System of atomic units

The international Système of units (S.I) is universal means of communication, but in counterpart, it is inappropriate with such or such scientific discipline. In particular, it is logically anthropomorphic, i.e. on human scale. However the rise of physics and chemistry was established on the concept of atom. And as Maxwell had prophesied it, would be built soon a rather atomic system.

System of Bohr

With each system of equations given (inevitably homogeneous), corresponds a dimensional Analyze given ( cf . the work of Barenblatt, dimensional analysis or that of Saint-Guilhem, analyzes dimensional ). This dimensional analysis provides a mechanical system of units if three parameters makes it possible to rebuild a length, a time and a mass.

Example 1: the system of Bohr, independently of his physical philosophy, is reduced to two equations with two unknown factors R and v :

$mvr = N \ hbar$ ,

$\ frac {mv^2} {R} = \ frac {Ze^2} {r^2}$ ,

depending on 2 parameters: $P = N \ hbar/m$ and $Q = \ frac {Ze^2} {m}$ .

One from of deduced easily: v = Q/P and R = P ² /Q, therefore a kinematic of the ions hydrogénoïdes, but not a system of units.

Example 2: the equation of Schrödinger of a hydrogénoïde utilizes 2 parameters $R = \ hbar^2/m$ and $S= Ze^2$ . Only possible energy is E = S ² /R and the length has = R/S; but not of system of units .

These considerations must make it possible to be warned against abusive assertions. The system of atomic units is built on electro (E ²) the - dynamic one (m) - quantum ( $\ hbar$ ), but it is seen that it is not registered in the two preceding problems.

System of atomic units

From the three preceding parameters, one sees that one can build a system of units, nevertheless one will never forget whom a particular theory can use only one particular part of this system.

It is interesting Mnémotechnique lies to retain some of these units compared to other system of units (for example that built on mechanics (m) quantum (H) relativistic (c)). It will be known how to pass from one system to the other: it will be enough to eliminate C to replace it by E ², via the size without dimension, known as constant of fine or constant structure of electrodynamic coupling, $\ alpha = \ frac {e^2} {\ hbar C} = 1/137.03599911$ (one asks the students only 1/137 or 1/137.036):

For example, from the energy of the second system E = mc ², one passes easily to the energy of the atomic system E=mc ². $\ alpha^2$ = 1 Hartree = 2 Rydbergs= 2*13.6 eV

Also we will build simultaneously the 2 tables:

Length : length of Compton $L_C= \ hbar/mc$ ; ray of Bohr $\ frac {L_c} {\ alpha} = \ frac {\ hbar^2} {me^2} = 53pm$ .

Speed : C = 299.792.458m/s; atomic speed: $c \ alpha = \ frac {e^2} {\ hbar}$ .

Note: not to forget that the system of Bohr gives much better: for example speed is $Ze^2/n \ hbar$ ; that gives relativistic correctionc for N = 1 and Z~137: and indeed, the theory of Hartree-Fock of the uranides must be relativistic.

Any kinematics from of deduced.

---- If one adds the mass m of the electron, all the system from of deduced :

Mass : m, mass of the electron in the two systems. The only important size without dimension is that of m/M = 1/1836.15267261 (with M = mass of u.m.a) that one currently does not include/understand: it remains contingent.

One from of deduced in particular Energy E= 2*13.6 eV

from where density of energy (E/L ³, i.e. pressure, or it Young modulus of materials), power, stiffness (E/L ² = 1556.892913 N/m) so important for the infra-red spectroscopy of the chemists, etc

Bonds with the S.I.

It remains to consider system MKS.A; i.e. the amp.

the electric units are a little particular in this moment in S.I because they are defined by report/ratio “in the vacuum”, by definition of the magnetic permeability of the vacuum, and C of the vacuum, from where the resistance of the vacuum (which leads to the 50 ohms of the cables BNC!), and via the unit of international time to all kinematics, therefore with RC, therefore with C, therefore with self inductance, etc . The vacuum is not too much liked metrologists, rightly (with which precision to carry out the vacuum?). And they prefer to refer to practical achievements which rather bring them closer to the system of atomic units by taking Q the elementary charge like definition than traditional intensity I of two parallel infinite wire. Between the Josephson effect which gives a unit of Voltage = h/q and the resistance of quantum Hall effect h/q ², which is 137 times that of the vacuum (modulo 2π), the metrologists has access with a precision of approximately 1/10¹⁰ to H and Q. Thus very logically, the S.I has a modified “practical realization”, while waiting for officialization as soon as a balance weighs an atom, or a given number (large) of atoms, or although a mechanical comparator of power and electric output will be invented.

Then the dream of Maxwell will be carried out; but while waiting, any physicist reasons on his system of equations and his Système of natural units, adapted.

See too

dimensional Analysis
Theorem of Buckingham
Constant fundamental
the international System of units
System of natural units
Model of Bohr
Meter, discussion.

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