The specific heat of magnetic linear chains

Physica 79B (1975) 427-466 ©North-Holland Publishing Company

THE SPECIFIC HEAT OF MAGNETIC LINEAR CHAINS H. W. J. BLOTE Kamerlingh Onnes Laboratorium der Ri/ksuniversiteit Leiden, Leiden, The Netherlands

(Commun. Suppl. No. 131) Received 18 February 1975

Energy spectra and thermodynamicat quantities of finite linear chains and polygons of axially symmetrically interacting spins have been numerically calculated. Extrapolation of the molar heat capacities versus the number of spins yielded estimates for infinite systems. Ground-state energies were also obtained. Data are presented for both ferromagnetic and antiferromagnetic chains. The influence of axially symmetric anisotropies in the exchange interaction was studied for S = ½and 1. This influence proved to be qualitatively the same in both cases. For S = 1 to 4, data are included for isotropic exchange in combination with several crystal field anisotropy parameter (D) values, both positive and negative. Further, the effect of the simultaneous presence of a D term and exchange anisotropy is studied for several cases. Agreement of the extrapolated data is found with some existing experimental and theoretical results.

1. I n t r o d u c t i o n The m e t h o d used b y B o n n e r and Fisher 1) has p r o v e d to be a p o w e r f u l tool f o r the calculation o f t h e r m o d y n a m i c a l quantities o f S = ½ m a g n e t i c linear chains. The m e t h o d consists o f the d e t e r m i n a t i o n o f the eigenvalues o f the h a m i l t o n i a n for finite systems and s u b s e q u e n t calculation o f e.g. the heat capacities. E x t r a p o l a t i o n o f the heat capacities C n versus the n u m b e r o f spins n o f a s y s t e m (to infinite n) gives an estimate o f the desired q u a n t i t y . F o r S = 1 chains, some data were calculated b y Weng2). These did not, however, include a n i s o t r o p y . E x p e r i m e n t a l results f o r S = 13) and S = ~ 4) systems w i t h single-ion a n i s o t r o p y (D t e r m ) , have b e e n o b t a i n e d b y Klaaysen during the last few years. These materials have the f o r m u l a M2X2L2, w h e r e M is Ni (S = 1) or Mn (S = ~ ), L is p y r a z o l e or p y r i d i n e , and X is C1 or Br. Since n o t h e o r e t i c a l results having sufficient n u m e r i c a l

428

H. W. J. Bl6te/Specific heat o f magnetic linear chains

precision were available for the specific heat of these systems, the present calculations were started and good agreement with the experimental results was obtained3'4). Results for S = ~ systems have already been given by Bonner and Fisher I ). More extensive numerical heat-capacity data for S = ½ are included in the present paper. Results for S = ½are also used in the section dealing with S -- ~ and ~ results. Recently some results for S = 1 with a D term have been published by De NeefS). No numerical specific-heat results are given, but his extrapolation procedure seems to give somewhat different results, as may be read from a plotS).

2. Theory The hamiltonian for a linear, axially symmetrically interacting chain (or polygon) with a D term is given by ae= -

< i,j>

{2JIl(si

Sjz) + J ± ( s i + s j _

- D ~ ( ~ S ( S + 1) - S~z ).

+ (1)

k

Each pair of interacting neighbours is counted once. For the determination of the eigenvalues of this hamiltonian we can make use o f a set of eigenfunctions o f S z - ~'k S k z belonging to only one eigenvalue of S z, since S z commutes with ~g. The eigenvalue problem is thus solved for each non-negative eigenvalue o f S z. This factorization of the eigenvalue problem greatly reduces the required computer time, which increases rapidly with the dimensions o f the hamiltonian matrix. Obviously it is the computer time and the storage capacity that put a limit to the n u m b e r of spins of the system, that can be handled. In the case of rings of n spins (for which a~' has n-fold rotational symmetry), Bonner and Fisher x) defined a translation operator Y" which assigns the spin vector at site i to the site i + 1. Eigenfunctions o f 5 c a n be written as n-I

Ip, I > = ~ e-2~rij l/ny-j[p

>,

(2)

/=o

where l is an integer 0 < l < n , and IP > arbitrary. For instance, for IP > we may choose eigenfunctions of the set o f all Siz. In general the exponential phase factor is complex. Since ~r commutes with at ~, a further factorization into smaller matrices is obtained (functions having different l are not

H. W. J. Bl6te/Specific heat of magnetic linear chains

429

coupled by aft). These functions were used by Bonner and Fisher. A further reduction o f the computer time can be obtained by introducing an operator qz, which interchanges the spin vectors at sites i and n - i + 1. ~ commutes with ~ , but not with 3"; instead we have q/3- = j r - 1q/. If we n o w define I p + , I > = (1 -+~')(IP, l > +

IP, n - 1 > ) ,

+

Ipi, l > = i ( 1

-+~)(Ip, l>-Ip,

n-l>),

then we obtain real coefficients, if the phase factors o f the I P, l > are properly chosen. Jgdoes n o t couple functions labelled by different signs. The a~ matrices are n o w about the same size as in the former case, but since all elements are real, a much more tractable form o f the eigenvalue problem is obtained. Moreover, the eigenvalues of a set o f IP}-, l > kets equal those of the set I p~, l > if l 4= 0 and l 4= n/2. Also in the case o f finite open chains, ~z can be used to advantage by introducing simply Ip-+> = Ip> +- ~gIp>.

3. Results (general) 3. 1. The ground-state energy o f isotropic chains

For all isotropic ferromagnetic polygons (J = JII = J±, D = 0) the expected ground-state energy per spin, - 2 J S 2, Le. the exact value for an infinite chain, was found. For isotropic antiferromagnetic polygons and chains of n spins, the ground-state energies per spin E n (O)/J are shown in table I for S = 1 to [. We may try to extrapolate these numbers to n = oo and to obtain thus the ground-state energies of infinite chains, E**(0), which will be denoted more shortly by E(0). The polygons (further to be referred to as rings) will be discussed first. The E n ( O ) / J S 2 values for rings are plotted in fig. 1 vs. 1In 2 (open symbols) or versus 1/n 3 (full symbols). Bonner and Fisher have already found that the E n (0) values o f antiferromagnetic S = ½ rings are approximately linearly dependent on 1/n 2 . For S = 1, Weng 2) found that the E n (0) values are approximately linearly dependent on 1/n a. Including the present result E a (0), we may n o w refine the estimate for the magnetic ground-state energy o f an infinite chain E(O)/J = 2.8059 for S = 1. Also for S = ], 2 and ~, ground-state energies (for smaller rings) were determined. These data cannot well be fitted by relations linear in

H. W. J. BlSte/Specific heat o f magnetic linear chains

430

TABLE 1 Ground-state energies per spin of finite antiferromagnetic Heisenberg rings and chains, for S = 1 to ~. The estimated values for infinite chains are shown below. Chains

Rings n

S=I

S=~

2 3 4 5 6 7 8

2 3 2.6125 2.8725 2.7349 2.8342

3.5 6 4.9731 5.7976

2.8059 ± 0.0002

S=2

6 10 8.4556

5.67 ± 0.02

8.5 15 12.4336

9.52 ± 0.04

10 8

7

6

S=I

S=a2

S=2

S={

2 2 2.3229 2.3321 2.4568 2.4670

3.75 4 4.5906 4.6873

6 6.6667 7.6105 7.8412

8.75 10 11.3812

S={

14.38 ±0.08

5

~

n

4

4

(l/n2scole)

~

3

2

En(O)

11

t

87

[

I

I

6

5

I

" n

4

(tin 3scale)

Fig. 1. Ground-state energies per spin of finite antiferromagnetic Heisenberg rings versus 1/n 2 (open symbols, upper scale) or 1/n 3 (fuU symbols, lower scale), for various spin values, o S = ~; 0 S = 1 ; z~S = ~; ~ S = 2; v S = ~.

