Fine-structure constant

From Wikipedia, the free encyclopedia

 

In physics, the fine-structure constant, also known as Sommerfeld's constant, commonly denoted by α (the Greek letter alpha), is a dimensionless physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It is related to the elementary charge e, which characterizes the strength of the coupling of an elementary charged particle with the electromagnetic field, by the formula 4πε₀ħcα = e². As a dimensionless quantity, it has the same numerical value whatever system of units is being used, which is nearly 1/137.

While there are multiple physical interpretations for α, it received its name from Arnold Sommerfeld introducing it (1916) in extending the Bohr model of the atom: α quantifies the gap in the fine structure of the spectral lines of the hydrogen atom, which had been precisely measured by Michelson and Morley.

Definition

Some equivalent definitions of α in terms of other fundamental physical constants are:

${\displaystyle \alpha ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{\hbar c}}={\frac {\mu _{0}}{4\pi }}{\frac {e^{2}c}{\hbar }}={\frac {k_{\text{e}}e^{2}}{\hbar c}}={\frac {c\mu _{0}}{2R_{\text{K}}}}={\frac {e^{2}}{4\pi }}{\frac {Z_{0}}{\hbar }}}$

where:

• e is the elementary charge (:= 1.602176634×10⁻¹⁹ C);
• π is the mathematical constant pi;
• ħ = h/2π is the reduced Planck constant (:= 6.62607015×10⁻³⁴ J⋅s/2π);
• c is the speed of light in vacuum (:= 299792458 m/s);
• ε₀ is the electric constant or permittivity of free space;
• µ₀ is the magnetic constant or permeability of free space;
• k_e is the Coulomb constant;
• R_K is the von Klitzing constant;
• Z₀ is the vacuum impedance or impedance of free space.

The definition reflects the relationship between α and the permeability of free space µ₀, which equals µ₀ = 2hα/ce². In the 2019 redefinition of SI base units, 4π × 1.00000000082(20)×10⁻⁷ H⋅m⁻¹ is the value for µ₀ based upon more accurate measurements of the fine structure constant.

In non-SI units

In electrostatic cgs units, the unit of electric charge, the statcoulomb, is defined so that the Coulomb constant, k_e, or the permittivity factor, 4πε₀, is 1 and dimensionless. Then the expression of the fine-structure constant, as commonly found in older physics literature, becomes

${\displaystyle \alpha ={\frac {e^{2}}{\hbar c}}.}$

In natural units, commonly used in high energy physics, where ε₀ = c = ħ = 1, the value of the fine-structure constant is

${\displaystyle \alpha ={\frac {e^{2}}{4\pi }}.}$

As such, the fine-structure constant is just another, albeit dimensionless, quantity determining (or determined by) the elementary charge: e = √4πα ≈ 0.30282212 in terms of such a natural unit of charge.

In atomic units (e = m_e = ħ = 1 and ε₀ = 1/4π), the fine structure constant is

${\displaystyle \alpha ={\frac {1}{c}}.}$

Measurement

Two example eighth-order Feynman diagrams that contribute to the electron self-interaction. The horizontal line with an arrow represents the electron while the wavy lines are virtual photons, and the circles represent virtual electron–positron pairs.

 

The 2018 CODATA recommended value of α is

α = e²/4πε₀ħc = 0.0072973525693(11).

This has a relative standard uncertainty of 0.15 parts per billion.

For reasons of convenience, historically the value of the reciprocal of the fine-structure constant is often specified. The 2018 CODATA recommended value is given by

α⁻¹ = 137.035999084(21).

While the value of α can be estimated from the values of the constants appearing in any of its definitions, the theory of quantum electrodynamics (QED) provides a way to measure α directly using the quantum Hall effect or the anomalous magnetic moment of the electron. The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant α (the magnetic moment of the electron is also referred to as "Landé g-factor" and symbolized as g). The most precise value of α obtained experimentally (as of 2012) is based on a measurement of g using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved 12672 tenth-order Feynman diagrams:

α⁻¹ = 137.035999174(35).

This measurement of α has a relative standard uncertainty of 2.5×10⁻¹⁰. This value and uncertainty are about the same as the latest experimental results.

