imported>John R. Brews |
imported>John R. Brews |
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| ===Ferromagnetism=== | | ==Tensor== |
| | In ''physics'' a '''tensor''' in its simplest form is a proportionality factor between two [[vector]] quantities that may differ in both magnitude and direction. Mathematically this relationship is: |
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| {{Image|Hysteresis loops.PNG|right|200px|Add image caption here.Magnetic flux density vs. magnetic field in steel and iron; the curve depends upon the direction of traversal, the phenomenon of [[hysteresis]]}} | | :<math> v_j = \sum_{k} \chi_{jk} w_k \ , </math> |
| For ferromagnetic materials, the self-interaction of the atoms tends to align them even when no external magnetic field is present. As a result, ferromagnetic materials create a net magnetic field (and magnetic flux density) in the space surrounding the material, and can form permanent magnets at temperatures below the [[Curie temperature]] of the material. At higher temperatures, the aligning interaction is inadequate to overcome the randomness introduced by thermal motions, and the material becomes paramagnetic.
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| The basis for cooperation between atomic magnetic moments is that electrons obey the [[Pauli exclusion principle]] that no two can occupy the same quantum state. That means configurations with aligned spins are energetically favored over misaligned spins by an ''exchange interaction'', favoring magnetization. The same idea underlies [[Hund's rules]] for atoms, namely, other things equal, electrons in atoms populate states to maximize their total spin. Not addressed here is the unanswered question: why do ferromagnetic materials profit from this effect more than other materials.<ref name=exchange>
| | where '''v''' is a vector with components {v<sub>j</sub>} and '''w''' is another vector with components {w<sub>j</sub>} and the quantity '''Χ''' = {χ<sub>ij</sub>} is a tensor. This example is a ''second rank'' tensor. The idea is extended to ''third'' rank tensors that relate a vector to a second rank tensor, as when electric polarization is related to stress in a crystal, and to ''fourth'' rank tensors that relate two second rank tensors, and so on. |
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| | Tensors can relate vectors of different dimensionality, as in the relation: |
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| For more about the exchange interaction, see {{cite book |title=Electrodynamics of continuous media |chapter=Chapter V: Ferromagnetism |pages=pp. 146 ''ff'' |author=LD Landau and EM Lifshitz |publisher=Pergamon Press |url=http://books.google.com/books?id=sxAJAQAAIAAJ&dq=editions%3AsxAJAQAAIAAJ&q=take+into+account+only+the+exchange+interaction#search_anchor |year=1960 |LCCN=60-14731}}, and for some simple examples {{cite book |title=Interacting electrons and quantum magnetism |author=Assa Auerbach |chapter=Chapter 2: Spin exchange |url=http://books.google.com/books?id=tiQlKzJa6GEC&pg=PA11 |pages=pp. 11 ''ff'' |isbn=0387942866 |year=1999 |publisher=Springer}}.</ref>
| | <math> \begin{pmatrix} |
| | | p_1\\ |
| The figure shows ''magnetization curves'' for two different ferromagnetic materials. The curves exhibit ''hysteresis'', that is, the curve is history dependent and, in particular, depends upon the direction in which the magnetic field increases. This complex behavior indicates that magnetization in such materials is not an equilibrium process. Larger samples break up into ''magnetic domains'' or sub-regions of different magnetization directions separated by ''domain walls''.<ref name=Mayergoyz>
| | p_2\\ |
| | | p_3 |
| {{cite book |title=The Science of Hysteresis, volume III |editor=Isaak D. Mayergoyz, Giorgio Bertotti, editors |author=F Fiorillo, C Appino and M Pasquale |chapter=§1.3 Energy in a magnetic system. Domain walls and domain structures |isbn=0123694337 |year=2005 |pages=pp. 29 ''ff'' |publisher=Elsevier Academic Press |url=http://books.google.com/books?id=88W3fMqNkRwC&pg=RA2-PA29&dq=%22process+is+an+out-of-equilibrium%22&hl=en&ei=yt8ITd3hM5SssAO51MneDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA#v=onepage&q=%22process%20is%20an%20out-of-equilibrium%22&f=false}} | | \end{pmatrix} |
| | | = |
| </ref> The magnetization curve is affected (in part) by the change in size of the various domains as they reluctantly adapt to changes in the external field.<ref name= Klüber0>
| | \begin{pmatrix} |
| | | T_{11} & T_{12} &T_{13}&T_{14}&T_{15}\\ |
| Same reference as previously, but p. 169:{{cite book |author=J Kübler |title=Theory of itinerant electron magnetism |publisher= Oxford |isbn=0199559023 |year=2009 |edition=Revised ed |url=http://books.google.com/books?id=ZbM0gHCcmaQC&pg=PA169 |chapter=§4.1.1 Stoner theory |pages =pp. 169 ''ff''}}
| | T_{11} & T_{12} &T_{13}&T_{14}&T_{15}\\ |
| | | T_{11} & T_{12} &T_{13}&T_{14}&T_{15}\\ |
| </ref>
| | T_{11} & T_{12} &T_{13}&T_{14}&T_{15}\\ |
| | | T_{11} & T_{12} &T_{13}&T_{14}&T_{15} |
| Today it is still impossible to predict from first principles that iron is ferromagnetic.<ref name=Graham>
| | \end{pmatrix} |
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| {{cite book |title=Introduction to magnetic materials |author=Bernard Dennis Cullity, Chad D. Graham |url=http://books.google.com/books?id=kk1el8vB4HoC&pg=PA131 |pages=p. 131 |isbn=0471477419 |year=2009 |publisher=Wiley-IEEE |edition=2nd ed |chapter=Chapter 4: Ferromagnetism}} | | \begin{pmatrix} |
| | | q_1\\ |
| </ref> However, some guidance can be obtained as to which metals are candidates, and which are not, based upon estimates of how exchange energy varies with atomic radii and spacing. "The theory of magnetism in solids is one of the central challenges in condensed matter physics, intrinsically involving many-body correlations, long range order and phase transitions..."<ref name=Martin>
| | q_2\\ |
| | | q_3\\ |
| {{cite book |title=Electronic structure: basic theory and practical methods |author=Richard M Martin |publisher=Cambridge University Press |pages=p. 24 |url=http://books.google.com/books?id=dmRTFLpSGNsC&pg=PA24 |year=2004 |isbn=0521782856}} | | q_4\\ |
| | | q_5 |
| </ref>
| | \end{pmatrix} </math> |
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| <references/> | |
Tensor
In physics a tensor in its simplest form is a proportionality factor between two vector quantities that may differ in both magnitude and direction. Mathematically this relationship is:
where v is a vector with components {vj} and w is another vector with components {wj} and the quantity Χ = {χij} is a tensor. This example is a second rank tensor. The idea is extended to third rank tensors that relate a vector to a second rank tensor, as when electric polarization is related to stress in a crystal, and to fourth rank tensors that relate two second rank tensors, and so on.
Tensors can relate vectors of different dimensionality, as in the relation:
