Magnetization and susceptibility studies on some antiferromagnetic linear chains

Magnetization and susceptibility studies on some

antiferromagnetic linear chains

Citation for published version (APA):

Botterman, A. C. (1976). Magnetization and susceptibility studies on some antiferromagnetic linear chains.

Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR91056

DOI:

10.6100/IR91056

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Published: 01/01/1976

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MAGNETIZATION AND SUSCEPTIBILITY STUDIES

ON SOME ANTIFERROMAGNETIC LINEAR CHAINS

MAGNETIZATION AND SUSCEPTIBILITY STUDIES

ON SOME ANTtFERROMAGNETIC LINEAR CHAINS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. IR. G. VOSSERS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN, IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 9 APRIL 1976 TE 16.00 UUR.

DOOR

ALBERT CORNELIS BOTTERMAN

Dit proefschrift is goedgekeurd door de promotoren prof. dr. M. J. Steenland en prof. dr. J. A. Cowen.

This investigation is part of the research program of the "Stichting Fundamenteel Onderzoek der Materie

(FOM)", which is financially supported by the

"Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (ZWO)".

CONTENTS CHAPTER I 1. Introduction CHAPTER II THEORY 2.1 2.2 2.2.1 2.2.2 2. 2. 3 2.2.4 2.2.5 CHAPTER 3.1 3.2 3. 2.1 3.2.2 3.3 3. 3.1 3.3.2 3. 3. 3 3.3.4 3. 3. 5 4.1 4.2 4.3 4. 3.1 4.3.2 4.4 4.5 5.1 5.2 5.3 III Introduction

Molecular field theory Perpendicular susceptibility

Phenomenological introduction of anisotropy The spinfZop transition

Critical hyperbola

Perpendicular susceptibility and anisotropy

EXPERIMENTAL APPARATUS Introduction

The Faraday balance method Description of the balance Measuring procedure

The dynamic susceptibility method The mutual inductance bridge The cryostat inductance Double pot system Single pot system

Sample holder and calibration of the apparatus

Introduction

Crystallography and preparation Magnetization results

Field and temperature dependence Angle dependence

Exchange p~rameters Susceptibility above TN

Introduction

Structure and preparation Experimental results 5 7 9 9 10 II 12 13 17 18 18 20 21 21 23 25 27 29 33 34 36 36 43 48

so

53 54 55

5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.4 5.5 6.1 6.2 6.3 6.4 6.5

Temperature dependence of magnetization and susc:eptibi Zi ty

AngZe and fieZd dependence of the magnetization

A possibZe modeZ

Derivation of the magnetic phase diagram Magnetization at high fieZds

Hami Ztonian and exchange parame te1's Discussion

Introduction

CPystaZZog1'aphy and p1'eparation High temperature susceptibility Low temperature behaviour A Zero field susceptibility

B Magnetization in the ordered state C The magnetic phase diag1'am

Interpretation and discussion

SUMMARY SAMENVATTING DANKBETUIGING LEVENSBERICHT 55 59 62 65 66 69

74

79 80 83 87 87 88 94 97 105 107 109 110

CHAPTER I

1 Introduction.

Compounds which exhibit lower dimensional magnetic behaviour offer an attractive field of experimental research, since a direct con-frontation with theoretical studies on model Hamiltonians in these systems is quite feasible. Generally such a confrontation is only significant if sufficient appropriate experiment·al data are avail-able, such as, for example, parameters concerning exchange and ani-sotropy. In view of this we decided to investigate some compounds which might possess a one-dimensional magnetic behaviour in view of

their crystallographic structure.

It is a common practice in the field of magnetism to study iso-structural salts which have either similar ligands and different magnetic ions or different ligands and similar magnetic ions. In this thesis the results of magnetization and susceptibility ex-periments on the series of isostructural linear chain compounds XMY3.2H20 (with X= Cs; Rb, M =

Mn;

Co andY= Cl; Br) and the Ni compound of the series MY2.2H20 (with M =Fe; Co; Ni andY= Cl; Br), NiC1 2.2H2

o,

will be presented. The research on these compounds forms a part of the program of our institute. For the interpretation and discussion of the experimental results we will also use information from specific heat, nuclear magnetic resonance (NMR), antiferromag-netic resonance (AFMR), neutron diffraction and X-ray experiments.

The organization of this thesis is as follows. In chapter II we will review some molecular field results needed for the

interpre-tation of the magnetization data in the ordered state. Chapter III describes briefly the experimental apparatus used in the magnetiza-tion and susceptibility experiments. In chapter IV the experimental results on the series of compounds XMnY3 .2H2

o

(with X= Cs, Rb and Y = Cl, Br) will be discussed. Chapter V is devoted to the Ising c_{hain compound CsCoc1 3.2H20, while in chapt}er VI our experiments on NiC1 2.2_H2

o

will be given.

CHAPTER II

THEORY

2.1 Introduction.

The exact solutions of the thermodynamic quantities of an infinite magnetic system with general interaction between the spins consti-tutes a problem which goes far beyond the numerical possibilities available at this moment. However, it has become a common practice ~n the last years to simplify the problem and search for the solutions of the so-called model Hamiltonians. The model behaviour of a system implies certain restrictions on the spatial dimensionality of the interactions (chains, layers) as well as on the dimensionality of the interaction-tensor itself (Ising, XY or Heisenberg).

Consider the Hamiltonian

_,.

_,.

H - 2 l: [ aJ .. S. S. + b (J .. S. S. + J .. S. S. ) ] - g]JBH. (l: S.),

i<j ~J ~Z JZ ~J ~X JX ~J ~y JY i 1

(2. I)

_,.

_,.

where J .. is the exchange parameter between spins Si and SJ., g 1s

1] _,.

the g-tensor, ]JB is the Bohr magneton and H the external magnetic field. The spatial dimensionality of the interaction depends on the arrangement of nearest neighbouring spins

S.

and

S.

in the ensemble.

1 J

The dimensionality of the exchange tensor can be categorized by the relative magnitudes of a and b.

When a = b = I we are dealing with the isotropic Heisenberg model, in which case no preferred spin direction is prescribed by (2.1).

When a= 0 and b = I equation (2.1) describes the XY model, where only x and y components of the spins play a role.

The case a

=

I and b = 0 represents the Ising model, where only spin components in the z direction have to be considered.

The thermodynamic quantities are generally deduced from the parti-tion funcparti-tion of the system under consideraparti-tion,

Z

= E

_.

exp (- E./kT), ~ ~

where E. are the eigenvalues of Hamiltonian (2.1). ~

Now, for instance, G - kT ln Z 3G

a

2G 32G

- <;m)r ;

Xas

= - (()H 3H )T ; CH = - T(W)H a a

e

(2.2) (2.3) where G is the Gibbs free energy, M the magnetization, X the

suscept-ibility and CH the magnetic specific heat.

