2 S 2 J 2 I 2 F 2 B Fl 2,F 1,F 2 1/2 LSJI allm L,S,J,I 1/2 LM L,SM S JM J JM J,IM I F i M F FM F,lm l L 1 L 2 L J 1 J 2 JJ1 J2 J J* I 1 I 2 I MJ M J,,0Y lml, F 1 F 2 FD 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;RI 1 A I 2 B IM I. 17 Not only does this channel function have the desired a- symptotic behavior characterized by Eq. 16, but one can also show from Eqs. 12 and 14 that it is an eigenstate of the total parity operator with an eigenvalue P P 1 P 2 1 l. 18 With channel functions given in terms of the quasimolecular states, we can now express the potential V BO ij (R) that enters the close-coupling equation 3 in terms of the adiabatic Born-Oppenheimer potentials. Some calculations lead to i M i P i 1 L 1 S 1 J 1i I 1 F 1i A 2 L 2 S 2 J 2i I 2 F 2i B F i l i j M j P j HBO 1 L 1 S 1 J 1j I 1 F 1j A 2 L 2 S 2 J 2j I 2 F 2j B F j l j i j Mi M j Pi P j 1 F i F jj 1i,J 2i,F 1i,F 2i,F i,l i,j 1j,J 2j,F 1j,F 2j,F j,l j 1/2 M L M S LSKIJi 1 J j J il,s,k,i,j i,j j J j l i K J i I F i i l j K J j I F j j L 1 L 2 L J 1i J 2i J il1 L2 L J2i Ji J2j Jj I 1 I 2 I I 1 I 2 I J 1j J 2j J jj1i F 1i F 2i F ij1j F 1j F 2j F j L S J i M L M S M L M S L S J j M L M S M L M S J i l i K M L M S 0 M L M S J j l j K M L M S 0 M L M SE ML SR, 19

5 2026 BO GAO 54 where Pi P j implies l i l j even cf. Eq. 18. If both atoms have zero nuclear spin, i.e., I 1 I 2 0, Eq. 19 reduces to V BO ij i j Mi M j Pi P j 1 J i J jj 1i,J 2i,J i,l i,j 1j,J 2j,J j,l j 1/2 M L M S LS L S J i M L M S M L M S L S J j M L M S M L M S J i l i L,SL 1 L 2 L J 1i J 2i J il1 L2 L J 1j J 2j J j M L M S 0 M L M S J j l j M L M S 0 M L M ML SR, 20 SE which agrees with the result of Zygelman et al. 18,19,51. o determine the potentials associated with H f and H hf, one should, in principle, calculate the matrix elements of these interactions with the fragmentation channel functions given by Eq. 17. At large R, they both become diagonalized and are given by V f hf ij V ij R E f i E hf i ij, 21 f hf where E i is the sum of fine-structure splitting for the two colliding atoms, and E i is the total hyperfine splitting. For simplicity, we will use this asymptotic potential for all R. he close-coupling equation 3 can now be solved subject to the physical boundary condition of Eq. 4. he S matrix, the scattering amplitude, and the differential cross sections are derived in the standard fashion see Appendix B and are given by S EIiK EIiK E 1, 22 where I represents a unit matrix, and f 1 L 1 S 1 J 1i F 1i M 1i A 2 L 2 S 2 J 2i F 2i M 2i B k i 1 L 1 S 1 J 1 j F 1 j M 1 j A 2 L 2 S 2 J 2 j F 2 j M 2 j B k j li m li l j m lj 2i k i k j 1/2 i l i l jy li * m li kˆiy lj m lj kˆj F 1j M 1j,F 2j M 2j F j M Fj F j M Fj,l j m lj S EI ji F i M Fi F j M Fj F 1i M 1i,F 2i M 2i F i M Fi F i M Fi,l i m li, d d 1L 1 S 1 J 1i F 1i M 1i A 2 L 2 S 2 J 2i F 2i M 2i B k i 1 L 1 S 1 J 1 j F 1 j M 1 j A 2 L 2 S 2 J 2 j F 2 j M 2 j B k j k j k i f i j 2, where f (i j) is a short-hand notation for the amplitude given by Eq. 23. B. he case of similar Z A Z B atoms with distinguishable nuclei An example of this case would be the collision between a 85 Rb atom and a 87 Rb atom. As long as Z A Z B, the Born-Oppenheimer states defined by Eq. 6 have inversion symmetry with respect to the center of the molecule, provided that nonadiabatic couplings are ignored 52. On the other hand, the quasimolecular state defined by Eq. 8 may or may not have this symmetry. From its corresponding two-atom state, one can show that the quasimolecular state behaves under inversion as 47,48 Î e 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R P 1 P 2 1 N 1 N 2 L 1 L 2 L 2 L 2 S 2 A where Î e refers to the inversion of electronic coordinates with respect to the center of the molecule. wo distinctive cases need to be addressed separately here L 2 S 2 1 L 1 S 1 he situation is this case does not differ very much from the case of Z A Z B discussed in the previous section, since the quasimolecular states of Eq. 8 already have the proper inversion symmetry. From Eq. 25, we have Î e 1 L 1 S 1 A 1 L 1 S 1 B LM L RˆSM S Rˆ;R with W LS 1 L 1 S 1 A 1 L 1 S 1 B LM L RˆSM S Rˆ;R 26 1 L 1 S 1 B LM L RˆSM S Rˆ;R, 25 W LS 1 LS. 27

