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the probable success of experiments aimed at producing magnetically trapped ultracold molecular samples depends heavily on the effects of collisional processes . for example , paramagnetic alkali dimers can be produced via photassociation of ultracold atoms @xcite , but the resulting molecules , typically in high - lying vibrational states , are subject to vibrational quenching collisions @xcite which can release a large amount of energy and dramatically affect the efficiency of the cooling . alternatively , cold molecules in their vibrational ground states can be produced either by thermal contact with a cold helium buffer gas @xcite or by stark slowing , for species that possess an electric dipole moment @xcite . collisions are of obvious importance to buffer - gas cooling ( bgc ) , as well as to forced evaporative cooling ( ec ) that will be required to lower the temperature of these gases further and achieve for instance bose - einstein condensation ( bec ) . both processes require large elastic collision rates to thermalize the gas . so far ec has not been realized in practice for molecules , but the success of the bgc technique for the production of cold cah @xcite and pbo @xcite molecules suggests that in the near future it will be possible to achieve bec using cold molecules . this would open the way for a number of new and fascinating experiments .    in order to be magnetically trapped , atoms or molecules must be in a weak - field seeking state , i.e. a state whose energy increases with the strength of the magnetic field . for each trappable weak - field seeking state there is in general a lower - energy untrapped strong - field seeking state , in which the molecules experience a force away from the center of the trap . collisions can drive transitions between trapped and untrapped states . these `` bad '' collisions can cause heating or atom loss . it is therefore important to assess the rate constants for the inelastic collisions .    in a series of previous papers the resilience of molecular oxygen against spin - changing collisions was investigated , in collisions of o@xmath1 molecules both with a helium buffer gas @xcite , and with other o@xmath1 molecules @xcite . these studies found that spin - changing rates due to spin - rotation coupling could be quite large . however , in the case of the @xmath0o@xmath1 molecule , where in the limit of zero field the only allowed exit channels are energetically degenerate with the incident channels , spin - flipping transitions require boosting the centrifugal angular momentum from @xmath4 = 0 to @xmath4 = 2 , meaning that these processes are strongly suppressed by the wigner s threshold laws at collision energies smaller than the height of the exit channel centrifugal barrier .    the results in refs . @xcite considered only the case of a vanishing external magnetic field , which is obviously not the case in experiments that trap molecules using spatially inhomogeneous magnetic fields . the present paper therefore explores the role that fields play in determining spin - changing collision rates . as we demonstrate , the presence of a magnetic field causes a zeeman asymptotic splitting between incident and exit channels , thus lifting the collision energy higher relative to the centrifugal barrier in the exit channel , and removing the wigner s suppression . studies of spin - changing ultracold collisions in the presence of an external magnetic field have been performed so far only for atomic species @xcite . in this paper , we present a detailed dynamical study at cold and ultracold temperatures for the atom - diatom system @xmath0o@xmath1 @xmath5 @xmath2he in a field . the basic model is described in sec . ii . in sec . iii we calculate elastic and inelastic rate constants for collisions involving the lowest - lying trappable state of @xmath0o@xmath1 over a wide range of field values ( from 0 up to 5000 gauss ) , then discuss the dependence of the rates on collision energy for several representative values of the field . this system is of direct relevance to bgc of molecular oxygen . generally , it allows us to quantify the removal of the wigner s law suppression as the field increases in strength . on this basis we determine a simple one - parameter fitting formula that reproduces the trend with field and energy of the loss rates . in sec . iv we use this formula to extend previous results on o@xmath1-o@xmath1 collisions to estimate the influence of the field on ec of this system . as mentioned in the introduction , we will consider in this paper molecules consisting of two @xmath0o atoms , whose nuclear spin @xmath6 is equal to @xmath7 . we assume that total spin * i * = * i*@xmath8 + * i*@xmath1 is conserved in the collision and polarized to its maximum value * i * = 5 , implying that the even molecular rotational states @xmath9 are separated from the odd ones @xcite . we limit the discussions of this paper to the `` even-@xmath9 '' manifold of molecular states , which is more appealing for cooling purposes @xcite having a spin 1 paramagnetic ground state . the electronic spin * s * of the o@xmath1 molecule has magnitude @xmath10 = 1 in the electronic ground state @xmath11 we are concerned with throughout this paper . the angular momentum * s * is coupled to the molecular rotation angular momentum * n * to give * j * , the total molecular angular momentum , which assumes the values @xmath12 , @xmath9 and @xmath13 for @xmath9 @xmath14 0 and is 1 for @xmath9 = 0 . the hamiltonian operator @xmath15 for molecular oxygen in the presence of an external magnetic field @xmath16 can be written as : @xmath17 where @xmath18 is the rotational constant . the @xmath0o@xmath1 molecule is considered to be a rigid rotor , with internuclear distance frozen to the equilibrium value of @xmath19 = 2.282 bohr ( the rigid rotor model has been shown to be very accurate for this system at the investigated collision energies @xcite ) . the fine - structure hamiltonian @xmath20 and the hamiltonian @xmath21 for the interaction of the molecule with the external magnetic field follow the treatment in ref . @xcite , disregarding the molecular hyperfine interaction . the fine structure hamiltonian is given as @xcite @xmath22 where @xmath23 is a unit vector parallel to the molecular axis and @xmath24 is a second - rank tensor @xcite . the fine structure parameters @xmath25 and @xmath26 have been taken from ref . @xcite , where they have been determined by microwave spectroscopy . the interaction of the field with the electronic spin can be expressed as @xcite : @xmath27 where @xmath28 indicates the external magnetic field , @xmath29 is the g - factor of the electron and @xmath30 is the bohr magneton . following @xcite we ignore a small interaction between the field and the rotational angular momentum . the matrix elements for @xmath20 and @xmath21 have been given in ref . @xcite , eqs . ( a5 ) and ( a6 ) , for a hund s case b basis set . we note here that the molecular rotational quantum number @xmath9 is no longer strictly a good quantum number for the molecular states , because different values of @xmath9 are coupled together by the fine - structure hamiltonian @xmath20 and by the interaction with the external field . the molecular total angular momentum quantum number @xmath31 is still a good quantum number with respect to @xmath20 , but not with respect to the field interaction @xmath21 term . however , its projection @xmath32 on the quantization axis is still conserved . consequently , our basis functions should be labeled as @xmath33 , where @xmath34 is a shorthand index denoting the pair of quantum numbers ( @xmath35 ) in the field dressed basis @xcite . however , the coupling between different @xmath9 s is weak ( the fine - structure coupling is small compared to the rotational separation ) and @xmath9 can be considered `` almost '' a good quantum number . similarly , @xmath31 is also approximately good for laboratory - strength magnetic fields , so that we can use without confusion the label @xmath36 . magnetic trapping is strongly related to the behavior of the molecules in a magnetic field . the low - energy zeeman levels of oxygen are displayed in fig . 1 for the even-@xmath9 species . ( throughout this paper we report energies in units of kelvin by dividing by the boltzmann constant @xmath37 . these units are related to wave numbers via 1 k = 0.695 @xmath38 ) . in order to be trapped in the usual magnetic traps a molecule must be in a weak - field - seeking state , i.e. , one whose energy rises with increasing magnetic field strength . thus the state @xmath39 = @xmath40 is the lowest - lying trappable state of the even-@xmath9 manifold , and this is the state on which we focus our attention below . this state is indicated by a heavy line in figure 1 . higher - lying states with @xmath9 @xmath41 2 are energetically forbidden at low temperatures . it is clear from fig . 1 that for any trapped state there is an untrapped , strong - field - seeking zeeman state at a lower energy . these states are not merely untrapped but antitrapped , experiencing a force away from the trapping region . in a magnetic trap collisions with buffer gas atoms , or more generally with other molecules , will therefore ultimately deplete the trap of its molecular population . the time available for cooling processes like bgc or ec , as well as the lifetime of a molecular bose - einstein condensate , is therefore limited and knowledge of the rate constants for spin - flipping collisions is essential to predict their feasibility .    in ref . @xcite the theoretical framework for atom - diatom scattering was derived in the limit of zero external field , along the lines of the model originally due to arthurs and dalgarno @xcite , and properly modified to incorporate the electronic spin of the oxygen molecule . here , the formulation of the scattering problem is further extended to account for the interaction with the external magnetic field . the full hamiltonian operator describing the he - o@xmath1 collision is given by @xmath42 + { \bf \hat{h}}_{{\rm o}_2 } + v(r,\theta)\ ] ] after multiplying the wave function by @xmath43 in order to remove first derivatives . here @xmath44 is the reduced mass for the he - o@xmath1 system , @xmath43 is the modulus of the jacobi vector joining the atom to the molecule s center - of - mass , @xmath45 is the centrifugal angular momentum operator , and @xmath15 is the molecular oxygen hamiltonian defined in ( [ o2ham ] ) . the potential term @xmath46 , depending on both the jacobi vector @xmath43 and on the bending angle @xmath47 that the molecule s axis makes with respect to @xmath48 , accounts for the he - o@xmath1 