1/n 3 f o r o d d a n d e v e n r i n g s s i m u l t a n e o u s l y ; t h e e x t r a p o l a t e d v a l u e s o b t a i n e d f r o m t h e o d d a n d e v e n r i n g s lie f a r a p a r t . T h e s a m e s t a t e m e n t c a n b e m a d e f o r 1In 2. F o r o d d n u m b e r s o f s p i n s , w e w o u l d e x p e c t classically a helical orientation of the spin vectors. Supposedly, the 180 ° m i s m a t c h is e v e n l y d i v i d e d i n t o n i n t e r v a l s o f m i s a l i g n m e n t , 7r/n, c a u s i n g

H. W. J. Bl6te/Specific heat of magnetic linear chains

431

an energy change o f - 2 J S 2 { 1 - cos 0r/n)} ~ - J S 2 7 r 2 / n 2 for n o t too small n. For S = 12, this kind of behaviour is observed; although the slope is found to be larger than 7r2. Remarkably, the S = 1 rings exhibit a different behaviour. For high spins at least one would expect the classical picture to be valid. For S = ], the odd points lie, using the 1In 2 scale, on a line having about the expected slope. The even rings agree with this extrapolation if 1/n 3 behaviour is assumed for even n. The above extrapolation method is also applicable to the data for S -- 2 and S = 4; for odd n and 1In 2 behaviour about the same slopes as for S = "] are found, while the extrapolation and the even point agree with 1/n 3 behaviour, with the slope gradually decreasing as a function of S. Thus the behaviour of the E n (0) values of S = 1 rings is found to be somewhat odd in comparison with the results for other S. Although the odd and even points for S = 1 rings seem to converge accurately to a mutual limit, it should be realized that the precision of the extrapolated ground-state energy critically depends on the assumption that E n (0) continues its behaviour linear in 1In 3 for n > 8. Further information about E(0) can be obtained from a comparison of the ground states of finite chains for various S (table I). Classically, we expect En(O)/J = E ( O ) / J - t~S2/n with the parameter a = 2. Therefore in fig. 2 En(O)/JS 2 - En(O)/JS 2 + 2/n is shown as a function of 1/n. An oscillating behaviour is found, but there is also a small overall dependence on 1In. For S -- ½a best fit is found with ~ slightly larger than 2, and for S -- 1, t~ smaller than 2. Therefore an averaging according to X n -~ [3 Xrt - 1 "J- (

1 - [3) X n ,

(3)

where x n is to be substituted by b o t h En(O ) and 1/n, has been used. The parameter ~ was chosen such that the oscillations were minimized. For S = 12the results are then found to be slightly convex upward, and for S = 1 convex downward. Extrapolations are obtained only with a modest precision, but the results agree with the values obtained from rings and hence support the extrapolation procedures used there. The final estimates are shown in table I, and for S > 1, agree well with the values from Kubo's 6) formula E(O)/JS 2= 2 + 0.726/S + 0.066/S 2. 3. 2. Accuracy o f the results

Estimates o f the accuracies o f the extrapolations can be obtained e.g. by changing the number o f rings or chains used in obtaining the best fit, or by changing weight factors. To restrict the number of digits in this paper, the estimated uncertainties are in general only implicitly indicated by the

H. IV. J. BI&te/Specific heat of magnetic linear chains

432 4.0

3.5 ( En.2

3.C

2.0

...............

IIIJ 10 8

I 7

I 6

I 5

I 4

~

I 3

(l/nscale)

1 2

Fig. 2. Ground-state energies per spin of finite antiferromagnetic Heisenberg chains versus 1/n for various spin values, o S = 12;0 S = 1 ;A S = ]; [] S = 2; v S = ~. A term 2/n has been added to the energy in order to make the data points fit better to horizontal lines. The extrapolated results from the finite rings are shown on the vertical scale. The dashed lines are for visual aid only. Note that the S = ½data show a (slight) convex upward tendency, and those for S = 1 convex downward. n u m b e r o f digits in e a c h n u m b e r , i.e. the i n a c c u r a c y a m o u n t s to 1 t o 10 in the last digit. F u r t h e r m o r e , n o t m o r e t h a n 6 digits are given, even in t h o s e cases w h e r e the e s t i m a t e d a c c u r a c y is b e t t e r .

3. 3. Analysis o f the heat-capacity data A n t i f e r r o m a g n e t i c rings in general were f o u n d t o p r o d u c e h e a t - c a p a c i t y values Cn (I3 oscillating as a f u n c t i o n o f the n u m b e r o f spins n e x c e p t at l o w t e m p e r a t u r e T, w h e r e Cn(T) generally increases m o n o t o n i c a l l y as a f u n c t i o n o f n, f o r n o t t o o large values o f n. E x t r a p o l a t i o n s t o C . ( T ) ( f u r t h e r to be d e n o t e d b y C(T)) were o b t a i n e d b y s t u d y i n g the r e l a t i o n b e t w e e n Cn(T) and Cn(T) - Cn_ 1 (13, w h i c h was f o u n d to be linear f o r s u f f i c i e n t l y large n and T. T h u s f o r high T c o n v e r g e n c e was o b t a i n e d , b u t f o r l o w T n o a d e q u a t e ring e x t r a p o l a t i o n s c o u l d be f o u n d . F o r f e r r o m a g n e t i c rings, n o oscillations (as a f u n c t i o n o f n) were f o u n d . Because Cn (T) at high t e m p e r a t u r e s was f o u n d to a p p r o a c h C(T) e x p o n e n t i a l l y as a f u n c t i o n o f n, c o n v e r g e n c e c o u l d still be o b t a i n e d f o r n o t t o o l o w T. F o r finite chains a n d high T, the c a l c u l a t e d h e a t - c a p a c i t y results closely

H. I4;.J. Bl6te/Specific heat of magnetic linear chains

433

obeyed relations

G(T)

=

C(T)

-

v(T)/n.

(4)

Hence, such expressions were fitted to the Cn(T) values, and extrapolations could be obtained to much lower temperatures than in the case o f rings. During the course o f these calculations, it became clear that, except at high temperatures, chain extrapolations o f the heat capacity show a marked superiority over ring extrapolations. With the exception o f high temperatures, q,(T) was in general not found to be equal to C(T), which would be true if Cn (T) were proportional to the n u m b e r o f interactions divided by the number o f spins. Further, in the case of antiferromagnetic chains at lower temperatures, an alternating behaviour o f Cn (T) as a function of T was observed. Therefore an averaging according to eq. (3) was applied to Cn(T) and 1In. This method allowed estimates o f C(T) to somewhat lower temperatures.

4. Heat capacity of isotropic

I-Ieisenbergchains

Numerical data for isotropic S = ½to ~ chains were already presented in a previous article7).* The heat-capacity plots are shown here again, together with additional information and discussion.

4. 1. Ferromagnetic Heisenberg chains For S = ½, heat capacities for chains up to n = 11 spins were obtained. For the other spin quantum numbers the values of n can be read from table I. The extrapolated results are plotted in fig. 3 versus 2kT/JS(S + 1), together with S = ~ results o f FisherS). The curves for small S extend to lower values o f this expression than those for large S; this is a consequence of the higher n values. The difference, however, is smaller than might be expected on the basis o f n alone. The data for large S are found to converge more rapidly as a function o f n than those for small S. For S = ½, extrapolations could be obtained down to k T / J = 0.03. At this temperature, the heat-capacity result was 26% lower than the prediction for the low-temperature asymptotical behaviour from ferromagnetic spin-wave theory,

*During the final stages of the preparation of the present article, also data of De Neefis) about that subject appeared.

434

H.W.J. BlSte/Specific heat of magnetic linear chains

oO'' oo ;

\\

c/qj

I

, )%

OOSi

feppomagnetic chains

O]

2 KT J s (S.l) __,,._

1

1/~

oo

j

1

lO

Fig. 3. Molar heat capacities of infinite ferromagnetic Heisenberg chains. The full lines for S = 12to ~ are the results of the present calculations. The curve for S = o~ was obtained by FisherS). The dashed lines indicate the ferromagnetic spinwave theory prediction for the low-temperature asymptotical behaviour of the heat capacity for several spin values.

C(T)/R = 0.3908 (kT/JS)~.

(5)

The slope of the low-temperature part of the S = 12curve in fig. 3 (logarithmic scale) is smaller than 0.5. Furthermore, entropy and energy considerations indicate that this slope, at least for a considerable temperature interval, must be smaller than 0.5 also below kT/J --- 0.03. All these data suggest that the spin-wave prediction [eq. (5) will be approached much closer than the data at kT/J = 0.03 do, and that C "~x/T behaviour is only slowly attained at low T. A s m o o t h extrapolation to behaviour according to eq. (5) at low T, accurately yielded b o t h the correct entropy and energy. For larger spins, a similar situation exists. At the lowest temperatures where good extrapolations were obtained, the results were from 20% (S = 1) to 12% (S = 4) below the spin-wave theory prediction. Furthermore, entropy and energy calculations showed that the heat capacities should decrease less rapidly than C " - x / T at still lower temperatures, hence the heat capacities there approach the behaviour according to eq. (5). In all cases smooth extrapolations to the spin-wave theory behaviour agreed accurately with b o t h the entropy and the energy. Thus, although no direct heat-capacity results have been obtained in the spin-wave limit, the data presented here support the ferromagnetic spin-wave theory result of eq. (5).

