Indian J Phys DOI 10.1007/s12648-012-0083-5
ORIGINAL PAPER
Radii of electrons and their a-quantized relations S Ghosh*, A Choudhury and J K Sarma Department of Physics, Tezpur University, Tezpur 784 028, Assam, India Received: 09 September 2011 / Accepted: 21 March 2012
Abstract: Varieties of experimental and theoretical considerations indicate eight different types of radii of an electron like classical electron radius, Compton radius (electron), electromagnetic radius (electron) etc. Here we attempt to discuss the a-quantized relations among different types of radii of electrons, where a is the fine structure constant. In addition, aquantized results for current and magnetic fields are also focused here. Keywords:
Electron radii; a-Quantization; Electron structure
PACS Nos.: 41.20.-q; 14.60.Cd
1. Introduction Since the discovery of electrons in 1897, various approaches have been made to explain its nature [1]. But the structure of electrons is yet to be discovered [2]. In fact to find the hadron and lepton structures works are going on. For hadrons, the structure functions have been calculated in theoretical ways [3]. An electron is stated as a point particle in the Standard Model of Physics. But it seems that anything point is an improbable entity. The different dynamic and static properties of an electron have revealed some facts about its extended size and shape [1, 2]. Various approaches have been made to give the exact structure of an electron. Thomson gave the idea of classical electron radius, R0 considering classical electrodynamics and later this was re-constructed by Lorentz, Abraham and Poincare [1]. Hence R0 is also known as Thomson-scattering length or Lorentz radius. With the help of Compton scattering experiment the Compton radius of an electron, RC has been introduced by equating Einstein energy equation and Planck–Einstein relation [1, 2, 4] and this is approximately 102 times larger than the classical electron radius. From the calculation of magnetic self-energy, the magnetic field radius RH is determined [4]. Application of electron self-energy is also used in a wider range [5]. QED equivalent electron radius RQED , which comes from Lamb shift [4], is approximately
*Corresponding author, E-mail:
[email protected]
equal to RC [4]. By considering the quantum mechanical formalism of angular momentum of electrons MacGregor has introduced a corrected version of Compton radius of electron as quantum mechanical Compton radius RQMC . Again considering Schwinger correction [6] of magnetic moment of electrons to RQMC , we have QED-corrected quantum mechanical Compton radius RaQMC [4]. Equating the total energy of the electron to the electrostatic contribution of the electron and its magnetic moment, we have arrived at the electromagnetic radius of electron Rem [7]. This is also known as Bohr radius of hydrogen atom [7]. The charge radius of the electron RE is yet to be calculated precisely. The Relativistic Spinning Sphere (RSS) model describes a point charge in an extended Compton-sized electron [4]. The Dynamical Spinning Sphere model also supports the same idea [8]. Recent LEP experiments in CERN give the signature of the size of the charge-radius of the electron as RE \1019 m (1017 cm) [8].
2. a-Quantized relations Classical electron radius is known as [1] R0 ¼
e2 mc2
ð1Þ
where, e is the charge of electron, m is the mass of the electron and c is the speed of light in free-space. The Compton radius of electron can be written as [1, 2, 4] Ó 2012 IACS