The partition function (2.2) of the ensemble described by (2.1) {even without field) is known for only a few cases. Generally one may state that the complexity of the calculation increases with spin and spatial dimensionality as well as larger spin quantum number S. In the majority of cases one has to deal with approximate solutions obtained from expansions of (2.2) and (2.3) in the limits of T ~~or T ~ 0. For more information the reader is referred to Stanley [1]. From the before mentioned calculations it can also be inferred that some of these model systems show long range order below a certain critical temperature Tc, This long range order is characterized by the fact that the correlation function <

S .. S

> f 0. One-dimensional Ising,

~ ~

one- and two-dimensional Heisenberg and one- and two-dimensional XY systems (without additional anisotropy) do not order in the sense noted above. In practice, however, these models are never completely realized. Usually perturbing, small, interchain or interlayer inter-actions trigger a three-dimensional ordered state at low temperatures. The general high temperature behaviour of the systems, however, re-flects roughly the low-dimensional characteristics, such as rounded maxima in specific heat and susceptibility, high spin-correlations and a large amount of magnetic entropy still to be gained above the ordering temperature T . Typical examples of this pseudo low-dimensional

c

behaviour are rather manifold. Among those are the samples studied ~n this thesis. However, our main attention will be focussed on the ordered state and in the interpretation we will frequently restrict ourselves to the mean field approximation. Some of the results of this mean field approximation will be reviewed in the following sections.

Here the exchange interaction of an arbitrary spin with its neighbours is treated as an effective field ~· acting on the spin under conside

r-ation. In spite of this rather crude approach, molecular field theory appears to be a helpful tool in the. explanation of the behaviour of many ordered magnetic systems.

2.2 Molecular field theory.

2.2.1

Perpendicular susceptibility.

In this section some aspects of a simple molecular field theory for an antiferromagnet with isotropic Heisenberg exchange (a

=

b

=

1 in equation (2. I)) will be discussed. The Hamiltonian can be written as

-+ -+

H - 2 l: J ..

s .. s.

i <j 1] 1 J (2.4)

In the ordered state of an antiferromagnet the lattice of the magnetic moments can be subdivided into a number of interpenetrating sublattices, so that magnetic ions from one sublattice have only nearest neighbours belonging to other sublattices. The simplest case consists of two sub-lattices, each containing half of the total number of magnetic moments, with the moments on each sublattice being mutually parallel. Summation of the magnetic moments of each sublattice yields two macroscopic mutually antiparallel magnetization vectors

M+

and

M-

with magnitude

~(N/Mw)g~B<S>. Here N is Avogadro's number, Mw the molecular weight and <S> the expectation value of the spin quantum number.

Considering only nearest and next nearest neighbours the effective field experienced by a spin in either of the sublattices is given by

(2.5)

Here A and B are constants representing nearest neighbour intersub-lattice and next nearest neighbour intrasublattice exchange inter-actions.

So far we have ignored anisotropy. In this case the, application of -++

-+--an external field will yield a magnetic array with M -M perpendicular

• • -++

+-to the external field. Balanc1ng the torques acttng on M and M

• -+ •

exerted by the effective exchange f1eld ~ and the external magnet1c field

H,

one then obtains for the susceptibility in the ordered state

x.l

A

1 (2.6) Only the exchange interaction acting between antiparallel aligned moments or sublattices enters in this expression. If we denote this interaction hy J, we can relate A in equation (2.6) to J through A=

4zjJj/(N/Mw)g

2

~i, where z is the number of nearest neighbours.

This yields

(2. 7)

Through (2.7) it is possible to obtain the value of the exchange con-stant J, by inserting the measured value of the perpendicular suscept-ibility. Since the above mentioned isotropic exchange interaction does not favour a certain direction, an additional anisotropy energy has to be introduced in order to fix the direction of the sublattice magneti-zation with regard to the crystal lattice at zero field. In the follow

-ing the cases with uniaxial and orthorhombic anisotropy will be dis-cussed.

2.2.2 PhenomenologicaZ introduction of anisotropy.

The simplest way to write an uniaxial anisotropy energy is as follows

(2. 8)

where K is a positive constant and a+ and a are the direction cosines of the sublattice magnetization vectors ~ and

M-

with the preferred x axis. Assuming that the magnitude of the sub lattice magnetization vectors lS not influenced by the anisotropy energy, a fictitious

ani-field, + introduced sot ropy _{HA' can be}

-+- -+-- H 6M

A (2.9)

++

Here oM- denotes an arbitrary spatial variation of the corresponding quantity, while the magnitude is kept constant. The components of the anisotropy field in the uniaxial case around the x axis can be deduced by equating the variations 6EA obtained from equation (2.8) and (2.9). This results in

+

0 HAx- = (K/M )a+

0 - (2. 10)

where M0 is the absolute value of the sublattice magnetization in vanishing external field; a± is the direction cosine with the x axis.

Now the case of orthorhombic anisotropy will be discussed. The anisotropy energy in this case will be written as

(2. II)

where

B±

and y± are the direction cosines of the sublattice magnetization vectors with the y and z axes respectively; _{K 1} and _K2 are constants for which we will assume _K2>K1>0. When the magnetic moments are oriented along the y or z axis, EA equals _{K 1} or _K2 respectively, whereas EA is zero along the easy x axis. One usually denotes the y and z axes as next preferred (medium) and hard directions respectively. Following a similar procedure as for the uniaxial case, one easily obtains the components of the anisotropy field for a~ orthorhombic system

0 (2. 12)

When the antiferromagnetic exchange energy is large with respect to the

-++

+-anisotropy energy and the external magnetic field is small, M and M will point almost antiparallel and equation (2. II) may be simplified to

(2. 13)

+

Here

B

and y are the direction cosines of the unit vector L, which

++

-+-defines the direction of M -M .

2.2.3 The spinfZop transition.

In the case of an antiferromagnet with uniaxial anisotropy the magnetic energy at low fields will be minimal if the magnetic moments are oriented along the easy, x, axis. When the external magnetic field, H, parallel to the easy axis, exceeds a certain critical value, H _cr , the magnetic energy will be lower if the magnetic moments point almost perpendicular

to the easy axis. At H

=

H the magnetic moments will move from the cr

to x, bending slightly towards the field direction. This phenomenon is called spinflop and H is the so-called spinflop field. At the spinflop

cr

field the parallel magnetization and the corresponding susceptibility suddenly increase; for H > H the magnetization increases linearly with

cr·

the field, extrapolating through zero magnetization at zero field ~n the magnetization versus field plot. The spinflop field is determined by the equilibrium between the decrease of the magnetic energy and the gain of anisotropy energy. In this way one obtains the following relation between the spinflop field H , the anisotropy field HA and the jump in the

sus-cr

ceptibility at H cr

(2. 14)

where M is the value of the sublattice magnetization given by !(N/Mw)gJlB<S>.

