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Transfer matrix in plane-rotator model
Volume 104A, number 6,7
PHYSICS LETTERS
10 September 1984
TRANSFER MATRIX IN PLANE-ROTATOR MODEL Daniel C. MATTIS
Department of Phys:cs, University of Utah, Salt Lake Oty, UT84112, USA Recewed 3 July 1984
We analyze the transfer matrix of the plane rotator model in d = 1 and 2 dimensions. In 1D we relate the magnetic field susceptibility of the ferromagnet and that of the antiferromagnet. In 2D, we produce some simple transformations that map the problem onto the s = 1/2 anisotropic Heisenberg chain, thereby reproducing all the known features of the KosterlitzThouless transiUon.
This paper concerns the statistical mechanics of the plane-rotator model (alias "classical X Y model"), the hamiltonian of which is: H =-J
~ c o s ( 0 i - 0j), (O)
(1)
the sum being over nearest-neighbor pairs (ij) on a regular lattice in d = 1 or 2 dimensions. (We shall treat d / > 3 by other methods in a separate publication.) The transfer-matrix formulation is particularly simple in 1D, where we obtain the magnetic susceptibility. In 2D, we reproduce the features of the Kosterlitz-Thouless phase transition [1 ], obtaining a critical temperature k r c = 0.883 J
(2)
(in excellent agreement with the best numerical simulations [2]), as well as the other interesting thermodynamic properties of the model. These are obtained by identifying the transfer matrix of (1) with the amsotropic XXZ model of the s = 1/2 linear chain antiferromagnet. For simplicity, let us start in 1D where the calculation of partition function Z and of free energy F = - k T In Z proceeds through an ordered sequence of integrations: Z = Tr ( e - all)
= ... ~ f dOn exp [-/~g/(n, n + 1)] . . . .
(3)
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
which is evaluated through the solution of an eigenvalue problem: I f d 0 ' exp [-/1H(0', 0)] t~i(O')= zlt~l(O ) . 2rr
(4)
The kernel is a generating function for Bessel functions of imaginary argument and integer order, lp: exp[r
cos(0' - 0)1 =
p(K)exp[
o(0'
P---®
- 0)l,
(5)
writing K =/~J. The lp(K) are thus the desired eigenvalues of the integral operator, ordered in magnitude (I0 > I 1 > 1 2 - - - ) and symmetric [(/._p(K) = lp(K)]. The corresponding elgenfunctions are plane waves, exp(ip0). Choosing the largest eigenvalue, we immediately obtain
f = FIN = - k T In Io(K ) .
(6)
This yields all the nonmagnetic thermodynamic functions. The magnetic susceptibility is obtained by perturbing H(O', 0) with a quantity: H ' = - ~ B ( c o s 0 + cos 0 ' ) ,
(7)
and calculating the resultant free energy to second order in B. Omitting details*l, we find for the susceptibility: ,1 Provided in the forthcoming book, ref. [3].
357
Volume 104A, number 6,7
PHYSICS LETTERS
(8)
XO = (C/T)(1 +/~)/(1 - / l ) ,
where C is Curie's constant, and/~ (/~d)~ (cos(0 - 0'))TA = I1 (K)/Io(K). t~, the zero-field short-range order parameter, is an odd function of the couphng constant J. For the Islng model linear chain, the same expression (8) applies with substitution of the appropriate/1 = tanh K, and the same is true for the classical Helsenberg model, with pt = coth K - 1/K [4,3]. This suggests that the "&mensionless susceptibility" X ( 3 J ) - XoT/C of any classical hnear chain should satisfy x(M)x(-M)
10 September 1984
(9)
= l,
with deviations therefrom ascribable to quantum fluctuations or higher-dimensional (e g. cross-hnk) couphngs. We turn next to that most interesting 2D problem, stu&ed by Kosterhtz and Thouless [I] and a number of others [5], o f the phase transition of (1) m 2D. The transfer-matrtx approach to th~s problem was pioneered by Luther and Scalapmo [6], their work being subsequently clarified and refined by den Nijs [7]. These papers showed the relevance of a certain quantum s = 1 chain, as well as the s = ~ anlsotropic Heisenberg antiferromagnetic chain, to this problem. We shall rederlve these results directly, without passing through the complications o f 6-vertex models and the hke. We generalize eq. (1) to allow for bonds J1 is one direction and J2 along the other, in an array o f N 1 N 2 sites. There is no loss of generality in assuming J1 ~
V ~ (01, 0 2 . . . . ON1 ) = exp(--/3fN1 ) ~ (01 .... ) .
