Fine-structure Constant, Anomalous Magnetic Moment, Relativity Factor and the Golden Ratio

Fine-structure Constant, Anomalous Magnetic Moment, Relativity Factor and the Golden Ratio that Divides the Bohr Radius

R. Heyrovska (1) and S. Narayan (2)

((1) J. Heyrovsky Institute of Physical Chemistry, Prague, Czech Republic and (2) Villa

Julie College, Stevenson, MD, USA )

Sommerfeld introduced the fine-structure constant (α) into physics, while he was taking into account the relativistic effects in the theory of the hydrogen atom. Ever since, it has puzzled many scientists like Eddington, Dirac, Feynman and others. Here the mysterious fine-structure constant, α (= λC,H/λdB = 1/137.036 = 2.627/360) is interpreted based on the finding that it is close to φ2/360 (= 2.618/360 = 1/137.508), where λC,H, the Compton wavelength for hydrogen is a distance equivalent to an arc length on the circumference λdB (= 2πa B) of a circle with the Bohr radius (a B) and φ is the Golden ratio, which was recently shown to divide the Bohr radius (a B) into two Golden sections at the point of electrical neutrality. From the data for the electron (e-) and proton (p+) g-factors, it is found that (360/φ2) - α-1 = 2/φ3 = (g p - g e)/(g p + g e), where 360/φ2 = λdB/(λC,H - λC,H,i), λC,H,i = φ2πrμ,H and rμ,H is the sum of the intrinsic radii of e- and p+ evaluated from the g-factors and that

α ? φ2/360 = 0.009/360 = λC,H,i/λdB = (1- γ)/γ, the factor for the advance of perihilion in Sommerfeld’s theory of the hydrogen atom, where γ is the relativity factor.

This world year of physics, WYP 2005, is also concerned with the mystery of the fine-structure constant (α) which puzzled Feynman [1]: “ ... is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the

"hand of God" wrote that number, ..." , see Gilson [2a]. α and its relation to the g-factors are recently rousing considerable interest [2 - 5]. The constancy (or not) of α defined as

2πe2/κhc, where e, κ, h and c are respectively, the charge, electric permittivity of free space, Planck’s constant and the speed of electromagnetic waves in vacuum, is being debated

mostly in terms of these parameters. The present relation between α and the magnetic

moment anomalies expressed as a long series of terms [6] is too complicated to comprehend. This work investigates α from a different point of view, on observing that it is close to φ2/360 (= 1/137.508), where φ is the Golden ratio [7], also called the Divine ratio (Feynman thought of π or e!), which was recently found [8,9] to play an important role in atomic physics.

φ is a mathematical constant which operates in a wide variety of geometrical workings in

the Universe ranging from mollusks in the sea to the spiral galaxies in the cosmos [7]. Geometrically, when the segments a and b of a line of length d are such that (a + b)/a = a/b,

the ratio a/b = (1 + 51/2)/2 = φ = 1.618033989... , and a (= d/φ) and b (= d/φ2) are called the Golden sections. Nature works through the remarkable and unique property of φ that the

power series, ..., 1/φ2, 1/φ, 1, φ, φ2, φ3, …, is at the same time also a Fibonacci series (0, 1, 1,

2, 3, 5, 8, etc.), where any Fibonacci number is related to its neighbors as follows: e.g., 3 =

1 +

2 = 5 – 2. Thus, φ = (1/φ) + 1 = φ2 – 1. Also, note that φ2/π = 0.833346 ~ (5/6).

In the case of a hydrogen atom, which consists of an electron (e-) and a proton (p+) (with magnetic moments μe and μp respectively) separated by the Bohr radius a B, it was shown recently [8] that the ground state energy (E H) is given by the difference,

E H = (1/2)(e2/κa B) = (1/2)(e2/κ)[(1/a B,p) - (1/a B,e)] = eI H = hνH(1a) 1/a B = (1/a B,p) - (1/a B,e); a B = a B,e + a B,p; (a B,e/a B,p)2 - (a B,e/a B,p) = 1 (1b-d)

where I H is the ionization potential of hydrogen, I e = -e/(κa B,e) and I p = e/(κa B,p) are the component potentials of (e-) and (p+) respectively, νH = c/λH is the frequency of light that ionizes hydrogen and λH is the corresponding wavelength. From Eqs. (1b,c), one obtains the Golden quadratic equation Eq. (1d) whose positive root is a B,e/a B,p = φ. This shows that a B is divided at the point of electrical neutrality P el into the two Golden sections, a B,e = a B/φ and

a B,p = a B/φ2 = a B,e - a B/φ3.

The above interpretation in terms of the Golden ratio was extended to other atoms and further, it was shown [8]that for any two atoms of the same kind, the covalent bond distance

is the sum of its Golden sections, the larger one being the anionic and the smaller one, the cationic radius of the atom. This enabled assignment of ionic radii for atoms, and many

bond lengths were shown to be additive. E.g., the crystal ionic distances of alkali halides

could be quantitatively reproduced by adding the respective ionic radii.

