19142 Phys. Chem. Chem. Phys., 2011, 13, 19142–19147 This journal is c the Owner Societies 2011

Cite this: Phys. Chem. Chem. Phys., 2011, 13, 19142–19147

Collisional relaxation of MnH (X7R+) in a magnetic field: effect of the

nuclear spin of Mn

T. Stoecklin* and Ph. Halvick

Received 6th May 2011, Accepted 25th August 2011

DOI: 10.1039/c1cp21466g

In the present study we investigate the role played by the hyperfine structure of manganese in the

cooling and magnetic trapping of MnH(7S+). The effect of the hyperfine structure of Mn on the

relaxation of the magnetically trappable maximally stretched low-field seeking state of MnH(7S+) in

collisions with 3He is deduced from comparison between the results of the present approach and our

previous nuclear spin free calculations. We show that our previous results are unchanged at the

temperature of the buffer gas cooling experiment but find a new resonance at very low collision energy.

The role played by the different contributions to the hyperfine diatomic Hamiltonian considered in this

work as well as the effect of an applied magnetic field on this resonance are also analyzed.

1. Introduction

Because of the recent advances in cooling and trapping experimental

methods of molecules,1–7 a wide range of theoretical studies are

dedicated to collisions in ultracold molecular gases.8–11 Potential

applications of the availability of ultracold molecular samples

are expected in many different fields like precision spectro-

scopic measurements12–14 or quantum information storage and

processing.15,16 A variety of molecular systems are currently cooled

down using photoassociation spectroscopy, magnetic tuning of

Feshbach resonances or Stark deceleration17–20 but 3He buffer

gas cooling is both the simplest and the most universal cooling

technique. It furthermore works for multiple degrees of

freedom simultaneously. The temperature which can be

achieved is still in the cold regime as it is limited by the vapor

pressure of the buffer gas which in this case is 400 mK. The

ultracold regime was shown to be then reachable for a few

particular cases by sympathetic cooling using ultracold atomic

gases or by evaporative cooling the buffer gas cooled

molecules. The latter technique was recently proved to even

allow the production of a Bose Einstein condensate.21 Ultracold

paramagnetic molecules can then be confined using a magnetic

field. The successful use of these two combined techniques has

been reported for CaH,22 CaF,23 NH,24 CrH and MnH25

which is the subject of the present study. On the theoretical

side, the efficiency of the buffer gas cooling of a given molecule

can be predicted by calculating the ratio of elastic to inelastic

collision rates with 3He while the calculation of its collisional

Zeeman relaxation rate submitted to a magnetic field allows

evaluating trap loss. Examples of such studies involving

molecules in a S state are He � O2(3S),26 He + CaH(2S+)

and Ar + NH(3S�),27 He � NH(3S�),28–31 He �N2

+(2S+)32–34 and He � MnH(7S+).35 In all these studies

the hyperfine structures of both 3He and the molecules were

neglected as the collision energies considered were larger than

the hyperfine structure splitting. In the case of He � MnH,

however, the calculated ratio of the elastic to inelastic cross

section was found to be larger than its experimental estimate.

In the present work we will check if the inelastic channels due

to the hyperfine structure could explain this discrepancy as the

hyperfine interaction couples the electron and nuclear spins

and may induce spin relaxation. This is the first theoretical

study of an inelastic atom–diatom collision submitted to a

magnetic field involving a 7S state molecule and taking into

account its hyperfine structure. As a matter of fact, the only

similar studies including the hyperfine structures of the colliders

at ultralow collision energy were dedicated to He � YbF(2S)36

and Rb � OH(2P)37,38 and the authors found the hyperfine

interactions to be important. However, if the magnitude of the

hyperfine constant bExpF of OH is comparable to the one of MnH,

YbF is a very peculiar case as the hyperfine constant bExpF of this

molecule is larger than its spin rotation interaction constant. In the

present calculations, we will also study the role played by each of

the diatomic hyperfine constants of MnH(7S) in its very low

collision energy dynamics with 3He and deduce which one is

critical in the cooling and trapping process of this molecule. In

Section 2 we describe the hyperfine diatomic terms included in

this study and recall the main steps of the Close Coupling

calculations. The results are discussed in Section 3.

