Hyperfine structures and the field effects of IBr molecule in its rovibronic ground state*

Defu Wang(王得富) Xuping Shao(邵旭萍) Yunxia Huang(黃云霞)Chuanliang Li(李傳亮) and Xiaohua Yang(楊曉華)

1School of Science,Nantong University,Nantong 226019,China

2Department of Physics,Taiyuan University of Science and Technology,Taiyuan 03324,China

Keywords: hyperfine structure,Zeeman effect,Stark effect,IBr molecule

1. Introduction

Molecule hyperfine structure provides additional degrees of freedom to manipulate cold molecules and new opportunities for sensitive probe of fundamental physics, and the molecular hyperfine structures can be investigated via molecular spectroscopy. However, the hyperfine shifts and separations are so small that they are usually concealed in the Doppler broadening due to the random thermal motion of the molecules.

Owing to the development of the molecular cooling in the past two decades[1-5]and the development of the laser single mode techniques,the hyperfine structures of molecules can be observed in laboratory precisely at present. Also the Zeeman and Stark effects of the molecules at hyperfine levels can be studied experimentally. Thus,the deduced field effects of the molecules may help experimentalists to manipulate or even to further cool cold molecules by means of the external magnetic and/or electric fields.

Polar molecules, due to that their anisotropic and longrange dipolar interactions add new ingredients to strongly correlated and collective quantum dynamics in many-body systems, are of great importance in the field of cold molecular physics, and alkali molecules are achieved coldest molecules so far. Therefore, heteronuclear (polar) alkali diatomic molecules have attracted many attentions.[6-10]Degenerate Fermi KRb ultracold molecules,associated from laser-cooled atoms, were achieved in the group of Ye by using Feshbach resonance followed by stimulated Raman adiabatic passage.[6]However, alkali heteronuclear molecules suffer from their short lifetimes due to chemical reaction,[6]which will undoubtedly limit their applications in the field of cold molecular physics. By contrast, heteronuclear halogen diatomic molecules are chemically stable, and thus, their ultracold molecules most probably have wider applications.[11-13]Thus,the study of the hyperfine structures together with their Zeeman and Stark effects of halogen diatomic molecules are essential. It is worth noting that heteronuclear halogen diatomic molecules could be slowed by our proposed near-resonantlaser-assisted Stark deceleration scheme[14]to the equivalent temperature at the order of microkelvin.

In addition, the electronic ground state of Iodine monobromide (IBr) is of1Σ symmetry, so that its ground state has neither Zeeman nor Stark effect at rotational level. However,it has both Zeeman and Stark effects at hyperfine level. So it is a good candidate for studying field effect of the hyperfine levels as well.

2. Theory

2.1. Hyperfine structure

The Hamiltonian of the IBr concerning its hyperfine structure consists of electronic,vibrational,rotational,and hyperfine terms. When studying the vibronic ground state, the first two parts can be treated as a constant.

The well-known rotational term is

whereBvandDvare the molecular rotational constant and its first order distortion,andJis the molecular total angular momentum excluding nuclear spins.

The hyperfine structures are caused by the interactions of nuclear spins with the molecular total angular momentum excluding the nuclear spins(Hsr)and of the nuclear quadrupoles with the gradients of the electric field of the electrons surrounding at the nuclei(HQ). The two interactions can be written as

whereIiis the nuclear spin andi=1 or 2 denotes iodine (I)or bromine (Br) atom, andCiis the nuclear spin-molecular rotation coupling constant and is expressed in Eqs.(1)-(3)in Ref. [15],eis the charge of an electron,T(2)(x) is the second rank tensor, andQand ?Erepresent the nuclear electric quadrupole and the gradient of the electric field of the electrons surrounding at nuclei,respectively.

The second rank tensors ofQand ?Eare as follows:[16]

with the selection rules ofJ'=J,(J±2)andF'1=F1,(F1±1),(F1±2).Here,eQqirepresents the nuclear electric quadrupole constant andC2is too small to be set at 0,and{}and()are the Wigner 6-jand 3-jsymbols,respectively.

