The Mössbauer effect and its applications at very low temperatures

The basic ideas and difficulties associated with nuclear gamma resonance absorption are discussed culminating with Mossbauer's great discovery of recoHess gamma emission and absorption. The extremely high inherent resolution of the Mossbauer ~f-rays provides a simple and powerful method of detecting extremely small changes in energy. In this review the applications of Mossbauer spectroscopy to problems in magnetism, superconductivity, Kondo effect, nuclear polarization and relativity are discussed, with particular regard to use of very low temperatures and the information thus obtained. Basic Mossbauer systems and cryostats which enable temperatures down to ~30 mK to be maintained for long periods are described. An interesting application discussed is where the M6ssbauer effect may itself be used as an absolute thermometer incorporating its own internal calibration.

The M6ssbauer effect and its applications at very low temperatures J. M. Williams Resonance fluorescence The process of resonance fluorescence is shown diagrammatically in Fig.l. At resonance, absorption occurs with subsequent re-emission of the resonant radiation in all directions, thus causing a net drop in the output of detector 1 and an increase in that of detector II. This illustrates the two fundamental methods of observing resonance fluorescent (1) the transmission method using detector I and (2) the scattering method using detector II.

17I

Firstly let us consider a cannon which, when fixed to the ground, fires projectiles a lnean distance E o. If a large number of such projectiles are fired there will be some distribution about E o due to straggling, as shown in Fig.2. We may say that there is some inherent 'natural width' F. If the caunon is WOWmounted on a small boat and the same experiment repeated, we find of course that the projectiles fall some distance R short of E o due to the fact that the boat recoils so as to conserve momentum. Again we observe a distribution with 'natural width' F as shown in Fig.3. The distance R by which they fall short o r e o may easily be calculated from energy and momentum considerations. In practice the surface of the lake may not be quiet as assumed in Fig.3. If rough, the boat will pitch and roll and the disThe a u t h o r is in the D e p a r t m e n t o f Physics, U n i v e r s i t y o f Sheffield, S h e f f i e l d $3 7 R H , UK. Received 24 March 1975.

CRYOGENICS

. JUNE

1975

i

Fq a

The phenomenon of atomic resonance fluorescence has been known for a long time 1 and accounts for such effects as the dark Fraunhofer lines observed when the sun is viewed through a spectrometer. In this case we have, for example, the optical photons emitted from the hot solar surface being resonantly absorbed by the cooler gases in the sun's atmosphere, and hence giving rise to the relatively sharp Fraunhofer absorption lines. A similar resonance absorption by nuclei of nuclear T-rays emitted by an excited nucleus of the same type as the absorber was thus predicted as long ago as 1929 2 but for reasons which we shall discuss, no appreciable nuclear resonance fluorescence was observed until M6ssbauer's Nobel prize winning discovery in 1958. 3 The reasons for tiffs are perhaps best illustrated by reference to the following classical analogue due to Debrunner and Fraunfelder. 4

. . . . . . . .

!°I

E] b Fig.1 D i a g r a m m a t i c representation o f resonance fluorescence a -- no resonance; b -- resonance Q = r a d i a t i o n source, A = absorber, I and II =~ radqation detectors

N(E) I

F

'No i

//

. ' " "~"

E

"~--.

I Eo

--~

I

~///~////////////////////////////////////.'//////7~//7 E 0

E°

Fig.2 Straggling in the distance o f projectiles shot by a f i x e d cannon. N(E) is the number of projectiles observed at the distance E

307

Nol---~ NE)

We note that R ~- E°2 2Mc 2

/

/!\e°

~ex~l

•

~

%~1

I o

_1_

,E

///////////////////'t/////

Fig.3 If the cannon is not fixed, it can recoil. As a result, projectiles fall short o f target by a distance R

N(E)

No

, ""

~E

lit is instructive to note that for atomic transitions E o eV and therefore D "" R giving appreciable overlap of emission and absorption spectra. Thus it is due to the much higher energies (E o ~ keV-MeV) of nuclear transitions that whereas the conditions necessary for resonance fluorescence in the atomic case are often satisfied, in the nuclear analogue they are normally not satisfied].

From the classical analogy it is obvious that to observe appreciable nuclear resonance absorption we must 'freeze out' the effects of recoil and Doppler broadening. In 1958 M6ssbauer 3 discovered that under certain conditions nuclei embedded in lattices at low temperatures were effectively 'frozen', and hence unable to recoil and experience Doppler broadening of their emission and absorption lines. This 'freezing' arises due to the quantized nature of the lattice vibrations and provided R < Ep, where Ep represents the energy spacing of these quantized levels, there exists a finite probability that the nucleus may not recoil alone, the recoil momentum being taken up by the whole crystal. As the mass of the whole crystal is effectively infinite the energy lost to recoil R "" E ~ / 2 M c 2 is zero. The probability for such 'recoil-free' emissions and corresponding absorp-

Fig.4 Shots fired f r o m a cannon mounted on a small boat in a rough sea experience both a recoil (R) and a Doppler broadening (D)

---f--;-_--4 ~__s----_--4~/#//////////~/////// 0

and that D >> P. Note that the energy required for absorption is higher than E o by an amount R, because momentum and energy must also be conserved in this process and some of the energy goes into causing the absorbing nucleus to recoil. For nuclear 7-rays it turns out that D < R at normal temperature and so the overlap of the emission and absorption spectra (which represents the possibility of resonance absorption) is negligible.

M6ssbauer's discovery

"

~-::

(2)

Eo

Fig.5 If the lake is frozen, Doppler broadening and recoil are avoided and aiming is simple; straggling still persists

tribution will suffer an additional broadening D which we may call the 'Doppler broadening', as shown in Fig.4.

T

B , / - / - / 7 " / - / ; t 7 / : / , -- E°

I(E )

The only way to avoid the recoil R and the Doppler broadening D is to wait until the lake is frozen (Fig.5). The analogous 'freezing' of a lattice containing radioactive nuclei is the essence of M6ssbauer's great discovery. In the nuclear case the 'natural width' of a 7-ray emitted with mean energy E o arises due to the Uncertainty Principle and is related to the nuclear lifetime 7- of the excited state by Pr

=

h

or

h P =-

0

Eo

S~

--¢2

- -

Fig.6 Natural line width. The energy o f the level B with a mean life r h a s a w i d t h F = h/r

(1)

T

The energy distribution from such a nuclear decay is shown in Fig.6. This of course assumes the nucleus to have infinite mass, and in that case the 7-ray could be resonantly absorbed by a similar nucleus of infinite mass which was in the ground state (that is, nuclear resonance fluorescence). When the nuclei are however embedded in real lattices the effect of recoil and Doppler broadening due to the thermal oscillations must be included and the emission and corresponding absorption energy distributions are as shown in Fig.7.

308

A

I(E)

l

I t

Overt°O--4-. Eo -R

l

I

Eo

Eo+R

Fig.7 Emission and absorption lines are Doppler broadened, and a small overlap exists. The Doppler broadened lines are much less peaked than the natural ones

CRYOGENICS

. JUNE 1975

tions, known as the Lamb--Mossbauer recoil free fraction, is given by s

Emission

spectrum

i"~-Ee Eg

.f= exp k 2 < x 2 >

(3)

where k = Eo/hc. On the Debye model this reduces to the following expression for the recoil-free fraction

E o= E. - Eg

~

Eo

0

Energy

0 D/T f=

exp"

1+4

2 k 0 t)

-0 I)

