Mössbauer effect studies of superparamagnetic (alfa)-FeOOH and (alfa)-Fe2O3

MOSSBAUER EFFECT STUDIES OF

SUPERPARAMAGNETIC a-FeOOH AND a-Fe^ O3

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M O S S B A U E R E F F E C T S T U D I E S O F

S U P E R P A R A M A G N E T I C a-FeOOH A N D a-Fe^ O3

PROEFSCHRIFT

ter verkrijging van de graad van doctor in

de technische wetenschappen aan de Technische

Hogeschool Delft, op gezag van de rector

magnificus ir. H. R. van Nauta Lemke, hoogleraar

in de afdeling der Elektrotechniek, voor een

commissie uit de senaat te verdedigen op

woensdag 26 april 1972 te 16.00 uur

door

ADRIANUS MICHIEL VAN DER KRAAN

natuurkundig ingenieur

geboren te Zoetermeer

/ ^ 2 / Sli^o

1972

Dit proefschrift is goedgekeurd door de promoter

Aan Lenie, Ronald en Nicolet

Aan mijn ouders.

CONTENTS

Page

Chapter I INTRODUCTION 9

REFERENCES 12

Chapter II MQSSBAUER SPECTROMETER 13

II. 1 Introduction 13

11.2 Survey of some possible constant acceleration systems 14

11.3 Description of spectrometer 17

.1 Binary counter and weighing network 17

.2 Velocity control system and velocity transducer 19

.3 Detection system 21

.4 The use of the multichannel analyzer 23

.5 Automatic controlling of the multiscaler 28

11.4 Cryostat for temperature dependent measurements 32

REFERENCES 36

Chapter III THEORY 37

111.1 Internal fields 37

.1 Isomer shift 38

.2 Electric quadrupole interaction 39

.3 Magnetic interactions 41

.4 Combined magnetic and quadrupole interaction 44

111.2 Lattice vibrations 45

.1 The recoilless fraction f 45

.2 Second-order Doppler shift 48

111.3 Superparamagnetism 48

111.4 Relaxation phenomena 53

REFERENCES 56

Chapter IV SAMPLE PREPARATION AND CHARACTERIZATION 58

IV.1 Introduction 58

IV.2 Preparation of a-FeOOH samples 58

IV.3 Preparation of a-Fe.O. samples 59

IV.4 Determination of the mean particle size 60

Page

Chapter V MEASUREMENTS AND RESULTS 65

V.l Introduction 65 V.2 Mossbauer effect in ultrafine particles of a-FeOOH 66

.1 Effective magnetic field 66 .2 Isomer shift and second-order Doppler shift 73

.3 Electric quadrupole interaction 73 .4 Measurements in an external magnetic field 75

V.3 Mossbauer effect in ultrafine particles of a-Fe.O, 77

.1 Effective magnetic field 77 .2 Electric quadrupole interaction 84 .3 Measurements in an external magnetic field 85

V.4 Contribution of surface ions to the Mossbauer spectra 88 of ultrafine particles of a-Fe 0

.1 Introduction 88 .2 The Morin transition 88

.3 Magnetic behaviour of surface ions 92 .4 Electric quadrupole interaction of surface ions 92

V.5 Mossbauer effect measurements at T = 4.2 K 97 V.6 Relative f measurements of ultrafine particles of 99

a-Fe^O^

REFERENCES 102

Chapter VI DISCUSSION 103 VI.1 Introduction 103 VI.2 Effective magnetic field 103

VI,3 Influence of an external magnetic field on the 108 relaxation behaviour of ultrafine particles of a-FeOOH

and a-Fe.O

VI.4 Electric quadrupole interaction 109 VI.5 Relative f measurements of ultrafine particles of 112

a-Fe^Oj

REFERENCES 115

SUMMARY 117

SAMENVATTING 119

CHAPTER I

INTRODUCTION

Since the discovery of the recoilless emission and absorption of the 129 keV 191

y-radiation of Ir by Rudolf L. Mossbauer in 1957 [1,2], many other isotopes amoung which Fe, have been found which show this effect. The nuclear resonance technique usually is called the Mossbauer effect, and the inventor has been rewarded the Nobel prize for Physics in 1961. The application of the Mossbauer effect has been very wide spread in the field of science, not in the least because of the wealth of information which can be obtained with relatively simple means. To-day the Mossbauer effect has become a standard experimental tool in the in-vestigation of solids. It is treated thoroughly in several books and review articles [3-91. Therefore we shall give only a short outline of the basic prin-ciples of the phenomenon of recoilless y-ray emission.

When a free nucleus at rest in an excited state with an energy E above the ground state emits a photon of energy E = p c, it will receive a recoil momentum

2 2 .- 2 2

-p and a recoil energy R = E /2 Mc - E /2 Mc where M is the mass of the nucleus and c the velocity of light. Conservation of energy gives for the photon energy E = E -R. Similarly for the absorption process it follows that to excite a

Y o ' ^ '^

nucleus from the ground state to the energy E , a photon energy E = E + R is necessary. For Fe, E = 14.4 keV and R " 2 x 1O" eV.

o

The excited state has an energy uncertainty F determined by its life time

57 —9 T , r = "h/T where 1i is Planck's constant divided by 2TT. For Fe, F = 4.6 x 10

n' n -^ ' eV. It is clear that resonant absorption of y-rays emitted by free nuclei is impossible under these circumstances. However when the nuclei are embedded in a completely rigid lattice the recoil momentum in the case of emitting or absorbing a y-ray is the same as for free nuclei, but it is taken up by the solid as a whole. In this case the inevitable recoil energy is drastically reduced and given by R ^

2 2

small compared to F, resonant emission and absorption occurs. In this case all the emission and absorption processes take place with negligibly loss of energy, so the recoilless fraction also called Mossbauer fraction or Debye-Waller factor f = 1. It is clear that f = 0 for free nuclei. In an actual solid f will have a value between these two extremes depending on the energy E of the photon, the mass of the nucleus, the rigidity of the lattice and the temperature T.

Since the decay of a nucleus from the first excited state to the ground state under emission of y-radiation E can be treated as a classically damped harmonic oscillator, the energy spectrum of the emitted photons is given by the Lorentz formula

^^^) = ^o "'2 2

° (E-E )^+ F^/4 o

where F is the natural linewidth.

In order to measure a resonant absorption spectrum a Mossbauer spectrometer IS used which consists basicly of a radioactive source, an absorber and a gamma ray detector. The energy of the emission spectrum of the source can be changed with respect to that of the absorption spectrum by moving the source towards and away from the absorber The gamma ray intensity measured with the detector m a transmission experiment located behind the absorber, plotted as a function of the Doppler velocity of the source is usually called a Mossbauer spectrum.

Due to the narrow natural linewidth F, the Mossbauer effect provides extremely high resolution. With the help of this technique one is able to measure very small energy shifts and splittings of the nuclear levels caused by the interaction of the atomic nucleus with electric and magnetic fields at the nuclear site. The observation time used in the Mossbauer effect is correlated with the nuclear Larmor precession time T . For an Fe nucleus m the first

_ Q

excited state and a typical effective magnetic field of 500 kOe, T - 2.5 x 10 s. So the Mossbauer effect with Fe is an useful tool to study relaxation processes which take place on such a small time scale.