(1 O)
NI
H(2) = - J 2 ~ .
n=l
c°S(0n - 0n-1 )
(13)
So far, the expressions are exact. If desired, the exponentlated operators m (1 I) may be combined to yield the simpler elgenvalue problem /t~/=flVl~/,
with
H = H (1) + H (2) .
(14)
This Introduces an error o(33") 3 owing to the neglect of a series in nested commutators. This sort of approxxmarion is often performed, with 11o sigmficant deletenous effects on the structure of the theory. We note ~t, but make no use of it in the following analysis, whteh ts exact until eq. (22). The "kinetic energy" operators W(Pn) are
w(p n) = k T in [I0 ( 3Jl )lip ( 3 J1 )] -
( 15 )
(At this point one obtains the s = 1 approximation, If one restricts the states of each rotator to the 3 values p = 0, -+1 and expresses the operators H (1) a n d H (2) in terms of the spin s = 1 operators Sx, Sny and S~z With suitable constants,
W(pEl) =* w(1)[S~] 2 . COS(0 n -- 0 m ) ~=} S ; S m + St7 S + '
(16)
yield the truncated hamiltonlan of Luther and Scalapino [6]. Rather than pursuing this approximation, we shall now show that the transfer matrtx is, in fact, analogous to the more familiar anisotroplc Helsenberg antlferromagnetlc chain.) Our procedure starts with a bond/site duality transformation on the one-dimensional transfer operators"
On - On-1
~ 13/3Xn
'
Pn ~ X n - X n - 1 ,
(17)
with the x n restricted to integers. The H (a) become.
Here V = [ V (1)11/2 [ V(2)] [ V(1)] 1/2
(11 )
is the transfer matrix in hermltean form, with V (a) = exp(-3H (a)) and N1 H (1)= ~
n=l
and
358
W(Pn),
with
Pn =i-la/bOn ,
(12)
H(1)-+ ~
El
w(x n -xEl_l),
H(2) ~ - ~ J 2 ~ [exp(O/BXn) + exp(-B/OXn)] , n (18) where H (2) is a translational operator for N 1 particles on a discrete chain, and H (1) is the two-body potential connecting particles which are nearest neighbors m the enumeration scheme, although not necessarily spatial nearest neighbors. We find it possible to transform
Volume 104A, number 6,7
PHYSICS LETTERS
H (1) into a more familiar form by first performing a shift operation:
Table 1 V(x)/kT as function of spacing x
(19a)
x n "+x n - 2n ,
I.e. W ( X n - X n - 1) -+ W ( X n - X n - 1 - 2),
10 September 1984
(19b)
then subtracting w(1) Ix n - x n _ 1 - 2] from each term in H (1), adding them back in the following way:
x
V(x)/kT
-1 0 1 2 3 4
3 ln(7 27 K~2) 2 ln(5 66 ~I- 2) 2 In(2 K i" 0 0 21n(2.21)
H (1) = ~ V ( x n - X n _ l ) + w(1)[L 1 - 2(N 1 - 1)] , n (20) where L 1 = X N I - x 1 and the V ( x n - x n _ 1) are discussed below. In the thermodynamic limit we treat L 1 as a c-number and set L 1 = 2(N 1 - 1). Hence the p a r t i c l e d e n s i t y is ½, and the coefficient of w(1) in (20) vanishes. To evaluate the V ( x ) we use the properties o f the I p at small K 1. V(x)= w ( x - 2) - w(1)[x - 2], 1.e. V(x) = k r { l x
- 2[ l n [ 2 k r ( I x
- 21 ! ) l / I x - 2 1 / j 1 ]
(21)
- (x - 2 ) l n [ 2 k T / J l ] ) .
