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==Vacuum (partial)==
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==Magnetic moment==
'''Partial vacuum''' refers to a realizable but non-ideal, or imperfect, vacuum. The '''partial pressure''' of a gas in a mixture of gases is the portion of the total gas pressure contributed by that gas. Laboratory vacuum historically was achieved by pumping down a vacuum chamber, and success was measured by the partial pressures of the residual gases. Because the gases cannot be completely removed, the result of pumping down is a partial vacuum. One instrument important in monitoring the success of pumping down is the [[mass spectrometer]], which ionizes the gases and then detects the ions as a current.<ref name=Carlson>
In physics, the '''magnetic moment''' of an object is a [[vector]] property, denoted here as '''m''', that determines the [[torque]], denoted here by '''τ''', it experiences in a [[magnetic flux density]] '''B''', namely '''τ'''&nbsp;=&nbsp;'''m&nbsp;×&nbsp;B''' (where '''×''' denotes the [[Vector product|vector cross product]]).  As such, it also determines the change in [[potential energy]] of the object, denoted here by ''U'', when it is introduced to this flux, namely ''U''&nbsp;=&nbsp;−'''m·B'''.<ref name=Bhatnagar>
 
{{cite book |title=A Complete Course in ISC Physics |author=V. P. Bhatnagar |url=http://books.google.com/books?id=2kh2LnCB6E4C&pg=PA246 |pages=p. 246 |isbn=8120902025 |year=1997 |publisher=Pitambar Publishing}}
 
</ref>
 
==Origin==
A magnetic moment may have a macroscopic origin in a bar magnet or a current loop, for example, or microscopic origin in the spin of an elementary particle like an electron, or in the [[angular momentum]] of an atom.
 
===Macroscopic examples===
The [[electric motor]] is based upon the torque experienced by a current loop in a magnetic field. The basic idea is that the current in the loop is made up of moving electrons, which are subect to the [[Lorentz force]] '''F''' in a magnetic field:
 
:<math>\mathbf F = -e \left( \mathbf {v \times B} \right) \ , </math>
 
where ''e'' is the [[Elementary charge|electron charge]] and '''v''' is the electron velocity. This force upon the electrons is communicated to the wire loop because the electrons cannot escape the wire, and so exert a force upon it. The electrons at the top of the loop move oppositely to those at the bottom, so the force at the top is opposite in direction to that at the bottom. If the magnetic field is in the plane of the loop, the forces are normal to this plane, causing a torque upon the loop tending to turn the loop about an axis along the direction of the field.<ref name=motor>
 
For a discussion of the operation of a motor based upon the Lorentz force, see for example, {{cite book |title=Drives and Control for Industrial Automation |author=Kok Kiong Tan, Andi Sudjana Putra |url=http://books.google.com/books?id=auGLxYZlvX4C&pg=PA48 |pages=pp. 48 ''ff'' |isbn=1848824246 |year=2010 |publisher=Springer}}
 
</ref>
 
The torque exerted upon a current loop of radius ''a'' carrying a current ''I'', placed in a uniform magnetic flux density '''B''' at an angle to the unit normal '''û<sub>n</sub>''' to the loop is:<ref name=Pramanik>
 
{{cite book |title=Electromagnetism: Theory and applications |author=A. Pramanik |url=http://books.google.com/books?id=gnEEwy12S5cC&pg=PT240 |pages=pp. 240 ''ff'' |isbn=8120319575 |year=2004 |publisher=PHI Learning Pvt. Ltd.}}
 
</ref>
 
:<math>\boldsymbol \tau = \mathit I \mathbf {S \times B } \ , </math>
where the vector '''S''' is:
:<math> \mathbf S = \pi a^2 \ \hat{\mathbf u} _n \ . </math>
Consequently the magnetic moment of this loop is:
:<math> \mathbf m = \mathit I \  \mathbf S\ . </math>
 
===Microscopic examples===
At a fundamental level, magnetic moment is related to the angular momentum of fundamental particles. In this discussion, focus is upon the electron and the atom.
 