In this thesis the Mossbauer effect will be studied in ultrafine

crystallites of a-Fe.O. and a-FeOOH. Ultrafine particles of ferro- and antiferro-magnetic materials behave like paramagnets, which is due to fast relaxation of the

ionic spins arising from thermal excitation. This effect is called superparamagne-tism by Neel [10]. The influence of the relaxation processes on the shape of the Mossbauer spectra of superparamagnetic particles will be investigated.

Since in ultrafine particles a significant fraction of the atoms is located near or at the surface, henceforth called surface ions, the question arises how the surface ions contribute to the Mossbauer spectrum. As the Mossbauer technique applies to individual nuclei, it is an excellent tool to probe the nuclei of surface ions by using particles of which the surface is enriched in the Mossbauer isotope Fe. So far in the literature not much attention has been paid to the role which surface ions of ultrafine particles may play.

In chapter II a short review is presented of the different types of Mossbauer spectrometers. In addition an account is given of the spectrometer developed, which is used for the measurements described in this thesis.

In chapter III a brief summary is given of the theory dealing with the interaction of a nucleus in a solid and its environment. Furthermore a description is presented of the phenomenon of superparamagnetism and the influence of fast electron spin relaxation on the shape of Mossbauer spectra.

In chapter IV the sample preparation and characterization will be described, while the results of the Mossbauer effect measurements with a-FeOOH and a-Fe 0, ultrafine crystallites as a function of temperature and external magnetic field will be presented in chapter V. Furthermore a description will be given of the method to enrich surface layers of ultrafine particles of a-Fe 0, in the isotope " F e .

In chapter VI a more thorough discussion of the results is presented leading to some final conclusions.

REFERENCES

1 R.L. Mossbauer, Z. Physik _1_51_, 124 (1958). 2 R.L. Mossbauer, Z. Naturforsch. J_4a, 221 (1959).

3 H. Frauenfelder, "The Mossbauer Effect", W.A. Benjamin, New York (1962).

4 A. Abragam, "L'effet Mossbauer et ses applications a 1'etude des champs internes", Gordon and Breach, New York (1964).

5 G.K. Wertheim, "Mossbauer Effect: Principles and Applications", Academic Press, New York (1964).

6 H. Wegener, "Der Mossbauer-Effekt und seine Anwendungen in Physik und Chemie", Bibl. Inst. Mannheim (1965).

7 A.J. Freeman and R.E. Watson, Magnetism IIA, Ed. G.T. Rado and H. Suhl, Academic Press, New York and London, 167 (1965).

8 J.R. de Voe and J.J. Spijkerman, "Mossbauer Spectroscopy", Anal. Chem. 38, 382R (1966).

9 V.I. Goldanskii and R.H. Herber "Chemical Applications of Mossbauer Spectroscopy", Academic Press, New York and London (1968).

CHAPTER II

MOSSBAUER SPECTROMETER

II. 1 Introduction

In Mossbauer spectroscopy a precision Doppler velocity spectrometer is required to produce a small energy shift in gamma-ray energy, thereby des-troying and fulfilling the condition for resonant absorption,

The maximum relative velocity between source and absorber needed for the study of the isomer shift, the quadrupole splitting, and the magnetic hyperfine splitting in Fe nuclei is of the order of 10 mm/s corresponding to a gamma-ray energy shift of 4.6 x 10 eV.

57 -9 As the natural linewidth of the 14.4 keV excited state of Fe is 4.6 x 10 eV,

corresponding to 0.1 mm/s the accuracy of the Doppler velocity should be much better to avoid instrumental line broadening of the resonant lines.

In general two different types of spectrometers have been used:

a) constant 'Oetoo'ity speatvometev. With a constant velocity system a single channel is needed to store the gamma-ray intensity. Counts are accumulated for each velocity and the velocity is changed stepwise. This method has the advantage that one is able to measure the interesting parts of a spectrum only and in addition that one can apply the technique of thermal scanning [ 4 ] . On the other hand any drift of the gamma-ray detection system may cause changes in the background during the experiment. The motion for such a system may be obtained either mechanically [1,2] or electromechanically [ 3 ] .

b) oonatant acoeleration spectrometer. With a constant acceleration system a sweep is made through the whole desired velocity range during each stroke of the velocity transducer. A multichannel analyser (MCA) is necessary to store the data in such a way that each channel corresponds to a certain in-crement of velocity. As the time of velocity scan is very short compared with the total measuring time of one point of a spectrum in the constant

velocity mode, long time drift of the detection system does not influence the spectral shape. A further advantage of the system is that in a relatively short time a qualitative picture of the Mossbauer spectrum can be obtained on the oscilloscope of the MCA.

For the measurements described in this thesis a Mossbauer spectrometer is used which differs from known constant acceleration systems. Before describing this system first a short survey of some constant acceleration systems will be given in section 11,2 after that the system developed will be treated in section 11,3. In addition a description is given in which way several spectro-meters can be run simultaneously completely independently, with one MCA for data collecting. Furthermore a control unit to run the MCA automatically is described. In section II.4 attention is paid to the equipment which is neces-sary to measure at different temperatures between 77 and 300 K.

II. 2 Survey of some possible constant acceleration systems

As mentioned before a MCA is necessary to store the gamma-ray counts by using a constant acceleration type spectrometer. However, there are two different ways in which a MCA can be used in order to obtain a Mossbauer spectrum. The first method is to modulate the pulses of the gamma-ray detector from a single-channel analyzer with a voltage proportional to the instantaneous velocity and using the MCA in the pulse-height analysis mode [1,5]. In this method the dead time to analyse a detector pulse is related to the channel number. The second method is to run the MCA in the multiscaler mode and to use a constant increment of the velocity for each channel [6,7]. This second method is far better than the first one and hence it will be described below,

Generally velocity transducers are employed which consist of two coupled loudspeakers or a system based on the same principle. One of the loudspeakers is used as the driving speaker, whereas the other one supplies a feedback signal proportional to the velocity.

signal for a power stage which produces the current for the driving coil. A block diagram of this system is shown in fig. 11,1 The total gain of the system is

forward gain g g^

1 + loop gain I + S^S2^

where g , g and h are the different transfer factors.

When the product g,g,h is much larger than 1 the velocity is proportional to the reference input signal even when the power stage and the driving coil are not linear. In order to obtain a constant acceleration the velocity versus time has to be a sawtooth or a triangle. Due to the constant increasing velocity each channel of the MCA used as multiscaler counts during an equal velocity range. The difference between the various systems lies in the way the reference signal is obtained.

The following methods are possible:

a) the reference signal is supplied by a function generator [81. Now it is ne-cessary to synchronise the address register of the MCA and the function gene-rator and to adapt the frequency of the channel stepping to each frequency of the function generator.

b) the reference signal is obtained by integrating the square wave of the address register of the MCA when the MCA switches from one memory half to the other [6,9], The reference signal is a triangle. The analog way of integrating introduces a possible drift and the chance that the increasing slope and the decreasing slope are not exactly equal.

c) the analog output from the address register to the scope of the MCA is ob-tained in a digital way and it can be used as a reference signal [10,11]. Now the reference signal is a sawtooth but this signal does not meet special requirements.

d) the BCD output of the address register of the MCA supplies the reference signal via a digital to analog converter [12]. In this case the reference signal is a very good sawtooth. However, the digital to analog converter is a

e) the reference signal is obtained in a digital way with a binary counter and a weighing network independent of the MCA. The channel stepping pulses to the MCA in the multiscaler mode also drive the binary counter with this net-work [13]. This method, developed in the interuniversitair reactor instituut, Delft, has the following advantages:

- a sawtooth or a triangle reference signal is obtained which is more precise than the analog output from the address register of a MCA.