2.2. Stark effect

Stark effect, due to the interaction of molecular electric dipole(μ)with an applied electric field(EZ)atZaxis, of the hyperfine levels can be expressed as[16]

whereT(1)(x)is the first rank tensor. The molecular total angular momentumFprojects toZaxis to form projection quantum numberMFin weak field limit. Therefore, the Hamiltonian matrices at the basis of|I1JF1I2FMF〉can be written as[18]

2.3. Zeeman effect

Zeeman effect, due to the interaction of the molecular total angular momentum excluding nuclear spin and nuclear spins with an applied magnetic field(BZ)atzaxis,of the hyperfine levels can be expressed as[16]

wheregris the molecular rotational Land′eg-factor,giis the nuclearg-factor,μN(yùn)is the nuclear magnetic dipole, andσiis the shielding coefficient of electrons surrounding to the nuclear spin. One can find the detailed expressions ofgrandσias Eq.(1.36)in Ref.[20]and as Eqs.(4)-(6) in Ref.[15],respectively.

Similarly, the molecular total angular momentumFprojects toZaxis to form projection quantum numberMFin weak field limit. Thus, the Hamiltonian matrices of the Zeeman effect can be written as[17]

whereδA,Bis the Kronecker delta,which represents the perturbation selection rules.

3. Results and discussion

3.1. Hyperfine structure

The molecular constants needed to compute the hyperfine structure of IBr in its vibronic ground state are listed in Table 1, and the nuclear spin-rotation coupling constant of I atom is calculated employing the Dalton program.[21]The spin of I and Br (both79Br and81Br) atoms are 5/2 and 3/2,respectively. Therefore, the hyperfine levels can be achieved by diagonalizing the Hamiltonian matrices of Eqs.(6)-(9),as plotted in Fig. 1. The quantum numbers and values (in unit MHz)of the levels are labeled as well. The assignment of the hyperfine levels is accomplished according to their eigenvectors. As evidenced in Fig. 1, the hyperfine structures of this two isotopologues only differ in value while the patterns appear almost the same. Thus,only the field effects of I79Br will be stated in the following Subsection 3.2 and Subsection 3.3.

Table 1. Molecular constants(in units of MHz)of the vibronic ground state of IBr.

aRef.[22],bRef.[23],ccalculated employing dalton.

Fig.1. Hyperfine levels(not scaled)of J=0 and 1 states in the rovibronic ground state of I79Br(a)and I81Br(b). The quantum numbers and the levels’values(in units of MHz)are labeled as well.

To verify our calculations, we try to predict the spectra to compare with the experimental values[22]as listed in Table 2. That the root-mean-squared error is 0.036 MHz, compared with the 83 experimental data,[22]e.g.,relative error of about 10?6,suggests that our methods are correct.

As described in Eqs.(6)-(9),the rotationally perturbation selection rule to hyperfine levels is ?J=0,±2. Thus,the hyperfine levels of theJ=0 state are directly perturbed by theJ=2 state and indirectly perturbed by theJ=4 and higher rotational states via intermediate states. Therefore,when studying the hyperfine levels of the rotational ground state, higherJstates should be included in the Hamiltonian matrix. The hyperfine levels’relative corrections,defined as the perturbation correction over the value of the level, of theJ=0 state due to the perturbation of theJ=4 rotational state are listed in Table 3. Due to the little corrections of theJ=4 state,the perturbations of higher rotational states are neglected and thus excluded in our Hamiltonian matrix. Therefore,the hyperfine levels are obtained by diagonalizing an 80×80 Hamiltonian matrix.

Table 2. Partial hyperfine spectra(in units of MHz)of the J=3 ←2 in the vobronic ground state of I79Br.

Table 3. Relative corrections of the hyperfine levels when the perturbation of J=4 state is included.

