Low-temperature heat capacity of the pseudo-one-dimensional magnetic systems CsMnCl3.2H2O, -RbMnCl3.2H2O, and CsMnBr3.2H2O

Low-temperature heat capacity of the

pseudoonedimensional magnetic systems CsMnCl3.2H2O,

-RbMnCl3.2H2O, and CsMnBr3.2H2O

Citation for published version (APA):

Kopinga, K. (1977). Low-temperature heat capacity of the pseudo-one-dimensional magnetic systems

CsMnCl3.2H2O, -RbMnCl3.2H2O, and CsMnBr3.2H2O. Physical Review B, 16(1), 427-432.

https://doi.org/10.1103/PhysRevB.16.427

DOI:

10.1103/PhysRevB.16.427

Document status and date:

Published: 01/01/1977

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Low-temperature

heat capacity

of

the

pseudo-one-dimensional

magnetic systems

CsMnC13

2H20,

a-RbMnC13

2H20,

and

CsMnBr3

2H20

K.

Kopinga

Department ofPhysics, Eindhoven University ofTechnology, Eindhoven, The Netherlands

(Received 1SMarch 1977)

The heat capacity of the isomorphic pseudo-one-dimensional magnetic systems CsMnC1, 2H20,

e-RbMnC1, 2H, O, and CsMnBr, 2H,Ohas been analyzed using recently developed estimates for the

lattice heat capacity ofanisotropic media. The magnetic contribution in the paramagnetic region could be

described very well by the S

=

S/2 linear-chain Heisenberg model. The intrachain interaction isfound as

J/k

= —

3,0Kforthe former two compounds and J/k

= —

2.6Kforthe latter compound. Theinterchain

interactions are smaller by two orders of magnitude. The results are compared with other experimental

evidence. A study ofthe critical behavior ofthe magnetic entropy increase yields a'

=

o.

=

+0.

10~'0.01

for all three substances.

I.INTRODUCTION

CsMnC1, ~_{2H,O (CMC), n-RbMnC1,} ~_{2H,O (n}

RMC) and CsMnBr, ~_{2H,O (CMB) belong to} a

series

of isomorphic compounds AMB,~_{2H,O with A =}

Cs,

Rb; M =

Fe,

Co,Mn;

B

=Cl,

Br.

Their

crystallo-graphic structure

is

orthorhombic with space

group

Pcca

and four formula units in a chemical

unit

cell.

"

Especially CMC has been the subject

of

a

very large number of experimental

investiga-tions.

The intrachain exchange interaction has

been determined from susceptibility, neutron

dif-fraction,

'

and EPR studies.

"

The magnetic space

group in the ordered state has been determined

from NMR experiments

as

P»c'ca'.

'

The

inter-chain interactions were estimated by

a

Green's-function method' and have been found

experimen-tally from neutron diffraction,

'

EPR line shape,

'

and NMR studies.

'

Recently, a fairly consistent

set

of interactions has been determined from a

spin-wave analysis of the susceptibility,

'

the mag

netic heat capacity,

"'"

and sublattice

magnetiza-tion'"

in the ordered

state.

The available evidence

indicates that the ratio ofthe interchain interactions to the intrachain interaction

is

on the order of 10

'-10

3.

In

a

previous

paper"

we reported the

experimen-tal

heat capacity of CMC in the temperature region

1.

1

—

52

K.

The data in the paramagnetic region

were interpreted by assuming that the magnetic contribution to the heat capacity above 9 K could

be represented by aHeisenberg S=-,''-

antiferro-magnetic linear-chain model.

It

was shown that the lattice contribution could not be described properly by the usual three-dimensional Debye

model. Given the chemically layered. structure

of this compound, the lattice heat capacity was

represented by

a

model proposed by Tarasov,

"

which was slightly modified to account

for

the dif-ferent modes ofvibration. The intrachain exchange

interaction Z/k was found from a simultaneous fit

of both the lattice and the magnetic contribution to

the data above 12 Kas Z/k

=-3.

3a

0.

3K.

One should note that the model ofTarasov only

offers

a

drastically simplified description of the

lattice-dynamical problem

—

even in the acoustic limit

—

which may readily affect the determination ofthe magnetic heat capacity

C„,

since in the pa-ramagnetic region C~&_{C~. Recently,}

a

more-

re-fined model has been proposed, '4 which

is

particu-larly suitable

for

the description of the low-temper-ature lattice heat capacity of very anisotropic

com-pounds, and involves only a small number of ad-justable

parameters.