2. Calculations

We use the analytical model of the potential energy surface

(PES) which we developed recently for the He–MnH collisions

and employed to obtain the binding energies of the 3He–MnH

Institut des Sciences Moleculaires, Universite de Bordeaux,CNRS-UMR 5255, 33405 Talence, France.E-mail: t.stoecklin@ism.u-bordeaux1.fr

PCCP Dynamic Article Links

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This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 19142–19147 19143

and 4He–MnH van der Waals complexes.39 We also utilized

the same PES in our nuclear spin free study of the collisional

Zeeman relaxation of the magnetically trappable maximally

stretched low-field seeking state MJ = 3 of MnH(X7S+)35

belonging to the septuplet associated with N = 0, J = S = 3,

where N, S and J designate the quantum numbers associated

with the rotational, the electronic spin and the total angular

momenta of MnH excluding nuclear spin while MN, MS and

MJ are the quantum numbers associated with their projections

along the Z space fixed axis. The methodology to describe

collision induced hyperfine transitions was developed long ago

by Alexander and Dagdigian40 for field free collisions of 2Smolecules with structureless atoms. Here we use a simple

extension of the method developed by Krems and Dalgarno27

to treat the collisional spin depolarisation of a diatomic

molecule in a 3S state in collision with a structureless atom

submitted to a magnetic field. The only difference with our

previous studies35,39 of this system lies in the fact that we

include hyperfine terms in the diatomic Hamiltonian. The rigid

rotor Hamiltonian is again written like:

H ¼ � �h2

2mA�BC

1

R

d2

dR2R

� �þ L

!2

2mA�BCR2þHeff

Mol þ VðR; yÞ

ð1Þ

where-

L is the angular momentum associated with the inter-

molecular coordinate-

R and HeffMol is the effective MnH(7S)

Hamiltonian given by Gengler et al.41

Heff

Mol ¼ BN!2 �DN

!4 þ gN

!S!þ 2

3l½3S2

Z � S!2�

þHeffHFðMnÞ þHeff

HFðHÞ þ gm0B!� S!

þ 1

2lD½N

!� S!;N!2� þ 10gST

3ðL2;NÞT3ðS;S;SÞ½ffiffiffi6phLjT2

0 ðL2ÞjLi�ð2Þ

with

Heff

HFðAÞ ¼ bF ðAÞI!A � S

!þ cðAÞ IAZ �

1

3I!A � S

!� �

þ eQq0ðAÞ4IAð2IA � 1Þ ½3I

2AZ � I

!2A�

þ 5ffiffiffiffiffi14p

bSðAÞT1ðI!AÞT1fT2ðL2Þ;T3ðS

!;S!;S!Þg

½3hLjT20 ðL2ÞjLi�

ð3Þwith A = Mn or H and

-

IA,-

S and-

N are, respectively, the

operators associated with the nuclear spin of the nucleus A

and with the electronic spin and rotational angular momentum

of MnH. The two last terms of expression (2) and the last term

of expression (3) are neglected as the constants lD = 6.1 �10�6 cm�1, gS = 1.33 � 10�5 cm�1 and bS(Mn) = 6.0 �10�6 cm�1 are by far the smallest. We furthermore selected

only the terms involving the Mn atom as the hyperfine

structure due to the Hydrogen atom would only add a very

small contribution to this Hamiltonian as can be seen in

Table 1 where the hyperfine structure constants of the 3He,

H and Mn atoms are presented. The contribution to the total

collisional Hamiltonian due to the hyperfine structure of the 3He

atom42 would give even smaller contributions and is then also

neglected. The diatomic Hamiltonian given by Gengler et al.41 is

defined in the molecular fixed frame whereas we are working in

the spaced fixed frame. Also, using the rotational transformation

properties of irreducible tensor operators43 we use instead the

following rotated effective diatomic Hamiltonian

Heff

Mol ¼ BN!2 �DN

!4 þ gN

!� S!þ 2

3l

ffiffiffiffiffiffi4p5

r ffiffiffi6p

�Xq

ð�ÞqY�q2 ðrÞ½S1 � S1�ð2Þq þ bF ðMnÞI!� S!

