Cold collisions of N atoms and NH molecules in magnetic fields
aa r X i v : . [ phy s i c s . c h e m - ph ] S e p Cold collisions of N ( S ) atoms and NH ( S )molecules in magnetic fields Piotr S. ˙Zuchowski ∗ and Jeremy M. Hutson † November 1, 2018
Abstract
We calculate the interaction potential between N atoms and NH molecules anduse it to investigate cold and ultracold collisions important for sympathetic cool-ing. The ratio of elastic to inelastic cross sections is large over a wide range ofcollision energy and magnetic field for most isotopic combinations, so that sym-pathetic cooling of NH molecules by N atoms is a good prospect. However, thereare important effects due to a p-wave shape resonance that may inhibit cooling insome cases. We show that scaling the reduced mass used in the collision is ap-proximately equivalent to scaling the interaction potential. We then explore thedependence of the scattering properties on the reduced mass and explain the reso-nant effects observed using angular-momentum-insensitive quantum defect theory.
At temperatures below about 1 mK, atoms and molecules enter a fully quantal regimewhere their de Broglie wavelength is large compared to molecular dimensions. Inthis regime, collision cross sections and reaction rates are dominated by long-rangeforces and resonance phenomena. It is likely to be possible to control reactionrates by tuning scattering resonances with applied electric and magnetic fields. At ∗ Department of Chemistry, Durham UniversityE-mail: [email protected] † Department of Chemistry, Durham UniversityE-mail: [email protected] m K, trapped atoms and molecules form quan-tum gases such as Bose-Einstein condensates and Fermi-degenerate gases, in whichevery molecule occupies the lowest allowed translational state in the trap. The quan-tum gas regime offers additional possibilities for a new form of quantum control, inwhich chemical transformations are carried out coherently on entire samples of ultra-cold atoms and molecules.There have been enormous advances towards these goals in the last few years. Inparticular, it is now possible to produce ultracold alkali metal dimers in their rovibronicground states from ultracold atoms both by photoassociation and by magnetoasso-ciation followed by stimulated Raman adiabatic passage (STIRAP).
However, foran alkali metal dimer even the ground rovibronic state has nuclear spin hyperfine struc-ture, and the resulting splittings are comparable to the kinetic energies involved inultracold collisions. For the case of K Rb, microwave transitions have been usedto transfer the ground-state molecules selectively between different hyperfine and Zee-man levels.
In a very recent development, Ospelkaus et al. have studied reactivecollisions of such state-selected molecules, both with one another and with ultracoldK and Rb atoms. They observed remarkable selectivity of the resulting reactions, inwhich flipping the spin of a single nucleus could cause dramatic changes in the outcomeof a collision. Methods that form molecules from ultracold atoms can be applied only in caseswhere the atoms themselves can be cooled. In practice this restricts them to the alkalimetals, the alkaline earths, and a few other elements. These species have a fairly lim-ited chemistry. In order to cool a wider class of molecules, including polyatomic ones,a number of direct cooling methods have been established over the last decade.Among these methods, buffer-gas cooling is based on the particularly simple idea ofcooling molecules by elastic collisions with cold He gas. If the molecules are param-agnetic and in low-field-seeking states, they can be confined in a magnetic trap. Thetemperatures which can be achieved in buffer gas cooling method are limited by thevapour pressure of the buffer gas (ca. 400 mK for He), but the method is particu-larly valuable for two reasons: (i) it is very general and can in principle be applied2o any paramagnetic species, provided that a detection scheme is available; (ii) it canproduce large numbers and high densities of cold molecules. Buffer-gas cooling hasbeen reported for a variety of molecules including CaH, CaF, NH , CrH andMnH and also for a number of paramagnetic atoms. Buffer-gas cooling followedby evaporative cooling has recently been used to achieve Bose-Einstein condensationwith no laser cooling for metastable helium. The direct methods established so far are limited to temperatures of 10 to 100 mKand above. To cool the molecules further, to the m K regime, second-stage coolingmethods must be developed. The most promising and conceptually the simplest methodis sympathetic cooling , in which the molecules are cooled by collisions with an atomicgas that can itself be cooled to the ultracold regime, such as an alkali metal. Themost robust trapping methods for molecules work for low-field-seeking states, whichare never the lowest possible state in an applied field. Inelastic collisions can there-fore occur, and either heat the trapped system or eject the molecules from the trap.Sympathetic cooling can thus be successful only if elastic collisions dominate inelasticones, and it is usually stated that the ratio of elastic to inelastic cross sections mustbe 100 or more. Sympathetic cooling was initially developed as a cooling methodfor trapped ions. More recently it has been used to achieve sub-Kelvin temperaturesfor polyatomic ions and has also been used to produce ultracold neutral atoms withscattering properties that are not suitable for evaporative cooling, such as K. Sympathetic