Absorption spectrum

dx ex o

1 (4)

where R is the free nucleus recoil energy, k is Boltzmann's constant, 0D is the Debye temperature of the lattice, and T is the absolute temperature. Such emission and absorption have an energy distribution having the natural width F and are centred at E 0, thus providing complete overlap and hence strong resonance absorption. The resulting spectra showing the recoil and recoil-free distributions are illustrated in Fig.8. Here the recoil-free fraction is shown as a delta function as F "~ D. Typically F / E o ~ 10 -li -10 "16 and so small relative velocities between source and absorber will destroy the resonance. Velocity modulation thus provides us with a method of sweeping the source energy through the corresponding absorption spectrum and measuring the relative absorption. For an identical source and absorber using the arrangement shown schematically in Fig.9, an absorption spectrum having twice the natural width U and centred at zero relative velocity is obtained. Table 1 shows a selection of the most commonly used M6ssbauer isotopes, the corresponding parent nuclei and their half-lives, the energies of the 3,-rays, and their natural, fractional and minimum observable widths. The scope offered by Mossbauer spectroscopy is evident from the large number of isotopes with suitable nuclear transitions. Of these the most often used are those of iron, tin, antimony, iodine, europium, and dysprosium, all of which are easy to work with, and which yield interesting results. They show strong resonant fractions, have narrow line widths suitable for hyperfine studies and represent different regions of the periodic table. Experimental

arrangements

Details of the experimental arrangements used for producing M6ssbauer velocity spectra have been described extensively in the literature, 6 9 and only a summary of the most common method will be given here. A Mossbauer experiment is essentially a measurement of the resonant absorption in the sample being studied as a flutction of energy, the energy being varied by using the Doppler effect. This amount to a measurement of counting rate at the detector as a function of velocity. The generally accepted method is to use a multichannel analyser whose memory stores are swept in synchronisln with the relative velocity normally imparted to the source. The source may then be moved cyclically so that the velocity (and hence energy) range required is covered several tinles per second. In order to ensure a linear energy scale the velocity of the source is generally arranged so that it is linear with respect to time, and the analyser spends equal times in each channel. Such a complete Mossbauer spectro-

('#VN6FI',IIN£

JUNE

1975

---" /

l---

0

",,2_--E0

r.-

Energy

Fig.8 Structure of gamma emission and absorption lines of nuclei bound in solids. Shown are line structures for t w o extreme cases: (1) the case of very high recoil energy where multiphonon processes dominate (dashed lines) and (2) the case of very low recoil energy, where only few phonons participate (solid line)

The process

The measurement

The result

Abs°r5

Emission

Resonance ) O obsorption / . / ~ ) / / - / - / Q - ~ / / -

i

o

( ~ ) Eo

iVelocity, v

Fig.9 Principle of resonance absorption measurements. Maximum resonance absorption, that is, minimum transmitted intensity, occurs whenever emission and absorption lines are centred at the same energy. The perfect resonance may be destroyed by means of the linear Doppler effect, if one applies various relative velocities between source and absorber. In this way one scans w i t h an emission line of natural width 1", an absorption line of natural width F, a procedure which gives rise to a transmitted line of width 21", for an infinitely thin absorber

meter is shown in Fig.l 0. This shows how the synchronism between velocity and MCA sweeping is obtahred, in addition to the normal 7-ray detection arrangement used. Most experiments, and almost all the exalnples used to illustrate this review, make use of the transmission geometry illustrated in Fig.10 with absorption producing a relative drop in counting rate. For some experiments, particularly those involving surface studies and materials which cannot be made thin enough, the use of the scattering geometry illustrated in Fig.1 is favoured. In these scattering experiments absorption corresponds to a relative increase in counting rate and the M6ssbauer spectra are simply inverted In this review we shall concentrate mainly on the low temperature aspects of the ME with a discussion of the experimental techniques used and the information obtainable from such low temperature studies. For a general introduction to the field of 3`-ray spectroscopy the reader is referred to the general references, i° i6

309

Table 1. Table of Mossbauer isotopes*

Absorbing nucleus Fe 57 Ni 61 Zn 67 K r 83 Sn 119 Sb 121 Re 125 1127

1129

Xe 129 Sm 149 Eu 151 D y 161 T m 169 Ta 181 Pt 195 A u 197 Np 237

Parent

Half-life

T-ray energy, keV

0057 0061 Ga 67 Kr 83m Sn 119m Sn 121m Te 125m me 127 Te 129 1129 Eu 149 Gd 151 T b 161 Er 169 W 181 A u 195 Pt 197 A m 241

270 days 1.7 h 78 h 2.4 h 250 days 5 yr 58 days 9 h 70 min 107 yr 106 days 120 days 6.9 days 9.3 days 140 days 192 days 20 h 458 yr

14.4125 67.4 93.31 9.3 23.8 37.2 35.5 57.6 27.75 39.6 22.5 21.6 25.7 8.41 6.23 98.8 77.3 59.5

Eo,

Natural w i d t h [', eV 4.66 8.77 4.91 3.10 2.57 1.30 2.93 2.40 2.72 4.52 6.41 4.70 1.62 1.17 6.71 2.64 2.41 6.84

x x x x x x x x x x x x x x x x x x

10 -9 1.0-8

10-11 10-9 10 .8 10-7 10-7 10-7 10-8 10-7 10 .8 10-8 10-8 10-7 10-11 10-6 10-7 10-9

Minimum experimental line w i d t h 2 [', mm s"1

F/Eo 3.2 1.3 5.3 3.3 1.1 3.5 8.3 4.2 9.8 1.1 2.9 2.2 6.3 1.4 1.1 2.7 3.1 !.1

X X X X X X X X X X X X X x x x x x

10 "13 10 "12 10 "16 10 "14 10 "12 10 "12 10 "12 10 "12 10 -13 10 "11 10 "12 10 "12 10 "13 10 "11 10 "14 10 11 10 "12 10 "13

0.1940 0.7805 3.152 x 10 .4 0.1980 0.6467 2.104 4.945 2.401 0.5868 6.843

1.708 1.303 0.3795 8.330 0.006457 16.28

1.871 0.06888

*compiledfromthe M0ssbauerEffectDataIndex39 =pply Absorber

l Feedback i Drive amplifier | Pickup

i

Sourc~ 2

B

Vibrator

I

+

)lifter I

I

control

Mainamplifier

Drivewaveform Multichannel store

Address pulsesadvance

I

I

Sel ectedpart ofspectrum

i

I analyser

I

I

I

Output I The block layout of a typical Mossbauer spectrometer

Fig.10 ioo%

6 2 keY IO%

O= 230 K Talel

~ 134 keV

0:31oK

0 I%

I

I { II

I0

i

10 2

~

\

I

10 3

II

I0 4

Temperature.K Fig.11 Temperature dependence of the f-factor for the gamma transitions of one of the lowest and one of the highest energies which have been observed as yet in measurements of recoilless resonance absorption

Low temperature requirements The reasons for using low temperatures in Mossbauer studies may be divided into three categories. 1. When using M6ssbauer nuclei for which the Lamb Mossbauer fraction is very small at high temperatures and

310

very strongly temperature dependent. From (3) it is seen that to increase f w e must reduce < x 2 >, the mean square thermal displacement. The temperature dependence o f f as calculated on the Debve theory is shown in (4} and in Fig. 11 we see how this varies for the two M6ssbauer isotopes Ta 181 and Re] 87. Fortunately the many isotopes including the commonly used Fe sv (for which f - ~ 0.7 at room tempera ture), f i s appreciable even at room temperature. For many other nuclei however the temperature dependence is very strong indeed and for W 182, for example, f i s only a fraction of a percent even at liquid helium temperatures. For such isotopes the use of low temperatures is absolutely essential. 2. When the physical properties of the source or absorber under study are only displayed at low temperatures. Such properties include magnetic effects, superconductivity, electronic properties, relativistic shifts, nuclear polarization, etc encountered in solids. Illustrative examples will be given following the discussion of hyperfine effects. 3. When the application of a very large applied magnetic field at the source or absorber is required, it is usual to use a superconducting solenoid. As Mossbauer experiments run typically for times of the order of a day, it is most con,)enlent to run such a magnet in the persistent mode. By cooling the filamentary NbTi magnets currently available to

CRYOGENICS.

JUNE

1975

Dewar. fixed

2~-~

I

/

"

-

I

I

Radiation

~//

shield

4 5 !i/ ,/j r

i

"

'°ur e"i t't

J

/ De'ec'or

1""," III

Absorber Fig.12 S c h e m a t i c d i a g r a m o f a t y p i c a l c r y o s t a t in w h i c h t h e absorber a l o n e is c o o l e d

temperatures well below 4.2 K, a significant increase in critical field may be obtained. Pumped helium cryostats are therefore becoming a not uncommon sight in Mossbauer laboratories.

