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Determination of Mossbauer Electric Field Gradient

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Determination of Mossbauer Electric Field Gradient 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 <rm3 >3d > O the sign determining factor is <3 cos20 - I >, which may be posi tive or negative depending on the man- ner in which the electrons are distributed among the d-orbitals. 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 <r-3> 2g is considerabl y smal ler. The effect of l igand symmetry is correspon- dingly enhanced. Consider two diamagnetic ferrous compounds in which no odd 3d electrons affect the EFG tensor. The very strong (greater + than12 CN-) 1 igand NO causes a spl i tting of 1.70 mm/sec. in sodi um ni- 12 tropruss ide. In Na4 1 F~(CN)5~~2 ( .2H20 the weaker 1 i gand (NO2) resul ts in a splitting of 0.86 mm/sec. 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.
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