The barrier shape IS given in table 1, with K 1 = J 1 / k T . Depending on the temperature, the potentials at x ~< 0 can be so large as to constitute an effective hard wall. Let us denote the temperature above which this app r o x i m a t i o n (our first) is valid, T O. Our second approximation neglects x 1> 4. The reason for this is the density ~, whxch implies that the important separations x n X n - 1 are now 1,2, and 3 with the average being 2. As there is small probability amphtude for excursions Xn - X n - 1 ~> 4, we can set V ( x ) equal to any convement value, such as V ( x ) = 0 for x ~> 4. We see the only significant remaining interaction is that at x n - X n _ 1 = l,i.e. -
H (1) = V(1) ~ ( C n C * n -- ~1 ) ( C n, - l e n n
1 -- I )
+ ~ V(1)L 1 ,
(22)
L1 rt=l
with the "hopping" part being, evidently,
(23)
(24)
We identify these as parallel and transverse parts of an Heisenberg 1D anisotroplc antiferromagnet of L 1 spins in zero external field (hence density ½). Its correlation functions are just those of the K - T phase transition [7]. For example, for T>~ T c (V(1) ~> J 2 in our notation), there is an energy gap in the spectrum of elementary excitations, A, which various studies of the Heisenberg model have given as follows [8]: A ~ 2rr exp (-- [2-3/2rr2(V(1) - J2) -1/2] } , at V(1) ~ J 2 , v(a),
atV(1)>>J 2.
(25)
This gap vanishes at V(1) = J2, thus the transxtion takes the form o f an essential singularity. The gapless Fermi liquid below the transiUon can be "bosonized" to yield the magnon properties. It remains to calculate Tc, the critical temperature, and to verify it is higher than TO, the temperature below which our approximations introduce a significant error. In the isotropic case, the equation V(1) = J2 = J1 = J yields (2/Kc)ln(2/Kc)
subject to a "hard wall" at x n - x n -1 = O, i.e. to (en) 2 = O. We have now defined a problem which is entirely expressible m terms o f s = } operators On: H (1) = V(1) ~[3 a n_o nzz I + ~ V ( I ) L 1 ,
L1
H(2) = - ~ J 2 ~ ( t ~+n a -n _ 1 + h.c.). n 1
= 1 ,
i.e.
kTc=
0.883J,
(26)
in excellent agreement with Monte Carlo cremates [2], which are in the range 0.89 + 0.02 J. On the other hand, TO (which is given by V(0) = J, i.e. by (Z/K0) ln(5.66K~ 2) = 1) i s k T 0 = 0.63 J, sufficiently below T c to be encouraging. The anisotropic cases are more favorable to the theory. For J 1 = 0.1 J and J 2 = 0 . 9 J , w e obtain k T c = 0.27 J and k T 0 = 0.08 J. The general formula for the critical temperature is: 359
Volume 104A, number 6,7
PHYSICS LETTERS
(2/K2c) ln(2/Klc ) = 1 with K l c ~
(27)
1 have benefited from many stimulatmg discussions with Sergey Rudln. Support for this research was provided by a grant DMR 81-06223 of the National Science Foundation. References
[1] J.M. Kosterlitz and D.J. Thouless, J. Phys C6 (1973) 1181, J.M Kosterlitz, J. Phys C7 (1974) 1046, V.L. Berezmskil, Soy Phys JETP 32 (1971) 493. [2] J. Tobochmk and G V Chester, Phys Rev. B20 (1979) 3761
360
10 September 1984
[3] D. Mattls, The theory of magnetism, II Thermodynamics and statlstxcal mechamcs (Springer, Berhn) to be published. [4] M.E Fisher, Am J Phys. 32 (1964) 343 [5] J Vlllam, J. Phys 36(1975)581, J Jos~et al,Phys Rev B16(1977) 1217, M N. Barber, Phys Rep. 59 (1980) 375, J. Kogut, Rev. Mod Phys. 51 (1979)659 [61 A Luther and D J Scalaplno, Phys Rev B16 (1977) 1153 [7] M.P denNlJs, Phys Rev B23 (1981) 6111, PhyslcalllA (1982) 273 [8] L R. Walker, Phys. Rev 116 (1959) 1089, R.J Baxter, Ann Phys (NY) 70 (1972) 323, C N Yang and C.P Yang, Phys Rev 150 (1966) 321, 327,151 (1966) 258, J D Johnson, S Krmsky and B.M McCoy, Phys Rev A8 (1973) 2526, HC Fogedby, J Phys Cll (1978) 4767
PHYSICS LETTERS
10 September 1984
TRANSFER MATRIX IN PLANE-ROTATOR MODEL Daniel C. MATTIS
Department of Phys:cs, University of Utah, Salt Lake Oty, UT84112, USA Recewed 3 July 1984
We analyze the transfer matrix of the plane rotator model in d = 1 and 2 dimensions. In 1D we relate the magnetic field susceptibility of the ferromagnet and that of the antiferromagnet. In 2D, we produce some simple transformations that map the problem onto the s = 1/2 anisotropic Heisenberg chain, thereby reproducing all the known features of the KosterlitzThouless transiUon.