6 54 HEORY OF SLOW-AOM COLLISIONS 2027 hus the potential is again given by Eq. 19 except that we can add one more quantum number to the notation by replacing E M L S by E M L SW LS. Everything else remains unchanged L 2 S 2 1 L 1 S 1 he situation becomes much different here since the quasimolecular state, 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R no longer has the proper inversion symmetry. However, its relationship with BO states of the proper symmetry is rather simple. From Eq. 25, one can choose a phase convention such that 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R 1 2 M LRˆSM S Rˆg;RM L RˆSM S Rˆu;R], 28a 2 L 2 S 2 A 1 L 1 S 1 B LM L RˆSM S Rˆ;R P 1 P 2 1 N 1 N 2 L 1 L 2 L 1 2 M LRˆSM S Rˆg;RM L RˆSM S Rˆu;R], 28b where the symbols g and u refer to gerade and ungerade states, respectively. A more important difference introduced by the similarity of the atoms is that there are now additional channels described by channel functions: P 2 L 2 S 2 J 2 I 1 F 2 A 1 L 1 S 1 J 1 I 2 F 1 B Fl J 1,J 2,F 1,F 2 1/2 LSJI all m L,S,J,I 1/2 LM L,SM S JM J JM J,IM I F i M F FM F,lm l L 2 L 1 L S 2 S 1 S J 2 J 1 JJ2 J1 J J* I 1 I 2 I MJ M J,,0Y lml, F 2 F 1 FD 2 L 2 S 2 A 1 L 1 S 1 B LM L RˆSM S Rˆ;RI 1 A I 2 B IM I, 29 which corresponds to interchanging the electronic states of the two atoms in Eq. 17. hese two sets of channels are coupled, and it is this coupling that gives rise to the resonance exchange. he complete potential V BO ij is now given in terms of the BO potentials by the following set of equations: i M i P i 1 L 1 S 1 J 1i I 1 F 1i A 2 L 2 S 2 J 2i I 2 F 2i B F i l i j M j P j HBO 1 L 1 S 1 J 1j I 1 F 1j A 2 L 2 S 2 J 2j I 2 F 2j B F j l j i j Mi M j 1 F i F jj 1i,J 2i,F 1i,F 2i,F i,l i,j 1j,J 2j,F 1j,F 2j,F j,l j 1/2 l i K J i I F i i l j K J j F 1i F 2i F il1 L2 M L M S LSIKJ i J j 1 J i J jl,s,i,k,j i,j j L2 L J2i Ji L J2j Jj I 1 I 2 I I 1 I 2 I I F j jl1 J 1i J 2i J ij1i J 1j J 2j J jj1j M L M S 0 M L M S F 1j F 2j F j L S J i M L M S M L M S L S J j J i l i K M L M S M L M S J j l j K M L M S 0 M L M S 1 2 E M L SM S ge ML SM S u, 30a i M i P i 2 L 2 S 2 J 2i I 1 F 2i A 1 L 1 S 1 J 1i I 2 F 1i B F i l i j M j P j HBO 2 L 2 S 2 J 1j I 1 F 1j A 1 L 1 S 1 J 2j I 2 F 2j B F j l j i j Mi M j 1 F i F jj 1i,J 2i,F 1i,F 2i,F i,l i,j 1j,J 2j,F 1j,F 2j,F j,l j 1/2 M L M S LSIKJ i J j 1 J i J jl,s,i,k,j i,j j

7 2028 BO GAO 54 l i K J i I F i i l j K J j L1 L J1i Ji L J1j Jj S 2 S 1 S I 1 I 2 I S 2 S 1 S I 1 I 2 I I F j jl2 J 2i J 1i J ij2i J 2j J 1j J jj2j M L M S 0 M L M S F 2i F 1i F il2 L1 L S J i M L M S M L M S L S J j M L M S M L M S J i l i K F 2j F 1j F j J j l j K M L M S 0 M L M S 1 2 E M L SM S ge ML SM S u, 30b i M i P i 1 L 1 S 1 J 1i I 1 F 1i A 2 L 2 S 2 J 2i I 2 F 2i B F i l i j M j P j 2 L 2 S 2 J 2j I 1 F 2j A 1 L 1 S 1 J 1j I 2 F 1j B F j l j j M j P j HBO 2 L 2 S 2 J 2j I 1 F 2j A 1 L 1 S 1 J 1j I 2 F 1j B F j l j i M i P i HBO 1 L 1 S 1 J 1i I 1 F 1i A 2 L 2 S 2 J 2i I 2 F 2i B F i l i i j Mi M j P 1 P 2 1 F i F j N 1 N 2J 1i,J 2i,F 1i,F 2i,F i,l i,j 1j,J 2j,F 1j,F 2j,F j,l j 1/2 1 J 1j J 2j J il,s,i,k,j i,j j M L M S LSIKJ i J j J1j Jj I 1 I 2 I J 1j J 2j J jj2j L 1 L 2 L l i K J i I F i i l j K J j L2 L J2i Ji I 1 I 2 I I F j jl1 J 1i J 2i J ij1i M L M S M L M S F 1i F 2i F i F 2j F 1j F j L S Ji M L M S M L M S L S J j J i l i K J j l j K M L M S 0 M L M SM L M S 0 M L M S 1 2 E M L SM S ge ML SM S u. 30c Note that the coupling which gives rise to the resonance exchange Eq. 30c is determined by the energy differences between the gerade and the ungerade states. he notations for the scattering amplitude and the cross section need to be changed from those in Eqs. 23 and 24 to f LS 1i J 1i F 1i M 1i A LS 2i J 2i F 2i M 2i B k i LS 1j J 1j F 1j M 1j A LS 2j J 2j F 2j M 2j B k j, d d LS 1iJ 1i F 1i M 1i A LS 2i J 2i F 2i M 2i B k i LS 1j J 1j F 1j M 1j A LS 2j J 2j F 2j M 2j B k j, to account for the possibility of resonance exchange. Here (LS) can either be 1 L 1 S 1 or 2 L 2 S 2. C. he case of identical nuclei he new element in this case is that the total wave function must have proper symmetry under the exchange of nuclei. From the corresponding two-atom state and Eq. 12, one can show that the quasimolecular state defined by Eq. 8 behaves as 47,48 Xˆ n 1 L 1 S 1 A 2 L 2 S 2 B LM L RˆSM S Rˆ;R 1 N 1 N 2 L 1 L 2 L 2 L 2 S 2 A 1 L 1 S 1 B LM L RˆSM S Rˆ;R 1 N 1 N 2 L 1 L 2 S 1 S 2 M L M Se im L M S 2 L 2 S 2 A 1 L 1 S 1 B LM L RˆSM S Rˆ;R 31 under the exchange of nuclei, an operation denoted by Xˆ n. From this equation and Eq. 12, it is straightforward to show that the channel function defined by Eq. 17 behaves as Xˆ P n 1 L 1 S 1 J 1 I 1 F 1 A 2 L 2 S 2 J 2 I 1 F 2 B Fl R 1 N 1 N 2 F 1 F 2 Fl P 2 L 2 S 2 J 2 I 1 F 2 A 1 L 1 S 1 J 1 I 1 F 1 B Fl R 32 keeping in mind that identical nuclei implies, of course, I 2 I 1 ). hus a channel function with the desired symmetry property, Xˆ P n 1 L 1 S 1 J 1 I 1 F 1, 2 L 2 S 2 J 2 I 1 F 2 Fl R 1 2I 1 P 1 L 1 S 1 J 1 I 1 F 1, 2 L 2 S 2 J 2 I 1 F 2 Fl R, 33

