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The reciprocal
space• Space of the wave vectors
• Fourier space
• Inverse• Orthogonal
Reciprocal Space:Geometrical definition
• Introduced by Bravais• Then used by Ewald (1917)
• Definition of basis vectors
• with volume of the cell
• Equivalent definition (2D, 3D...)
• is orthogonal to et but NOT in gal to
• Reciprocal space: vector space basis • Reciprocal lattice: set of points
DL
RL
ab
b* a*
integers
𝒂∗=2𝜋𝒃∧𝒄𝑣
;𝒃∗=2𝜋𝒄∧𝒂𝑣
;𝒄∗=2𝜋𝒂∧𝒃𝑣
𝑸h𝑘𝑙=h𝒂∗+𝑘𝒃∗+𝑙𝒄∗
Definition by plane waves
• belongs to RS iff:
• Reciprocal space
• Set of wave vectors of the plane waves with the direct space periodicity
• Si mwlvkuhuvwhkl 2)(2.RQ
• Si on pose
entiers.
muvwuvw 2.RQR *** cbaQ zyxhkl
*** cbaQ lkhhkl
zy,x,zyx, hklhklhkl 2.,2.2. cQbQaQ
𝒒
𝑸h𝑘𝑙=h𝒂∗+𝑘𝒃∗+𝑙𝒄∗
𝒒
∀𝑹𝑢𝑣𝑤𝑒𝑖𝑸 ∙𝑹𝑢𝑣𝑤=1⟺∀𝑹𝑢𝑣𝑤𝑸 ∙𝑹𝑢𝑣𝑤=2𝜋𝑚
Properties of the RS • Symmetry
The reciprocal spacethe same point symmetry
as the direct lattice
Let be a symmetry operator of the RL and a point of the RL.
Thus belongs to the RL
• Duality• The reciprocal lattice of the RL is the direct lattice:
• RL of the RL consists in points such that:
• If the relation is verified• Conversely if , it satisfies ,
thus , and are integers and belongs to the RL
b* a*DL RLab
∀𝑸h𝑘𝑙𝑸h𝑘𝑙 ∙𝑹=2𝜋𝑚
• The nodes of a lattice are regrouped in equally spaced planes:
The lattice planes• Family of planes
Lattice planes, rows
[100]
[001]
[010]
<100>
• Row : series of nodes in the direction Ruvw • Notation [uvw], u, v, w relatively prime
• Symmetry equivalent directions are noted: <uvw>
Lattice planes
c
1/3
1/4 1/2b
a
h, k, l Miller indices• Family of lattice planes • Famillies of planes equivalent by symmetry
dhkl
• Distance between planes
• If N(hkl) is the planar node density, N(hkl)/dhkl is the volumic node density • The most dense planes are the more distant• Crystals facets are planes with small indices
The lattice plane closest to the origin, intersects the cell axes in:
(0,0,1) (3,2,4)
𝒂h
,𝒃𝑘
,𝒄𝑙
Lattice planes and RS
Q010=d*
Q020
d010=2p/Q0102p/Q020
• The lattice plane closest to the origin satisfies:
• It intersects the axes in: h, k, l Miller indices (mutually prime)
To each family of lattice planes of period corresponds
A reciprocal space row of period • This row is orthogonal to the lattice planes
• The smallest vetor of this row has a magnitude
𝒉𝒖+𝒌𝒗+𝒍𝒘=𝟏𝒂h
,𝒃𝑘
,𝒄𝑙
𝒅=𝟐𝝅 /𝑸
𝑹𝑢𝑣𝑤
is a RL vector
cannot be shorter, it is the row period
Miller indices:
𝑑
?