H. W. J. BlSte/Specific heat of magnetic linear chains

435

A remarkable phenomenon is the occurrence of two inflexion points in the S ~> 1 curves of fig. 3. One o f them immediately follows from the numerical data; the other inflexion point is inferred from the reasonable assumption that a s m o o t h change to behaviour according to eq. (5) occurs at low temperatures. Below the lower inflexion point, the spin-wave limit is then approached from below.

4. 2. An tiferromagnetic Heisenberg chains Also in the antiferromagnetic case the results for small S are somewhat better than those for large S. Convergent results are found not to extend to such low temperatures as in the ferromagnetic case. The extrapolated heat-capacity data are shown in fig. 4 versus 2kT/JS(S + 1). For all curves, the heat capacities are found to decrease more rapidly than C "- T in the temperature range below the corresponding heat-capacity maximum. Only in the case of S = ½, a decrease o f the slope suggesting a transition to C "" T behaviour, is clearly observed. Postulating this behaviour in the low temperature limit, and using entropy and energy evaluations, we estimate C(T)/R = ( - 0 . 3 3 + O.02)kT/J, which is a factor 3.2 below Kubo's 6) antiferromagnetic spin-wave theory result 1.0

5/2 \

0.5

0.2

, "~ 5~

0

1 - 3/2 1

005 C/RI 0'.02

0.1

~/2

ant ant 2 kT

I

££I

1

10

Fig. 4. Molar heat capacities of infinite antiferromagnetic Heisenberg chains. The full lines for S = 12 to ~ are the results of the present calculations. The curve for S = ~ was obtained by Fisher8). The dashed lines indicate Kubo's 6) spin wave theory prediction for the low-temperature asymptotical behaviour of the heat capacity for several spin values.

436

H.W.J. Bl6te/Specific heat of magnetic linear chains C(T)/R = -Tr kT/SJS.

(6)

This conclusion agrees with the results o f Bonner and Fisher 1). For S = 1, an even more rapid decrease o f the heat capacity below the temperature of the m a x i m u m is found (fig. 4). Here we estimate (supposing that C "~ T behaviour is eventually attained), C(T)/R -~ - 0 . 1 8 k T H , which is lower than Weng's 2) estimate (C(T)/R ~ - 0.35 kT/J) and a factor 2.8 below eq. (6). In a similar way we estimate C(T)/R = - a k T / J with c~ ~ 0.21, 0.18 and 0.17 for S - 3, 2, and ~, respectively. These a coefficients are smaller by factors 1.66, 1.45 and 1.23 respectively than those (7r/6S) following from eq. (6). The values o f a obtained for S t> 1 should be considered as tentative, since the accuracy is estimated to be only 20%. In spite of the limited accuracy, however, if we assume C "-~ T behaviour at low temperatures, we can conclude that Kubo's formula (6) holds much better for large spins than for small ones.

5. Results for chains having axially symmetric exchange and zero D Data are presented f o r S = ½and 1 only. In practice, zero D is not believed to occur frequently in combination with anisotropic exchange for high spins. The influence o f exchange anisotropy for S > 1 can, however, be qualitatively estimated from the data presented here; S = and S = 1 chains are found to behave very similarly when exchange anisotropy is introduced. Further, see also the footnote in section 7. The discussion given below is written for the case Jjl/Jl >I O, while in practice also Jll/Ji < 0 occurs (in cerium ethylsulfate 9) it has been found that JII < 0 and J l > 0). The results do, however, also apply to the case Jii/Ji < 0 since only the sign of JiL determines whether the heat-capacity curve is o f the ferro- or antiferromagnetic type; the sign o f J 1 is irrelevant here. Results are obtained both for axial (Jii/J± > 1) and equatorial (Jll/Ji < I ) anisotropy, and for ferromagnetic as well as for antiferromagnetic interaction. 5.1. Results for spin Extrapolated heat-capacity data for chains o f anisotropically interacting spins ~ are shown in figs. 5 - 8 and tables II to V. Especially for Jii > J ± ferromagnetic chains, even small anisotropies strongly influence the heat

437

H. I¢. J. Bl6te/Specific heat of magnetic linear chains

05 ~

.

] I

.

.

.

~ ~ \

o5~o

~~[/~

\\

I o,, F~ = ~ ~ / / ~ . ~ 0£)5 t

001

J"

/

/

/

0.02 T.__. k ~ 0.05 J.

\

//

H;Im.

0.1

02

~

0.5

10

2

5

10

Fig. 5. Heat capacities of ferromagnetic S = ½chains with interactions intermediate between Heisenberg and Ising (H-I). The numbers in the figure indicate the values of

]±~Ill.

i

0.5

i

I

03

i

I

F

0.5

~.o°,~~-~

0.2 0.1 -0.8

0.05

-0.5

0.02 I 0.01

"

01

02

kT J.

05

1

2

10

Fig. 6. Heat capacities of antiferromagnetic S = ½chains with interactions intermediate between Heisenberg and Ising. The curves are labelled by their J±/JII values.

H. l~. J. Bl6te/Specific heat o f magnetic linear chains

438 05

I

[

I

I

I

I

I

I

o 0.2

0.1

0.0~. C/R

0.01

\\o

/

0.0:

I / I Q02 k._~T'D" OO5 J~

1

I

I

I

I

0.1

Q2

0.5

1

2

Ol~k~ 5

lO

Fig. 7. Heat capacities of ferromagnetic S = ½chains with interactions intermediate b e t w e e n Heisenberg and XY. The curves are labelled by their Jii/J± values. For JII = 0, the exact result of Katsura 1°) is shown.

0.5

i

,

0.2

JJ. 0.1

005

c/Rl 0.02

0.01

I

0.1

I

0.2 ~ J~

L

~

h

L

05

1

2

5

~L~

10

Fig. 8. Heat capacities of antiferromagnetic S = 12chains with interactions intermediate between Heisenberg and XY. The curve with Jii = 0 is obtained from the exact expression given by Katsural°).

H. W. J. Bli~te/Specific heat o f magnetic linear chains

439

TABLE II Heat capacities of ferromagnetic S = ½chains with interactions intermediate between Heisenberg and Ising, as a function of temperature. For J±/JII = 0 see table III, and for J L/JII = 1 see table IV. /JII 0.5

0.8

0.9

0.95

0.98

0.015 0.039 0.081 0.200 0.290 0.316 0.289 0.2262 0.1895 0.17020 0.152846 0.145747 0.142126 0.139508 0.136756 0.133395 0.129320 0.119528 0.108668 0.097852 0.087715 0.078532 0.059935 0.046577 0.036960 0.029914 0.024642 0.020617 0.015000 0.011378 0.008914 0.007167 0.005884 0.004168 0.003104 0.002401 0.001911 0.001558 0.001007 0.000704

0.03 0.08 0.16 0.23 0.262 0.237 0.204 0.1726 0.1501 0.14231 0.13932 0.137281 0.136556 0.136153 0.135499 0.134147 0.131882 0.128712 0.120274 0.110307 0.100024 0.090173 0.081108 0.062428 0.048787 0.038865 0.031545 0.026041 0.021822 0.015915 0.012091 0.009483 0.007631 0.006270 0.004445 0.003313 0.002563 0.002041 0.001664 0.001077 0.000753

0.08 0.18 0.19 0.175 0.160 0.142 0.130 0.1255 0.1238 0.12575 0.12830 0.130380 0.132822 0.133921 0.134447 0.134392 0.133510 0.131653 0.128859 0.121032 0.111488 0.101471 0.091758 0.082746 0.063992 0.050169 0.040053 0.032562 0.026914 0.022574 0.016487 0.012536 0.009839 0.007921 0.006510 0.004618 0.003443 0.002665 0.002123 0.001730 0.001120 0.000784

kT/J,\ 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.15 0.20 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

0.000009 0.000114 0.002403 0.013642 0.040789 0.112297 0.262969 0.372115 0.414306 0.386188 0.325457 0.274574 0.235849 0,205839 0.181740 0.161804 0.130539 0.107184 0.089277 0.075296 0.064218 0.044985 0.033088 0.025286 0.019916 0.016075 0.013239 0.009417 0.007033 0.005449 0.004344 0.003544 0.002486 0.001840 0.001416 0.001123 0.000913 0.000587 0.000409

0.0003 0.0026 0.0098 0.0507 0.1294 0.2224 0.3284 0.3613 0.31548 0.26772 0.208500 0.179972 0.164747 0.155012 0.147381 0.140388 0.133516 0.119818 0.106564 0.094356 0.083466 0.073934 0.055365 0.042500 0.033442 0.026902 0.022060 0.018391 0.013315 0.010064 0.007866 0.006312 0.005176 0.003658 0.002721 0.002102 0.001672 0.001362 0.000880 0.000615

H. W. J. B1fte/Specific heat of" magnetic linear chains

440

TABLE III Heat capacities of antiferromagnetic S = ½chains with interactions intermediate between Heisenberg and Ising, as a function of temperature. For JJJtl = 1 see table V.