cooling for molecules has not yet been achieved, but several propos-als have been explored. It was initially proposed for NH molecules colliding with Rbatoms and studied in more details by Lara et al. for OH colliding with Rb. BothOH and NH molecules interact very strongly with Rb and the anisotropy of the in-teraction potential is large compared to the molecular rotational constant. The largeanisotropy implies large couplings between channels with different n (monomer ro-tation angular momentum) and L (end-over-end angular momentum) quantum num-bers, and Lara et al. showed that this resulted in large inelastic cross-sections in theultracold regime. The remedies they suggested to improve sympathetic cooling anddecrease inelastic cross sections were: (i) to use light atoms as coolants, in order to3ncrease the heights of centrifugal barriers and suppress inelastic channels; (ii) to findatom-molecule system with much smaller anisotropy in the interaction potential.Soldán et al. considered the possibility of reducing the anisotropy by using alkaline-earth atoms (Ae) as collision partners for NH molecules. They showed that the neutralstates of Ae–NH systems are coupled to ion-pair states Ae + NH − , with crossings be-tween the neutral and ion-pair surfaces at linear geometries. For Sr and Ca atoms thecrossings occurs at energies below the atom-molecule threshold, so will be accessiblein low-energy collisions. However, for Be-NH and Mg-NH the crossings occurs atenergies more than 1000 cm − above the atom-molecule threshold. In these systems,the ion-pair state is likely to be inaccessible, so it is reasonable to carry out collisioncalculations on a single covalent surface. In addition, the potential energy surface forMg–NH turned out to be only weakly anisotropic. Wallis and Hutson carried outquantum scattering calculations of spin relaxation collisions (in magnetic fields) andshowed that sympathetic cooling of NH by collisions with Mg atoms should be achiev-able if the molecules can be precooled to about 10 mK.Sympathetic cooling has also been considered for NH and ND . In this case themolecules are initially slowed in a Stark decelerator. ˙Zuchowski et al. surveyed theinteraction potentials for NH interacting with alkali-metal and alkaline-earth atoms.˙Zuchowski and Hutson then carried out quantum scattering calculations on collisionsof ND with Rb atoms and showed that molecules that are initially in the upper compo-nent of the ammonia inversion doublet are likely to undergo fast collisional relaxationto the ground state, and that this is likely to prevent sympathetic cooling of moleculestrapped in low-field-seeking states in an electrostatic trap. However, there is a goodprospect for sympathetic cooling of ammonia molecules in high-field-seeking states,even with magnetically trapped atoms, because the terms in the hamiltonian that mightcause spin-changing collisions of the Rb atoms are very small. High-field-seekingstates of ND can be confined in an alternating current trap. Recently, Hummon et al. demonstrated buffer-gas cooling and trapping of N ( S)atoms and simultaneous co-trapping of NH molecules. Subsequent work has demon-strated N atom densities around 5 × cm − and lifetimes around 10 s. This offers4he possibility of cooling the atoms further with atomic evaporative cooling, which hasalready been achieved for metastable helium and Cr atoms. A gas of N atoms is potentially an excellent coolant for a sympathetic coolingexperiment. The N atom has a very small polarizability compared to Group I andGroup II elements and this results in low C coefficients and small anisotropies ofthe interaction potentials with molecules. The N atom also has a relatively low mass,which results in higher centrifugal barriers and stronger suppression of inelasticity forparticles scattered with L > N or N. We assume that both N and NH are initiallyin their magnetically trappable spin-stretched states, with the maximum possible valuesof the electron spin projection numbers. For such states only spin relaxation (and notspin exchange) can occur and only the sextet interaction potential contributes. Wereport calculations of the sextet potential for N–NH and explore the behaviour of crosssections as a function of collision energy and magnetic field. We discuss the sensitivityof the scattering results with respect to uncertainties in the interaction potential. Finally,we analyze the behaviour of the shape resonances in terms of angular-momentum-insensitive quantum-defect theory (AQDT). The total spin of the N( S) + NH( S − ) system can be , or . The chemical reactionN + NH → N + H, which occurs principally on the doublet surface, has been studied indetail by Varandas and coworkers and by Francombe and Nyman. It was shownthat the doublet N–NH system forms an N H complex without a potential barrier alongthe minimum energy path. A very small barrier exists between the N H complex andN + H products and overall the reaction of forming N +H yields 6.33 eV of energy.5able 1: Basis-set dependence of the N–NH interaction energy at the global minimumfor F12 calculations. The complete basis set (CBS) extrapolation was obtained with thecorrelation energy functional E ( X ) = A + BX − where X is the maximum angularmomentum of electronic basis set.basis set E int (cm − )aug-cc-pVTZ − . − . − . − . − . − . To obtain the sextet interaction potential we applied the recently developed ex-plicitly correlated, unrestricted coupled-cluster method with single, double and non-iterative triple excitations [UCCSD(T)].