8 54 HEORY OF SLOW-AOM COLLISIONS 2029 can be written in terms of channel functions given by Eq. 17 as P 1 L 1 S 1 J 1 I 1 F 1, 2 L 2 S 2 J 2 I 1 F 2 Fl R C FF P 1 L 1 S 1 J 1 I 1 F 1 A 2 L 2 S 2 J 2 I 1 F 2 B Fl R 1 N 1 N 2 2I 1 F 1 F 2 Fl P 2 L 2 S 2 J 2 I 1 F 2 A 1 L 1 S 1 J 1 I 1 F 1 B Fl R, 34 in which C FF is a normalization constant given by C FF 21 2 L 2 S 2 J 2 I 1 F 2, 1 L 1 S 1 J 1 I 1 F 1 1/2. Equation 34 implies that if 2 L 2 S 2 J 2 I 1 F 2 1 L 1 S 1 J 1 I 1 F 1 implying, of course, N 1 N 2 ), only states satisfying Fleven 35 are possible 53, regardless of whether the nuclei are bosons or fermions. he potential V BO ij is given in this case by i M i P i 1 L 1 S 1 J 1i I 1 F 1i, 2 L 2 S 2 J 2i I 1 F 2i F i l i j M j P j HBO 1 L 1 S 1 J 1j I 1 F 1j, 2 L 2 S 2 J 2j I 1 F 2j F j l j 2C FF i C FF j i M i P i 1 L 1 S 1 J 1i I 1 F 1i A 2 L 2 S 2 J 2i I 1 F 2i B F i l i j M j P j 1 N 1 N 2 2I 1 l j F 1j F 2j F j i M i P i 1 L 1 S 1 J 1i I 1 F 1i A 2 L 2 S 2 J 2i I 1 F 2i B F i l i HBO 1 L 1 S 1 J 1j I 1 F 1j A 2 L 2 S 2 J 2j I 1 F 2j B F j l j j M j P j HBO 2 L 2 S 2 J 2j I 1 F 2j A 1 L 1 S 1 J 1j I 1 F 1j B F j l, j where the matrix elements on the right-hand side are given by Eq. 30 for ( 1 L 1 S 1 )( 2 L 2 S 2 ), and by Eq. 19 for ( 1 L 1 S 1 )( 2 L 2 S 2 ). he differential cross section in the center-of-mass frame is given by see Appendix B 36 d d LS 1iJ 1i F 1i M 1i,LS 2i J 2i F 2i M 2i k i LS 1j J 1j F 1j M 1j,LS 2j J 2j F 2j M 2j k j FF 2 k j C j k i C FF i 2 fi j,k j1 N 1 N 2 2I 1fi j,k j 2, 37 where f (i j,k j ) is a short-hand notation for the full expression f LS 1i J 1i F 1i M 1i,LS 2i J 2i F 2i M 2i k i LS 1j J 1j F 1j M 1j,LS 2j J 2j F 2j M 2j k j, which is related to the S matrix by Eq. 23. Equation 37 implies that d d k j d d k j in the center-of-mass frame is generally applicable to collisions between atoms with identical nuclei, which may be in different states or carry different charge. D. LS coupling In previous sections, we have established the proper identification of fragmentation channels and the close-coupling equations to be solved for different types of collisions. he rest of this paper deals basically with the question of how to solve these equations effectively. he potentials given by Eqs. 19, 30, and 36 are complicated because the fragmentation channel functions do not reflect the symmetry of H BO. A much simpler representation of V BO can be obtained in the LS coupled condensation channels 43, which are defined, in the case of distinguishable nuclei, by the asymptotic behavior P 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B LlLSKI R all m L 1 M L1,L 2 M L2 LM L LM L,lm l LM L S 1 M S1,S 2 M S2 SM S LM L,SM S KM K KM K,IM I 1 L 1 M L1 S 1 M S1 A 2 L 2 M L2 S 2 M S2 B Y lml,i 1 A I 2 B IM I, where all the magnetic quantum numbers are quantized along a space-fixed axis. A channel function with such an asymptotic behavior can be written in terms of the quasimolecular states defined by Eq. 8 as 38

9 2030 BO GAO 54 M P 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B LlLSKI all LM L,lm l LM L LM L,SM S KM K m L* KM K,IM I D ML M L S*,,0D M S M S,,0Y lml, 1 L 1 S 1 A 2 L 2 S 2 B LM L Rˆ SM S Rˆ ;RI 1 A I 2 B IM I. 39 he potential V BO becomes much simplified in this coupling scheme and is given by M P 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B L l L S K I M P HBO 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B L l L S K I M M P P L L L L S S K K I I l,l 1/2 M L L l L M L 0 M L L l L M L 0 M LE ML S L R, 40 which is block diagonal in all quantum numbers except l. his potential is consistent with Eq. 22 of Ref. 19. In the case of similar atom (Z A Z B ) with distinguishable nuclei and ( 1 L 1 S 1 )( 2 L 2 S 2 ), there are additional channels characterized by M P 2 L 2 S 2 I 1 A 1 L 1 S 1 I 2 B LlLSKI L* all LM L,lm l LM L LM L,SM S KM K KM K,IM I D ML M m L S*,,0D M S M S,,0 Y l m l, 2 L 2 S 2 A 1 L 1 S 1 B LM L Rˆ SM S Rˆ ;RI 1 A I 2 B IM I, 41 which is a result of interchanging the electronic states in Eq. 39. With the help of Eq. 28, we obtain the potential V BO : M P 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B L l L S K I M P HBO 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B L l L S K I M P HBO 2 M P L 2 S 2 I 1 A 1 L 1 S 1 I 2 B L l L S K I 2 L 2 S 2 I 1 A 1 L 1 S 1 I 2 B L l L S K I M M P P L L L L S S K K I I l,l 1/2 M L L l L M L 0 M L L l L M L 0 M L 1 2 E M L S gl RE ML S ul R, 42a M P 1 L 1 S 1 I 1 A 2 L 2 S 2 I 2 B L l L S K I M P HBO 2 L 2 S 2 I 1 A 1 L 1 S 1 I 2 B L l L S K I M M P P L L L L S S K K I I P 1 P 2 1 N 1 N 2 L 1 L 2 L l,l 1/2 M L L l L M L 0 M L L l L M L 0 M L 1 2 E M L S gl RE ML S ul R. 42b In the case of identical nuclei, the channel functions must have a proper symmetry under the exchange of nuclei. From Eqs. 12 and 31, we have Xˆ P n 1 L 1 S 1 I 1 A 2 L 2 S 2 I 1 B LlLSKI R 1 N 1 N 2 L 1 L 2 L2I 1 Il P 2 L 2 S 2 I 1 A 1 L 1 S 1 I 1 B LlLSKI R. 43 hus a channel function with proper symmetry under the exchange of nuclei is given by P 1 L 1 S 1 I 1, 2 L 2 S 2 I 1 LlLSKI R C LS P 1 L 1 S 1 I 1 A 2 L 2 S 2 I 1 B LlLSKI R 1 N 1 N 2 L 1 L 2 LIl P 2 L 2 S 2 I 1 A 1 L 1 S 1 I 1 B LlLSKI R, 44