0

0.3

0.5

0.8

0.000001 0.000016 0.000582 0.004540 0.016684 0.056418 0.166201 0.282604 0.369543 0.438148 0.419975 0.371192 0.318262 0.270477 0.230034. 0.196610 0.146647 0.112571 0.088707 0.071500 0.058751 0.038442 0.027020 0.019997 0.015383 0.012195 0.009901 0.006896 0.005076 0.003891 0.003077 0.002494 0.001733 0.001274 0.000976 0.000771 0.000625 0.000400 0.000278

0.018 0.046 0.091 0.161 0.2230 0.2779 0.35220 0.37711 0.366575 0.337595 0.302056 0.266434 0.233656 0.179739 0.140080 0.111163 0.089849 0.073860 0.048173 0.033691 0.024813 0.019005 0.015007 0.012144 0.008414 0.006167 0.004712 0.003717 0.003006 0.002082 0.001527 0.001167 0.000921 0.000745 0.000476 0.000330

0.04 0.06 0.106 0.152 0.1956 0.2739 0.32818 0.35191 0.350736 0.333723 0.308592 0.280549 0.226486 0.181544 0.146455 0.119500 0.098765 0.064694 0.045195 0.033190 0.025338 0.019946 0.016094 0.011094 0.008099 0.006167 0.004851 0.003914 0.002702 0.001976 0.001507 0.001187 0.000960 0.000611 0.000423

0.05 0.08 0.11 0.137 0.196 0.2582 0.3066 0.33663 0.34891 0.346997 0.335236 0.296558 0.252813 0.212603 0.178455 0.150385 0.101153 0.071429 0.052666 0.040243 0.031661 0.025514 0.017532 0.012757 0.009686 0.007599 0.006118 0.004207 0.003067 0.002335 0.001836 0.001481 0.000941 0.000650

- k T/JII " ~

0.05 0.06 0.08 0.10 0.12 0.15 0.20 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

1-1. W. .L BliSte/Specific heat o f magnetic linear chains

441

TABLE IV Heat capacities of ferromagnetic S = 12chains with interactions intermediate between Heisenberg and X Y , as a function of temperature. ForJif/J ± = 0 see table V.

0.5

0.8

0.9

0.95

1

0.067 0.082 0.1110 0.1411 0.1696 0.2021 0.22436 0.22158 0.210220 0.187091 0.171172 0.160942 0.153761 0.147828 0.142205 0.136477 0.124491 0.112300 0.100619 0.089884 0.080273 0.061024 0.047317 0.037492 0.030314 0,024953 0,020865 0.015169 0.011499 0.009005 0.007238 0.005941 0.004207 0.003132 0.002422 0.001928 0.001571 0.001016 0.000710

0.06 0.075 0.095 0.114 0.149 0.1752 0.1878 0.1900 0.1788 0.16714 0.15884 0.149211 0.144309 0.141512 0.139403 0.137101 0.134195 0.130568 0.121547 0.111225 0.100721 0.090717 0.081545 0.062701 0.048973 0.038997 0.031644 0.026118 0.021884 0.015957 0.012121 0.009506 0.007649 0.006284 0.004455 0.003320 0.002568 0.002045 0.001667 0.001079 0.000755

0.08 0.10 0.120 0.137 0.157 0.160 0.1560 0.1486 0.14159 0.138674 0.137519 0.136682 0.136304 0.136045 0.135522 0.134327 0.132246 0.129279 0.121202 0.111502 0.101397 0.091625 0.082577 0.063796 0.049981 0.039886 0.032414 0.026785 0,022463 0,016401 0,012468 0.009785 0.007876 0.006473 0.004591 0.003423 0.002649 0.002110 0.001720 0.001113 0.000779

0.071 0.078 0.084 0.089 0.0975 0.1037 0.1086 0.1144 0.12107 0.12555 0.128587 0.131903 0.133379 0.134090 0.134198 0.133479 0.131796 0.129178 0.121668 0.112363 0.102496 0.092862 0.083875 0.065061 0.051111 0.040864 0.033256 0.027509 0,023088 0.016877 0.012840 0.010082 0.008119 0.006675 0.004737 0.003532 0.002734 0,002178 0.001776 0.001150 0.000804

kT/J± 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.15 0.20 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1,0 1,2 1,4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

0.04 0.047 0.063 0.081 0.102 0.1320 0.1846 0.22968 0.259607 0.277769 0.266084 0.245953 0.225504 0.206801 0.190007 0,174917 0.148892 0.127398 0.109605 0.094852 0.082596 0.060050 0.045246 0.035153 0.028021 0.022819 0.018922 0.013592 0.010219 0.007955 0.006365 0.005207 0.003668 0.002721 0.002099 0.001667 0.001356 0.000875 0.000611

H . W . J . Blbte/Specific heat of magnetic linear chains

442

TABLE V Heat capacities of antiferromagnetic S = 12 chains with interactions intermediate between Heisenberg and XY, as a function of temperature. The result~ for Jtl/J± = 0 are obtained from Katsura's 1°) exact expression.

0

0.5

0.8

1

0.010479 0.015733 0.021002 0.026295 0.031616 0.042370 0.053325 0.064560 0.082181 0.114687 0.151677 0.191017 0.261998 0.307308 0.325044 0.322423 0.307543 0.286520 0.263272 0.218064 0.179514 0.148479 0.123905 0.104457 0.071190 0.051197 0.038426 0.029834 0.023801 0.019413 0.013604 0.010050 0.007722 0.006116 0.004963 0.003454 0.002541 0.001947 0.001540 0.001248 0.000799 0.000555

0.090 0.115 0.142 0.206 0.2671 0.3126 0.33842 0.34634 0.34077 0.326352 0.285112 0.241239 0.202014 0.169206 0.142476 0.095913 0.067894 0.050196 0.038457 0.030328 0.024491 0.016889 0.012324 0.009378 0.007371 0.005943 0.004096 0.002992 0.002280 0.001795 0.001450 0.000923 0.000638

0.080 0.100 0.126 0.178 0.237 0.288 0.3237 0.3437 0.34939 0.344217 0.314351 0.274349 0.234698 0.199490 0.169679 0.115766 0.082350 0.060961 0.046689 0.036782 0.029663 0.020397 0.014844 0.011270 0.008840 0.007115 0.004891 0.003565 0.002712 0.002132 0.001720 0.001092 0.000755

0.070 0.090 0.111 0.162 0.218 0.269 0.3092 0.3353 0.34786 0.34899 0.328549 0.293395 0.255295 0.219739 0.188649 0.130569 0.093557 0.069525 0.053358 0.042083 0.033958 0.023358 0.016997 0.012901 0.010115 0.008138 0.005590 0.004072 0.003097 0.002434 0.001962 0.001245 0.000860

-kT/J±~ 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.15 0.20 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 !0 !2 14 16 18 20 25 30

H. W. J. Bliite/Specific heat o f magnetic linear chains

443

capacity at low temperatures (fig. 5). The occurrence of a second maximum in several curves for small anisotropy can be related to the deformation of the spin-wave spectrum by the axial anisotropy. The heat capacity (which for a ferromagnetic chain is relatively high at low temperatures) vanishes when k T is much below the gap energy, while at high T the heat capacity is not much affected. Since entropy is conserved, the heat capacity at intermediate temperatures is increased, leading to the second maximum in the heat capacity curve. For J±/JII = 0 (Ising interaction), the chain extrapolations gave, as expected, exact heat-capacity results, equal to that of a Schottky anomaly for two levels. For antiferromagnetic chains, with Jii/J± > 1, the anisotropy has a less drastic influence than in the ferromagnetic case, which may be related to a steeper dispersion relation of antiferromagnets at low energies. Further, in the case ofJii/J ± < 1, (preferred plane) the influence is smaller than for Jii/J± > 1 (preferred axis), both for ferromagnets and antiferromagnets. This can be seen very clearly from the small differences among the curves in fig. 8, which represent antiferromagnetic Heisenberg and X Y chains, and two intermediate cases. For JII/J ± = 0 ( X Y model) exact specific-heat results were obtained by Katsural°). The present extrapolated data agreed well with the exact results (better than 0.01% above the maximum, to about 5% at k T / J 1 = 0.15). The data shown in figs. 7 and 8, and table V for JII = 0 are those obtained from Katsura's expression. Further, the energy at T = 0 was obtained for several values of the anisotropy parameter. Good agreement was found with results calculated from the exact expressions given by Orbach 11), Walker ~2) and Yang and Yang 13) for S = ~. The latter results are given in table VI and fig. 9. TABLE VI Ground-state energies per spin of infinite S = ½and S = 1 chains for several values of the exchange anisotropy. The S = ½data are obtained from exact expressions given in refs. 11 to 13. The S = 1 results are extrapolated from finite systems. The estimated inaccuracy in the last digit is given between brackets.