The discussion splits naturally into two parts: [[kinematics|kinematic]] and [[dynamics|dynamic]].
====Kinematics====
The kinematical discussion, which does not enter upon the physical origins of magnetism and its effects upon mechanics, deals with the classification of atomic states based upon symmetry. Although these ideas apply to nucleii and other particles, here attention is focused on electrons in atoms. The symmetry analysis leads to the identification of spin '''S''' and orbital angular momentum '''L''' and its combination '''J'''&nbsp;=&nbsp;'''L&nbsp;+&nbsp;S'''.<ref name=Weyl>
 
The mathematics of this classification is explained masterfully in {{cite book |title=The theory of groups and quantum mechanics |author=Hermann Weyl |isbn=0486602699 |year=1950 |publisher=Courier Dover Publications |edition=Reprint of 1932 ed |url=http://books.google.com/books?id=jQbEcDDqGb8C&pg=PA185 |pages=pp. 185 ''ff'' |chapter=Chapter IV A §1 ''The representation induced in system space by the rotation group''}}. The application to atomic spectra is explained in great detail in the classic {{cite book |title=The theory of atomic spectra |author=EU Condon and GH Shortley |isbn=0521092094 |year=1935 |publisher=Cambridge University Press |chapter=Chapter III: ''Angular momentum'' |pages=pp. 45 ''ff'' |url=http://books.google.com/books?id=hPyD-Nc_YmgC&pg=PA45}}.
 
</ref>
 
The electron has a spin.  The resultant total spin '''S''' of an ensemble of electrons in an atom is the vector sum of the constituent spins '''s<sub>j</sub>''':
 
:<math> \mathbf {S} = \sum_{j=1}^N \ \mathbf{s_j} \ . </math>
 
Likewise, the orbital momenta of an ensemble of electrons in an atom add as vectors.
 
Where both spin and orbital motion are present, they combine by vector addition:
 
:<math>\mathbf{J = L +S} \ .</math>
 
The mathematical underpinning of these matters is the ''infinitesimal rotation'' from which finite rotations can be generated. If the three coordinate axes are labeled {''i, j, k ''} and the infinitesimal rotations about each of these axes are labeled {''R<sub>i</sub>, R<sub>j</sub>, R<sub>k</sub>''}, then these infinitesimal rotations obey the ''commutation relations'':
:<math> R_i R_j - R_jR_i = i \varepsilon_{ijk} R_k \ , </math>
for any choices of subscripts. Here ε<sub>ijk</sub> is the [[Levi-Civita symbol]] which equals one if ''ijk = xyz'' or any permutation that keeps the same cyclic order, or minus one if the order is different, or zero if any two of the indices are the same.
 
These commutation relations now are viewed as applying in general, not restricted to a three dimensional space, and the question opened as to what this generalization might imply. In particular, one might construct sets of square matrices of various dimensions that satisfy these commutation rules; each set is a so-called ''representation'' of the rules. One finds that there are many such sets, but they can be sorted into two kinds: irreducible and reducible. The reducible sets of matrices can be shown to be equivalent to matrices with smaller irreducible matrices down the diagonal, so that the rules are satisfied within these smaller constituent matrices, and the entire matrix is not essential. The irreducible sets cannot be arranged this way.<ref name=representations>
 
For a discussion see Weyl, cited above, or {{cite book |title =Rotational spectroscopy of diatomic molecules |author=John M. Brown, Alan Carrington |url=http://books.google.com/books?id=TU4eA7MoDrQC&pg=PA143 |chapter=§5.2.4 Representations of the rotation group |pages=pp. 143 ''ff'' |isbn=0521530784 |publisher=Cambridge University Press |year=2003}}
 
</ref>
 
The matrices of dimension 2 are found from observation to be connected to the spin of the electron. One set of these matrices is based upon the [[Pauli spin matrices]]:<ref name= Reiher>
 
{{cite book |title=Relativistic quantum chemistry: the fundamental theory of molecular science |author=Markus Reiher, Alexander Wolf |url=http://books.google.com/books?id=u47v2YmR-P8C&pg=PA141 |pages=p. 141 |isbn=3527312927 |year=2009 |publisher= Wiley-VCH}}
 
 
</ref>
 
:<math>\sigma^x = \begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix} \ ; \ \sigma^y = \begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix} \ ; \ \sigma^z = \begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix} \ , </math>
 
which satisfy:
:<math> \frac{1}{2}\sigma^{\alpha} \frac{1}{2}\sigma^{\beta} -\frac{1}{2}\sigma^{\beta} \frac{1}{2}\sigma^{\alpha}  =i \ \varepsilon_{\alpha \beta \gamma} \frac{1}{2}\sigma^{\gamma} \ , </math>
with αβγ any combination of ''xyz''.
The higher dimensional irreducible sets of matrices are found to correspond to the spin of assemblies of electrons, or to the orbital motion of electrons in atoms, or a combination of both.
 