- the binary counter and the address register are running independently, so that it is possible to wait some time after the flight back of the sawtooth. In this way each channel of the MCA is effectively used,

- the sawtooth or triangle voltage is obtained in a digital way, so the drift is small,

- the velocity step is synchronized with a certain channel,

- with a triangular reference signal the positive and the negative slope are exactly identical,

The spectrometer described in section II.3 is of the constant acceleration type of which the reference signal is obtained following method e ) . The

MCA is used in the multiscaler mode.

REF INPUT FEEDBACK SIGNAL CONTROLLED VELOCITY

II. 3 Description of spectrometer (fig. II.2) II.3.1 Binary counter and weighing network

In order to get a triangular or sawtooth reference signal to drive a velocity transducer the principle of a binary counter connected with a weighing network is used. Any pulse generator which supplies a pulse larger than + 2V triggers a Schmitt-trigger. The pulses from the Schmitt-trigger go to a binary counter which consists of 10 dividers. The output of each divider is connected with an emitter follower and an electronic switch in order to provide a low output impedance and to fix the output voltage at a reference voltage. The re-ference voltage is supplied by a battery with the result that the output voltage of the dividers is absolutely free from ripple and noise.

Each electronic switch of the binary counter is connected with the

cor-PULSE OENERATDR

JUUL

T

Fig, II.2 Block diagram of the Mossbauer spectrometer.

responding input U to U of a (2R - R) weighing network. This network of resistors is shown in fig. II.3. Using the principle of superposition it is clear that the output voltage U of the weighing network is given by the following expression:

4 "9 4 "8 ^ IT "7 ^ k "6 ^ is "5 " k "4 ^ Tl2 "3 '^ 314 "2 *

768 " l

* 1536

UO UI U2 U3 U4 US U6 U7 US U9

1

-1—0

Fig. II.3 (2R-R) resistor weighing network.

The voltages U. to U. can be + E or - E, which are the reference battery voltages, depending on the position of the corresponding divider. It directly

2 2 follows that U is a constantly stepwise increasing voltage from - -^rE to + -^E when pulses of a certain frequency enter the binary counter. Since E = 9V the peak to peak voltage of the signal is 12V in 1024 steps,

In this way a sawtooth reference signal is obtained. In order to get a triangular reference signal the binary counter has to switch from the adding to the subtracting position and back when the signal reaches certain levels, To make this possible gates are introduced between the dividers of the binary counter which are controlled by a flip-flop. This flip-flop is set in the adding or subtracting position when the binary counter is in position " 0 " or in position "912" respectively. When the sawtooth mode is used the "912" gate triggers a "one shot" which after 10 ps resets the binary counter in position " 0 " ,

Now the sawtooth and triangular reference signals consist of 912 and 1824 steps respectively. However since a MCA of Intertechnique type Didac 800 is used which can be divided in I, 2 or 4 subgroups, the number cf pulses to the address register has to be divided in the sawtooth mode by 1, 2 or 4 and in the triangular mode by 2, 4 or 8 respectively,

During the reversal of the velocity the data storage is interrupted. This is done by some electronic gates. In the sawtooth mode a gate supplies a "start storage" pulse to the MCA when the binary counter is in position "56". So the MCA is waiting 5% of the total sweep time before and after the flight back of the velocity transducer. In the triangular mode the same "start storage" pulse is used, however when the binary counter is in position "856" the pulses to the address register of the MCA are inhibited by a gate controlled by the

"Inhibit flip-flop" which receives a pulse at position "856".

In position "912" the binary counter is set in the subtraction mode and when it arrives in position "856" the "inhibit flip-flop" is reset and the channel stepping pulses to the address register of the MCA go on. A complete diagram of

the binary counter and weighing network system is shown in fig. II.4.

11.3,2 Velocity control system and velocity transducer

In order to obtain reliable Mossbauer spectra with a constant acceleration system it is necessary to avoid experimental line-width broadening. It means that the velocity has to be a perfect linear function of the channel number, in case a MCA is used as a multiscaler. This can be realized by using an ideal reference signal and a feedback system in which the measured velocity is compared with that reference signal. In section 11,3,1 a description is given how the reference signal and the absolute synchronisation between a certain velocity and a given channel number are obtained. Since the methods for controlling the velocity reported in the literature are not essentially different, the present method will be described very briefly,

The easiest way to drive a Mossbauer source is to use a loudspeaker, In order to be able to control the real velocity it is necessary to measure

Fig, II.4 Diagram of binary counter and weighing network in order to obtain

a sawtooth or triangular reference signal. The outputs of the binary counter

are connected via an emitter follower (E) and electronic switch (S) with the

inputs of the weighing network. The numbers in the different gates indicate at

which position of the binary counter the gate is triggered,

this velocity in the first place. In fact this is the inverse problem of gene-rating the velocity and it can be solved in the same way. The present velocity transducer essentially consists of two loudspeakers, whose coils are mechani-cally coupled by a perspex rod. The source is attached to this bar. The driving speaker is of the type 9710 M of Philips with a low coil impedance of 8 ohm, while the feedback speaker is of the type 9710 AM with a high coil impedance of 300 ohm. The high impedance coil has many windings and it produces a large feedback signal. The sensitivity is 15 mV per """/s.

A diagram of the whole velocity control system is shown in fig. ii.5. The voltage which is proportional to the velocity generated in the feedback

POWER

^STAGE DRIVE FEEDBACK COIL COIL

COUPLING ^->r^ CAP

SUMMING OP AMP

Fig. 11,5 Block diagram of the velocity control system.

coil, is fed via a LM 201 operational amplifier of National Semiconductors to the operational amplifier LM 201 which is used as summing point of the reference signal and the feedback signal. The amplified difference between these two signals is fed into a low drift amplifier with a power of 35 W. This is achieved by using silicon power transistors suitable for a high vol-tage. The coupling capacitor in the system is used to decrease the influence of dc-drift. The system is adjusted in such a way that at maximum reference signal input (controlled by potentiometer P) the maximum obtainable velocity is 10 / s , which is sufficient for experiments on Fe.

II.3.3 Detection system

In order to perform a Mossbauer transmission experiment a y-ray detection

system behind the absorber is needed. For the detection of the 14.4 keV-radiation of Fe either a proportional counter or a scintillation counter is used. In practice there is an important difference between both counters,

Proportional counters have the advantage of better energy resolution and low background due to the rapidly decreasing efficiency for higher energy y-rays, A second advantage of a proportional counter is that the amplification is hardly influenced by an external magnetic field. On the other hand these types of gasfilled detectors have a relatively long dead time and the overall lifetime is limited. Since in our experiments we use counting rates up to 50.000 pulses/s in the present spectrometer a scintillation counter of Harshaw Chemical Company is used, which consists of an EMI 6096 S photomulti-plier and a Nal(Tl) scintillation crystal 0.1 mm thick and 37.5 mm in diameter. By choosing the thickness of the crystal 0.1 mm the efficiency for counting the 14.4 keV radiation is still 8 0 % , in contrast to the detection of the higher energy y-rays of 120 keV and above, thus reducing the counting rate due to background considerably. Only in case of measurements in an external magnetic field a proportional counter of Amperex type 303 PC 421 is used. The pulses from the counter are amplified by a preamplifier and an Ortec 486 main amplifier combined with a single channel analyzer to select the 14.4 keV radia-tion, In fig, 11,6 a typical y-spectrum of Fe is shown obtained with the scintillation counter by using a source of 25 mCi Co diffused into a Palladium foil and a 4 mm thick paraffine filter in order to stop the 6.6 keV X-rays, The source is of the Nuclear Science and Engineering

Corporation. The spectrum was obtained with the window of the single channel analyzer as indicated in fig. II.6,

The way in which the pulses finally are stored in the MCA, will be des-cribed in the next section,

1.