IE(O)/Jtll

IE(0)/J±I

IJ±l/Jll S=½ -0.9 -0.8 -0.5 0 0.5 0.8 0.9

0.857217 0.829058 0.750000 0.636620 0.549038 0.512679 0.504546

S=I 2.710 2.633 2.460 2.2325 2.0835 2.021 2.0064

S=½ (5) (5) (8) (5) (5) (1) (1)

-1 -0.9 -0.8 -0.5 0

0.886295 0.827737 0.770561 0.617222 0.5

S=I 2.8059 2.610 2.483 2.172 2

(2) (8) (2) (1)

444

H. W. J. B l 6 t e / S p e c i f i c

5

I

I

heat of magnetic linear chains

I

I

[

I

\\\ 23_

\ \ \ \ \ ~ \ \

-E(O)I[

,..,. 0 L -2

I

I

IJJ.I

-1

_~_

i

".,",,. / / ,, "~'"

.."

/"

,'

l

o

I 1

Fig. 9. Extrapolated ground state energies per spin of S = ~2and S = 1 chains as a f u n c t i o n of the anisotropy in the exchange interaction. For Jl[:>[Jl_[ one obtains the "classical" energy. For smaller J , , the results (full lines) no longer coincide with the classical prediction (dashed lines).

5. 2. Spin 1 chains with anisotropic exchange and zero D Heat-capacity results are given in figs. 1 0 - 1 3 and tables VII to X. A behaviour similar to that of the S -- ½curves is found. Also for S = 1, small anisotropies strongly influence the ferromagnetic curve at low temperatures. For J±/Jll = 0 very good agreement was found with the exact expression of Suzuki et al. 14) for the heat capacity of a S = 1 Ising chain. As a further check o f the present S = 1 results, at high temperatures agreement was found with the series expansion for a S = 1 Heisenberg chain, 16J 2 C(T)/R- 3k2T 2

16J a 320J 4 - - + 3 k 3 T 3 - 9k 4 T4

....

Further, also the energy at T-- 0 was extrapolated from the results for finite chains and rings. These data are given in table VI and fig. 9 together with the S = 12results.

6. Results for isotropic exchange and non-zero D Data are presented for S = 1 to 4. In particular S -- 1 and S = ~ chains are relevant for comparison with experimental data (refs. 3 and 4, Ni and Mn chains respectively).

H. W. J. Bliite/Specific heat of magnetic linear chains

445

1 '

0.5'

0

'

'

~

2

5

10

0.~

0.2

0.1

0.05

0.02

0011 0.1

i Q2

=

kT J,

Q5

1

20

Fig. 10. Heat capacities of ferromagnetic S = 1 chains with interactions intermediate between Heisenberg and Ising. The extrapolated results for Jx = 0 agree very accurately with the exact expression of Suzuki 14) for the heat capacity of an S = 1 lsing chain.

!

I

I

0.5

Q2

0.1

O0 °o

O0

r

0.2 _ kT J,

0.5

1

I

I

2

5

10

20

Fig. 1 1 . Heat capacities of antiferromagnetic S = 1 chains with interactions intermediate between Heisenberg and Ising.

H. W. J. BlSte/Specific heat o f magnetic linear chains

446

I

I

I

I

I

I

I

0.5

0.2

J"- 09

°s6

so,

\•

H -XY

0.0~, -

ferrom.

C/R 1

0.0;

0.9

0.5

08

o,o:

' 01

0.2 kT

P

J

'

'

'

0.5

1

2

5

' °°~ 10

20

J~. Fig. 12. Heat capacities of f e r r o m a g n e t i c S = 1 chains w i t h i n t e r a c t i o n s i n t e r m e d i a t e between Heisenberg and XY.

I

i

I

E

I

I

0.5

0.2 Q5 0.1

j~S=1

C/R~

antiferrom.

~k~l

0.o;

~ O

0.01

I Q2-kT J~

i 0.5

~ 1

i 2

i 5

__

0~ -

10

20

Fig. 13. Heat capacities of a n t i f e r r o m a g n e t i c S = 1 chains w i t h i n t e r a c t i o n s i n t e r m e d i a t e between Heisenberg and XY.

H. W. J. Bliite/Specific heat of magnetic linear chains

447

TABLE VII Heat capacities of ferromagnetic S = 1 chains with interactions intermediate between Heisenberg and Ising, as a function of temperature. For J±/JII = 0 see table VIII, and for J±/JII = 1 see table IX. /JII 0.5

0.8

0.9

0.95

0.0007 0.004 0.013 0.067 0.184 0.353 0.533 0.6801 0.7699 0.8040 0.7619 0.6642 0.56600 0.48233 0.414141 0.294306 0.219434 0.169665 0.134928 0.109748 0.090937 0.065262 0.049032 0.038148 0.030506 0.024939 0.017551 0.013011 0.010027 0.007961 0.006473 0.004170 0.002908

0.04 0.08 0.15 0.35 0.518 0.575 0.561 0.523 0.483 0.4502 0.4039 0.3738 0.35076 0.33029 0.310685 0.263921 0.222030 0.186567 0.157422 0.133750 0.114540 0.086054 0.066604 0.052890 0.042924 0.035482 0.025354 0.018984 0.014730 0.011754 0.009594 0.006220 0.004354

0.17 0.32 0.40 0.452 0.416 0.372 0.341 0.3245 0.3165 0.3132 0.3127 0.31274 0.31000 0.30381 0.294552 0.263350 0.228766 0.196452 0.168370 0.144712 0.125028 0.095106 0.074211 0.059264 0.048293 0.040043 0.028735 0.021575 0.016772 0.013402 0.010950 0.007112 0.004984

0.27 0.34 0.35 0.32 0.28 0.263 0.254 0.2541 0.2597 0.26699 0.27461 0.28788 0.29637 0.29940 0.297440 0.291442 0.265492 0.233520 0.202330 0.174549 0.150773 0.130767 0.100024 0.078335 0.062719 0.051204 0.042517 0.030570 0.022981 0.017882 0.014298 0.011687 0.007596 0.005326

kT/Jll~ 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

448

H. W. J. B16te/Specific heat of magnetic linear chains TABLE VIII

Heat capacities of antiferromagnetic S = 1 chains with interactions intermediate between Heisenberg and Ising, as a function of temperature. The results for J±/JIt = 0 are obtained from Suzuki's 14) exact expression. For J±/JII = 1 see table X.

0

0.5

0.8

0.9

0.000002 0.000086 0.000865 0.013750 0.065918 0.176020 0.336676 O. 519951 0.691758 0.825164 0.939319 0.893174 0.776778 0.650268 0.538149 0.341583 0.230212 0.164428 0.123051 0.095509 0.076297 0.051959 0.037696 0.028616 0.022475 0.018126 0.012515 0.009162 0.006998 0.005521 0.004467 0.002854 0.001980

0.014 0.063 0.156 0.282 0.420 0.544 0.6384 0.6991 0.7365 0.7039 0.6421 0.57313 0.50685 0,370859 0.276207 0.211096 0.165453 0.132633 0.108416 0.076011 0.056054 0.042966 0.033945 0.027476 0.019041 0.013958 0.010665 0.008411 0.006802 0.004339 0.003006

0.07 0.13 0.20 0.26 0.33 0.395 0.444 0.485 0.5405 0.5616 0.55786 0,53815 0.509172 0.423553 0.343343 0.277733 0.226346 0.186498 0.155503 0.111853 0.083723 0.064762 0.051463 0.041816 0.029104 0.021378 0.016348 0.012898 0.010431 0.006651 0.004605

0.13 0.18 0.23 0.29 0.353 0.404 0.445 0.505 0.5377 0.5466 0.5381 0.51813 0.445936 0.370070 0.304290 0.250887 0.208469 0.174907 0.126853 0.095421 0.074042 0.058959 0.047973 0.033444 0.024587 0.018811 0.014845 0.012007 0.007657 0.005301

kT/IJtt[~ 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

H. W. J. BliJte/Specific heat of" magnetic linear chains

449

TABLE IX Heat capacities of ferromagnetic S = 1 chains with interactions intermediate between Heisenberg and X Y , as a function of temperature. For Jii/J± = 0 see table X.