Cryostats Mossbauer cryostats are divided into two types, those which cool either the source or absorber, and those which cool both. For most applications it is necessary to cool only the absorber, the need to cool both source and absorber being restricted mainly to isotopes for which f is only appreciable at low enough temperatures. In the first type, shown schematically in Fig.12, the absorber is cooled either by liquid nitrogen for temperatures down to 77 K, or by helium when temperatures down to 4 K are required. The vacuum and radiation shield windows are made from a material which has a low atomic absorption coefficient for the "/-ray energies used and which is known not to contain any of the resonant nuclei being studied. Aluminized Mylar is very often used for such windows and may be glued successfully onto metal or glass surfaces. When both source and absorber cooling is required a system such as that shown in Fig.13 is often used. In such a system the source (which in some cases may be held at a different temperature from that of the absorber), is given the necessary Doppler velocity via a stainless steel drive tube which is connected to the vibrator outside the cryostat itself. To change the absorber (or source) the whole velocity drive unit may be lifted out from the cryostat. The -),-radiation is again extracted via Mylar windows at the bottom and detected by a suitable y-detector. Fig.14 shows an actual complete system used in the author's laboratory. Above is the arrangement for lifting out the drive unit and below is seen a cooled high resolution germanium semiconductor 7 detector. With such a He 4 system temperatures down to 1.5 K are readily obtainable by pumping the coolant. By incorporating a pumped He a insert this limit may be extended down to 0.3 K. Th6 price paid for such improvements however, is a loss of simplicity and much longer times for sample change. This is often unimportant, in Situations where individual spectra require several days counting, but for shorter runs may become a significant loss. Superconducting magnets may also easily be incorporated into such systems.

CRYOGENICS. JUNE 1975

The recently introduced continuous flow (CF) type cryostats may overcome these problems and are gaining in popularity due to the simplicity of their design and use. Gas or liquid (N 2 or He) is drawn from a storage container and used to cool the sample via a heat exchanger as shown in Fig.15. These cryostats have the advantages of rapid sample change, cool-down, and warm-up. By reducing the pressure of the liquid helium used temperatures down to 3 K are readily obtained with sample changing times of less than a minute and initial cool-down times of the order of 30 seconds. The liquid.helium consumption in cooling to 3 K may be as low as 0.5 litres. With minor modifications these cryostats can be used to cool both source and absorber if necessary, the drive again being provided from above by an external vibrator. These CF cryostats consume more coolant when running at the lowest temperatures whereas for the conventional bath-type cryostat consumption is greatest at high temperatures when heat has to be pumped into the sample. When temperatures well below 0.3 K are required it is necessary to use a He3/He 4 dilution refrigerator. Such a cryostat was first used for Mossbauer studies by Kalvius et al 17 and is shown schematically in Fig.16. In principle either source or absorber may be cooled to ~0.03 K by placing it in the mixing chamber, the other remaining at a temperature of around 1.5 K. Fig.17 shows the construction of the mixing chamber, in this case for a cooled source experiment. In this experiment the absorber would be moved below the source by attaching it via a fork to the

5

6 9 I0

12

16

15 17

Fig.13 H e l i u m - t e m p e r a t u r e glass c r y o s t a t 1 -- V e l o c i t y d r i v e u n i t ; 2 -- h e l i u m e x c h a n g e gas inlet, 3 - T e f l o n gasket, 4 -- g r o u n d glass surface, 5 - p u m p i n g a r m , 6 - stainless steel heat e x c h a n g e r t u b e , 7 -- stainless steel t u b i n g f o r absorber s u p p o r t , 8 -- stainless steel push rod, 9 -- h e l i u m e x c h a n g e gas, t 0 -- l i q u i d n i t r o g e n , 1 1 -- h i g h - v a c u u m valve f o r seal o f f , 12 -- l i q u i d h e l i u m , 13 source s u p p o r t e d b y f l e x u r e plates, 14 -- a b s o r b e r and heating coil, 15 -- g r o u n d glass flange, w i t h e p o x y c o a t e d M y l a r seal, 16 -- metal heat shield, 17 -- g r o u n d glass flange f o r O-ring seal, 18 -- metal b o t t o m flange w i t h M y l a r e x i t w i n d o w . T h e v e l o c i t y d r i v e u n i t a n d source-absorber m o u n t i n g can be l i f t e d o u t o f t h e c r y o s t a t f o r c h a n g i n g absorbers (or sources)

311

being studied, the Fe s7 M6ssbauer nucleus accounts for approximately 50% of the data published so far.

Isomer shift Due to the finite probability of s-electrons penetrating the nucleus, the M6ssbauer transition energy is modified due to the electrostatic interaction by an amount which depends

r ~ : : ~ : ~ ~

Sample access port Combined evacuation and safety valve

"-~

~

Helium inlet

Helium outlet

Exchange

/ nH Window

Heat exchanger,temperature sensor and heater

i/1

I

Fig.14 Photograph of an actual Mossbauer cryostat used in the author's laboratory, together w i t h drive and detector systems. The source drive system is about to be loaded into the cryostat f r o m above

gasspace

.•ays \

I I

L

-

Alternative transverse or longitudinal arrangements possible

~

7"rays

vibrating rod coming down from the vibrator above. Sample temperatures down to below 0.03 K have been achieved for operating times up to 24 hours with only one filling of the He 4 pot.

Fig.15 A typical continuous f l o w Mossbauer cryostat which allows the absorber alone, or source and absorber to be cooled, Note the alternative geometry possible in each case

Hyperfine interactions Applications of the Mossbauer effect make use of its extremely high energy resolution. The natural line width of the M6ssbauer radiation expressed as a fraction of the energy (F/Eo), may be as low as 5 x 10 "16 (for example, the 93 keV ?-radiation from Zn 67, which has a lifetime ~13/2s) and is typically ~ 1 0 "13. The extremely high inherent resolution of the M6ssbauer process is illustrated dramatically in Fig.18. These line widths are often comparable with or smaller than the interactions between the nuclei and their atomic electrons -- the so-called hyperfine interactions. Thus one of the most useful M6ssbauer applications is to the study of the strength and nature of these interactions in different electronic, chemical, and magnetic states of atoms. This has opened up vast new possibilities for the study of condensed matter, but in this article we shall concentrate on their application to studies at low temperatures. The three hyperfine interactions normally measured are the isomer shift, magnetic dipole interaction, and electric quadrupole interaction. These will be briefly reviewed using the commonly used Fe sv nucleus as a particular examples with which to illustrate their effect. Although, as seen from Table 1 many other nuclei are now

312

~L_._.~Vibrator

~

Pumping head

Syphonfor filling the He4 pot He4 line Tube for filling the He' reservoir To vacuum pump

in vacuum

Vacuum sealed amphenol connector He3 line Vacuum seals He4 return line He3 line Thermal anchor at 80 K Vibrating rod Safety valve

r

Vacuum
Operating space Cone at 7 K i

Helium reservoir

--Copper shield at 80 K

=-II[t]

Constriction AIshield

Superconducting

solenoid

He pot a t l 2 K Still at 0 8 K ~:~1111 i,z,4,5,6 = Sintered heat ~-~:JJ exchanges 3= Concentric tube heat exchanger ~ S o u r c e "'-Mixing chamber Absorber pws \

1

I~.~ L~:

Radiation shield ~

Be 0 I

I

I0 I

20 cm I

I

I

Fig.16 Schematic illustration of the He3/He 4 Mossbauer cryostat used b y Kalvius et al 17

CRYOGENICS. JUNE 1975

Nylon support

[]

A

Stainless steel

He ~n Stycost seal Be window

OCoer ~

known, the magnetic field H may be determined. In the absence of any external fields, the hyperfine field Hhf is given b y

[ [(37. (s. 7)

-._>

Beryllium

Hhf

2J3

= --

E

LD He3 liquid

+-r3

•

Indium seal

•

Thermometers

,Hg solder

~

Hea/He4 solution

It

Soft solder joint

Fig.17 One typical construction of the mixing chamber in a He3/He 4 Mossbauer cryostat

\ .... ;s

electrons

Sintered copper

s I~(0)12

(9)

3

"['he three terms in (9) represent the orbital, dipole, and Fermi contact contributions respectively, ~3being the Bohr

g~l' ~ "'~";"~iff ~'C~' '."'i ~ ~ !:~'~ ' ~"'"." '.:~i ~~ i..~¢'~ ''":"' ~w""'~>' i ......... ...~-'t~A~.......7~a'~, ~77" ~1

on the size of the nucleus and the density of s-electrons at the nucleus I~(0)l 2. This change in energy relative to a poinl nucleus may be written as

,

q 0

.

