Revista Brasileira de Física, Vol. 10, N. O 3, 1980 Determination of Mossbauer Electric Field Gradient
V. K. GARG Departamento de F/sica e Quimica, Universidade Federal do Espírito Santo, 29W Vitdria, ES, Brasil
Recebido em 12 de Maio de 1980
There are severa1 reports of the electric quadrupole interac- tions available in the literat~re?-~The present discussion is a short survey, introducing the electric quadrupole up to the experimental pola- rised studies.
Existem várias publ icaçÓes sobre interações de quadrupolo elé- tri co na 1 i teratura -" A presente discussão é um resumo, introduzindo o quadrupolo elétrico até o nivel de recentes estudos experimentais.
1. INTRODUCTION, (t STANDARD FORM OF EFG
The magnitude of the quadrupole splitting is pronortional to the Z componerit of the electric field gradient (EFG) tensor which inte- racts with the quadrupole moment of the nucleus. For I = 3/2, the qua- drupole spl itting can be expressed as
Where Q is the quadrupole moment of the nucleus
eq = V = - the Z component of the EFG ZZ e = proton charge and 0 is the asymmetry parameter Either q or Vzz is referred to as the field gradient. The EFG tensor has nine cornponents.
The potential at the Mtlssbauer nucleus (located at the ori- 1/2 gin) due to point charge q at (z,z~,z)a distance r = (x2 + y2 + z2) from the origin is given by
V =q/r ( 3)
The negative gradient of this potent ial, -$V, is the elec- -i tric field, E, at the nucleus, with components
++ the gradient of the electric field at the nucleus, TE, is given by
the second partia1 der i vati ve tensor components are
Vxx = q ( 3x2 - r2)r-' ,
V =q(3y2 - r2)r-5 , YY vzz = (3z2 - r2)r-' , - 5 v =v =3qq r> , w Y*C - 5 VXZ = vzx = 3 qxz r2 , - 5 v =Vzx=3qyz r . Y '2
The EFG cornponents can be expressed in spherical coordinates as - vxZ - = 3 4'3 sino cor0 cosa - 3 V = V = 3 qr sino coso siri@ YZ ZY
The total contribution to each EFG components is obtained by simply adding the individual contributions. If the coordinate axes are properly chosen, the above tensor can be reduced to diagonal form so that the EFG can be completely specified by the three components Vxx , V and VzZ. These components are also not independent and are related YY by Laplace's equation (8) vxx + vyy + vzz = 0 reducing the number of independent parameter to two, I'ZZ and ri where
The: value of EFG tensor components obviously depend upon the choice of the coordinate axes. For this reason a standard form of EFG tensor is expressed by def ining a unique set of axes known as the prin- cipal axes of the EFG tensor. This unique coordinate system is the one for which the off-diagonal components are zero and diagonal components are ordered, such that this restricts O é ri I. Now, only two of the five independent para- meters of the EFG tensor seem to be eft. Actually, however, three in- dependent off-diagonal elements have been replaced by the Euler angles (a,$,y) necessary to describe the re ative orientation of the princi- pal and initial axes. The orientation parameters affect the line in- tensities in single crystal spectra.
Many properties of the EFG tensor can be deduced from the symmetry properties of the mlecule or crystal. Often the Z axis of the EFG coincides with the highest symmetry axis of the rnolecule or crys- tal. The Z axis of EFG, and the X and Y axes, do not always coincide with metal - ligand bond directions. In such cases, it is necessary to choose an arbi trary set of axes, work out the EFG components, diagona- lise the tensor, and then choose 1vZ21 3 ]vyy1 3 /V_\.
If the electronic configuration has a symmetry lower than cubic, electrons about the ~gssbaueratorn also contribute to the EFG tensor.
2. CORE POLARISATION
The field gradient which is final ly produced at the~&sbauer nucleus, say i ron, is modif ied by the presence of the inner closed- -shell elec rons. While these core electrons themselves do not contri- bute to the field gradient, they are polarisec by the field gradient of the latt ce and valence electrons. This eff'ect is accounted for by the Sternhe mer5 antishielding parameter,s. The total field gradient at the nucleus
lattice
The subscript ' ion' and 'lattice' st:and for the charge dis- tribution of the spherica 1 3d valence electroris belonging to the fer-
646 rous (50 3tS6) and the neighbouring ion in the crystall ine lattice res- pectively. (I-R) and (1 -ym) are the Sternheimer factors which have been introduced to correct for the polarisation of the ferric like (6S2 3d5) core by the EFG of the ion and the lattice charge distribu- 3+ tion. For ~e"R = 0.32~. ym = - 10 .68 and for Fe ym = - 9. 19. The other distribution to the EFG tensor conçists of the uncompensated 3d electrons. In octahedral ~ymrnetr~'~,the 3d electrons are spl i t into two groups. Those transforming as T are largel y unaffected, since 2g these do not ordinarily participate in bonding to ligands. These are usual ly termed as dxy, &z and dyz and represent electric charge con- centrated in regions other than along the 1igand directions. The E 9 ele'ctrons, dz2 and 2, couple wi th 1igand orbi tals produce anti - bonding wavefunctions. These concentrate the charge along the bond di- rections. As a resul t, these E electrons occupy a higher energy leve1 9 than T electrons. 2g
The magnitude of spl i tting of these groups of electrons has a profound effect on the magnetic and ~Ossbauerproperties of these compounds. deak 1igands, such as OH-, H20, NH3 and the halogens produ- ce relatively small splitting. The five or six 3d electrons of ferric or ferrous compounds f i 11 by ~und'srule for maximum mul tiplici ty. Strong liyands such as CN, pyridine or - O - phenanthroline cause a greater spl itting and lead to the fitting of T Shell. 2g
For an electron in a state Y the electric f ield gradient has the exptxtat ion va l ue.