We used the aug-cc-pVTZ basis set ofPeterson et al. , which is designed specifically for use with explicitly correlated cal-culations. The results from the explicitly correlated (F12) calculation are comparedwith those from UCCSD(T) calculations with uncorrelated basis sets in Table 1: it maybe seen that the explicitly correlated approach dramatically reduces the error caused byusing unsaturated basis sets. A fixed NH bond length of 1.0367 Å was used in all thecalculations.The potential energy surface was obtained by carrying out explicitly correlatedUCCSD(T) calculations on a grid in Jacobi coordinates ( R i , q j ) , where R is the in-termolecular distance measured to the NH center of mass and q is the angle betweenthe NH bond vector and the vector from the NH center of mass to the N atom. The ra-dial grid R i was from 2.5 to 10 Å in 0.25 Å steps and the angular grid q j was a set of 11Gauss-Lobatto quadrature points, which include the two linear geometries. All inter-action energies were corrected for basis-set superposition error using the counterpoiseprocedure. Radial interpolation is carried out using the reproducing kernel Hilbert space (RKHS)method to evaluate V ( R , q j ) for arbitrary R and given q j . At each distance R , the6able 2: Van der Waals coefficients for N–NH ( E h a n ) from density-functional calcula-tions. n , l C n , l , , , , , , P l ( cos q ) for l up to 8, V ( R , q ) = (cid:229) l V l ( R ) P l ( cos q ) . (1)The coefficients V l ( R ) are obtained by integrating the ab initio potential using Gauss-Lobatto quadrature. To provide an improved description of the long-range interaction, we impose ananalytical representation on the long-range part of the components of the projectedpotential, V lr l ( R ) = − (cid:229) n = (cid:229) l = C n , l R − n . (2)The Van der Waals coefficients are given in Table 2 and were calculated with therestricted open-shell coupled Kohn-Sham method with asymptotically corrected PBE0 functional. We connect the long-range function smoothly to the supermolecu-lar potential using the switching function f ( R ) = +
14 sin p x (cid:0) − sin p x (cid:1) , (3)where x = R − b + R − ab − a with a = b =
11 Å. f ( R ) = R < f ( R ) = R >
11 Å.The potential energy surface for N–NH is shown in Fig. 1. It has two minima ofcomparable depths at linear geometries: 89.1 cm − at N–NH and 76.4 cm − at N–HN. The two minima are separated by a saddle point near the T-shaped geometry. Theanisotropy of the potential near the Van der Waals minimum is about 40 cm − , and the7able 3: Characteristic points on the N–NH potential energy surface. Energies aregiven in cm − , R in Å.Global minimum Saddle point Secondary minimum R , q ◦ ◦ -89.1 -39.2 -76.4 − − − − − −60 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − q / (cid:176) R / A Figure 1: The ab initio interaction potential of N–NH. Contours are labeled in cm − .The angle q = V ( R ) . An important problem in electronic structure theory is the estimation of error boundsfor calculated interaction energies. Since scattering calculations at very low energiesdepend strongly on the details of the interaction, in this section we discuss the uncer-tainty of the N–NH interaction potential obtained here.The largest contributions to the uncertainty of the interaction potential arise fromthe approximate treatment of electronic correlation and the incompleteness of the elec-tronic basis set. We expect that the effect of the neglecting vibrations of the NHmolecule is much less important, as are relativistic and nonadiabatic effects.First we need to estimate how well the UCCSD(T) method works for sextet N–NH.To explore this, we performed 7-electron full configuration-interaction (FCI) calcula-8ions of the interaction energy. We included all electrons arising from the H atom andthe 2 p electrons of N atoms. With FCI it was possible to use only a very small basis set(6-31G, augmented with spd midbond functions with an exponent 0.4). For this smallbasis set, we compared the contribution to the correlation part of the interaction energy(i.e., the intramonomer correlation and the dispersion energy) with the UCCSD(T) re-sults for several linear geometries N–NH and N–HN . The FCI correlation energy to belarger than the UCCSD(T) correlation energy by 1 to 1.5%, for a wide range of R and atboth linear geometries. To a good approximation we expect the ratio E FCIcorr / E UCCSD ( T ) corr to be constant in different basis sets. This suggests that the global minimum energyobtained with the coupled-cluster method is underestimated by ca. 1.5 cm − .The basis set convergence pattern shown in Table 1 yields a complete basis-setlimit of the global minimum depth of 90.47 cm − . This is 1.37 cm − more than inthe method used for the complete surface here. We also performed test calculationsincluding additional core-valence basis functions that are absent in the basis set used forthe complete surface potential. The interaction energy at the global minimum obtainedwith aug-cc-pCVTZ is approximately 1 cm − s maller than for basis sets with no core-valence functions.In summary we can set the error bounds on the interaction potential at the globalminimum between − − , which is approximately between −
1% and +3%.
The Hamiltonian of the NH molecule may be written H NH = b NH ˆ N + g ˆ N · ˆ S + (cid:20) p (cid:21) l SS (cid:229) q ( − ) q Y , − q ( ˆ r ) [ S ⊗ S ] ( ) q . (4)The three terms are, respectively, the rigid rotor Hamiltonian, the spin-rotation interac-tion and the intramonomer spin-spin interaction. The numerical values of the constantsused in the present work are b NH = .
343 cm − , g = − .