This paper concerns the statistical mechanics of the plane-rotator model (alias "classical X Y model"), the hamiltonian of which is: H =-J
~ c o s ( 0 i - 0j), (O)
(1)
the sum being over nearest-neighbor pairs (ij) on a regular lattice in d = 1 or 2 dimensions. (We shall treat d / > 3 by other methods in a separate publication.) The transfer-matrix formulation is particularly simple in 1D, where we obtain the magnetic susceptibility. In 2D, we reproduce the features of the Kosterlitz-Thouless phase transition [1 ], obtaining a critical temperature k r c = 0.883 J
(2)
(in excellent agreement with the best numerical simulations [2]), as well as the other interesting thermodynamic properties of the model. These are obtained by identifying the transfer matrix of (1) with the amsotropic XXZ model of the s = 1/2 linear chain antiferromagnet. For simplicity, let us start in 1D where the calculation of partition function Z and of free energy F = - k T In Z proceeds through an ordered sequence of integrations: Z = Tr ( e - all)
= ... ~ f dOn exp [-/~g/(n, n + 1)] . . . .
(3)
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
which is evaluated through the solution of an eigenvalue problem: I f d 0 ' exp [-/1H(0', 0)] t~i(O')= zlt~l(O ) . 2rr
(4)
The kernel is a generating function for Bessel functions of imaginary argument and integer order, lp: exp[r
cos(0' - 0)1 =
p(K)exp[
o(0'
P---®
- 0)l,
(5)
writing K =/~J. The lp(K) are thus the desired eigenvalues of the integral operator, ordered in magnitude (I0 > I 1 > 1 2 - - - ) and symmetric [(/._p(K) = lp(K)]. The corresponding elgenfunctions are plane waves, exp(ip0). Choosing the largest eigenvalue, we immediately obtain
f = FIN = - k T In Io(K ) .
(6)
This yields all the nonmagnetic thermodynamic functions. The magnetic susceptibility is obtained by perturbing H(O', 0) with a quantity: H ' = - ~ B ( c o s 0 + cos 0 ' ) ,
(7)
and calculating the resultant free energy to second order in B. Omitting details*l, we find for the susceptibility: ,1 Provided in the forthcoming book, ref. [3].
357
Volume 104A, number 6,7
PHYSICS LETTERS
(8)
XO = (C/T)(1 +/~)/(1 - / l ) ,
where C is Curie's constant, and/~ (/~d)~ (cos(0 - 0'))TA = I1 (K)/Io(K). t~, the zero-field short-range order parameter, is an odd function of the couphng constant J. For the Islng model linear chain, the same expression (8) applies with substitution of the appropriate/1 = tanh K, and the same is true for the classical Helsenberg model, with pt = coth K - 1/K [4,3]. This suggests that the "&mensionless susceptibility" X ( 3 J ) - XoT/C of any classical hnear chain should satisfy x(M)x(-M)
10 September 1984
(9)
= l,
with deviations therefrom ascribable to quantum fluctuations or higher-dimensional (e g. cross-hnk) couphngs. We turn next to that most interesting 2D problem, stu&ed by Kosterhtz and Thouless [I] and a number of others [5], o f the phase transition of (1) m 2D. The transfer-matrtx approach to th~s problem was pioneered by Luther and Scalapmo [6], their work being subsequently clarified and refined by den Nijs [7]. These papers showed the relevance of a certain quantum s = 1 chain, as well as the s = ~ anlsotropic Heisenberg antiferromagnetic chain, to this problem. We shall rederlve these results directly, without passing through the complications o f 6-vertex models and the hke. We generalize eq. (1) to allow for bonds J1 is one direction and J2 along the other, in an array o f N 1 N 2 sites. There is no loss of generality in assuming J1 ~
V ~ (01, 0 2 . . . . ON1 ) = exp(--/3fN1 ) ~ (01 .... ) .
(1 O)
NI
H(2) = - J 2 ~ .
n=l
c°S(0n - 0n-1 )
(13)
So far, the expressions are exact. If desired, the exponentlated operators m (1 I) may be combined to yield the simpler elgenvalue problem /t~/=flVl~/,
with
H = H (1) + H (2) .
(14)
This Introduces an error o(33") 3 owing to the neglect of a series in nested commutators. This sort of approxxmarion is often performed, with 11o sigmficant deletenous effects on the structure of the theory. We note ~t, but make no use of it in the following analysis, whteh ts exact until eq. (22). The "kinetic energy" operators W(Pn) are
w(p n) = k T in [I0 ( 3Jl )lip ( 3 J1 )] -
( 15 )
(At this point one obtains the s = 1 approximation, If one restricts the states of each rotator to the 3 values p = 0, -+1 and expresses the operators H (1) a n d H (2) in terms of the spin s = 1 operators Sx, Sny and S~z With suitable constants,
W(pEl) =* w(1)[S~] 2 . COS(0 n -- 0 m ) ~=} S ; S m + St7 S + '
(16)
yield the truncated hamiltonlan of Luther and Scalapino [6]. Rather than pursuing this approximation, we shall now show that the transfer matrtx is, in fact, analogous to the more familiar anisotroplc Helsenberg antlferromagnetlc chain.) Our procedure starts with a bond/site duality transformation on the one-dimensional transfer operators"
On - On-1
~ 13/3Xn
'
Pn ~ X n - X n - 1 ,
(17)
with the x n restricted to integers. The H (a) become.
Here V = [ V (1)11/2 [ V(2)] [ V(1)] 1/2
(11 )
is the transfer matrix in hermltean form, with V (a) = exp(-3H (a)) and N1 H (1)= ~
n=l
and
358
W(Pn),
with
Pn =i-la/bOn ,
(12)
H(1)-+ ~
El
w(x n -xEl_l),
H(2) ~ - ~ J 2 ~ [exp(O/BXn) + exp(-B/OXn)] , n (18) where H (2) is a translational operator for N 1 particles on a discrete chain, and H (1) is the two-body potential connecting particles which are nearest neighbors m the enumeration scheme, although not necessarily spatial nearest neighbors. We find it possible to transform
Volume 104A, number 6,7
PHYSICS LETTERS
H (1) into a more familiar form by first performing a shift operation:
Table 1 V(x)/kT as function of spacing x
(19a)
x n "+x n - 2n ,
I.e. W ( X n - X n - 1) -+ W ( X n - X n - 1 - 2),
10 September 1984
(19b)
then subtracting w(1) Ix n - x n _ 1 - 2] from each term in H (1), adding them back in the following way:
x
V(x)/kT
-1 0 1 2 3 4
3 ln(7 27 K~2) 2 ln(5 66 ~I- 2) 2 In(2 K i" 0 0 21n(2.21)
H (1) = ~ V ( x n - X n _ l ) + w(1)[L 1 - 2(N 1 - 1)] , n (20) where L 1 = X N I - x 1 and the V ( x n - x n _ 1) are discussed below. In the thermodynamic limit we treat L 1 as a c-number and set L 1 = 2(N 1 - 1). Hence the p a r t i c l e d e n s i t y is ½, and the coefficient of w(1) in (20) vanishes. To evaluate the V ( x ) we use the properties o f the I p at small K 1. V(x)= w ( x - 2) - w(1)[x - 2], 1.e. V(x) = k r { l x
- 2[ l n [ 2 k r ( I x
- 21 ! ) l / I x - 2 1 / j 1 ]
(21)
- (x - 2 ) l n [ 2 k T / J l ] ) .