- Z e 2 (R2x 5

R2d)l~(0)l 2

(5)

where Rex and Rg d represent the nuclear excited and ground state mean charge radii, and Z is the atomic number. As this change may be different in source and absorber, to maintain the resonant absorption condition the difference in the two transition energies 2/T E,{

l:" s =

-

Z e 2 ( R2e x

t~s(0)12 [

2xlO -I°

.

4xlO -I° 6 xlO-IO Gamma- ray energy, eV

Resonance energy Eo-" 93000 eV

Fig.18 The sharpness of Mossbauer ?,-ray lines (here that of Zn (~7) is such that a plot on a linear scale that displays the resonance o n l y 1/8 inch wide must stretch f r o m Earth to Pluto if the origin is to be included. Note that resonance energy ( ~ 9 3 keV) is not k n o w n to 15 places: Mossbauer techniques have great sensitivity not great accuracy.

R2gd ) [l@a(0)12 Source

(6)

C

1:s

6E

5

Ze2ixp(0)l 2 R 2 (relatwe to point nucleus)

&

=

' 2~r Eo + _ Ze2i,l,s(0)12lR~x

I£a

=

Eo + - -

Isomer shift

=

Ea

IS

=

(7)

This difference m energy, ~ (normally measured in velocity units v), is known as the isomer shift of the absorber and, of course, may only be defined or measured relative to a particular source. [The standard reference source now being generally agreed for Fe s7 is sodium nitropruside (SNP)]. The origin and effect of such an isomer shift on the absorption spectrum of a Ferricinium bromide absorber is shown in Fig. 19.

(a)

2~T

v Eo = t : .

Absorber

{s)

nlust be compensated for by' producing a relative velocity' r between source and absorber such that

=

width

,

211

6l:" =

jJ De,o~ce p° ,0ev

:

ng2dl

5

2/r

5

21T

5

Rg2al

Ze2ig~a(O)t2[R2ex

E~ Ze2[blq(0)l 2

, q ' j 0 ) 21[R~x

'

R2 1

gd

Magnetic dipole interaction If the nucleus exp~iences a magnetic field, the interaction between the field H and the nuclear magnetic dipole m o m e n t /x causes a splitting of each nuclear level o f spin I into 2 / + I sublevels corresponding to the allowed values of

.=_o 2

g6 8 I0

z_XE'M = .

i

H. I

(s)

The effect of such a field on a Fe s7 absorber used in conunction w i t h an unsplit emitter is shown in Fig,20. From analysis of such spectra, provided the nuclear moments are

CRYOGENICS,

J U N E 1975

Ferrieinium bromide at 20 K vs £r Co5z I

-04

1

-03

]

-02

1~1 I

I

I

__

I

I

-OI O OI 02 O5 04 Velocity, cm s-I Fig.19 Isomer shift. The effect of t h e e l e c t r l c rnonopole interaction is to shift nuclear levels w i t h o u t lifting the spin degeneracy. The shifts are small compared to the total energy of the gamma ray. The ferricinium bromide spectrum, measured usinga Co 57 in Or source is f r o m reference 41

313

m,

Table 2. Core electron exchange polarization Magnetic field at nucleus, kG Mn 2+ (3d s) 1sl"

Fe 3+ (3d s)

\

Fe 2+ (3d 6)

2502840[

- 2 5 0 2 8 7 0 ~ - 30 )

1sT 2s1"

226670 ! _ 1410

2sT

-228080

3s1'

3121O

- 50

- 30

- 1790

- 1310

shift

3_

TIIII

A2111L_.

-E.(m~) : ~,Hm/ t -Z~EM = p . H / I

Mo~letic

diRole

stblit tir~

96

~+740 3s$

+ 1215

+ 790

•

0"2

~so

-30470

fVvvlV

e

- 700

Total From

the

calculations of

Watson

- 630 and

Freeman

- 550 82

40

8C

-12.

|

- ,' o• - o 8i - 6 6 - 6 4 - ~ z

o

Velocity,

magneton and ~ the electron spin. The sum is carried out over all the atomic electrons. It is worthwhile noting that the contribution due to the contact term may be very large, a single 1s electron, for example, producing a field of " 3 x 109 Oe. We must remember however that the ls, 2s, and 3s electrons are 'spin paired' so that in non-magnetic atoms the main contribution would result from unpaired outer s-electrons. In magnetic atoms such as iron however an effect known as core-polarization, resulting from the exchange interaction between the 'magnetic' d-electrons and the core s-electrons, results in a partial unpairing of the spins which then produces an appreciable effective field at the nucleus. The results of computer calculation of such contributions for Mn 2÷, Fe 3+, and Fe 2+ are listed in Table 2. Electric quadrupole interaction

Because the nuclear charge may be non-spherical (that is, having a quadrupole moment Q), the existence of an electric field gradient eq = d 2 V/dz 2 may also split the energy levels. The interaction energy is given by eq2Q ZkEQ - *"2I-,+lt 1) [ 3 ~ - I ( I +

1)]

(10)

in the simple case of an axially symmetric field gradient. Its effect on the energy levels and on the resulting Fe s~ M6ssbauer absorption spectrum measured relative to an unsplit emitter is shown in Fig.21 (note that Q = 0 for the ground state I = ~Alevel). Such an electric field gradient arises either from asymmetrically disposed ligand ions or from partly filled electron shells on the ion itself. Applications Magnetic studies

As seen from (9) the nucleus of a magnetic atom experiences a magnetic hyperfine field. The magnitude of the field, as measured in a M6ssbauer effect experiment, depends on the strengths of the various contributions and on the time average over a time comparable with the nucleur Larmor precession period.

314

l

i

J

oz 04 06

,

:

08 ,o

t

iz

Crn s-I

For paramagnetic materials this average is usually zero, the paramagnetic relaxation being typically much faster than the nuclear Larmor precession rate. The resulting M6ssbauer spectrum shows no magnetic hyperfine splitting and in the absence of a quardupole interaction yields a single absorption line. When cooled through the magnetic ordering temperature (Tc or TN) however the time average Of Hhf becomes non-zero and is usually closely proportional to the magnetization, that is, <

U,oca >

= M

(73

(11)

The M6ssbauer spectrum is hence split, the magnitude of the splitting increasing with decreasing temperature until the saturation value is obtained at very low temperatures compared with the ordering temperature. A series of such spectra for FeF 2 is shown in Fig.22, and the extracted hyperfine field variation in Fig.23. It is evident from (9) that the effective field is the algebraic sum of a number of terms, some positive and some negative. The resultant hyperfine field may thus be positive or negative. [Note: A positive/negative hyperfine field is defined as being parallel/antiparaUel to the bulk magnetization vector|. The determination of the sign of the field thus gives additional information regarding its origin. By observing the effect on the M6ssbauer spectrum of an external applied field large enough to align the magnetic domains, this sign may easily be determined. If the nuclear Zeeman splitting increases, then the applied field must add to the internal field thus showing it to be positive. A decrease in the observed splitting obviously indicates a negative internal field. The cryostat shown in Fig.14 incorporates a large superconducting magnet (6 T) used for such hyperfine field sign determinations in the author's laboratoryJ a In this photograph the vibrator and source drive assembly is being loaded into the cryostat from above. Although normally in the paramagnetic state the electron spin relaxation rate is too fast to observe a non-zero time average for the hyperfine field, under certain conditions