𝒏
Distance between lattice planes dhkl
• General case
• Hexagonal system:
• Cubic system :
• dhkl distance between planes (hkl)
Qhkl smallest vextor of the row
𝑑h𝑘𝑙=2𝜋
√h2𝑎∗ 2+𝑘2𝑏∗2+𝑙2𝑐∗ 2+2 h𝑘𝑎∗𝑏∗ cos𝛾∗+2𝑘𝑙𝑏𝑐∗cos𝛼∗+2 h𝑙𝑎∗𝑐∗ cos 𝛽∗
𝑑h𝑘𝑙=2𝜋𝑄h𝑘𝑙
𝑑h𝑘𝑙=𝑎
√ 43
(h¿¿2+𝑘2+h𝑘)+ 𝑙2(𝑎𝑐
)2
¿
𝑑h𝑘𝑙=𝑎
√h2+𝑘2+ 𝑙2
Multiple unit cells
• Body centered cell
• The condition implies
1) integers (Reciprocal space of lattice 2)
• Reflection conditions
cbaR
cbaR
0.5)w0.5)v0.5)u
wvu
uvw
uvw
(((I
F
PIFA
PFIA
Conditions
abA
B
b* a*
A*
B*
a
a*
• Hexagonal lattice• A = a-b; B=a+b; C=c
*)*(2
1
2
)(2
22*
*)*(2
1
2
)(2
22*
abbacAC
B
bacbaCB
A
vv
vv
2nkh
h+𝑘+ 𝑙=2𝑛
∀𝑹𝑢𝑣𝑤𝑸 ∙𝑹𝑢𝑣𝑤=2𝜋𝑛
same parity
• Definition• Fonction or distribution
• Direct lattice described by a ’’node density’’ function:
Fourier transform of the RS
• The Fourier transform of direct lattice is the reciprocal lattice
‘‘node density’’ of RL
• The reciprocal space is the FT of the Direct space
𝑆 (𝒓 )=∑𝑢𝑣𝑤
𝛿(𝒓−𝑹𝑢𝑣𝑤)
𝑇𝐹 (𝑆 (𝒓 ) )=𝐹 (𝒒 )=∫∑𝑢𝑣𝑤
𝛿(𝒓 −𝑹𝑢𝑣𝑤)𝑒−𝑖𝒒 ∙𝒓 𝑑3𝒓
𝐹 (𝒒 )=𝑣∗∑h𝑘𝑙
𝛿(𝒒−𝑸h𝑘𝑙)
¿ ∑𝑢𝑣𝑤
𝑒−𝑖𝒒 ∙𝑹𝑢𝑣𝑤=∑𝑢
𝑒− 2𝑖 𝜋 𝑞𝑥𝑢∑𝑣
𝑒−2 𝑖 𝜋 𝑞𝑦 𝑣∑𝑤
𝑒− 2𝑖 𝜋 𝑞𝑧𝑤
¿∑h
𝛿 (𝑞𝑥− h )∑ 𝛿 (𝑞𝑦 −𝑘 )∑𝑙
𝛿 (𝑞𝑧− 𝑙)
∑h
𝛿 (𝑞−h𝑇 )= 1𝑇 ∑
𝑛=− ∞
+∞
𝑒− 2𝑖 𝜋 𝑛 𝑞
𝑇
Série de Fourier duPeigne de Dirac
Properties of the FT
uvwuvw
hklhkl vTF )())((1 RrQq
• Duality of RS and DS
• RS and DS have the same point symmetry• Let O be a symmetry operator of the DS
…then O is a symmetry operator of RS
)(')'('))'((
)()())((
3'.3'.
3)(.3).( 1
qFrrrrO
rrrrq
rqrq
rOqrq
∫∫∫∫
deSdeS
deSdeSOF
ii
iiO
• Convolution• Convolution of f and g is f * g
∫ udurur 3)()())(( gfgf
)()()2()(
)()()(3 gTFfTFπfgTF
gTFfTFgfTF
Application to low dimenbsion objects
2p/a
a
a
• 1D : chain )()u()(S
u// rarr
• 2D : planes )()()( // rbarr
uvw
vuS
hk
yx kqhqF )()()(q
h
x hqF )()(q
** baq yx qq
*aq xq
Set of parallel plane
Lattice of lines
a*
b b*
a*
Relation with diffraction• Bragg’s law
• Diffraction on lattice planes, spacing
Vecteur de diffusion• q normal to the lattice planes
Diffraction
belongs to RS(to the direction planes)
ki kdqq
d
𝒒=2𝜋𝑑
𝒏 𝒒=22𝜋𝑑𝒏
2𝑑 sin𝜃=𝑚𝜆 𝒒=𝒌𝑑−𝒌𝑖
𝒒=2𝑘 sin𝜃=4𝜋𝜆
sin𝜃=2𝜋𝑑𝑚