These expectati on val ues have been calculated using (O,@). r3~(r)ymII The values of <3 cos20 - 1> are independent of R(r) and are easily foundll frorn various p, d - orbitals. TAB L E
Expectation values of qvalence for p and d electrons
Wave Wave Symmetry funct ion function representa- t ion
it is clear from symmetry considerations that as long as d- orbi tals are degenerate, = O. Even imder the influence of 'q'vaience a cubic crystalline field, the field gradient at the nucleus will va- nish since the degenerate T and E states both retain spherical sym- 2g g metry. Distort ions f rom cubic symmetry, however, remove the degeneracy of the states and if the non-degenerate d-orbitals one unequally popu- lated, a field gradient is produced.
\ FREE ION \ \ dxz*dyz Teg',, -y --
- L cueic CRYSTALLINÈ FIEL0 ~XY AXI AL DISTORTION l'he two situations which can produce a non zero < q > valence are spin free (six d-electrons) and spin paired k3+(fived-elec- trons) in an axial ly distorted octahedra field (wi th negative 1igands).
At OOK the electrons populate only the lowest possible states and one obtains the limiting field gradient for these two cases. lngallsll has treated the field gradient in spin - free ferrous complexes in much more detai 1. He obtains a reduction factor to include thg effects of finite temperature (~oltzmandistribution) and of spin - orbit cou- pling. Now
since (I-R) >O and
The relative magnitude of these two effects, 1igand disposi- rion and uncompensated 3d electrons change compl etely wi th 1igand strength. Li gands, suff iciently weak that the ~e~+high spin electron confi guration is obtai ned, do not cause observable EFG. The uncompen- 2+ sated 3d-electron in Fe high spin compounds causes a large quadrupo- le spl itting. The mixed strong - weak ligand case, Fe3+ (O-phenthroli- 3 ne)2C13- which has no 3delectron contribution to the EFG does not yield a resolved doublet. The cyanides lie at the other extreme, 2+ K~IF~(CN-)6 1, has a zero spl i tting in consequence of i ts balanced 1i - gands and fi 1 led T structure. K3 ~F~~+(cN-)~1 3 H20 has one T e- 2g 2g lectron missing, but yields only a small âplitting (0.3 mm/sec). This i1lustrates the nephelauxetic effect of strong 1 igands, in which the ligarids force the T electrons away from the nucleus where the
The choice of basis for 3d electrons is normal ly ei ther dz2, dx2-y2, h,&z, dyz or do, dkl, dk2. Each of the wave functions in these alternate bases has zero asymmetry parameter. There exists, ho- wever, other 3d - wavefunct ions wi th asymmetry parameter di ffering from zero and even attaining the values of uni ty. Consider the wavefunc- ti on
where X is a real number, the EFG tensor is
'1 - 2X2 O O - 3 2 EFG = - - O -2 + i2 0 (14) 7 1 + ~2 o o 1 + A~ , /
The mixing coefficient X depends on the details of the dis- tribution and the nature of metal - l igand interaction. It might be supposed that such a wave function are idle speculation. However, the EFG tensor in ferrous ammonium sulphate has been explained only by re- couri.e to such a wavefunction by Col l ins et al. 14.
3. QUADRUPOLE SPLITTING
The principal relation giving the hyperfine spl itting of nuclear energy due to electric quadrupole interaction is given by
where I+ = I 2 I . Now, eigenval ues B and EQ appropriate to the -XY Q * nuclear exci ted state and the ground state rnus t be cal culated. The resonance energies are now given by the difference between the ei- genval ues
4. SIGN C)F EFG AND REASONS FOR MEASURING IT
Sinse the electric field gradient is actually a second rank tensor, what real ly is meant by the sign of VzZ or of q? The various theoretical methods employed in calculating the electric field gradi- ents a11 suffer because of inadequacies of the wavefunctions available for iron ions. Therefore, theoretical estimates of the field gradients are usually not very rel iable. However, the sign of EFG is completely determined by the angular part of the wavefunction. The angular part of the wavefunctions is much more reliable than the radial part, and hence, calculations of the sign can be made with some confidente if a real i stic method i s chosen. Thus the experimental determination of the sign of Vzz is reliable for differentiating between two theoretical models when they lead to different signs.
5. MAGNETIC PERTURBATION
One of the difficulties in studying quadrupole effects in ~e~~ i s the presence of onl y one measurable parameter in powdered, zero magnetic field samples. Ruby and Fl inn15 f irst suggested an interesting possibil ity of applying an external magnetic field in order to display more inforniation in the ~Ossbauerspectrum. It was called magnetic per- turbation and its theory was worked out by Coll ins16, and Gabriel and ~ub~l~.Theoretical studies, analytical by collins16 and numerical by Gabriel and ~ub~l~,verified that two normally identical lines become markedly different in presence of external magnetic field. By identi- fying the lines means that the sign of VZz may be measured in this way. In addition, the fine structure of the perturbed spectra may be exami - ned to yield rough values of asymmetry parameter, within limits deter- mined by magnetic anisotropy in paramagne~.~CLJ -,\~!nds 2nd by vibratio- na1 isotropy of the sample14. For the l imit of .-I-. , L ;y parameter and for appl ied magnetic fields of about 25 Koe csr more, in the ~e~~ 1 rni = + 3 l ines appear as triplet and the rnl = f - 1 ines appear as dou- 2 2 blet. For determining the sign of VZZ of ~e~~ doublet occurrs at the more positive velocity indicating V positive ;ind vice-versa. As the ZZ asymmetry parameter increases, the l ines begin tci look more a1 ike to the point of becoming identical triplets for an asymmetry parameter of uni ty, although the exact appearence of the l ine~depends on whether the magnetic field is parallel or perpendicular to the experimental axis, the direction of the incident radiation.