4

°0 ioo MO ~ 300

Fig. II.6 y-ray spectrum of Fe obtained with a scintillation counter.

II.3.4 The use of the multichannel analyzer

In the present constant acceleration spectrometer the y-ray pulses from a single channel analyzer are stored as a function of time in a MCA of Intertechnique type Didac 800. The memory of this MCA can be divided in 2 or 4 subgroups of 400 and 200 channels each respectively. Since in the case of Mossbauer measurements on Fe 400 discrete points in one particular spectrum are enough, the MCA is used to store the data of two separate Mossbauer spectrometers at the same time. However, the MCA has only one address register so that the two spectrometers have to be synchronized and the address register has to be arranged in a certain manner.

The Didac 800 consists of three main parts: a counting register, an address register and a memory. By using the MCA as a multiscaler, one can change the address by a pulse on the input of the address register. The y-ray pulses are

counted in the counting register which is connected with a certain memory place, determined by the position of the address register. By changing the position of the address register a memory cycle is made. The address register of the Didac 800 consists of one "200-counter" and two "2-counters" while the sequence of these counters determines the division of the memory of the multi-scaler. The two "2-counters" are called "bit a" and "bit b". In order to measure more spectra simultaneously, there are two methods both of which will be discussed now.

A. The routing system method.

According to this method the channel stepping pulses are fed to the serial input of the "200-counter" of the address register while the "bits a and b" are controlled by so called "routing" pulses. In the relevant case of dividing the memory in 2x 400 channels the block diagram is shown in fig. II.7.

MEMORY ADDRESS" CHANNEL STEPPING 200 COUNTER I—I Q E|» AMPLIFIER ANALYZER DETECTOR (9 LI K

s

¥ D

Fig. II.7 Block diagram of the routing system in order to measure two spectra simultaneously on a Didac 800 multichannel analyser of Intertechnique.

After amplification and discrimination the pulses from the detectors I and II are fed into the "routing" inputs E and F controling "bit b" while "bit a" is connected with the output of the "200-counter". The pulses from both detectors are also fed into a mixing unit. Each pulse from the mixing unit is counted in the counting register and it commands the MCA to make a "memory cycle". In this way each detector pulse determines his subgroup of the memory by his routing pulse and it is put into the corresponding memory place as soon as possible.

This method is the easiest way to store data of more than one spectro-meter in one multiscaler at the same time. Only a very small and simple modi-fication of the MCA is necessary to complete the routing system; in addition a pulse mixing unit has to be built. However, this method has an important disadvantage. After receiving a pulse from one of the detectors the MCA makes a memory cycle which takes 5 ys, so a dead time of 5 us is introduced. Due to this overall dead time the different measurements in the various sub-groups of the memory influence each other. When for instance the counting rate in one subgroup is decreasing due to resonant absorption, there is a chance that the other subgroups count more y-rays. As a result a decrease in pulse rate in some channels due to resonant absorption of one measurement will introduce peaks in the corresponding channels of the other measurements. This effect becomes more important the higher the counting rates, thus limiting the overall counting rate for the MCA to about 10.000 pulses/s.

In fig. II.8 the results are shown of two different measurements carried out at the same time. In order to illustrate this effect no absorber was placed between source and detector in measurement B. The counting rates were

4 4

1 X 10 p/s and 1.2 x 10 p/s for measurements A and B respectively.

B. External counting registers method.

When the memory is divided in two equal parts of 400 channels, a block diagram as shown in fig. II.9 is used. The sequence of the "200-counter" and "bits a and b" are arranged in such a way that each channel stepping pulse

ai7

% 014

.-<'••>•.

-iis-

—^

Doppiv NMooty

Pig. II.8 The Mossbauer spectra of two experiments A and B carried out simul-taneously on one MCA following the routing system method.

4 A : with an absorber of a-Fe 0 , counting rate 1x10 p/s,

4 B : without an absorber , counting rate 1.2x10 p/s.

switches the connection between memory and counting register from the first half to the second half of the memory. It means that the channel stepping takes place in the following sequence: 1 - 401 - 2 - 402 etc. The y-detector pulses are not fed directly into the "counting register" of the MCA, but each detector has at its disposal two external counting registers in which the y-ray pulses are buffered in turn. When the detector pulses are counted in one of the external registers, the data content of the other register can be emptied into the serial input of the "counting register" of the MCA, The switching between the two external registers also happens by the channel

Fig, 11,9 Block diagram of the external counting registers system in order to measure two spectra simultaneously on a Didac 800 multichannel analyser of Intertechnique,

stepping pulses.

The register is switched by the channel stepping pulse in the mode "down", which means that each pulse entering the register, lowered the original content. The emptying of the data content of the external registers which are ready to read out takes place in the following way, A simple generator supplies read-out pulses with a frequency of 2 Mc/s.These pulses are fed into the desired

register and via an "Or" gate to the "counting register" of the MCA until this register reaches the position 0. The different registers to be read out get the opportunity to drop their content in the MCA in turn. This read-out process of each register has to take place within the time between two channel stepping pulses, and this process has to be finished for all registers within the time between two velocity steps of the spectrometer. The external registers consist of four bits which means that 15 pulses can be detected within the time of a velocity increment,

Since the velocity increment has a time width of 140 y s , and taking into account the statistical nature of the y-ray pulses a counting rate up to 70.000 pulses/s can be used for each experiment.

By using this method to measure more spectra with one multiscaler simul-taneously, the different measurements are completely independent. There is no dead time problem at all since there is no loss of any detector pulse during the time the MCA is busy. In fig. 11.10 the spectra are given for the same measurements as illustrated in fig, 11,8, however, this time the method of the external counting registers is applied. For these reasons this method is used in the measurements described in Chapter V. A complete diagram of this system is given in [14].

11.3,5 Automatic controlling of the multiscaler

Since accurate measurements of resonant gamma absorption spectra generally require rather long counting times it is worthwhile to use a strong source in order to shorten the measuring time. In addition it is worthwhile to run the spectrometers automatically in order to use the total time available effecti-vely.

Therefore a system is developed according to which both spectrometers can be run automatically by following a pre-set programme. For instance it is desirable to be able to measure the resonant absorption spectra of different absorbers at room temperature one after the other without interruption or to

• ^ Q M 0 2 0 ^.flVftf

_{V " * " " — • *}

.v-'^-\ .'^

•••X

, v . V

ni^

Tir

Tikr

Fig. 11.10 The Mossbauer spectra of two experiments A and B carried out simul-taneously on one MCA following the external counting registers method.

