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STUDIES ON QUASILATTICES : MISTAKES AND LONG RANGE CORRELATION IN ONE
DIMENSION
J. Lu, J. Birman
To cite this version:
J. Lu, J. Birman. STUDIES ON QUASILATTICES : MISTAKES AND LONG RANGE CORRE- LATION IN ONE DIMENSION. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-251-C3-258.
�10.1051/jphyscol:1986327�. �jpa-00225739�
STUDIES ON QUASILATTICES : MISTAKES AND LONG RANGE CORRELATION IN ONE DIMENSION
J . P . LU and J.L. BIRMAN
P h y s i c s D e p a r t m e n t , C i t y C o l l e g e of the C i t y U n i v e r s i t y o f N e w Y o r k . N Y 10031, U . S . A .
Abstract- Mistakes in one dimensional quasilattices are studied. The effect of mistakes on the diffraction intensity is calculated exactly: we show that the diffraction pattern is changed qualitatively. Detailed calculations of Fourier transform and configuration average are given. Also the density-density correla- tion for both perfect and imperfect quasilattices have been studied. We show explicitly that quasilattices are infinitely correlated, and that introducing mistakes does not break this correlation. Mistakes are shown to be generic for the quasilat- tice: they cost little energy, and increase the entropy.
0. Introduction
Defects in quasicrystals have been a major theme of this workshop. Elsewhere we reported results for a particular class of defect in the quasilattice, namely "mistakes"
'.
In this paper we present the detailed mathematical analysis of the Fourier transform and the configuration average for this class of defect. Also we will examine the long range order of a quasilattice from the point of view of the density-density correlation function. Our results show: 1) there is infinitely long range correlation in the quasilat- tice, and 2 ) this correlation is preserved even if mistakes are introduced. Therefore mis- takes in a quasilattice are qualitatively different from the stacking faults in a normal crystal, where the long range correlation in a crystal is lost when faults or mistakes are introduced. In section 3 we give the exact calculation of the effect of mistakes on the diffraction intensity. In section 4 we use a simple pairwise interaction model to show that mistakes are generic for quasicrystals.
1. Perfect One Dimensional Quasilattice
A simple one dimensional quasilattice with two basic lengths is defined by giving the location of the n t h atom as
where [ z ] represents the largest integer which is smaller than or equal to z , and a, J3 E (0.1). The lattice defined by Eq.1 has only two nearest neighbor spacings: S = 1
1 1
and L = 1
+ -.
The average spacing is a = 1+ -.
P PC+
The lattice can also be obtained by a projection method from a two dimensional rectangular lattice
'.
Let the basic units of the rectangular lattice be b and b , along x and y directions respectively. Draw a line with slope k . Take a unit cell and slide it along the direction of the line: one obtains a strip. Project all lattice points inside thisArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986327
C3-252 JOURNAL DE PHYSIQUE
strip onto the line: one gets the lattice defined by Eq.1. The relations between the parameters are:
b 1 1 1
(-1' = (1
+
- ) ( u - 1 ) ; k 2 = ( 1+ -I/
( u - 1) .b 2 P P
Therefore there is an exact one-to-one correspondence between Eq.1 and the projection method.
Using z = [ z ]
+
{ z ), where ( z ) is the fractional part of z , one can rewrite Eq.1 as Here f ( z ) is a periodic function of z with period ua :The Fourier transform of the lattice of delta functions a t positions x ( n ) defined by Eq.1 is denoted as Fo(q ). that is
where
5,
= na +$. Since exp(iq f((1)
is a periodic function of (i with period u a.
onecan perform the Fourier expansion
Theref ore
Now the summation over n can be carried out in the limit N4-. According t o the Poisson s u m formula this gives the A function. After some algebra one obtains
sinz,,,, / 2 Fo(q =
C
exp(i $,, )Z,nn 2 A(9a - q,,, a )
mn
with
and A(q) = 0 except A(0) = 1. If u is an irrational number, qrnn will be dense every- where in q space. Eq.8 has been obtained by several workers using different approaches. The advantage of the present formalism is that the generalization to addi- tional quasiperiodic or almost periodic lattices is straightforward, as one can see from Eqs. 1, 5 , and 6* .
* After we finished this work we learned of the work by E. Bombieri and J. Taylor, presented at this workshop, where they discussed such generalizations.
city we will consider this case in the following, but this is not necessary.
The lattice defined by Eq.1 has perfect long range correlation. To prove this w e examine the pair correlation function. Assume a t every lattice site there is an atom with identical mass and the total mass of system is normalized t o 1. Then the mass density is
Here the summation is over all integers, and N + w is understood. The t w o particle correlation function is defined as
K ( x l . x 2 > = < p ( x 1 ) p ( x 2 )
> .
(11)The pair correlation function is defined as G ( x ) = ~ K ( x ' , x'
+
x )dx'.