055 cm − and l SS = . − . The NH molecule is assumed to be in its ground vibrational state.The Hamiltonian of the N–NH collision system in a magnetic field may be written H = − ¯ h m R d dR R + ˆ L m R + H NH + H Z + V SS + V int ( R , q ) . (5)Here ˆ L is the operator for the end-over-end angular momentum of N and NH aboutone another, H Z represents the Zeeman interaction of N and NH with the magneticfield, V SS is the (anisotropic) intermolecular spin-spin interaction, and V int ( R , q ) is theintermolecular potential.The convention for quantum numbers in this paper is as follows: L and M L denotethe end-over-end angular momentum and its projection onto the space-fixed Z axisdefined by the magnetic field. Monomer quantum numbers are indicated with lower-case letters to avoid confusion with those of the collision system as a whole. Thespins and spin projections of the N and NH molecules are denoted by s A , s B and m s A ,and m s B , respectively. The rotational quantum number of the NH molecule and itsprojection are denoted n B and m n B The projection of the total angular momentum, M tot = M L + m n B + m s B + m s A , (6)is rigorously conserved in a collision, but the total angular momentum itself is not,except at zero field. It is convenient to carry out scattering calculations is a fully un-coupled basis set, | s A m s A i| s B m s B i| n B m n B i| LM L i . We have written a plug-in routine forthe MOLSCAT scattering program, implementing all the matrix elements requiredfor scattering calculations in this basis set.set.The total spin S of a system made up of an open-shell atom and an open-shellmolecule can take values between | s A − s B | and s A + s B . For N–NH the allowed valuesare S = , and , corresponding to doublet, quartet and sextet, respectively. The10nteraction potential V int ( R , q ) may be written in terms of projection operators, V int ( R , q ) = s A + s B (cid:229) S = −| s A + s B | | S i V S ( R , q ) h S | (7)and the general matrix element of V int ( R , q ) in our basis set is h s A m s A s B m s B n B m n B LM L | V int ( R , q ) | s A m ′ s A s B m ′ s B n ′ B m ′ n B L ′ M ′ L i = (cid:229) S ( − ) s A + s B − m s A − m s B − M L ( S + ) h n B m n B LM L | V S ( R , q ) | n ′ B m ′ n B L ′ M ′ L i s A s B Sm s A m s B − m s A − m s B s A s B Sm ′ s A m ′ s B − m ′ s A − m ′ s B . (8)The three interaction potentials V S ( R , q ) differ only by short-range Pauli exchangeterms. They have the same long-range coefficients, so become degenerate once theN atom and NH molecule are far enough apart that their valence shells do not over-lap. The doublet surface has a potential well several hundred times deeper than theVan der Waals sextet state, so that full quantum calculations including the doubletstate would require very large basis sets of rotational functions and could not be con-verged. In the present work we therefore approximate the operator V int ( R , q ) operatorby taking V S = V / for all spin states. This approximation is legitimate because weare primarily interested in N–NH collisions between magnetically trapped atoms andmolecules, with m s A = s A = and m s B = s B =
1. These are spin-stretched states,and V / and V / have no matrix elements (diagonal or off-diagonal) involving spin-stretched states. When this approximation is made, orthogonality relations for the 3 j symbols reduce Eq. 8 to a form diagonal both in m s A and m s B . The explicit expres-sion for h n B m n B LM L | V S ( R , q ) | n B m ′ n B L ′ M ′ L i is the same as for scattering of NH from aclosed-shell atom, with the addition of factors d m s A m ′ s A .11he intermolecular spin-spin interaction has matrix elements h s A m s A s B m s B n B m n B LM L | V SS | s A m ′ s A s B m ′ s B n ′ B m ′ n B L ′ M ′ L i = √ l ( R ) d n B n ′ B d m n B m ′ n B ( − ) s A + s B − m s A − m s B − M L [ s A ( s A + )( s A + ) s B ( s B + )( s B + )( L + )( L ′ + )] L L ′ (cid:229) q q L L ′ − M L − q − q M ′ L q q − q − q s A s A − m s A q m ′ s A s B s B − m s B q m ′ s B . (9)The spin-spin coupling constant l ( R ) is E h a a / R , where E h is the Hartree energyand a is the fine-structure constant.The matrix elements for NH monomer operators are the same as for scattering ofNH from a closed-shell atom, with the addition of factors d m s A m ′ s A .If one or both of the colliding species is not in a spin-stretched state (with the high-est possible value of m S ), the system will undergo very fast spin exchange driven by the difference between the S = , and S = potentials. For spin-stretched states, how-ever, spin exchange cannot occur and only spin relaxation is possible. There are twomechanisms for spin relaxation. The first is similar to the well-known mechanism ofspin relaxation for spin-stretched states of alkali metal atoms, and arises through directcoupling of the initial state m s A = + , m s B = + n B = m n B = m s A and/or m s B reduced by 1 by the intermolecular spin-spin interaction term and M L increased to conserve M tot . Such transitions are relatively slow, because the in-termolecular spin-spin interaction is weak. The second mechanism is that describedby Krems and Dalgarno. For the same initial state, the intramonomer spin-spin in-teraction mixes n B = n B =
2, and even in a magnetic field it mixes m n B = m n B = ± , ±
2. The states with m n B = + , + m s B = , −
1. The states with m n = , n = m n = M L but the same m B (which is lower than in the initial state). Thismechanism is also expected to be fairly weak for a low-anisotropy system such as N–NH: the n = n = − , while the12otential anisotropy V ( R ) that couples them is a short-range interaction that is nevergreater than 40 cm − in the energetically accessible region. For both mechanisms,spin relaxation is suppressed for s-wave scattering ( L = , M L =
0) at low energies andfields because the conservation of M tot requires M ′ L = L ′ >
0, producinga centrifugal barrier in the outgoing channel.We carry out scattering calculations with the MOLSCAT package. The coupledequations are solved using the hybrid log-derivative/Airy propagator of Alexander andManolopoulos. We used the fixed-step log-derivative propagator from 2.8 Å to 70Å with an interval size of 0.08 Å, followed by a variable-step Airy propagation outto 400 Å. We carried out convergence tests on state-to-state cross sections both in thes-wave regime and at energies up to E = B =
200 G, 1000 G and2 T. In all cases a basis set with n = . . . L = . . . Fig. 2 shows the Zeeman energy levels of the noninteracting N+NH system. In thebuffer-gas cooling experiment, both atoms and molecules are trapped in their low-field-seeking state with m s A = and n B = m n B = m s B =
1. The experiment hasalready achieved temperatures around 550 mK, and at this temperature atoms with anenergy of 5 kT sample magnetic fields up to 2 T in a quadrupole trap. However, as thetemperature decreases, so too will the magnetic fields sampled. We therefore considercollision energies from 10 m K to 1 K and fields from 10 G to 2 T.The N atom is considerably less polarizable than alkali metal or alkaline earthatoms. As a result, the dispersion coefficient C , for N–NH is considerably lowerthan for most metal atom – molecule systems that have been considered previously ascandidates for sympathetic cooling. Together with a low reduced mass, this results inrelatively high centrifugal barriers for L > L =