Because,

moreover, reliable estimates

for

the heat capacity of Heisenberg

mag-netic linear-chain systems have been reported,

""

we thought

it

worthwhile to consider the

experi-mental data on CMC in more

detail.

Secondly,

specific-heat measurements were performed onthe

isomorphic n-RMC and CMB. The results on the

latter

two compounds may yield additional

infor-mation about the applicability ofthe theoretical

model

for

the lattice heat capacity, since the

various isomorphs involve rather different atomic

masses.

On the other hand, by comparison of the magnetic behavior of CMC, n-RMC, and CMB the influence ofboth the intermediate alkali ion and

the halide ions onthe various magnetic interactions

may be studied.

In the following analysis ofthe experimental data

on CMC, n-RMC,

as

well as CMB, the lattice

heat capacity will be described by the simplified

form of the expression

for

the heat capacity of

a

layered structure given by'

Cz(T)

=Fi(28t,

8„T)+Fi(8),

8~, T)

+_F2(8O,

28„T)

.

It

was assumed that in the temperature region of

interest

the H,Omolecules vibrate as awhole and

428

K.

KOPINGA 16 CsMnCl3.2H20 ~O O VYÃYEYY/XPYXPXXPXXXXPXEA o~Q

~-o

-2.0 -2.5 -3.0 JIk(K) I -3.5

FIG.

1.

rms error offits tothe experimental data on

CsMnCls 2H&O between 9 and 52K. The shaded bar

de-notes the region in which the total magnetic entropy

in-crease amounts to Rln6 within the experimental uncer-tainty.

that rotational states

are

of no importance. This yields a total of seven vibrating units in aformula

unit.

II.CsMnC13 '28&O

The data on CMC above 9Kwere fitted

simul-taneously varying the three parameters

e„e„e,

in the expression

for

the

lattice-specific

heat as

well as the parameter J'/k in the expression

for

the magnetic-specific heat, which was represented

by the Heisenberg S=—,

'

antiferromagnetic

linear-chain model.

"

This yielded

J/k

=-2.

85+

0.

02 _K,

_e,

=28V +

1.

0K,

6,

=204+1.

5K,

e

=56.

0+0.

5

K.

The quoted

error

bounds correspond to the

un-certainty of the parameter values assuming that the deviations

are

purely

statistical

in nature.

The actual uncertainty, however, may be

larger.

Since we

are

primarily interested in the value of the exchange interaction

J/k,

we performed

sev-eral

fits in which this parameter was fixed at a certain value. In

Fig.

1 the rms

error

of these

fits is

plotted as afunction of

J/k.

Since in the neighborhood of the minimum the

error

varies rather slowly with the parameter

J/k,

the

uncer-tainty was estimated independently by considering

the total magnetic entropy associated with these

fits.

The magnetic entropy increase

~S

was

de-termined in a similar way as described in

Ref.

12.

Itappeared that

~S

did agree with the

theo-retical

value

B

ln6 within the experimental

error

for

-2.8)J/k)-3.

5 K, which

is

indicated by the

shaded bar in

Fig.

1.

The intrachain interaction

was estimated

as

aweighed average of both

pro-cedures outlined in this figure as J'/k

=-3.

_0~",

K.

The experimental magnetic heat capacity

C,

„„

-C~

„„

is

represented by open

circles

in

Fig. 2.

The drawn curve denotes the theoretical estimate

for

a S=—,

'

Heisenberg linear-chain system with

J'/k= —

_3.

0

K.

The

error

bars

reflect the

uncer-tainty in the determination of the tota/ heatcapacity

(-1%).

Inspection ofthis figure shows agood

agree-ment between experiment and theory above 9

K.

The magnitude of the intrachain exchange inter-action in CMC inferred from the present analysis

is

somewhat smaller than the magnitude given in

Hef.

12.

Both results, however,

are

well within

the range of values reported in the

literature.

