A P ro to n M agnetic R eson an ce S tu d y o f th e O ne
D im ensional A ntiferrom agnetic S y stem
ACCEPTED
C sFeC l3-2H20
F A C U L T Y OF
GRADUATE STUDIESby
-T3— ... Michael J. Rensing
/CQn B.Sc., University of Victoria, 1982
^ATE— ... ■ ... ■■■— ■■■■ - M.Sc., University of Toronto, 1984
A D issertation Subm itted in P artial Fulfillment of the Requirements for the Degree of
D O C TO R O F PHILOSOPHY
in the D epartm ent of Physics and Astronomy
We accept this dissertation as conforming to the required stan dard
Dr. A. W atton, Supervisor (D epartm ent of Physics and Astronom y)
Dr. H.W. Dosso, D epartm ental M ember (D epartm ent of Physics and Astronomy)
Dr. Pl.M. Clements, D epartm ental M emfefif^Pepartment of Physics and Astronomy)
Dr. K.R. Dixon, Outside-NRmBer (D epartm ent of Chem istry)
Dr. D.E. Hewgill, O utside fyjfem^er (D epartm ent of M athem atics)
Dr. E.D. Rogak, A d d itio n ^ M em b er (D epartm ent of M athem atics)
Dr. H. Peemoeller, External Exam iner (University of W aterloo) © M ICHAEL J. RENSING, 1990
University of V ictoria
All rights reserved This thesis may not be reproduced in whole or in p a rt, by mimeograph or other means, w ithout the permission of th e author.
A bstract
Proton line shapes and sp in-lattice relaxation tim es were acquired for th e easy- axis antiferrom agnet CsFeCl3-2 H 2 0 . T he line shapes were acquired at '250 X for a complete 360° rotation abo ut th e b axis of the crystal, and a t various tem peratures between 250 K and 23 K, for tw o orientations.
A m odel for th e line shapes h as been created based on the tem p eratu re dependance of the interaction between protons and th e local m agnetic field differences a t each proton site. The field differences lead to sym m etric line shapes whose peaks have a separation which increases as th e tem perature decreases. T his model accounts well for the m ajor com ponent of tile observed line shapes, provided th a t th e proton-proton separation is increased over th e separation obtained from published d a ta . T h e residual portion of the observed line shapes appears to be a single line which results from water molecules which behave as though they are not p art of th e crystal stru ctu re.
A simple stochastic model for the behavior of 7r-kink solitons has been devel oped and used to predict values for the proton spin-lattice relaxation tim es. In this model, th e solitons are tre a te d as infinitesimal non-interacting particles travelling along straight chains of Fe2+ ions at constant speed. T he arrival of solitons at a random ly chosen point on th e chain is a Poisson process which leads to an expression for the spin-lattice relaxation whic agrees well w ith observations.
A B S T R A C T iii
Examiners:
Dr. A. W atton, Supervisor (D epartm ent of Physics and Astronomy)
Dr. H.W . Dosso, D epartm ental M ember (D epartm ent of Physics and Astronomy)
Dr. R.M. Clem ents, D epartm ental Member (D epartm ent of Physics and Astronomy)
Dr. K.R. Dixon, O utside M em b ertD ep artm en t of C hem istry)
Dr. D.E. Hewgill, O utside M em ber (D epartm ent of M athem atics)
. . . .-_7
Dr. E.D. Rogak, A dditional M em ber (D epartm ent of M athem atics)
A bstract
ii
Table o f Contents
iv
List o f Tables
vii
List o f Figures
viii
List of Symbols
ix
A cknowledgem ents
x
1
Introduction
1
1.1 M agnetic S y s te m s ... 1 1.2 CsFeCl3-2H20 ... 2 1.3 S o l i t o n s ... 5 1.4 M o tiv a tio n ... 72
Experim ental M ethods
8
2.1 Growing th e C r y s t a l s ... . 82.2 C W E x p e r im e n ts ... 11
2.2.1 C alculating th e Line S h a p e s ... 15
2.3 P ulse E x p e r i m e n t s ... 15
2.4 C h ap ter Sum m ary ... 20
C O N T E N T S v
3
Experimental! R esults
21
3.1 CW Line Shapes ... 21
3.1.1 T he O rien tation Dependance of the Line S h a p e s ... 21
3.1.2 T he T em p eratu re Dependance of the Line S h a p e s ... 23
3.2 T he T, V a lu e s ... 26
3.3 C h ap ter Sum m ary ... 28
4
Theoretical A nalysis and Discussion
30
4.1 T h e Line Shapes ... 304.1.1 Fine S tru c tu re o f th e Proton L i n e s ... 30
4.1.2 T he M agnetic Interaction ... 35
4.1.3 Line B roadening ... 38
4.1.4 T he A dditional W a t e r ... 42
4.1.5 The Total Line S h a p e s ... 44
4.2 T h e Spin L attice R e l a x a ti o n ... 45
4.2.1 Soliton R e l a x a t i o n ... ... 45
4.2.2 T he Arrival Frequency of The S o l i t o n s ... 51
4.2.3 T he F lu c tu a tin g Local F i e l d ... 52
4.2.4 T he M agnon M o d e l ... 54
4.3 C h ap ter Sum m ary ... 54
5 Conclusions
50
5.1 T h e Line Shapes ... 565.2 T h e Spin-Lattice R e la x a tio n ... 58
Preferences
60
A The Programm able P u lse Sequencer
62
B The Jackknife M ethod
64
List o f Tables
1.1 S tru ctu ral param eters of CsFeCl3-2H2 0 ... 3 2.1 X -Ray analysis of th e crystal stru ctu re of C s F e C ls ^ I^ O ... 9 4.1 F its to C \ / T + C2/ T 2 for the proton line splitting and th e line shift. . 38 4.2 P aram eters from model f i t ... 40
1.1 Schem atic representation of th e crystallographic stru c tu re... 4
1.2 A rray of th e m agnetic m om ents of CsFeCl3-2H20 in th e ordered state. 4 1.3 Morphology of single crystals... 5
2.1 Block diagram of the CW experim ent ap paratus... 12
2.2 T h e observed line shapes... 13
2.3 Block diagram of the pulse experim ent apparatus... 16
3.1 T h e 0° orientation line shapes... 24
3.2 T h e 63° orientation line shapes... 25
3.3 T h e 90° orientation line shapes... 26
3.4 T h e T\ d a ta for th e 0° o rien tation... 27
3.5 T h e Tj d a ta for th e 90° o rientation... 28
4.1 Line displacem ents for fixed a, and various choices of 6... 32,
4.2 T h e spherical coordinate system ... 33
4.3 Line separations... 34
4.4 T h e tem perature dependance of th e line splitting... 37
4.5 Two line shape com ponents for th e 0° orientation at two tem peratures. 42 1.6 Tw o model fits for th e 0° orien tation... 44
4.7 Two model fits for th e 90° o rien tatio n ... 45
List o f Sym bols
D . . . m agnetic anisotropy param eter G(t) . . . correlation functionh . . . P lanck’s co n stan t divided by 2tt
Hchain • ■ • H am iltonian of the iron spin system 7 i2 . . . Zeem an H am iltonian
Ho . . . applied m agnetic field J , Jij .. .exchange integral J ( u ) .. .sp e c tra l density k . . . B oltzm ann’s con stant M (t) . . . m agnetization
r,j . . . inter-nuclear distance S, S,' . . . nuclear spin vector
Six, Siv, S{t • • • com ponents of the i th nuclear spin T\ . . . nuclear spin-lattice relaxation tim e
/? . . . Bohr m agneton
7 . . . nuclear gyrom agnetic ratio A . . . arrival frequency of solitons ft . . . m agnetic m om ent
<f>m . . . canting angle of th e iron m agnetic moments
1 would like to thank a num ber of people who have m ade it possible to com plete this dissertation.