þ cðMnÞffiffiffiffiffiffi4p5

r ffiffiffi6p

3

Xq

ð�ÞqY�q2 ðrÞ½I1 � S1�ð2Þq

þ eQq0ðMnÞ4I1ð2I1 � 1Þ

ffiffiffiffiffiffi4p5

r ffiffiffi6p X

q

ð�ÞqY�q2 ðrÞ½I1 � I1�ð2Þq

þ gm0B!� S!

ð4Þ

The matrix elements of the nuclear spin free part of this Hamil-

tonian in the uncoupled basis set ji = |NMNi|SMSi|IMIi, whereI and MI stand for the quantum numbers associated with the

nuclear spin operator-

I of Mn and for its projection along the Z

space fixed axis, are given in ref. 27. Some of those of the

hyperfine contributions to the Hamiltonian are also given in

ref. 36. We give here the expressions of all these contributions

using our notations:

hNMN jhSMSjhIMI jhlMl jbF ðMnÞI!�S!jN 0MN0 ijSMS0 ijIMI 0 ijl0Ml0 i

¼bF ðMnÞdN;N0dMN ;MN0 dl;l0dMl ;Ml0 ½dMS ;MS0 dMI ;MI 0MIMS

þ 12dMI ;MI 0 �1dMS ;MS0 �1a�ðI ;MI 0 Þa�ðS;MS0 Þ� ð5Þ

with a�ða;bÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia aþ1ð Þ�b b�1ð Þ

pand

hNMN jhSMSjhIMI jhlMl jcðMnÞffiffiffiffiffiffi4p5

r ffiffiffi6p

3

Xq

ð�ÞqY�q2 ðrÞ

½I1 � S1�ð2Þq jN 0MN0 ijSMS0 ijIMI 0 ijl0Ml0 i

¼ cðMnÞffiffiffiffiffi30p

3dl;l0dMl ;Ml0 ½ð2S þ 1ÞSðS þ 1Þð2I þ 1Þ

� IðI þ 1Þð2N þ 1Þð2N 0 þ 1Þ�12ð�Þ½SþI�MF �

�N 2 N 0

0 0 0

!N 2 N 0

�MN �qN MN0

!1 1 2

qS qI �qN

!

�S 1 S

�MS qS MS0

!I 1 I

�MI qI MI 0

!

ð6Þ

Table 1 Comparison of the hyperfine constant values of the 3He, Hand Mn atoms. The nuclear spin of each nucleus is also reminded

I bExpF /cm�1 cExp/cm�1 eQq0/cm�1

Mn41 5/2 0.009284 0.001254 �0.0054H41 1/2 0.00061 0.000693He42 1/2 0.00003 0.000035

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19144 Phys. Chem. Chem. Phys., 2011, 13, 19142–19147 This journal is c the Owner Societies 2011

where MF = MN + MS + MI = MJ + MI is the projection

along the z space fixed axis of-

F =-

N +-

S +-

I, the total

angular momentum of the molecule excluding the nuclear spin

of hydrogen and qN = MN0 � MN, qS = MS � MS0, qI =

MI � MI0 with qN = qS + qI.

This last equality only expresses the fact that the projection

MF of the total angular momentum of the molecule is

conserved.

hNMN jhSMSjhIMI jhlMl jeQq0ðMnÞ4Ið2I � 1Þ

ffiffiffiffiffiffi4p5

r ffiffiffi6p

�Xq

ð�ÞqY�q2 ðrÞ½I1 � I1�ð2Þq jN 0MN0 ijSMS0 ijIMI 0 ijl0Ml0 i

¼ eQq0ðMnÞ4Ið2I � 1Þ

ffiffiffiffiffi30p

dMS ;MS0 dl;l0dMl ;Ml0 IðI þ 1Þð2I þ 1Þf g

� ½ð2N þ 1Þð2N 0 þ 1Þ12ð�Þ½IþMN�MI �

1 1 2

I I I

( )

�N 2 N 0

0 0 0

!N 2 N 0

�MN �q MN0

!I 2 I

�MI q MI 0

!