10 54 HEORY OF SLOW-AOM COLLISIONS 2031 which satisfies Xˆ P n 1 L 1 S 1 I 1, 2 L 2 S 2 I 1 LlLSKI R 1 2I 1 P 1 L 1 S 1 I 1, 2 L 2 S 2 I 1 LlLSKI R. 45 he constant C LS is a normalization constant: C LS 21 2 L 2 S 2, 1 L 1 S 1 1/2. Equation 44 implies that if 2 L 2 S 2 1 L 1 S 1 implying, of course, N 1 N 2 ), only states satisfying LSIleven 46 are possible 53. he same condition holds whether the nuclei are fermions or bosons. he potential V BO is now M P 1 L 1 S 1 I 1, 2 L 2 S 2 I 1 L l L S K I M P HBO 1 L 1 S 1 I 1, 2 L 2 S 2 I 1 L l L S K I 2C LS C LS M P HBO 1 M P L 1 S 1 I 1 A 2 L 2 S 2 I 1 B L l L S K I 1 L 1 S 1 I 1 A 2 L 2 S 2 I 1 B L l L S K I 1 N 1 N 2 L 1 L 2 L I l M P 1 L 1 S 1 I 1 A 2 L 2 S 2 I 1 B L l L S K I H BO M P 2 L 2 S 2 I 1 A 1 L 1 S 1 I 1 B L l L S K I, 47 with the matrix elements on the right-hand side given by Eq. 42 if ( 2 L 2 S 2 )( 1 L 1 S 1 ) and by Eq. 40 if ( 2 L 2 S 2 )( 1 L 1 S 1 ). It is clear that the representation of V BO is much simpler in the LS coupling than in the FF coupling. he catch is, of course, that H f and H hf are no longer diagonal. he more serious problem is that since the channel couplings due to H f and H hf do not vanish at large R, the physical K matrix cannot be defined in the LS coupling unless H f and H hf are strictly zero. his is where frame transformation and MQD come into play. E. Frame transformations and MQD It can be shown see Appendix A that the FF coupled channel functions Eq. 17 are related to the LS coupled channel functions Eq. 39 by a frame transformation where j P U i li l 1 F i L S K I U j P R, J 1i,F 1i,J 2i,F 2i,F i,l,l,s,k,i 1/2 Jx 1 J xj x l i K J x S L L l i K J x I F i 48 1 L 2 L J2i Jx L I 1 I 2 I 49a J 1i J 2i J xj1i F 1i F 2i F i. From Eqs. 34 and 44, it is clear that in the case of identical nuclei, the properly symmetrized channel functions are related by the frame transformation U i C FF i /C LS U i 1 2 L 2 S 2, 1 L 1 S 1 1J 2i F 2i,J 1i F 1i 1/2 U i. 49b () For the sake of simplified notation, U i will also be denoted by U () i, with the understanding that the proper transformation is to be used for different cases. We emphasize that the normalization factor in Eq. 49b is important. he transformation would not be unitary without it. he goal of this section is to present a simple method which gives complete information about a hyperfine collision from either the existing LS coupled calculations 34,8,23 or a JJ coupled calculation discussed in Appendix A. It is an approximate method that combines the use of frame transformation and MQD. he concept of this method is well known, and has been very successful in interpreting the spinorbital effects on photoionization It leads in our case to a multichannel effective range theory which is the same as the one developed previously for electron-atom scattering 54. I will first present the method. he underlying assumptions and future refinements will then be discussed. Briefly summarizing the method, we first solve the closecoupling equations in the LS coupled condensation channels ignoring both the spin-orbital and the hyperfine interactions, or in the JJ coupled channels ignoring only the hyperfine interactions. An energy-insensitive short-range K matrix (K 0() in LS coupling; K 0() xy in JJ coupling is extracted from this set of solutions, from which the energyinsensitive short-range K matrix in the fragmentation channels, K 0() ij, is obtained through a frame transformation. MQD is then applied to give the physical K matrix K () ij in terms of K 0() ij. Specifically, K 0() is defined by the large R expansion of the wave function in LS coupled condensation channels as follows:

11 2032 BO GAO 54 P P Rf 0 g 0 K 0 /R. 50 Here f 0 and g 0 are a pair of linearly independent solutions corresponding to the asymptotic potential in a condensation channel. hey differ from the usual f,g pair in that they are required to have energy-independent normalization at the origin. If the interaction in a condensation channel can be treated as a short-range interaction, f 0 and g 0 are specifically a pair of analytic functions of energy see Appendix C: (1) l is an N c N c diagonal matrix with elements (1) l i. K 0() oo, K 0() oc, K 0() co, and K 0() cc, are submatrices of K 0() corresponding to open-open, open-close, close-open, and close-close channels, respectively. It should be remembered that the K 0() matrix referred to here is the one in fragmentation channels. It is related to the one in condensation channels by Eq. 53. If all channels are closed, the physical boundary conditions can only be satisfied at discrete energy levels determined by f 0 1 k l Rj l k R, 51a det1 l l1/2 K 0 l1/2 0, 55 where g 0 k l 1 Ry l k R, EE, k 2 1/2 /. 51b Note that the expansion Eq. 50 with an R-independent K 0 at large R is possible only when the spin-orbital and the hyperfine interactions are ignored this is why the physical K matrix cannot be defined in LS coupling unless there are no such interactions. he energy-insensitive K 0() ij is defined in a similar fashion by a large R expansion of the wave function in fragmentation channels: j P i P i Rf 0 i ij g 0 i K 0 ij /R. 52 After solving for K 0() in the condensation channels, K 0() ij is obtained by using the frame transformation Eq. 49: K 0 ij U i K 0 U j. 53 he wave functions satisfying the proper physical boundary conditions are then constructed as linear superpositions of solutions specified by Eq. 52. his is a standard procedure in MQD 26,28, and lead in our case to the multichanneleffective-range equation 54, which relates the physical K matrix to K 0 ij : K Ek l1/2 K 0 oo K 0 oc l1/2 1 l l1/2 K 0 cc l1/2 1 l1/2 K 0 co k l1/2. 54 Here k l1/2 is an N o N o diagonal matrix with elements l k i 1/2 i, with N o being the number of open channels at energy E. l1/2 is an N c N c diagonal matrix with elements l i 1/2 i, with N c being the number of closed channels at energy E, and i 2E i E 1/2 /. which gives the relationship between bound state energy levels and the scattering matrix. Specializing to the case of collision between alkali-metal atoms in their ground states, Eqs. 49, 53, and 54 enable one to generate complete information about the collision from single channel calculations such as those of 34,8,23. Numerical results of this sort will be presented later. If the spin-orbit interaction is significant, the K 0 matrix in Eq. 53 should be replaced by K 0 xy calculated by solving the close-coupling equations in the JJ coupling scheme see Appendix A, and the frame transformation U () i is replaced by U () ix given by Eqs. A6 and A15. he theory presented in this section is, of course, only an approximation with a certain range of validity. It relies on Eq. 53, which is strictly valid only when there are no spinorbit and hyperfine interactions. he corresponding transformation from JJ coupling requires the hyperfine interactions to be zero. However, they are useful approximations even in the presence of such interactions. First, it is obviously a good method to account for the hyperfine effects at energies much greater than the hyperfine splitting. his covers a wide energy range that is of great importance in many applications 2. Second, the degree to which these approximations fail is a direct measure of the effects of long-range interactions. his is because Eq. 53 is always going to be valid inside an R 0 where the LS coupled (JJ coupled potential is much greater than the spin-orbital hyperfine interactions. hus the frame transformation method presented in this section basically assumes that after R 0 the electrostatic and the exchange interactions are sufficiently weak or that they drop off so quickly that their effects can be ignored. hird, it can be systematically improved upon. Specifically, the effect of long-range interactions in the outside region where the hyperfine interaction is dominant can be incorporated by a proper choice of the f 0 and g 0 pair. For a 1/r 6 potential, the proper pair can be obtained either by the usual perturbation theory or by a method of Cavagnero 55. Any error after such a correction reflects the evolution of the wave function through the intermediate region where the electrostatic and the exchange interaction is comparable to the hyperfine interactions. his region can be treated by solving the closecoupling equation in the FF coupled channels, if needed. It will be interesting to identify the significance of this contribution, especially its energy dependence.