CRYOGENICS . JUNE 1975

this relaxation may be much slower, and the relaxation time may become comparable with the Larmor period. This effect is explained schematically in Fig.24. Sufficiently long relaxation times are obtained when the distance between the magnetic ions is large, thus reducing the strength of the spin-spin inter-action which contributes to the relaxation process. These times usually increase rapidly as the temperature is reduced and it is possible to observe the effect of the change in relaxation rate by observing the M6ssbauer spectrum as the temperature is reduced. A typical set of Fe sv spectra obtained at low temperatures in an iron dithiocarbamate is shown in Fig.25. At 1.5 K the relaxation is sufficiently slow for the full hyperfine field to be exhibited in the paramagnetic state. Detailed studies of such relaxation spectra provide valuable tests of relaxation theories. Another relaxation phenomenon of interest and capable of study using the M6ssbauer effect is the converse case where under certain conditions the relaxation times for magnetically ordered materials can become comparable with and shorter than the Larmor period. This occurs when material which is ferromagnetic in bulk is divided into very fine particles. Due to thermal fluctuations associated with the thermal energy such particles may behave as paramagnetic particles and are labelled superparamagnetic. In this case the relaxation times are determined by the size of the particle, the anisotropy energy and the temperature. Fig.26

103.--r-

,

--x

r

,

IOO1 ,,~ .':-

~-

- - ~

,

"."

88

[7o2 Kv

"i

.Q

g O

mI I l

3/2 l ~ E Q ( ' ~

3/2)

= 3/2

AE

,~

±

1/2

88 112

~ Isomer shift

0 g 5

2O

Cr:C°~'

-04

-03

',

I

o z

04

X, ,' 6

' ,,,,

eqQ

Fig.22 Hyperfine structure of Fe 57 in FeF 2 well below the Neel temperature (Wertheim and Buchanan 43)

shows a series of Fe s7 M6ssbauer spectra obtained in very fine particles of Cu 10% Co as a function of particle size. 19

= E Q ( 3 / 2 ) -- E Q ( I / 2 ) = - -

2

-02

-01 Velocity, cm s-I

Fig,21 Electric quadrupole splitting in Fe sT. The M0ssbauer spectrum of biferrocenyl measured at 20 K relative to a Co s-/ in Cr source is that obtained by Wertheim and Herber 41

CRYOGENICS.

I_

o

Doppler velocity, crn s q

--qr --L

t

o z

=112

~ io 15

l

o,i

Quadrupole splitting

~m~ 1(1+1)] ~'Q

t.

oa

JUNE

1975

Superconductivity

The question whether superconductivity and magnetic order can co-exist has been under discussion for some time. 2° On the simple BCS theory co-existence is clearly impossible, but the early work of Matthias and his group z1-2s seemed to indicate that such a co-existence might be possible. Following this suggestion Gorkov and Lusinov 26 took into account the scattering of the conduction electrons by the magnetic spins and showed that in prificiple a co-exislence of superconductivity and ferromagnetism is possible. The early idea of co-existence was based on observations similar to those shown in Fig.27 which shows the effect on the superconducting transition temperature Tc of substituting the magnetic rare earth Gd for Ce in the superconducting Laves compound Ce Ru z.27 Tc is little affected by substantial Gd Ru 2 concentrations, and an extrapolation of the smooth curve suggests that superconductivity may exist in Ce I - x Gdx Ru 2 up to x ~ 0.16. On the other hand the magnetic

315

550

anomalous behaviour which could be due to magnetic order, although no direct evidence is available. However by introducing Co s7 as an impurity, the M6ssbauer effect in Fe s7 can be used to detect directly magnetic spin order transferred from the RE sites in the form of a hyperfine magnetic field. Figs 28 and 29 show the Ce 1- x Gdx Ru2 M0ssbauer spectra obtained at 3.95 K and ~1.4 K for a range o f x values. 28 The Co 57 parent atom occupies Ru sites and, it is believed, senses the magnetic ordering only if they are nearby aligned Gd sites. The spectra are therefore interpreted as the superposition of a pure quadrupole doublet and a combined quadrup01e and magnetic dipole six line spectra (for the site influenced by the magnetic order). The hyperfine field obtained in each case is shown in the figure. It is thus demonstrated dearly that magnetic order exists, the observed hyperfine field increasing with increasing Gd content and with decreasing temperature as

50C

25(;

20(7

15(7

IOC

5(7

I

00

I0

I

20

I

30

I

40

I

50

I

I

60

I

70

30

90

I00

Temperature, K

99

Fig.23 Temperature dependence o f the magnetic hyperfine field in FeF 2 (Wertheim and Buchanan 43)

98 Relaxation rote

Slow

I00

Instantaneous hyperfine field

Electron

~ Up Down -------

99:

J

+550

i

- 550

]3me tO

2

Medium

+5500 t"lime - ~

-550

"6 n~

+550

fl FIFIFIFII-I fl I1 FIIllGflflflfo

,....,

WU UII U U U U UU UUU U

"rime

"rime histogrom of hyperfine field seen by nucleus

I~

Fig.24 Origin o f magnetic hyperfine structure in paramagnetic materials with slow and medium relaxation. If the electron spin relaxation time is very long, the nucleus 'sees' a magnetic field o f well defined absolute value and produces a sharp hyparfine pattern. As the electronic relaxation becomes more rapid, the effective field 'seen' by the nucleus has a distribution o f values. A t very high relaxation rates, this distribution becomes narrow and lies about zero field, thus yielding a single absorption line (after Cohen 44)

,-0

1.5 K 99 98 97

susceptibility of the Gd rich alloys show ferromagnetic order with a smoothly varying magnetic transition temperature 0 intersecting the Tc curve at a finite temperature The question remained, does co-existence occur in the region labelled S + M? Susceptibility and magnetization measurements cannot, however, be used directly to study the magnetic order in this region once the material becomes superconducting because of the diamagnetic shielding. Specific heat and thermal expansion measurements show

316

t

-10

'.

-08

-04

0 0.4 Velocity, cm sd

08

1.0

Fig.25 Spectra of Fe 57 at low temperatures in an iron dithiocarbamate. A t 1.5 K, the electronic relaxation is slow, and a s h a r p hyperfine pattern with large splitting is obtained. As the temperature is increased, the electronic relaxation speeds up and the lines become broadened and collapse into a single line f o r fast electronic relaxation (data f r o m Wickman and Wagner 45 )

CRYOGENICS. JUNE 1975

,..

e,g

.....

dependence of the hyperfine field should follow the Brillouin function

( g..ku q_o ) T

Hhf =Hsat Bs \

(12)

where Hsat is the saturation field when the Brillouin function B s = 1, and is proportional to the total impurity moment < S > . H o is the applied magnetic field and T is the absolute temperature. Mossbauer measurements of the Fe s7 impurity hyperfine fields in Pt and Pd, as functions of Ho/Tfor temperatures of 3.95 K, 1.1 K, and 0.5 K, are summarized in Fig.30. Fits to the Brillouin function yield values o f g and S appropriate to the local magnetic lnontent. The 'local magnetic moments' obtained from such experiments with Fe s7 impurities in 4d transition metals are shown in Fig.31. We note the sharp appearance of a moment between Nb and Mo, and the presence of the socalled 'giant moments' near Pd. These 'giant moments' are of considerable theoretical interest and arise due to the polarization of the surrounding host atoms by the impurity moment.

-7 -6

-5

-4 -5 -2

-I

0

I

2

3

4

5

6

7

Relotive velocity, rnrn s-I " Fig.26 A selection of Fe 57 Mossbauer spectra obtained at a single temperature (300 K) for different mean particle volumes of fine particles of Cu 10% Co V (cm 3) - 4.35 x 1 0 q T ( A ) , 1.2 x 10 q'~ (B), 3 x 1 0 q g ( c ) 7.5 x 10-2°(C), 3.75 x 10-2°(E). (from Krop and Williams 19)

expected. It should also be noted that as x increases the higher probability of an Fe atom being near a Gd atom is reflected in the increased intensity of the magnetic six line spectrum relative to the pure quadrupole doublet. Further experiments to look directly as the RE sites using a suitable Mossbauer isotope are envisaged.