0.5

0.8

0.9

1.0

0.07 0.09 0.117 0.142 0.1932 0.2510 0.3112 0.36708 0.41322 0.44706 0.468711 0.482479 0.470742 0.446218 0.416437 0.385380 0.312989 0.253836 0.207551 0.171622 0.143604 0.121547 0.089814 0.068756 0.054183 0.043727 0.035993 0.025567 0.019068 0.014755 0.011750 0.009575 0.006191 0.004327

0.10 0.13 0.162 0.196 0.262 0.311 0.340 0.353 0.3572 0.3577 0.3564 0.3516 0.34495 0.33603 0.32482 0.311673 0.274036 0.235856 0.201411 0.171988 0.197448 0.127162 0.096496 0.075183 0.059980 0.048842 0.040476 0.029025 0.021782 0.016928 0.013524 0.011047 0.007173 0.005026

0.13 0.17 0.20 0.23 0.272 0.285 0.286 0.286 0.288 0.2909 0.2949 0.3025 0.30716 0.30757 0.30375 0.29643 0.268463 0.235443 0.203653 0.175503 0.151489 0.131325 0.100382 0.078585 0.062901 0.051344 0.042627 0.030643 0.023033 0.017921 0.014328 0.011712 0.007612 0.005337

0.120 0.133 0.146 0.157 0.177 0.194 0.209 0.2234 0.2365 0.2488 0.25995 0.27794 0.28973 0.295439 0.295774 0.291730 0.269312 0.239246 0.208844 0.181200 0.157221 0.136839 0.105198 0.082670 0.066349 0.054263 0.045117 0.032499 0.024461 0.019049 0.015241 0.012464 0.008107 0.005687

k T/J~\ 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

H. W. J. Bl6te/Specific heat of magnetic linear chains

450

TABLE X Heat capacities of antiferromagnetic S = 1 chains with interactions intermediate between Heisenberg and X Y as a function of temperature.

0

0.5

0.8

1

0.081 0.098 0.137 0.1849 0.2424 0.3072 0.37323 0.43400 0.48503 0.550713 0.571143 0.559352 0.528827 0.489535 0.387346 0.302630 0.238793 0.191483 0.156154 0.129364 0.092499 0.069159 0.053553 0.042643 0.034732 0.024293 0.017926 0.013763 0.010895 0.008837 0.005668 0.003940

0.17 0.23 0.290 0.352 0.415 0.4681 0.5414 0.5731 0.57375 0.55427 0.523515 0.431371 0.346239 0.277861 0.225106 0.184675 0.153497 0.109969 0.082137 0.063464 0.050403 0.040945 0.028498 0.020939 0.016019 0.012643 0.010229 0.006528 0.004523

0.16 0.22 0.28 0.340 0.392 0.441 0.506 0.5422 0.5522 0.5439 0.52387 0.45061 0.373643 0.307019 0.253000 0.210141 0.176254 0.127778 0.096098 0.074558 0.059366 0.048303 0.033674 0.024757 0.018942 0.014948 0.012091 0.007711 0.005339

0.10 0.14 0.20 0.26 0.320 0.374 0.420 0.491 0.530 0.547 0.545 0.5305 0.4695 0.3978 0.33215 0.276900 0.231967 0.195808 0.143176 0.108240 0.084257 0.067236 0.054788 0.038264 0.028157 0.021554 0.017014 0.013764 0.008779 0.006078

-kT/}Jj_l~ 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

6.1. Data f o r S -- 1 S p e c i f i c - h e a t r e s u l t s are s h o w n i n figs. 1 4 - 1 7 a n d t a b l e s X I t o X I V . F o r c l a r i t y , r e s u l t s f o r e a c h c o m b i n a t i o n o f t h e signs o f J a n d D are shown separately.

H. W. J. Bl6te/Specific heat o f magnetic linear chains 1

I I

I

I

I

I

-1 -2

0

0

0'~11

I

I

I

I

5

"i0

20

50

-5

2

Q2 k.J_T" 05 J

451

~

1

2

100

Fig. 14. Heat capacities of ferromagnetic S = 1 chains with isotropic interaction and negative D terms (crystal field anisotropies). The effect of a small negative D term resembles that of a small axial anisotropy (fig. 10).

1

I

I

I

_2

I

f

I

i~

l

-5

0.5

.10 0.2

0.05

0021

L

J

~

i

i

0.2 _ k ~ " 05 J

1

2

5

10

20

\

i 50

100

Fig. 15. Heat capacities of antiferromagnetic S = 1 chains with isotropic interaction and negative D terms. For large D, the extrapolated results approach the sum of a Schottky anomaly (due to the D term) and an Ising anomaly for magnetic interaction in the lower doublet.

H. W. J. BlSte/Specific heat of magnetic linear chains

452

10

20

5 0.5

1

2

Q2

0.1

/

S=1

0.05 J

/

/

c/R QO~

I

0.01 Q1

0.2 k T J

I

I

I

05

1

2

I

5

20

10

50

100

Fig. 16. Heat capacities of ferromagnetic S = 1 chains with isotropic interaction and positive D terms. The effect of a small D term resembles that of a small anisotropy of the X Y type.

I

I

I

]

I

~,--,

20

I

I

50

0.5

2 --0-=0 0.2

0.1

0.05

c/R' Go

I

0.(3 Q2 . k T " 0.5 J

1

2

5

I

10

20

50

100

Fig. 17. Heat capacities of antiferromagnetic S = 1 chains with isotropic interaction and positive D terms. For large D, the heat capacity approaches a Schottky anomaly for a higher doublet and a lower singlet.

H. I¥. J. Bl6te/Specific heat o f magnetic linear chains

453

TABLE XI Heat capacities of ferromagnetic S = 1 chains with isotropic exchange and a negative D term.

-0.5

-1

-2

-5

-10

0.0006 0.0030 0.011 " 0.055 O. 153 0.293 0.441 0.563 0.639 0.6712 0.6521 0.5915 0.53050 0.47914 0.437150 0.358809 0.301291 0.255540 0.218240 0.187611 0.162354 0.124038 0.097178 0.077861 0.063613 0.052854 0.038041 0.028617 0.022278 0.017820 0.014569 0.009474 0.006644

0.000004 0.000100 0.000787 0.009292 0.03734 0.08938 0.16087 0.24349 0.32876 0.40970 0.54031 0.61789 0.647874 0.644794 0.622640 0.539339 0.458055 0.390477 0.335484 0.290506 0.253393 0.196595 0.156099 0.126460 0.104254 0.087263 0.063510 0.048155 0.037705 0.030292 0.024853 0.016256 0.011444

0.000002 0.000055 0.000489 0.006591 0.0283.70 0.070097 0.127173 0.191109 0.254109 0.310937 0.397712 0.450043 0.477425 0.489320 0.492238 0.481712 0.461321 0.437131 0.411108 0.384425 0.357964 0.308106 0.264301 0.227098 0.195989 0.170107 0.130575 0.102704 0.082575 0.067672 0.056380 0.037862 0.027096

k T/J \ 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

0.04 0.10 0.20 0.30 0.45 0.473 0.434 0.390 0.360 0.3404 0.3295 0.3205 0.31765 9.31494 0.31009 0.302707 0.275830 0.243699 0.212171 0.183821 0.159346 0.138612 0.106488 0.083651 0.067120 0.054885 0.045628 0.032862 0.024731 0.019258 0.015407 0.012599 0.008194 0.005748

0.020 0.047 0.088 0.233 0.417 O. 544 0.583 0.569 0.533 0.4944 0.4344 0.3962 0.37083 0.35141 0.334420 0.294589 0.256489 0.221700 0.191319 0.165458 0.143709 0.110216 0.086501 0.069367 0.056701 0.047125 0.033929 0.025528 0.019875 0.015899 0.013000 0.008454 0.005929

H. W. J. Bl6te/Specific heat o f magnetic linear chains

454

TABLE XlI Heat capacities of antiferromagnetic S = 1 chains with isotropic exchange and a negative D term.

-2

-5

-10

-20

-50

0.00002 0.00027 0.00414 0.01953 0.05172 0.09902 0.15534 0.21398 0.26974 0.361965 0.425005 0.464152 0.486764 0.498634 0.502914 0.489308 0.467092 0.440392 0.411713 0.382719 0.327677 0.279475 0.238857 0.205169 0.177356 0.135245 0.105856 0.084786 0.069276 0.057578 0.038499 0.027472

0.000001 0.000025 0.000251 0.004057 0.019582 0.052350 0.100590 0.157911 0.217255 0.272924 0.361248 0.413846 0.435959 0.436729 0.424499 0.373516 0.327265 0.298251 0.284699 0.281473 0.283859 0.293119 0.298574 0.297032 0.289399 0.277565 0.247865 0.216915 0.188524 0.163823 0.142812 0.103594 0.077753

0.000001 0.000026 0.000264 0.004255 0.020412 0.054243 0.103677 0.162019 0.221994 0.277957 0.365782 0.417021 0.437406 0.436179 0.421521 0.360654 0.296658 0.242532 0.199630 0.166459 0.141277 0.109558 0.096653 0.097264 0.107098 0.122567 0.159767 0.194469 0.220838 0.237857 0.246631 0.243927 0.223618

-kT/J 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

0.05 0.10 0.18 0.25 0.33 0.404 0.464 0.552 0.597 0.6123 0.6071 0.58880 0.517000 0.437643 0.366200 0.306364 0.257625 0.218262 0.160615 0.122043 0.095387 0.076364 0.062391 0.043749 0.032287 0.024769 0.019586 0.015866 0.010144 0.007034

0.005 0.023 0.0592 0.1125 0.1781 0.2500 0.3226 0.45348 0.55075 0.61095 0.63956 0.644984 0.602986 0.532164 0.460915 0.397366 0.343086 0.297477 0.227391 0.177919 0.142275 0.115993 0.096172 0.068948 0.051694 0.040127 0.032019 0.026125 0.016919 0.011830

H. W. J. Bl6te/Specific heat o f magnetic linear chains

455

TABLE XllI Heat capacities of ferromagnetic S = 1 chains with isotropic exchange and a positive D term.