Collins16 applied this magnetic pertiirbation to determine the sign of ferrocene at 4.2~. The sign of q in some ionic ferrous com- pounds has also been determiried18. It must be remarked that this method is appl icable to powder samples and most samples are powders rather than single crystals.
6. GOLDANSKII-KARYAGIN EFFECT
The intensities for a single crystal absorber are general- ly not equal . Occasional ly in powder sample absorbers also there is asymmetry in the two 1ines of a ~e~~ quadrupole spl it spectrum.The most obvious reason for intensi ty asymmetry in powdei-ed samples is simpl y the preferential orientation. This sort of preferential orientation in powdered samples may be reduced or eliminated by improving sample grin- ding and ar mixinç with noninterfering powder such as chalk dust, sugar etc. If stiil the sample shows the intensity asymmetry it may be due to Goldanski i Karyagin effect.
~ar~a~in'~has derived an expression for the ratio of two lines as a function of the difference of the mean - square vibrational ampl itudes of the nucleus along and perpendicular to the Vzz axis. Sin- ce the recoi lless fraction is related to the mear square vibrational am- plitude in the direction of the incident gamma, the contribution of each microcrystal in the sample is dependent upon i t s orientation, yielding a preferential intensity having an effect similar to preferen- tia1 orientation. The Goldanski i - Karyagin effect may, sometimes, be employed to deduce the sign of Vzz. The reportedZ0 dependence of in- tensity ratio upon (
7. IMPORTANCE OF EFG INFORMATION
Generally, the quadrupole splitting has been used in the literature as a sort of measure of distortion of a system from spherical symmetry. Same quadrupolesplittingcould be shared by two compounds, for exarnple, iron sulphide, (pyri te) and iron sulphide (marcasite); but thei r strength would differ.
Ingal lsl1 has shown that the l igand and valence contribu- tions to V invariabl y have opposite s ign for some systems. Thus at the z.z low temperature limit, where the valence contribution depends neither on distortion nor temperature, increased ligand distortion would decrease VZz and heiice the quadrupole splitting, since the valence contribution would be doininant. At the high temperature 1 imi t, the problem would be ligand only and be subject to the difficulties, but would, within those splitting iiidicates increased distortion. In the intermediate temperatu- re range, where the valence contribution is both temperature and distor- tion sensitive the story becomes very complicated. The splitting of the d - orbitals, which determine the valence contribution of distortion, whereas the ligand contribution to the EFG corresponds to q/r3 type of distortion. Since both contributions increase, in opposite senses in res- ponse to two different types of distortion, the meaningful assignment of a distortion - quadrupole splitting relationship for such system at in- termediate ternperatures requi re some prior assumptions. 8. SINGLE CRYSTAL METHOD FOR EFG DETERNIINATION
From the discussion of 'Goldanskii Karyagin effect' it is apparent that peak intensity area ratio may also be a measurable parame- ter (of iron and tin) of compounds by the use 01' single.crystals. Howe- ver, to obtain accurate values of EFG parameters it is essential to f ind a theoretical expression for the area in term 01' these parameters.
Following ~an~*l,let us introduce the rate fSo, the number of recoi 1 less 14 4 Kev photons emi tted per secorid into the sol id angle subtended by the detector; (I-f)so being the nori -recoilless rate. Since the emi tted reco lless photons are not actually monoenergetic, but show a distribution o energies centered about E0 = '14.4 Kev, we introduce the function S(E-E0) , the number recoi 1less photons emi tted per second per unit energy interval into the same detector ang'le. If the source is mo- ving with velocity V relative to absorber, then recoilless photon dis- tribution i s actual 1y centered about E o + (V/C)E~= E. + E and S(E - E,,) -+ S(E - E. - E). S(E -Eo - E) is a resonance type function with ~(FWHM)= lo7?Z, therefore
the photon arrival rate R(E) at the detector is m m R(€) = e-'e S(E -EO -E) exp[fl '(E) W ] + (1-f ') So e-' e (19) where e = average surface density of atom m = surface density of nuclei capable of resonance f' = recoilless absorption probability '(E) = energy dependent resonance absorption cross section per nu- cleus in the presence of EFG when no perturbing fields are present; the resonance when no perturbing fields are present, cross section is given bYz2
whe re ao = natural l inewidth of the exci ted state a = interna1 cross-section coefficient
However, i n the sol i d, the resonance cross-section becornes more compl i ca- ted due to energy dependent spl i ttings, shifts and broadenings. Since the actual resonance is strictly a nuclear process, the condition
is always fulfilled. Consequently u(E), in general, may be written in the form U(E) = n u K(E) (22) -200 r where
If the nuclear energy levels are spl it by H or/and EFG fields so that the various energy transi tions can be resolved in a M8s- sbauer type experirnent, then K(E) may be written in the form,
where Kn(E-E n ) = a resonance type function which is chosen to have nor- malisation
P (O) = relative angular dependent absorption probability satisfying the n condi tion
P,(Q,Q) = 1 since / K(E)& = 1 (24)
The directional dependence of Pn has its origin in the fact
655 that H and/or EFG interaction destroys the spherical symmetry of the nu- clear environment.