12 54 HEORY OF SLOW-AOM COLLISIONS 2033 III. DISCUSSIONS A. Sensitivity to the potential It is well recognized and understood that cold-atom collisions are very sensitive to the potential 4,7,8. We give here a brief discussion of some issues of interest in our present work. he total phase shift of a collision comprises contributions from both short-range and long-range interactions. We will assume that the asymptotic potentials are known precisely and look at the dependence of the short-range phase on the potential. For atom-atom collisions, the short-range phase can be estimated by the integral of kr2 1/2 /EVR 1/2, the uncertainty of which is related to the uncertainty in V by k/2 2 1/2 EVR 1/2 V. If the distance over which the potential may be in error is denoted by d, the condition that the phase is determined within is roughly kd. For slow-atom collisions, the corresponding requirement on the uncertainty in potential can be estimated by V d 22 1/2 V min 1/2, 56 where V min is the depth or some typical value of the potential. hus we see that the sensitive dependence of slow-atom collisions is due mainly to the large mass of an atom. Since atoms are thousands times heavier than electrons and the distance over which the potential may be incorrect is about a few times bigger, it follows that the potential has to be about 10 4 times more accurate if the atom-atom phase shift is to be determined to the same degree of accuracy as the electronatom scattering. his is why quantities such as the angular distribution are so much more difficult to determine for atom-atom collision as opposed to electron-atom collisions. What is unique about cold-atom collisions is that if the last bound state happens to be very close to the threshold, the error in the short-range phase gets further amplified in the determination of cross sections 4. Note that a sensitive dependence on potential also implies a sensitive dependence on energy, since k/2 2 1/2 EVR 1/2 E. Fortunately, it is easily verified that ignoring hyperfine interaction over a range of 10 a.u. does not change the phase significantly, which is actually an assumption that makes the approximation in the preceding section useful. B. MQD and frame transformation as a numerical tool We point out here that the method of MQD and frame transformation is useful even when the goal is to solve the FF coupled close-coupling equations numerically. MQD provides an elegant and systematic way of imposing the physical boundary conditions, and the frame transformations lead us to basis sets in which the potential is approximately block diagonalized in certain regions of the configuration space. For example, the LS coupling block-diagonalizes the Born-Oppenheimer part of the potential. he spin-orbit and hyperfine interactions are not diagonalized. Specifically, if we take this part of the interaction to be that of two separate atoms, their matrix elements are given by f V x V hf i U x E f x U x, U i E hf i U i, f where E x is the total fine-structure splitting for the two atoms in a JJ coupled channel x see Appendix A. U () x is the LS to JJ frame transformation given by Eqs. A4 and A14. E hf i is the total hyperfine splitting for the two atoms in a fragmentation channel i, and U () i is the LS to FF frame transformation given by Eq. 49. It is obvious that in the region where the electrostatic and exchange interactions dominate, the LS coupled channel functions provide a better basis set for numerical integration 21. hus, depending on the strength of the spin-orbital interaction, one can separate the space into three or four regions. he outermost region is handled by MQD; the innermost region is integrated in LS coupling. he middle region is handled in FF coupling or a combination of JJ and FF coupling. Matching at various boundaries is facilitated by the frame transformations. Such a scheme fits especially well with Gordon s method of solving the close-coupling equations 56. IV. CONCLUSIONS A general theory of slow-atom collisions has been presented with special emphasis on the effects of nuclear spin statistics and atomic fine and/or hyperfine structures. It reduces to the theory of Stoof et al. 20 for collisions between alkali-metal atoms in their ground state, and to the theory of Zygelman et al. 18 if there are no hyperfine interactions and if the nuclei are either distinguishable, or indistinguishable in which case the atoms must be in the same electronic states. By employing the frame transformation and MQD techniques, we arrived at an approximate method which gives complete information about hyperfine collisions from much simpler LS coupled or JJ coupled calculations. he method can be systematically improved upon, and it tells one where each contribution comes from. he next natural step is to compare the results obtained by solving the close-coupling equations in the FF coupled channels and the results obtained by various approximations discussed in Sec. II E. Reference pair functions f 0 and g 0 for 1/r n potentials, especially for n6 and n3, also deserve a closer look. Getting back to the question of whether a 85 Rb atom in the ground F3 state is distinguishable from a 85 Rb atom in the ground F2 state, the answer is that it is an ill-posed question that can be deceiving. If the atoms are given one at a time implying that their wave functions do not overlap, one can of course tell which one is in F2 and which one is in F3. But that is missing the point of indistinguishability.