0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

0.5

I

2

5

10

0.13 0.17 0.20 0.23 0.28 0.300 0.305 0.306 0.3055 0.3057 0.3068 0.31024 0.31277 0.312497 0.308862 0.302115 0.275829 0.243831 0.212314 0.183948 0.159457 0.138697 0.106548 0.083693 0.067150 0.054906 0.045644 0.032872 0.924738 0.019262 0.015410 0.012601 0.008196 0.005748

0.09 0.12 0.15 0.177 0.232 0.286 0.328 0.3555 0.3701 0.3761 0.3769 0.37177 0.36342 0.353644 0.354521 0.330050 0.294601 0.257503 0.222825 0.192328 0.166311 0.144415 0.110695 0.086832 0.069604 0.056874 0.047256 0.034007 0.025578 0.019909 0.015923 0.013018 0.008463 0.005934

0.10 0.12 0.170 0.215 0.260 0.3057 0.3475 0.3829 0.41034 0.44229 0.450647 0.444441 0.430174 0.411728 0.359640 0.308928 0.264081 0.225984 0.194192 0.167830 0.127786 0.099787 0.079726 0.064982 0.053885 0.038663 0.029020 0.022552 0.018014 0.014713 0.009548 0.006687

0.006 0.016 0.049 0.093 0.138 0.180 0.219 0.257 0.2939 0.36503 0.431040 0.487213 0.530036 0.558323 0.573785 0.538877 0.483761 0.425316 0.370916 0.323037 0.247074 0.192555 0.153180 0.124225 0.102489 0.072856 0.054255 0.041888 0.033278 0.027055 0.017404 0.012115

0.000006 0.000115 0.000763 0.002866 0.007583 0.015939 0.028581 0.067234 0.121566 0.186921 0.258151 0.330573 0.494269 0.609990 0.671845 0.689059 0.675190 0.642486 0.553975 0.463946 0.385795 0.321704 0.270164 0.195642 0.146721 0.113483 0.090098 0.073116 0.046746 0.032340

H. W. J. Bl6te/Specific heat of magnetic linear chains

456

TABLE XIV Heat capacities of antiferromagnetic S = 1 chains with isotropic exchange and a positive D term.

2

5

10

20

50

0.10 0.14 0.19 0.245 0.301 0.357 0.452 0.5186 0.5559 0.56975 0.56609 0.51458 0.442469 0.372920 0.313001 0.263535 0.223322 0.164193 0.124574 0.097215 0.077715 0.063411 0.044367 0.032687 0.025042 0.019780 0.016009 0.010218 0.007077

0.004 0.010 0.036 0.070 0.106 0.141 0.175 0.209 0.244 0.3180 0.39324 0.46297 0.521207 0.564713 0.609796 0.589253 0.537235 0.475867 0.416089 0.362274 0.275724 0.213364 0.168486 0.135689 0.111247 0.078238 0.057771 0.044300 0.034999 0.028325 0.018066 0.012501

0.000006 0.000105 0.000697 0.002631 0.006992 0.014756 0.026561 0.062938 0.114603 0.177444 0.246713 0.317969 0.482371 0.602460 0.669897 0.692160 0.682058 0.651763 0.564970 0.474388 0.394893 0.329341 0.276486 0.199949 0.149713 0.115619 0.091663 0.074292 0.047377 0.032714

0.000002 0.000010 0.000113 0.000637 0.002294 0.006129 0.013306 0.051791 01122859 0.219365 0.327689 0.434416 0.529679 0.666961 0.731137 0.738993 0.711404 0.665008 0.555561 0.453769 0.370078 0.303967 0.252235 0.165954 0.116040

0.000002 0.000042 0.000307 0.001330 0.004063 0.009745 0.034690 0.082165 0.151328 0.236059 0.328121 0.504440 0.639429 0.721135 0.755305 0.754075 0.669585 0.553175

-kT/J 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 6 7 8 9 10 12 14 16 18 20 25 30

H. W. J. Blrte/Specific heat of magnetic linear chains

457

Negative D values in the hamiltonian [eq. (1)] lead to a preferred axis and thus strongly influences the heat capacity at low temperatures (fig. 14) especially for ferromagnetic chains, even if ID/JI is small. The effect is similar to that o f a small anisotropy in the exchange (Jii/J± > 1, fig. 10). For large ID/J[, however, there is a considerable difference with anisotropic exchange. Two anomalies can be clearly distinguished when ID/Ji increases and these can be interpreted as a Schottky anomaly due to the D term, and an Ising anomaly due to Ising interaction in the lower effective S' = ½ doublet. For D/l J[ = - 5 0 , the extrapolated data shown here come close to the sum of these t w o anomalies. For large positive D the extrapolations approach a Schottky anomaly for a higher doublet and a lower singlet. For S = 1, the influence o f D on the ground-state energy has also been estimated from the results for finite chains and rings. A non-classical energy reduction is found in all cases, except for D < 0, J > 0. The results are given in table XV and figs. 18a and b.

6.2. Data for S = ], 2 and Z2 The heat capacity o f S = ] chains is shown in figs. 1 9 - 2 2 . These results show an influence o f D similar to that of S = 1, with the exception of the curves for large positive D. Since, for S = ~, we have a doublet lowest, a separate magnetic ordering anomaly occurs on the low-temperature side. This doublet can be described by an effective spin S' = ½, and an anisotropic i n t e r a c t i o n J[] = J, J]. = 4J. These S' = z2 curves have been separately calculated, and were found to fit well to the curves for large D/IJI. Since the S = ½ data are accurate down to lower temperatures than those for S = ~, for D/IJI = + 50 the S' = ½results are shown in figs. 21 and 22 at the low T side. These S' -- ½ results are expected to be a good approximation for high D/IJI and low T. Results for S = 2 are plotted in figs. 2 3 - 2 6 , and for S = ~2in figs. 2 7 - 3 0 . These data behave similarly to those for S = 1 and S = 3, respectively. The accuracy is, of course, less than for lower spin values. The curves for S = { and D/I J[ = + 50, just as for S = ~, consist partly of data derived for S' = 12 systems. For S = z, we have Jill = J, J~ = 9J.

7. Linear chains having both kinds of

anisotropy

In real systems o f interacting spins > ½ there will often be a considerable anisotropy b o t h in the exchange and in the crystal field parameter (D term). Since there exist many possible combinations of these anisotropy parameters,

458

H. W. J. Bl6te/Specific heat of magnetic linear chains TABLE XV Extrapolated ground-state energies of S = 1 chains with isotropic exchange and non-zero D terms. The estimated inaccuracy in the last digit is given between brackets. D<0

}D/JI 0.5 1

2 5 10 20 50

D>0 J <0

J <0 2.826 2.887 3.098 3.9270 5.49077 8.754694 18.704592

6 5 4

(4) (4) (2) (6) (4) (1) (<1)

2.825 2.878 3.061 4.2068 7.09870 13.542591 33.414895

jJ

D>0 J >0 (4) (6) (8) (3) (1) (<1) (<1)

2.103 2.229 2.540 3.9769 7.02568 13.523035 33.411707

(2) (2) (2) (1) (1) (<1) (<1)

3>o //// //

/" I" -8

2 7

I -6

i -4

I~)

I -2

~

0

2

/I

J
4

I 8

6

//~//

6

~ 5

iiII ~'.~

iiI

\'x

//

J

2

I

I

I

-8

-6

-4

I "",1 -2

0

I 2

/

I

I

L

4

6

8

Fig. 18. Extrapolated ground-state energies per spin of S = 1 chains with isotropic interactions as a function of the crystal field anisotropy parameter D. a) Ferromagnetic case. For D ~< 0 the "classical" energy - 2 J + ½D is found. For large positive D the extrapolated energy (full line) approaches -23D (dashed line). b) Antiferromagnetic case. For both signs of D the energy (full line) behaves nonclassically. For large negative D the energy is found to approach - 2 J -t- ~D. For large positive D the results approach -~3D, but they approach the ferromagnetic results even more rapidly there.