If B is the detector efficiency for 14.4 Kev protons and RB, som velocity independent background count ing rate, then M(E) = BR(E)+R~ the experimental data of interest is actua lly M(E)/c where c is the th count rate independent of source velocity. The area under the n peak 1 m (A,) is for tn c< 1 and ($ f Bo S9 exp(-O e ) = D given by
where Pn = relative absorption probabi 1 ity for transi tion f' = recoi 1 less absorption probabi l ity.
lf X, Y and Z are the principal axes of the EFG at an i roi? nucleus in the absorber both Pn and fr will vary with the orientation angle of the incident photon bearn relative to the axes. The angular de- pendente of ffresul ts from an anisotropy in the rnean square d i spl ace- ment of the vibrating nucleus.
For a single crystal absorber wi th two or more possible orientations for EFG'S relative to the incident bearn, the area ratio is given by In case the absorber is a powder, the expression for the ratio of the area is
The importante of the Debye-Wal ler factor ( or Mussbauer fraction) in deterrnining the intensi ties of the absorption l i nes was pointed out oy Goldanski i et al. (G.K.E) .
~or~~~and ~ien~~have di scussed the anal ys is of EFG in s in- gle crystal absorbers containing ~e~~,whi le ~ar~a~ln~~'~~ has discussed the more general case of a nuclear transition of arbitrary multipolari- ty, all in the thin absorber limit. ~or~~~appl ied his analysis to F~cI~.~H~Oand eval uated a1 1 the EFG parameters, whi le Danon and lanna- rei la27 used the same analysis to confi rm thei r expectations concernino the f ield gradient parameters in sodi um n i troprusside, N~~~F~(cN)5~01.~H,o. Ingalls, Ono and ~handler~~and Chandra and ~uri~~have also applied a similar analysis for the deterrnination of EFG as a function of tempera- ture in Ferrous ammoniurnsulphate. Chandra and puri2' also analysed FeS04.7H20. There are reports, in the 1iterature, of EFG using similar 31-33 analysis for Fe(KS04) 2.6H20 30, FeS04 .4H20 , FeSQk 34, FeSiF6 .6H20 (~ef.35-37), FeTi03 38, FeS2(p) 39, ~eS~(rn)40, KfeS241 -42, ~~~0~ 43~44, ~errocene'~,Sideri te45 etc.
In order to use the above equation let i t be expressed in -+ -+ -+ nuclear EFG parameters. Considering the crystal areas a, b, and c as mu-
A A tua1 ly orthogonal, the uni t vectores Ri, Yi, Ti) locate the principal axes of EFG of ithsi te, the three Euler angles (ai, Bi, yi) designating -+ -+ -+ their orientations relative to (a,b,c) being unknown. Angles ( Oi, 0;) are the orientation angles of the incident beam relative to (xi,Yi,Zi). Let us introduce (O,@) the polar and azinuthal angles of the incident +++ photon beam relative to (a,b,c) axes.
The expressions for relative transi tion probabi l ities P3and P1 were derived utilizing the fact that 14.4 Kev garnma rays is a magne- tic dipolar radiation and given by 2 Expressing cos20i and sin Oi cos2~i in terms of known ex- perimental angles (e,+) and the known Euler angles relating site to axes ((;,8,3,from equations (above) assuming isotropy of ff(o ,Q ) one gets
where
K = sin28[(cos2+)za,2 + (sin2+)~:] + (cos2e)z + (si& sin %)zazc b
K'= sin28 L(cos2+) (X ,2- y2) + (si n2+) (xcZ - y2)] + (cos20) ( ~2-y2) + a a C bb
- + The syirbols x,,, x etc denote the directioti cosines X.a, b ?.% etc. These di rection cosines can be expressed as a function of the same Euler angles (a,f3,y). The schematics of the absorption of 14.4 Kev gama rays be ~e~~ in single crystals in shown in the figure.
In the above analysis, unfortunately, it has been necessary to use samples of sufficient thickness that the thin absorber anal ysis can not be completely justi fied; because the saturation effects can not be neglected. Therefore, i t is necessary to have a method of analysis in which saturation is taken into account. Because of polarization depen- dente of the individual nuclear cross secti6ns the inten~ities~~or a - rea~~~of the absorption 1ines can not be calculated in the thin absor- ber approximation. Polarization effects must also be taken into account in making a detai led interpretation of spectra obtained by appl y i ng a magnetic field to a thick, randomly oriented powder of a material exbi- biting a quedrupole doublet15. Blume and ~istner~~calculated the inten- sities transmitted through an absorber of arbitrary thickness in the ca- se where a11 parameters describing the individual cross sections are known. Housley; Grant and ~onser~'general ised the Bl ume and ~i~tner~~ treatment, for more than one resonant si te per uni t cell by introducing an average scattering ampl itude per unit cell obtained by summing the properly weighted scattering ampl itudes from a1 1 si tes, a11 scattering amplitude matrices being expressed in terms of same basis polarisations. Housley et a14' obtained formulas relating absorptions as a function to the individual nuclear cross sections for radiation propagating along special directions in the monocrystal which are predictable from crystal symmetry cons iderations. In these cases the gamma ray beam can be thought of as consisting of two orthogonal intensities each of which propagates through the crystal with its own index of refraction. Housley et a~~~-~~ has shown by means of several exarnples, the importante of polarisation effect i n analysis of data on single crystals or magnetised materiais and determina~ion~~of the sign of the magnetic hyperf ine f ield. Hous 1 ey, Gonser and ~rant~~determined the Debye Waller factor in Siderite taking into consideration the polarlsation terms. Siderite has been of conside- rable interest and there exist many reports45.
9. FINITE THICKNESS ANALYSIS
The complex index of refraction for the material is given by53
where k and k' are the wave vectors of the incident the scattered radiat- ion, respectively, IV is the number of resonant nuclei per uni t v01 ume, and F(k,kr) is the coherent scattering amplitude for a single nucleus.The quantities E and n may be replaced by a 2x2 matrix. Generalising, for a compl icated crystal having several resonant nuclear sites per uni t cell, Sum runs over a11 the sites. It is assumed that the electromagnetic inte- raction is so weak that the index of refraction has a negl igible influen- ce on an incident beam in a distance of order of the unit cell dimnsions. Now expressing n in terms of the densi ty matrix rotation.
