Difference between revisions of "Reciprocal space"
From Online Dictionary of Crystallography
BrianMcMahon (talk | contribs) (Tidied translations and corrected German and Spanish (U. Mueller)) |
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| − | = | + | <font color="blue">Espace réciproque</font> (''Fr''). <font color="red">Reziproker Raum</font> (''Ge''). <font color="black">Spazio reciproco</font> (''It''). <font color="purple">逆空間</font> (''Ja''). <font color="green">Espacio recíproco</font> (''Sp''). |
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| − | Espace réciproque (''Fr''). | ||
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(1) | (1) | ||
| − | + | <center> | |
'''a*'''. '''a''' = 1; '''b*'''. '''b''' = 1; '''c*'''. '''c''' = 1; | '''a*'''. '''a''' = 1; '''b*'''. '''b''' = 1; '''c*'''. '''c''' = 1; | ||
'''a*'''. '''b''' = 0; '''a*'''. '''c''' = 0; '''b*'''. '''a''' = 0; '''b*'''. '''c''' = 0; '''c*'''. '''a''' = 0; '''c*'''. '''b''' = 0. | '''a*'''. '''b''' = 0; '''a*'''. '''c''' = 0; '''b*'''. '''a''' = 0; '''b*'''. '''c''' = 0; '''c*'''. '''a''' = 0; '''c*'''. '''b''' = 0. | ||
| − | + | </center> | |
(2) | (2) | ||
| − | + | <center> | |
'''a*''' = ('''b''' × '''c''')/ ('''a''', '''b''', '''c'''); | '''a*''' = ('''b''' × '''c''')/ ('''a''', '''b''', '''c'''); | ||
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'''c*''' = ('''b''' × '''c''')/ ('''a''', '''b''', '''c'''); | '''c*''' = ('''b''' × '''c''')/ ('''a''', '''b''', '''c'''); | ||
| − | + | </center> | |
where ('''b''' × '''c''') is the vector product of basis vectors '''b''' and '''c''' and ('''a''', '''b''', '''c''') = ''V'' is the triple scalar product of basis vectors '''a''', '''b''' and '''c''' and is equal to the volume ''V'' of the cell constructed on the vectors '''a''', '''b''' and '''c'''. | where ('''b''' × '''c''') is the vector product of basis vectors '''b''' and '''c''' and ('''a''', '''b''', '''c''') = ''V'' is the triple scalar product of basis vectors '''a''', '''b''' and '''c''' and is equal to the volume ''V'' of the cell constructed on the vectors '''a''', '''b''' and '''c'''. | ||
The reciprocal and direct spaces are reciprocal of one another, that is the reciprocal space associated to the reciprocal space is the direct space. They are related by a Fourier transform and the reciprocal space is also called ''Fourier space'' or ''phase space''. | The reciprocal and direct spaces are reciprocal of one another, that is the reciprocal space associated to the reciprocal space is the direct space. They are related by a Fourier transform and the reciprocal space is also called ''Fourier space'' or ''phase space''. | ||
| − | The '''vector product''' of two direct space vectors, | + | The '''vector product''' of two direct space vectors, <math>{\bold r_1} = u_1 {\bold a} + v_1 {\bold b} + w_1 {\bold c}</math> and <math>{\bold r_2} = u_2 {\bold a} + v_2 {\bold b} + w_2 {\bold c}</math> is a reciprocal space vector, |
| − | + | <center> | |
| − | + | <math> | |
| − | + | {\bold r*} = {\bold r_1} \times {\bold r_2} = V (v_1 w_2 - v_2 w_1) {\bold a*} + V (w_1 u_2 - w_2 u_1) {\bold b*} + V (u_1 v_2 - u_2 v_1) {\bold c*}. | |
| − | + | </math> | |
| + | </center> | ||
| + | Reciprocally, the vector product of two reciprocal vectors is a direct space vector. | ||
As a consequence of the set of definitions (1), the '''scalar product''' of a direct space vector '''r''' = ''u'' '''a''' + ''v'' '''b''' + ''w'' '''c''' | As a consequence of the set of definitions (1), the '''scalar product''' of a direct space vector '''r''' = ''u'' '''a''' + ''v'' '''b''' + ''w'' '''c''' | ||
| − | by a reciprocal space vector '''r*''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''' is simply: '''r''' . '''r*''' = ''uh'' + ''vk'' +''wl''. | + | by a reciprocal space vector '''r*''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''' is simply: |
| − | + | <center> | |
| − | In a ''' | + | '''r''' . '''r*''' = ''uh'' + ''vk'' +''wl''. |
| + | </center> | ||
| + | In a '''coordinate system change''', the coordinates of a vector in reciprocal space transform like the basis vectors in direct space and are called for that reason ''covariant''. The vectors in reciprocal space transform like the coordinates in direct space and are called ''contravariant''. | ||
== Geometrical relationships == | == Geometrical relationships == | ||
| − | The '''volume''' ''V*'' = ('''a*''', '''b*''', '''c*''') of the cell constructed on the reciprocal vectors '''a*''','''b*''' | + | The '''volume''' ''V*'' = ('''a*''', '''b*''', '''c*''') of the cell constructed on the reciprocal vectors '''a*''', '''b*''' and '''c*''' is equal to 1/''V''. |
| − | and '''c*''' is equal to 1/''V''. | ||