For a perfect quasilattice one finds, from Eq.10 and Eq.11, that- 1 I m n
- F Z 6 [ x
-
(m-n )a+
?( iT+ PI -
{ _+ PI 11
I2 ,171
In the last step we have used the property of the irrational number 7 : that
(-1
n is uni-7
formly distributed in the interval (0.1) when n takes all integer values. So G o ( x ) is still a sum of 6 functions. This is clear evidence that the positions of lattice points are infinitely correlated. From Eq.12 one also finds that the distance between t w o lattice points separated by I lattice sites can only take two values, namely either
1 1
x o I = I
+
[--] or nol+-. From the point of view of statistical symmetry 3 , Eq.127 7
clearly shows that a quasicrystal belongs to the class of a crystal.
2. Mistakes And Long Range Correlation
The mistake we have considered is a special class of defect in the quasicrystal.
namely it is a transposition of two different segments ( L - S ) in the lattice struc- ture. A similar defect in the crystal is the stacking fault 4. However mistakes and stacking faults are qualitatively different, since stacking faults break the long range correlation of the crystal. but mistakes do not break the long range correlation in the quasicrystal case. We will now show this.
When we study the local configurational order in the perfect Fibonacci lattice, it is evident there is no SS and LLL in the sequence (Fig.l). The mistakes we have con- sidered do not violate this constraint. This leads to the result that these transpositions can only occur in the clusters as: LLSLS - + L S L L S . More importantly the positions where such a transposition can occur are a subset of all lattice sites, and these positions follow the Fibonacci sequence. Therefore the positions of possible mistakes also form the Fibonacci lattice, but with inflated unit segments: S' = 74S and L' = 'T4L (Fig.l) l .
C3-25 4 JOURNAL DE PHYSIQUE
Fig.1 A segment of a perfect Fibonacci lattice. All clusters of type LLSLS are underlined. The distance between these clusters is either S' = 3L +2S = r4S or L' = 5L 4-3s = T ~ L . These new basic lengths also follow the Fibonacci sequence, hence form a new Fibonacci lat- tice with basic length S' = r 4 S .
Fig.2 a) The cluster LSLLSLSL whose position is labeled by ym in the perfect quasiperiodic lattice. b) A mistake occurred at site y,: and the cluster became LSLSLLSL. The atomic site where a mistake occured is shifted by an amount 1/ 7 . But the remaining sites are the same.
Let x, label all lattice sites, let y,,, be the possible sites for.mistakes and let a, be a configuration variable associated with y,,,
.
am = 1 represents a mistake a t y,: that is the atom initially a t y, is shifted by(--I
1 (see Fig.2). am = 0 represents no mistake7
a t y,,
.
For a specific configuration { a ) , the mass density isAssume an independent and uniform distribution for every a;,
Then the average density is F(x ) = < p ( x . b I ) >
I t follows that the two particle correlation function is K (x .x' ) = < p ( x . { d ) p ( x 8 .{a)) >
= p(x )p(xl )
+
p [p(x I d (x' )+
p(x' )d (x 11+
+
< d ( x , ( a } ) d (x' .{a)) >< d ( x . ~ ~ I ) d ( x ' . { c r l ) > =
F < x [ S ( X - ( y m - - ) ) - 6 ( x - y m ) 1
ym J'l 7
Therefore
K ( x .x' ) = p(x )p(x' )
+
D ( x .x' ) .Hence the pair correlation function G ( x ) is also split into two parts G ( x ) = G ( ~ ) + G = ( ~ )
.
where
G ( x = l ~ ( x ' )p(x
+
x' )dx'rn 1 m 2
- ( { ? I ) ~ ( x
-
yo,,,+
-) 7-
( I - ( - - } ) 6 ( x 7 --
- ) ] 7 . ( 2 0 )The integration and summation are carried out similarly to the steps leading t o Eq.12.
And
E(x)=
~ D ( x ' . x+
x 3 ) d x 'I n .
Where x On = n
+ -[-I
1s the Fibonacci lattice with CY = /3 = 0, and is the subset7 . 7 .
of x O , where the mistakes can occur.
From Eq.19-Eq.21 one sees that the pair correlation function is a sum of 6 func- tions even through we have introduced mistakes into the system. In other words the long range correlation of the quasilattice is preserved. This also qualitatively explains the results we had obtained previously
':
the diffraction intensity is still a sum of Bragg peaks. Detailed proof of this result will be given in the next section.C3-256 JOURNAL DE PHYSIQUE
3. Effect Of Mistakes On T h e Diffraction Intensity
After the lengthy mathematics of section 1 and 2, the calculation of the effect of mistakes on the diffraction intensity is quite straightforward. The diffraction intensity is
Where H ( q .q' ) is the Fourier transform of the two particle correlation function K ( x . x ' ) . In the previous paper
',
w e have taken I ( q ) as F ( q ) F ( q ), where F ( q ) is the Fourier transform of the average mass density p(x ). This is equivalent to taking K ( x .x' ) = p(x )p(x' ). We now prove that this is exact.From Eq.18 one sees the difference is
D ( X .XI ) = K ( x .x' )-P(x )F(X' )
- p ( l - P )
- 1
N 2 [ S ( x - ( y m
- -1)
- S(x -ymI
x7 ym
From this expression it follows that D ( x ,x' ) is of order of - 1 smaller than p(x )P(x' ).
