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Description of the angular dependence of dipole transitions by intensity tensors
Volume 34, number 3
DESCRIPTION BY INTENSITY
Reinhart
CHEMICAL
OF THE AiWULAR
1 Au@st I975
PHYSICS LETTERS
DEPENDENCE
OF DIPQLE TFXNSIT’IQNS
TENSORS
ZIMMERMANN
Physikaliwhes Instifut II der ClniversitZtEhngen-NCmberg,
D-8520 Erlongen, Ger!norzy
Received 21 Mach 1975 Rcvisod manuscript
rcceivrd 28 April I975
It is shown that, ti the case of a dipole transition (e.g. 57F~, L’gSn 61Ni, . . . h16ssbauer nuclei), the angular dependence of the intensity of on absorption line for a non-polarized T-beam can b’c described by an intensity tensor. Advantages of the
formulation are discussed.
The evaluation uf the angular dependence of y-ray absorption intensities is a fundamental problem in ‘y-ray spectroscopy. 1; provides information on the hyperfine The simplification of the evaluation of dipole transition cussed in the present work. Method and results can be Dipole transitions occur beWee nuclear multiplets are usually split by hyperfine fields at the nucleus and plets ja and jb are arbitrary linear colnbinations of the
fields at the nucleus and has been treated extensively [l-7]. intensities by introduction of an intensity tensor is disapplied to single crystal Mijssbauer spectra. of nuclear spin j0 and jb with jb = ja f 1, The multiplets by external fields. The eigenstates IA) and IL?>of the multistates Ii,, ma> and Ijb, mb), respectively, i.e.,
c
IA)=
(1)
flla=-ja,...,ja In absence
behveen
I=
of thickness
effects
and an anisotropic
the states IA) ntid IS> is calculated
Debye-Wailer
by the formula
factor
the relative
line intensity
lcc,x’“12
(2)
)
M
where m has the value 1, 0, -1, Xm are vector
If we use cartesian
components
Xx = -2-1’2(x1_x-1) cx = -2-“2(q-c_1),
for a transition
[6]
(c, ,
spherical
harmonics
for L = 1 [8] and
and Xm transform contragredient) XY = Ts1j2 i(X’+Xml)
,
X2 =x0
;
= -2;1’2i(q W-1) 1 c, = co I cv the intensity1 is sxpreaed by a formula ident_ica! to that in (2) just with m = x,y,z. Specifying& = j, t 1, applying Racah’s f+rnuIa for CIebsch-Gordan coefficients [?I and ornittiag a common factor [(2j,)!/2(2j,+3)!] I”
CHEMICAL
Volume 34, number 3
PHYSICSLETTERS
1 August 1975
we find for cX, cu and cz
(4) It now can be shown that the angular dependence of the relative intensity I for a transition involving a nonpolarized r-beam with polar angles 0, Q and direction cosines
ex=sinBcos@,
e,=sin@sinQ,
e,=cose,
takes the simple form (5)
where I pQ = 2(c, c_;+cy c;+c= cz‘) 6,,
- (cp c;icq c;> .
The quantities Ipp4, p. q = x,y, z can be interpreted as components of an intensity tensor which is real, symmetric and of second rank. Thus the intensity describes an ellipsoid in the three-dimensional space. The components of the intensity tensor, Ip4 are not iudependent of each other since the multiplication of cr, cy , c, by an arbitrary phase does not change their values. With Tp4 = Ip4 - UXX+1,,+I,,)6,, the interrelation
(6)
>
between the components
is expressed as (7)
If two transitions are degcntrate the total line intensity is characterized by an intensity tensor which is the sum of the intensity tensors for each transition. Since eq. (7) is, in general, valid only for !ines which result from a tnnsition between nondegenerate states, it is possible to differentiate between degenerate and nondegenerate lines. One of the advantages of the intensity tensor formulation arises from the close rzlaticnship between a particular symmetry at the nucleus and properties of the intensity tensor of every transition line between the rntdtiplets. In the presence of a nuclear electric field gradient (EFG), for instance, the intensity tensors have to be diagonal within the principal acres system of the EFG. Even if t!!ere is an effective magnetic field along one of the principal axes v XX v yy’ V,, of the EFG the components IXy, I xZ, I yc have to vanish. in both cases the principal axes system of the EFC can bc found by determination of that of the intensity tensor. The particu!ar values for the non-vanishing components
of the intensity tensor cannot be obtained by syrn-
metry arguments but have to be calculated by d.iagonahzing the hyperfine field hamiltonian and applying formula (3). A comp!ete discussion is beyond the scope of the present work and resu!ts are presented only for transitions withj, = &,jb = $ e7Fe, llgSn , ___)where the total intensity tensor of ezch of ihe two quadrupolar lines obeys 417
Volume 34, number 3 the
CHEMICAL
PHYSICS
LETTERS
1 August 1975
relation
I - 3,, = ypq . .pq In this case the electric anisotropy at the nucleus is directly reflected by the anisotropy of each line intensity. The asymmetry parameter T) of the traceless intensity tensor Ip4 - $8,, is equal to that of the EFG tensor. This equality can he successfully applied to the evaluation of paramagnetic single crystal 57Fe Mijssbauer spectra [lo] _ Further advantages of the utilization of an intensity tensor in nuclear solid state physics arise from its analogy to the susceptibfity tensor x, e.g.: (i) The powder mean value of the inter.sity is one third of the trace of the intensity tecsor Ip”‘v = (I,,+ly,,+1,,)/3
_
[ii) Formulae for .iextured material can be taken over from those of the x-tensor, (iii) For more than one equivalent lattice site the macroscopic intensity I can be described tensity
tenser
which
is the mean
value
of the local
intensity
tensors
of each
lattice.