I \
I 'i i j u p pij N .f! (x-x. .) 11 12 JJ - -o 1 2k (x-xij2 + 1 ij ij P 22 , here j and i represent si tes and hyperf ine coniponents. The quanti ty x is 2E related to the energy B by x = r and xijis correspond to the resonant e- nergies at the different sites. The Debye - Waller factors are no longer assumed i sotropic but are the ones appropriate for the di rection k.
In the special case where n is diagonal, an incident radiation beam may be decomposed into two components with polarisations correspon- ding to the basis polarisations in which n is diagonal. Each of these beams may be regarded as separatel y propogatinci through the crystal wi th its own index of refraction.
An effective thickness matrix u may be defined from the equa- tion (32) and will be diagonal when n is diagomal
n.f! 2 --2 (33) (x-x. .) + 1 23
where ni are the number of resonant nuclei of type j per uni t area. When
660 n is diagonal, the matrix elements of o can be extracted from experimen tal !ine intensities, if the spectra are well resolved.
Assuming that the Y ray beam can be described as a plane wavc incident on a slab of homogeneous absorbing material of uniform thickness and that the absorption may be descri bed wi thout di spersion. Descri binf the polarisation and intensity of the unshifted component of the sourci radiation at any energy by a density matrix
where I is the total intensi ty due to the transi tion of interest and r is the fract ion of the em; tting nuclei in site A. The source resonance e nergies xkz and si te positions L will not generally correspond to those the absorber. The quantity 0 describes the Doppler shift in the resonanc energies due to relative motion of the source and absorber. The genera expression for the transmitted intensity at a particular energy x whe radiation from a source described by Eq. (34) is incident on an absorbe described by equation (33) can easi ly be found since Sllgives the pro jection of the transrnitted source intensity on one of the basis polariza tions and that of the other.
This express ion is s imi lar to the corresponding expression foi- the unpo larised radiation except For containing two terms instead of one on th right. It may be substracted from the incident intensily Sll+S22and th resiilt integratea over x to obtain the fractional absorption dip at an Dop~iervelocity v. Integrating over u to obtain the dimensionless areai the absorption spectrum.
Thi- unknown parameters of the experiment are contained in th o's and hopefully can be extracted from measurements made in differen di rection and/or wi th di fferent thickness absorbers. If the absorption 1i nes are well resolved individual areas can be defined. Assuming source and absorber resonance lines to be of natural width, then after Bykov and t h ~ein~~,for the overlap of the kth source 1 ine from si te wi th the i absorber line from site j, this gives ,
, (36)
where TO, and II are zeroth and f irst - order Bessel functions of imagi- nary arguments. When different absorber ines have same energy, thei r con- tributions are added in the arguments of functions of equation above.
The di mensionless .ares Ak'ij is elated to the bõckground correc- ted experimental area B by
For relatively thin absorbers, power series expansions in equation (36) are useful. If we restrict ourselves for simplicity to a single line un- polarised source Si1 = S22 = f/2 and we define the average cross section on resonance p = 1uii(xij) + ~22(4~)1/2 and the fractional polarisation
"11 - "22 a = of the absorption we may write O11 + "22
1 1 5 A =p- (i + a2)p2 +ib (I + 3a2)p3 - 384 (1 + 6a2 + a4)p4 + ... on inverting
1 1 g =AiF (1 +a2)A2 +T(1 +a2 + 2a4)A3+ (1 +a4 +6a6)d4 384 ... Therefore for a relatively thin absorber, the effects of polarisations may be quite small unless fractional polarisations become large.
Equations for the area ratios which have been previously used in analysing the experimental data can be obtained by dropping all but the f i rst term in equation (38).
Grant, Housley and ~onserl~appl ied the above corrections to determine the electric field gradient and the mean square displacement tensor parameters at the Fe si tes in sodium ni troprusside and found that the principal axes of two tensor do not coincide. Similar analysis was also appl ied in the case of FeS04.4H20 33 in the temperature range of 300 -77K.
10. USING IPOLARISED RADIATION
The polarisation of y rays has been reported54-58 by many wor- kers, and it can take place by the Stark and Zeeman effect that are al- ready familiar in the optical polarisation st~dies~~,If the recoilfree y rays are brought into resonance the matching of polarisation compo- nents in the source and the absorber nuclear transitions is required in addition to the usual appropriate Doppler mtion. For example, circular- ly polarised y rays from a source can only be resonantly absorbed if a transi tion with the same hel icity is offered (conservat.ion of angular mmentum). By observing the hel icity of ci rcularly polarised y rays, the sign of magnetic hyperfine field can be determined .
Linearly polarised y rays60 -62 are of particular interest. Some methods for producing 14.4 Kev linearly polarised y rays are brie- fly described below;
One simple method is to use a magnetical ly source rnatri x (iron foil) and to magnetise it in a fixed direction. With iron foil it is convenient to magnetise perpendicular to the di rection of emi ssion. The emitted radiation is then not monochromatic but consists of six indivi- dual coniponents with intensity ratios 3:4:1:1:4:3 each of which has a
definite polarisation. Iri the case we are considering t3/2 + 11/2 and
?1/2 + f1/2 transitions have a linear polarisation with the electric vec- tor vihration parallel with respect to the magnetisation, and 11!2 ++-1/2 transitions are similar but perpendicularly polarised. In a single crys- tal absorber, of the source and absorber are directionall y polarised by a magnetic field which can be interna1 in origin or by an electric field such as is associated with the electric field gradient tensor, the pola- risation of each emission line becomes important. The relative orienta- tion of the principal di rections in the source and the absorbcr will cause variation of the !ine intensity. Detailed theory of the line in- tensi ties has been reported63'04. One of the first appl i cati on of an i ron foi l source was in the determination of the sign of the quadrupole coupting constant in a single crystal of ~eSiF~.6~,065.