The barrier shape IS given in table 1, with K 1 = J 1 / k T . Depending on the temperature, the potentials at x ~< 0 can be so large as to constitute an effective hard wall. Let us denote the temperature above which this app r o x i m a t i o n (our first) is valid, T O. Our second approximation neglects x 1> 4. The reason for this is the density ~, whxch implies that the important separations x n X n - 1 are now 1,2, and 3 with the average being 2. As there is small probability amphtude for excursions Xn - X n - 1 ~> 4, we can set V ( x ) equal to any convement value, such as V ( x ) = 0 for x ~> 4. We see the only significant remaining interaction is that at x n - X n _ 1 = l,i.e. -
H (1) = V(1) ~ ( C n C * n -- ~1 ) ( C n, - l e n n
1 -- I )
+ ~ V(1)L 1 ,
(22)
L1 rt=l
with the "hopping" part being, evidently,
(23)
(24)
We identify these as parallel and transverse parts of an Heisenberg 1D anisotroplc antiferromagnet of L 1 spins in zero external field (hence density ½). Its correlation functions are just those of the K - T phase transition [7]. For example, for T>~ T c (V(1) ~> J 2 in our notation), there is an energy gap in the spectrum of elementary excitations, A, which various studies of the Heisenberg model have given as follows [8]: A ~ 2rr exp (-- [2-3/2rr2(V(1) - J2) -1/2] } , at V(1) ~ J 2 , v(a),
atV(1)>>J 2.
(25)
This gap vanishes at V(1) = J2, thus the transxtion takes the form o f an essential singularity. The gapless Fermi liquid below the transiUon can be "bosonized" to yield the magnon properties. It remains to calculate Tc, the critical temperature, and to verify it is higher than TO, the temperature below which our approximations introduce a significant error. In the isotropic case, the equation V(1) = J2 = J1 = J yields (2/Kc)ln(2/Kc)
subject to a "hard wall" at x n - x n -1 = O, i.e. to (en) 2 = O. We have now defined a problem which is entirely expressible m terms o f s = } operators On: H (1) = V(1) ~[3 a n_o nzz I + ~ V ( I ) L 1 ,
L1
H(2) = - ~ J 2 ~ ( t ~+n a -n _ 1 + h.c.). n 1
= 1 ,
i.e.
kTc=
0.883J,
(26)
in excellent agreement with Monte Carlo cremates [2], which are in the range 0.89 + 0.02 J. On the other hand, TO (which is given by V(0) = J, i.e. by (Z/K0) ln(5.66K~ 2) = 1) i s k T 0 = 0.63 J, sufficiently below T c to be encouraging. The anisotropic cases are more favorable to the theory. For J 1 = 0.1 J and J 2 = 0 . 9 J , w e obtain k T c = 0.27 J and k T 0 = 0.08 J. The general formula for the critical temperature is: 359
Volume 104A, number 6,7
PHYSICS LETTERS
(2/K2c) ln(2/Klc ) = 1 with K l c ~
(27)
1 have benefited from many stimulatmg discussions with Sergey Rudln. Support for this research was provided by a grant DMR 81-06223 of the National Science Foundation. References
[1] J.M. Kosterlitz and D.J. Thouless, J. Phys C6 (1973) 1181, J.M Kosterlitz, J. Phys C7 (1974) 1046, V.L. Berezmskil, Soy Phys JETP 32 (1971) 493. [2] J. Tobochmk and G V Chester, Phys Rev. B20 (1979) 3761
360
10 September 1984
[3] D. Mattls, The theory of magnetism, II Thermodynamics and statlstxcal mechamcs (Springer, Berhn) to be published. [4] M.E Fisher, Am J Phys. 32 (1964) 343 [5] J Vlllam, J. Phys 36(1975)581, J Jos~et al,Phys Rev B16(1977) 1217, M N. Barber, Phys Rep. 59 (1980) 375, J. Kogut, Rev. Mod Phys. 51 (1979)659 [61 A Luther and D J Scalaplno, Phys Rev B16 (1977) 1153 [7] M.P denNlJs, Phys Rev B23 (1981) 6111, PhyslcalllA (1982) 273 [8] L R. Walker, Phys. Rev 116 (1959) 1089, R.J Baxter, Ann Phys (NY) 70 (1972) 323, C N Yang and C.P Yang, Phys Rev 150 (1966) 321, 327,151 (1966) 258, J D Johnson, S Krmsky and B.M McCoy, Phys Rev A8 (1973) 2526, HC Fogedby, J Phys Cll (1978) 4767