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Energy of a non-ideal hexagonal close packed lattice
Volume
32A.
number
ENERGY
1
OF
PHYSICS
A
NON-IDEAL
LETTERS
1
HEXAGONAL
CLOSE
PACKED
June 1970
LATTICE
Roger HOWARD
Received
25 April
1970
The energy of a static assembly of inert gas atoms in an almost hexagonal close packed structure has been calculated as a function of the ratio c/n of :I unit cell dimensions. The minimum occurs at 99.986% of the ‘ideal’ value of J8/3.
The properties of inert gases in both fluid and solid states are consistent 1221 with the use of the ‘6-12 potential’ ~E{(cT;Y)[1 - (a,‘~)~} for the interaction energy of a pair of atoms separated by a distance Y. The total energy of a crystal is simply the sum of this expression over all pairs of atoms. It can conveniently be considered in three parts: E,, the energy of a crystal-in which all atoms are at rest: E,, the energy of the zero point motion: and the thermal energy. which is the excess of the total over E, t E, at temperatures above absolute zero. E,. E, and the lattice spacing that minimises E. have previously been obtained [3] for cubic and (ideal) hexagonal close packing. The symmetry of an hcp lattice is unchanged by expansion or contraction parallel to the hexagonal axis; it is therefore of interest to investigate the energy dependence on such deviations from the ‘ideal’ geometry. A hexagonal lattice is generated by basic primitive translations as follows: B and b which are of equal magnitude and at angle of 1200 both in the plane perpendicular to the hexagonal axis; and c which is parallel to the axis. In the hcp structure there are two such lattices, one displaced from the other by ia+ Q b ++c; the ‘ideal’value for c/a (found for packing hard spheres) is q. Writing c/‘a = yq for non-ideal structures, one can define reduced coordinates ?‘= f/u for each y; E, is then minimised by the condition
summing over all neighbors of an arbitrarily chosen site. Computer calculations of the reduced lattice sums have been made; they are given by summation over some 450 shells of nearest neighbors,
Data table Y
O.Y’JS 0.996
_____
CY.-12
x7-_”
34.601 026
12.380 076 6
0.997 0.998
14.571 556 14.542209 14.512 983
12.329 12.279 12.230
693 4 727 1 174 1
0.999 0.999 86 1.000 1.001
14.483 14.458 14.454 14.426
880 948 898 03G
12.181 12.139 12.132 12.083
030 092 293 959
8 6 8 5
1.002 1.003 1 .(I04 1.005
14.397 14.368 14.340 14.311
294 672 168 783
12.036 024 11.988 485 11.941339 11.894 581
6 5 1 8
CL’BIC
14.453
922
12.131880
2
Y
a/u
_E”/E per atom
0. 993 0.996 0.997 0.998
0.915 738 92 0.916 053 02 0.91636525 0.916 675 61
8.610 203 4 8.610 523 9 8.610 770 7 8.6109440
0.999 0.000 86 1.000 1.001
0.916 984 12 0.91724795 0.917 290 76 0.917 595 54
8,611043 9 8.611 0710 8.6110704 8.6110232
1.002 1.003 1.004 1.005
0.917 898 45 0.918 199 50 0.918 498 68 0.918 796 00
8.610902 8.610 709 8.610 442 8.610 102
0.917 285 65
8.610
8 3 3 4
200 9
and approximating the remaining terms by an integral over an appropriate volume [4]. The accuracy of this procedure is found by considering fluctuations in the result when different numbers of shells are summed. The results given in the data table have errors of the order of one or two in the last place. For comparison, results for cubic close packing are also included. The rela37
Volume 32A. number 1
PHYSICS
LETTERS
tionship between E, and y is closely parabolic (second differences vary by at most 5 in the last place), and minimum energy occurs for y = 0.999 86 k 0.0004. The behavior of the dominant nearest neighbour terms indicate that no other minima occur.
1 June 1970
Re_ferences [I] G. Boato. Cryogenics 4 (1964) 6s. [Z] J. Grindlay nnd R. Howard. J. Phys. Chem. Solids. supp. l(1961) 129. [3] R.Ho\vard. Phvs. Letters 29A (1969) 53. [41 1\?1. Born and K. Hunng. Dynamical theory of crystal lattices.
O.U.P.
(1954).
appendix III.
I wish to thank Hans-Jiirgen Schirlitzki, whom I have been unable to locate. for a letter suggesting this problem. *****
IMAGING SYSTEMS OF NON-EQUALLY
USING LINEAR ARRAYS SPACED ELEMENTS
F. GORI and G. GUATTARI Istitlrto
di Fisicn.
Facolt?i
di Ingegneria. Received
liniversity
11 March
oJ~Romc.