ð7Þ

The diagonalisation of the diatomic Hamiltonian (4) [CHeffMol

C�1]ab= xadab in the uncoupled basis setji=|NMNi|SMSi|IMIigives the diatomic energies represented in Fig. 1 as a function of

the magnitude of the applied magnetic field. Each energy level xaof the diatomic molecule is associated with now a single value of

MF denoted as MF(a). As F characterizes the multiplets in the

absence of the magnetic field, we can clearly see for B = 0 and

N = 0 the six points corresponding to its possible values: 1/2,

3/2, 5/2, 7/2, 9/2, 11/2. This figure can be compared to Fig. 1 of

our previous work which did not include the hyperfine

interactions. Each curve which was previously associated with

a given value of MJ is now split into 6 curves corresponding

asymptotically to the 6 possible values of MI(Mn) ranging

from �5/2 up to 5/2. We in fact also included in the present

figure the hyperfine structure of the hydrogen atom which

again should split each of these curves into two new ones

associated with the two possible values of MI(H) = �1/2,1/2but this splitting is so small that it is hardly visible. The new

elements of the present figure are the many avoided crossings

which appear between the curves associated with different values

of MF for values of the field lower than 1500 G. For a given

valueMI of the projection of the total angular momentum along

the direction of the z space fixed axis and for a given value of

MF(a), the projection of the relative angular momentum L along

the z space fixed axis is then simply ML = MI � MF(a). Thebasis set describing the collision process is obtained by adding the

possible values of the quantum number L for each value of a.This basis set is denoted by the quantum numbers a,ML, and L.

In this basis set, the close coupling equations which have to be

solved take the form

d2

dR2� LðLþ 1Þ

R2þ 2m½E � xa�

� �Fa;MLðaÞ;LðRÞ

¼ 2mX

a0 ;M0Lða0Þ;L0

½CTUC�a0;M

0Lða0Þ;L0

a;MLðaÞ;L Fa0;MLða0Þ;L0 ðRÞ ð8Þ

demonstrated in ref. 27. In order to be able to compare our results

with those of our nuclear spin free previous calculations we used

the same propagation and basis set parameters.35 As the nuclear

spin of manganese is 5/2 we study the relaxation cross sections of

the spin stretched state MF ¼MJ þMI ¼ 3þ 52¼ 11

2. These

calculations are heavier than those neglecting the nuclear spin.

However, as the van der Waals well associated with the He–MnH

complex is shallow and the rotational constant of MnH is quite

large the size of the coupled system which is propagated remains

reasonable. Depending on the value of MI and of the parity, it is

ranging between approximately 300 and 1000.

3. Results

We consider the collisions between 3He and MnH in its

rotational, fundamental and maximally stretched state MnH

(N= 0,MF ¼ 112 ). The calculations are performed forM= ¼ 11

2

as the s wave incident channel which is dominant at ultra-low

collision energy has L = ML = 0. Fig. 2 shows the

resulting relaxation transition cross sections as a function of

collision energy. For a 7S state molecule each diatomic state has

Fig. 1 Diatomic eigenenergies ofMnH(7S+) as a function of the applied

magnetic field calculated using the effective Hamiltonian (2). Both the

hyperfine structures of manganese and hydrogen are taken into account.

Fig. 2 Cross sections for the relaxation transitions in MnH(N = 0,

MF ¼ 112)–3He collisions as a function of collision energy calculated for

M= ¼ 112. Only the transitions giving appreciable contributions are

represented. The values of DF = F0 � F and DMF = MF0 � MF are

reported for each curve and the total relaxation cross section is also

shown and compared to its value when the hyperfine structure is not

included (i.e. for MJ = 3).