3.2. Stark effect

In weak field limit,the total angular momentumFincluding nuclear spin projects to the applied electric field atZaxis to form projection quantum numbersMF,whereMF=0,1,...,F. Besides, the electric dipole of the I79Br of the rovibronic ground state isμ=0.737 Debye.[24]We find that the eigenvectors begin to show anomalous behavior when the applied field increasing more than 1000 V/cm,which implies that the angular momentum coupling sequence has become to change and our method does not work anymore. Figure 2 plots the Stark sublevels of the hyperfine levels varying with the applied electric field up to 1000 V/cm. Evidently,the|J,F1,F,MF〉=|0,2.5,4,4〉is perturbed downwards sharply at high electric field,and is found that it is perturbed by the|1,2.5,4,4〉level.

Fig. 2. Stark effects of the hyperfine levels of the rovibronic ground state(J=0,v=0,X1Σ)of I79Br varying with the applied electric field.The quantum numbers of F and MF are labeled as well.

3.3. Zeeman effect

Similar to that in the electric field,the total molecular angular momentumFprojects to applied magnetic field to formMF, whereMF=?F,(?F+1),...,0,...,(F ?1),F. Zeeman effect at the hyperfine levels comes from the interactions of the rotational magnetic moment and the nuclear spins to the applied magnetic field as expressed in Eq. (12). Nuclear magnetic dipole isμN(yùn)=eˉh/(2MP),whereˉhis reduced Planck constant,andMPis the mass of a proton.Twog-factors and the shielding coefficients are absent experimentally, and thus are calculated with the Hatree-Fork method in the basis of SadlejpVTZ employing the Dalton program.[21]The calculation of nuclear shieldings using London Atomic Orbitals[25]to ensure fast basis set convergence and gauge origin independent results. The natural connection is used in order to get numerically accurate results. By default,the center of mass is chosen as gauge origin. The obtained parameters of I79Br are listed in Table 4,and those(both of ours and literature[26])of I35Cl are also listed. The differences of the shielding coefficients are due to the latest vision program we used while those of the literature were obtained employing the previous vision. Nevertheless,we calculate the sublevels using two set of shielding coefficients and find that they only result in a relative error of about 10?4.

Table 4. Molecular parameters needed in the studying of Zeeman effect at hyperfine level of I79Br and also those of I35Cl are listed for comparison.

Fig.3. Zeeman effects of the hyperfine levels of the rovibronic ground state (J =0, v=0, X1Σ) of I79Br varying with the applied magnetic field. The quantum numbers of F and MF are labeled as well.

Consequently,the Zeeman sublevels of the hyperfine levels can be achieved by diagonalizing the Hamiltonian matrices of Eqs.(13)-(15).Figure 3 plots the Zeeman sublevels varying with the applied magnetic field up to 500 Gs(1 Gs=10?4T).It shows that the splitting of the Zeeman sublevels decreases with the increasing ofF. Unlike those of ICl,[26]the Zeeman sublevels with differentMFof I79Br does not cross each other due to large hyperfine separations. Thus, the perturbations of the Zeeman sublevels do not apparently exhibit. However,when we get insight into the sublevels of theF=3 level as shown in Fig.4,we can also observe the slight perturbations of the sublevels due to their nonlinear responses to the applied field, even that the|0,2.5,3,3〉and|0,2.5,3,?3〉levels are slightly perturbed by the|0,2.5,4,3〉and|0,2.5,4,?3〉levels,respectively. The perturbation selection rules are summarized as: ?J=0,±1;?F1=0,±1;?F=0,±1;and ?MF=0.

Fig.4. Zeeman sub-levels of the|J,F1,F〉=|0,2.5,3〉level in the vobronic ground state of I79Br varying with the applied magnetic field.

4. Conclusion

Hyperfine structures of I79Br and I81Br in their rovibronic ground states are theoretically studied by diagonalizing the 80×80 effective Hamiltonian matrix, and our results are in good agreement with the experimental.[22]Thereafter,the Zeeman and Stark effects of the rovibrionic ground state of I79Br at hyperfine level are studied,perturbations of the sub-Zeeman and sub-Stark levels are observed. The results are essential in the field of manipulation and even further cooling of cold molecules via applying electric and/or magnetic fields.