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The Heisenberg ferromagnet with second neighbour interactions for general spin
Volume 17, number 3
PHYSICS
case, for which each mode tends asymptotically to its resonance frequency, the curves for the delta-function case oscillate about their appropriate resonance frequencies (ne+). This is due to the function @/an)[Ji(n)]. We see by putting w = nwc that its zeros must coincide with passage of the curves through the harmonic frequencies, i.e., for all values of (w2/w2), the modes pass through the points define 8 by Fw = nwc, J,(p) = 0 and [w = nwc, J;(p) = 01. d As (w,/w,$ increases from zero, the loops above a given harmonic approach the loops below the harmonic immediately following it. The points at which the loops can couple must always lie between (Y and o( +l rn’ where o! represents the mth ze%? of J (~7. l!he first points at which two loops touch Ge (w2 w2) = 6.62 (?z= 3 touches 2 mode) and (w$/w,) = 6.81. (n = 2 touches n = 1 mode). After coupling has occurred there are ranges of ~_rin which purely real solutions for w do not exist. The resulting complex conjugate roots have been studied, and in the figure their real parts their real parts are indicated by fine lines and their imaginary parts are shown dotted. These imaginary frequency components can become very strong indeed. For example, when the imaginary component of (w/we) reaches unity, growth rates of the order of 50 dB per cyclotron period (=27r/wc) are implied. Study of the n = 1 mode indicates that an instabili can occur with zero real part. For (ws/wc)? a 17.02 the downward loops can cut the w = 0 axis. Byond this value, purely imaginary solutions can be found. It is easily shown [4,5]
n=
p4c
LETTERS
15 July 1965
from eq. (1) that the loops touching the w = 0 axis will lie between alternate pairs of roots of
J&d ‘&a Jib-d.
*tar?lhe computed curves indicate that in a warm infinite plasma immersed in a magnetic field, an infinite series of perpendicularly-propagating modes can exist. Infinite plasmas growing waves can be expected either due to the presence of boundaries or structures supporting slow waves, or to direct mode coupling and absolute instability at sufficiently large values of (w2/wz>. The considerations made so far for elec % ons extend to positive ions also , ;mc! ::linik- ior, cyclotron harmonic microinstabilities can be predicted. The mechanism may consequently be relevant to ion cyclotron harmonic oscillations observed in thermo-nuclear fusion study plasmas [l]. Applications can be envisaged in which these growth phenomena could be used in microwave oscillators and amplifiers employing either low-pressure discharges, or internally-rotating electron beams under high vacuum conditions.
References 1. F. W.Crawford, Nucl.Fusion 5 (1965) 73. 2. I.B. Bernstein, Phys. Rev. 109 (1958) 10. 3. F. W.Crawford, J. R.es.NBSRadio Science, to be publishedJune X965). 4. F. W. Crawford and J. A. Tataronis, J. Appl. Phys. ~ to be published. 5. R.A.Dory, G.E.GuestandE.G.Harrie, Phys.Rev. Letters 14 (1965) 131. 6. A.Bere and S.Gruber, Appl.Phys.Letters 6 (1965) 27.
*****
THE HEISENBERG FERROMAGNET NEIGHBOUR INTERACTIONS FOR
WITH SECOND GENERAL SPIN
K. PIRNIE and P. J. WOOD University of Newcastle upon Tyna
Department qf Physics,
Received 3 June 1965
We wish to report on a calculation of the magnetic susceptibility in zero field for spin S, including second neighbour interactions, as a function of temperature, using the Heisenberg model. The Heisenberg Hamiltonian including second neighbour interactions may be written &‘=
-2
c
Jij
,@I.s(f) _ go
2
sf) ,
(0) 241
Volume 17, number 3
PHYSICS
LETTERS
15 July 1965
where 51 if i and j are first neighbour sites JQ = J2 if i and j are second neighbour sites 0 otherwise.