• -+ -+

This relation can also be obtained by balanc~ng the torques of H, HA and -+

~· In the following section the more complicated case for orthorhombic anisotropy will be discussed.

2.2.4 Critical

hyperbola.

The behaviour of the sublattice magnetizations in an external magnetic

field for an orthorhombic system in case of large exchange interaction is described by the theory of Nagamiya [2]. This theory is valid for T ~ 0 K. If the magnetic field points in an arbitrary direction, the free energy G can be written as

G (2. IS)

where

X;;

and X1. are the parallel and perpendicular susceptibility with

-+ .

respect to L, respect~vely; H// and Hl. are the components of the

mag--+ -+

netic field H parallel and perpendicular to L respectively (see figure

2.1). Substituting Hcos~ and Hsin~ for H// and Hl. with~ as the angle -+ ....

between Hand L, equation (2.15) can be written as a function of a,

8

and y. The equilibrium positions of the sublattice magnetizations can be derived through minimalization of this free energy with respect to a,

8

andy, subject to the condition a 2 +

8

2 + y 2 = I . This leads to a

I I L I M+ I I H I HJ. y,z

· z

a·

z

h +r -+- .... ....

Fig. 2.1 Geometnca t.sp ay of t e vectors M , M , L and H. HI/ and

H~

are the components of the field

H

parallel and perpendicular to

L

respectively.

Since this equation is not easily solvable for arbitrary field directions, only the cases in which the field is restricted to the xz and xy planes will be reviewed.

~

When the external magnetic field H is located in the xz plane, a spinflop transition can be observed when the field is parallel to the easy x axis. Under certain conditions a spinflop-like transition will also be observed if

H

deviates from x. The components of the field at which this transition occurs, satisfy the following equation

(2. 16)

This is the so-called critical hyperbola, which has been derived by Gorter et al. [3] for T = 0 K.

When the external field intersects this hyperbola, the directions of the sublattice magnetizations change over from the neighbourhood of the easy, x, axis to the neighbourhood of the next preferred, y, axis.

This is accompanied by a sudden increase of the magnetization measured along the field direction. As long as the .magnetic field remains inside the shaded area of the hyperbola in the H -H plane (see figure 2.2),

X Z

Fig. 2.2 Critical hyperbola in the H - H plane. If H en+ ters X y

the shaded area a spinfZop Zike transition occurs.

the magnetic moments remain oriented nearly perpendicular to the xz plane, bending only slightly towards the field direction. If the magnetic field vector moves away from the H axis, the measured magnetization

X

does not change until the shaded area is left. Then the magnetization suddenly drops. This behaviour leads to plateau-like rotation diagrams of the magnetization centered around the x axis.

If the field is parallel to the easy x axis equation (2.16) can be simplified to

(2. 17)

where H is the field strength at which the transition, called spinflop, cr

occurs. Comparing this equation with equation (2.14}, one obtains at T

=

_{0 K: K1}

=

M0HA. With equation (2.12) HA can be relate~ to they component of the orthorhombic anisotropy field by HA =

I

HAy

I

.

When the field is confined to the xy plane, the solution of the above mentioned eigenvalue problem yields a result quite different from the one obtained for the xz plane. Now a critical field only occurs with the field parallel to the easy axis. When the external magnetic

field exceeds the so-called spinflop field, H , given by equation cr

(2. 17), the sublattice magnetizations move from the neighbourhood of the x axis to the neighbourhood of the y axis, similar to the former case. In this case, however, no plateaus are present in the magneti-zation as a function of the angle of the field with the easy axis. Around this axis peaked max1ma will be found if the external field exceeds the spinflop field.

Finally, when the field is restricted to the yz plane, no such behaviour as described before will occur and the magnetic moments

remain oriented almost along the x axis, except for very high fields.

2.2.5 Perpendicular susceptibility and anisotropy.

The perpendicular susceptibility appearing 1n the above theory is

+

only related to the vector L and not to a specific crystal field direction. The existence of anisotropy, however, influences the perpendicular susceptibility. It tends to reduce the value of X~

with respect to the isotropic situation, since it becomes more diffi-cult to turn the magnetic moments from the preferred direction. In general, however, the anisotropy energy is much smaller than the exchange energy, so that anisotropy will not influence the perpendicular suscept-ibility very much. In accordance with equation (2.7), the value of X~

is mainly determined by the value of the antiferromagnetic exchange parameter.

The case of uniaxial anisotropy will be treated first. Balancing the

+

torques exerted by the effective exchange field ~· ~he anisotropy field

H!

and the external field

H,

in the case that

H

is perpendicular to the easy axis, one obtains

(2. 18)

where X~(O) is given by equation (2.7) and a= HA/~. If the external field is parallel to the easy axis and larger than the spinflop field, one obtains

For orthorhombic anisotropy the magnitudes of the perpendicular susceptibilities measured along the next preferred and hard directions are different. Balancing of torques yields

References Chapter II

I. H.E. Stanley, Introduction to Phase Transitions and CriticaZ

Phenomena (Oxford University Press, London, 1971). 2. T. Nagamiya, Prog. Theor. Phys. ~ (1954) 309. 3. C.J. Gorter and J. Haantjes, Physica ~ (1952) 285.

CHAPTER III

EXPERIMENTAL APPARATUS

3.1 Introduction.

Most magnetization measurements reported in this thesis were performed with a Faraday balance in combination with a conventional iron magnet,

from which a maximum field strength of 22.8 kOe* could be obtained. The direction of the external magnetic field with respect to the sample could be varied over 360 degrees by rotating the magnet in the horizon-tal plane. The static susceptibility is indirectly obtained from the

l1M

slope of the magnetization versus field curves: X= !;H' The Faraday method provides absolute values of the magnetization of a sample in contrast with the dynamic susceptibility technique, which will be dis-cussed more extensively in section 3.3.

The dynamic susceptibility, given as Xd = x' - ix", was measured using yn

a mutual inductance method. The change of the mutual inductance of a coil system consisting of primary and secondary coils, due to the presence of a magnetic sample was measured with a mutual inductance bridge des-cribed by Wiebes et al. [I]. The dynamic susceptibility is obtained in bridge units, which have to be calibrated in absolute units by measure-ments on a well known paramagnetic salt. Measuremeasure-ments are possible with and without an external magnetic field. Part of the measurements on CsCoC1 3.2H20 were performed using a copper coil wound around the helium cryostat and cooled with liquid nitrogen. With this coil magnet a maximum field of 4 kOe was reached, which was high enough to study the metamagnetic phase transition. In a later stage a superconducting solenoid was used, ~n which a maximum field of about 22 kOe could be obtained. Recently an improved superconducting magnet was installed,

* The CGS system of units will be used throughout this thesis since this system is prevalent in the literature on magnetic materials at this time.

in which a max~mum field of 95 kOe could be reached. This field range enabled us to measure the magnetic phase diagram of NiC1 2 .2H20.