The assignment of hyperfine components ir1 a complex spectrum would be easier if the source was effectively polarised and monochroma- tic so that the resonan: absorption would only be observed in those tran- sitions wi th the sane polarisation. This mthod was first demonstrated by ~ousle~~~using a paramagnetic single crystalline source exhibiting a quadrupole spl i: spectrum. If the ZZprincipal axis of the EFG is per- pendicular to the direction of observation the y rays corresponding to the ?3/2 transition are totally 1 inearly palarised and the y rays cor- responding to ?1/2 transi tion are partia1 ly l inearly polarised. This mo- re intense line can be selectively filtered out. The asymmetry parameter should be zero to ensure that the nuclear wavefunction are pure Im> sta- tes.
An equivalent mthod which does not require a splIt source is to use a polariser. This was first demonstrated using iron foils mag!ie-
tised perpendicular to the propagation67 and more recently using a po- lycrystall ine sample of Yattridm iron garnet (Y IG) .
When source and absorber exhibit a hyperfine icterac,tion, the polarisatior? of the y rays has to De taken into account in deriving the relative line intensi ti~s~~'~~.By analysing the line intensities ob- tained from a source emitting linearly polarised recoilfree y rays and a single crystal absorber cut in an arbitrary direction, one can deter- mine the spin orientation and the principal axes of the EFG with respect to the rrystal axesS'1ós17s65?70171.~aniels~~ has considered the pro- b!em of calculating line intensities in ~Ussbauerabsorption using the polarised gamma rays polarised absorbers with especial reference to ~e~~.
The polarisation dependence has been reported i n a number of theoretical studies49s57963t73-79and various technfques have been used tc carry out polarimetry experiwnts based on t ransver! y magnet i sed C057(crFe) source51,52s~8,60t61:64268,71?7ht80-88 or by fi1 ter techni - que56'60'"2'4'89. Kultipl icity resul ting from the si x 1inearly polari- sed l ines or inefficiency of the selective excitation or of the filters in terms of intensity, degree of polarisation and non Lorentzian lines are some of the disadvantages of the above methods. Sources exhibiting quadrupole hyperfine interaction will circumvent some of the problems. The utility of a quadrupole spl it coS7 source for 2 polarimeter depends on fulfillment of following conditions.
The material can be grown as a single crystal and cutting of thin plates in various directions is possible.
~0~~can be incorporated with a unique charge state arid a crys- tallografically miqe resonance site.
The !attice site should have axial symmetry with unique orien- tations.
The quadrupole splitting should be large or at least well re- sol ved .
Gonser, Sakai and ~eune'~reported a qradrupole ~e~~polarime- ter ccnsisting of single crystal of L~N~O~:CO~'as source (,polariser) and FeC03 (si deri te) as a absorber (anal yser) . The possibi 1 i ty of using two identical single crystals as polariser and analyser has been mentio- ned by Goldaiiski i et al.45 and ~;ibbe6 used i t as a means of determining tne directl~~naldata normal to the observation axis in the case of fer- rous armoni iiin sulphate and the EFG was determined by using crystals grown on one particular face (301) so that the observation direction was nor- mal to the b axis and at 47.3' to the c axis in the ac plane. This type of measuremen ts have d if f icul ty in measuring an accurate thickness cur- ve and because of the large time envolved. The choice of the parameters to fi t the data is also not unambiguous. Gibbe7 made further experiments on ferrous ammonium sulphate using a coS7/~efoi 1 source magnetised in the plane of the foi 1 and perpendicular of observation by mounting the source in between the poles of a permanent magnet. Thus, most inner and outer pai r of l ines were plane polarised whi le para1 lel to the appl ied magnetic field, and the other two lines are polarised perpendicular to the f ield.
Because of the self absorption in the source matrix, and, also due to incomplete polarisation of the foi1 in absence of the applied magnetic field is not strong enough, the predicted intensity ratios (3:4:1:1:4:3) are unlikely to be achieved. Gibb91sg2 redetermined the e- lectric field gradient tensor in FeCl .4H O using a polarised ~gssbauer source in the same method. Spiering and vogelg3 calculated the absorp- tion cross section for a polarised single crystal absorber of f ini te thickness using a intensity matrix method and applied it in the case of Fec12.4H20. 2immermanng4 also made experiment on Fec12.4H20 us i ng the intensi ty tensor method.