4 A : with an absorber of a-Fe.O , counting rate 1x10 p/s.

L

B : without an absorber , counting rate 1.2x10 p/s.

measure the spectra of one absorber at different temperatures. Also it should be possible to choose the desired time for each measurement in advance. From these requirements it follows that when a measurement is finished the results

should be stored in some way. Furthermore provisions should be made so that the absorber is changed automatically or that the same absorber is adjusted to an other temperature before a new measurement is started. The multiscaler control unit supplies the required signals. A block diagram of the multiscaler control unit is given in fig, 11.11. A complete diagram is given in [15].

rnwcn

l r i > - - * JO

Tl

n\

NUM1 INT1 NUM2 IN12 ». STOWAGE "•RESET MEMC«T .-PCS ERASE p-INT READ OUT " N U M READOUT

Fig. 11.11 Block diagram of the multiscaler control unit.

The multiscaler steps through the channels in the way described in section I1.3.4.B. A cycle through all channels is called one sweep. It is possible to adjust the multiscaler so that it stops after a preset number of sweeps. The number of sweeps is counted in channel number 0. The time duration of one sweep depends on the frequency of the velocity transducer and it is in this case 456 X 140 ys = 63.84 ms. The maximum number of adjustable sweeps is (10 -1) corresponding with a time of 17.5 hour. Since sometimes a longer measuring time is required, it is desirable to have the possibility to choose different measuring times for the two spectrometers, called MB I and MB II, Therefore for each spectrometer the stop signal of the multiscaler is fed into a

"4-counter", By means of a switch it can be determined whether the measurement is finished after one, two or three times the adjusted number of sweeps,

For MB I there are ten such switches combined with a "10-counter", This "10-counter" selects which switch determines that the measurement on MB I has

to be finished, while this "10-counter" receives a pulse after each measuring cycle. Since this pulse also triggers an absorber changer it is possible to ad-just in advance the measuring time by a factor of one, two or three times the selected number of sweeps for ten different measurements one after the other on MB I, For MB II one can select the time by one switch only, so the measuring times for all measurements are the same,

When a measurement is finished on either one of the spectrometers MB I or MB II the data collected in the MCA have to be read out in a certain manner. Therefore it is necessary to switch the MCA in a subgroup concerned with this spectrometer and after that to the command "read out". The subgroup selection occurs by the signal "MB I-ready" of "MB Il-ready" from the time programmer, des cribed before. The data can be read out with a paper tape puncher of Tally type 420 PR in two different ways; either in a numerical way or in an integrated way. The integrated way means that the content of a channel is added to the sum of the contents of all previous channels. These conditions for read out can be ad-justed by simple switches in advance.

When the preset number of sweeps is reached it is necessary to examine which conditions adjusted beforehand are satisfied and to follow also the corresponding instructions. These instructions may have the following sequence:

1) to read out one of the subgroups according to one or both manners mentioned above

2) to erase the corresponding memory part

3) to erase the content of channel 0 in which the number of sweeps are counted 4) to give the command to change an absorber or temperature, and finally to

start the multiscaler again.

The sequence of these instructions is determined by an automatic cycling switch or "8-counter". This "8-counter" is reset and deblocked by the signal of the MCA when the number of sweeps is reached. This signal starts also a generator the pulses of which drive the "8-counter". The positions 2, 3, 4 and 5 of the "8-counter" are used to read out and erase the desired parts of the

memory, while position 6 is used to erase channel 0. When the "8-counter" arrives in position 7, a start signal is given to the MCA while the "8-counter" is blocked until the multiscaler stops again.

II.4 Cryostat for temperature dependent measurements

In order to use the resonant gamma-ray absorption technique to study problems of solid state physics it is necessary to have the possibility to vary the temperature of the absorbers to be investigated. Therefore a liquid nitrogen bath cryostat is developed, which permits measurements to be performed at any temperature between 77 and 500 K. The cryostat is illustrated in fig. 11.12.

10 -m

Fig. 11.12 Cross section of the liquid nitrogen cryostat. 1,2 Outer and inner vessel of brass 5 cryostat tail of brass 3 stainless steel jacket 7 windows of mylar 4 vacuum flange 8 sample holder 5 nylon spacer 9 vacuum transits

10 y-ray direction.

The main parts of the cryostat are made of brass. The inner vessel is fixed by two stainless steel jackets at the top and nylon spacers at the side wall. The

bottom of the inner vessel is of copper, while in the middle of the bottom an absorber holder is placed in such a way that it can easely be changed. The outer jacket of the cryostat has a removable tail. In this tail two holes are made which are closed by mylar foils to transmit the low energy y-rays. The cryostat is evacuated by a small oil diffusion pump. The tail is used in order to provide a sufficiently small distance between source, absorber and detector to ensure a counting rate which is as high as possible. Moreover experiments with the cryostat placed between the pole gaps of an electromagnet are possible as well.

The absorber holder is provided with a heater coil, so the temperature can be adjusted continuously in the above mentioned region. The temperature is measured with a copper-constantan thermocouple and it is controlled by a tempera

ture control system which is described below. When no heater current is applied the lowest temperature obtainable is 66 K by pumping off the liquid nitrogen and using a copper sample holder. The power one needs to get higher temperatures depends on the thermal resistance between the absorber and the liquid nitrogen bath. With a sample holder which consists of a brass rod of 50 mm length and 3 mm diameter, 2 W is needed to hold the absorber at room temperature. The con-tent of the cryostat is 2 1 while the liquid nitrogen consumption without any heater current is 0.04 liter per hour,

The cryostat can be refilled automatically in a very simple way, A diagram of the refilling system is shown in fig, 11,13, This system uses maximum and minimum liquid nitrogen level sensors which are simple germanium diodes. The

forward resistance of the diodes changes drastically when they touch the liquid nitrogen. The large voltage drops across the diodes are used after electronic adjustment to switch a flip-flop via a gating circuit. This flip-flop controls a relay which closes or opens a magnetic valve in the nitrogen supply to the cryostat. When the minimum level sensor is out of the liquid nitrogen the relay which opens the magnetic valve is closed. By the same relay a heater current is switched on to build up the pressure in the dewar. When the maximum

Fig. 11.13 Diagram of the refilling system of the liquid nitrogen cryostat.

«

level sensor is cooled down by the liquid nitrogen the relay is opened, the heater current is switched off, and the magnetic valve is closed. The conditions that the relay only can be opened or closed when both sensors are in or out of the liquid nitrogen respectively are given by the gating circuit. The refilling of the cryostat can also be started or stopped by hand.

The temperature measured with a thermocouple at the position of an absorber in the cryostat is stabilized by a temperature control system. The diagram of the control system is shown in fig. 11.14. From the thermocouple voltage a reference voltage V ^ is subtracted. The difference between these voltages is supplied to a John Fluke nuldetector type 845 AB which is used as a low drift dc amplifier. The amplification factor can be selected between zero and

6 . . 4 10 . In the system mentioned here the amplification is about 3 x 1 0 .

Cryostat

Fig. 11,14 Block diagram of the proportional plus integrating temperature control system.

The amplified signal is via an operational amplifier fed into a power stage which supplies the required power to the heater coil of the sample holder, The temperature is varied by changing the reference voltage. So far the tempe-rature is proportionally controlled. However, it is well known that in a purely proportional control system the stabilized temperature is ultimately influenced by whatever changes may occur in the process (such as changes in thermal isolation between absorber and surrounding). This influence can be avoided by using a proportional plus integrating control system. The difference between the desired temperature given by the reference voltage, and the

control system.

With such a system the temperature stability is better than 0.025 degree.