1, 71 mK for L =
2, 120 mK for L =
3, etc. The high centrifugal barriers also mean that quite small13 e n e r gy / K B / Gauss m sA =3/2 m sB =1 m sA =3/2 m sB =0,m sA =1/2 m sB =1 m sA =1/2 m sB =0,m sA =-1/2 m sB =1 m sA =3/2 m sB =-1 m sA =-1/2 m sB =0,m sA =-3/2 m sB =1 m sA = 1/2 m sB =-1 m sA =-3/2 m sB =0,m sA =-1/2 m sB =-1 m sA =-3/2 m sB =-1 Figure 2: Energy levels of the noninteracting N+NH system in a magnetic field. Thedotted red lines show the energy obtained by adding the d-wave centrifugal barrierheight (71 mK) to the levels with M tot = and M tot = . The crossings between thered lines and the initial-state energy indicate the fields above which s-wave inelasticcross sections are no longer suppressed by centrifugal barriers.number of partial waves are needed to converge cross sections: for example, includingcontributions from L up to 4 is sufficient to obtain convergence up to about 0.5 K. Calculated elastic and inelastic cross sections for the four different isotopic combina-tions are shown as a function of collision energy E in Fig. 3, for representative magneticfields of 200 G, 300 G, 1000 G and 2 T. In a simple hard-sphere model of sympatheticcooling, neglecting inelastic collisions, the temperature relaxes towards equilibriumand reaches a 1/ e point after ( m + m ) / m m collisions, where m and m are themasses of the two species. For sympathetic cooling to be successful we need the ratioof elastic to inelastic cross sections to be much larger than this. The calculated ratiosare shown in Fig. 4: for the most part they are more than 50 at collision energies above14 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH 200 G 300 G1000 G 2 T 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH p d 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH p d 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K N− NH p d
Figure 3: Elastic and inelastic cross sections for N–NH scattering for N– NH, N– NH, N– NH and N– NH for different magnetic fields. The elastic cross section(black line) is almost independent of field. The positions of the p- and d-wave exit-channel barriers are marked with vertical lines.15 m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH 200 G 300 G1000 G 2 T 1 10 100 1000 10000 100 m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH 1 10 100 1000 10000 100 m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH 1 10 100 1000 10000 100 m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH 1 10 100 1000 10000 100 m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH 1 10 100 1000 10000 100 m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH 1 10 100 1000 10000 100 m K 1mK 10 mK 100 mK 1K e l a s t i c / i ne l a s t i c collision energy / K N− NH Figure 4: The ratio of elastic to inelastic cross sections for N – NH, N – NH, N – NH and N – NH systems for different magnetic fields. The value of 100typically required for sympathetic cooling is marked with a dotted horizontal line.about 1 mK, indicating that sympathetic cooling of NH by N is likely to be feasible.Several different effects are evident in Fig. 3. The first is that the cross sectionsenter the s-wave regime, where they are proportional to E − / , at quite different en-ergies for different isotopic species. This occurs because of p-wave resonant effects.Once in the s-wave regime, however, the inelastic cross sections generally decrease atmagnetic fields below about 300 G, because of the centrifugal barriers in the outgoingchannels. Since atoms and molecules in a quadrupole trap sample lower and lowerfields as the temperature is decreased, this indicates that sympathetic cooling will be-come increasingly effective as the temperature is lowered, as predicted for Mg-NH. Lastly, inelastic collisions are also suppressed for very high magnetic fields. All theseeffects will be discussed in more detail below.Spin relaxation collisions can change m s A for the N atom, m s B for the NH molecule,or both. Fig. 5 shows the state-to-state cross sections for the most important final statesfor N– NH as a function of energy at two different fields. It may be seen that domi-16 m K 1mK 10 mK 100 mK s / Å collision energy / K 0.1 T total inelastic|3/2,0>|1/2,0>|3/2,−1>|1/2,1> s / Å
10 G
Figure 5: State-to-state inelastic cross sections for N – NH for weak (10 G) andstrong (0.1 T) magnetic fields. 17ant final states are those in which m s A and/or m s B has changed by 1. These collisionsare driven by the intermolecular spin-spin interaction. Transitions that change m s B by2 can occur only by the second mechanism described in Section 3 above, involving thepotential anisotropy, and are seen to be very much weaker except in a small resonantregion.The N– NH system shows behaviour quite different from the others, with a largepeak in the inelastic cross sections near 10 mK which reduces the elastic-to-inelasticratio to around 10. This ratio may not be high enough for effective sympathetic coolingfrom an initial temperature of tens of milliKelvin. The peak appears at the same energyfor all values of the field. It arises from a p-wave shape resonance in the incomingchannel, as discussed in section 4.3 below. For the larger reduced masses of the otherisotopic combinations, the quasibound state responsible for the shape resonance dropsbelow threshold and becomes a true bound state. Thus the other isotopic combinationsdo not exhibit this feature and have more favourable properties for sympathetic cooling.The N– NH system exhibits d-wave shape resonances for collision energies of50 to 70 mK, but they are much weaker than the p-wave resonance for N– NH andtheir presence does not strongly affect the total inelastic cross section.The L = L =
0, spin relaxation requires outgoing L ≥