REFERENCES
1. M.H.Cohen and R.Reif in SoZid State Physics, ed. F.Seitz and D.Turn- bul 1 (~cademic Press, New York 1957) . 2. S. P. Puri , Bocd. NucZear Physics and SoZid State Physics Symposiwn December 28-31, 1969, Roorkee, India, 1, 71 (1969). 3. J .C .Trav i s i n An Introduction to ~&sbauerSpectrocospy, ed, Leopol d May (Plenum Press, New York, 1971). 4. B.D. Dunlap in ~b'ssbawrEffect Data Index (1970) ed. John G.Stevens and Virginia E. Stevens (~lenumPress, New York 1971). p. 25. 5. R.M.Sternheimer, Phys. Rev. 80, 102 (1950). 6. Ref. 1 page 324. 7. A.J.Freeman and R.E.Watson, Phys. Rev. 131, 2566 (1963). 8. R.lngalls, Phys. Rev. 128, 1155 (1962). 9. R.M.Sternheimer, Phys. Rev. 130, 1433 (1963) . 10. F.A.Coi:tan and G.Wi lkinson in Advanced Inorganic Chevfstry, ( Int. pu- blishers, New York 1963) p.563. 11. R.lngalls, Phys. Rev. 133, A787 (1964). 12. R.W.Graint, R.M.Housley and U.Gonser, Phys.Rev.178, 523 (1969). 13. E.Fluck., W.Kerler and W.Newirth, Agnew Chem. 20, 277 (1963).(lnter- national ecli tion) . 14. R.L.Col1 ins and J.C.Travis in MJssbauer Effect ~ethodoZogy,Vol .3,ed. I. J.Gruverman (~lenumPress, New York 1967) . 15. S.L.Ruby and P.A.Flinn, Rev.Mod.Phys. 36, 351 (1964). 16. R.L.Collins, J.Chem. Phys. 42, 1072 (1965). 17. J.R.Gabrie1 and S.L.Ruby, Nucl. lnstru. Methods 36, 23 (1965). 18. R.W.Grant, H.Widersich,A.H.Muir,Jr., U.Gonser and W.N.Delgass, J. Chem. Phys. 45, 1015 (1966). 19. S.V.Karyagin, Proc. Acad. Sci. USSR, Phys. Chem. Sec. 148, 110 (1964). (~ngli sh Translat ion) .
ZO.P.A.Flinn, S.L.Ruby and W.L.Keh1, Scientific paper 63-128.117, P .6, Westinghouse Research Laboratories, Beulah Road, Churchi I1 Borough , Pit- tsburgh, Pennsylvania, U.S.A. 21. G.Lang, Nucl. Instru. Methods 24, 425 (1963).
22. J.D.Jackson, Can. J. Phys. 33, 595 (1955). 23. P.Zory, Phys. Rev. 140, ~1401(1965). 24. P.Z.Hien, Soviet Phys. JETP 22, 1080 (1966). 25. S.V.Karyagin, Soviet Phys. Sol id State 8, 1387 (1966). 26. S.V.Karyagin, Soviet Phys.Solld State 9, 1979 (1968).
27. J.Danon and L.lannarella, J.Chern.Phys. 47, 382 (1967). Ref. 12 also. M. H. Clear, J.F.Duncan and D.McConchie, J.Roya1 Soc.New Zealand 1, 71 (1971). 28. R. lngalls, K.Ono and L.Chandler, Phys. Rev.172, 295 (1968). 29. K.Chandra and S.P.Puri, Phys. Rev. 269, 272 (1968). 30. V.K.Garg and S.P.Puri , J. 1norg.Nucl .Chem. 35, 2'795 (1973). 31. V.K.Garg and S.P.Puri, J.Chem.Phys. 54, 209 (1971). 32. V.K.Garg and Y.S.Liu, J.Chem.Phys. 62, 56 (1975). 33. M.S. Nascimento and V.K.Garg, J.Chem. Phys., 66, 5798 (1977). 34. H.N.Ok, Phys. Rev. B, 4, 3870 (1971). 35. C.E.Johnson in Hyperfine Structwe and NucZear Radiatwns ed.E. Mat- thias and D.A. Shi rley (~orthHol land publ ishing co., Amsterdarn, 1968) p.226. 36. V.K.Garg arid K.Chandra, Phys.Stat.Sol idi (b) 50, ~49(1972) . 37. J.Chappert, G.Jenhanno, F.Varret, in Procd. Int. Conf. kf8ssbat~er Spectroscopy, Cracow, Poland 30 Augcist 1975 Vol. 1,2C-9, p.161. 38. V.K.Garg and S.P.Puri, Phys.Stat. Solidi (b) $4, Y44 (1971). 39. 'd.K.Garg, Y.S.Liuand S.P.Puri, J.Appl.Phys. 45, 70 (1974) and re- ference therein. P.A.Montano and M.S.Seehra, Sol idi State Comm. 20, 879 (19?6). 40. R.Garq and V.K.Garg, 4ppl .Phys. 16, 75 (1978). 41. R.Garg, A.C.Oliveii-a and V.K.Garg, Phys. Stat. Solidi (5) 84, K17 (1577). 42. R.B.ScorzelIi, C.A.Taft, J.Danon and V.K.Garg, J.Phys. C. Solid Sta- te Physics 12, 1379 (1978). 43. T.Ruskov,T.TomovandSGeorgiev. Phys. Stat. Solidi (a) 37, 295 (1976) and reference there i n. 44. A.N.Salugin, V.A.Povi tskii, M.V.Fi lin, V.M.Erkin and V.V. Dudoladav, Sovier Phys. SolidState 26, 792 (1974). 45. R.M.Hous ley, U.Gonser and R.W.Grant, Phys .Rev.Letts. , 1279 (1968). D. L. Nagy, I .Dezsi and U.Gonser, Neues .Jahrb.Mineral .Monatsh 3, 101 (1975). I.P.V~nogradov, A.M.Pritchard, E.F.Makarov and I.P.Suzdalev, So- lid State Comm. 8, 965 (1970). V. I .Goldanskii, E.F.Makarov, 1 .P. Suzda- lsv and I.P.Vinogrddov, Phys. Rev. Letts. 20, 137 (1968) JETP 27, 44 (1968). 46. J.G.Dash, 8.D.Taylor. D.E.Naqle, P.P.Craig snd W.M.Visscher, Phys. Rev. 122, 11 16 (1961). 47. G.A.Bykov and Pham Zey Hien, Zh. Eksperirn. i Teor. Fiz. 43, gOj(1962). Çoviet Physics JETP 16, 646 (1963). 45. M.B!ume and O.C.Kistner, Phys. Rev. 171, 417 (1968). 43. R.M.Housley, R.W.Grant. and U.Gonser, Phys.Rev. 178, 514 (1969). 50. R.M.Housley in Mbssbacnnr Effect Met&odoZogy, Vo1 .5, ed. I. J. Gruver- nan, Plenum Press, New York (1970). 51. R,M.Housley and U.Gonser, 2hys.Rev.171, 480 (1968). 52. U.Gonser and R.M.Housley, Phys.Rev. Letts. 266, 157 (1968). 53. M.Lax, Rev. Mod. Phys. 23, 287 (1951). 54. G.J.Perlow, S.S.Hanna, M.Harnermesh, C.Littlejohn, D.H.Vicent, R.S. Preston and S.Heberle, Phys. Rev. Letts. 4, 74 (1960) . 55. W.A.Shurc1 i ff - ?oZar;.Ysed Light product-Yon and Use (Oxford Uni~ersi- ty Press, London, 1962). 56. S.S.Hanna, J.Heberle, C.Littlejohn, G.J.Perlm, R.S.Prestori, and D. H.Vicent, Phys. Rev. Letts, 4, 177 (1960). 57. H.Wegener and F.E.Gbenshain, Z.Physik 163, 17 (1961). 58. A.Kastler, Compt. Rend. 255, 3397 (1962). 59. N.Blum and L.Grodzins, Phys. R~.J. 136, A133 (1964). 60. P.lmbert, J.Physique 27, 429 (1966).