On

the other hand, the present estimate for the

mag-netic contribution in the ordered

state,

which is

10 CsMnCl3 2HPO 8-, theoretical estimate E lg QJ with 3/k= -3.0K

FIG.2. Magnetic heat capacity ofCsMnC13'2II&O. The open circles corres-pond to C _t CI

error bars reflect the

un-certainty in _C~~t. The drawn curve denotes the

'

theoretical estimate for a

S=_~Heisenberg

linear-chain system with J/k

=-3.

0K.

10 20

T(K)

based upon extrapolation of

_C~,

_~,

down to T

=0,

agrees within 2% with

earlier

estimates,

"

and

hence the spin-wave analysis given in aprevious

paper" is

still

correct.

Inview of these results

we

feel

confident to state that the behavior of the magnetic heat capacity of CMC

is

rather well

es-tablished, both above and below the ordering

tem-perature.

III. n-RbMnC13 2H20 AND CsMnBr3 2H20

The magnetic properties of n-RMC and CMB have been studied

less

extensively than those of the

isomorphic CMC. a-RMC has been found to order

antiferromagnetically at

T„=

4.

56 K,

"

with the same magnetic space group

as

CMC,

i.e.

,

P»c'ca',

"

while CMB

orders

antiferromagnetically at

T„

=

_5.

_{75 K,"withmagnetic space group}

_Pc'c'a'.

"

From

the corresponding magnetic arrays

it

is

obvious that in CMB the interchain interaction

along the crystallographic b direction is positive,

in contrast toboth CMC arid

e-RMC.

Single

crystals

of n-RMC were grown by cooling

a saturated solution of MnCl, 4H, O and RbCl in

molar ratio

5:1

in 8M HCl from 50to 5

C.

The

crystallized mixture of n and P modification

grad-ually transforms into the n modification after

a

few weeks at 5

C.

The crystals were more

or

less

needle-shaped with average dimensions of

10&&1&&1_{mm. Single crystals of CMB were grown}

by slow evaporation of a saturated solution of

MnBr, ~_{4H,Q and}

CsBr

_in molar ratio _of

6:1

_at

room temperature. The

crystals

were rather large (typical dimensions 3

x

8x 15mm) and showed roughly the same morphology as CMC.

Specimens consisting of

-0.

1 mol of small

crys-tals

were measured with avacuum calorimeter

of conventional design, which was fitted with

a

temperature-controlled heat

screen

to enable very accurate measurements at temperatures up to

about 50

K.

Temperature readings were obtained

from a calibrated germanium thermometer that was measured with an

ac

resistance

bridge

oper-ating at -1VO

cycles/sec.

For

both compounds, the total heat capacity showed the same behavior

as

the data on CMC. The heat capacity of CMB,

however, appeared to be considerably

larger

than

that of the two chlorine isomorphs.

The separation of the magnetic and the lattice

contribution tothe heat capacity has been achieved

by analyzing the data between 9and 52 Kaccording to the procedure described above. The intrachain exchange interaction in o.-HMC isfound as Z/k

=

-3.

0",

_{'4K, the corresponding interaction in CMB}

is

found as

J/k

=-2.

6",

)

K.

Due to the relatively

high lattice heat capacity, the uncertainty in the value ofZ/k in CMB

is

somewhat larger than

that in both chlorine isomorphs. The lattice

con-tribution in n-RMC

is

represented by

6,

=254 K,

9,

=232 K,

6,

=63 K;

for

CMB the parameters

9,

=204 K,

6,

=165

_K,

_9,

=52 K

are

obtained. If these values

are

compared with the values

_9,

= 277 K,

9,

=219

_K,

_9,

= 53 K obtained

for

CMC

from the fit with

J/k

=-3.

0K, it

is

obvious that the

lattice-specific

heats of these compounds

can-not be related to each other by

a

simple

tempera-ture-

independent scaling

factor.

The experimental magnetic heat capacity

C,

_,

—

_C~

„„

is

shown by open

circles

in

Figs.

3and

4

for

n-RMC and CMB, respectively. The drawn

curve denotes the corresponding theoretical

esti-mate. The m8.gnetic heat capacity of n-RMC

ap-pears

to be basically identical tothat ofCMC, 10 O E fg a—Rb MnCl3 2H20 theoretical estimate

with 3ik=-3.0K FIG.

3.

Magnetic

heatca-pacity ofe-HbMnC13 ~_2H20.