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components on different orientations of both the electronic and

nuclear spins. Also, we did not try to differentiate the transitions

corresponding to the full reorientation of both the electronic

and nuclear spins from those conserving approximately the

nuclear spin projection, both anyway resulting in reorientation

of the electronic spin and then to trap loss. If we first compare

the total relaxation cross section which is also represented, to

its value when the hyperfine structure is not included for

MJ = 3 we see that the two curves are very similar for energies

higher than the resonance at 0.08 cm�1 which we discussed in

our previous work. Conversely the very low collision energy

regime is completely different and a new resonance appears at

around 0.025 cm�1. On the same figure the main contributions

to the relaxation cross section are represented. Not surprisingly

they correspond to the transitions DMF = 1 and 2 for both

resonances. At the lowest collision energy the magnitude of the

contributions is as usual sorted in ascending order as a function

of the kinetic energy available in the final channel. This kinetic

energy is roughly equal to the difference of energy between the

initial and final F multiplets. Consequently, in the limit of zero

collision energy, the transitions inside the initial multiplet

(DF=0) give negligible contributions while the largest component

is associated with DF=2 and is followed by those associated with

DF=1. Another interesting feature lies in the fact that conversely

to their nuclear spin free counterparts these partial cross sections

do not follow the E2 threshold laws predicted by Krems and

Dalgarno44 in such a case. This is clearly the effect of a zero

energy resonance due to the hyperfine interactions. In order to

further analyse the origin of the new resonance at 0.025 cm�1

we performed new calculations changing the values of the three

hyperfine constants bExpF , cExp and eQq0. The corresponding

relaxation cross sections are presented in Fig. 3. The manganese

nuclear electric quadrupole interaction is clearly less influential

as even when putting the corresponding constant eQq0 equal

to zero the results seem to be almost unchanged. If we now

consider the two other parameters of the manganese magnetic

hyperfine interactions we see that the electron–nuclear dipolar

constant cExp plays a role only in the very low collision energy

regime where it enhances the magnitude of the relaxation cross

section. On the other side, the Fermi contact constant bExpF is

clearly responsible for the 0.025 cm�1 resonance as it is

sufficient to divide its value by 9 to completely remove this

resonance. It is also responsible with cExp of the very low

energy behaviour as the relaxation cross section is divided by

almost three orders of magnitude when varying its value in the

same range. The domination of the Fermi contact interactions

can be understood by comparing the matrix elements of these

three hyperfine contributions to the diatomic Hamiltonian

given in (5), (6) and (7). Actually, the Fermi contact interaction

is the only hyperfine contribution to the diatomic Hamiltonian

to give non-zero matrix elements between the levels of the

N=0multiplet as the matrix elements (6) and (7) are clearly zero

for N = N0 = 0. This does not mean that the intermolecular

coupling between the molecular states of MnH follows a direct

mechanism. Actually the intermolecular potential is independent

of both the electronic and nuclear spins. The matrix elements

of the intermolecular potential must comply to the rules

DMS =MS –MS0 = DMI=MI –MI0 =0 which means

that DMF = DMN = MN – MN0 (9)

As the main component of the eigenvector associated with

the initial level MF ¼ 112which we consider in this study is

along N = MN = 0, we obtain by applying the simple rule (9)

to any DMF a 0 transition that DMN =MN�MN0 a 0 which

implies that N0 a 0. In other words, the relaxation process is

mediated by the excited level N = 2. This result was already

shown in our previous nuclear spin free study of the same

system35 which also demonstrated the crucial role played by

the electronic spin–spin interaction. In order to check if other

partial waves than the s wave were also modified by the

hyperfine interactions, we performed new calculations of the

relaxation cross sections for MI = 9/2, 7/2 and 5/2 as

illustrated in Fig. 4. As expected, the curve calculated for

M= ¼MF ¼ 112

which is associated with the s wave clearly

gives the highest contribution to the relaxation cross section of

the (N= 0,MF ¼ 112) state of MnH at very low energy but the

values of MI = MF � 1, MF � 2 give also important

contributions in the region of the 0.08 cm�1 resonance. The

resonance due to the nuclear spin interactions at 0.025 cm�1 also

Fig. 3 Comparison as a function of collision energy of the cross

sections for the total relaxation in MnH(N= 0,MF ¼ 112)–3He collision

calculated for M= ¼ 112 and using different sets of hyperfine constants

for the manganese atom. The curve computed using the experimental

values of the Mn hyperfine constants bExpF (Mn) = 0.009284 cm�1,

cExp(Mn) = 0.001254 cm�1, eQqExp0 (Mn) = �0.0054 cm�1 is compared

to those obtained when changing the value of one of these constants.

