The Spinning Period of a Free Electron and the Periods of Spin and Orbital Motions of Electron in Atomic States

The spinning period for a free electron and the periods of spin and orbital motion of the electron in an atomic state have been calculated. We have shown that for a free electron the spinning period is: ( ) s free T 20 1.9 10 s − = × . But in the atomic case we show that, both the spin and the orbital periods depend on the quantum numbers n, l m , s m and the effective Landé-g factor, g ∗ which is a function of the quantum number l of the atomic state , , , l s n l m m given in Dirac notation. We have also calculated these periods for the ground state and some excited states—hydrogen and hydrogen-like atoms. For atomic states the approximate values of spinning period are ( ) s atomic T 21 10 s − ≅ and the related orbital periods are: ( ) ( ) atomic T 16 15 0 10 10 s − − = . Therefore atto-second processes which are related to the pulse of 10−18 s will filter the orbital motion of the electron but will be long enough to detect the details of the spin motion, such as flip-flops.


Introduction
To calculate the periods of spin and orbital motions of an electron in an atomic state , , , l s n l m m in Dirac re-presentation, we consider the total magnetic moment of an electron in the presence of a magnetic field in the z direction.The z-component of the total magnetic moment of electron is given by [1] ( ) where B µ is the Bohr magneton, which is given by 2 e mc  and * g is the effective Landé-g factor which takes the values * 1 1, 2,3, 4,5 g l = + = depending on the values of the outermost electrons and 0,1, 2,3, 4 l = (corresponding to the so called , , , s p d f states respectively).To calculate the spin period of an electron, we will use the magnetic top model which was first introduced by Barut et al. [2].For calculating the period of the orbital motion we will use the current loop model [3]- [7].

Period of the Spinning Motion of Electron
From Equation (1) the z-component of magnetic moment associated with the spinning motion is: To proceed further, we calculate the intrinsic magnetic moment of electron with a semiclassical, magnetic top model which was first introduced by Barut et al. [2].
In the magnetic top model, the spin angular momentum of electron is produced by the spinning of the electronic charge (−e) which is assumed to be uniformly distributed inside a sphere of a radius R. We denote the spin angular frequency of the rotating charged sphere by s ω , then the magnitude of the magnetic moment of this sphere can be calculated(Appendix I) to be In the presence of the magnetic field , the z-component of the magnetic moment of the spinning sphere becomes: Which defines the radius of electron as below: For a free electron * 2 g = substituting other related variables in Equation (7) gives us the radius of a free electron, free R : Substitution of Equation ( 8) in Equation (5b) gives us the spinning period for a free electron: ( ) 20 1.9 10 s which is in good agreement with the semiclassical calculation of Olszewski [8].
For an electron in an atom, we cannot calculate the radius directly from Equation (7) n l m m .
In the following section we find an expression for the period of orbital motion, ( ) ( T for the outermost electron in hydrogen and hydrogen-like atoms: which is given by Equation ( 14): When we take the ratio of the periods given in Equation (5b) and Equation ( 14), we find: It is known that when there is no quantum entanglement, for a free electron, the Landé-g factor is equal to 2. For an electron in an atom the Landé-g factor is given by: ( ) ( ) ( ) ( ) which varies is in range of min max 1 2 g g g = < < = . Recently, Saglam et al. [1] showed that because of the quantum entanglements in an atom the Landé-g factor is replaced by the effective g-factor, * g which takes the values *  1 1, 2,3, 4,5 g l = + = depending on 0,1, 2,3, 4 l = (corresponding to the so called , , , s p d f states respectively) values of the outermost electrons together with the unfilled shells respectively.So the maximum values of the effective Landé-g factor, * g can be as high as 5. Therefore * * * min max 1 5 g g g =< < =.If we calculate the effective g-factor, * g for the ground state hydrogen atom, 1, 0, 0,1 2 , we found that for the states: 1, 0, 0,1 2 , 2,1, 0,1 2 , 3, 2, 2,1 2 , 4,3,1,1 2 and 4,3, 2,1 2 in Table 1.

Period of the Orbital Motion of Electron
From Equation (1) the z-component of the total magnetic moment is:  in the current loop model [2]: we assume that the magnetic moment associated with the orbital motion of electron is produced by the fictitious point charge (−e) rotating in a circular orbit with the angular frequency 2π j j T ω = and the radius n r in x-y plane.In this model the z-com- ponent of the magnetic moment will be We note that the quantum number (l) gets involved through the effective Lande-g factor, * g which takes the values * 1 g l = + .For example, for the ground state of hydrogen atom 1,0,0,1/ 2 , substituting * 1 g = and the other related parameter in Equation (15), we find: Similarly for the state 2,1, 0,1 2 the corresponding period is: where we put: where ( ) is the charge of the spherical shell with the radius r and thickness dr .First we want to calculate the magnetic moment of this spherical shell with the surface charge density ( 2 4π q r σ = ).Let us assume that the spinning is about z-axis with the angular frequency, s ω .Let us consider the charge element dq in the area of the band with the radius ( sin r θ ) and the thickness ( d r θ ) in spherical coordinates: The current element dI produced by the rotating band charge with the angular frequency, s ω will be: The magnetic moment element of this band current will be: ( ) the atomic case we show that, both the spin and the orbital periods depend on the quantum numbers n, l m , s m and the effective Landé-g factor, g * which is a function of the quantum number l of the atomic state , , , l s n l m m given in Dirac notation.We have also calculated these periods for the ground state and some excited states-hydrogen and hydrogen-like atoms.For atomic states the approximate values of spinning period are ( ) is the spinning period.Let us consider the equatorial velocity of this spinning sphere,

* 1 g
= .For this value we calculate the period of ground state orbit and find: g = and other related parameters in Equation (11), we find the spinning period of electron in 1, 0, 0,1 2 state, Now we find another expression for total µ m ; here the subscript (0) stands for orbital motion.Now we can find the values of ( ) ( spherical shell gives us the magnetic moment of this shell, shell µ : V) and integrate over the spherical volume, we find the total magnetic moment of the sphere of radius R: -I) in (A-VI) we find: , because we need to know the effective values of * g .For the same reason we must take the effective values of * l s