The presence of a localized magnetic moment is also an essential component of the Kondo effect. Kondo 3] showed that the resistance minimum seen in alloys such as Fe in Cu at low temperatures could be interpreted as being due t o a higher order scattering between t h e magnetic Fe impurity and the Cu conduction electrons, lie showed that this higher order scattering increased the resistivity as log T at low temperatures and was the result of a strong correlation between the magnetic impurity and the conduction electron. These spin correlations, which exist at temperatures well below the characteristic Kondo temperature TK (TK ~ 10 K for Fe in Cu), should considerably modify the low temperature spin behaviour both on the impurity site and on the host metal electrons and ions in the impurity neighbourhood. The Mossbauer effect in Fe 57 is thus an ideal microscopic probe to study the Formation and effects of these low temperature correlations. Fig.32 shows the measured hyperfine fields at Fe s7 nuclei in Cu as a function of Ho/Tfor three different values of the applied field H 0. The hyperfine field is primarily due to

Similar measurements 29 have been performed on the alloy system Eu x La 1 - x . Using the Mossbauer effect in Eu lsj they observed magnetic ordering temperatures down to 200 mK in the co-existent region. These workers also point out that the possibiliD of their hyperfine magnetic fields arising from slow relaxation in a paramagnetic state may be ruled oul. Local m a g n e t i c m o m e n t s -

Kondo effect

Another problem in magnetism which has engaged both theoretical and experimental physicists extensively is that labelled the 'localized magnetic moment problem'. 3° The problem is essentially t h a t of describing the formation and behaviour of a magnetic moment on an impurity atom in a non-magnetic host. By observing the hyperfine interaction at the impurity nucleus, the Mossbauer effect provides a useful method of studying this problem. For a transition metal impurity such as Fe there are several contributions to the hyperfine field all of which are proportional to the impurity magnetization < S z > . For an impurity which behaves like an isolated magnetic moment, the temperature

CRYOGENICS.

JUNE

1975

Ce

I-x

12

Gd Ru x

2

/

I0

8I k"

6~-'

' M

S

L

ff / /

o L

_.

z__ 005

,,-

/

\\ S,M

l olo Mole fraction,

X

\

i \. ois

h o2o

d25

.~

Fig,27 The magnetic-superconducting phase diagram f o r Ce! - x G d x Ru2. Tc is the superconducting transition temperature and 0 c is the magnetic ordering temperature. The data points and solid curves represent the magnetic susceptibility results of Wilhelm and Hillenbrand 27. The dashed curves represent extrapolations which define a region of possible coexistence of ferromagnetism and superc o n d u c t i v i t y (S + M)

317

Ce0.92 Gdo.08 Ru2

C e o . 9 6 Gdo.04 Ru 2 I

1

I

1

I

l

I.O

l

t

J

1

I

C~0.885 Gdo.H5 Ru2

I

I

I

i

T

i

i

1

i

i

0.9 T : 3.95 K Hi:O

vI c ,OJ

t-

08

E

.~

Hi : 5 . 7 T

V V "'-°

i.0

O

tic 0.9

- . ":._,:o: HI --3.OT

VVH' :o

O.e -3

I

1

1

I

L

-2

-I

0

I

2

I

3

-3

Velocity, m m s-t

I

I

I

-i o ~ Velocity, m m s-t

-2

EL_

I

~

~

-3

I

-2

I

I

I

-~ o ~ Velocity, m m $-I

I

I

2

3

Fig•28 Low temperature Fe 57 Mossbauer spectra for Ce l _ x Gdx Ru2 for x = 0.04, 0.08 and 0.115• The curves through the data represent least squares fits. The dashed component curves denote pure quodrupole sites in a two site model for the Fe 57 nuclei• The solid component curves represent quadrupole plus magnetic sites• The internal magnetic field H 1 (in tesla) determined for these second sites is listed for each spectrum (after Erickson et a128)

C e o 8 7 Gdo.i 3 Ru 2 I

I

I

I

I

Ceo.B55Gdo.14 5 Ru 2 I

I

1

1

I

I

Ceo.75 Gdo.25 Ru 2 1

1

1

1.00

I

1

I

~r~-.-- ~ . . . .

•

T

~ ~--

0.95

'~ 0.90 E ¢k) C

Hi:6"6T

Hi=6.8 T

T--3.95 K H:8.I T

~ _j.

T= 1 . 4 0 K

]

H,= 7.7 T

Hi:7.9 T

H i =8.5T

T = 3.95 K V

-- 0 8 5 ctJ

•-> t o o

0.9~

09C . l 3

,V I -2

1 -I

I O

I I

l 2

I 3

-3

Velocity, mrn S-I Fig.29

I

-2

I

-I

1

I

Velocity, mm s -t

I

2

I

3

-3

-2

I -I

I 0

1 I

', 2

I 3

Velocity, m m $ - I

As for Fig.28 but with x = 0.13, 0.145, and 0•25

the core polarization produced by the moment on the Fe and the spin polarization associated with the moment. Note that the me,,sured hyperfine field does not follow a Brillouin function and that the saturation value as T-* 0 is not constant, but increases with increasing applied field Ho.. The growth of asa t with Ho is interpreted as being due to the breaking up of the spin compensated state and the growth of the moment on the Fe impurity. A detailed review of MOssbauer studies on the Kondo effect may be found in the review paper by Heeger. 3°

318

I

O

Nuclear polarization -- the M~issbauer t h e r m o m e t e r

The introduction of the large cooling capacity of He3/He 4 dilution refrigerators to Mossbauer experiments has considerably extended the range of investigations, such as those concerning low lying magnetic transitions, paramagnetic relaxation, localized magnetic moments, Kondo effect, etc, and which have already been discussed. In addition, the availability of source and/or absorber temperatures as low as 0.03 K enable other experiments to be

CRYOGENICS.

J U N E 1975

a 10 mCi Co s7 source (a typical strength used for M6ssbauer experiments) the total heating power is 80 erg s q which if absorbed by the mixing chamber is sufficient to prevent the refrigerator cooling below 0.07 K. By careful design of the mixing chamber however, the heating effect may be minimized. In absorber experiments the radioactive heating is much smaller and an order of magnitude decrease in temperature is in principle possible provided certain precautions are taken, such as the shielding of the source in such a way as to make sure that most of the heat is absorbed by the still and that radiation does not strike the mixing chamber outside the active area of the absorber.

..-7r---

?

~l

;

•

/"

25 9,.5 K

• IlK •

•

05K

I00 t t

0

I

I

I

a Fe in Pd

300

I

I

i

IIIIIIIII 7 II

I

2 5 -I o 250 H/T, kOe K

15

.

As described in the section on hyperfine interactions, in the presence of a magnetic hyperfine field Hi the nuclear states separate into a series of sublevels. For Fe s7 (Fig.20) the ground state (I = ½) splits into two sublevels and the excited state (I = 3/2) into four sublevels, each characterized by a nmgnetic quantum number m I. The magnetic interaction energy is - / l ! H i mi/I where/.t! : gl/~N I is the

14 aa

1

200

•

:t

• IlK

I

i05K

~o

rn

lOC

8

c_

Eo

6 4

I

O

I

l__

t

I

2

b

__l

I

I

3

4

IIIIIIIIII 7 II

15

Fig.30 The magnetic hyperfine field of dilute Fe in Pt and Pd as a f u n c t i o n of H 0 / T . B e l o w 4 K there seems to be some deviation f r o m a simple Brillouin function. This deviation may in part be due to strong correlation effects at low temperatures (data f r o m references 46 and 47)

performed which were not previously possible. These include experiments concerning orientated nuclei which can, tor example, provide an absolute thermometer. Previously such experiments were normally restricted by the degree of cooling possible using He 3 crystals (T >~ 0.3 K) although temperatures down to 0.025 K have been obtained using magnetic cooling. 32 For such Mossbauer experiments the He 3 temperatures are not low enough and the adiabatic demagnetizalion technique suffers from the serious drawback of being a 'one-shot" method, which means that the temperature cannot be kept constant for the relatively long periods (~ many hours) necessary to collect sufficient Mossbauer data. The He3/He 4 Mossbauer cryostat described by Kalvius et a117 (see Figs 16 and 17)enables temperatures of around 0.03 K to be held for up to 24 hours with only one fill of the He 4 pot. In principle it should be possible to improve such a cryostat to give continuous temperatures as low as 0.01 K. Calibrated germanium and carbon resistance thermometers measure the temperature and form part of a feed-back stabilizing system. One interesting problem encountered is that of radioactive heating in the source. For

CRYOGENICS.