Using the notation of a previous paper [l] we write y@= @(S+ 1) 5 aa 0-n , n=O
a0 = 1 ,
where X is a reduced susceptibility, and 0 = kZ’/Jl. In the previous work [l] the coefficiencts a, were functions of the spin S and of the lattice. In the present work they are also functions of the ratio of the two exchange integrals, (Y= Jz/Jl. Wojtowicz [2] has given the coefficients al, ~2, a3 and a4 as functions of the lattice parameters and Dalton and Wood have recently given a5 for S = $ for the three cubic lattices. We have calculated a5 for general spin in terms of lattice parameters. These expressions are rather cumbersome and we therefore give here a5 only for the three cubic lattices (the full results are available on request). Using the abbreviation X = S(S+ 1) as in [l] we have: Face centred cubic lattice [(34 402 464X4 - 13 511984X3 + 2 105 ‘760X2 - 120 240X + 1728) +
a5 = s5
+ Q!(101245 312X4 - 26 256 832X3 + 1976 184X2 - 29 736X) + + (~~(110157 824X4 - 21078 624X3 + 1620 108X2 - 39 312X) + + 03(53 820 480X4 - 11994 080x3 + 1116 360X2 - 42 840X) + + ~~(11303 040x4 - 4 438 560X3 + 606 060X2 - 18 900X) + + (~~(752240X4 - 552 200x3 + 188 016X2 - 25 974X + 864) Body centred cubic lattice
I
[ (5 980 000X4 - 3 124 880X3 + 726 312X2 - 70 848X + 1728) +
a5 = s5
+ o(25790 688X4 - 9 844 128X3 + 1317 456X2 - 35 154X) + + (~~(42630 336X4 - 12 591936X3 + 1160 7 12X2 - 40 068X) + + (r3(32 508 000X4 - 9 245 040x3 + 1138 410X2 - 45 360X) + + (~~(11303 040X4 - 4 438 560X3 + 606 060X2 - 18 900x) + + (~~(1128 360X4 - 828 300x3 + 282 024X2 - 38 961X + 1296) Simple cubic lattice
1
“X (1128 360X4 - 828 300X3 + 282 024X2 - 38 96lX + 1296) + a5 = 42525 i + cr(14477 568X4 - 7 230 048X3 + 1263 276X2 - 44 604X) + + (~~(71129 856X4 - 22 455 216X3 + 1791 342X2 - 58 968X) + + a3(161 461440x4 - 32 793 600X3 + 2 906 820X2 - 64 260X) + + or4(167700 960X4 - 33 193 440X3 + 2 014 740X2 - 28 350X) +
Dr.
+ (y5(51603 696X4 - 20 267 976X3 + 3 158 640X2 - 180 360X + 2 592)
1 .
J. Eve of the University of Newcastle Computing Laboratory has developed a programma for the evaluation of lattice counts including second neighbour links. The lattice counts used in the above work although initially determined in terms of lattice parameters have been checked against the VaheS ob-
242
PHYSICS
Volume 17, number 3
LETTERS
15 July 1965
tained from the computer. This programme will enable us to calculate QSfor general spin and this will be published in the near future. We should like to thank Professor
G. S. Rushbrooke for his encouragement.
References 1. G.S.Ruehbrooke and P.J.Wood, Molecular Phys.1 (1958) 257. 2. P. J. Wojtowicz and R.I. Joseph, Phye. Rev. 135 (1964) A1314. 3. N. W.Dalton and D. W. Wood, Phye. Rev. 138 (1965) A779.
*****
THREE-BODY
BOUND
STATES
IN INERT
GASES
W. ZICXENDRAHT and H. STENSCHXE Institut fllr Mathematische
Physik,
Technische Hochschule,
Received
Xenon, krypton, argon and neon form threeatomic compounds the binding energy of which is being calculated by means of a variational procedure using Lennard- Jones potentials and special three-body coordinates [l-3]. The value of the binding energy of Xe3 turns out to be 16 times larger than a value calculated recently by Buluggiu and Foglia [4]. The coordinates, y, o, 6, are related to the distances of the three atoms, rik, by the following relations: 1 ‘23 = +jcZy[l - sin (Y - sin p]’ ,
‘31=
&Kzy[l-
Y12 = iDy[l
sinasin(@$r)]~,
(1)
- sine! sin@-&.