First, I have to thank my supervisor, Dr. A rthur W atton, for financial and m oral support. A rthur always was encouraging, and our many discussions were always helpful.
K athy Beveridge and Dr. Gordon Bushnell were of invaluable assistance by ob taining and analysing the X-Ray d ata o f th e crystals.
Neil Honkanen of the Physics and A stronom y Electronics Shop built and tested th e Program m able Pulse Sequencer.
The te x t of this dissertation has been typeset using lATgX on the Apollo com puter system belonging to th e University of V ictoria High Energy Physics Group.
C hapter 1
Introduction
1.1
M a g n etic S y stem s
B oth ferrom agnetic and antiferrom agnetic m aterials are system s of m agnetic spins a: ranged w ith some kind of order. The interaction between th e spins wh’ch creates this order is known as th e exchange interaction. The Heisenberg model for the excharge interaction between two spins i and j gives an energy
U = -Uj Si -Sj ,
where is called the exchange integral, and is related to th e overlap of th e wave functions of th e electrons on atom s i and j . Both ferrom agnetic and antiferrom agnetic systems can be described by this equation. For ferrom agnetic system s, J,j > 0, and th e m inim um energy configuration is then achieved when th e spins all align parallel to each other. For antiierrom agnetie systems, < 0, th e m inim um energy configuration is achieved when the spins are aligned in an antiparallel fashion, so th a t the net m agnetic moment for th e system is zero.
Some m agnetic systems are more easily magnetized along certain axes th an along others. T he energy which causes this is called the anisotropy energy. One model which is used to describe systems with anisotropy is
u = d '£i s]z, )
where D is th e anisotropy param eter. This can describe two types of system . If D > 0, the Ham iltonian is a m inim um for S jz —» 0, and we have an easy plane of magnetization. If D < 0, the H am iltonian is m inim um w hen is m axim ized, and we have an easy axis (the z axis, in this case). T he te rm “easy plane” is used to describe a m agnetic system whose m agnetization is g reatest when th e applied field lies anywhere in a certain plane. T he term “easy axis” is used to describe a m agnetic system whose m agnetization is greatest when th e field is applied along r>ne axis.
T he m agnetic system produces local fields which vary throughout th e crystal. These local fields give rise to th e fine structure of line shapes of nuclei within th e crystal. This will be discussed in detail in Sec. 4.1
There are a num ber of dynam ic effects which reduce th e order of perfectly aligned magnetic systems. One effect, whose quantization is known as a magnon, is caused by th e motions of the spins in a wave. This is an oscillation of th e relative orientation of th e spins on a lattice. T he term is analogous to th e te rm phonon, used to describe the quantization of the oscillations of th e atoms on a lattice. Since the magnons in a lattice are associated w ith changes in the spin orientations, they produce fluctuating fields w ithin th e crystal, which in tu rn can produce relaxation. A nother excitation associated with th e motion of th e spins is the soliton. T h b is a non-linear d isturbance which causes th e orientation of th e spins to change by large am ounts over a relatively small distance. T he motions of th e solitons also produce fluctuating fields which can produce spin-lattice relaxation. Solitons will be discussed in Sec. 1.3.
1.2
C sFeC l3-2H20
The crystal CsFeCl3'2H2 0 belongs to the series of isomorphic transition-m etal halides A M B y2a.q (A = Cs,Rb; A f=M n,Co,Fe; i?= C l,B r; aq = H 2O, D2O). A great deal of work has been done on CsFeCl3-2H2 0 , and on the related com pound RbFeCl3*2H 2 0 , to study Mossbauer line widths (Le Fever et al., 1981b], spin-cluster resonance [Nijhof
C H A P T E R 1. IN T R O D U C T IO N 3 300 K 4.2 K i y z X y z Cs " l — 4 0 0.1481(17) l 4 0 0.1970(15) Fe 0 0.4695(9) 4l 0 0.4694.9) 4l Cl(i) 41 1 2 0.1506(9) 14 1 2 0.1514(7) Cl(2) 0.0894(6) 0.2263(7) 0.3880(6) 0.878(4) 0.2267(6) 0.3893(6) 0 0.0680(10) 0.6864(14) 0.3650(12) 0.0681(15) 0.6829(10) 0.3684(10) D ( l ) 0.0262(11) 0.6873(13) 0.4404(11) 0.0267(8) 0.6882(11) 0.4424(8) % ) 0.1716(12) 0.8903(13) 0.3842(11) 0.1721(10) 0.6962(11) 0.3317(9) a 8.9809(7)A 8.9466(6)A b 7.2132(5)A 7.1247(4)A c 11.3867(10)A 11.324c', 10)A
Table 1.1: S tru ctu ral param eters of CsFeCl3 2DpO. T h e values for C'sFeCl3-2H20 are assumed to be sim ilar enough h at these values can be used for theoretical calcula tions. 1 a stan d ard deviations based on statistics only are given in parentheses, in units of th e least significant digit. From Basten et al. [1978].
et al., 1983] [van VTmmeren and de Jonge, 1979] [van V lim m eren et al., 1980], specific heats [Kopinga, et al., 1985], neutron diffraction [Basten e t a l , 1978] [Smeets et al., 1985], and nuclear spin lattice relaxation tim es (Ti) [Tinus e t al., 1982].
T he crystallographic stru cture of C sF eC ls^ I^ O belongs to th e orthorhom bic space group Pcca (£>2/l), with the lattice param eters given in Table 1.1. There are four for mula units p er un it cell. The stru ctu re consists of cis-octahedra which are coupled along th e a axis by a shared chlorine ion (Fig. 1.1). T h ere is a strong exchange coupling betw een the neighbouring Fe2+ ions in this direction. The exchange inter actions in th e b and c directions are a t least two orders of m agnitude reduced from the exchange interaction in the a direction [Le Fever et al., 1981a]. T his means th a t the system behaves as a quasi-one-dimensional m agnetic system , so we can consider the Fe2+ system as chains of spins aligned along th e a axis. T h e chains of m agnetic Fe2+ ions are separated in the b direction by layers of Cs+ ions. In the c direction the chains are separated by layers of w ater molecules.