Physical interpretations

The fine-structure constant, α, has several physical interpretations. α is:

• The square of the ratio of the elementary charge to the Planck charge

${\displaystyle \alpha =\left({\frac {e}{q_{\text{P}}}}\right)^{2}.}$

• The ratio of two energies: (i) the energy needed to overcome the electrostatic repulsion between two electrons a distance of d apart, and (ii) the energy of a single photon of wavelength λ = 2πd (or of angular wavelength d; see Planck relation):

${\displaystyle \alpha =\left.{\frac {e^{2}}{4\pi \varepsilon _{0}d}}\right/{\frac {hc}{\lambda }}={\frac {e^{2}}{4\pi \varepsilon _{0}d}}\times {\frac {2\pi d}{hc}}={\frac {e^{2}}{4\pi \varepsilon _{0}d}}\times {\frac {d}{\hbar c}}={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}}.}$

• The ratio of the velocity of the electron in the first circular orbit of the Bohr model of the atom, which is 1/4πε₀ e²/ħ to the speed of light in vacuum, c. This is Sommerfeld's original physical interpretation. Then the square of α is the ratio between the Hartree energy (27.2 eV = twice the Rydberg energy = approximately twice its ionization energy) and the electron rest energy (511 keV).
• The two ratios of three characteristic lengths: the classical electron radius r_e, the Compton wavelength of the electron λ_e, and the Bohr radius a₀:

${\displaystyle r_{\text{e}}={\frac {\alpha \lambda _{\text{e}}}{2\pi }}=\alpha ^{2}a_{0}}$

• In quantum electrodynamics, α is directly related to the coupling constant determining the strength of the interaction between electrons and photons. The theory does not predict its value. Therefore, α must be determined experimentally. In fact, α is one of about 20 empirical parameters in the Standard Model of particle physics, whose value is not determined within the Standard Model.
• In the electroweak theory unifying the weak interaction with electromagnetism, α is absorbed into two other coupling constants associated with the electroweak gauge fields. In this theory, the electromagnetic interaction is treated as a mixture of interactions associated with the electroweak fields. The strength of the electromagnetic interaction varies with the strength of the energy field.
• Given two hypothetical point particles each of Planck mass and elementary charge, separated by any distance, α is the ratio of their electrostatic repulsive force to their gravitational attractive force.
• In the fields of electrical engineering and solid-state physics, the fine-structure constant is one fourth the product of the characteristic impedance of free space, Z₀ = μ₀c, and the conductance quantum, G₀ = 2e²/h:

${\displaystyle \alpha ={\tfrac {1}{4}}Z_{0}G_{0}}$.

The optical conductivity of graphene for visible frequencies is theoretically given by πG₀/4, and as a result its light absorption and transmission properties can be expressed in terms of the fine structure constant alone. The absorption value for normal-incident light on graphene in vacuum would then be given by πα/(1 + πα/2)² or 2.24%, and the transmission by 1/(1 + πα/2)² or 97.75% (experimentally observed to be between 97.6% and 97.8%).

• The fine-structure constant gives the maximum positive charge of an atomic nucleus that will allow a stable electron-orbit around it within the Bohr model (element feynmanium). For an electron orbiting an atomic nucleus with atomic number Z, mv²/r = 1/4πε₀ Ze²/r². The Heisenberg uncertainty principle momentum/position uncertainty relationship of such an electron is just mvr = ħ. The relativistic limiting value for v is c, and so the limiting value for Z is the reciprocal of the fine-structure constant, 137.
• The magnetic moment of the electron indicates that the charge is circulating at a radius r_Q with the velocity of light. It generates the radiation energy m_ec² and has an angular momentum L = 1 ħ = r_Qm_ec. The field energy of the stationary Coulomb field is m_ec² = e²/4πε₀r_e and defines the classical electron radius r_e. These values inserted into the definition of alpha yields α = r_e/r_Q. It compares the dynamic structure of the electron with the classical static assumption.
• Alpha is related to the probability that an electron will emit or absorb a photon.
• Some properties of subatomic particles exhibit a relation with α. A model for the observed relationship yields an approximation for α given by the two gamma functions Γ(1/3) |Γ(−1/3)| ≈ α⁻¹/4π.