Since the field of the solenoid is coaxial with the measuring inductance, a special apparatus was constructed to rotate the sample with respect to the field direction. As the inner diameter of the measuring inductance is only 14 mm, the samples which can be used in this apparatus are

rather small '(about 4 and 8 mm in radial and axial directions respect-ively).

3.2 The Faraday balance method.

3.2.1 Description of the balance.

Using the Faraday balance the static magnetization is obtained from the force exerted on a magnetic sample in an inhomogeneous magnetic field. The vertical component of this force can be written as follows

F z

I

ClH ClH (M ~ + M ____L + M X az y dZ Z

v

aH

_z) dV dZ •

Here the z axis points vertically, M. is the i component of the ~

magnetization per unit volume and dV is a volume element of the sample. For

ClH ClH

_J_ _ z

dZ ' dZ

small samples and field

H

in the x direction and 0, equation (3.1) can be simplified to

F z ClH X mcr -x dZ (3. I) (3.2) where ox is the x component of the magnetization per unit mass and m

the mass of the sample.

The construction of the Faraday balance is illustrated in figure 3.1.

The balance was constructed by Dr. H.M. Gijsman, who kindly put it at our disposal. A complete description of this balance is given in his thesis [2], but for the sake of completeness a survey of the construction will be given. The numbers used in the description of the balance refer

to the details of figure 3.1.

The sample is fixed to a long thin quartz tube (16) (I m long and 2 mm ~n diameter), which transfers the vertical force to the beam (II) of an electrically compensated balance. The tube is attached to a scale (12),

b---~ .---.---7 ~---8

--- -- - -16

~---17

.--- -- - - 18

Fig. 3.1 Schematic picture of the Faraday balance. Most of the details are discussed in the text. 7 is a diaphragm; 14 is a stainless steel tube; 15 is a helium cryostat; 17 is a conic joint to which a glass tube can be fixed for experiments in liquid

which hangs on one side of the balance beam, while the other side holds an adjustable counter weight (4). The beam hangs on two, 0.05 mm thick, torsion strips of Cu-Be alloy (2).

Two permanent magnets (1), with their fields in opposite direction to cancel the influence of stray fields of the big magnet, are placed around a pair of compensation coils

(3),

which are fixed to the balance beam. A beam of light produced by a lamp (8) falls on a small mirror (5) mounted on the balance beam; The reflected beam is focussed by two lenses (9,6) on a differential photovoltaic cell (10). In this way voltages are generated in the upper and lower parts of the photocell, which are fed to the input of an electronic amplifier system, whose

output current passes through the compensation coils. A small displace-ment of the mirror, due to a force acting on the sample holder, results in a current through the coils, counteracting the movement of the balance beam. If the amplification factor of the electronic system is high enough, the sample will remain in the original position. In this way the com-pensation current is proportional to the torque on the balance beam produced by the vertical force acting on the sample. The proportionality constant is determined by putting small standard weights (13) (from 50 mg to I g) on the scale. The sample is located 7.5 mm above the

<lH

edges of the pole pieces (22), where Hx <lzx has a smooth max1mum. Thus, a small inaccuracy in the position of the sample will be of minor 1n-fluence.

A flat coiled spring (20) of a Cu-Be alloy is mounted at the bottom of the quartz tube just above the sample holder, preventing a possible displacement of the sample towards one of the magnet poles. The coiled spring is manufactured by a photographic etching procedure, which ensures rigidity in the horizontal plane and flexibility in the verti-cal direction. The spring is soldered on a platinum pin (19), which sticks out of the glass tube (18) surrounding the quartz tube. The single crystals can be glued directly under the quartz cylinder (21)

at the end of the quartz tube, or placed in a holder made of quartz or epoxy.

3.2.2 Measuring procedure.

ethylene and propane. The sample holder is directly immersed in liquid helium, but separated from the liquid by a glass tube if one of the other cooling agents is used, since the violent boiling of these liquids prevents a proper functioning of the balance. A disadvantage of this apparatus is that measurements are only possible in limited temperature ranges. These could be extended in some cases by solidifying hydrogen and nitrogen.

The smallest observable variation of the compensation current amounts to 10-S A, corresponding to a force of about 10-6 N. The maximum force for which the balance can be compensated amounts to about 3x10-2 N. The absolute accuracy of the magnetization measurements was approximately one per cent, whereas the relative accuracy was about 0.1 of a per cent. The main error in the measurements originates from the inaccuracy in the determination of the gradient of the magnetic field at the sample. This gradient is measured with a rotating pick-up coil, 1n combination with a calibrated integrating voltmeter. Because of the poor stability of the magnet power supply, the field gradients could not be determined with an accuracy of better than one per cent.

In liquid helium, the evaporation of the liquid causes the upwards force on the sample holder to decrease linearly with time. Corrections must be made for this effect and for the magnetic contribution of the empty sample holder.

3.3 The dynamic susceptibility method.

3.3.1 The mutual inductance bridge.

The dynamic susceptibility of some compounds treated in this thesis was obtained using a mutual inductance method. The dynamic susceptibility is defined as Xd

=

ClM/()H, where the

yn

and the oscillating field H (given as of phase. The dynamic susceptib{lity,

magnetization per unit volume M iwt

H

=

H0e ) are generally out

which can be simplified in case of small fields to _. xd _yn

=

M/H, will be ~ complex quantity, writteh as

x _dyn

=

x' - ix"· (3. 3)

~n vacuum be M . When a magnetic sample is inserted, the mutual inductance 0

can be written as M. d =11M , with 11:: 1+4rrA.(x'-iX"), where A. is the

~n o

volume fraction of the secondary coil occupied by the sample. The sus-ceptibility measurements are performed with current dead secondary coils. Then the relation V

s - M. d dl /dt for the voltage V ~n p s secondary coil and the primary current I may be used

p

v

s - iwr M [1 + 4nA.p 0 x'l - 4nwr M A.x'' p 0 To enable sensitive x' measurements Cx'<<1), the factor

across the open

(3.4)

in equation (3.4) is cancelled by using two identical empty "bucking" coils on each side of the secondary coil. When the mutual inductance of each of these coils with the primary coil is M /2, a compensating voltage iwl M

0 p 0

is obtained in the secondary bridge Circuit (see also 3.3.2). Thus the total voltage across the secondary coil containing the sample and the bucking coils is

v

s -4niwr M A.x' - 4nwi p 0 p M A.x0 " (3.5) For these dynamic susceptibility measurements a mutual inductance bridge after Wiebes et al. [1] was used (fig. 3.2). The bridge consists

of a current source in combination with two current dividing networks and a standard mutual inductance (M ) constructed in an optimal way to

s reduce the effect of stray capacitances.