Figure 1.1: Schem atic representatio n of the crystallographic structure. This is the general stru ctu re of A M B 3• 2Ha0 . In this case, A = Cs, M = Fe, and B = Cl. Only one set of hydrogens and hydrogen bonds is shown. (From Le Fever e t al. [1981a])
Figure 1.2: A rray of the m agnetic m om ents of C sF eC la^H jO in th e ordered state. All m om ents are located in th e ac plane a t th e canting angle <f>m from th e a axis. (From B asten e t al. [1978])
C H A P T E R 1. IN T R O D U C T IO N
b
CFC RFC
Figure 1.3: Morphology of single crystals. T h e R bFeC l3*2H 2 0 (RFC) crystal shape is from Le Fever et al. [1981b], while th e CsFeCl3-2 H2 0 (C FC ) crystal shape is the observed shape of our crystal.
tem p eratu re of T/v = 12.6 K [Le Fever et al., 1981b]. For the tem p eratu re range in which all of these experim ents were conducted (30 K to 300 K ), the crystal is param agnetic, w ith th e m agnetic m om ents arranged as in Fig. 1.2. T he spins do not align along a principal axis. Instead, they lie in th e ac plane at a constant canting angle <f>m from the a axis. This will have th e effect of reducing the sym m etry so th a t th e m agnetic sym m etry will be near to P cca, bu t is expected to deviate somewhat. T h e canting angle is <j>m = 15° for C sF e C ls^ I^ O [Basten et al., 1978]. This canting angle m eans th a t each antiferrom agnetic chain has a n et ferrom agnetic moment in th e c direction, which is com pensated by th e m om ents of neighbouring chains.
Fig. 1.3 shows th e morphology of a single crystal of RbFeCl3*2H20 , as well as the observed m orphology of our single crystal of CsFeCl3-2 H20 . T h e sim ilarity of the two types of cry stal helped confirm th e identification of th e grown crystal.
1.3
S o lito n s
In the p ast th irty years, a great deal of interest has been generated in non-linear sys tem s. T hese systems exhibit stable disturbances which have th e following properties [Drazin 1983]:
1. T he disturbance is localized.
2. T he disturbance exists w ithout dispersion, i.e. th e disturbance is perm anent. 3. A num ber of disturbances interact so th a t after the interactions each disturbance
retains its form , w ith only a phase shift to show th a t the interactions have occurred.
These properties have led to th e coining of th e word “soliton” as a descriptive name for this ty pe of disturbance.
Physically, solitons can be observed in a num ber of situations. T he first recorded observation of a soliton was th a t of a solitary wave of w ater in a canal. J. Scott Russell [1844] reported th a t in 1834 he observed a boat stop suddenly, which resulted in “a rounded, sm ooth and well-defined heap of w ater, which continued its course along the channel apparently w ithout change of form or dim inution of speed” . Soliton behavior has since been observed in o th er shallow w ater system s, as well as in atm ospheric systems.
Since solitons exh ibit particle-like properties, they have been proposed as models for elem entary particles. Indeed, a num ber of particle field theories have soliton solutions, b u t these solitons would have such large masses th a t they are not yet observable.
Of particu lar in terest to us are antiferrom agnetic system s of spins arranged in chains which do no t in teract strongly with neighbouring chains. W ithin th e chains, the spins are coupled so th a t they form sequences of spins which altern ate in align m ent. Given certain assum ptions, Mikeska [1080] has shown th a t th e m otion of the spins can be described by w hat is known as a sine-Gordon equation. If the orientation of the spins is described in term s of a field <j>, then th e sine-Gordon equation in one dimension is of th e form
1 d2<f> 2 . .
C H A P T E R 1. IN T R O D U C T IO N
where c is th e characteristic speed, and m is th e mass. T he soliton solutions of this equation a /e known as kink solitons because th e orientation of the spins changes through ir over a relatively small num ber of atom ic spacings, creating a kink in the alignm ent of the spins.
1.4
M o tiv a tio n
This stu d y was undertaken because there were no studies of th e proton line shapes of CsFeCl3-2H2 0 , and it was felt th a t such a stu d y combined w ith T\ measurements would a d d to th e understanding of the interactions w ithin the crystal. It was hoped th a t th e soliton m odel for relaxation tim es [Mikeska, 1980] [Tinus et al., 1982] could be expanded, and th a t this model could be te sted further by comparing it to the m agnon m odel [Goto and Yan aguchi, 1981].
Since protons have spin S — 1/2, th ere are no quadrupolar interactions. This would have m ade th e analysis of th e line shapes extrem ely complex. The lack of possible q uadrupolar interactions simplifies th e problem by limiting th e possible in teractio ns to dipolar types of interactions. A nother advantage to observing protons is th a t th e y are th e nuclei which give th e largest N M R response, which is of great use in o btaining d a ta w ith a good signal to noise ratio in a reasonable experim ental time.
E xperim ental M ethods
2.1
G row ing th e C rystals
The crystals of CsFeCl3-2H20 were grown by slow evaporation of a solution of CsCl and FeCl2’5H2 0 dissolved in distilled w ater [Basten et al., 1978] [Smeets e t al., 1985]. The solutions were kept in an enclosed cham ber filled w ith nitrogen to prevent for m ation of iron oxides, which precipitate from the solution.
Several days after the solutions were first mixed, small crystals began to appear. These were allowed to grow until crystals of about one m illim etre in th e smallest dim ension appeared. T he most regular shaped of the crystals were th e n removed from th e solutions and kept for seed stock to grow larger crystals.
Some of th e seed stock crystals were sent to a commercial laboratory for chemical analysis, and proved to have too little chlorine for th e correct stoichiom etry. Also, a large num ber of these crystals did not conform to the description of th e macroscopic shape given by Le Fever e t al. [1981b], and a brief X -Ray analysis using a Nonius CAD 4-F diffractom eter showed th a t these crystals were triclinic. These crystals were dissolved in distilled w ater and allowed to form “second generation” crystals, again by slow evaporation. T h e new crystals did conform to th e macroscopic shape described by Le Fever et al. (see Fig. 1.3). Leaving the crystals to grow larger on th e bottom of the beakers m erely resulted in large masses of small crystals growing random ly on
C H A P T E R 2. E X P E R I M E N T A L M E T H O D S 9 M easured Expected a 8.94(3)A S.9809(7)A b 7.196(3)A 7.2132(5)A c 11.346(9)A 11.3867(10)A a 90°.22(5) 90° ft 90°.12(15) 90° 7 90°.18(12) 90°
Table 2.1: X-Ray analysis of th e crystal stru ctu re of CsFeCl3-2H20 . Expected values are from B asten e t al. [1978]. l a standard deviations based on statistics only are given in parentheses, in units of the least significant digit. T he measured values are based on a poor quality crystal. They are accurate enough to confirm the identity of th e crystal, but were not used in any calculations.
each other.
All of th e clean crystals from the “second generation” solutions were kept until a large enough supply was obtained to allow th e m to be re-dissolved in distilled water and produce a useful volume of solution from which more crystals can be grown. Sam ples of th e “second generation” crystals were sent for chlorine analysis, and proved to have th e correct percentage of chlorine. An X -ray analysis was performed on a Non ius CAD 4-F diffractom eter. A unit cell stru ctu re was determ ined using the routine SEA RC H , after which a transform ation of th e un it cell into meaningful param eters was done using th e routine TRANS. T h e final results agreed well w ith the published param eters (see Table 2.1), confirming the id en tity of th e crystal.
A la te r X-Ray analysis using a W eissenberg rotation photograph was not com pletely successful, b u t it did confirm th a t th e principal axes of th e crystal were per pendicular to th e crystal faces. A ttem pts were m ade to cleave the crystal along the macroscopic crystal axes. Only one plane proved to have easy cleavage, agreeing with Sm eets e t al. [1985], and van Vlimmeren and de Jonge [1979].