S. Ghosh et al.
RC ¼
h : mc
ð2Þ
Hence from Eqs. (1) and (2) we have R0 ¼ aRC
ð3Þ 2
where fine structure constant a ¼ hec. The electromagnetic radius is given by [7] 2
Rem ¼
h : me2
ð4Þ
Relating Eqs. (2) and (4) we have RC ¼ aRem :
ð5Þ
R0 ¼ a2 Rem : ð6Þ In Eq. (6) we arrive at the next order of difference in a. a-Quantized behavior of R0 , RC and Rem are shown in Eqs. (3), (5) and (6). Out of all eight radii of electrons Rem is the largest one. Being concerned with numerical values of the electron radii, one can find that RC , RQMC , RaQMC , RQED and RH are close in results. Remaining is charge radius of the electron and this radius is expected according to LEP results of 1017 cm (1019 m) order or even less than that [6]. It is to be mentioned that 1017 cm (1019 m) is another a-quantized state and we have a2 R0 1017 . Hence it can be written that ð7Þ
ð8Þ
Using Eqs. (6) and (7) (taking RE ¼ a2 R0 only), we obtain RE Rem ¼ R20 :
Fig. 1 a-Quantized behavior of the different electron radii
a-Quantized results of electron radii inspire us to examine the other properties of the electron. Magnetic moment of the electron is invariant for any radii and it is given by [4] eh : 2mc
ð10Þ
From Eqs. (1) and (10) we obtain l ¼ a1
eR0 : 2
ð11Þ
Using the a-quantization property of radii of electron from Eqs. (3), (6) and (7) in Eq. (11) we can write l ¼ a1
eR0 eRC eRem eRE ¼ ¼a ¼ a3 : 2 2 2 2
ð12Þ
Again the rotating charge produces current and for the different electron radii the current contribution come out as [9] ec IC ¼ ; ð13aÞ 2pRC ec ; ð13bÞ I0 ¼ a1 2pRC ec ð13cÞ Iem ¼ a 2pRC and
From Eqs. (3) and (6), we get R0 Rem ¼ R2C :
3. a-Quantization of current and magnetic field
l¼
Equations (3) and (5) describe two a-quantized steps amongst three different radii of electron which will be clear in Fig. 1. Using Eq. (5) in Eq. (3) we get
RE a 2 R0 :
We observe that Eqs. (3), (5), (6) and (7) give the relations between two radii, but Eqs. (8) and (9) give those of three radii.
ð9Þ
IE ¼ a3
ec : 2pRC
ð13dÞ
Current contributions resulting from Eqs. (13a–d) are distinctly shown as a-quantized where factor of a is multiplied with IC to produce the currents for other radii. Hence the a-quantization in current is very distinct here.
Radii of electrons Table 1 a-Quantized radii of electron and corresponding current and magnetic field
4. Conclusions
Radius
Current
Magnetic field
R0
I0 ¼ a1 IC
B0 ¼ a2 BC
RC
IC
BC
Rem
Iem ¼ aIC
Bem ¼ a2 BC
RE
IE ¼ a3 IC
BE ¼ a6 BC
Our calculations show the a-quantization among several radii of electrons, their current contributions and the magnetic fields. It is shown that all the eight radii of electrons are a-quantized. a-Quantized relations also enable us to connect the different electromagnetic phenomena, which are responsible for the origin of those electron radii. Corresponding current and magnetic fields are also related accordingly. The results for a-quantized factors for different radii of the electron strengthen our proposal of the form of charge-radius RE . Interestingly the value for that radius is consistent with LEP results [8]. The fine-structure constant deals a role of connector between any two radii and this reflects the fact that those different phenomena are also connected through a only.
Table 2 a-Quantized factors of radii of electron and corresponding current and magnetic field Radius
a-Quantized factor for radii
a-Quantized factor for I
a-Quantized factor for B
R0
a
RC
1
a1 1
B2 1
Rem
a1
a
a2
RE
a
3
a
3
Acknowledgments Authors are grateful to DAE-BRNS for financial support to the work under a project. Authors are thankful to M. H. MacGregor for useful discussion.
a6
Similarly, magnetic field and current are related [10] as B¼
2I : cR
ð14Þ
Putting the results of Eqs. (13a–d) in Eq. (14), (also following the relations among the current-loops from [9]) the magnetic fields are obtained as BC ¼
2IC ; cRC
ð15aÞ
B0 ¼ a2
2IC ; cRC
ð15bÞ
Bem ¼ a2
2IC cRC
ð15cÞ
2IC cRC
ð15dÞ
and BE ¼ a6 .
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