When perturbation theory is applied to quantum electrodynamics, the resulting perturbative expansions for physical results are expressed as sets of power series in α. Because α is much less than one, higher powers of α are soon unimportant, making the perturbation theory practical in this case. On the other hand, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong nuclear force extremely difficult.

Variation with energy scale

In quantum electrodynamics, the more thorough quantum field theory underlying the electromagnetic coupling, the renormalization group dictates how the strength of the electromagnetic interaction grows logarithmically as the relevant energy scale increases. The value of the fine-structure constant α is linked to the observed value of this coupling associated with the energy scale of the electron mass: the electron is a lower bound for this energy scale, because it (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore, 1/137.036 is the asymptotic value of the fine-structure constant at zero energy. At higher energies, such as the scale of the Z boson, about 90 GeV, one measures an effective α ≈ 1/127, instead. 

As the energy scale increases, the strength of the electromagnetic interaction in the Standard Model approaches that of the other two fundamental interactions, a feature important for grand unification theories. If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole—this fact undermines the consistency of quantum electrodynamics beyond perturbative expansions.

History

Sommerfeld memorial at University of Munich

 

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley, Arnold Sommerfeld extended the Bohr model to include elliptical orbits and relativistic dependence of mass on velocity. He introduced a term for the fine-structure constant in 1916. The first physical interpretation of the fine-structure constant α was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum. Equivalently, it was the quotient between the minimum angular momentum allowed by relativity for a closed orbit, and the minimum angular momentum allowed for it by quantum mechanics. It appears naturally in Sommerfeld's analysis, and determines the size of the splitting or fine-structure of the hydrogenic spectral lines. 

With the development of quantum electrodynamics (QED) the significance of α has broadened from a spectroscopic phenomenon to a general coupling constant for the electromagnetic field, determining the strength of the interaction between electrons and photons. The term α/2π is engraved on the tombstone of one of the pioneers of QED, Julian Schwinger, referring to his calculation of the anomalous magnetic dipole moment.

Is the fine-structure constant actually constant?

Physicists have pondered whether the fine-structure constant is in fact constant, or whether its value differs by location and over time. A varying α has been proposed as a way of solving problems in cosmology and astrophysics. String theory and other proposals for going beyond the Standard Model of particle physics have led to theoretical interest in whether the accepted physical constants (not just α) actually vary. 

In the experiments below, Δα represents the change in α over time, which can be computed by α_prev − α_now. If the fine-structure constant really is a constant, then any experiment should show that

${\displaystyle {\frac {\Delta \alpha }{\alpha }}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\alpha _{\mathrm {prev} }-\alpha _{\mathrm {now} }}{\alpha _{\mathrm {now} }}}=0,}$

or as close to zero as experiment can measure. Any value far away from zero would indicate that α does change over time. So far, most experimental data is consistent with α being constant.

Past rate of change

The first experimenters to test whether the fine-structure constant might actually vary examined the spectral lines of distant astronomical objects and the products of radioactive decay in the Oklo natural nuclear fission reactor. Their findings were consistent with no variation in the fine-structure constant between these two vastly separated locations and times.

Improved technology at the dawn of the 21st century made it possible to probe the value of α at much larger distances and to a much greater accuracy. In 1999, a team led by John K. Webb of the University of New South Wales claimed the first detection of a variation in α. Using the Keck telescopes and a data set of 128 quasars at redshifts 0.5 < z < 3, Webb et al. found that their spectra were consistent with a slight increase in α over the last 10–12 billion years. Specifically, they found that

${\displaystyle {\frac {\Delta \alpha }{\alpha }}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\alpha _{\mathrm {prev} }-\alpha _{\mathrm {now} }}{\alpha _{\mathrm {now} }}}=\left(-5.7\pm 1.0\right)\times 10^{-6}.}$

In other words, they measured the value to be somewhere between −0.0000047 and −0.0000067. This is a very small value, nearly zero, but their error bars do not actually include zero. This result either indicates that α is not constant or that there is experimental error that the experimenters did not know how to measure. 