A concentrated liquor was obtained by dissolving th e small “second generation” crystals, and seed crystals were suspended from a thread in the liquor. The seed crystals in the “th ird generation” solutions were allowed to grow for several months,
while th e solutions were filtered and more “third generation” solution was added as required. Only one crystal grew successfully from four startin g seeds, but this one reached dim ensions of roughly 6 x 3 x 1 cm3 . One other crystal reached dimensions of roughly Ix 0 .5 x 0 .5 cm3 before it would not continue to grow consistently. This proved to be an excellent test crystal for the prelim inary work.
A few prelim inary experim ents were performed using this test crystal, in order to determ ine tu n in g procedures and experim ental param eters. These experim ents produced prelim inary line shapes which had th e sym m etry expected for a reduced sym m etry Pcca stru c tu re (see Sec. 4.1.1). Also, prelim inary T\ m easurem ents agreed with th e results published by Tinus et al. [1982].
To sum m arize, th e results from a num ber of tests confirmed th a t th e crystals were CsFeCl3-2H20 :
1. T h e morphology of th e crystals of CsFeCl3-2H20 agreed with published descrip tions of th e isomorphic crystal RbFeCl3-2H20 . This was a good indication th a t th e crystal had the same basic stru ctu re as RbFeCl3-2H20 , as it should have had.
2. T h e crystals cleaved easily along only one plane, agreeing w ith Smeets et al. [1985], and van Vlimmeren and de Jonge [1979]. If we oriented th e crystal on th e basis of Le Fever et al.’s picture, th e direction of cleavage was along th e ab plane, again agreeing with these au th ors’ descriptions.
3. rif i e X-Ray analysi-5 produced lattice param eters which agreed w ith th e p ub lished data.
4. T h e chemical analysis gave a chlorine content which agreed w ith th e expected content for CsFeCl3'2H20 .
5. C W experim ents indicated th a t the line shapes followed the p attern s expected for CsFeCl3-2H20 when the orientation was changed.
C H A P T E R 2. E X P E R IM E N T A L M E T H O D S 11
6. Pulse NM R experim ents (see Sec. 3.2) gave T x values which agreed well with those of Tinus et al. [1982].
It was found th a t the ciystalline stru c tu re was quite easy to destroy. Drying a crystal by enclosing it in a container over a desiccant, by placing it in a vacuum, or by blowing dry nitrogen gas over it all caused th e crystal to become an opaque yellow colour. W hen the opaque crystals were cut open after a short drying tim e, it was found th a t th e interiors were still clear, and apparently crystalline. Longer drying tim es produced crystals which were opaque throughout. CW experim ents were perform ed on these crystals before and after drying. T h e line shapes from the crystal showed definite stru ctu re before drying, whereas after com plete drying of the crystal, only one narrow line was observed, indicating th a t th e crystalline structure was destroyed by th e drying process.
2.2
C W E xp erim en ts
P roton line shapes for CsFeCl3-2H20 were obtained using a Spin-Lock MO-lOO m ar ginal oscillator operating at 30 MHz. T he resonant m agnetic field was applied by a Varian electrom agnet. A low level 100 Hz m agnetic field was added to the resonant field by a pair of Helmholtz coils. This caused a m odulation a t the sam e frequency in the response from the sample. This response was used as input to an Ithaco D ynatrac 391A lock-in amplifier, which d etected and filtered th e signal. Because of th e m odulation field, the signal measured was actually th e derivative of the line shape centred about the tim e average of th e to tal applied field. T he signal was recorded using two different systems.
The prelim inary d ata were recorded using an eight-bit analogue to digital converter interfaced to a PC -X T. The com puter would record severa' scans, and store th em on disk. As there was concern th a t there would be some drift in the field a t which the scans would begin, th e experim enter would align the scans using s o r e clearly defined
Magnet Helmholtz coila Oacillator Power supply Scan S ta rt Senaor Lock-In Amplifier Marginal Oacillator Sample
and Coil Computer
Figure 2.1: Block diagram of th e CW experim ent app aratus.
feature of th e derivative line shape. T he com puter would then average th e d a ta and store it on disk.
The final d ata were recorded using the Tektronix 2230 D igital Storage Oscil loscope. An accurate trigger was built to detect th e s ta rt of each scan, and the oscilloscope external clock function was used. T he P rogram m able Pulse Sequencer (see A ppendix A) was used to generate an external tim e base for th e oscilloscope, so th a t each scan of th e m agnetic field corresponded to one scan of th e oscilloscope. At th e end of each scan, the derivative line shape d a ta were read from th e oscilloscope and stored on a m agnetic disk. At some later tim e, th e experim enter would use a com puter program to align and sum the scans, and th en in teg rate th e derivative line shape to produce th e actual line shape. A block diagram of th e experim ental setup is shown in Fig. 2.1.
Using th e test crystal, proton line shapes were obtained w ith the applied m ag netic field in the ac :ne. The crystal was rotated abo ut th e b axis, and a line shape was obtained every ten degrees for 360 degrees startin g from an arb itrary orientation (Fig. 2.2). T h e sym m etries observed (see Sec. 4.1.1) agreed w ith th e expected behav ior for th e theoretical line shapes of CsFeClau'HjO, sup porting our identification of the structure.
Several experim ents were performed by reversing th e direction of the scan to confirm th a t there were no distortions of th e line shapes resulting from th e scan
C H A P T E R 2. E X P E R I M E N T A L M E T H O D S
380
360
340
320
300
280
^ 2 6 0 tn8 240
Z * 2002 180
d 160
I 140
S 120
100
80
60
40
20
20
30
- 1 0
- 3 0
-2 0
A Field (Gauss)
Figure 2.2: T he observed line shapes. T he orientations are relative to an arb itrary 0° orientation. T he line shape baseline corresponds to th e orientation, while the am plitude is in arbitrary units.
direction. No such distortions were observed.
To ensure th a t the rate of change of th e m agnetic field did no t cause distortions of the line shape, the scan rate was increased until distortions were observed. The scan rate used was one order of m agnitude less than th e ra te a t which th e distortion first appeared. A similar m ethod was used to ensure th a t th e filtering tim e constant did not introduce distortions. T he tim e constant was increased u ntil distortions were observed. The tim e constant used was one order of m agnitude less th a n th e value for which the distortions were first observed. These checks were perform ed before beginning the acquisition of each line shape.
Once a num ber of prelim inary experim ents had been perform ed on th e small sample crystal to determ ine some of th e experim ental param eters and procedures, a sample was cut from the large crystal so as to allow th e sam ple to be ro tated ab o u t th e b axis. X-Ray back scattering confirmed th a t th e sam ple was cut so th a t th e sam ple could be aligned to w ithin 5° of the b axis, b u t it was not possible to shape th e crystal in order to orient it more precisely, because of its b rittle n atu re . A large enough piece of the crystal survived the a tte m p t a t shaping to be an excellent sam ple for fu rther experim ental study. Unfortunately, th e small te st crystal used previously, which was also an excellent sample, was destroyed in an a tte m p t to rem ove un attach ed w ater molecules. These were suspected to be present on th e surface and a t defect sites in the crystal because of otherw ise unexplained features in th e observed line shapes, (see Sec. 3.1 and 4.1.4)
Line shapes were obtained from th e cut sam ple for a tem p eratu re range of 250 K to 23 K. At 23 K th e signal disappeared. This disappearance can p/obably be a ttrib u te d to the signal broadening enough for th e intensity to fall below th e noise level of th e spectrom eter. T he discussion of the CW line shapes will be continued in Sec. 3.1.
