原子物理-原子在磁场中的行为

Atoms in magnetic field

Abstract

In magnetic field, a electron which has spin and resolves the nucleus is acted by magnetic torque. That results in the magnetic energy and also the change of angular momentum. The electron spin resonance is a situation of the overturn of angular momentum. Zeeman effect and Paschen-Back effect are the energy shift and splitting caused by magnetic field. The three of them is an comprehensive picture to describe an atom in magnetic field.

1 Electron Spin Resonance ]1[

The electron spin resonance is actually a method to study the behavior of electron and is a phenomenon that when a new oscillatory magnetic field with a certain frequency is added to a static one would cause the electron which was

processingaround the original field turn over its spin direction.

In practical use, one direction for the horizontal

oscillatory field 1B is sufficient, saying the x-direction.

However, to illustrate clearly, the horizontal field

here is given in the x-y plane. The phase-difference

between x B and y B is set as π/2 and the two

components have same amplititude so that 1B is a

circular field.

If we ignore the horizontal field first, the magnetic moment would be precessing along the static field

0B . The precession angular velocity is 0B g B p

μω=, where g is the Landé g-factor, B μ is the Bohr magnetic moment. Now, if the perpendicular magnetic field 1B has the very same angular frequency. Then the torque generated by it would be like a wrench precessing around 0B with the μsynchronously, as showed in Fig 1.1. After a short time μ’s direction would be changed a little and revolving on another circle. Finally, the direction of it will be turned over. Besides, from the view of energy conservation, the change of magnetic moment’s energy is 02B g m E B s μω⋅==∆ , just enough for an electron to flip.

Nevertheless, we didn’t use some peculiar property of spin, which means it’s also applicable for orbital magnetic moment, and when that is turned into consideration, j μis in place of s μ.

Fig 1.1

2 Zeeman effect

The Zeeman effect is the effect of splitting a spectral line into several components in the presence of a static and weak magnetic field.]2[

The calculation of j μ can be obtain both from the vector model or the calculation of operator. Although, it’s quite visual to use the former one, it can also bring some problems and make some concepts vague and unclear. For example, we couldn’t

understand why s l j ±=since we imagine and assume s ,j and l are vectors but

actually they ’re not and we also don’t know why j μ

is precessing around j . So we must solve the problem through Schrodinger equation.

After all, that’s the original and essential way. To be simple, we assume ψ is the eigenstate with quantum number l, s and j so that the operators exerting on it turn into numbers.

We need the matrix of various spin and orbital matrix to calculate the energy, luckily, others have finished that before. The Schrodinger equation is

ψ+∆=ψ⎥⎦

⎤⎢⎣⎡++)(ˆˆ2others E E others s B m e l B m e (1) where e and m are the charge and mass of electron respectively and B denotes the

external magnetic field. However, we need to express l or s in the form of one

another so that the calculation can resume. Assume s l j ±=.

y y z y z x x z x z z z y x z j s j j s j s j j s j s j j j j s j )()()()(s 2222z -+-+⋅=++= (2)]3[

Where the last two terms on the right side will disappear when applied to wavefunctions with same j and even in other cases its value is very small for the weak external field. Thus, apply (2) on ψ then we could finally get,

ψ⎥⎦⎤⎢⎣⎡++++-++=ψ∆)1(2)1()1()1(12j j s s l l j j j m eB E z , (3) and

)1(2)

1()1()1(1++++-++=j j s s l l j j g . (4) The thing must be clear is that s l j ±= is a equation of eigenvalue which is got