In 2004, a smaller study of 23 absorption systems by Chand et al., using the Very Large Telescope, found no measurable variation:

${\displaystyle {\frac {\Delta \alpha }{\alpha _{\mathrm {em} }}}=\left(-0.6\pm 0.6\right)\times 10^{-6}.}$

However, in 2007 simple flaws were identified in the analysis method of Chand et al., discrediting those results.

King et al. have used Markov chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine Δα/α from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for Δα/α for particular models. This suggests that the statistical uncertainties and best estimate for Δα/α stated by Webb et al. and Murphy et al. are robust. 

Lamoreaux and Torgerson analyzed data from the Oklo natural nuclear fission reactor in 2004, and concluded that α has changed in the past 2 billion years by 45 parts per billion. They claimed that this finding was "probably accurate to within 20%". Accuracy is dependent on estimates of impurities and temperature in the natural reactor. These conclusions have to be verified.

In 2007, Khatri and Wandelt of the University of Illinois at Urbana-Champaign realized that the 21 cm hyperfine transition in neutral hydrogen of the early universe leaves a unique absorption line imprint in the cosmic microwave background radiation. They proposed using this effect to measure the value of α during the epoch before the formation of the first stars. In principle, this technique provides enough information to measure a variation of 1 part in 10⁹ (4 orders of magnitude better than the current quasar constraints). However, the constraint which can be placed on α is strongly dependent upon effective integration time, going as t^{−​1⁄₂}. The European LOFAR radio telescope would only be able to constrain Δα/α to about 0.3%. The collecting area required to constrain Δα/α to the current level of quasar constraints is on the order of 100 square kilometers, which is economically impracticable at the present time.

Present rate of change

In 2008, Rosenband et al. used the frequency ratio of
Al⁺ and
Hg⁺ in single-ion optical atomic clocks to place a very stringent constraint on the present-time temporal variation of α, namely α̇/α = (−1.6±2.3)×10⁻¹⁷ per year. Note that any present day null constraint on the time variation of alpha does not necessarily rule out time variation in the past. Indeed, some theories that predict a variable fine-structure constant also predict that the value of the fine-structure constant should become practically fixed in its value once the universe enters its current dark energy-dominated epoch.

Spatial variation – Australian dipole

In September 2010 researchers from Australia said they had identified a dipole-like structure in the variation of the fine-structure constant across the observable universe. They used data on quasars obtained by the Very Large Telescope, combined with the previous data obtained by Webb at the Keck telescopes. The fine-structure constant appears to have been larger by one part in 100,000 in the direction of the southern hemisphere constellation Ara, 10 billion years ago. Similarly, the constant appeared to have been smaller by a similar fraction in the northern direction, 10 billion years ago.

In September and October 2010, after Webb's released research, physicists Chad Orzel and Sean M. Carroll suggested various approaches of how Webb's observations may be wrong. Orzel argues that the study may contain wrong data due to subtle differences in the two telescopes, in which one of the telescopes the data set was slightly high and on the other slightly low, so that they cancel each other out when they overlapped. He finds it suspicious that the sources showing the greatest changes are all observed by one telescope, with the region observed by both telescopes aligning so well with the sources where no effect is observed. Carroll suggested a totally different approach; he looks at the fine-structure constant as a scalar field and claims that if the telescopes are correct and the fine-structure constant varies smoothly over the universe, then the scalar field must have a very small mass. However, previous research has shown that the mass is not likely to be extremely small. Both of these scientists' early criticisms point to the fact that different techniques are needed to confirm or contradict the results, as Webb, et al., also concluded in their study. 

In October 2011, Webb et al. reported a variation in α dependent on both redshift and spatial direction. They report "the combined data set fits a spatial dipole" with an increase in α with redshift in one direction and a decrease in the other. "Independent VLT and Keck samples give consistent dipole directions and amplitudes...."

Anthropic explanation

The anthropic principle is a controversial argument of why the fine-structure constant has the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were much different. For instance, were α to change by 4%, stellar fusion would not produce carbon, so that carbon-based life would be impossible. If α were greater than 0.1, stellar fusion would be impossible, and no place in the universe would be warm enough for life as we know it.