2. If the energy difference between the incoming and outgoing channels is smallerthan the height of the L = N– NH at a collision energy of 50 m K. The inelasticcross sections generally decrease at magnetic fields below about 500 G, though thereis a dip in each state-to-state component between 100 and 300 G. These dips are dueto suppression of the inelastic cross sections in the wings of resonances, as describedby Hutson et al. ; in this case the resonances concerned are shape resonances in thed-wave outgoing channels.The suppression of inelastic scattering by the centrifugal barrier is clear in the totalinelastic cross section only for systems with reduced mass larger than for N– NH.18 s / Å magnetic field / G total inelastic D M tot =2 D M tot =1 0 50 100 150 200 0 200 400 600 800 1000 1200 1400 s / Å magnetic field / G Figure 6: Inelastic s-wave cross section as a function of magnetic field for N– NHat a collision energy of 50 m K. The dashed vertical lines correspond to the fields forwhich the kinetic energy release from M tot = to M tot = and M tot = states areequal to the height of the d-wave barrier.For the N– NH system itself, the p-wave contribution is very strong even at very lowenergies. In fact, the p-wave enhancement of the inelastic cross section between 50 m Kand 1 mK is so strong that the total inelastic cross section does not follow the E − power-law dependence expected from the Wigner threshold laws at these energies.Suppression of inelastic collisions due to barriers in the outgoing channels de-creases as the magnetic field increases (so that the kinetic energy release increases).Eventually, however, the inelastic cross section reaches a maximum and starts to de-crease again. This occurs for all partial waves, and the total cross sections at a field of2 T are typically reduced by a factor of about 10 from their values at 0.1 T. Since forsome isotopic combinations the low-field ratio of elastic to inelastic cross sections maynot be large enough at temperatures of 1 to 10 mK, the application of a strong bias fieldto a magnetic trap offers a possible way way to improve the ratio.Suppression of inelastic cross sections at high fields has been observed for O( P)-He collisions, for OH-OH and for collisions of Cr atoms. For small inelasticity,19
5 10 15 20 25 R − f i n f ou t / a r b i t r a r y un i t s R / Å 0.5 K1 K3 K Figure 7: The integrand of the distorted-wave Born approximation (Eq. 10) for N– NH at a collision energy of 1 mK for different kinetic energy releases. The incomingwavefunction is calculated for L = L = s i → f = p k − i (cid:12)(cid:12)(cid:12)(cid:12) Z y i ( R ) U i f ( R ) y f ( R ) dR (cid:12)(cid:12)(cid:12)(cid:12) , (10)where y i and y f are energy-normalized wavefunctions in the initial and final channels, U i f is the coupling between the channels, and k i is the wave vector in the incomingchannel. Fig. 7 shows the integrand of Eq. 10 for the intermolecular spin-spin term( R − ) at kinetic energy releases of 0.5 K, 1 K and 3 K, corresponding to fields of3800 G, 7600 G and 2.3 T for transitions with D m s A + D m s B = −
1. It may be seen thatthere is significant oscillatory cancellation in the integral at high fields, when y i ( R ) and y f ( R ) oscillate out of phase with one another in the interaction region, and thiscombines with the effect of the resonances in the d-wave outgoing channels to producethe maxima in Fig. 6. The oscillatory cancellation occurs for arbitrary partial waves.20
100 1000 100000.95 1.00 1.05 1.10 s / Å l elastic 1 10 100 1000 10000 s / Å inelasticV−> l V m −> lm Figure 8: Elastic and total inelastic cross sections for N– NH at E = B =
50 G as a function of both reduced mass and potential scaling factors.
Because of the crucial role played by the p-wave shape resonance for N– NH atenergies up to 10 mK, it is important to investigate the influence of uncertainties in theinteraction potential on the cross sections. The shape of the 2D interaction potential iscomplicated and the scattering properties might in principle depend on many param-eters. However, Gribakin and Flambaum showed that for single-channel scatteringthe scattering length a behaves as a = ¯ a h − tan (cid:16) F − p (cid:17)i , (11)where, for a potential with long-range form − C R − , the mean scattering length ¯ a is0 . ( m C / ¯ h ) and F = Z ¥ (cid:0) m V int ( R ) / ¯ h (cid:1) dR . (12)Although N–NH is a many-channel scattering problem, it is elastically dominated andEq. 12 with V int ( R ) replaced by V ( R ) reproduces the major features of the elastic scat-tering. Thus scaling m is approximately equivalent to scaling the entire interaction21otential, and either of these scalings provides a good way to explore the variation ofscattering length as a function of potential. Fig. 8 shows the elastic and total inelasticcross sections for N– NH at E = B =
50 G as a function of both reducedmass and potential scaling factors. It may be seen that the two scalings have a verysimilar effect, apart from a small shift in the Feshbach resonance around l = . − m → lm fora collision energy of 5 mK. A scaling factor l = N– NH. The result is shown in Fig. 9. There is a strong maximum in the totalinelastic cross section near l = . elastic channel. A change of 1.2% in the potential is within the estimated error bound of ourcalculations. Enhancement of the cross sections due to the p-wave resonance mightthus occur for heavier isotopic combinations than N– NH if our potential is slightlytoo deep. Although we believe it is more likely that our potential is too shallow thantoo deep, this cannot be ruled out. However, it is quite unlikely that enhancement dueto the p-wave resonance would occur for the heaviest system, N– NH.
In this section we consider the N–NH scattering in the context of angular-momentum-insensitive quantum defect theory (AQDT). The upper part of Fig. 10 shows thepositions of (quasi)bound states for L = . . . l for values between 0.8 and 1.2. The L = N– NH, well outside theestimated error bounds for the potential. There is thus no s-wave resonance in thescattering for any of the systems considered here.The p-wave shape resonance in the cross sections for N– NH arises from thequasibound state with L =
1, which is 6 mK above threshold for l =
1. As the reducedmass increases above this, the L = l = .
024 and22 s / Å l error of the interaction potential elasticinelastic 1 10 100 1000 10000 1000000.95 1.00 1.05 1.10 s / Å l error of the interaction potential s / Å l error of the interaction potential Figure 9: Elastic and inelastic cross sections for a collision energy of 5 mK as a functionof scaled reduced mass. The dashed vertical lines indicate the values of m for thesystems concerned here, from left to right, N– NH, N– NH, N– NH and N– NH, respectively. The solid vertical line shows the value l = .