A special feature of this bridge is the fact that the imaginary part of the voltage in the secondary circuit is compensated by an adjustable resistive network (A in fig. 3.2) in combination with a fixed mutual inductance. The bridge operates at several frequencies between about 80 Hz and 10 kHz, whereas the commercially available variable mutual inductances used in other bridge systems (e.g. Hartshorn bridge [3]) limit the experiments to a narrow frequency range around the resonance frequency of that inductance (about a few hundred Hz).

By means of the purely resistive network A a fraction I /I of the

s p

primary current I passes through the primary of the standard mutual

p

inductance M . P is a precision potentiometer reading a fraction p,

s

while a further stepwise current reduction q is obtained by means of the network

Q.

The transformed voltage in the secondary of the standard mutual inductance is used to compensate for the imaginary part of the voltage across the secondary coil of the cryostat inductance. To keep the primary currents through the standard and the primary cryostat coils in phase, the standard primary is tuned with the variable capacitor cs at the required angular frequency

w,

making the impedance of the standard primary purely resistive.

The real part of the voltage across the secondary of the cryostat coils is compensated by passing a fraction IR/Ip of the primary current through the resistance R (2n) by means of the current dividing network B. Here V is a precision potentiometer reading a fraction v and W provides a

stepwise reduction factor w.

The variable capacitors c 1 and c 2 serve to compensate phase shifts in the voltages across the primary and secondary coils of the cryostat inductance. Thus the effects of parasitic capacitances are made as low as possible. The output of the bridge is connected to a lock-in amplifier, whose output is fed to the vertical axis of an oscilloscope, while the

horizontal plates are fed with a reference voltage from the primary current. Thus a Lissajous figure is .obtained which can be reduced to a straight horizontal line by adjusting the current dividing networks A and B. The readings of the current dividers provide a relative measure of the dynamic susceptibility, as will be shown below.

The fraction I /I of the primary current through the standard inductance, s p

adjusted by the current divider A, is approximated by the product pq. The transformed voltage in the secondary circuit becomes -iwM I

s s -iwM pqi . Equating this to the imaginary part of equation (3.5) we

s p obtain

>..x'

M s 41fM pq. 0 (3. 6)

The susceptibility per unit mass~ is given as~= x'/p, where pis the density of the sample. Thus equation (3.6) transforms into

M s

41f(M

/v )

pq, 0 0

(3.7)

where m is the mass of the sample and V0 is the volume of the central part of the secondary winding.

The real part of equation (3.5) is compensated by the voltage due to the fraction of I which passes through the resistance R.

p

This fraction is approximated by IR/Ip = vw/2 and leads to

>..x"

_4muM R vw ₂

0

(3. 8)

In this way the relative values of the real and imaginary parts of the dynamic susceptibility are directly obtained from the settings of the current dividers A and B.

3.3.2 The cryostat inductance.

The cryostat inductance (fig. 3.3) consists of a primary winding (7) on a Celoron former (I) and a secondary winding (4) directly wound on the primary. In order to take full advantage of the measurement range of the mutual inductance bridge it is necessary to make the mutual inductance of the cryostat coils very close to zero in the absence of a sample. The secondary winding is therefore constructed 1n three parts, where the two outer sections (3,6), which are opposite-ly wound and used as "bucking" coils, each contain

!

and the centre-part (4), meant to contain the sample, contains ~ of the total number

of turns. The "bucking" coils are separated from the centre part by

I mm thick spacers. By removing (or adding) a few turns of the second-ary winding the total mutual inductance is made as close to zero as possible. Before this adjustment the cryostat coils are cooled with liquid nitrogen. t I. 5 6 7 ~~ "'

..

..

~~ 1 - - - - --- - -

r-~

~ -- --- ---

-

11

-~

f--II. 28 14 ~ ₇₈

Fig. 3.3 The cryostat mutual inductance. Dimensions are given

in mm.

At both ends of the primary winding a few extra turns (2) are applied in order to compensate for the reduction of the field near these ends. Thus the homogeneity of the field in the secondary winding is improved

3

(better than 1:10 over the secondary centre-part).

The primary coil consists of about 1200 turns of Povin D insulated copper wire of 0.17 mm diameter (chosen because of its rather thick insulation), wound in three layers. The secondary winding contains about 15.000

turns of similar wire with a diameter of 0.07 mm.

3.3.3 Double pot system.

With the double pot system (fig. 3.4) the dynamic susceptibility can be measured at temperatures between 1. 2 K and 50 K.· The cryostat induc-tance (9) and sample holder (12) are mounted inside a cylindrical copper pot (13) containing helium contact gas. In order to isolate this pot from the surroundings it is suspended freely by cotton threads (IS) from a framework (II), inside a second pot (14) surrounded by a helium cryostat. To minimize the heat leak through the electrical connectors, the wiring from the feedthroughs (2) on the covers of the pots is made of thin coils of constantan wire (0. 10 mm thick and I m long) with an almost temperature independent resistance of 60

n.

Copper wires are

used for the primary current to reduce the heat production. The sample can be thermally isolated from the He bath by evacuating the space

between both pots through the stainless steel tube (1) from which the whole apparatus is suspended. The experimental procedure will be sketched below.

[\

•

'

'

(

'

'

(

•

'

'

'

1---' - - - -- 8 --1-1-l-l--- - - -- - 9 b ~~~---10 1-l-+1--- - - - - -11 ~ fl.H-+-Ifl--l~-H---12 ~ ~~~~---13 ~ ~ p 1!--- - - -- 11.

~

"- 1-1-- - - ---- ----15

Fig. 3.4 The double pot system. 1- stainless steel tube; 2 - stycast feedthrough; 3 - brass cover; 4 - teflon ring; 5 - brass flange; 6 - radiation shield;

7 - spacing block; 8 - Ge thermometer; 17 - evacuation tube. Further details are discussed in the text.

First the sample is cooled down to the temperature of the He bath by letting some He gas in the intervening space; after the evacuation of this space the temperature of the sample can be raised by means of the bifilar wound heater (10) on the outer surface of the inner pot. When thermal equilibrium is reached, X is measured assuming that the temperature of the sample and thermometer are equal. The temperature sensor is a CryoCal CR 1000 germanium resistor (8), which was cali-brated between 1.5 K and 100 K by the manufacturing company. The pots are made vacuum tight by tightening bolts through a brass cover, with a 0.10 mm thick teflon washer (4, 16) used as a vacuum seal between flange and cover (5). This seal remains vacuum tight below the \-point of helium and the same seal can be used for several coolings from room temperature to liquid helium temperature. Indium 0-rings could not be used as a vacuum seal, because of the superconductive properties of In below 3.4 K.