REFERENCES

1 "The Mossbauer effect" Proc. Second Mossb. Conf. 1961, Edited by: D.M.J. Compton and A.H. Schoen. John Wiley and Sons, New York (1962), Part III: Experimental problems p. 37.

2 P.A. Flinn, Rev. Sci. Instr. 34, 1422 (1963).

3 J. Lipkin, B. Schechter, S. Shtrikman and D. Treves, Rev. Sci. Instr. 35, 1336 (1964).

4 R.S. Preston, S.S. Hanna and J. Heberle, Phys. Rev. J_28, 2207 (1962). 5 W. Kerler and W. Neuwirth, Z. Physik 167, 176 (1962).

6 E. Kankeleit, Rev. Sci. Instr. 35, 194 (1964). 7 T.E. Cranshaw, Nucl. Instr. Meth. 30, 101 (1964). 8 Laben spectrometer, Milan, Italy (1967).

9 H. de Waard, Rev. Sci. Instr. 36, 1728 (1965).

10 Y. Reggev, S. Bukshpan, M. Pasternak and D. Segal, Nucl. Instr. Meth. 52_, 193 (1967).

11 F.C. Ruegg, J.J. Spijkerman and J.R. de Voe, Rev.Sci. Instr. 36., 356 (1965). 12 Elron spectrometer, Haifa, Israel, (1967).

13 J, de Blois and A.M. van der Kraan, I.R.I, report 132-68-01, Delft (1968). 14 J. de Blois, I.R.I, report 132-70-06, Delft (1970).

CHAPTER III

THEORY

111,1 Internal fields

As described in chapter I the resonant gamma-ray absorption technique has

a very high energy resolution. Due to this high resolution one is able to measure

very small energy shifts and splittings of nuclear levels caused by the

inter-action of the atomic nucleus with electric and magnetic fields. Many surveys

give the theory describing the interaction between a nucleus and its

environ-ment [1-4].

Generally the interaction of a nucleus in a solid can be represented by

the Hamiltonian:

H^H^^B^.H^.H^,

(3.1)

where H represents the intra-nuclear forces, H the energy change due to

penetration of electrons into the nuclear volume, ft. the interaction of the

nuclear electric quadrupole moment with an electric field gradient at the

nuclear site, and ft the interaction of the nuclear magnetic dipole moment with

a magnetic field at the nuclear site. The term ft in equation (3.1) is larger

than the other terms by a factor of about 10 or more and it is responsible for

the nuclear levels as they are observed by normal y-ray spectroscopy. Without

interaction between the nucleus and its environment, the eigenstates of ft will

be degenerate in the spin quantum number m,. This degeneracy will be removed or

reduced by the terms ft and ft of equation (3.1) respectively if the eigenstate

of ft has a spin or a quadrupole moment Q. The Hamiltonian ft is spherically

symmetric [3] and has no influence on the degeneracy of the eigenstates of ft .

The only effect of ft is a small shift in energy of the eigenvalues of ft ,

independent of the spin and quadrupole moment of the eigenstates of ft

III.1.1 Isomer shift

The electrostatic energy shift of a nuclear level, due to the finite size of the nucleus and penetration of electronic charge into the nuclear volume, is usually calculated by taking the difference between the electrostatic interaction of a hypothetical point nucleus and one of an actual radius R. The nuclear charge distribution is assumed to be uniform within a sphere with radius R and zero outside this sphere. The electronic charge density is assumed to be uniform over the nuclear volume. Only s-electrons have a finite density at the nuclear site which is equal to \ii(o) \ . ft can now be given by [4]:

ftp = I IT Z e V |iJj(o)l^ , (3.2)

where Ze is the nuclear charge. The nuclear radius will in general be different for each nuclear energy level and one can only observe the energy difference between two such levels. The expression for the change in the energy of the y-ray due to the nuclear electrostatic interaction is therefore the difference of two terms like equation (3,2), For the actual case of Fe in which the Mossbauer transition takes place between the nuclear ground state and the first excited state the energy change AE is given by:

AE = I TT Ze^ |iJj(o)|^ (R^^ - R ^ ) , (3.3)

R and R are the nuclear radii in the excited and ground state respectively.

e g

In a Mossbauer experiment one measures the energy difference between the y-ray emitted by the source and the y-ray absorbed by an absorber.

2 2

As R - R IS a constant for a given Mossbauer transition an energy shift can only be observed if source and absorber have a different value for |il)(o) | . This shift is usually called the isomer shift 6 which is given by:

6 = AE - AE = I TT Ze^ (R ^ - R ^) [ U (o) I " I'l' (o) I ] . (3.4)

a s 5 e g '-'^a ' s ' ' -> ' ^ '

2 2

where the subscripts a and s refer to absorber and source. Since R < R for

e 8

the 14.4 keV level in Fe [ 5 ] , 6 is positive when the s-electron density in the

absorber is smaller than in the source, and the center of the measured spectrum

is then shifted towards a positive velocity. Due to the shielding effect of the

3d - valence electrons of iron, |ip(o)| increases with the ionicity of the iron

ion [ 5 ] . More information about the systematics of isomer shifts is given in

[ 6 , 7 ] .

III.1.2 Electric quadrupole interaction

The second term in the multipole expansion of the electrostatic interaction

of a nucleus with its surrounding electronic charge is the quadrupole coupling.

This is the result of the interaction of the nuclear quadrupole moment Q with the

electric field gradient produced by the electronic charges in the crystal. Due

to this interaction a Mossbauer spectrum can show a quadrupole splitting. The

field gradient is a 3 by 3 tensor, which can be reduced to a diagonal form so

that it can be specified by the three components:

•S^V ,, 6^V , „ 6^V

V = — = • , V = — ^ and V = — = • ,

X X . 2 yy s: ^ Z Z . 2

ox oy oz

where V(x,y,z) is the electrostatic potential. These three components are not independent, since they must obey the Laplace equation:

A V = V + V + V = 0 . (3.5) XX yy zz

In equation (3.5) the charge density of the s-electrons is neglected since the s-electrons have a spherically symmetric distribution and do not contribute to the field gradient. The electric field gradient is often expressed in terms of V or eq and the asyrranetry parameter ri, defined as:

V - V

r,=-^ i l . ( 3 . 6 )

V

zz

ft of e q u a t i o n ( 3 . 1 ) may be w r i t t e n as [ 4 ] :

*Q = 4 1 7 ^ f3 I^ - I d . 1) . 1 Ti al - l ! ) ] . ( 3 . 7 )

where I. = I + il , I is the quantum number of the nuclear state considered. ^^ x — y

Assuming ft = 0 in equation (3.1) and since ft >> ft , ft is applied as a first order perturbation of ft . In this case the exact solution of (3.7) is given by:

3 m^ - 1(1+1) 2 1/2

e = e ^ Q i [ 1 + 5 - ] , ( 3 . 8 )

I 41(21 - 1 ) ^

where m is the magnetic quantum number of the nuclear state considered. In the case of Fe the Mossbauer transition takes place between the nuclear ground

1 . 3 state with spin I = -T, and the first excited state with spin I = —. The ground state with spin — has a Q = 0, while the first excited state has a Q = 0. The resulting splitting of the levels is shown in fig. III.l under the assumption q Q > 0. The quadrupole splitting is then given by:

1/2 2e = £3/2 - =1/2 = 2 ^ ^

2 -1 1 2

Q[-^^

Equation (3.8) represents the first order term of the perturbation calculations. However, in some cases the second order terms can be of the same order of magnitude as the first order ones as calculated by Hafner et al. [ 8 ] . Since

these terms are not relevant in the experiments described in chapter V, they will not be treated here.