024 for which thescattering length a = a . 23ecomes a "real" bound state. This explains why we see no p-wave shape resonancesin the cross sections for collisions with reduced mass larger than 1 . m , , i.e. forall systems containing at least one N atom. For large values of m the L = N– NH cross sections.The lower panel of Fig. 10 shows the s-wave scattering length as a function ofscaling factor l . As expected, there is a pole near l = . L = a variesbetween 22.3 and 15.2 Å, so that the elastic cross section for small E coll (in the Wignerregime) varies between 6250 and 2900 Å .In angular-momentum-insensitive quantum defect theory, the scattering prop-erties of a system for arbitrary L can be predicted from only a few parameters: thes-wave scattering length a , the dispersion coefficient C and the reduced mass m (andthus ¯ a ). The positions where L > a and ¯ a . In particular, when a = a there is an L = a slightlylarger than 2 ¯ a have a p-wave shape resonance at a collision energies below the heightof the p-wave centrifugal barrier. This is the case for the N– NH system here. For a = ¯ a there is an L = N– NH is 1 .
54 ¯ a , which is close enough above ¯ a to produce a d-wave shape resonance atfinite energy. The energies at which the p-wave and d-wave resonances appear can beread off the L = C L ( E ) functions introduced by Mies. These functions give the connection be-tween a semiclassical JWKB description of scattering states (valid at large collision24 ene r g y / K l p−wave barrierd−wave barrier L=0 L=1 L=2 −20 0 20 40 60 80 1000.8 0.9 1.0 1.1 1.2 sc a tt e r i ng l eng t h / Å l Figure 10: Near-threshold bound states of sextet N–NH for L = . . . m (upper panel), and the m -dependence of the s-wave scattering length (lower panel).Field-free calculations with a structureless atom and diatom were used to obtain thebound states here. The dashed vertical lines indicate the values of m for N– NH, N– NH, N– NH and N– NH, respectively, from left to right. The red dottedline on the lower panel indicates 2 ¯ a and the green dotted line indicates ¯ a .25 m K 1mK 10 mK 100 mK C − ( E ) collision energy / K N− NH m res14 N− NH Figure 11: The p-wave transmission function C − ( E ) for N–NH systems with reducedmasses corresponding to: N– NH ( a / ¯ a = . N– NH ( a / ¯ a = . l = . a / ¯ a → + .energies) and the near-threshold behaviour. The function C − L ( E ) can be viewed as anenhancement factor in the short-range part of the wavefunction due to the presence ofthe long-range potential, including any resonant effects in the incoming channel. InFig. 11 we show the C − ( E ) functions for p-wave scattering with l =
1, 1 .
023 and1.031 (the last of these values corresponding to the reduced mass of N– NH). Asthe scattering length decreases and reaches 2 ¯ a , the height of the peak in C − ( E ) goesto + ¥ , as shown in Fig. 12, and the energy at which the peak occurs approaches zero.The width of the resonance decreases, corresponding to increasing the lifetime of thequasibound state. The intensity of the resonance rapidly decreases once the scatteringlength is larger than 2 ¯ a , corresponding to a bound state below threshold. We have calculated a potential energy surface for N atoms ( S ) interacting with NHmolecules ( S − ) in the spin- (sextet) state, using unrestricted coupled-cluster calcula-tions with an explicitly correlated basis set. This is the surface that governs collisions26 l Figure 12: The height of the peak in the C − ( E ) function as a function of reduced-massscaling factor l .of cold N atoms and NH molecules in a magnetic trap. We have used the surface tocarry out quantum scattering calculations of cold collisions for different isotopic com-binations of N and NH, as a function of collision energy and magnetic field.The sextet potential energy surface is weakly anisotropic, with an anisotropy of ap-proximately 40 cm − in the well region. The anisotropy is dominated by the P ( cos q ) Legendre, which mixes states with D n = ± n . Sincethe anisotropy is smaller than the separation between the n = N– NH. The scaling revealed that there are ma-27or effects arising for a p-wave shape resonance, which produce enhanced inelasticscattering for N– NH at low energies on our best potential surface. We have usedangular-momentum-insensitive quantum defect theory (AQDT) to understand how theresults change for different isotopic combinations.Scaling the potential energy surface, or equivalently the reduced mass, is a veryuseful tool for understanding cold collision calculations. In combination with AQDT,it can provide powerful insights into low-energy scattering for low-energy collisionswhere where only a few partial waves contribute to the scattering.
Acknowledgments
We are grateful to John Doyle for interesting us in this problem, and to Alisdair Wallisfor valuable discussions. This work is supported by EPSRC under collaborative projectCoPoMol of the ESF EUROCORES Programme EuroQUAM.