| − | The '''lengths''' ''a*'', ''b*'', ''c*'' of the reciprocal basis vectors and the '''angles''', α*, β*, γ*, between | + | The '''lengths''' ''a*'', ''b*'', ''c*'' of the reciprocal basis vectors and the '''angles''', α*, β*, γ*, between the pairs of reciprocal vectors ('''b*''', '''c*'''), ('''c*''', '''a*'''), ('''a*''', '''b*'''), are related to the corresponding lengths and angles for the direct basis vectors through the following relations: |
| − | the pairs of reciprocal vectors ('''b*''', '''c*'''), ('''c*''', '''a*'''), ('''a*''', '''b*'''), are related to | ||
| − | the corresponding lengths and angles for the direct basis vectors through the following relations: | ||
''a*'' = ''b'' ''c'' sin α/''V''; ''b*'' = ''c'' ''a'' sin β/''V''; ''c*'' = ''a'' ''b'' sin γ/''V''; | ''a*'' = ''b'' ''c'' sin α/''V''; ''b*'' = ''c'' ''a'' sin β/''V''; ''c*'' = ''a'' ''b'' sin γ/''V''; | ||
| − | cos α* = (cos βcos γ - cos α)/ | + | cos α* = (cos βcos γ - cos α)/(sin β sin γ);<br> |
| − | cos β* = (cos γcos α - cos β)/ | + | cos β* = (cos γcos α - cos β)/(sin γ sin α);<br> |
| − | cos γ* = (cos αcos β - cos γ)/ | + | cos γ* = (cos αcos β - cos γ)/(sin α sin β). |
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| − | + | == History == | |
| − | + | The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs [(1881). ''Elements of Vector Analysis, arranged for the Use of Students in Physics''. Yale University, New Haven]. | |
| − | + | == See also == | |
| + | *[[Reciprocal lattice]] | ||
| + | *[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ ''The Reciprocal Lattice''] (Teaching Pamphlet No. 4 of the International Union of Crystallography) | ||
| + | *Chapter 1.1 of ''International Tables for Crystallography, Volume B'' | ||
| + | *Chapter 1.1 of ''International Tables for Crystallography, Volume C'' | ||
| + | *Chapter 1.1.2 of ''International Tables for Crystallography, Volume D'' | ||
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| − | + | [[Category:Fundamental crystallography]] | |
Revision as of 16:50, 17 November 2017
Espace réciproque (Fr). Reziproker Raum (Ge). Spazio reciproco (It). 逆空間 (Ja). Espacio recíproco (Sp).
Definition
The basis vectors a*, b*, c* of the reciprocal space are related to the basis vectors a, b, c of the direct space (or crystal space) through either of the following two equivalent sets of relations:
(1)
a*. a = 1; b*. b = 1; c*. c = 1;
a*. b = 0; a*. c = 0; b*. a = 0; b*. c = 0; c*. a = 0; c*. b = 0.
(2)
a* = (b × c)/ (a, b, c);
b* = (c × a)/ (a, b, c);
c* = (b × c)/ (a, b, c);
where (b × c) is the vector product of basis vectors b and c and (a, b, c) = V is the triple scalar product of basis vectors a, b and c and is equal to the volume V of the cell constructed on the vectors a, b and c.
The reciprocal and direct spaces are reciprocal of one another, that is the reciprocal space associated to the reciprocal space is the direct space. They are related by a Fourier transform and the reciprocal space is also called Fourier space or phase space.
The vector product of two direct space vectors, [math]{\bold r_1} = u_1 {\bold a} + v_1 {\bold b} + w_1 {\bold c}[/math] and [math]{\bold r_2} = u_2 {\bold a} + v_2 {\bold b} + w_2 {\bold c}[/math] is a reciprocal space vector,
[math] {\bold r*} = {\bold r_1} \times {\bold r_2} = V (v_1 w_2 - v_2 w_1) {\bold a*} + V (w_1 u_2 - w_2 u_1) {\bold b*} + V (u_1 v_2 - u_2 v_1) {\bold c*}. [/math]
Reciprocally, the vector product of two reciprocal vectors is a direct space vector.
As a consequence of the set of definitions (1), the scalar product of a direct space vector r = u a + v b + w c by a reciprocal space vector r* = h a* + k b* + l c* is simply:
r . r* = uh + vk +wl.
In a coordinate system change, the coordinates of a vector in reciprocal space transform like the basis vectors in direct space and are called for that reason covariant. The vectors in reciprocal space transform like the coordinates in direct space and are called contravariant.
Geometrical relationships
The volume V* = (a*, b*, c*) of the cell constructed on the reciprocal vectors a*, b* and c* is equal to 1/V.
The lengths a*, b*, c* of the reciprocal basis vectors and the angles, α*, β*, γ*, between the pairs of reciprocal vectors (b*, c*), (c*, a*), (a*, b*), are related to the corresponding lengths and angles for the direct basis vectors through the following relations:
a* = b c sin α/V; b* = c a sin β/V; c* = a b sin γ/V;
cos α* = (cos βcos γ - cos α)/(sin β sin γ);
cos β* = (cos γcos α - cos β)/(sin γ sin α);
cos γ* = (cos αcos β - cos γ)/(sin α sin β).
History
The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs [(1881). Elements of Vector Analysis, arranged for the Use of Students in Physics. Yale University, New Haven].
See also
- Reciprocal lattice
- The Reciprocal Lattice (Teaching Pamphlet No. 4 of the International Union of Crystallography)
- Chapter 1.1 of International Tables for Crystallography, Volume B
- Chapter 1.1 of International Tables for Crystallography, Volume C
- Chapter 1.1.2 of International Tables for Crystallography, Volume D