The open circles corres-pond toC,

~t

—CI,

«„

the error bars reflect the

un-certainty in C~~t. The drawn curve denotes the

theoretical estimate for a

9=

~Heisenberg

linear-chain system with J/k

=—

3.

0K. The anomaly at

2.19Kisduetoa small

frac-tionofP-HbMnC13 '2H20.

480

K.

KOPINGA

10

CsMnBr& 2H20

t eor stimate

FIQ.4. Magnetic heat capacity of CsMnBr3.2820. The open circles

corres-pond to Q~ t—Q& c&,the

error bars reflect the

un-certainty inC,

»t.

The drawn curve denotes the

theoretical estimate fora

S=~ Heisenberg

linear-chain system with J/k

=-2.

6K. The anomaly at

2.8Kis due to a small fraction ofCs2MnBr4~_2H20.

10 20 30 40 50

TABLE

I.

Values for the intrachain exchange interaction J&and the interchain interactions

J2 and J3 for CsMnC13 ~_{2H20, e-RbMnC13} ~_{2H&O, and CsMnBr3 2H20. The} last _{column gives}

the fraction ofthe total magnetic entropy increase removed below the ordering temperature.

CsMnC13 2H&O

Neutron diffraction ESR line shape

Paramagnetic NMR

Proton spin-lattice ralaxation Oguchi's formula

Susceptibility below Tz Specific heat~

Jq/k = 2.6&K, (Z2+J3(=7x&0 (Jq[

between /Z2J =JJ'3/=2x10 fJ&J

»d

l&2I=too(Z,[=2.6xto-']Z,)

~~,~

=6«0-'~~,

~,~,=0

&g/k=-& 2 K,

Ill

=6x«~l&il =61&31

&i/k

=-6

0K I»+&31

=6x«

'1&iI ~crit=i3

e-RbMnC13 ~_2H20

Susceptibility" above T~

below T&

Specific heat plus Oguchi s formula

J(/k

=-2.

9K

Ji/k

=-3.

8K

~i/k =

-2.

_{0K. I&~I}=l&sl =7xto-'I&&[ it=

|3.

6%

CsMnBr3 ~_2H&O

Susceptibility

"

above TN below _Tz

Specific heat plus Oguchi's formula

J(/k =—

3.

0/—3.2 K Ji/k =—

3.

7 K Jg/k =—2.6K, f

j,

/=(z3J =

t.

4x10~fJ'zf ~crit =20.9% ~Reference

4.

Reference 6. c_{Reference 8} Reference 23. ~Reference 20. Reference S. ~Reference 10. "Reference 22.

except

for

the three-dimensional ordering, which

occurs

at slightly lower temperatures. The small peak at T

=2.

19

K

is

due to a small fraction

(1%-2/o) of P-HbMnCi, ~_2H,

_O.

_{Unfortunately, this peak}

precludes adetermination ofthe interchain inter-actions from the magnetic heat capacity at low

temperatures by linear spin-wave theory.

"

There-fore,

these interactions were estimated from Green's function

theory"

using the values

J/k

=

-2.

_{0Kand T~}=

_4.

_55Kas

I

J'/J

I=gx].0

'.

A

similar procedure

for

CMC yields l

J'/J

l= 8x 10

In CMB, an analysis of the low-temperature

mag-netic heat capacity with linear spin-wave theory

is

hampered by the small peak at T

=2.

8 K, w'hich

is

due to

a

small fraction of

Cs,

MnBr4~_2H,

_Q.

"

Substitution of the values

J/0=

-2.

6Kand Tz=

5.

'l5

Kin Oguchi's

expression"

yields l

J'/J

l

=1.

4

x

10

'.

In view ofthe results given above, one might be

tempted to conclude that the magnitude of the

in-terchain interactions

increases

going from n-RMC

via CMC to CMB. We wish to emphasize,

how-ever,

that the observed change of the ordering temperature may also be explained by a variation of the (relatively small) magnetic anisotropy, which has been found to

increase

going from

n-RMC via CMC to CMB.

"

Since the influence of anisotropy

is

not included in Oguchi s theory, the

estimated values of l

J'/Jl

should be considered

with some

reservations.

A survey ofsome

repre-sentative results obtained from various

experi-mental techniques

is

presented in Table

I.

with the additional conditions

n'=z,

T„=T„',

and

E

=E'.