interaction . we use the @xmath49 potential energy surface ( pes ) by cybulski et al . @xcite , which approximates the true well depth to @xmath50 20% . the full multichannel calculation requires casting @xmath51 in an appropriate angular momentum basis . our field dressed basis for close - coupling calculations is then latexmath:[\[\label{basis }     number @xmath10 is not explicitly indicated being always equal to 1 in the problem treated here . the quantum number @xmath4 stands for the partial wave representing the rotation of the molecule and the he atom about their common center of mass , @xmath53 is the projection of @xmath54 onto the laboratory axis , and @xmath55 is the laboratory projection of the total angular momentum , @xmath56 . at the collision energies of interest we assume that the oxygen electronic state and the helium atom state are preserved and therefore we suppress the first two kets in equation ( [ basis ] ) in the following . we note here that , at variance with the formulation in the zero field limit , the total angular momentum of the system @xmath57 ( equal to * n * + * s * + * l * ) is no longer a good quantum number , because different @xmath58 s are coupled together by the interaction with the external field . this means that the dynamical problem is no longer factorizable for different values of @xmath58 , thus requiring larger numbers of channels to be treated simultaneously . the problem is still factorizable for @xmath55 , but in general the number of channels to be included for each calculation is much larger than in the previous case . numerical details of the calculations will be given in the next section . the coupled - channel equations are then propagated using a log - derivative method @xcite and solved subject to scattering boundary conditions to yield a scattering matrix s : @xmath59 as already noted , the projection of the total angular momentum @xmath55 is still a good quantum number , implying that @xmath32 @xmath60 @xmath53 @xmath61 @xmath62 @xmath60 @xmath63 . note that in general each of the quantum numbers @xmath9 , @xmath31 , @xmath32 , @xmath4 , and @xmath53 are subject to change in a collision , consistent with conserving @xmath55 . following ref . @xcite , the state - to - state cross sections are given by : @xmath64 and the corresponding rate coefficients are given by @xmath65 where @xmath66 is the relative velocity of the collision partners before the collision . for notational convenience we will in the following refer to collisions that preserve the incident molecular quantum numbers as `` elastic , '' and those that change the quantum numbers as `` loss . '' in this section we consider elastic and state - changing , inelastic rate constants for the incident channel @xmath67 = @xmath68 . at the investigated collision energies ( from 1 @xmath44k up to 10 k ) there are two open inelastic channels , namely @xmath67 = @xmath69 and @xmath70 , both of which are untrapped ( see fig . we begin by computing rate constants for the elastic and spin - flipping transitions @xmath71 @xmath72 @xmath73 and @xmath71 @xmath72 @xmath74 in the low - field limit . results in this section refer to 1 @xmath44k collision energy and are converged using partial waves up to @xmath4 = 6 and including the rotational states @xmath9 = 0 , 2 , 4 and 6 for the oxygen molecules . the maximum value of @xmath43 to which the coupled - channel equations are propagated depends on the strength of the field , ranging from 600 bohr in the case of smallest field to 450 bohr for highest values . these parameters assure rate constants convergent within less then 5 % , which is adequate for our purposes .    in principle , the scattering matrix should be determined for each possible value of the projection of the total angular momentum @xmath55 . however , we know that s - wave collisions dominate the incident channel at ultralow collision energies , which corresponds for our incident channel to the value @xmath75 . we have verified that including in the calculation only the @xmath55 = 1 channels changes the results by less than 1 % at @xmath44k energies ; in this section we therefore include only this contribution . the number of channels to be propagated according to the given convergent quantum numbers is then only 205 . 2 shows elastic and inelastic rate constants at energy @xmath76k and for low values of the field . the inelastic rate constants are non zero even at zero field , as shown in ref . . however , in this limit the final states are degenerate with the initial state , and inelastic transitions are strongly suppressed by the presence of a @xmath77-wave centrifugal barrier in the exit channel , whose height is about 0.59 k. this effect is able to suppress the molecule loss , at least as long as the collision energy does not exceed the barrier height @xcite . as soon as a field is applied , the thresholds are no longer degenerate in energy , so that the energy in the exit channel is not as far below the centrifugal barrier . as a consequence , inelastic transitions are no longer as strongly suppressed , even in the limit of very low collision energy . rather , they increase dramatically even in a weak field , with rates being boosted by 5 or 6 orders of magnitude in a 1 gauss field . on the other hand , elastic scattering is nearly unchanged by the field . this sudden increase of the