Rome.
Ital)
1970
A linear array of n equally spaced elements gives an image with 2~2-1 resolved points. A number of points much greater than 2n-1 can be obtained using arrays with a suitable arrangement of n non-equally spaced elements.
Image forming devices using arrays of elements are often encountered in radio astronomy; they are also of interest in optics [l]. In radio astronomy the simplest devices of this kind use linear arrays of equally spaced aerials [2]. In this letter we want to observe that this kind of array is highly redundant. We will show that the redundancy can be reduced by using arrays with non-equally spaced aerials. Let us briefly review the principle of operation of such devices. Let us consider an extended incoherent quasi-monochromatic source whose intensity distribution is u(x) and an imaging system with a pupil function (describing the array) p(5). The instrument has a spread function S(X) given by 1F[p(5)]12 where F means Fourier transformation. As is well known, the Fourier transform of the intensity distribution I(X) in the image plane is given by [3]: F[l(x)]
= F[o(@l+‘[s(x)]
(1)
According to the Van Cittert-Zernike theorem gives, in the far-field approximation, the mutual intensity J(t) for two points at a distance 5 from each other: in fact, in this approximation 5(<) is spatially stationary, that is, it depends only on the distance between the two points [4]. F[ a(x)]
38
Furthermore Fls (x)i is given bv the autocorrelation function A(i) = j (t) ;@ (5). So we can write eq. (1) as: F[Z(x)]
= J(S).A(<)
(2)
Let us now observe that the pupil function p(t) for a linear array is given by a series of spikes (6like functions); the same is true for A(5) which is different from zero only for those values of 5 (regardless of their sign) that correspond to distances between two aerials in the array. It then follows from eq. (2) that we can think of F[l(x)] (and then of the image Z(X)) as obtained from the measurement of J(5) for every pair of aerials. It should be noted that this argument also applies to bidimensional arrays [5]. We consider now a linear array of n equally spaced aerials at a distanced from each other; in such an array there are n-k pairs of aerials with a distance kd(k = 0, 1 . . . ,. n). From this we see the origin of the redundancy: as 5(<) is a stationary function, we are repeating (n-k) times the measurement of J&d). A method for reducing this redundancy is that of using a suitable arrangement of non-equally spaced aerials. An example will explain the method. In fig. la) and lb) the pupil functions p(s), p’(t)
32A.
number
ENERGY
1
OF
PHYSICS
A
NON-IDEAL
LETTERS
1
HEXAGONAL
CLOSE
PACKED
June 1970
LATTICE
Roger HOWARD
Received
25 April
1970
The energy of a static assembly of inert gas atoms in an almost hexagonal close packed structure has been calculated as a function of the ratio c/n of :I unit cell dimensions. The minimum occurs at 99.986% of the ‘ideal’ value of J8/3.
The properties of inert gases in both fluid and solid states are consistent 1221 with the use of the ‘6-12 potential’ ~E{(cT;Y)[1 - (a,‘~)~} for the interaction energy of a pair of atoms separated by a distance Y. The total energy of a crystal is simply the sum of this expression over all pairs of atoms. It can conveniently be considered in three parts: E,, the energy of a crystal-in which all atoms are at rest: E,, the energy of the zero point motion: and the thermal energy. which is the excess of the total over E, t E, at temperatures above absolute zero. E,. E, and the lattice spacing that minimises E. have previously been obtained [3] for cubic and (ideal) hexagonal close packing. The symmetry of an hcp lattice is unchanged by expansion or contraction parallel to the hexagonal axis; it is therefore of interest to investigate the energy dependence on such deviations from the ‘ideal’ geometry. A hexagonal lattice is generated by basic primitive translations as follows: B and b which are of equal magnitude and at angle of 1200 both in the plane perpendicular to the hexagonal axis; and c which is parallel to the axis. In the hcp structure there are two such lattices, one displaced from the other by ia+ Q b ++c; the ‘ideal’value for c/a (found for packing hard spheres) is q. Writing c/‘a = yq for non-ideal structures, one can define reduced coordinates ?‘= f/u for each y; E, is then minimised by the condition
summing over all neighbors of an arbitrarily chosen site. Computer calculations of the reduced lattice sums have been made; they are given by summation over some 450 shells of nearest neighbors,
Data table Y
O.Y’JS 0.996
_____
CY.-12
