Temperature dependence of atom-atom interactions H. Wennerstro ¨ m, 1 J. Daicic, 2,3 and B. W. Ninham 1,4 1 Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE-22100 Lund, Sweden 2 Institute for Surface Chemistry, P.O. Box 5607, SE-11486 Stockholm, Sweden 3 Department of Chemistry, Surface Chemistry, Royal Institute of Technology, SE-10044 Stockholm, Sweden 4 Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia ~Received 16 February 1999! The interaction potential at finite temperature between two atoms in the asymptotic limit of large separa- tions, R @\ c / k B T , is V ( R ) 523 k B T a 2 (0)/ R 6 , identical to the expression for two classical polarizable par- ticles. This is in contrast to the zero-temperature, large-separation Casimir-Polder retarded potential V ( R ) 5223\ c a 2 (0)/4p R 7 . The result is demonstrated by two independent methods, so as to facilitate a reinter- pretation of retardation effects in interatomic interactions. Here, we argue that the conventional mechanistic interpretation of retardation at large separations in terms of the loss of interatomic correlations due to the finite velocity of light is too simplistic, and relies on nonquantum, time-dependent concepts to explain an equilib- rium, quantum effect. We offer an alternative picture in terms of the nature of the thermal excitations of the electromagnetic field, which results in a remarkable manifestation of the correspondence principle at suffi- ciently large interatomic distances. @S1050-2947~99!07409-0# PACS number~s!: 34.20.Cf, 03.70.1k, 11.10.Wx The nature of atom-atom interactions is a fundamental conceptual issue impinging on broad areas of physics and chemistry. Our understanding of the stability of matter is a principal example. It was established by London @1# that an attractive dispersion force exists between all atoms due to mutual fluctuation-induced dipole creation, mediated by the electromagnetic field, and that the atom-atom interaction po- tential is given to leading order by V ~ R ! 52 3 4 \ v 0 a 2 ~ 0 ! R 6 , ~1! where a (0) is the static polarizability of the ~identical! at- oms and v 0 their principal absorption frequency. Casimir and Polder @2# later demonstrated that at sufficiently large separations R, such that R @c / v 0 , the interaction potential is modified to V ~ R ! 52 23 4 p \ c a 2 ~ 0 ! R 7 ~2! and thus a more rapid decay of the interaction with increas- ing R. Presumably because of the appearance of the speed of light c in Eq. ~2!, this large-separation modification of the interaction potential was mechanistically interpreted as a re- tardation effect due to the finite velocity of the photons me- diating the interaction. This has been the accepted picture of pair atomic interac- tions since 1948. The above two formulas have been used extensively in calculation in different contexts @3#, such as the theories of dilute gases, colloidal stability, and atomic spectroscopy, but arguably their greater significance lies in the mechanistic interpretation that they suggest: a softening of the potential when the distance is sufficiently large for the finite value of c to induce a significant loss of interatomic correlations in the dipole fluctuations or, in other words, when R @c / v 0 . The original calculations leading to Eqs. ~1! and ~2! were related to the ground state of the atom-field system, implying zero temperature. The frequent use of these expressions since suggests a wide spread assumption that finite values of T would furnish only weak higher-order corrections to them. In a recent paper, Ninham and Daicic @4# reexamined the nature of dispersion forces at finite temperature using the Lifshitz theory @5# of interactions between continuous media. In the atom-atom case, obtained by taking the dilute limit, we in- deed concluded that in the nonretarded regime, R !c / v 0 , Eq. ~1! applies for all T with only very weak T corrections. In contrast, we demonstrated that the Casimir-Polder result, Eq. ~2!, cannot be the correct asymptotic large-R interaction po- tential for any T 0. In other words, asymptotically there is a nonanalyticity in T which fundamentally alters the scaling of the interaction potential. In fact, the interaction potential for R @\ c / k B T and at finite T is, to leading order, V ~ R , T ! 523 k B T a 2 ~ 0 ! R 6 , ~3! which we recognize as the interaction between two classical, isotropic, polarizable particles. A result similar in form to Eq. ~3! has been published previously @6#. 1 However, the conceptual consequences arising therefrom were not dis- cussed, and it seems to have largely been ignored. It should also be pointed out that Milonni and Smith @7# have shown that a result identical to Eq. ~3! also applies in the short- distance ( R !\ c / k B T ), high-temperature ( \ v 0 / k B T !1) re- gime, recovering the classical result of Boyer @8#. This is an observation of the ‘‘conventional’’ correspondence principle, induced by raising the thermal energy well above the atomic absorption energy in the short-separation regime. It is to be expected that a quantum system goes over to its classical analog at high temperatures. What is remarkable, however, is 1 The result of Eq. ~3! in actual fact first appeared in A. D. McLachlan @24#. PHYSICAL REVIEW A SEPTEMBER 1999 VOLUME 60, NUMBER 3 PRA 60 1050-2947/99/60~3!/2581~4!/$15.00 2581 ©1999 The American Physical Society