With the above described apparatus it is difficult to obtain absolute

values for the dynamic susceptibility of a sample. To this end, cali-brations with a known paramagnetic salt are necessary to obtain the sensitivity of the bridge. Moreover, the setting of the bridge for the empty apparatus has to be known. This setting is not reproducible

as it depends on how the cryostat inductance is cooled. The dynamic

susceptibility of a sample can also be calibrated if the static sus-ceptibility is known in the paramagnetic region and if it is assumed

that X'

=

X and X" = 0 in that region. In addition, corrections stat

must be made for the magnetic contribution of the empty apparatus, which was temperature independent below about 20 K.

A disadvantage of the system is that the orientation of the sample cannot be changed during an experiment. Moreover, the replacement of a sample is a very laborious and time-consl)ming job since both pots

have to be opened and several wires have to be removed. To overcome

the disadvantage of this method, another apparatus has been developed which operates with only one vacuum pot.

3.3.4 Single

pot system.

The apparatus (fig. 3.5) consists of a single cylindrical copper can

flange (7) is mounted, forming the upper boundary of the cryostat

volume. Through the central hole of this flange a long cylindrical

sample holder (see below) can be inserted. The pot is made vacuum

I ~ 2 -

r;--r

'

]

I I = 6

r

/

~

_,,

~ _h ~ f-l 7

r~

(j'

-

~v

~

~ '

-!ff

Erl~

8 10 11 !"""

_';"

12 ~

\

-~ '\ 13 '\ I 11. 15

I

16 :

T

17 18

Fig. 3.5 The single pot system. 1- scale for vertical

position of sample holder; 2 - rotating cylinder

with screw thread for sample holder adjustment; 3 - rubber 0-rings; 4 - transfer tube hole;

5 - feedthrough; 8 - stycast feedthrough; 9 - cover; 10 - teflon washer; 11 - flange. Further details

tight as was already described in 3.3.3. Under the cover a spacing block (14) is mounted, made from a material with poor heat conducting properties. The measuring mutual inductance (15) is fixed with a screw coupling to the bottom of the spacing block. About it a long cylindri-cal copper shield (17) is mounted, on which a bifilar heater (16) is wound. A calibrated germanium thermometer (12) of the same type as described before and an Allen-Bradley carbon resistor (13) are placed on the spacing block. The Allen-Bradley resistor, which is calibrated in zero field against the germanium thermometer, is used because of its weak field dependence to measure the temperature in applied field. The measuring procedure LS as described above. The helium contact gas pressure can be reduced by pumping through the small tube (6). When

this pressure is low enough the temperature can be raised by the heater (10). Normally a pressure of about I mm Hg is used.

Lower values may not be used, as the helium gas serves to equalize the temperature of the sample and the thermometers. Since the heat transport to the helium bath is larger than in the double pot system, the maximum attainable temperature will be lower. The whole apparatus was placed inside a superconducting magnet.

3.3.5 Sample holder and calibration of the apparatus.

The sample holder (fig. 3.6) consists of a long stainless steel tube (4), which terminates in a rod of synthetic material (6). A second thin stainless tube (3), located inside this tube and sealed with two

rubber 0-rings (2), fits to a rod of synthetic material (5), indented at the end. Rotation of this rod is transferred to a vertical gear-wheel (7), on which the sample is attached. The rotation of this gear-wheel can be read from the dial (I) on top. One revolution of the rod corresponds to a rotation of the gear-wheel of 180 degrees. Those parts of the sample holder which may possibly enter the measuring coils are made of a synthetic material to avoid spurious interactions with the coils. The accuracy in the angle setting of the gear-wheel was about 2°. With this apparatus the preferred direction of a magnetic crystal can be directly determined by measuring the susceptibility for different orientations with respect to the direction of the alternating field. In the case of NiC1₂.2H₂0, for example, the location of the preferred

direction could be detected easily and the angle dependence of the critical fields has been studied with this apparatus. The empty coil reading is now directly obtained by pulling the sample holder out of the secondary coils. Thus the net effect of the sample expressed in

6

Fig. 3.6 The sample holder. Dimensions are given in mm.

bridge units is obtained, which was impossible with the double pot system. Corrections for the empty sample holder effect remain, of course, necessary. The bridge must be calibrated by means of a known paramagnetic salt.

The measured effect depends on the position of the sample in the mutual inductance, as is illustrated in figure 3.7. When the sample

is located in the middle of the central secondary coil, the measured effect is maximal. The effect of the coil system (taken as I at

h

=

0) decreases as the sample is moved up or down, goes through zero somewhere between the central part and the outer parts of the secondary coil and then becomes negative. A 116.5 mg powder sample of the paramagnetic salt FeNH4

(so

4) 2.12H2

o

[6) was used for the calibrations which were performed at T

=

4.2 K. From figure 3.7 it can be seen that if a long sample is used, the measured effect averaged over the sample will decrease. To obtain high accuracy the calibration salt must occupy the same volume as the single crystal. To avoid calibrating the apparatus for each sample, the crystals are kept small. Because of the small 'dimensions of the samples the absolute accuracy is not better than 5%. At the start of the experiment, the position of maximum effect is determined to ensure a proper calibration. If the sample holder is used in the single pot apparatus it is possible to replace the sample

1.0 0.8 Q6 0.4

u

Q2 ~ (ij

"'

0 -~ ;;; ~-02 -0.4 -o.5 -0.8 -6 -4 -2. 0 h!cml FeNH4<S0d~2~ T-4.20K 6

Fig. 3.? The effect of the single pot apparatus as a function of the vertical position of a 116.5 mg powder sample

of FeNH_4(S04J_2.12H20 (v = 1195Hz).

during the measurements at liquid He temperature. An overpressure of He gas in the pumping line prevents pollution of the interior

of the pot with water, oxygen and nitrogen during removal and

References Chapter III

I. J. Wiebes, W.S. Hulscher and H.C. Kramers, Appl.Sci.Res. ~ (1964-1965) 213.

2. H.M. Gijsman, Thesis Leiden (1958). 3. L. Hartshorn, J.Sci.Instr.

l

(1925) 145. 4. CryoCal, Inc. Riviera Beach, Florida, U.S.A.

5. D.N. Astrov and L.B. Belyanskii, Cryogenics 7 (1967) II 1. 6. J. van den Handel, Thesis Leiden (1940).

CHAPTER IV

4.1 Introduction.

From a crystallographic viewpoint, the series of compounds denoted by

XMnY 3.2H20, with X = Cs, Rb and Y = Cl, Br, are fairly good examples of linear chain compounds. A strong indication of magnetic lower di-mensional behaviour in CsMnC1 3 .2H20 has been given by the dynamic sus-ceptibility results of Smith and Friedberg [1]. The susceptibility measured at hydrogen temperatures appears to increase with increasing temperature. Since the susceptibility measured in the liquid nitrogen range increases with decreasing temperature, a broad maximum in the

susceptibility must be present in the intermediate temperature range.