x7-_”
34.601 026
12.380 076 6
0.997 0.998
14.571 556 14.542209 14.512 983
12.329 12.279 12.230
693 4 727 1 174 1
0.999 0.999 86 1.000 1.001
14.483 14.458 14.454 14.426
880 948 898 03G
12.181 12.139 12.132 12.083
030 092 293 959
8 6 8 5
1.002 1.003 1 .(I04 1.005
14.397 14.368 14.340 14.311
294 672 168 783
12.036 024 11.988 485 11.941339 11.894 581
6 5 1 8
CL’BIC
14.453
922
12.131880
2
Y
a/u
_E”/E per atom
0. 993 0.996 0.997 0.998
0.915 738 92 0.916 053 02 0.91636525 0.916 675 61
8.610 203 4 8.610 523 9 8.610 770 7 8.6109440
0.999 0.000 86 1.000 1.001
0.916 984 12 0.91724795 0.917 290 76 0.917 595 54
8,611043 9 8.611 0710 8.6110704 8.6110232
1.002 1.003 1.004 1.005
0.917 898 45 0.918 199 50 0.918 498 68 0.918 796 00
8.610902 8.610 709 8.610 442 8.610 102
0.917 285 65
8.610
8 3 3 4
200 9
and approximating the remaining terms by an integral over an appropriate volume [4]. The accuracy of this procedure is found by considering fluctuations in the result when different numbers of shells are summed. The results given in the data table have errors of the order of one or two in the last place. For comparison, results for cubic close packing are also included. The rela37
Volume 32A. number 1
PHYSICS
LETTERS
tionship between E, and y is closely parabolic (second differences vary by at most 5 in the last place), and minimum energy occurs for y = 0.999 86 k 0.0004. The behavior of the dominant nearest neighbour terms indicate that no other minima occur.
1 June 1970
Re_ferences [I] G. Boato. Cryogenics 4 (1964) 6s. [Z] J. Grindlay nnd R. Howard. J. Phys. Chem. Solids. supp. l(1961) 129. [3] R.Ho\vard. Phvs. Letters 29A (1969) 53. [41 1\?1. Born and K. Hunng. Dynamical theory of crystal lattices.
O.U.P.
(1954).
appendix III.
I wish to thank Hans-Jiirgen Schirlitzki, whom I have been unable to locate. for a letter suggesting this problem. *****
IMAGING SYSTEMS OF NON-EQUALLY
USING LINEAR ARRAYS SPACED ELEMENTS
F. GORI and G. GUATTARI Istitlrto
di Fisicn.
Facolt?i
di Ingegneria. Received
liniversity
11 March
oJ~Romc.
Rome.
Ital)
1970
A linear array of n equally spaced elements gives an image with 2~2-1 resolved points. A number of points much greater than 2n-1 can be obtained using arrays with a suitable arrangement of n non-equally spaced elements.
Image forming devices using arrays of elements are often encountered in radio astronomy; they are also of interest in optics [l]. In radio astronomy the simplest devices of this kind use linear arrays of equally spaced aerials [2]. In this letter we want to observe that this kind of array is highly redundant. We will show that the redundancy can be reduced by using arrays with non-equally spaced aerials. Let us briefly review the principle of operation of such devices. Let us consider an extended incoherent quasi-monochromatic source whose intensity distribution is u(x) and an imaging system with a pupil function (describing the array) p(5). The instrument has a spread function S(X) given by 1F[p(5)]12 where F means Fourier transformation. As is well known, the Fourier transform of the intensity distribution I(X) in the image plane is given by [3]: F[l(x)]
= F[o(@l+‘[s(x)]
(1)
According to the Van Cittert-Zernike theorem gives, in the far-field approximation, the mutual intensity J(t) for two points at a distance 5 from each other: in fact, in this approximation 5(<) is spatially stationary, that is, it depends only on the distance between the two points [4]. F[ a(x)]
38
Furthermore Fls (x)i is given bv the autocorrelation function A(i) = j (t) ;@ (5). So we can write eq. (1) as: F[Z(x)]
= J(S).A(<)
(2)
Let us now observe that the pupil function p(t) for a linear array is given by a series of spikes (6like functions); the same is true for A(5) which is different from zero only for those values of 5 (regardless of their sign) that correspond to distances between two aerials in the array. It then follows from eq. (2) that we can think of F[l(x)] (and then of the image Z(X)) as obtained from the measurement of J(5) for every pair of aerials. It should be noted that this argument also applies to bidimensional arrays [5]. We consider now a linear array of n equally spaced aerials at a distanced from each other; in such an array there are n-k pairs of aerials with a distance kd(k = 0, 1 . . . ,. n). From this we see the origin of the redundancy: as 5(<) is a stationary function, we are repeating (n-k) times the measurement of J&d). A method for reducing this redundancy is that of using a suitable arrangement of non-equally spaced aerials. An example will explain the method. In fig. la) and lb) the pupil functions p(s), p’(t)