C H A P T E R 2. E X P E R IM E N T A L M E T H O D S 15
2.2.1 Calculating th e Line Shapes
In order to observe th e line shapes directly, th e derivative line shapes were integrated using th e formula
where A is th e spacing between consecutive points in th e d ata, and N is th e number of points in the derivative line shape. W hen y; goes to zero at ± o o , this is equivalent to
K - A ( 7 + S " + t ) -
( 2 .2 )Eq. (2.1) was used instead of th e m ore usual Eq. (2.2) because th e noise in th e signal causes th e error in Yn calculated from Eq. (2.2) to increase as n increases. If Eq. (2.2) was used, the baseline of the in teg rated line shape showed a definite slope because of the errors in the d a ta points of th e derivative line shape. T he errors result from two sources; the random m easurem ent errors, and the system atic errors resulting from the imprecision in th e definition of th e baseline. If Eq. (2.2) is used, th e random errors create a random walk which m anifests itself as an offset in th e baseline of the integrated line shape. W hen Eq. (2.1) is used, th e random walk problem is elim inated, the offset of th e baseline of the integrated line shapes is greatly reduced, and the slope of the baseline will be the result of th e baseline im precision of the derivative line shape only.
2.3
P u lse E x p erim en ts
Spin-lattice relaxation tim es (Tt ) for th e proton system in CsFeCla ^HaO were ob tained using a Spin-Lock coherent pulse spectrom eter w ith a 30 MHz probe. A block d> igram of th e ap paratu s is shown in Fig. 2.3. T he pulse sequences used were con- tolled by the Program m able Pulse Sequencer (P P S) described in A ppendix A. The
Pulse Sequencer Computer Spin-Lock Spectrom eter Power Supply
Figure 2.3: Block diagram of th e pulse experim ent apparatus.
free induction decay (FID) d a ta were recorded using a T ektronix 2230 Digital Storage Oscilloscope.
T he signals were detected using th e diode d etect m ode instead of th e phase detect mode. This m eant th a t the signals were rectified, thereby improving the stability of the experim ents by reducing th e sensitivity of the system to small drifts in the m agnetic field. This also elim inated th e phase coherence inform ation in th e signal, simplifying th e tuning procedure. T he gain of th e system was adjusted so th a t the signal was m easured in the linear portion of th e diode d etecto r response curve. The dead tim e for the system was 6/rs.
It was decided th a t a § -t- | pulse sequence would be used to produce a mag
netization recovery curve ( M ( r ) ) from which T\ could be m easured. (A | pulse is a pulse which causes the m agnetization in th e ro tatin g fram e to ro tate through an angle of f radians.) The reason was th a t, because the m easured values of T\ were so short, a continuous | pulse satu ratio n sequence would n ot allow us to measure the FID baseline accurately. A fter some experim entation w ith pulse lengths, it was found th a t pulse lengths greater than 5/ts introduced a d isto rtio n in th e shape of th e measured m agnetization recovery curve. For this reason it was decided to use a pulse length of 5 /j s. Shorter pulse lengths did not produce any m easurable change in the m agnetization recovery curve. This m eant th a t th e pulses used may have produced less th a n a \ ro tatio n of the m agnetization b u t because of th e w idth of the resonance,
C H A P T E R 2. E X P E R I M E N T A L M E T H O D S 17
creating a | pulse for all th e spins was not possible anyway.
As a result of th e complex n a tu re of th e proton line shapes, the FID curves of the protons were also complex. T h a t is, they were not simple exponentials. Although there was some evidence th a t different p arts of the FID relaxed with different tim e constants, th e differences were so slight th a t it was impossible to separate any dif ferences from th e noise in th e signals. It was decided to treat the entire FID as the response of a nuclear spin system relaxing to the lattice as an exponential w ith a single tim e constant, I \ . It was assum ed th a t the pulse lengths were negligible with respect to th e spin-spin relaxation tim e, so th a t there was no evolution of th e spin system during th e tim e th a t th e pulses were applied. In other words, it was assumed th a t th e m agnitude of the m agnetization did no t change while the pulses were being applied.
T he sho rter-th an -1 pulses produced a m agnetization, measured im m ediately after the end of th e second pulse, given by
M ( r ) = (M (oo) - ' ' - e ' T/Tl) + M(0)
where r is the tim e between th e two pulses, M (t) is the measured m agnetization, M( oo) is th e equilibrium m agnetization for the system , and Af(0) is the residual m agnetization. If th e pulses were exactly | , we would have Af(0) = 0. T x is the spin-lattice relaxation tim e for th e system.
It was found th a t the PPS. com bined w ith the signal averaging capabilities of the digitizing oscilloscope, allowed m easurem ents of T\ to be m ade over a much larger range and to a g reater accuracy th a n was previously possible. However, two problem s im m ediately becam e apparent.
T he first problem was a result of th e discrete n atu re of the d a ta from th e oscil loscope. If the noise level of th e averaged signal is low enough to be less th a n one digitization interval, th e error is no longer normally distributed. Instead, th e m axi mum error for each point is a co nstant equal to one digitization interval. T his can
lead to problem s when try in g to estim ate the overall erro r for param eters which have been determ ined using curve fitting routines, since these routines are used with the assum ption th a t th e errors in th e d a ta are distributed according to a normal distribu tion. A fter some study, this problem was ignored because th e error was so small th a t no significant consequences occurred by assuming a norma) distribution of errors.
A suggestion has been m ade th a t this problem could be circum vented by generat ing artificial electrical noise w ith known param eters and adding it to th e signal being digitized. This m etho d m ay be im plem ented if it becomes necessary, b ut to date it has not been used.
The second problem was much m ore significant. U ntil com puter acquisition was added to th e experim ental procedure, the m ethod of finding values for 2\ consisted of p lo ttin g th e m agnetization as a function of r on sem i-logarithm ic paper, and then fitting a line by eye. This leads to a (visually) weighted fit, where the slope of th e line was 1 / T \ . As a num erical fit is much more accurate, th e com puter was programm ed to find T\ autom atically. This requires a more rigorous approach to the error analysis, so v/e will proceed as follows.
M ( oo), M ( r ) , and M (0) all have m easurem ent errors associated w ith them , but th e error in Af(oo) and M ( 0 ) will be constant and independant of r for each mea surem ent of T\. M (t), however, is rem easured for every value of r . This means th at
the error in M (t) is the only one which introduces an error which depends on r . This could affect a least squares fit for Ti. If we w rite the erro r in A f(r) as e ^ , we have
- In
I
* (0)J
U M (o o ) - M (
0 )J
L M(oo)-= [M (o o )- M ( r ) ] [ _____sm;____
C H A P T E R 2. E X P E R I M E N T A L M ETH O D S 19
~ 1n p / ( o o ) - M ( r ) 1 eM ~ UW(oo) - Jlf(0)J M ( o o ) - M ( r ) ’
where th e last te rm is th e portion of the error in y ( r ) which depends on r.