Since the de Broglie wavelength, λdB (= 2πa B) is equal to the circumference of the circle

with the radius a B, it is also the sum of the circumferences of two Golden circles, λdB,e =

2πa B,e = (2πa B)/φ and λdB,p = 2πa B,p = (2πa B)/φ2 with the radii a B,e and a B,p. These are also

equal to the Golden arc lengths corresponding to the Golden angles, (2π/φ) and (2π/φ2)

radians or 3600/φ (= 222.492o) and 3600/φ2 (= 137.508o) respectively. Note: 2π radians (=

2πr/r)are divided into 360 degrees where both the radian and the degree are dimensionless.)

α (=2πe2/κhc = 1/137.036 = 2.627/360) is the ratio of wavelengths,

α = λC,H/λdB = λdB/(λH/2) (2)

where λC,H = λC,e + λC,p, the sum of the Compton wavelength shifts for the electron and proton, corresponds to an arc length, subtended by a central angle of 360/137.036 =

2.6270 (= φ2 + 0.009), of the circumference λdB of the circle. Since α = λC,H/λdB = r C,H/a B, it

is the ratio of r C,H (= λC,H/2π), the Compton radius, which is related to the magnetic

moments and a B, which is associated with the electrical charges of e- and p+. r C,H is given by,

μB + μN = (e/2μH,red)h bar = (ec/2)r C,H(3a)

h bar = h/2π = cm e r C,e = cm p r C,p = cμH,red r C,H(3b)

μB/r C,e = μN/r C,p = (μB + μN)/r C,H = ec/2 (3c)

where μB = (ec/2)r C,e = (e/2m e)h bar is the Bohr magneton and μN = (ec/2)r C,p = (e/2m p)h bar is

the nuclear magneton, r C,H = r C,e + r C,p, m e and m p are the rest masses of the electron and

proton, respectively, and μH,red = m e m p/(m e + m p) is the reduced mass of hydrogen. Eq. (3c)

shows that the magnetic pole strength is ec/2.

The g-factor (g e/2) for the electron is defined as the ratio μe/μB, and is given by,

μe = (g e/2)μB = (1 + a e)μB = μB + μe,intr = (ec/2)Rμ,e (4a)

g e/2 = Rμ,e/r C,e = (c/h bar)m e Rμ,e = Iμ,e/I (4b)

where μe,intr = a eμB = (ec/2)rμ,e can be considered as the intrinsic magnetic moment of the

electron, rμ,e is the intrinsic radius, a e = rμ,e/r C,e, is the electron magnetic moment anomaly

[6b] and Rμ,e = r C,e + rμ,e. Eq. (4b) shows that g e/2 is the ratio of the moments Iμ,e = m e Rμ,e

(due to the magnetic moment) and I = h bar/c.

Similarly, μN and μp are related by g p/2 as shown,

μp = (g p/2)μN = (1 + a p)μN = μN + μp,intr = (ec/2)Rμ,p (5a)

g p/2 = Rμ,p/r C,p = (c/h bar)m p Rμ,p = Iμ,p/I (5b)

where μp,intr = a pμN = (ec/2)rμ,p is the intrinsic magnetic moment of the proton, rμ,p is the

intrinsic radius, a p = rμ,p/r C,p is the proton magnetic moment anomaly and Rμ,p = r C,p + rμ,p.

Eq. (5b) shows that g p/2 is proportional to the moment, Iμ,p = m p Rμ,p.

On dividing a B into the sections aμ,e and aμ,p at a point, Pμ [8,9] such that the moments of

inertia (due to the magnetic moments), Iμ,e aμ,e = Iμ,p aμ,p, one obtains,

Iμ,e/Iμ,p = m e Rμ,e/m p Rμ,p = m eμe/m pμp = aμ,p/aμ,e = g e/g p = 0.3585 (6a) (aμ,e - aμ,p)/a B = (g p – g e)/(g e + g p) = 2/φ3 =2(a B,e - a B,p)/a B = 360/φ2 - α-1(6b)

In Eq. (6b), use of the data from [6c] for the g-factors gives (g p – g e)/(g e + g p) = 0.472 = 2/φ3

and therefore, g e/g p = (φ3 – 2)/(φ3 + 2); see also [9].

From Eqs. (2) and (4) - (6), one finds that 360/φ2 stands for the ratio,

360/φ2 = λdB/(λC,H - λC,H,i) (7)

where, λC,H,i = φ2πrμ,H and rμ,H = rμ,e +rμ,p. The distances, λC,H, (λC,H - λC,H,i) and λC,H,i

correspond to small arc lengths on the circle of circumference λdB, subtended by central

angles of 2.6270, 2.6180 (= φ2) and 0.0090 (= 2.627 - 2.618) respectively.