Numerological explanations and multiverse theory

As a dimensionless constant which does not seem to be directly related to any mathematical constant, the fine-structure constant has long fascinated physicists.

Arthur Eddington argued that the value could be "obtained by pure deduction" and he related it to the Eddington number, his estimate of the number of protons in the universe. This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately the integer 137, but precisely the integer 137. Other physicists neither adopted this conjecture nor accepted his arguments but by the 1940s experimental values for 1/α deviated sufficiently from 137 to refute Eddington's argument.

The fine-structure constant so intrigued physicist Wolfgang Pauli that he collaborated with psychoanalyst Carl Jung in a quest to understand its significance. Similarly, Max Born believed that would the value of alpha differ, the universe would degenerate. Thus, he asserted that 1/137 is a law of nature.

Richard Feynman, one of the originators and early developers of the theory of quantum electrodynamics (QED), referred to the fine-structure constant in these terms:

There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: 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, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!

— Richard Feynman, Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 978-0-691-08388-9.

Conversely, statistician I. J. Good argued that a numerological explanation would only be acceptable if it could be based on a good theory that is not yet known but "exists" in the sense of a Platonic Ideal.

Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. However, no numerological explanation has ever been accepted by the community.

In the early 21st century, multiple physicists, including Stephen Hawking in his book A Brief History of Time, began exploring the idea of a multiverse, and the fine-structure constant was one of several universal constants that suggested the idea of a fine-tuned universe.

Quotes

The mystery about α is actually a double mystery. The first mystery – the origin of its numerical value α ≈ 1/137 – has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.

— Malcolm H. Mac Gregor, M. H. MacGregor (2007). The Power of Alpha. World Scientific. p. 69. ISBN 978-981-256-961-5.

Relativistic quantum chemistry

From Wikipedia, the free encyclopedia

Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to explain elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example of such an explanation is the color of gold: due to relativistic effects, it is not silvery like most other metals.

The term relativistic effects was developed in light of the history of quantum mechanics. Initially quantum mechanics was developed without considering the theory of relativity. Relativistic effects are those discrepancies between values calculated by models that consider and that do not consider relativity. Relativistic effects are important for the heavier elements with high atomic numbers. In the most common layout of the periodic table, these elements are shown in the lower area. Examples are the lanthanides and actinides.

Relativistic effects in chemistry can be considered to be perturbations, or small corrections, to the non-relativistic theory of chemistry, which is developed from the solutions of the Schrödinger equation. These corrections affect the electrons differently depending on the electron speed relative to the speed of light. Relativistic effects are more prominent in heavy elements because only in these elements do electrons attain sufficient speeds for the elements to have properties that differ from what non-relativistic chemistry predicts.

History

Beginning in 1935, Bertha Swirles described a relativistic treatment of a many-electron system, in spite of Paul Dirac's 1929 assertion that the only imperfections remaining in quantum mechanics "give rise to difficulties only when high-speed particles are involved, and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass and velocity and assumes only Coulomb forces between the various electrons and atomic nuclei."

Theoretical chemists by and large agreed with Dirac's sentiment until the 1970s, when relativistic effects were observed in heavy elements. The Schrödinger equation had been developed without considering relativity in Schrödinger's 1926 paper. Relativistic corrections were made to the Schrödinger equation to explain the fine structure of atomic spectra, but this development and others did not immediately trickle into the chemical community. Since atomic spectral lines were largely in the realm of physics and not in that of chemistry, most chemists were unfamiliar with relativistic quantum mechanics, and their attention was on lighter elements typical for the organic chemistry focus of the time.

Dirac's opinion on the role relativistic quantum mechanics would play for chemical systems is wrong for two reasons. First, electrons in s and p atomic orbitals travel at a significant fraction of the speed of light. Second, relativistic effects cause indirect consequences that are especially evident for d and f atomic orbitals.

Qualitative treatment

Relativistic γ as a function of velocity. For a small velocity, the ${\displaystyle E_{\rm {rel}}}$ (ordinate) is equal to ${\displaystyle E_{0}=mc^{2}}$ but as ${\displaystyle v_{\rm {e}}\to c}$ the ${\displaystyle E_{\rm {rel}}}$ goes to infinity.