We can see th a t these errors are not constant for each value of r , and th a t th e errors in M ( r ) should b e weighted by l/(A /(o o ) — A /(r)) in order to obtain an accurate fit for y (r ) . T he weighting results in errors for y ( r ) which do not have th e same normal distribution. T h e least squares method used to make a linear fit to th e d ata depends on all of the errors having th e same normal distribution for different r in order to calculate the variance in th e fit. Even if the in pu t errors are weighted to obtain the correct value for 1 /1 \, a correct value for the variance of 1/T\ will not be obtained. Because of th e increased range of pulse spacings for which it was possible to make m eaningful observations, this became a significant problem . A lthough th e values for 1 /T i appeared to be accurately fitted, a visual exam ination of the plots of th e d ata showed large deviations from th e fitted line for long values of r .
A d a ta resam pling scheme known as the jackknife fit (A ppendix B) allows this problem to be avoided, b u t th e visual fit is still not inform ative. For this reason it was decided to use a x 2 fit t ° th e un-transformed data. T he actual curve fit to the d a ta was
y = A[ 1] ( l - e - ^ W ) + A[3] (2.3)
where A [l] = (M (oo) - M (0 )), A[2] = Ti, and A[3] = A/(0). This m ethod has the benefit of producing a m eaningful covariance m atrix for th e three param eters being found, as th e errors of th e in p u t d ata have the same norm al distribution for all r . Also, as th e errors should all have the same distribution, th e visual fit of the plot shows any problem s in th e inp ut d a ta more clearly. Unfortunately, since th e m agnitude of the in p u t error is not known, th e actual variance of T\ cannot be calculated from the fit, and th e jackknife fit m ust still be used to get a realistic estim ate of the error.
Ti values were found for CsFeCl3-2H2 0 at various orientations and tem peratures. The discussion of th e T\ d a ta will be continued in Sec. 3.2.
2.4
C hapter Sum m ary
• Tw o useful crystals of CsFeCl3-2H2 0 were grown. The small one was used as a te s t crystal for prelim inary work. T he large one was cut to provide th e sample on which th e m ajor portion of this thesis is based.
• T h e identification of the crystals was confirmed by a num ber of methods. • C W experim ents were performed to o btain derivative line shapes at a num ber
of orientations at 250 K, and at a num ber of tem peratures for two orientations. T h e derivative line shapes were integrated using a numerical m ethod which allowed for statistical errors in th e data.
• P ulse experim ents were performed to o b tain protofa spin-lattice relaxation tim es a t a num ber of tem peratures for two orientations.
Chapter 3
Experim ental R esults
3.1
C W L ine S h ap es
Im m ediately after th e te st crystal was grown, a few line shapes were acquired as described in Sec. 2.2, b u t w ithout th e digitizing recorder or signal averaging. The inform ation obtained was enough to confirm th a t the crystal appeared to be a sin gle crystal, and th a t the line shapes did have some structure. The crystal was then stored for about a year while th e large crystal was grown, and the new d a ta acquisi tion equipm ent was developed. W hen experim ents were resumed, the new line shapes appeared to have been modified by the addition of another peak. T h e protons pro ducing this peak were assum ed to have been absorbed from the environm ent while the crystal was stored. This is why a num ber of attem p ts were made to d ry th e crystal. T he w ater molecules containing th e protons producing this additional peak will be referred to as “additional w ater” . This will be discussed a t a num ber of points in the following text.
3.1.1
The Orientation Dependance of the Line Shapes
T he first set of line shapes were acquired from the test crystal as described in Sec. 2.2. T he crystal was ro tated abo ut its b axis with th e m agnetic field applied in th e ac plane. Line shapes were acquired every ten degrees starting from an arb itrary orientation.
T h e tem perature was held at about 250 K since th e oscillator would n ot operate above this tem perature. T h e line shapes are shown in Fig. 2.2. From these line shapes, a num ber of observations can be made:
• T he line shapes are quite complex, and can change dram atically over quite small ranges of orientation.
• T here is a two-fold rotational sym m etry, in th a t th e line shapes 180° ap art appear to be th e same. T he two-fold sym m etry is predicted by th e Pcca sym m etry as a reduction of a four-fold reflection sym m etry, bu t th e full reflection sym m etry is not apparent. This could be a ttrib u te d to an internal magnetic field created by th e canting of the spins as described in Sec. 1.2, which reduces th e symmetry. Not enough d ata have been acquired to d ate to prove this, but if one examines Fig. 2.2 closely, one can see sim ilarities in line shapes which indicate th a t th e m agnetic stru cture has near four-fold sym m etry. This agrees w ith the fact th a t th e canting angle is small.
• T he num ber of peaks appearing in the line shapes varies between one (e.g. 80°) and five (e.g. 30°). W hile it is not surprising th a t there are orientations where an odd num ber of peaks is o b se r d, it is su rp rsin g th a t the m aximum num ber of peaks is odd. If there are no quadrupolar interactions (and there are none), the dipole-dipole interaction aicne would produce pairs of peaks. These could blend together to produce an odd , .umber of peaks, b u t it would be very im probable for the maximum num ber of peaks observed to be the result of such a blend. One explanation which is more plausible is th e additional water further discussed in Sec. 4.1.4.
• An asym m etry can be observed in th e line shapes. To verify th a t this was not an artifact of the experim ental techniques, th e direction of th e magnetic field scan was reversed, and the equipm ent settings were checked. This asym m etry is of particular interest because, as will be shown later, th e absorption line
C H A P T E R 3. E X P E R IM E N T A L R E SU L T S 23
shape should be sym m etrical about th e centre of th e line. Again, a plausible explanation is th e additional water discussed in Sec. 4.1.4.
T he last two points are complications which indicated th a t the dependance of the line shape?, on tem p eratu re would have to be studied next. Because of the great length of tim e which would have been required to repeat th e orientation study at a num ber of different tem peratures (each line shape requires ten hours), it was decided to lim it th e tem p eratu re experim ents to three orientations:
1. T h e orientation for which the line profile is th e sim plest (a single peak). For reasons which will become apparent in the theory section, this was called the 0° orientation.
2. T he orientation perpendicular to the 0° orientation, where the line shape has only two peaks. T his was called the 90° orientation.
3. An orientation 27° from the two peaks orientation, w here th e next sim plest line shape appears. This was called the 63° orientation.
3.1.2
The Tem perature D ependance o f th e Line Shapes
The 0° orientation line shapes at various tem peratures are plo tted in Fig. 3.1. The single peak which was observed at 250 K first divided into two peaks of unequal am plitude as th e tem p eratu re was decreased to ab out 215 K, then into th ree peaks as the tem p eratu re was decreased further. W hen three peaks were observed, th e outer two were of nearly th e sam e amplitude. A t 32 K this p a tte rn was sim ilar, but the low field peak split into two on either side of th e position where a single peak was expected. The 23 K line shape showed a single peak. T he peak expected on either side had decreased in am p litu de enough to be indistinguishable from th e noise, and hence were unobservable. Because there were no o ther peaks present to provide reference positions, it was not possible to calculate th e shift of th is peak relative to its high
350 300 250 200 u i* 3 to hi 0) O. 150
s
v t-100 50 -2 0 0 -1 5 0 -1 0 0 - 5 0 0 50 A F ie ld ( G a u s s ) 100 150 200Figure 3.1: The 0° orientation line shapes. T h e baseline corresponds to th e experi m ental tem perature, while the am plitudes are in arb itrary units. T he dashed portions of each line shape indicate d ata which were not recorded as p a rt of th e line shape. tem p eratu re position. This is why it is p lo tted w ith zero field shift. O ur theoretical explanation for these line shapes will be given in Sec. 4.1.