On noting that 0.0090 = 360(1 - γ)/γ = 0.009(6)0, the advance of the perihelion in Sommerfeld’s theory [10] of the hydrogen atom, where γ = (1- α2)1/2 = 0.99997(3) is the

relativity factor when the angular momentum p = h bar, one obtains the following relations,

λC,H,i/λdB = α ? φ2/360 = (1 - γ)/γ and γ = λdB/(λdB + λC,H,i) = 0.99997(5) (8a,b)

Thus the Golden ratio provides a quantitative link between the various known quantities in atomic physics.

Figure 1 shows the role of φ in the hydroegn atom and the caption gives all the details.

One of us (R. H.) is thankful to the Ministry of Industry and Trade of the Czech Republic (project No. 1H-PK/42) for the financial support.

References:

[1] S-I. Tomonaga, J. Schwinger and R. P. Feynman,

https://www.sodocs.net/doc/f47529941.html,/physics/laureates/1965/

[2] a) J. G. Gilson, https://www.sodocs.net/doc/f47529941.html,/~ugah174/and b)

https://www.sodocs.net/doc/f47529941.html,/wiki/Fine-structure_constant

[3] P. Tzanavaris et al., Phys. Rev. Lett. 95, 041301 (2005); and the literature cited therein.

[4] C. Seife, Science, 306, 793 (2004).

[5] a) P. Davies et al., Nature, 418, 602 (2002) and b) T. Davis,

https://www.sodocs.net/doc/f47529941.html,.au/sao/guest/davis/

[6] a) P. J. Mohr and B. N. Taylor, Phys. Today, 54, 29 (2001) and b)

https://www.sodocs.net/doc/f47529941.html,/cuu/constants/

[7] a) M. Livio, M. The Golden Ratio, the story of phi, the world’s most astonishing number. (Broadway Books, New York, 2002) and b) https://www.sodocs.net/doc/f47529941.html,

[8] R. Heyrovska, Mol. Phys., 103, 877 (2005) and the literature therein.

[9] R. Heyrovska and S. Narayan, 2nd IUPAP International Conference on Women in Physics, Rio de Janeiro, Brazil, May 2005, Poster. Abstract in:

https://www.sodocs.net/doc/f47529941.html,.br/eventos/wp/ii/sys/resumos/R0100-2.pdf

[10] A. E. Ruark and H. C. Urey, Atoms, molecules and quanta, Volume I, (Dover Publications, New York, 1964), p. 134.

Figure 1. The Golden ratio, point of electrical neutrality (P el) and magnetic center (Pμ) of the hydrogen atom.

AB = aΒ (= 5.29465x10-11m, from the data in [6c]), the Bohr radius, is the distance between the electron (e-) with magnetic moments μe at A and proton (p+) with magnetic moments μp at B, and O is the center of AB. The (auxiliary) right-angled triangle ABC is constructed for locating the point of electrical neutrality, P el, such that BC = BO = a B/2 = CD. This makes, AC = √5a B/2 and AD = a B/φ, where φ = 1.618034. P el is then marked on AB so that AP el = AD = a B/φ = a B,e and BP el = a B,p = a B/φ2.

EA = AP el = a B/φ = 0.618a B and BG = BP el = a B/φ2 = 0.382a B. F is the center of EG (=

2a B) and 2FO = 2OP el = FP el = (a B,e - a B,p) = a B/φ3 = 0.236a B, The circumference of the circle EHGJE with F as the center = 2πa B = λdB, the de Broglie wavelength. The Golden angles HFJ = 360/φ2 (= 137.5080) and 360/φ (= 222.4920). The Golden arcs HGJ = (2π/φ2)a B = λdB,p and HEJ = (2π/φ)a B = λdB,e. Note that a common tangent to these two circles from a point beyond G (not shown in Fig. 1), will be inclined to AB at an angle sin-1(a B,e - a B,p)/a B = sin-1(1/φ3).

Pμ, the magnetic center, is located such that (g e/g p) = aμ,p/aμ,e, 2P el Pμ = FP el = a B/φ3, APμ = aμ,e = a B/2 + (a B/φ3) = (φ3 + 2)/2φ3 =0.736a B, BPμ = (g e/g p)aμ,e = aμ,p = a B/2 – (a B/φ3) = (φ3 - 2)/2φ3 = 0.264a B and (aμ,e - aμ,p)/a B = 2/φ3 = 360/φ2 - α-1. Note: The chord HJ intersects AB at I close to Pμ and BI = 0.255a B.

The sum of the Compton wavelengths of (e-) and (p+), λC,H (= αλdB), (λC,H - λC,H,i) [= (φ2/360)λdB] and λC,H,i {= φ2πrμ,H = (α ? φ2/360)λdB = [(1 - γ)/γ]λdB, where rμ,H is the sum of the intrinsic radii of (e-) and (p+) and γ is the relativity factor,} correspond to arc lengths, on the circle EHGJE of circumference λdB, which subtend at the center F, the angles, 2.627o, 2.618o (= φ2) and 0.009o respectively.

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