13 2034 BO GAO 54 he point is that when the wave functions overlap, we cannot tell which one is in F2. We only know that one of the atoms is in it and the other one is in F3. hus the system of two 85 Rb atoms, one in F2 and one in F3, should be treated as two identical atoms in different component states, just as the two different spin orientations of an electron. he same conclusion is, of course, applicable to any two atoms with the same constituents and in any states. However, for a system made up of an ion and an atom of the same species or ions of the same species but of different charge, one needs to think in terms of nuclear statistics. Finally, it seems clear that the ultimate theory of coldatom collisions should be an R-matrix type of approach in which short-range interactions are treated by a complete basis set expansion including both the electronic and the nuclear degrees of freedom. Close-coupling equations, such as the ones presented here, are used only beyond a certain R 0. Such sophistication may not be necessary at high temperatures, but may be needed for cold collisions due to their sensitive dependence on the potential. ACKNOWLEDGMEN I would like to thank Professor Anthony F. Starace for careful reading of the manuscript and for helpful suggestions. APPENDIX A: FRAME RANSFORMAIONS AND JJ COUPLING he frame transformation Eq. 49 is derived by going through JJ coupled channel functions, which are defined, for distinguishable nuclei, by the asymptotic behavior P 1 L 1 S 1 J 1 I 1 A 2 L 2 S 2 J 2 I 2 B JlKI R all m J 1 M 1,J 2 M 2 JM J JM J,lm l KM K KM K,IM I 1 L 1 S 1 J 1 M 1 A 2 L 2 S 2 J 2 M 2 B Y lml,i 1 A I 2 B IM I, A1 where all magnetic quantum numbers are quantized along a space-fixed axis. From Eq. 8, it can be shown that the JJ coupled channels functions can be written in terms of the quasimolecular states as P 1 L 1 S 1 J 1 I 1 A 2 L 2 S 2 J 2 I 2 B JlKI J 1,J 2 1/2 LS L2 L all L,S 1/2 LM L,SM S JM J JM J,lm l KM K KM K,IM I L1 m J 1 J 2 J J* D MJ M J,,0Y li m l, 1 L 1 S 1 A 2 L 2 S 2 B LM L Rˆ SM S Rˆ ;RI 1 A I 2 B IM I. A2 his channel function, to be identified by an index x, is related to the LS coupled channel functions cf. Eq. 39, identified by the indices, by the frame transformation 9,18,19 where i P R U ix P x R, A5 x P R U x P R, A3 U ix li l x J1i J 1x J2i J 2x 1 l i F i K x I x where U x lx l Kx K Ix I 1 l x J x L S J 1x,J 2x,J x,l,l,s 1/2 l x K J x S L L L1 L2 L J 1x J 2x J x. A4 An FF coupled channel function cf. Eq. 17, identified by the index i, is related to the JJ coupled channel functions by a similar frame transformation, F 1i,F 2i,F i,j x,k x,i x 1/2 l i K x J x I x F i J1i J2i Jx I 1 I 2 I x F 1i F 2i F i. A6 he LS to FF frame transformation is then obtained through which gives Eq. 49. U i x U ix U x, A7

14 54 HEORY OF SLOW-AOM COLLISIONS 2035 In cases where spin-orbit interaction is significant, the starting point for the frame transformation and MQD method described in Sec. II E is the solution of closecoupling equations in the JJ coupling. For a collision between two different atoms or a collision between similar atoms with distinguishable nuclei and ( 2 L 2 S 2 )( 1 L 1 S 1 ), the potential V BO xy that enters into the close-coupling equation in JJ coupling is determined by x M x P x 1 L 1 S 1 J 1x I 1 A 2 L 2 S 2 J 2x I 2 B J x l x K x I x y M y P y HBO 1 L 1 S 1 J 1y I 1 A 2 L 2 S 2 J 2y I 2 B J y l y K y I y x y Mx M y Px P y Kx K y Ix I y 1 J x J yj 1x,J 2x,J x,l x,j 1y,J 2y,J y,l y 1/2 M L M S LS L,SL 1 L 2 L J 1x J 2x J x L 1 L 2 L J 1y J 2y J y L S Jx M L M S M L M S L S J y M L M S M L M S J x l x K y M L M S 0 M L M S J y l y K y M L M S 0 M L M SE ML SR. A8 In the case of