This equation did satisfactorily

describe

thedatafor7&10

'&-a&10

'and7~10

'&z&5x10

',

with the parameter values n

=+

0.

11a

0.

01,

A =

2.

01

+0.

.

05 J'/mol Kand A'

=3.

05

+0.

05J/mol

K.

Un-fortunately, attempts to refine the

fit

by including

a

constant term

8

in

Eq.

(2)

or

by removing the condition

n'

=_{n were unsuccessful, since they all}

resulted in rather strong correlations between the

parameters.

The systematic deviations

for larger

values of

l&l may partly be caused by the

fact

that the

heat-capacity anomaly has

a

rather weak divergence,

and hence correction terms to the limiting behavior

C„=A

la'l

™

may be important already

close

to the

ordering temperature. As there seems to be some

evidence that the relative importance ofthese

cor-rection terms

is

smaller

for

quantities having a

larger

value of the

critical

exponent, we shall next consider the

critical

behavior ofthe magnetic entropy increase

_8,

given by

lim(lnrS(c) —S «]/inc

f=1

—n .

I

I:

CsMnCl3. 2H20 2

IV. CRITICAL BEHAVIOR

Next, we shall give some attention to the

critical

behavior of the magnetic heat capacity. Inthe. neighborhood of the three-dimensional ordering temperature, very accurate measurements were performed with typical temperature increments

4T

«0.

5 mK, in order to avoid experimental

"rounding" of the specific heat anomaly.

For ~T

&2 mK, the experimental rounding appeared to be

negligible. The magnetic specific heat

C„was

obtained by subtracting the lattice contribution

presented above from the experimental data.

As a

first

attempt to obtain the

critical

exponents

c.' and

_n,

we plotted log»Cs vs

_log„

lel

for

several

values of

T~.

For

all three compounds these plots

showed a rather pronounced curvature of the data,

both

for

l&l&5x 10

'

_and

_for

lel&5x 10

',

which

could not be removed by

a

physically acceptable

readjustment ofthe value of

T„.

Since the data on

n-RMC showed the smallest rounding, we

per-formed

least-squares

fits

of the magnetic heat

capacity ofthis substance to the more general ex-pression (2) I o 0)

—

0 -3 t I ij) O 0) 0 -3 L. I -3 O CO 0 -3 -2 iog„[i-iilN[

FIG.5. Double logarithmic plot of S—

S~«vs

~1

—_T/Tz( for CsMnC13 2H20, o.-RbMnC13~_{2H20, and}

CsMnBr3 2H&O. The open circles represent the data

432

K.

KOPINGA 16

We evaluated both Sand

_S.

.

.

from the experimen-tal data by numerical integration of C~/T. The

results were plotted doublelogarithmically

for

several values of

T~.

For

all three compounds the data

for

T&

T„could

be represented

surpris-ingly well by straight lines over more than two

decades of e if

T„was

chosen afew mK above the heat capacity maximum. The

fact

that the actual value of

_T„

is

found at somewhat higher tempera-tures than C seems to be a rather common

fea-ture,

caused by the rounding ofthe asymmetric

heat capacity anomaly.

""

The results

are

shown

in

Fig.

5

for

CMC, n-RMC, and CMB. The data

for

T&

T„reveal

some curvature, but the results

suggest that

z'

=

a.

I

_{ox all three comPoumds the}

critical

exponent

is

found to be equal, and amounts

to n

=+0.10+0.01.

Although astudy of the

critical

behavior using the magnetic entropy increase

is

somewhat unconventional, the value of n

is

con-sistent with the results from renormalization group theory

'

for

a three-dimensional ordering

of asystem with spin dimensionality

close

to

1,

and agrees very well with the value found from

a

direct fit

to the data on Q.-RMC.

ACKNOWLEDGMENTS

The author wishes to acknowledge the stimulating

discussions with

Dr.

W.

J.

M.de Jonge. The help

of

J.

P.

M. Smeets and

J.

P.

A. M.Hijmans in the

analysis of the

critical

behavior

is

greatly

appre-ciated.

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J.

Jensen,

P.

Andersen, and S.

E.

Rasmussen, Acta Chem. Scand. 16,1890 (1962).

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J.

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