 

One of the most important and familiar results of relativity is that the relativistic mass of the electron increases by

${\displaystyle m_{\rm {rel}}={\frac {m_{\rm {e}}}{\sqrt {1-(v_{\rm {e}}/c)^{2}}}}}$

where ${\displaystyle \displaystyle m_{e},v_{e},c}$ are the electron rest mass, velocity of the electron, and speed of light respectively. The figure at the right illustrates the relativistic effects on the mass of an electron as a function of its velocity. 

This has an immediate implication on the Bohr radius (${\displaystyle \displaystyle a_{0}}$) which is given by

${\displaystyle a_{0}={\frac {\hbar }{m_{\rm {e}}c\alpha }}}$

where ${\displaystyle \hbar }$ is the reduced Planck's constant and α is the fine-structure constant (a relativistic correction for the Bohr model).

Arnold Sommerfeld calculated that, for a 1s electron of a hydrogen atom with an orbiting radius of 0.0529 nm, α ≈ 1/137. That is to say, the fine-structure constant shows the electron traveling at nearly 1/137 the speed of light. One can extend this to a larger element by using the expression v ≈ Zc/137 for a 1s electron where v is its radial velocity. For gold with (Z = 79) the 1s electron will be going (α = 0.58c) 58% of the speed of light. Plugging this in for v/c for the relativistic mass one finds that m_rel = 1.22m_e and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%. 

If one substitutes in the relativistic mass into the equation for the Bohr radius it can be written

${\displaystyle a_{\rm {rel}}={\frac {\hbar {\sqrt {1-(v_{\rm {e}}/c)^{2}}}}{m_{\rm {e}}c\alpha }}}$

Ratio of relativistic and nonrelativistic Bohr radii, as a function of electron velocity

It follows that

${\displaystyle {\frac {a_{\rm {rel}}}{a_{0}}}={\sqrt {1-(v_{\rm {e}}/c)^{2}}}}$

At right, the above ratio of the relativistic and nonrelativistic Bohr radii has been plotted as a function of the electron velocity. Notice how the relativistic model shows the radius decreasing with increasing velocity. 

When the Bohr treatment is extended to hydrogenic atoms, the Bohr radius becomes

${\displaystyle r={\frac {n^{2}\hbar ^{2}4\pi \varepsilon _{0}}{m_{\rm {e}}Ze^{2}}}}$

where ${\displaystyle n}$ is the principal quantum number and Z is an integer for the atomic number. From quantum mechanics the angular momentum is given as ${\displaystyle mv_{\rm {e}}r=n\hbar }$. Substituting into the equation above and solving for ${\displaystyle v}$ gives

${\displaystyle r={\frac {mv_{e}rn\hbar 4\pi \varepsilon _{0}}{mZe^{2}}}}$

${\displaystyle 1={\frac {v_{\rm {e}}n\hbar 4\pi \varepsilon _{0}}{Ze^{2}}}}$

${\displaystyle v_{\rm {e}}={\frac {Ze^{2}}{n\hbar 4\pi \varepsilon _{0}}}}$

From this point atomic units can be used to simplify the expression into

${\displaystyle v_{\rm {e}}={\frac {Z}{n}}}$

Substituting this into the expression for the Bohr ratio mentioned above gives

${\displaystyle {\frac {a_{\rm {rel}}}{a_{0}}}={\sqrt {1-\left({\frac {Z}{nc}}\right)^{2}}}}$

At this point one can see that for a low value of ${\displaystyle n}$ and a high value of ${\displaystyle Z}$ that ${\displaystyle {\frac {a_{\rm {rel}}}{a_{0}}}<1 annotation="">$. This fits with intuition: electrons with lower principal quantum numbers will have a higher probability density of being nearer to the nucleus. A nucleus with a large charge will cause an electron to have a high velocity. A higher electron velocity means an increased electron relativistic mass, as a result the electrons will be near the nucleus more of the time and thereby contract the radius for small principal quantum numbers.

Periodic table deviations

The periodic table was constructed by scientists who noticed periodic trends in known elements of the time. Indeed, the patterns found in it is what gives the periodic table its power. Many of the chemical and physical differences between the 6th period (Cs–Rn) and the 5th period (Rb–Xe) arise from the larger relativistic effects for the former. These relativistic effects are particularly large for gold and its neighbors, platinum and mercury.