Susceptibility experiments performed by Kobayashi et al. [2] revealed

a maximum in X at about 30 K. Specific heat measurements by Kopinga et al. [3] show ordering temperatures of 4.89 K, 4.56 K and 5.75 K

for CsMnc1 3.2H20, a-RbMnC1 3.2H2

o

and _{CsMnBr 3.2H20 respectively.}

For _{CsMnc1 3.2H20, for example,} 87 per cent of the theoretical value

of the entropy (Rln(2S+I)) has been removed above the ordering tempera-ture, indicating the existence of a rather large amount of short range order.

Nuclear magnetic resonance (NMR) experiments by Spence et al. [4,5]

have shown that below TN = 4.89. _{K, CsMnC1 3. 2H20} orders

antiferromagnetic-ally with P2bc'ca' as the magnetic space group. This magnetic space group

has also been found for a-RbM_{nC1 3.2H2}

o

by De Jonge et al. [6]. It des

-cribes an antiparallel arrangement of neighbouring magnetic moments

along the three crystallographic axes. NMR experiments by Swuste et al.

[7,8] _{on CsMnBr 3 .2H20} reveal a different magnetic space group, namely

Pc'c'a', which describes an· antiferromagnetic ordering between nearest

neighbour spins along the ~ and ~ directions but a ferromagnetic arrange-+

ment along b.

linear chain systems t,he magnetization of small single crystals of

XMnY3

.zH

20 with X= Cs, Rb andY= Cl,Br has been measured [9].

This series of isostructural compoundsoffers a possibility to study the

influence of a change in the surroundings of the magnetic Mn ion on the

magnetic properties.

4.2 Crystallography and preparation.

The crystallographic structure of _XMnY3.2H2

o

with X = Cs, Rb and Y = Cl

has been determined by Jensen et al. [10,11]. The structure of the Cs and

Rb compound is described by the orthorhombic space group Pcca, with four formula units per unit cell. The lattice parameters are given in table 4.1. From single crystal and powder X-ray experiments by Swuste et al. [7,8] a similar crystallographic space group has been found _{for CsMnBr 3.2H20.} The lattice constants are also given in table 4.1. Since these parameters differ only slightly from the values for the chlorine compounds, we assume that CsMnBr 3.2H20 is isostructural with the two chlorine compounds.

a(~) b(~) c(~)

CsMnc1 3 .2H20 9.060 7.285 I I .455

a-RbMnC1 3• 2H20 9.005 7.055 11.340

CsMnBr 3. 2H20 9.61 7.49 II. 94

Table 4.1. Lattice constants of three

compounds of the series XMnY3.2H20.

The Mn ions are surrounded by distorted octahedra, containing four

chlorine or bromine and two oxygen ions (fig. 4. 1). The octahedra of

neighbouring manganese ions share one chlorine or bromine ion, thus

forming chains along the ; axis. These chains are separated from n

eigh-bouring chains by layers of Cs or Rb ions and water molecules in the

...

...

b and c directions respectively.

Considering that the crystallographic structure shows that neighbouring

Mn ions along the chain direction share one halogen ion, we expect a

strong exchange interaction between these ions. Since at least two non

...

~directions, the interchain exchange interactions will probably be weak.

Single crystals of CsMnC1 3.2H20 were grown at room temperature by

Fig. 4.1

CrystaUographic structure

of

_XMnY3

.2Hl.

evaporation from an aqueous solution, containing _{CsCl and MnC1 2 .4H20} of pro analysi quality in equimolar ratio [10]. In this way large single crystals are grown very easily. The crystals are usually shaped as rectangular pink coloured platelets, which are poorly

....

developed along the c direction (see figure 4.2). The crystals cleave easily perpendicular to this direction.

c

Fig.· 4.2

Morp

hology

of a sing

le cr

ystal of CsMnCl

₃

.2H

₂

0.

Single crystals of _CsMnBr3

.2H

₂₀ were grown at room temperature fxom an aqueous solution, containing _{CsBr and MnBr 2.4H20} in the molar ratio 1:6

[7,8]. The crystals display the same morphology as CsMnC1 3 .zH2

o.

Single crystals of a-RbMnC1 3.2H20 were obtained at 4°C from a solution of RbCl and MnC1 2.4H20 in hydrochloric acid in the molar ratio 1:5 [II]. Two crystal modifications precipitate from this solution, an orthorhombic and a triclinic one called a-RbMnC1_{3.2H20 and B-RbMnC13.2H2}

o

respectively. Large crystals of the a compound are obtained by a slow transformation of the B phase into the a phase, which is stable at about 0°C. The crystals grow in the shape of thick needles, frequently twinned and sometimes

hollow. Moreover, the crystals are afforescent.

4.3 Magnetization Pesults.

4.3.1 Field and tempePatUPe dependence.

The magnetization has been measured with a Faraday balance at liquid helium temperatures at constant field strengths and for various direct-ions of the external field in the ab and be planes. The rotation dia-grams of the magnetization measured at low fields, show minima at the

b

axis and maxima at the~ and~ axes. This angle dependence indicates ~

that the b axis is the preferred direction of the magnetic moments.

The _{magnetization of CsMnC1 3.2H20} at T

=

1.1 K, measured with the external

~ ~ ~

field, H, along the a, b and c axes, is presented in figure 4.3. The ~

magnetization, ob, measured with H along the b axis exhibits a spinflop transition. Taking the spinflop field, Her' as the field at which the slope in the ob versus H curve is maximal, we obtain Her 17.3 kOe at T

=

1. I K. For H > H , ab lies below the perpendicular magnetization

cr ~ ~

measured with the field parallel to the a and c axes. Due to lack of data far above Her' the high field magnetization cannot be extrapolated properly through zero at zero field. The perpendicular magnetization

~

measured along the a axis is about five per cent larger than that ob-tained with H along

~.

From this the conclusion is drawn that the

~

and ~

a axes are the next preferred and hardest directions for the magnetic moments respectively.

~ ~

The magnetization of _{a-RbMnC1 3.zH20} measured at I. I K along a and b axes, is shown in figure 4.4. ob exhibits a spinflop transition at about 14.6 kOe. The fact that a does not properly extrapolate through zero at zero

a

field may be due to misorientation of the sample, since the irregular surface of the crystals prevents proper mounting.

10 2.5 E ~ ~2.0

"

E

"'

b 1.5 1.0 0.5 0 5 20 H ( kOel

Fig. 4.3 The magnetization, a, of a 87.7 mg single crystal + 7 + of CsMnCl 3.2H20 measured along the a, o and c axes at T

=

1.1 K.

H//~triangles;

H/Jb-circles; H/1;-squares.

wr

I -J.S

f-!

i

10~ I

I

25~

I j

~

~ ' . -; 2or E i ~ b 0 10 15 20 H!kOe)

Fig. 4.4 The magnetization,

o,

of a 15.9 mg single crystal

of a-RbMnC_{l 3.2H2}

o

measured along the~ and b axes

The magnetization of CsMnBr 3.2H2

o

at T

=

1.1 K obtained along the prefer-red and next preferprefer-red directions is shown in figure

4.5.