H. W. J. Bl6te/Specific heat of magnetic linear chains

459

0.1

0.05

~,~T////2`0

o&\ \ \

°?1111 Q01/

0.5 i

\

;~,\ \ \

1i

2i ~

I % 1

5i

10 i

20 I

50 i

100 i

20 'l 0

J

Fig. 19. Heat capacities of ferromagnetic S = ] chains with isotropic exchange interaction and negative D terms. The influence of negative D is qualitatively the same as for S = 1.

D=

I

-2

_

1

-

I

~

I

I

I

I

I

100

200

0

0.5

Q2

Q1

/I/

so~

I/

J <:0

QO~ C/R 00'

0.01

I

I

I

I

I

I

1

2 ~

5

10

20

50

J

Fig. 20. Heat capacities of antiferromagnetic S = ] chains with isotropic exchange interaction and negative D terms.

H. W. J. Bl6"te/Specific heat of magnetic linear chains

460 1

[

I

I

'

I

.

I

~

10

°FJ

,,

°°51c,q

~;~o o.o

°°1 oo,

o~

,

I

I

I

I

I

20

\ o ~\ \ \\

\

\\

~~ J

5

,'o

/o

\

\

\\\\ ~o

,oo

\

~oo

Fig. 21. Heat capacities of ferromagnetic S = ] chains with isotropic e x c h a n g e interaction and positive D terms. The influence of a large positive D t e r m is different f r o m that in the case o f S = 1 since we n o w have a doublet lowest. Below kT/J = 2.5, the D/J = 50 curve is o b t a i n e d f r o m S' = I results (see text).

1

I

I

I

I

I

I

I

I

I

O1

o05 5o

s4

\\ \

\

I 20

50

\

\

002I 0.01 ~

I 0.5

I 1

I 2

_kr"

J

I 5

I 10

100

I 200

Fig. 22. Heat capacities of a n t i f e r r o m a g n e t i c S = ~ chains with isotropic exchange interaction and positive D terms. Below kT/~[ = 2.5, the D/[JI = 50 curve is obtained f r o m S' = ~ results (see text).

461

H. 14/.J. BlSte/Specific heat o f magnetic linear chains i

I

I

i

-2

I

I

-o.~Z._/ ",/,. ~ 0.5

I

I

I

-5

-20

~

0.2 __=

0.1 0O5 CIR QO; 0.01 Q5

1

2 kT J

5

10

20

50

100

200

Fig. 23. Heat capacities of ferromagnetic S = 2 chains with isotropic exchange interaction and negative D terms. The influence of D is similar to that in the S = 1 case (fig. 15).

I

I

I

-21

--5

I

I

I

I

0.05

~,~T/ 0011

//

D,~

F // I

I I

2 ~

\ \

\\\

I

I

I

5

10

20

I

\\\\\ I

50

\\

1\

100

\ I

\

200

J

Fig. 24. Heat capacities of antiferromagnetic S = 2 chains with isotropic exchange interaction and negative D terms.

H. W.J. B/rite~Specific heat of magnetic linear chains

462 1

I

i

i

,

2

05

1

2 ~

'

'

,

20

50

i

,

02 =0 0.1

•

5

10

100

200

Fig. 25. Heat capacities of ferromagnetic S = 2 chains with isotropic e x c h a n g e interaction and positive D terms.

[

02

I

I

I

I

I

I

IJ~=0

~,~TI / O.OI

I

I

/ !

=

2

-k'r

"

5

o~\\\ , 10

, 20

50

100

200

J

Fig. 26. Heat capacities of antiferromagnetic S = 2 chains with isotropic e x c h a n g e interaction and positive D terms. Note that the l o w - t e m p e r a t u r e sides of these curves are a p p r o x i m a t e l y p r o p o r t i o n a l to T over a wider range of D values than in the S = 1 case.

H. W. J. Bl6te/Specific heat o f magnetic linear chains T

f

T - - T - - T

-2

T

T

1

50

100

200

463

-5

-0.5 " ~ X ~ Q ~

- 20

-Q2 0.5

0.2

0.1

QO~ J

C/R

0.0;

0.01

t

1

2 ~ J

5

10

20

Fig. 27. Heat capacities of ferromagnetic S = ~2chains with isotropic exchange interaction and negative D terms. This combination of the signs of D and J gives also for S = ~2results similar to those for S = 1. i

i

_

L

i

1

I

-20

i

I

5O

O5

~=o 02

G1

~

s=~ D~
0.05 C/R

-5

0

00~

0.01

2 - k"l" J

P

,5

10

20

50

100

200

Fig. 28. Heat capacities of antiferromagnetic S = ~ chains with isotropic exchange and negative D terms.

H . W . J . BlSte/Specific heat of magnetic linear chains

464 1

i

i

i

i

I

J

5

10

~

I

20

50

i

i

~_

Q5

Q2 0.1 0.05 CIR1 0.02 0.01

I

~

2 ~ J

/

I

"~\

\

100

I

200

Fig. 29. Heat capacities of ferromagnetic S = s chains with isotropic exchange and positive D terms. Below kT/J= 6, the curve forD~J= 5 0 i s obtained from S' = { results (see text).

05

Q2 0.1 0.0."

S=~

50

0

D>~O

~

~

~

C/R

0.02 0.01

I

2 -kT J

.

I

5

I

10

I

20

I

50

100

200

Fig. 30. Heat capacities of antiferromagnetic S = ~ chains with isotropic exchange and positive D terms. Below kT/IJI = 6, the curve for D/IJI = 50 is o b t a i n e d f r o m S' = 12 results (see text).

11. w. J. Blate/Specific heat of magnetic linear chains

465

only an incomplete and overall description is given here*. a) ]DI >~ IJII + J l l . This is the simplest situation. If a singlet is lowest (D > 0 and S integer), anisotropy in the exchange has little influence. If a doublet is lowest the Jil and J~. in the S' = 12description of the lower temperature anomaly vary proportional to JIt and Jib) [D[ ,~ IJII + Jll, In this case the anisotropy can be interpreted as the sum o f two anisotropy fields, one coming from the exchange anisotropy and the other from D. Extrapolated heat-capacity data for S = 1, D = -+0.5 and Jii/J± values between 0.8 and 1.25, showed that the two kinds o f anisotropy are additive in a first approximation and that they can even approximately cancel out if they are o f opposite sign. c) [D[ ~ IJII + J±l. General remarks cannot very well be made in this case. The influence of a moderate anisotropy in the exchange (10%) was studied for S = 1, J ~ 1, D = -+2. The two kinds o f anisotropy were still found to be approximately additive, Le. the low-temperature side o f the curves was found to shift to higher temperatures when the anisotropies were of the same character (Ising, or X Y like). However, 10% exchange anisotropy was found to have only a small influence when compared to the D ~ 0 case

Acknowledgements The author thanks Professor W. J. Huiskamp for suggesting this problem and Dr. R. C. Thiel for a critical reading of the manuscript.

References 1) J. C. Bonner and M. E. Fisher, Phys. Rev. 135 A (1964) 640. 2) C.-Y. Weng, Thesis, Carnegie-Mellon University, 1968. 3) F. W. Klaaysen, H. W. J. B1/Ste,H. den Adel and Z. Dokoupil, Solid State Commun. 14 (1974) 607. 4) F. W. Klaaysen, Thesis, University of Leiden, 1974, and to be published, Physica, 1975. 5) T. de Neef, Phys. Letters 47 A (1974) 51. 6) R. Kubo, Phys. Rev. 87 (1952) 568. 7) H. W. J. B16te, Physica 78 (1974) 241. 8) M. E. Fisher, Am. J. Phys. 32 (1964) 343. 9) H. W. J. Bl6te, Physica 61 (1972) 361 (Commun. Kamerlingh Onnes Lab., Leiden No. 393C). 10) S. Katsura, Phys. Rev. 127 (1962) 1507. 11) R. Orbach, Phys. Rev. 112 (1958) 309. *Requests for tabulated specific-heat data for linear chains with specified S, JII, J±, and D may be directed to the author.

466 12) 13) 14) 15)

H. W. J. Bl6te/Specific heat o f magnetic linear chains

L. R. Walker, Phys. Rev. 116 (1959) 1089. C. N. Yang and C. P. Yang, Phys. Rev. 150 (1966) 327. M. Suzuki, B. Tsujiyama and S. Katsura, J. math. Phys. 8 (1967) 124. T. de Neef, J. Phys. A 7 (1974) L 171.