Atomik birimler

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Atomik birimler (İngilizcesi: Atomic units) (au veya a.u.), atom fiziği hesaplamaları için uygun olan doğal birimlerin bir sistemini oluşturmaktadır. Atomik birimlerin iki ayrı türü bulunmaktadır: Hartree atomik birimleri[1] ve Rydberg atomik birimleri. Bu birimlerdeki farklılık yük ile kütle seçimindeki farklılıktan ileri gelmektedir. Bu makalede, Hartree atomik birimleri ile ilgilenilmiştir. Atomik birimlerdeki, aşağıdaki dört temel fiziksel sabitin sayısal değerlerinin tanımı aşağıda gösterildiği gibidir:

Atomik birimler genellikle "a.u." veya "au" biçiminde kısaltma ile ifade edilir, değişik bağlamlardaki astronomik birimler, keyfi birimler ve soğurma birimleri için de kullanılan aynı biçimli kısaltma ile karıştırılmamalıdır.

Kullanım ve notasyon

Atomik birimler SI birimleriyle benzerlik gösterir. Uzunluğun ya da kütlenin birimi bu benzerliğe örnek olarak gösterilebilir. Bu benzerliklere karşın kullanımı ve gösterimi SI birimlerinden farklılıklar taşır.

"m" kadar kütleye sahip bir parçacığın kütlesinin elektronun kütlesinin 3.4 katı olduğunu varsayalım. Bu "m" değeri üç biçimde yazılabilir:

Temel atomik birimler

Atomik birimlerin 4 ana sabiti bulunmaktadır(yukarı bakınız). Bu nedenle bunların sayısal değerleri tanımı gereği atomik birimlerde birleşmektedir.

Temel Atomik Birimler
Boyut Ad Simge/Tanım SI birimindeki değeri[5]
kütle elektronun dingin kütlesi 9.10938291(40)×1031 kg
yük temel yük 1.602176565(35)×1019 C
açısal momentum Planck sabiti 1.054571726(47)×1034 J·s
elektrik sabiti Coulomb gücü sabiti 8.9875517873681×109 kg·m3·s-2·C-2

İlgili fiziki sabitler

Dimensionless physical constants retain their values in any system of units. Of particular importance is the fine-structure constant . This immediately gives the value of the speed of light, expressed in atomic units.

Some physical constants expressed in atomic units
Name Symbol/Definition Value in atomic units
speed of light
classical electron radius
proton mass

Derived atomic units

Below are given a few derived units. Some of them have proper names and symbols assigned, as indicated in the table. kB is Boltzmann constant.

Derived atomic units
Dimension Name Symbol Expression Value in SI units Value in more common units
length Bohr radius 5.2917720859(36)×1011 m 0.052918 nm=0.52918 Å
energy Hartree energy 4.35974417(75)×1018 J 27.211 eV=627.509 kcal·mol−1
time 2.418884326505(16)×1017 s
velocity 2.1876912633(73)×106 m·s−1
force 8.2387225(14)×108 N82.387 nN=51.421 eV·Å−1
temperature 3.1577464(55)×105 Şablon:Convert/ScientificValue/LoffAonSoffT
pressure 2.9421912(19)×1013 Pa
electric field 5.14220652(11)×1011 V·m−1 5.14220652(11) GV·cm−1=51.4220652(11) V·Å−1
electric dipole moment 8.47835326(19)×1030 C·m 2.541746 D

There are two common variants of atomic units, one where they are used in conjunction with SI units for electromagnetism, and one where they are used with Gaussian-CGS units.[6] Although the units written above are the same either way (including the unit for electric field), the units related to magnetism are not. In the SI system, the atomic unit for magnetic field is

1 a.u. = = 2.35×105 T = 2.35×109 G,

and in the Gaussian-cgs unit system, the atomic unit for magnetic field is

1 a.u. = = 1.72×103 T = 1.72×107 G.

(These differ by a factor of α.)

Other magnetism-related quantities are also different in the two systems. An important example is the Bohr magneton: In SI-based atomic units,[7]

a.u.

and in Gaussian-based atomic units,[8]

a.u.

Atomik birimde Bohr modeli

Atomic units are chosen to reflect the properties of electrons in atoms. This is particularly clear from the classical Bohr model of the hydrogen atom in its ground state. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model):

Non-relativistic quantum mechanics in atomic units

The Schrödinger equation for an electron in SI units is

.

The same equation in au is

.

For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:

,

while atomic units transform the preceding equation into

.

Comparison with Planck units

Both Planck units and au are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. It should be kept in mind that au were designed for atomic-scale calculations in the present-day universe, while Planck units are more suitable for quantum gravity and early-universe cosmology. Both au and Planck units normalize the reduced Planck constant. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant G and the speed of light in a vacuum, c. Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and, as a result, the speed of light in atomic units is a large value, . The orbital velocity of an electron around a small atom is of the order of 1 in atomic units, so the discrepancy between the velocity units in the two systems reflects the fact that electrons orbit small atoms much slower than the speed of light (around 2 orders of magnitude slower).

There are much larger discrepancies in some other units. For example, the unit of mass in atomic units is the mass of an electron, while the unit of mass in Planck units is the Planck mass, a mass so large that if a single particle had that much mass it might collapse into a black hole. Indeed, the Planck unit of mass is 22 orders of magnitude larger than the au unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.

Ayrıca bakınız

Kaynakça

  1. Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods". Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University Press) 24 (1): s. 89–110. DOI:10.1017/S0305004100011919. http://journals.cambridge.org/action/displayAbstract?aid=1733252.
  2. 1 2 Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. s. 155. ISBN 978-0-486-41464-5. http://books.google.com/books?id=XpGM7r69LdkC&pg=PA155.
  3. Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. s. 217. ISBN 978-0-486-67355-4. http://books.google.com/books?id=l4zv4dukBT0C&pg=PA217.
  4. Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd bas.). Springer. s. 5. ISBN 978-0-387-20802-2. http://books.google.com/books?id=Jj-ad_2aNOAC&pg=PA5.
  5. "The NIST Reference on Constants, Units and Uncertainty". National Institute of Standard and Technology. http://physics.nist.gov/cuu/index.html. Erişim tarihi: 1 April 2012.
  6. "A note on Units". Physics 7550 — Atomic and Molecular Spectra. University of Colorado lecture notes. 11 Ekim 2012 tarihinde kaynağından arşivlendi. http://web.archive.org/web/20121011113525/http://www.colorado.edu/physics/phys7550/phys7550_sp07/extras/Appendix_1.pdf.
  7. Chis, Vasile. "Atomic Units; Molecular Hamiltonian; Born-Oppenheimer Approximation". Molecular Structure and Properties Calculations. Babes-Bolyai University lecture notes. 31 Ekim 2015 tarihinde kaynağından arşivlendi. http://web.archive.org/web/20151031031622/http://phys.ubbcluj.ro/~vchis/cursuri/cspm/course2.pdf.)
  8. Budker, Dmitry; Kimball, Derek F.; DeMille, David P. (2004). Atomic Physics: An Exploration through Problems and Solutions. Oxford University Press. s. 380. ISBN 978-0-19-850950-9. http://books.google.com/books?id=GW6pclAk-JcC&pg=PA380.

Dış bağlantılar

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