The matrices can be viewed as acting upon vectors in a space. (For example, a space of dimension (2ℓ+1) can be constructed from the [[spherical harmonics]] ''Y''<sub>ℓ</sub><sup>m</sup>, and their transformations under rotations lead to matrices of dimension (2ℓ+1) that satisfy the commutation rules.<ref name=Hladik>
 
{{cite book |title=Spinors in physics |author=Jean Hladik |url=http://books.google.com/books?id=25eTtXqLW8UC&pg=PA83 |pages=pp. 83''ff'' |chapter=§3.3.2 Spherical harmonics |isbn=0387986472 |year=1999 |publisher=Springer}}
 
</ref>) If the general infinitesimal rotation is labeled '''J''' where '''J''' = '''S''' or '''L''' or '''L + S''', for example, then the basis vectors in this space can be labeled by the integers ''j'' and ''m'' where ''m'' is restricted to the values {&nbsp;−''j'', −''j''+1, ... , ''j''−1, ''j''&nbsp;}. Denoting a basis vector by |''j, m''>, one finds:
 
:<math>J^2 |j, \ m> = j(j+1) |j, \ m> \ , </math>
:<math>J_z|j, \ m > = m |j, \ m > \ . </math>
 
Of course, the formalism has application to other elementary particles as well.
 
====Dynamics====
The dynamic aspect introduces the proportionality between magnetic moment and angular momentum, the [[gyromagnetic ratio]], and attempts to explain its origin based upon [[quantum electrodynamics]].


[[Angular momentum (quantum)|Angular momentum]] is introduced as proportional to an infinitesimal rotation, and is related to the same commutation relations, but with a proportionality factor of ℏ. Thus, in general ℏ'''J''' is an angular momentum, which clearly extends the idea of angular momentum far beyond the intuitive classical concept that applies in only three-dimensional space.
{{cite book |title=Vacuum physics and technology |editor=G. L. Weissler, Robert Warner Carlson |chapter=Chapter 3: Partial pressure measurement |url=http://books.google.com/books?id=tfLWfAx1ZWQC&pg=PA81 |pages=pp. 81 ''ff'' |isbn=0124759149 |publisher=Academic Press |year=1979 |edition=2nd ed}}


The magnetic moment '''m'''<sub>S</sub> of a system of electrons with spin '''S''' is:<ref name=experiment>
The measured magnetic moment of an electron differs slightly from the value ''g''=2 due to interaction with the quantum vacuum. See Newton, for example.
</ref>
</ref>
:<math>\mathbf{m_S} = 2m_B \mathbf S \ , </math>
and the magnetic moment '''m'''<sub>L</sub> of an electronic orbital momentum '''L''' is:
:<math>\mathbf{m_L} = m_B \mathbf{L} \ . </math>
Here the factor ''m<sub>B</sub>'' refers to the [[Bohr magneton]], defined by:
:<math>m_B = \frac{e \hbar}{2 m_e} \ , </math>
with ''e'' = the [[Elementary charge|electron charge]], ℏ = [[Planck's constant]] divided by 2π, and ''m<sub>e</sub>'' = the electron mass. These relations are generalized using the ''g''-factor:
:<math>\mathbf{m_J} = g m_B \  \mathbf J \ , </math>
with ''g''=2 for spin ('''J''' = '''S''') and ''g''=1 for orbital motion ('''J''' = '''L''').<ref name=Poole>


{{cite book |title=Electron spin resonance: a comprehensive treatise on experimental techniques |author=Charles P. Poole |url=http://books.google.com/books?id=P-4PIoi7Z7IC&pg=PA4 |isbn=0486694445 |year=1996 |publisher=Courier Dover Publications |edition=Reprint of Wiley 1982 2nd ed |pages=p. 4}}
To determine the properties of the ideal vacuum, the idea was that measurement of properties as pumping down took place could be fitted to theoretical expressions and extrapolated to zero pressure to find the behavior of "true" vacuum. That is an empirical approach to defining vacuum. Unfortunately, the theory calculating the properties of vacuum is rather complicated today (see [[vacuum (quantum electrodynamics)]]), and measurements are too inaccurate to verify the theory at extremely low pressures. Consequently, this strategy for defining vacuum is of limited accuracy, and cannot be relied upon to check experimentally the behavior of "true" vacuum.