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Phys. Rev. A , 1993, , 546.[76] B. Gao, Phys. Rev. A , 1998, , 1728.[77] B. Gao, Phys. Rev. A , 2000, , 050702.[78] F. H. Mies, J. Chem. Phys. , 1984, , 2514.[79] F. H. Mies and P. S. Julienne, J. Chem. Phys. , 1984, , 2526.33 { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K Swave elasticSwave inelastic B=50Swave inelastic B=1000inelastic B=50inelastic B=1000tot elastic m K 1mK 10 mK 100 mK C − ( E ) collision energy / K N− NH m res N− NH { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K inelastic B=50inelastic B=1000inelastic B=5000inelastic B=10000inelastic B=20000tot elastictot elastic m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 50 G 0.1 T 2 T 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D s / Å collision energy / K 200 G 200 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 1000 G 1000 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 20000 G 20000 G m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D m K 1mK 10 mK 100 mK 1K s / Å collision energy / K 100 G, potential x 1.02 elasticinelasticS−wave inelasticS−wave elasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K elasticinelasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K P D F ’barr’ m K 1mK 10 mK 100 mK 1K s / Å2
Phys. Rev. A , 1993, , 546.[76] B. Gao, Phys. Rev. A , 1998, , 1728.[77] B. Gao, Phys. Rev. A , 2000, , 050702.[78] F. H. Mies, J. Chem. Phys. , 1984, , 2514.[79] F. H. Mies and P. S. Julienne, J. Chem. Phys. , 1984, , 2526.33 { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K Swave elasticSwave inelastic B=50Swave inelastic B=1000inelastic B=50inelastic B=1000tot elastic m K 1mK 10 mK 100 mK C − ( E ) collision energy / K N− NH m res N− NH { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K inelastic B=50inelastic B=1000inelastic B=5000inelastic B=10000inelastic B=20000tot elastictot elastic m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 50 G 0.1 T 2 T 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D s / Å collision energy / K 200 G 200 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 1000 G 1000 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 20000 G 20000 G m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D m K 1mK 10 mK 100 mK 1K s / Å collision energy / K 100 G, potential x 1.02 elasticinelasticS−wave inelasticS−wave elasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K elasticinelasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K P D F ’barr’ m K 1mK 10 mK 100 mK 1K s / Å2 collision energy / K 100 G , 15N − NH elasticinelastics−wave elastics−wave inelasticp−wave inelasticp−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å2
Phys. Rev. A , 1993, , 546.[76] B. Gao, Phys. Rev. A , 1998, , 1728.[77] B. Gao, Phys. Rev. A , 2000, , 050702.[78] F. H. Mies, J. Chem. Phys. , 1984, , 2514.[79] F. H. Mies and P. S. Julienne, J. Chem. Phys. , 1984, , 2526.33 { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K Swave elasticSwave inelastic B=50Swave inelastic B=1000inelastic B=50inelastic B=1000tot elastic m K 1mK 10 mK 100 mK C − ( E ) collision energy / K N− NH m res N− NH { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K inelastic B=50inelastic B=1000inelastic B=5000inelastic B=10000inelastic B=20000tot elastictot elastic m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 50 G 0.1 T 2 T 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D s / Å collision energy / K 200 G 200 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 1000 G 1000 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 20000 G 20000 G m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D m K 1mK 10 mK 100 mK 1K s / Å collision energy / K 100 G, potential x 1.02 elasticinelasticS−wave inelasticS−wave elasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K elasticinelasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K P D F ’barr’ m K 1mK 10 mK 100 mK 1K s / Å2 collision energy / K 100 G , 15N − NH elasticinelastics−wave elastics−wave inelasticp−wave inelasticp−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å2 collision energy / K 100 G , 15N − NH P D F m K 1mK 10 mK 100 mK 1K s / Å collision energy / K 1000 G, potential x 1.02 elasticinelasticS−wave inelasticS−wave elasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K elasticinelasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K P D F ’barr’ m K 1mK 10 mK 100 mK 1K s / Å2
Phys. Rev. A , 1993, , 546.[76] B. Gao, Phys. Rev. A , 1998, , 1728.[77] B. Gao, Phys. Rev. A , 2000, , 050702.[78] F. H. Mies, J. Chem. Phys. , 1984, , 2514.[79] F. H. Mies and P. S. Julienne, J. Chem. Phys. , 1984, , 2526.33 { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K Swave elasticSwave inelastic B=50Swave inelastic B=1000inelastic B=50inelastic B=1000tot elastic m K 1mK 10 mK 100 mK C − ( E ) collision energy / K N− NH m res N− NH { / S y m bo l s } / A ^ {/Times−Italic E}_{kin} / K inelastic B=50inelastic B=1000inelastic B=5000inelastic B=10000inelastic B=20000tot elastictot elastic m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 50 G 0.1 T 2 T 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N− NH P D s / Å collision energy / K 200 G 200 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 1000 G 1000 G0.01 0.1 1 10 100 1000 10000 1mK 10 mK 100 mK 1K s / Å collision energy / K 20000 G 20000 G m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK s / Å collision energy / K N −− NH P D m K 1mK 10 mK 100 mK 1K s / Å collision energy / K 100 G, potential x 1.02 elasticinelasticS−wave inelasticS−wave elasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K elasticinelasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K P D F ’barr’ m K 1mK 10 mK 100 mK 1K s / Å2 collision energy / K 100 G , 15N − NH elasticinelastics−wave elastics−wave inelasticp−wave inelasticp−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å2 collision energy / K 100 G , 15N − NH P D F m K 1mK 10 mK 100 mK 1K s / Å collision energy / K 1000 G, potential x 1.02 elasticinelasticS−wave inelasticS−wave elasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K elasticinelasticS−wave inelasticS−wave elasticP−wave inelasticP−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å collision energy / K P D F ’barr’ m K 1mK 10 mK 100 mK 1K s / Å2 collision energy / K 1000 G , 15N − NH elasticinelastics−wave elastics−wave inelasticp−wave inelasticp−wave elastic 0.1 1 10 100 1000 10000 100000 100 m K 1mK 10 mK 100 mK 1K s / Å2