The 63° orientation line shapes are p lo tted in Fig. 3.2. T h e d a ta were acquired only for tem peratures above 77 K because it soon became ap paren t th a t th is was a complex line shape as a result of th e behavior of th e internal m agnetic fields, hence th e study of this orientation was discontinued and has been left as a topic for fu tu re study.
C H A P T E R 3. E X P E R I M E N T A L R E S U L T S 25 350 300 200 100 -3 0 -2 0 - 1 0 20 A F ie ld (G a u s a )
Figure 3.2: T h e 63° orientation line shapes. T he baseline corresponds to th e experi m ental tem p eratu re, while th e am plitudes are in arb itrary units.
are no d ram atic changes to th e line shape over a large range in tem p eratu re. The m ain feature is th a t th e difference between th e am plitudes of th e two peaks becomes greater as th e tem p eratu re decreases. At th e same tim e, th e line is broadening, so the m inim um between th e two peaks is becoming less pronounced, un til th e line blends into one asym m etrical peak a t 32 K. O ur explanation for this behavior is given in Sec. 4.1.
350 300 250 o 200 0.150 100 -3 0 -2 0 - 1 0 20 A F ie ld ( G a u s s )
Figure 3.3: T he 90° orientation line shapes. T h e baseline corresponds to th e experi m ental tem peratu re, while th e am plitudes are in arb itrary units.
3.2
T h e Ti V alues
As with the CW line shapes, th e T\ values were acquired for th e 0° and 90° orienta tions. T he oth er orientations will be left for fu tu re study.
A series of m easurem ents was m ade a t th e 0° orientation between 53.5 K and 294 K. Significant variations in Ti were found a t each tem p eratu re. Because of the spread in local fields, the signal response is spread over a range of m agnetic field strengths. We will call this spread in th e field strengths th e signal response range. It was discovered th a t the variations in T\ occurred as th e field strength was varied
C H A P T E R 3. E X P E R I M E N T A L R E S U L T S 27
10
0.015 0.02
0.005 0.01
l / T e m p e r a t u r e (K '*)
Figure 3.4: T h e T\ d a ta for the 0° orientation. T he vertical bars are the range of Ti values across th e signal response range. T he line is th e theoretical curve predicted by th e stochastic soli ton model.
from one side of th e signal response range to th e other. As a result, measurements of Ti were m ade a t a num ber of points across th e signal response range. The results of these experim ents are plotted in Fig. 3.4 for th e 0° orientation. A similar set of experim ents was perform ed for th e 90° orientation. T h e insults are plotted in Fig. 3.5. In both plots, th e bars at each tem p eratu re give th e m easured range of Ti values for th a t tem p eratu re.
For b o th orientations, the m easured values for T\ agree well with those of Tinus et al. [1982]. I t is to be noted th a t th e range of Ti values observed across the signal
— 10
0.005 0.01
1 / T e m p e r a t u r e ( K ’1)
0.015 0.02
Figure 3.5: The T\ d ata for th e 90° orientation. The vertical bars are th e range of Ti values across th e signal response range. T he line is the theoretical curve predicted by the stochastic soliton model.
response range corresponds to th e range of scatter present in th e d a ta of Tinus et al.. The theoretical explanation of these results is given in Sec. 4.2.
3.3
C hapter S u m m a ry
• W ith the te st crystal at a tem p eratu re of 250 K, line shapes were acquired by ro tating the crystal ab ou t its b axis, with the magnetic field in th e ac plane. • Line shapes were acquired for a num ber of tem peratures between 250 K and
C H A P T E R 3. E X P E R I M E N T A L R E SU L T S 29
23 K for th e 0°, 63°, and 90° orientations.
• S pin-lattice relaxation times were acquired for th e 0° and 90° orientations at a num ber of tem peratu res between 53.5 K and 273 K.
Theoretical Analysis and
D iscussion
4.1
T h e Line S hapes
To analyse the line shapes observed in the CW experim ents, th e interaction between two protons on the same w ater molecule will be studied Each of th e two protons will be in a different m agnetic field as the result of the presence of th e Fe2+ ions. In addition to th e Zeeman interaction with th e two different fields, th e dipole-dipole interaction between protons on the same water molecule can lead to resolved stru ctu re in the line shape. T he m agnitude of any interaction between protons of different w ater molecules will be reduced by 1/ r 3, so th a t th e effect will only be to broaden th e line shapes. In addition, th e line shapes appear to be affected by the presence of w ater molecules which we have called additional w ater, as they behave as though they are not p a rt of the crystal structure.
4.1.1
Fine Structure of the Proton Twines
As th e crystal contains ferrom agnetic centres, it will be assumed th a t each proton is in a slightly different local field (c.f. [Bloembergen 1950], [Poulis 1951]). T he source of the local fields will be discussed in Sec. 4.1.2. Assuming different local fields leads to th e following line positions (in units of Gauss) and intensities:
C H A P T E R 4. T H E O R E T I C A L A N A L Y S I S A N D DI SCUSSION 31 position h = c + 2a + y /a 2 + b2 h = c — 2a — y /a 2 + b2 h = c + 2a — y /a 2 + b2 h = c - 2a + + b2 intensity a + b - y /a 2 + 62) a 2 + b2 — b y /a 2 + b2 ( - a + b + y /a 2 + b2^ a 2 + b2 + b y /a 2 + b2 ' (4.1) where a = -I 7jjh 4 r 3 (1 — 3 c o s 2 ^) b an d c
In these equations, 7 // is the gyromagnetic ratio of th e protons, r is the separation of the protons, and 9 is the angle between the proton-proton vector and the applied m agnetic field. The direction of the applied m agnetic field is considered to be th e 2 direction. Hi is the z com ponent of the total m agnetic field at one proton, and H 2 is th e z com ponent of th e total m agnetic field a t th e other.
A com plete derivation of these equations parallels the work of Bloembergen very closely, a n d is given in Appendix C. The result gives a spectrum of two pairs of lines, where th e two lines in each pair are of equal intensity.