The S&r&linger-equation three interacting atoms is:
for an s-state of
+-La-+2 a2+psb ( sin Y aY y2 aa
a zi +
Germany
2 June 1965
The constants E and o are taken from Hirschfelder et al. [5]. We used the trial function
J/ = N(cos ar)myn eeky
(5)
(n, m integers; n, m a 4) with n, m and k as parameters. This function is suggested by the following considerations: (1) Configurations, in which two atoms are close to each other compared to the distance of the third atom, are improbable because of the strong repulsive core in the potential. Such configurations would belong to values of 01which are close to $r. Q is in fact small around a! = $r. (2) Configurations, in which all three atoms are close together, are also forbidden as a consequence of the repulsive core in the potential. This property of the wave functions is assured by the factor y? For large values of n and m in eq. (5) a high probability results for finding the atoms at the corners of an equilateral triangle. In fact it turned out that the lowest values of the binding energies are obtained for rather large values of n and m. The parameter k in $J was adjusted to give a maximum for the expectation value of the potential energy. This simplifies the calculations considerably and does not cause a large error because the kinetic energy is rather insensitive against changes of k while the potential energy has a very sharp maximum when plotted as a function of k. The results for the different inert gases are (m = 10, n = 47 [5]): Xe: E = -1.94 X 10m2eV, Kr: E = -1.49 X 10m2 eV, k E = -1.65 X 10-Z eV, Ne: E = -2.6 X 10-3 eV. The calculations of Bu-
(2) I$s=O* -4+E-Iqy,a,p) a2
+Gf%
Karlsruhe,
ap
Where M is the mass of one atom and v(Y, a,p) = v23(‘23) + v3l(y31) + “12(7-12)
(3)
(4)
243
PHYSICS
case, for which each mode tends asymptotically to its resonance frequency, the curves for the delta-function case oscillate about their appropriate resonance frequencies (ne+). This is due to the function @/an)[Ji(n)]. We see by putting w = nwc that its zeros must coincide with passage of the curves through the harmonic frequencies, i.e., for all values of (w2/w2), the modes pass through the points define 8 by Fw = nwc, J,(p) = 0 and [w = nwc, J;(p) = 01. d As (w,/w,$ increases from zero, the loops above a given harmonic approach the loops below the harmonic immediately following it. The points at which the loops can couple must always lie between (Y and o( +l rn’ where o! represents the mth ze%? of J (~7. l!he first points at which two loops touch Ge (w2 w2) = 6.62 (?z= 3 touches 2 mode) and (w$/w,) = 6.81. (n = 2 touches n = 1 mode). After coupling has occurred there are ranges of ~_rin which purely real solutions for w do not exist. The resulting complex conjugate roots have been studied, and in the figure their real parts their real parts are indicated by fine lines and their imaginary parts are shown dotted. These imaginary frequency components can become very strong indeed. For example, when the imaginary component of (w/we) reaches unity, growth rates of the order of 50 dB per cyclotron period (=27r/wc) are implied. Study of the n = 1 mode indicates that an instabili can occur with zero real part. For (ws/wc)? a 17.02 the downward loops can cut the w = 0 axis. Byond this value, purely imaginary solutions can be found. It is easily shown [4,5]
n=
p4c
LETTERS
15 July 1965
from eq. (1) that the loops touching the w = 0 axis will lie between alternate pairs of roots of
J&d ‘&a Jib-d.