61. U.Gonser, p.343 i r; Hyperfine Structure and Nuclear Radiations ed. E.
Matthias and D.A. Shi rley, North Hol land Publ ishing CO., Amsterdam, (1968). 62. J.P.Starnpe1, P.A. Fl irin in Mgssbauer Effect MethodoZogy Vo1.6, ed. I .J.Gruvermen, Plenuni Press, New York, (1971). 63. H.Fraunfelder, D.E.Nagle, R.S.Taylor, D.R.F.Cochran and W.M.Visschar, Phys.Rev. 1,?6, 1065 (1962). 64. J.T.Dehri, J.G.Marzolf arid J.F.Salmon, Phys. Rev. 135, B 1307 (1964). 65. C.E.Johrison, W.Marshal1 and G.J.Perlow, Phys.Rev. 126, 1503 1962) . 66. R.M.You:;ley, Nucl . Instr. Methods, 6 2, 321 (1968) . 67. S.Shtrikman, Solid State Commun. 5, 701 (1967). 68. S.Shtrikman and S. Somekn, Rev. Sci. Instru. 40, 1151 (1969) 69. U.Gonser in Hyperfzne Interactions ed. A.J.Freeman and R.B. rankel , (~cademicPress, New York, 1967) . 70. U.Gonser, R.W.Grant, H.Wiedersich and S.Gel ler, Appl . Phys. Letts. 9 , 18 (1966). 71. U.Gonser and R.W.Grant, Phys. Stat. Sol idi 21, 331 (1967). 72. J.M.Daniels, Nucl. Instru.Methods 128, 483 (1975). 73. Y.M.Aizazyan, V.A.Belyakov, Sov.Phys.So1 id State 13, 808 (1971). 74. A.V.Mitin, Phys. Stat. Solidi (b) 53, 93 (1972). 75. J.P.Hanrion, N.J.Carron, G.T.Trammel1, Phys.Rev. B, 9, 2791 (1974). 76. J.P.Hanrion, N. J.Carron, G.T.Tramme1 I, Phys .Rev. B, 9, 2810 (1974). 77. D.Barb, M.Rogalski, J.Chim. Phys. Phys.Chim.Bio1. 72, 470 (1975). 78. D.Barb , Frocd. Intem. Conf. Mbrssbauer Spectroscopy, Cracow, Pol and (1975) p. 3'79, Vol. 2. 79. R.W. Grent i n MJssbauer Spectroscopy - Topics in AppZied Physics, Vol. 5, ed. U.Goriser (Springer Verlag, Berlin, 1975). 80. C.E.Johrison, J.Phys. C. 2, 1996 (1969). 81. R.W.Grarit and S.Geller, Sol id State Commun. 7, 129 (1969). 82. S.Gel lei-, R.W.Grant, L.D.Fu1 lmer, J.Phys.Chem.So1 ids 31, 793 (1970) . 83. R.W.Grant, J.Appl .Phys. 42, 1619 (1971). 84. H.D.Pfanner, U.Gonser, Nucl. lnstru.Methods 124, 297 (1974). 85. S.Morup, J. Physique, Colloq. C6, 683 (1974). 86. T.C.Gibb, J.Phys. C.Solid State Phys. 7, 1001 (1974). 87. T.C.Gibb, J.Phys. C.Solid State Phys. 8, 229 (1975). 88. H.Fischer, U.Gonser, H.D.Pfannes and T.Shinjo, Procd. Intern. Conf. M8ssbauer Spectroscopy, Cracow, Pol and p. 463 (1 975) Vol . 1 . 89. F.Varret, P. lmbert, G.Jehanno and R.Saint-James, Phys .Stat. Sol idi (a) 27, K99 (1975). 90. U.Gonser, H.Sakai and W.Keune, J.Physique C6, 709 (1976). 91. T.C.Gibb, Chem.Phys. 7, 449 (1975). 92. T.C.Gibb, J.Chem.Soc.Dal ton Trans. 743 (1968). 93. H.Spiering and H.Vogel, Hyperfine Interac. 3, 221 (1977). 94. R.Zimmermann, Chem. Phys. Letts, 34, 416 (1975).