inelastic transition rates can be reproduced semiquantitatively by applying the distorted wave born approximation ( dwba ) @xcite , as has been successfully done for the magnetic dipolar interaction of cold alkali atoms @xcite as well as in a number of problems in cold collisions @xcite . the first order dwba is a simple two state perturbation approximation applicable in cases where inelastic scattering is weak in comparison with elastic scattering . the dwba expression for the off - diagonal k - matrix elements is : @xmath78 where @xmath79 and @xmath80 represent the energy normalized scattering wave function of the initial and final states calculated on the diabatic potential corresponding to the states involved in the inelastic transition . @xmath81 is the corresponding diabatic coupling term of the hamiltonian ( [ totalh ] ) . the first - order dwba result is shown as a dotted line in fig.2 . the dwba also yields information on the threshold dependence of the loss rates on energy and field . to see this , first note that the spin - changing processes we are considering are strongly dominated at low collision energy by s - waves in the incident channel , and by d - waves in the exit channel . this change of partial wave is necessary to conserve angular momentum during a collision that changes the molecule s spin . for small values of the magnetic field ( for which the zeeman splitting does not exceed the height of the exit channel centrifugal barrier ) the exit channel is still in the threshold regime , whereby the wave functions @xmath79 and @xmath80 in ( [ kmatr ] ) can be approximated by the small - argument limit of energy - normalized spherical bessel functions , @xmath82 where @xmath83 and @xmath84 are the incident and final wave numbers and @xmath85 and @xmath86 are the incident and final partial waves . assuming a small-@xmath43 cutoff to insure convergence of the integral in ( [ kmatr ] ) with respect to @xmath87 singularity in the coupling potential @xmath88 , the energy - dependence of the k - matrix element is @xmath89 by considering the relationship between @xmath90 and the effective rate constant @xmath91 , it is straightforward to show that the rate constant behaves approximately as @xmath92 where @xmath93 is the collision energy , and we have taken into account that the final kinetic energy in the exit channel is incremented by an amount @xmath94 = @xmath95 @xmath29 @xmath96 corresponding to the linear zeeman shift . @xmath95 (= @xmath32 @xmath5 @xmath62 ) stands for the difference between the initial and final values of @xmath32 in the two channels involved in the transition . the actual value of @xmath94 is modified by quadratic zeeman shift , but the linear approximation is adequate for achieving a simple fitting formula . in the present case these shifts amount to less than 10% changes in the approximated rate constants . from eq . ( [ prope ] ) , considering that in our case @xmath97 = 0 and @xmath98 = 2 , a simple expression for the rate constants can be derived : @xmath99 where @xmath100 represents an overall scaling constant and @xmath101 is conveniently chosen as the height of the centrifugal barrier in the exit , d - wave channel . in the limit of very low collision energy , the @xmath102 term obviously dominates over @xmath93 , leading to a nonzero rate constant , as is the case for exothermic collisions . this simple expression allows us to interpret the threshold behavior of the rates with the field , explaining the 5/2 exponential dependence on @xmath16 found in our calculation and shown explicitly in the bilogarithmic plot for the rate constants ( fig . the dashed lines represent a fit to the rate constants in the limit of zero magnetic field , yielding coefficients @xmath100 = 2.73 @xmath103 10@xmath104 cm@xmath2 sec@xmath106 and @xmath100 = 1.45 @xmath103 10@xmath104 cm@xmath2 sec@xmath106 for the transition to the final states @xmath73 and @xmath74 , respectively . apart from zeros in the actual rate constants , the overall trend is indeed @xmath107 . the zeros in the real rates arise from interferences between the s - and d - wave radial wave functions , as we have verified qualitatively by the dwba . nevertheless , the simple one - parameter expression ( [ ratek ] ) provides a reasonable upper bound to the complete calculation which , it will be recalled , requires a calculation involving 205 coupled channels . the simple formula ( [ ratek ] ) holds , of course , only when both incident and final channels are in the threshold regime . assuming low incident energies , this restriction therefore limits the size of magnetic field for which ( [ ratek ] ) applies . namely , this expression is only useful when the zeeman energy splitting between incident and final states remains smaller than the height of the d - wave centrifugal barrier . for the channels considered here , these fields are 2430 gauss and 4860 gauss for the @xmath108 and @xmath109 final states , respectively ; vertical arrows in fig . 3 indicate these field values . relation ( [ ratek ] ) serves as a useful quick fitting formula for data at smaller fields , and allows us to generalize the results obtained in this paper also to different systems . possible applications are discussed in section iv .    