*tar?lhe computed curves indicate that in a warm infinite plasma immersed in a magnetic field, an infinite series of perpendicularly-propagating modes can exist. Infinite plasmas growing waves can be expected either due to the presence of boundaries or structures supporting slow waves, or to direct mode coupling and absolute instability at sufficiently large values of (w2/wz>. The considerations made so far for elec % ons extend to positive ions also , ;mc! ::linik- ior, cyclotron harmonic microinstabilities can be predicted. The mechanism may consequently be relevant to ion cyclotron harmonic oscillations observed in thermo-nuclear fusion study plasmas [l]. Applications can be envisaged in which these growth phenomena could be used in microwave oscillators and amplifiers employing either low-pressure discharges, or internally-rotating electron beams under high vacuum conditions.
References 1. F. W.Crawford, Nucl.Fusion 5 (1965) 73. 2. I.B. Bernstein, Phys. Rev. 109 (1958) 10. 3. F. W.Crawford, J. R.es.NBSRadio Science, to be publishedJune X965). 4. F. W. Crawford and J. A. Tataronis, J. Appl. Phys. ~ to be published. 5. R.A.Dory, G.E.GuestandE.G.Harrie, Phys.Rev. Letters 14 (1965) 131. 6. A.Bere and S.Gruber, Appl.Phys.Letters 6 (1965) 27.
*****
THE HEISENBERG FERROMAGNET NEIGHBOUR INTERACTIONS FOR
WITH SECOND GENERAL SPIN
K. PIRNIE and P. J. WOOD University of Newcastle upon Tyna
Department qf Physics,
Received 3 June 1965
We wish to report on a calculation of the magnetic susceptibility in zero field for spin S, including second neighbour interactions, as a function of temperature, using the Heisenberg model. The Heisenberg Hamiltonian including second neighbour interactions may be written &‘=
-2
c
Jij
,@I.s(f) _ go
2
sf) ,
(0) 241
Volume 17, number 3
PHYSICS
LETTERS
15 July 1965
where 51 if i and j are first neighbour sites JQ = J2 if i and j are second neighbour sites 0 otherwise.
Using the notation of a previous paper [l] we write y@= @(S+ 1) 5 aa 0-n , n=O
a0 = 1 ,
where X is a reduced susceptibility, and 0 = kZ’/Jl. In the previous work [l] the coefficiencts a, were functions of the spin S and of the lattice. In the present work they are also functions of the ratio of the two exchange integrals, (Y= Jz/Jl. Wojtowicz [2] has given the coefficients al, ~2, a3 and a4 as functions of the lattice parameters and Dalton and Wood have recently given a5 for S = $ for the three cubic lattices. We have calculated a5 for general spin in terms of lattice parameters. These expressions are rather cumbersome and we therefore give here a5 only for the three cubic lattices (the full results are available on request). Using the abbreviation X = S(S+ 1) as in [l] we have: Face centred cubic lattice [(34 402 464X4 - 13 511984X3 + 2 105 ‘760X2 - 120 240X + 1728) +
a5 = s5
+ Q!(101245 312X4 - 26 256 832X3 + 1976 184X2 - 29 736X) + + (~~(110157 824X4 - 21078 624X3 + 1620 108X2 - 39 312X) + + 03(53 820 480X4 - 11994 080x3 + 1116 360X2 - 42 840X) + + ~~(11303 040x4 - 4 438 560X3 + 606 060X2 - 18 900X) + + (~~(752240X4 - 552 200x3 + 188 016X2 - 25 974X + 864) Body centred cubic lattice
I
[ (5 980 000X4 - 3 124 880X3 + 726 312X2 - 70 848X + 1728) +
a5 = s5
+ o(25790 688X4 - 9 844 128X3 + 1317 456X2 - 35 154X) + + (~~(42630 336X4 - 12 591936X3 + 1160 7 12X2 - 40 068X) + + (r3(32 508 000X4 - 9 245 040x3 + 1138 410X2 - 45 360X) + + (~~(11303 040X4 - 4 438 560X3 + 606 060X2 - 18 900x) + + (~~(1128 360X4 - 828 300x3 + 282 024X2 - 38 961X + 1296) Simple cubic lattice
1
“X (1128 360X4 - 828 300X3 + 282 024X2 - 38 96lX + 1296) + a5 = 42525 i + cr(14477 568X4 - 7 230 048X3 + 1263 276X2 - 44 604X) + + (~~(71129 856X4 - 22 455 216X3 + 1791 342X2 - 58 968X) + + a3(161 461440x4 - 32 793 600X3 + 2 906 820X2 - 64 260X) + + or4(167700 960X4 - 33 193 440X3 + 2 014 740X2 - 28 350X) +
Dr.