Fig. 4 Cross sections for relaxation in MnH(N = 0, MF ¼ 112)–3He

collisions calculated for the different values of MI written on each

curve.

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appears on the two curves associated with these contributions

but is less pronounced. We then did new calculations of the

relaxation cross sections of MnH (N = 0, MF ¼ 112 ) in

collisions with 3He collisions for MI = 11/2 as a function of

the magnitude of a magnetic field applied along the Z space

fixed axis. The resulting curves are presented in Fig. 5. When

the field increases the resonance due to the hyperfine interaction

moves monotonously to lower collision energy and increases in

magnitude before vanishing completely for values of the field

higher than 500 G. This behavior shows that it is a Feshbach

resonance whereas the shape resonance at 0.08 cm�1 does not

change much as a function of the magnitude of the applied

magnetic field. As we have seen in Fig. 2 that the main

component of the resonance at 0.025 cm�1 is corresponding

to a transition from the level MF ¼ 112

towards the level

MF ¼ 92, the levels involved in this transition belong mainly

to the two higher multiplets F = 11/2 and F = 9/2 which are

represented in Fig. 1. This figure shows clearly that when the

magnetic field increases the mixing and then the coupling

between these levels first increases and decreases again in

agreement with the tuning of the Feshbach resonance as a

function of the magnetic field observed in Fig. 5.

We conclude this study by calculating the ratio of the elastic

to inelastic cross section which was recently measured in a

buffer gas cooling experiment by Stoll et al.25 Its calculated

value summed over the values of MI giving appreciable

contributions is represented in Fig. 6. Its minimum around

the resonance at 0.1 cm�1 is still three orders of magnitude too

high compared to its experimental value. The discrepancy

between the experimental and calculated values of this ratio is

not reduced when including the nuclear interaction suggesting

that our proposed argument based on the accuracy of the PES

is the sole remaining explanation.

4. Conclusion

We presented the first theoretical study of an inelastic atom

diatom collision submitted to a magnetic field involving a7S state molecule and taking into account its hyperfine structure.

The comparison of the present results with those obtained

previously when neglecting the hyperfine structure of manganese

allowed us to investigate the role played by the hyperfine

structure of manganese in the cooling and magnetic trapping

of MnH(7S+). We found that the hyperfine interaction needs

to be included in a theoretical approach only for collision

energies below 0.1 cm�1. It can then be neglected to model a

Buffer gas cooling experiment but could be important for

sympathetic cooling using ultra cold atoms as a new resonance

was found at very low energy. This new resonance can be

completely removed by applying a magnetic field and was

showed to be due to the Fermi contact interaction suggesting

that the magnitude of the nuclear spin constant bExpF can be

used to evaluate the possible importance of the hyperfine

interaction for similar molecules. We also found that the

discrepancy between the experimental and calculated value

of the elastic to inelastic cross section ratio is not reduced

when including the nuclear interaction suggesting that our

previously proposed argument based on the accuracy of the

PES is the sole remaining explanation.

References

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Fig. 5 Cross sections as a function of collision energy for the relaxa-

tion in MnH(N = 0, MJ = 11/2)–3He collisions. Each curve is

associated with a given value in Gauss of the applied magnetic field.

Fig. 6 Ratio of the elastic to inelastic cross sections of MnH(N = 0,

MF ¼ 112 , MJ = 3) in collisions with 3He as a function of collision

energy using our model PES. It is made out of the sum of theMI =9/2,

7/2 and 5/2 contributions which are the most important.

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