ie^qQ

-«-

tVz

t'U

Fig, III.l Representation of the two lower levels of Fe and the magnetic dipole transition between these in the case of an electric quadrupole splitting only,

III.1.3 Magnetic interactions

The fourth term ft of equation (3.1) represents the interaction between an effective magnetic field at the nucleus and the magnetic dipole moment of a nuclear level. The Hamiltonian of the interaction can be written as:

-^^•\ = -

8 ^ H^ • I.

where g is the nuclear gyromagnetic factor of the nuclear state considered, I the nuclear state spin operator, H the effective magnetic field at the

-24

nucleus and y ~ 5.05 x 10 erg/Oe, the nuclear magneton. By using first

order perturbation calculations one obtains the Zeeman hyperfine splitting which results in an energy shift:

for a particular nuclear state with magnetic spin quantum number m . In the case of Fe the first excited state has a spin I = 3/2 and the ground state has a spin I = 1/2, resulting in a splitting of four and two sublevels of the excited and ground state respectively. Since the y-ray transition in Fe is a magnetic dipole transition, the allowed transitions are characterized by Am = 0, +^ 1. This leads to six possible transitions shown in fig. 111,2 a,

T-f

%

'A

-L

+'/2

^

'

1

'

ie»qQ

Fig, III.2 Representation of the two lower levels of Fe and the magnetic dipole transitions between these in two different cases:

a) with a magnetic hyperfine splitting only and

b) with a magnetic hyperfine and an electric quadrupole splitting.

In order to measure this six different transitions in the absorber of a Mossbauer experiment usually a source is applied in which neither an effective magnetic field nor an electric field gradient is present at the nucleus. When the g-values are known, H is determined experimentally from the separation of the resonant absorption lines. The individual transition probabilities are given by the squares of Clebsch-Gordon coefficients and they depend on the angle 9

between H and the direction of propagation of the y-rays. For Fe the relative intensities are given in (3.12).

Transitions 3/2 1/2 -3/2 -1/2 1/2 1/2 -1/2 -1/2 -1/2 1/2 1/2 -1/2

Am

0

0

3

2

1

The effective magnetic field H is given by [1,2]:

H„ = J: 2y„ 1. 3 r.(s, . r.) + 4 TT Si 6 (r-) •i J (3.12) (3.13)

The summation must be taken over all electrons of the atom or ion, surrounding the nucleus, y is the Bohr magneton while T. and s. represent the orbital- and the spinmoment of the i-th electron with position vector r. respectively. The first term

r.

2

1

(3.14)

of equation (3.13) gives the orbital contribution to H , which is zero for e

closed and half-filled shells with spherical symmetry. The second term of (3.13)

^ 2 u 3 r.(s. . r .) s. 1 1 1 1 _(3.15)

is due to the dipolar interaction of the nuclear magnetic moment with the magnetic moment of the electron spin. This contribution to H is again zero for closed shells and half-filled shells with spherical symmetry. The last

term of (3.13) gives the Fermi-contact interaction [9] H which can be c written as; H = r |-JTy i k . ( o ) | ^ (3.16) C . D D 1 1 1 2

where |ljJ.(o)| is the density of the i-th electron wave function at the nucleus. The spin density arises partly from the localized s-electrons and partly from the conduction electrons. The contribution of expression (3.16) is small when no spin polarization of the s-electrons would occur. However, due to the spin polarization of the s-electrons by the unpaired 3d-electrons in the case of Fe [10] a different electron density results at the nucleus for electrons with spin parallel or antiparallel to the spin of the 3d-electrons. As a result the Fermi contact interaction H , becomes the most important contribution to H .

c e Watson and Freeman [2] have calculated the resulting H using the

Unrestriced Hartree-Fock method and the results are in very good agreement with the experimentally determined values.

111.1,4 Combined magnetic and electric quadrupole interaction

When the magnetic interaction as well as the electric quadrupole interaction are present at the same time there is no longer a unique quantization axis and the problem can in general not be solved in closed form as described in the previous sections,

The sum of ft and ft from equation (3,1) has to be considered as the total perturbation Hamiltonian on ft , In the case that ft >> ft., ft may be considered as a perturbation on ft + ft , When a coordinate frame (X,Y,Z) is chosen in which Z // H , this will in general not coincide with the (x,y,z)-frame used in expres-sion (3.7). The energy shift due to the quadrupole interaction follows from the first order perturbation theory and is given by [ 1 ] :

(3 cos^e - 1) + ri sin^ecos2(|) , (3,17)

where 6 and ()) are the polar angles of the direction of H with respect to the e

(x,y,z)-frame in which the electric field gradient tensor is diagonal. The first order energy shifts are shown in fig. III.2 b.

In the case that ft is of the same order of magnitude as ft in general numerical methods have to be used to solve the problem. For Fe a computer program is described in ref. [11].

III.2 Lattice vibrations

Due to lattice vibrations the total intensity of the resonant absorption lines in a Mossbauer spectrum depends on the temperature of both absorber and source. Furthermore the vibrational velocity of the nuclei leads to a second shift in the center of gravity of a Mossbauer spectrum which is unrelated to the isomer shift described in section III. 1,1, The first influence of the lattice vibrations is expressed by the recoilless fraction f or Debye-Waller factor while the second one is given by the second-order Doppler shift.'

III.2.1 The recoilless fraction f

The recoilless fraction f is defined as the fraction of the y-ray emission or absorption processes which take place without any energy loss. It is clear from chapter I that f=0 for free nuclei, while f=l when the y-ray transitions take place in a completely rigid lattice. In general the value of f will be between these two extremes, depending on the nuclear transition energy E , the mass M of the nucleus, the rigidity of the lattice characterized by the Debye temperature 9 and the temperature T. For a lattice in which all nuclei occupy equivalent lattice sites it can be derived from classical

mechanics [12] as well as from quantum mechanics [13,14] that f is given by the expression:

f = exp ( - k^<x^>), (3.18)

where k is the wave vector of the nuclear radiation, and <x > the mean square displacement at temperature T of the nucleus considered in the direction of propagation of the y-rays. Using the Debye approximation for the lattice vibrations expression (3.18) can be rewritten as [ 3 ] :

% / T f = exp 3 E,

R

where E is the recoil energy of the free atom, k the Boltzmann constant,

R B 9 the Debye temperature and T the absolute temperature. It follows from

(3.19) that only when E << k9 an appreciable recoilless fraction occurs. The recoilless fraction f of an absorber is directly related to the area of a Mossbauer spectrum corresponding with resonant absorption. The normalized area A of a spectrum can be given by:

1

I(-)

dv

1

-nf'a(E)

d E , (3.20)

where v is the relative Doppler velocity between source and absorber, I(v) the y-ray intensity at a Doppler velocity v, I('") the y-ray intensity at such a velocity where no resonant absorption occurs. In expression (3.20) f and f are the recoilless fractions of the source and absorber respectively,

2

n the number of resonant atoms per cm with a resonant cross section o ( E ) . The resonant cross section a(E) is given by:

a(E)

2TT^2 ( i ^ + i ) ( ^ r ^ )

(ir^)