Along the

+

b axis a spinflop-like transition takes place at about 21.3 kOe. Since

E

I.S ~ rn ~ E ~ b tO

ns

15 H<kOel

Fig. 4.5 The magnetization, a, of a 46.2 mg single crystal of _{CsMnBr3.2H20 measured a}long the~ and b axes

at T

=

1.1 K. H//~-triangles; H//b-aircles.

the behaviour of crb with the magnetic field for H > H could not be

cr

observed in the limited field range of the magnet, more information about the nature of Her can be obtained from the temperature dependence

of the critical field. The static susceptibility _{of XMnY3}

.

z

H

_{20, obtained}

shows an antiferromagnetic behaviour (see figure 4.6). The open circles are the data points for Xa and the solid circles correspond to Xb· The curves given by I are the results for _{a-RbMnC1 3.2H20. Xb} is rather high

200 180 160 140 E ~ 120 :> E !!! 100 X (LJ ~ 80 60

::t

0 T (Kl

Fig. 4.6 The static susceptibility,

x.

_{of XMnY3.2H20 (}with

X

=

Cs, Rb and Y

=

Cl, B~) as a function of temp

er-atu~e, meas~ed along the ; and

b

axes (open and solid symbols respectively).

The data denoted by 1, 2 and 3 are the results fo~

a-RbMnCl3.2H20, CsMnCl3.2H20 and _{CsMnBr 3.2H20}

respectively.

with respect to curves 2 and 3 and does not extrapolate through zero at

zero temperature. This may be due to the already mentioned possibility of slight misorientation. The perpendicular susceptibility of _{CsMnC1 3.2H20,}

Xa' (curve 2, open circles) increases about 10 per cent as the temperature

decreases from 4.2 to 1.1 K.

is presented in figure 4.7. H increases with increasing temperature

cr '

which is in agreement with the usually observed temperature dependence of a spinflop transition. For CsMnC1 3.2H20 the data points between I K and 4 K can be presented by H

=

16.4 + 0.84 T, where H is given in

cr cr 25

----

---Fig. 4.7 I !Kl

Temperature dependenee of the eritieaZ fields, H , er for XMnY 3. 2H

i·

kOe and T in degrees K. This temperature dependence agrees rather well with the results of Butterworth and Woollam [12], who applied

magneto-thermal techniques to obtain the. magnetic phase diagram.

Susceptibility and critical field values are summarized in table 4.2. Since the spinflop field of CsMnBr 3.2H20 could only be observed below 2.6 K with the Faraday balance, because of the limited field range of the magnet, the temperature dependence of the spinflop field above 2. 6 K has been obtained from dynamic susceptibility experiments in a super-conducting magnet. In this way the spinflop field, observed as a peak in the

x'

versus H curves, has been followed to about 5 K.

A phenomenological expression for the spinflop field has been derived from the equilibrium between the decrease of the magnetization energy

TN(K) H (kOe) 6 6 cr 10 X;;<emu/gram) 10 X.L(emu/gram) CsMnC1 3.2H20 4.89 17.3 8.4 142 a.-RbMnC1 3.2H20 4.56 14.6 38 175 CsMnBr 3.2H20 5.75 21.3 8.2 110 Table 4.2. ....

Experimental values for Her'

X;;

and X.L (H//a) at T I. I K. In the first column TN is given.

and the gain in anisotropy energy at the spinflop field (see equation

2. 14)

(4. I)

Here HA is the anisotropy field and M the sublattice magnetization given by !CN/Mw)g~B<S>, where N is Avogadro's number and Mw the mole-cular weight. Substituting the values for x.L' X;; and Her at T

=

1.1 K from table 4.2 in equation 4.1, the corresponding values for the ani-sotropy fields for XMnY 3.2H20 have been calculated. They are presented in table 4.3. In these calculations the full magnetic moment of five Bohr magnetons has been taken.

HA(Oe) zjJj/k(K) ~(kOe) a

CsMnC1 3.2H20 474 7.99 297 ! J.6xl0 -3 a-RbMn<:l_{3 • 2H2}0 290 7.58 282 I.OxiO -3

CsMnBr 3.2H20 770 7.33 273 2.8xl0 -3

j

Table 4. 3.

Values for HA~ zjJj/k, ~ and a HA/HE, where the experimental values from table 4.2 are used.

4.3.2 AngZe dependence.

The angle dependence of the magnetization of XMnY3.2H2

o

has been studied at several magnetic field strengths. Rotation diagrams measured at T

=

1.1 K with the magnetic field in the ab plane are shown in figures

:J E -"! b 10 1.0 0 -120 a -90 -60 -30 b 0 30 0 <degrresl 60 H <kOeJ I 16.92 2 17.71 3 19.08 4 20.23 5 21.20 6 22.00 a 90 120

Fig. 4.8 Rotation diagrams of the magnetization, a, of the 87.7 mg singZe arystaZ of CsMnCZ 3.2H20 with the fieZd, H, in the ab plane at T

=

1.1 K.

4.8, 4.9 and 4.10. The three compounds show a similar angular behaviour. The magnetization behaves cosine-like for H < Her' with maxima and minima at the ~ and

b

axes respectively. See for example the curves denoted by I in the figures 4.8, 4.9 and 4.10.

I f the magnetic field exceeds the spinflop field, a relatively peaked maximum in the magnetization appears at the low fields and 8 0 (figures 4.8 and 4.9, curve 2; fig. 4. 10, curve 6), transforming into an essentially flat maximum if the magnetic field strength is much larger than Her" In that case the magnetization remains constant over a large angular

inter--> val around the b axis.

At a certain critical angle, 8cr'measured ~rom the

b

axis, taken as the angle at which the slope in the

a

versus 8 curve is maximal, the

mag-H<kOe> 1 12.86 2 15.16 3 16.61 I. 18.56 5 19.22 6 20.37 7 21.31. 8 22.47 a b a 0 -120 -00 -60 -30 0 30 60 90 120

a

(degrees)

Fig. 4.9 Rotation diagrams of the magnetization, a, of the 15.9 mg single crystal of a-RbMnCZ_3.2H20 in the

ab

plane at T

=

1.1 K.

~

·,----.---.---,----.---,----.---.---.---,----,---,--, 2.5. HCIIQe) 60 90 0 !degrees/ 120 I 6.90 2 IO.IS l 1S.16 I. 16.S6 I 10.37 6 21.34 7 21.74 8 22.12 9 22.81

Fig.4.10 Rotation diagram of the magnetization,

a,

of the 46.2 mg single crystal of CsMnBr_3.2H2

o

in the

ab

plane at T

=

1.1 K.