Eo 2

0.25OH~T, kOe K -I

JUNE

1975

j

0

Y

4 Zr

~

t

5 Nb

6 Mo

%__~ 7 Re

I 8 Ru

9 Rh

I

I

I0 Pd

II Ag

Electron concentration The magnetic moment per Fe i m p u r i t y associated w i t h 1% Fig.31 Fe as a f u n c t i o n of the 4d transition metal alloy concentration. Note the sharp appearance of an i m p u r i t y m o m e n t between Nb and Mo and the presence of the 'giant moments' near Pd (from reference

48)

8060

/I

\

8o%.%.01~.

4,...-4,---• H = 136 kOe •

H o

/

=66 kOe

.

. .. .. ..

s

/],,~'"

"

T

"-~

Z

__

2C

,

t

I

1

lO

L

I

I

I

I00

I

J

I

I000

Ho/ T Fig.32 The magnetic hyperfine field at Fe 57 nuclei in Cu as a f u n c t i o n of Ho/T for three different values of the applied field H 0. Note that Hsat(H 0) depends on H o (after Frankel et al 49)

319

moments are small and the largest hyperfine felds obtainable in iron compounds are around 60 T. This produces a ground state separation of only 0.004 K (measured in temperature units) thus requiring very low temperatures to produce observable polarization effects. In Fig.33 we see the spectrum of Fe203 measured at 0.043 K and showing a small polarization of about 8%.

,°,.. I00 I

0 95~

0 9011 I

-10

1 I

I

I -5

I

I

I

I [ I I I 0 Velocity, mm s -I

I

I I 5

I

I

I0

Fig.33 Mossbauer spectrum of Fe s7 in Fe 2 0 3 at 0.043 K, the source being Co 57 in Cu at 1.5 K, the polarization is ~ 8 % ( E n h o l m et al 5o)

nuclear magnetic moment of the state with spin I and ON is the nuclear magneton. Thus neighbouring levels are separated by an energy e given by (13)

e = igl ON Hi l whilst the total splitting of each state 1 will be 2 O[ Hi.

The relative population of each sublevel is determined by the Boltzmann factor

P(mi) =

exp [ - m le/kT]

(14)

2 exp[--mle/kT ]

m |

and so when the thermal energy kT becomes comparable with the separation of the extreme levels, that is, when

k T ~ 201Hi

(15)

the individual sublevels will have unequal populations. Normally the hyperfine energy is so small that the levels may be considered as equally populated, but at the very low temperatures now available this may not be the case. In a M6ssbauer absorption experiment the relative intensity I of a transition from one of the ground state levels mg to one of the excited state levels m e is given by I (mg -~ me) ac p (mg) C (mg, me)

For Au 197 a hyperfine field Hi = - 1 2 8 T is obtained when placed in a metallic iron matrix. This produces a total ground state splitting of 0.014 K. This should result in a large polarization effect when the Mossbauer spectrum is measured at temperatures below this value. Fig.34 shows the nuclear levels obtained and indicates the strengths of the relative intensities expected at 0.043 K. The resulting polarization effects are clearly visible in the Mdssbauer absorption spectrum obtained at 0.043 K and shown in Fig.35. Here we also see that at 4.2 K (where kT>> 2 01 Hi) no polarization is evident. In the Au 197 spectrum the eight individual absorption lines are not all resolved but are bunched into two groups of four.

Other M6ssbauer nuclei which have larger hyperfine splittings may prove to be better thermometers, working up to higher temperatures. Examples would be Dy 161 which in dysprosium metal experiences a hyperfine field H i = 710 T thus producing a splitting 2 Oi Hi = 0.240 K, and Np 237 which in Np AI2 experiences a field of 300 T with a corresponding splitting 2 Ol Hi = 0.620 K. It should be noted that in principle an electric quadrupole splitting would do equally well as a means of producing nuclear polarization, but in practice the magnitudes of such splittings are small compared with those of magnetic interactions. In practice it is best to measure the area, rather than intensity, associated with each absorption line, but care must be taken in allowing for correction factors which arise due to the fact that transmitted intensity at resonance is also a function of the effective absorber thickness. 33, 34 Thus although there are inherent practical difficulties associated with the widespread use of such a Mossbauer thermometer it does have the advantage of being absolute with its own internal calibration, and so may prove useful for certain specialized applications, and calibration purposes. AJ97in Fe / ~r

mp

/

(16)

where C (mg, me) is the square of the relevant ClebschGordon coefficient and P (mg) is the Boltzmann factor of (14). When nuclear polarization occurs it is therefore reflected in the relative intensities of the M6ssbauer absorption lines. By choosing two lines for which the Clebsch-Gordon coefficients are equal [for example, C (mg, me) = C ( -mg, - m e ) ] we may therefore define the absolute temperature by

-I/2

(- 2mge'~

I(+mg~+me) _P(+mg) , P=l(-mg-+-me) P(--rag) e x p \

)c-T ,]

P(m~)

(17)

The hyperfine energy e is obtained from the separation of the absorption line, thus providing an absolute thermometer with its own internal calibration. For Fe sT, the nuclear

320

IIH

3

~

'

/

~

~

* 3/2 , I/2 - I/2 - 3/2

0.22 024 O26 0.28

O0~4K

Fig.34 Hyperfine transitions of Au 197 in a magnetic field. The widths of the arrows indicate their relative line intensities at 0.043 K (after Kalvius et al 17)

CRYOGENICS. JUNE 1975

absorption spectrum would experience a shift which could be comparable with the line width. Even the zero phonon 3,-transitions are associated with a small change in the internal energy of the crystal at the expense of the photon energy, on account o f the difference in mass of the nucleus in its excited and ground states. The resulting 3,-energy displacement is given by

E 8E

1,,2 - - - Eo 2 ¢2

=

8

An equivalent expression is obtained if we regard the energy displacement as arising from the relativistic time dilation caused by the vibrational motion o f the nuclei in the lattice (the second order Doppler effect). When source and absorber are at the same temperature these displacements become equal and absorption still occurs symmetrically about zero relative velocity. However, when the temperatures are different, a shift in the absorption spectrum is obtained as a result o f v 2 being different in source and absorber. Such a fractional change in the energy of the Fe s7 7-rays emitted by a source at different temperatures is shown in Fig.36. This change was determined by observing the shift in the M6ssbauer absorption spectrum measured relative to an absorber at room temperature.

g.

-I0

0

(18)

I0

Velocity, mms q Fig.35 M6ssbauer spectra of Au 197 f o r an absorber o f 1 at % gold in iron at 4.2 K and 0.043 K. The source is Pt 19~ in p l a t i n u m at 1,5 K. Polarization effects are clearly visible in the lower spect r u m (data f r o m Kalvius et al 17)

The importance of this temperature shift first became apparent during attempts to use the Mossbauer effect in Fe s7 to measure the gravitational red s h i f t ) 7, 38 It was realized that the gravitational shift expected

Relativistic temperature shift We have seen already that the Mossbauer absorption spectrum is expected to become symmetrical about zero relative velocity when both source and absorber are subject to the same environment, that is, there is no isomer shift. This symmetry is only observed however if the equality of environment also includes temperature. Pound and Rebka 3s and Josephson 36 were the first to point out that if source and absorber were held at different temperatures, the

~K

gh

= Eo

(19) c2

could easily be masked by a temperature shift o f this magnitude resulting from a temperature difference of only 1 K between source and absorber. By using a careful arrange-

Table 3. Mbssbauer periodic table*

Inert Gases

1A

Number of observed M6ssbauer transitions

..,d

11A

,~

2~"

IliA

IVA

VA

VIA

VIIA

Number of isotopes in which the Mossbauer effect has been observed ~

~

IIIB IVB

VB

VIB

VIII

VIIB

IB

Fe

Ni 1 1

1

1 ~

Zn

~

Ge

1

~

1 1 Cs

1 ~

4Hf 4

I Ba

2 1 Ta

1

**

F

1

Th

5 4W

1

1

Pr Pa

1 1 Re

1

Nd

1

1

U

1 1

2 Os

Pm

1 3

2

Sm

,NP 1 ~

Ir

2 1

3 2

Eu

Pt

4 ~7

Sn

~

y//////

1

1 1

4

1 ~/~///~

1 1

Sb

1 1

-I"e

Kr

2 2

I

1

2 2

Xe

1 Au 1

Gd

8 1

Tb

1 4

Dy

6 il

Ho

1 5

Er

5

Tm

1

1 5

Yb

6///////£

, A m l ~ ~ ~ ~ / / ~ / / / ~ / , ~ / ~ / f ~ ~ .