N Let us calculate its Fourier transform explicitly.
b
( q ,q' ) = ( x ,r' ) e'9" + i9'" dxdx'7 7
t S ( x - y,, )S(x' - y, ) ] e'q" + '9'"' d xdx'
We see in the l i m i t N -+ m, t h i s i s zero compared with F o . For the contribution to the diffraction intensity I ( q ) we take q' = -q
The Fourier transform of the average density is easily calculated from Eq.15 to be F ( q ) = j p ( x ) e i q " d x
7
i$,,,,,
.
= ~ n n . 4,nn , $, Z m nsln- - psin-
2 sin-
F ( q ) = - I -27 27 Aka - qm, a )
Where the various parameters are
From Eq.27 and 28 we see that the diffraction still consists of Bragg peaks, whose location is the same as if there were no mistakes. However the intensity of each peak is modified. Depending on the value of q and the phases the intensities could be either decreased a s in the case of a normal crystal when disorder is present, or it could be enhanced. This latter result is new and significant. We have reported elsewhere on the diffraction pattern of a three dimensional quasilattice, illustrating these changes pro- duced by mistakes
'.
4. Energy Cost Associated W i t h Mistakes
In this section we use a simple model to demonstrate the small energy required to produce a mistake. Therefore mistakes are a generic disorder for a quasicrystal.
Assume the potential energy of the whole system is the sum of all pairwise interactions
Suppose a mistake occurs a t site y,. As we pointed out. mistakes can only occur in terms of a cluster transposition: LLSLS -t LSLLS. Owing to the constraint (no LLL or S S ) , the cluster must have the form LSLLSLSL. It is clear from Fig.2 that a mistake only changes the position of the atom a t y,. not the neighboring sites in the cluster.
Therefore the terms changed in Eq.22 are those associated with y,. Up to the 4th nearest neighbor the contribution to the total potential energy from these terms are
for the perfect lattice. And
v"
=v t s ) + v ( s +
L )+
V ( 2 S+
L )+
V ( 2 S+
2 L ) + V ( 3 L + S ) + V ( ~ L + S ) + V ( 2 L ) + V ( L )for a lattice with a mistake. We see that V ' =
v".
So if the interation is a decreasing function of distance, which is physically plausible, w e can say that mistakes cost very little energy. On the other hand, by allowing mistakes, the system gains configurational entropy. Therefore the free energy of the whole system could be decreased. We believe this is one of reasons that mistake could be often observed in quasicrystals.C3-258 JOURNAL DE PHYSIQUE
The essential physics is contained in the imposed constraint on local order. This is unique for a quasilattice, since the cluster after the transposition: LSLLS, is allowed somewhere in the perfect lattice as a building block
.
Generally one can impose more than just nearest neighbor order. Then mistakes are still possible, but with rarer fre- quency (the scaling r4 in last section will be replaced by a higher power of 7 ) . Also the energy cost of mistakes will be even smaller, that is to say t h e equality V = V" will be satisfied even when one considers more then the 4th nearest neighbor interactions. This result has no exact counterpart in the normal crystal case. Consider for example stack- ing faults in the CCP layer structure. The perfect sequence will be...
ABCABCABC...
o r ... ACBACBACB ... . Two types of fault could occur: ... ABC(B)ACB...
or ... ABC ( B )CAB ... . where the layer in the brackets is t h e layer a t fault. One immedi- ately sees that in both cases new clusters: BCB o r CBC, are introduced. These new clusters are not allowed in the perfect sequences. therefore they must cost energy.Further there is no w a y one can impose local order constraints to avoid this, as we did for the case of a quasilattice.
5. Conclusions
In summary. we presented a detailed analytical calculation of the diffraction intensity for one dimensional quasiperiodic lattices with and without mistakes. The long range correlation of a quasilattice was studied using the pair correlation function.
We showed explicitly the infinite correlation of a quasilattice. We also show t h a t this correlation is not broken by mistakes. The physical origin of mistakes is examined.
Results show that mistakes are generic for a quasilattice. Hence mistakes in quasilattices are qualitatively different from normal defects in crystals such as: stacking faults and dislocations. Further investigations are in progress.
We thank Dr. K. Arya for constant discussions. Discussions with Dr. M. Jar& and Dr. C. Henley are acknowledged. The work was support in part by PSC-FRAP-6-65280 and NSF-DMR-830398 1.
6. References
[I] Jian Ping Lu and Joseph L. Birman. Preprint
.
submitted for publication.[2] M. Duneau and A. Katz. Phys. Rev. Lett. 54, 2688(1985). V. Elser and C. Hen- ley. Phys. Rev. Lett. 54. 1730(1985). R. Zia and W. Dallas. Joun. of Phys. A18.
L314( 1985).
[3] J. L. Birman and H.-R. Trebin. Joun. of Stat. Phys. 38. 371(1985).
141 A. Wilson. X-Ray Optics 2nd Edn, London:Methuen (1962). T. Welberry, Rep.
Prog. Phys. 48. 1543(1985).