1+a

(i^^iT-(E-E ) ^ + (ir)2 ° • (i^^iT-(E-E^)^ + (i n ^

(3.21)

where ^ is the wavelength of the resonant radiation divided by 2TT, I and I the spin quantum numbers of the excited and ground state of the nucleus

respectively, E and T the energy and natural line width of the nuclear excited state and a the coefficient of internal conversion. The expression (3.20) for A is rewritten by Lang [15] in a power expansion which is given by:

A = i TT f r ? ~ r - • [ll-l]':' ^°o f'"^H'' ^ ^ ^^"^^ (^•22)

where t = a f'n, named the effective thickness of the absorber. For values

o

of t < 1, F(t) is approximately equal to t, that means that the normalized area of a Mossbauer spectrum is proportional to f . By determining this absorption area as a function of temperature one gets information about the lattice vibrations and thus about 9 . However, for determining the absolute value of f it is necessary also to know the recoilless fraction of the source. The latter can be measured using the black absorber technique [16],

The expression for f given in equation (3.19) is only valid when the

lattice vibrations of the Mossbauer nuclei are isotropic. Then the recoilless fraction f is independent of the propagation direction of the y-radiation. However, anisotropy of the vibrations of atoms can exist which results in an anisotropic recoilless fraction f. In the case of axial symmetry of the re-coilless fraction, f is given by [17]:

f(e) = exp - k^ {<x^> + (<x^//> - <x^>)cos^e} (3.23)

where 9 is the angle between the wave vector k and the axis of symmetry. 2 2

The terms <x > and "^x .. > are the mean square displacements perpendicular and parallel to the symmetry axis respectively. The anisotropy of f leads to an asymmetry of the quadrupole splitting components in a Mossbauer spectrum of a randomly oriented powdered sample. This effect is known as the Goldanskii-Karyagin effect.

III.2.2 Second-order Doppler shift

In contrast to the isomer shift the second-order Doppler shift is a relativistic effect [18,19], When a nucleus with a velocity of v emits a y-ray of energy E , this energy undergoes a Doppler shift given by:

E = E o 1 + — cos a r '• ^

[ • - 7 J

where a is the angle between the propagation direction of the y-radiation and the velocity v,

For small velocities v << c, (3,24) can be approximated by:

2

1 + — cos a

1

2 c ^

-(3.25)

The first term op (3.25) in brackets denotes the classical Doppler shift while 2

V . • > the second term — ^ describes the second-order Doppler shift. As the vibration

2c

time of nuclei in a lattice is much shorter than the life time of the nuclear state, the linear term — cos a in eq (3.25) averages out. The quadratic term ^2

—i r will remain and causes a very small shift in the energy of the emitted or

2c

absorbed y - r a y . This energy s h i f t i s temperature dependent and i t can be given

by:

i ( J ^ ) = - - ^ . (3.26)

where C is the lattice specific heat. In the high temperature limit, where C = 3k , this coefficient is equal to 2.44 x 10 per degree for Fe.

L B

k is the Boltzmann constant. B

III.3 Superparamagnetism

A particle of ferromagnetic material below a critical size consists of a single magnetic domain. This is first predicted by Frenkel and Dorfman [20] in 1930. Kittel [21] and others estimated for a spherical sample of the common

ferromagnetic materials roughly a critical particle size of 150 A. The magnetic behaviour of a single domain particle changes drastically in comparison with the bulk material with decreasing particle size. An assembly of ultrafine par-ticles can reach in some way a state of thermodynamic equilibrium when no

external field is applied. This means, that the magnetic moments of the particles are oriented randomly so there is no macroscopic magnetization and hence no remanence. In the case of a vanishingly small anisotropy the magnetic behaviour of an assembly of single domain particles in thermodynamic equilibrium can be calculated very easily.

When an external field H is applied to an assembly of such particles each with a total magnetic moment y, there will be a Boltzmann distribution of the orientations of y with respect to H. The fraction of the saturation magnetiza-tion that has been aligned by the field is calculated by averaging the inter-action energy - yH cos 9 of a particle moment which makes an angle 9 with the field over the Boltzmann distribution. The result is the wellknown Langevin function

yH k„T -I M = M

^°^^

kTT

-

W

where M is the resulting magnetization and M the saturated magnetization and T the absolute temperature. The behaviour is the same as for classical para-magnets. Only the magnetic moment y is not that of a single atom, but rather of a single domain ferromagnetic particle which may contain more than 10 atomic moments coupled together. The approximations of equation (3.27) for low and high fields are given by:

M/M^ = 3 ^ for yH « k^T , (3.28) B

k T

M/M = 1 - - 5 - for yH » k T . (3.29) s yH B

However, generally speaking single domain particles will have anisotropic contributions to their total energy. This anisotropy energy can be due to the crystal structure but it can also be due to the particle shape or to internal stresses. The anisotropy contributions cause a change in the Boltzmann distri-bution of 6's in thermal equilibrium, and the magnetization curve will no longer be a single Langevin function. However, in the case of a randomly oriented assembly of uniaxially anisotropic particles and m the case that the anisotropy IS of cubic symmetry the relations (3.28) and (3.29) are still valid, indepen-dently of the direction of the external field [22].

It is quite obvious that the magnetic properties of an assembly of single domain particles m thermal equilibrium are totally different from those of stable single domain particles. So it is necessary to consider the conditions for approaching thermodynamic equilibrium of an assembly of single domain particles. The simplest manner in order to reach thermodynamic equilibrium is that the individual particles of the assembly rotate which is possible if the particles are suspended in a liquid medium [23]. In a solid this relaxation mechanism will in principle not be available. However, m 1949, Neel [24] pointed out that if a single domain particle is smaller than a critical volume its direction of magnetization undergoes a fluctuation between the different possible easy directions of magnetization due to the thermal excitation energy. In the case of an assembly of single domain particles with uniaxial anisotropy that is fully magnetized in one of the easy directions, the total magnetization will decrease as:

-t/T

M = M e ° , (3.30) s

where M is the saturation magnetization, t the time after removal of the field and T the so-called relaxation time of the process. It is clear that

o

T IS correlated with the number of magnetic moments which change their o

direction per unit time. Since the uniaxial anisotropy energy E in first approximation is given by:

E = KV sin^9 , (3.31)

where K is the anisotropy energy per unit volume, V the particle volume and 6 the angle between the symmetry axis and the direction of magnetization, the relaxation time T is given by:

o

—KM

T = f exp ( /k T) , (3.32)

0 O D

9 1 2 - 1

where f is a frequency factor of the order of 10 - 10 s . The frequency factor is calculated in different ways by Neel [24] and Brown [25], both using random thermal effective torques. From the expression (3.32) it follows that the relaxation time T is very strongly dependent on the particle volume and the temperature. In general there is a narrow range of particle sizes only in which the relaxation times will be of the order of experimentally accessible observation times. When T is smaller than the observation time in an

experi-o

ment, an assembly of such particles magnetized up to saturation will reach thermal equilibrium within the observation time and no remanent magnetization will be observed.

So the magnetization behaviour of single domain particles in thermodynamic equilibrium is identical with that of atomic paramagnetism except that an extremely large moment is involved. Because of these similarities this behaviour has been called "Superparamagnetism". Following expression (3.27) the magnetiza-tion curve for an isotropic sample is temperature dependent in such a way that curves taken at different temperatures superimpose when plotted versus H/T after correcting for the temperature dependence of the spontaneous magnetization. However, it is not clear until now wether the temperature dependence of the spontaneous magnetization is the same for ultrafine particles as for the bulk material.