+ (y5(51603 696X4 - 20 267 976X3 + 3 158 640X2 - 180 360X + 2 592)
1 .
J. Eve of the University of Newcastle Computing Laboratory has developed a programma for the evaluation of lattice counts including second neighbour links. The lattice counts used in the above work although initially determined in terms of lattice parameters have been checked against the VaheS ob-
242
PHYSICS
Volume 17, number 3
LETTERS
15 July 1965
tained from the computer. This programme will enable us to calculate QSfor general spin and this will be published in the near future. We should like to thank Professor
G. S. Rushbrooke for his encouragement.
References 1. G.S.Ruehbrooke and P.J.Wood, Molecular Phys.1 (1958) 257. 2. P. J. Wojtowicz and R.I. Joseph, Phye. Rev. 135 (1964) A1314. 3. N. W.Dalton and D. W. Wood, Phye. Rev. 138 (1965) A779.
*****
THREE-BODY
BOUND
STATES
IN INERT
GASES
W. ZICXENDRAHT and H. STENSCHXE Institut fllr Mathematische
Physik,
Technische Hochschule,
Received
Xenon, krypton, argon and neon form threeatomic compounds the binding energy of which is being calculated by means of a variational procedure using Lennard- Jones potentials and special three-body coordinates [l-3]. The value of the binding energy of Xe3 turns out to be 16 times larger than a value calculated recently by Buluggiu and Foglia [4]. The coordinates, y, o, 6, are related to the distances of the three atoms, rik, by the following relations: 1 ‘23 = +jcZy[l - sin (Y - sin p]’ ,
‘31=
&Kzy[l-
Y12 = iDy[l
sinasin(@$r)]~,
(1)
- sine! sin@-&.
The S&r&linger-equation three interacting atoms is:
for an s-state of
+-La-+2 a2+psb ( sin Y aY y2 aa
a zi +
Germany
2 June 1965
The constants E and o are taken from Hirschfelder et al. [5]. We used the trial function
J/ = N(cos ar)myn eeky
(5)
(n, m integers; n, m a 4) with n, m and k as parameters. This function is suggested by the following considerations: (1) Configurations, in which two atoms are close to each other compared to the distance of the third atom, are improbable because of the strong repulsive core in the potential. Such configurations would belong to values of 01which are close to $r. Q is in fact small around a! = $r. (2) Configurations, in which all three atoms are close together, are also forbidden as a consequence of the repulsive core in the potential. This property of the wave functions is assured by the factor y? For large values of n and m in eq. (5) a high probability results for finding the atoms at the corners of an equilateral triangle. In fact it turned out that the lowest values of the binding energies are obtained for rather large values of n and m. The parameter k in $J was adjusted to give a maximum for the expectation value of the potential energy. This simplifies the calculations considerably and does not cause a large error because the kinetic energy is rather insensitive against changes of k while the potential energy has a very sharp maximum when plotted as a function of k. The results for the different inert gases are (m = 10, n = 47 [5]): Xe: E = -1.94 X 10m2eV, Kr: E = -1.49 X 10m2 eV, k E = -1.65 X 10-Z eV, Ne: E = -2.6 X 10-3 eV. The calculations of Bu-
(2) I$s=O* -4+E-Iqy,a,p) a2
+Gf%
Karlsruhe,
ap
Where M is the mass of one atom and v(Y, a,p) = v23(‘23) + v3l(y31) + “12(7-12)
(3)
(4)
243