Mercury

Mercury (Hg) is a liquid down to −39°C. Bonding forces are weaker for Hg–Hg bonds than for its immediate neighbors such as cadmium (m.p. 321°C) and gold (m.p. 1064°C). The lanthanide contraction is a partial explanation; however, it does not entirely account for this anomaly. In the gas phase mercury is alone in metals in that it is quite typically found in a monomeric form as Hg(g). Hg₂²⁺(g) also forms and it is a stable species due to the relativistic shortening of the bond. 

Hg₂(g) does not form because the 6s² orbital is contracted by relativistic effects and may therefore only weakly contribute to any bonding; in fact Hg–Hg bonding must be mostly the result of van der Waals forces, which explains why the bonding for Hg–Hg is weak enough to allow for Hg to be a liquid at room temperature.

Au₂(g) and Hg(g) are analogous, at the least in having the same nature of difference, to H₂(g) and He(g). It is for the relativistic contraction of the 6s² orbital that gaseous mercury can be called a pseudo noble gas.

Color of gold and caesium

Spectral reflectance curves for aluminum (Al), silver (Ag), and gold (Au) metal mirrors

 

Alkali metal coloration: rubidium (silvery) versus caesium (golden)

The reflectivity of aluminum (Al), silver (Ag), and gold (Au) is shown in the graph above. The human eye sees electromagnetic radiation with a wavelength near 600 nm as yellow. Gold appears yellow because it absorbs blue light more than it absorbs other visible wavelengths of light; the reflected light reaching the eye is therefore lacking in blue compared to the incident light. Since yellow is complementary to blue, this makes a piece of gold under white light appear yellow to human eyes. 

The electronic transition from the 5d orbital to the 6s orbital is responsible for this absorption. An analogous transition occurs in silver, but the relativistic effects are smaller than in gold. While silver's 4d orbital experiences some relativistic expansion and the 5s orbital some contraction, the 4d-5s distance in silver is still much greater than the 5d-6s distance in gold. The relativistic effects increase the 5d orbital's distance from the atom's nucleus and decrease the 6s orbital's distance.

Caesium, the heaviest of the alkali metals which can be collected in quantities sufficient for viewing, has a golden hue, whereas the other alkali metals are silver-white. However, relativistic effects are not very significant at Z = 55 for caesium (not far from Z = 47 for silver). The golden colour of caesium comes from the decreasing frequency of light required to excite electrons of the alkali metals as the group is descended. For lithium through rubidium this frequency is in the ultraviolet, but for caesium it enters the blue–violet end of the spectrum; in other words, the plasmonic frequency of the alkali metals becomes lower from lithium to caesium. Thus caesium transmits and partially absorbs violet light preferentially while other colours (having lower frequency) are reflected; hence it appears yellowish.

Lead-acid battery

Without relativity, lead would be expected to behave much like tin, so tin-acid batteries should work just as well as the lead-acid batteries commonly used in cars. However, calculations show that about 10 V of the 12 V produced by a six-cell lead-acid battery arises purely from relativistic effects, explaining why tin-acid batteries do not work.

Inert pair effect

In Tl(I) (thallium), Pb(II) (lead), and Bi(III) (bismuth) complexes a 6s² electron pair exists. The inert pair effect is the tendency of this pair of electrons to resist oxidation due to a relativistic contraction of the 6s orbital.

Other effects

Additional phenomena commonly caused by relativistic effects are the following:

• Metallophilic interactions
• The stability of the gold anion, Au⁻, in compounds such as CsAu
• The crystal structure of lead, which is face-centered cubic instead of diamond-like
• The striking similarity between zirconium and hafnium
• The stability of the uranyl cation, as well as other high oxidation states in the early actinides (Pa-Am)
• The small atomic radii of francium and radium
• About 10% of the lanthanide contraction is attributed to the relativistic mass of high velocity electrons and the smaller Bohr radius that results
• In the case of gold, significantly more than 10% of its contraction is due to relativistically heavy electrons, and gold (element 79) is almost twice as dense as lead (element 82)