Fig. 4.1 shows th e relative positions and intensities for various ratios of 6 : a. The line p a tte rn s are each centred a t c. As c is th e average of the two iocal fields, it causes a shift in th e centre of the system. This is not depicted in the figure because these shifts w ere not directly observable with our equipm ent, as we had no fixed reference signal to determ ine th e absolute field of th e observed line shapes. Note th a t for b < a, th ere are two lines separated by 6a. These are th e two lines which result from the
1 I _ _ 1 1 b = 1 0 a ! I 1 ! b = 8 a 1 i 1 1 b = 6 a 1
1
. . . ! 1 b = 4 a 1 n 1 b = 2 a i i 1 b = a b = a / 2 i b = a / 4 b = a / 8 11 11 111111 i 11 i 111 i 111 i M 1 11111 i 1111111 1111 11 11 i i i - 2 0 0 - 1 6 0 - 1 2 0 - 8 0 - 4 0 0 40 80 120 160 200 A F ie ld ( G a u s s )Figure 4.1: Line displacem ents for fixed a, and various choices of 6. In this figure we have shifted the scale so th a t c = 0 .
dipole-dipole interaction. W hen b^> a there are four lines which can be grouped into two pairs, one of which is a t fields greater th an c, and one of which is a t fields less than c. The two lines in each pair are separated by 4a, while th e centres of th e pairs are separated by 26. As a special case, if a = 0 we will see one line for 6 « 0 and two lines separated by 26 for 6 > 0 .
We now have enough inform ation about th e line displacements to identify the orientations of th e crystal w ith respect to th e applied m agnetic field, when th e line displacements are com pared to th e relative positions of the peaks in th e observed
C H A P T E R 4. T H E O R E T IC A L A N A L Y S I S A N D D ISC U SSIO N 33
Figure 4.2: The spherical coordinate system.
data. For high tem peratures it can be assum ed th a t rapid therm al fluctuations of the local field inhom ogeneities arising from th e m agnetic centres will tend to average these inhom ogeneities to zero, leaving only the dipole-dipole interaction as a mech anism for creating fine structure in the lines of CsFeCl3-2H2 0 . Using the coordinate system illu strated in Fig. 4.2, Fig. 4.3 shows th e theoretical separations of th e lines a ttrib u te d to th e dipole-dipole interactions of th e eight pairs of protons in one unit cell of CsFeCl3-2H20 . Eight separations were calculated and plotted. There are a t most two non-equivalent dipole-dipole interactions, because of th e sym m etry of th e crystal stru ctu re. Exam ination of Fig. 4.3 leads to th e identification of th e orientations of our crystal when com pared with the line shapes plotted in Fig. 2.2.
We know th a t th e observations were m ade w ith the applied field in a plane per pendicular to th e b axis (i.e. = 0°). In Fig. 4.3b, we note th a t at 0 — 0° and 0 = 180°, all of th e lines have the same small separations, so th a t if there is a rea sonable am ount of broadening, we expect to see only a single wide peak. At 6 = 90° and 0 = 270° all of th e lines again have the sam e separations, out now the separation is large enough th a t we expect to see two distinct peaks in th e line shape. At 27° on either side of each of 0 = 90° and 0 = 270°, one of th e pairs of lines is at m axim um separation, while th e other pal*; is at zero separation. In this case, we would expect to see th ree peaks, where the centre peak has twice the am plitude of the outer two
0.76 R o ta tio n a b o u t th e a axis (^ -9 0 ° ) a) 46 to 196 ISO 270 31S 960 -0.26 -0.76 0.76 R o ta tio n a b o u t th e b ax is (p=0°) 0.26 b) 916 360 196 160 270 00 -0.76 0.76 R o ta tio n a b o u t th e c ax is (4=90°) 0.26 c) 960 270 100
Figure 4.3: Line separations. This is only the result of th e dipole-dipole interaction of all non-equivalent proton pairs. The separations are in arb itrary units.
peaks. Studying Fig. 2.2, we do indeed find th is p attern . T h e single peaks occur at orientations of 80° and 260°, th e double peaks occur a t 170° and 350°, and th e triple peaks occur about 30° on either side of these double peak orientations. Hence we have made the identification of the orientations of th e crystal from the line shapes. Note th a t the asym m etry of th e observed line shapes is not predicted by th e interactions discussed to this point.
T he m ain reasons for choosing the 0° and 90° orientations in th e tem p erature dependant studies are also m ade clear in Fig. 4.3b. For th e 0° orientation, we have only one unique separation of th e lines attrib u te d to th e dipole-dipole interaction, and the lines should be close enough together to appear as one line if th e broadening is
C H A P T E R 4. T H E O R E T IC A L A N A L Y S I S A N D D ISC U SSIO N 35
sufficient. This implies th a t any stru ctu re in th e line shape should be a ttrib u ted to the interaction of th e protons with th e m agnetic centres only. At th e 90° orientation, we again have only one separation of th e lines a ttrib u te d to th e dipole-dipole interaction, b u t in this case, th e two lines should be far enough ap art not to appear as a single line. In this case we would expect to see four lines whose positions vary (as b varies) as in Fig. 4.1.
4.1.2
The M agnetic Interaction
Kopinga et al. [1985] use a H am iltonian for th e iron spin system given by Hchain = - J E S,.Si+1 + D £ S i ,
i i
where S,- represents the full spin of th e ith iron nucleus. In this system, S = 2. In the case of CsFeCl3-2H2 0 , Kopinga et al. find
— = -6 .0 ± 0.5 K k
and ^ = -4 0 ± 20 K. k
By following the explanation in Sec. 1.1, we can see th a t this crystal is antiferrom ag netic, with an easy axis of m agnetization.
We need to add th e Zeeman interaction of th e iron spins with the applied m agnetic field,
H0,
in order to have a com plete H am iltonian for th e Fe2+ ions:Hz = - X > S , - g , - H o , I
where is th e Bohr m agneton, and g,' is th e g-tensor for the i lh spin. The total H am iltonian for the Fe2+ ions is then
H
=
H z+
HchainIn A ppendix D, we show th a t u k, which is the tim e averaged m agnetic moment of the k th spin, is given in dyadic notation for term s up to 0 ( T ~ 2) by
~Pk = ^ f S (S + 1)g*-g.fHo
+
-j ^ S ( S + l ) g * .{ ^ 5 ( 5 + 1 )
( J g k - i+
Dgk+
J g k+1)- D \ [ ( 5 ( 5 + 1) + ( i t + 3 3) + (35(5 + 1) - l)k k ] - g * } H 0 .
We know 5 , <7, and D for this system, and if we make th e assum ption th a t the g- tensors do not change w ith position within th e crystal, th en g t - i = g* = g/t+i = g, and we have th a t
_ 2/3
P k = —^ - g - g - H o + j j ^ g - { —10 4 g + [52(11 + 3 3) + 1 2 2 k k ] - g } - H 0 .
Given a com plete knowledge of g, we could predict values of Jik for any Ho, but this is not possible, nor do we have enough d ata on Jik to calculate g.
One essential feature of th e magnetic m om ent which follows from this equation is th a t the general dependance of th e m agnitude of th e m ean m agnetic m om ent on tem perature is of th e form
V’k = C\—A
C2y2- (4.3)If this m agnetic m om ent theory is correct, we would expect to see such a tem perature dependance of th e local fields resulting from th e iron spins.
If we study Fig. 3.1, we see th a t for the 0° orientation th ere is a definite increase in the separation of th e peaks as th e tem perature decreases. We note th a t there is no evidence of a dipole-dipole line splitting, except perhaps in one peak of the 32 K line shape. T he broadening of the peaks results in th e dipole-dipole line pairs being unresolved, as we had hoped, and we can consider a in Eq. (4.1) to be effectively zero. This means th a t th e peak to peak separation of th e o u te r peaks will be equal