</ref>
Some experiments affected by vacuum are examined in [[quantum electrodynamics]], such as [[spontaneous emission]] and natural spectral linebreadths, the [[Lamb shift]] the [[Casimir force]], and [[quantum beats]] between spontaneously emitting systems in vacuum.<ref name= Prigogine>
As mentioned earlier, where both spin and orbital motion are present, they combine by vector addition:<ref name=Newton>
 
{{cite book |title=Quantum physics: a text for graduate students |author=Roger G. Newton |url=http://books.google.com/books?id=uPAzSaCuAk4C&pg=PA162 |pages=p. 162 |isbn=0387954732 |publisher=Springer |year=2002}}
 
</ref>
 
:<math>\mathbf{J = L +S} \ .</math>
 
The magnetic moment of an atom of angular momentum '''J''' is
:<math>\mathbf {m_J} = g m_B \mathbf J \ , </math>
 
with ''g'' now the ''Landé ''g''-factor'' or spectroscopic splitting factor:<ref name=Singh>
 
{{cite book |title=Introduction To Modern Physics |author=R. B. Singh |url=http://books.google.com/books?id=fd3jyXM-vfMC&pg=PA262 |pages=p. 262 |year=2008 |isbn=8122414087 |publisher=New Age International}}
 
</ref>


:<math> g = \frac {3}{2} + \frac{S(S+1)-L(L+1)}{2J(J+1)} \ . </math>
See, for example, {{cite book |author=WM Brubaker |year=2001 |author=Alexander S Shumovsky |editor=MW Evans & I Prigogine |pages=pp. 396 ''ff'' |url=http://books.google.com/books?id=EjCK4zqTqmYC&pg=PA396 |isbn=0471389307 |title=Modern nonlinear optics, Part 1 |edition=2nd ed |publisher=Wiley}}


If an atom with this associated magnetic moment now is subjected to a magnetic flux, it will experience a torque due to the applied field.
</ref> So far these experiments tell us more about atoms than about the vacuum.


==Notes==
==Notes==
<references/>
<references/>
http://books.google.com/books?id=2zypV5EbKuIC&pg=PA374&dq=kinematics+dynamics+%22angular+momentum%22&hl=en&ei=55IRTdaFMYS4sAPdwaCnDw&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFoQ6AEwCTge#v=onepage&q=kinematics%20dynamics%20%22angular%20momentum%22&f=false

Revision as of 11:47, 5 January 2011

Vacuum (partial)


Partial vacuum refers to a realizable but non-ideal, or imperfect, vacuum. The partial pressure of a gas in a mixture of gases is the portion of the total gas pressure contributed by that gas. Laboratory vacuum historically was achieved by pumping down a vacuum chamber, and success was measured by the partial pressures of the residual gases. Because the gases cannot be completely removed, the result of pumping down is a partial vacuum. One instrument important in monitoring the success of pumping down is the mass spectrometer, which ionizes the gases and then detects the ions as a current.[1]

To determine the properties of the ideal vacuum, the idea was that measurement of properties as pumping down took place could be fitted to theoretical expressions and extrapolated to zero pressure to find the behavior of "true" vacuum. That is an empirical approach to defining vacuum. Unfortunately, the theory calculating the properties of vacuum is rather complicated today (see vacuum (quantum electrodynamics)), and measurements are too inaccurate to verify the theory at extremely low pressures. Consequently, this strategy for defining vacuum is of limited accuracy, and cannot be relied upon to check experimentally the behavior of "true" vacuum.

Some experiments affected by vacuum are examined in quantum electrodynamics, such as spontaneous emission and natural spectral linebreadths, the Lamb shift the Casimir force, and quantum beats between spontaneously emitting systems in vacuum.[2] So far these experiments tell us more about atoms than about the vacuum.

Notes

  1. ↑ (1979) “Chapter 3: Partial pressure measurement”, G. L. Weissler, Robert Warner Carlson: Vacuum physics and technology, 2nd ed. Academic Press, pp. 81 ff. ISBN 0124759149. 
  2. ↑ See, for example, Alexander S Shumovsky (2001). MW Evans & I Prigogine: Modern nonlinear optics, Part 1, 2nd ed. Wiley, pp. 396 ff. ISBN 0471389307. 
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