when stronger values of the magnetic field are considered , the dwba is no longer able to reproduce the full calculation , because the coupling is no longer a weak perturbation . strong field interaction mixes up different channels leading to a much more complicated picture which can not be explained in terms of a simple two state model . however , in this limit the loss rate constants seem to be unacceptably large anyway , owing to the effectiveness of spin - rotation coupling in changing spins . for example , for @xmath16 @xmath50 4800 gauss , the inelastic rate constant for the @xmath67 = @xmath73 exit channel exceeds that for the elastic one . this indicates that the high field limit fine structure changing collisions represent a potential limitation for the success of the collisional cooling processes .      in this section , the dependence of elastic and inelastic rate constants on the collision energy are analyzed for the zero field limit and for three representative values of the field , namely 10 , 200 and 4500 gauss .    in fig . 4.a , we report our results for collision energies 1@xmath44k - 1 k , for the four mentioned values of the field . elastic scattering , which is largely determined by s - waves in both incident and final channels , is weakly affected by the strength of the field . for spin - changing collisions , however , the energy dependence changes dramatically , in accordance with ( [ ratek ] ) , which is shown in the figure using dashed lines . again the trends are well - represented , although the formula overestimates the rates due to the zeros in the real rates described in the previous section .    as a general trend , we observe that when the field is increased , the low - energy inelastic rate constants are substantially pushed up towards the elastic ones , but at higher energies the rates are less sensitive to the field . this is better illustrated in fig . 4.b , where the total loss rate constants ( that is the sum of the inelastic scattering in the two exothermic channels ) for the different values of the field are plotted together with the elastic channel results in zero field ( which , as stressed before , are essentially independent of the @xmath16 value ) .    in the range of collision energies from 1@xmath44k to 1 k and for low magnetic fields the rates for elastic collisions remain significantly higher than the rate for the spin - flipping . in particular , at buffer - gas - cooling energies of @xmath110k and below , the inelastic rate constants are in the @xmath111 cm@xmath2/sec range , and remain so even in the presence of a field . this result verifies the suitability of the @xmath0o@xmath1 molecule for bgc . given the comparatively small uncertainties in the pes @xcite , the results shown here are probably fairly realistic for the he - o@xmath1 system . we have continued the analysis in the range of collision energies from 1 to 10 k. we note that for energies larger than the height of the centrifugal barrier , the approximation of including only @xmath55 = 1 in the calculation is no longer accurate within a few percent , as was the case at lower energy . we checked that in this energy range the full calculations including all the possible @xmath55 values provides results which differ ( in the worst case ) by about a factor of 3 for the inelastic channels and by about a factor of 5 for the elastic one . the full calculation is computationally very expensive for @xmath16 @xmath14 0 , as opposed to the case of zero field where a total-@xmath58 basis can be adopted . results indicate that inelastic rates for high collision energies are not much lower than elastic rates , and of course become higher still at energies where resonances exist .    at collision energies above 1k both feshbach and shape resonances appear in the cross sections , as noted in @xcite . we find that these resonances move somewhat as a function of magnetic field . nevertheless , they are sufficiently narrow that they are completely `` washed out '' by thermal averaging in a gas . we therefore present these results as a function of temperature rather than energy . to this end we assume a maxwellian velocity distribution of the collision partners characterized by a kinetic temperature @xmath112 . the thermally averaged rate constants are then expressed as : @xmath113 where @xmath37 is the boltzmann constant and @xmath114 stands for cross sections . to compute this average , the values of the cross sections for @xmath93 @xmath14 10 k are extrapolated from their values at 10 k.    averaged rate constants are shown in fig . 5 for the same set of field values as in fig . 4 . the condition for magnetic trapping to be successful is usually expressed as @xmath115 @xmath14 10 @xmath116 @xcite , so that we can conclude that for collisions of @xmath0o@xmath1 with @xmath2he this condition is fulfilled at least for temperatures up to 1 kelvin and values of the field for which the asymptotic zeeman splitting does not exceeds the height of the exit channel centrifugal barrier . even in the absence of detailed information on cold collisions of a particular molecule , the fitting formula ( [ ratek ] ) can be used as an approximate guide to what the rates might be . for example , we can inquire about the prospects for evaporatively cooling @xmath0o@xmath1 molecules once they have been successfully cooled in a first stage of bgc . for this system , the d - wave centrifugal barrier height is @xmath117 mk , meaning that the threshold law is expected to hold for field values smaller than @xmath50 53 gauss for transitions that produce one or more molecules in the @xmath73 final state .    