From MossbauerEffect Data Index39

CRYOGENICS. JUNE 1975

321

12

I 4

13 14 15 16

_~o O

17

18 0

100

200 Temperoture, K

500

Fig.36 The temperature shift in Fe 57 relative to a r o o m temperature source. The curve is calculated f o r a Debye temperature 0 D = 4 2 0 K ( f r o m Pound and Rebka 37)

19 20 21 22

ment employing helium gas to equalize the temperatures, the measurements confirmed Einstein's prediction for the shift produced when source and absorber were separated by a distance h in gravitational field g, to within about 5% statistical accuracy.

Discussion In this review the discussion has been restricted to applications of the Mossbauer effect at very low temperatures, and the Fe s7 isotope has been the main one used to illustrate these applications. Table 3 shows the extensive range of elements which have shown Mossbauer resonances in at least one isotope. These are distributed throughout the periodic table and have been used in applications covering the broad spectrum of physics, chemistry, biology, geology, metallurgy, and even archaeology. Low temperatures are not essential in all applications, but are generally found to greatly extend the usefulness of this most valuable spectroscopic tool in the ways outlined in this review.

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

References l 2 3 4 5 6 7 8 9 10 11

322

Wood, R.W. Physical Optics (Macmillan, NY, 3rd edn, 1934) Ch 18 Kuhn, W. Phil Mag 8 (1929) 625 Mossbauer, R. L. Z Physik 151 (1958) 124 Debrunner, P. G., Fraunfelder, H. An Introduction to Mossbauer Spectroscopy [May, L. (ed)] (Adam Hilger, London, 1971) 1 Boyle, A. J. F., Hall, H. E. Repts Prog Phys 25 (1962) 441 Spijkerman, J . J . 'Instrumentation' An Introduction to Mossbauer Spectroscopy [May, L. (ed)] (Adam Hilger, London, 1971) Ch 2 Bunbury, D. St. P. J Scihtst 43 (1966) 783 Cranshaw, T. E. JPhysE. Scilnst 7 (1974)497 Kankeleit, E. Rev Scihlst 35 (1964) 194 Frauenfelder, H. The Mossbauer Effect (Benjamin Publ, NY, 1962) Wertheim, G. K. The M6ssbauer Effect: Principles and Applications (Academic Press, NY, 1964)

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M6ssbauer, R. L., Clauser, M.J. Hyperfine Interactions [Freeman, A. J., Frankel, R. B. (eds)] (Academic Press, NY, 1967) Danon, J. Lectures on the M6ssbauer Effect (Gordon and Breach, NY, 1968) Greenwood, N. N., Gibb, T. C. Mossbauer Spectroscopy (Chapman and Hall, London, 1971) May, L. An Introduction to M6ssbauer Spectroscopy (Adam Hilger, London, 1971) Gruverman, I. J., Seidel, C. W., Dieterly, D. K. (eds) MOssbauer Effect Methodology, Vols 1-9 (Plenum, NY, 1966-74) Kalvius, G. M., Katila, T. E., Lounasma, O. V. 'Mdssbauer experiments with He3/He 4 dilution refrigerator' in M6ssbauer Methodology [Gruverman, I. J. (ed)[ Vol 5 (Plenum Press, NY and London, 1970) 231 Brooks, J. S., Williams, J. M. JPhysF(MetalPhysics) 4 (1974) 2033 Krop, K.,Williams, J. M. JPhysF(MetalPhysics) 1 (1971) 938 Fischer, O., Peter, M. in Magnetism [Rado, G. T. and Suhl, H. (eds)] (Academic Press, 1973) 327 Matthias, B. T., Corenzwit, E., Suhl, H. Phys Rev Lett 1 (1958) 92 Matthias, B. T., Corenzwit, E., Suhl, H. Phys Rev Lett 1 (1958) 449 Suhl, H., Matthias, B. T., Corenzwit, E. J Phys Chem Solids 11 (1959) 346 Hein, R. A., Folge, R. L., Matthias, B. T., Corenzwit, E. Phys Rev Lett 2 (1959) 500 Philips, N. E., Matthias, B. T. Phys Rev 121 (1961) 105 Gorkov, L. P., Rusinov, A. 1. Soy Phys JETP 19 (1964) 922 Wilhelm, M., Hillenbrand, K. Z Naturforsch 26a (1971) 141 Erickson, D. J., Olsen, C. E., Taylor, R. D. M6ssbauer Effect Methodology, [Gruverman, 1. J., Seidel, C. W. (eds)] Vol 8 (Plenum Press, NY and London, 1973) Gumprecht, D., Steiner, P., Hufner, S. Phys Lett 48A (1974) 269 Heeger, A.J. Solid State Physics [Seitz, F., Turnbull, D., Ehrenreich, H. (eds)] Vol 23 (Academic Press, NY, 1969) 284 Kondo, J. Prog Theor Phys (Japan) 32 (1964) 37 Shinohara, M., lto, A., Fujita, T., lshigaki, A., Ono, K. JapJApplPhys 6 0 9 6 7 ) 682; 7 (1968) 170 Margulies, S., Ehrman, J. R. NuclnstMeth 12 (1961) 131 Bykov, G. A., Pham Zuy Hien SovPhysJETP16(1963) 646 Pound, R. V., Rebka, G. A. Jr PhysRevLett4(1960) 274 Josephson, B. D. Phys Rev Lett 4 (1960) 341 Pound, R. V., Rebka, G. A. Jr Phys Rev Lett 4 (1960) 337 Cranshaw, T. E., Schiller, J. P., Whitehead, A. B. Phys Rev Lett 4 (1960) 163 Stevens, J. G., Stevens, V. E. (eds) MCSssbauer Effect Data Index (Adam Hilger, London, 1972) (This is now an annual publication covering the year's literature) Watson, R. E., Freeman, A. J. Phys Rev 123 (1961) 2027 Wertheim, G. K., Herber, R. H. J Chem Phys 38 (1963) 2106 Wertheim, G. K. The Mossbauer Effect: Principles and Applications (Academic Press, NY, 1964) 73 Wertheim, G. K., Buchanan, D. N. E. Phys Rev 161 (1967) 478 Cohen, R. L. Science 178 (1972) 828 Wickman, H. H., Wagner, C. F. J Chem Phys 51 (1969) 435 Craig, P. P., Perisho, R. C., Segman, R., Steyert, W. A. PhysRev 138 (1965) A1460 Mayley, M. P., Taylor, R. D., Thompson, J. L. J Appl Phys 38 (1967) 1249 Clogston, A. M., Matthias, B. T., Peter, M., Williams, H. J., Corenzwit, E., Sherwood, R. C. Phys Rev 125 (1962) 541 Frankei, R. B., Blum, N. A., Schwartz, B. B., Kim, D. J. Phys Rev Lett 18 (1967) 1050 Enholm, G. J., Katila, T. E., Lounasmaa, O. V., Reivari, P. Phys Lett 25A (1967) 758

CRYOGENICS.

JUNE

1975