a collision between two similar atoms with distinguishable nuclei and ( 2 L 2 S 2 )( 1 L 1 S 1 ), there are additional channels characterized by P J1 2,J L 2 S 2 J 2 I 1 A 1 L 1 S 1 J 1 I 2 B JlKI 2 1/2 LSJI all L,S 1/2 LM L,SM S JM J JM J,lm l KM K KM K,IM I m 2 L 1 L J* L S 2 S 1 S MJ M J,,0Y li m l, 2 L 2 S 2 A 1 L 1 S 1 B J 2 J 1 JD LM L Rˆ SM S Rˆ ;RI 1 A I 2 B IM I. A9 he complete potential V BO xy is specified in this case by x M x P x 1 L 1 S 1 J 1x I 1 A 2 L 2 S 2 J 2x I 2 B J x l x K x I x y M y P y HBO 1 L 1 S 1 J 1y I 1 A 2 L 2 S 2 J 2y I 2 B J y l y K y I y x y Mx M y Px P y Kx K y Ix I y 1 J x J yj 1x,J 2x,J x,l x,j 1y,J 2y,J y,l y 1/2 M L M S LS L,SL 1 L 2 L J 1x J 2x J x L 1 L 2 L J 1y J 2y J y L S Jx M L M S M L M S L S J y M L M S M L M S J x l x K y M L M S 0 M L M S J y l y K y M L M S 0 M L M S 1 2 E M L S gl RE ML S ul R, A10a x M x P x 2 L 2 S 2 J 2x I 1 A 1 L 1 S 1 J 1x I 2 B J x l x K x I x y M y P y HBO 2 L 2 S 2 J 2y I 1 A 1 L 1 S 1 J 1y I 2 B J y l y K y I y 1 J 1x J 2x J x J 1y J 2y J y x M x P x 1 L 1 S 1 J 1x I 1 A 2 L 2 S 2 J 2x I 2 B J x l x K x I x y M y P y HBO 1 L 1 S 1 J 1y I 1 A 2 L 2 S 2 J 2y I 2 B J y l y K y I, y A10b

15 2036 BO GAO 54 x M x P x 1 L 1 S 1 J 1x I 1 A 2 L 2 S 2 J 2x I 2 B J x l x K x I x y M y P y HBO 2 L 2 S 2 J 2y I 1 A 1 L 1 S 1 J 1y I 2 B J y l y K y I y x y Mx M y Px P y Kx K y Ix I y P 1 P 2 1 N 1 N 2 J 1y J 2y J xj 1x,J 2x,J x,l x,j 1y,J 2y,J y,l y 1/2 L,SL 1 L 2 L L2 L xl1 M L M S LS J 1x J 2x J J 1y J 2y J y L S Jx M L M S M L M S L S J y J x l x K y J y l y K y M L M S 0 M L M SM L M S 0 M L M S 1 2 E M L S gl RE ML S ul R. A10c M L M S M L M S For a collision between two atoms with identical nuclei, a JJ coupled channel function with proper symmetry under the exchange of nuclei is P 1 L 1 S 1 J 1 I 1, 2 L 2 S 2 J 2 I 1 JlKI R where C JJ P 1 L 1 S 1 J 1 I 1 A 2 L 2 S 2 J 2 I 1 B JlKI R 1 N 1 N 2 J 1 J 2 JIl P 2 L 2 S 2 J 2 I 1 A 1 L 1 S 1 J 1 I 1 B JlKI R, A11 C JJ 21 2 L 2 S 2 J 2, 1 L 1 S 1 J 1 1/2 is a normalization constant. From Eq. A11, it is clear that if only states satisfying 2 L 2 S 2 J 2 1 L 1 S 1 J 1, JIleven A12 are possible. his is true regardless of whether the nuclei are fermions or bosons. he potential V BO xy for the case of identical nuclei is therefore x M x P x 1 L 1 S 1 J 1x I 1, 2 L 2 S 2 J 2x I 2 J x l x K x I x y M y P y HBO 1 L 1 S 1 J 1y I 1, 2 L 2 S 2 J 2y I 2 J y l y K y I y 2C JJ x C JJ y x M x P x 1 L 1 S 1 J 1x I 1 A 2 L 2 S 2 J 2x I 2 B J x l x K x I x y M y P y HBO 1 L 1 S 1 J 1y I 1 A 2 L 2 S 2 J 2y I 2 B J y l y K y I y 1 N 1 N 2 J 1y J 2y J y I y l y x M x P x HBO 1 y M y P y L 1 S 1 J 1x I 1 A 2 L 2 S 2 J 2x I 2 B J x l x K x I x 2 L 2 S 2 J 2y I 1 A 1 L 1 S 1 J 1y I 2 B J y l y K y I, A13 y where the matrix elements on the right-hand side are given by Eq. A10 for ( 1 L 1 S 1 )( 2 L 2 S 2 ), and by Eq. A18 for ( 1 L 1 S 1 )( 2 L 2 S 2 ). he LS to JJ frame transformation for identical nuclei is U x C JJ x /C LS U x 1 2 L 2 S 2, 1 L 1 S 1 1J 2x,J 1x 1/2 U x. A14 he JJ to FF frame transformation for identical nuclei is U ix C FF i /C JJ x U ix 1 2 L 2 S 2 J 2i, 1 L 1 S 1 J 1i 1F 2i,F 1i 1/2 U ix. A15 he JJ coupled channel functions diagonalize the asymptotic spin-orbit interactions, i.e., f V xy R E f x xy, A16 f where E x is the total fine-structure splitting for the two atoms. he asymptotic hyperfine interactions are, however, not diagonalized in JJ coupling. Specifically, hf V xy R i U ix E hf i U iy, A17 hf where E i is the total hyperfine splitting for the two atoms in a fragmentation channel i, in which the asymptotic hyperfine interactions are diagonalized. U () ix is the JJ to FF frame transformation given by Eqs. A6 and A15.