in the case of @xmath0o@xmath1 cold collisions we have access to the zero - field calculations of ref . we have fit the energy dependence of all the inelastic rate constants to yield the dashed curve labeled `` @xmath118 '' in figure 6 , which represents the total loss . the full calculation ( solid line ) has some additional features due to scattering resonances near zero energy , but this will not affect our conclusions here . based on the zero - field fit , we use ( [ ratek ] ) to estimate the loss rates in nonzero field . the general trends are the same , namely , the rates rise sharply at low energy , but are roughly field - independent at larger energies .    for evaporative cooling to be successful requires , roughly speaking , that the ratio of elastic to inelastic collision rates , @xmath119 , exceed 100 @xcite . for the estimated results shown in figure 6 , this condition holds only at energies below @xmath110mk for fields as low as 10 gauss , and not at all for near - critical fields of @xmath120 gauss . thus evaporative cooling from an initially buffer - gas - cooled sample may prove trickier than previously expected . on the other hand , bgc is characterized by a large number of molecules cooled in the initial step ; it is possible that a certain loss can be tolerated , and that a final sample at sub-@xmath44k temperatures will still hold enough molecules to reach critical phase space density for bec . detailed kinetic simulations are required to determine if this is so . alternatively , a recent proposal suggests that nh molecules could be cooled via stark slowing to temperatures as low as 1 mk @xcite . in this case ec may work quite well . for many systems of interest to ultracold studies , there does not exist any information on spin - changing collisions at low enough temperatures . in such cases it may be possible to make order - of - magnitude guesses anyway . for example , suppose a rate constant is known for higher temperature collisions . in the absence of any other information we could simply assert that the rate has the same value at the energy @xmath101 corresponding to the height of the centrifugal barrier . the fitting formula ( [ ratek ] ) then gives the behavior of this rate at lower temperatures . for example , the zero - field rate constant in the first panel of fig . 5 has a value of @xmath121 cm@xmath2/sec at a temperature of 4 k. extrapolating this value down to @xmath122 k yields a coefficient @xmath123 cm@xmath2/sec for the total loss , just a factor of two higher than the fit to the low - energy calculation . one of the main aims of this paper is to understand in a broad sense collision of paramagnetic molecules in the limit of ultralow temperatures in the presence of a magnetic field . this is a part of our effort in showing that molecules with nonzero spin in the lowest energy state ( such as @xmath0o@xmath1 investigated here ) can be successfully cooled and used for bec purposes . elastic and inelastic scattering in presence of a magnetic field for the specific system @xmath2he - @xmath0o@xmath1 has been characterized in detail . our attention has been focused on the lowest lying trappable state of the molecule . this information is immediately relevant for bgc of molecular oxygen , and suggests that the presence of the field does not particularly hinder the bgc process . this work extends previous predictions which referred to the oxygen molecule in zero field @xcite , and definitely assess the theoretical possibility for trapping this species . moreover , we have illustrated the simple underlying physics of spin - changing rates in general . for fields low enough that both the incident and exit channels are in the threshold regime , the rate constants vary according to the wigner - law dependence ( @xmath93 + @xmath102)@xmath124 . this insight allows us to make estimates of rate constants beyond the ones calculated in detail . in particular , a small field was found to have a dramatic effect on the evaporative cooling of @xmath0o@xmath1 , which must be taken into account in future experiments aimed at quantum degenerate molecular gases .
we investigate the collisional stability of magnetically trapped ultracold molecules , taking into account the influence of magnetic fields . 
 we compute elastic and spin - state - changing inelastic rate constants for collisions of the prototype molecule @xmath0o@xmath1 with a @xmath2he buffer gas as a function of the magnetic field and the translational collision energy . 
 we find that spin - state - changing collisions are suppressed by wigner s threshold laws as long as the asymptotic zeeman splitting between incident and final states does not exceed the height of the centrifugal barrier in the exit channel . 
 in addition , we propose a useful one - parameter fitting formula that describes the threshold behavior of the inelastic rates as a function of the field and collision energy . 
 results show a semi - quantitative agreement of this formula with the full quantum calculations , and suggest useful applications also to different systems . as an example , we predict the low - energy rate constants relevant to evaporative cooling of molecular oxygen .    # 1@xmath3#1