Symmetries
by a macroscopic of the crystal
inare
reflected by the macroscopic intensity tensor in the same way as by the x-tensor. In a rhombic crystal, for instance, the principal The author
axes of the macroscopic thanks
Professor
intensity
tensor have to be parallel
to the twofold
axes.
H. Wegener, Dr. D. Nagy, Dr. H. Spiering and Dr. C. Ritter for helpful and enassistance of the Deutsche Forschungsgemeinschaft is much appreciated.
couraging discussions. The financial
R~fekxms [l]
H. Frauenfelder,
DE.
Naslr-, R.D. Taylor,
D.R.F.
Cochran
and W.M. Visschcr,
Phys. Rev. 126 (1962)
‘f?] J.R. Gabriel and S.L. Ruby, Irlucl. Instr. Methods 36 (1965) 23. [3] P. Zpry, Phys. Rev. 140A (1565) 1401. [4]
S.V. Karyagin,
Soviet Phys. Solid State 8 (1966)
1387. 961.
[5] G.R. Hoy and S. Chandra, J. (3hem. Phys. 47 (1967) [6] 171 [8 j [9] [IO]
W. Kiindig, Nucl. Instr. Methods 48 (1967) 219. A.J. Stone and W.L. Pitlinger. Phys. Rev. 165 (1968) 1319. S. De Bendetti, Nuclear inte.ractions (Wiley, New York, 1964). A. Messiah, Quantum mechanics, Vol. 2 (Not&-Holland, Amsterdam, R. Zimmermann, Nucl. Instr. hiethods, submitted for publication.
1970)
p. 1059.
1065.
DESCRIPTION BY INTENSITY
Reinhart
CHEMICAL
OF THE AiWULAR
1 Au@st I975
PHYSICS LETTERS
DEPENDENCE
OF DIPQLE TFXNSIT’IQNS
TENSORS
ZIMMERMANN
Physikaliwhes Instifut II der ClniversitZtEhngen-NCmberg,
D-8520 Erlongen, Ger!norzy
Received 21 Mach 1975 Rcvisod manuscript
rcceivrd 28 April I975
It is shown that, ti the case of a dipole transition (e.g. 57F~, L’gSn 61Ni, . . . h16ssbauer nuclei), the angular dependence of the intensity of on absorption line for a non-polarized T-beam can b’c described by an intensity tensor. Advantages of the
formulation are discussed.
The evaluation uf the angular dependence of y-ray absorption intensities is a fundamental problem in ‘y-ray spectroscopy. 1; provides information on the hyperfine The simplification of the evaluation of dipole transition cussed in the present work. Method and results can be Dipole transitions occur beWee nuclear multiplets are usually split by hyperfine fields at the nucleus and plets ja and jb are arbitrary linear colnbinations of the
fields at the nucleus and has been treated extensively [l-7]. intensities by introduction of an intensity tensor is disapplied to single crystal Mijssbauer spectra. of nuclear spin j0 and jb with jb = ja f 1, The multiplets by external fields. The eigenstates IA) and IL?>of the multistates Ii,, ma> and Ijb, mb), respectively, i.e.,
c
IA)=
(1)
flla=-ja,...,ja In absence
behveen
I=
of thickness
effects
and an anisotropic
the states IA) ntid IS> is calculated
Debye-Wailer
by the formula
factor
the relative
line intensity
lcc,x’“12
(2)
)
M
where m has the value 1, 0, -1, Xm are vector
If we use cartesian
components
Xx = -2-1’2(x1_x-1) cx = -2-“2(q-c_1),
for a transition
[6]
(c, ,
spherical
harmonics
for L = 1 [8] and
and Xm transform contragredient) XY = Ts1j2 i(X’+Xml)
,
X2 =x0
;
= -2;1’2i(q W-1) 1 c, = co I cv the intensity1 is sxpreaed by a formula ident_ica! to that in (2) just with m = x,y,z. Specifying& = j, t 1, applying Racah’s f+rnuIa for CIebsch-Gordan coefficients [?I and ornittiag a common factor [(2j,)!/2(2j,+3)!] I”
CHEMICAL
Volume 34, number 3
PHYSICSLETTERS
1 August 1975
we find for cX, cu and cz
(4) It now can be shown that the angular dependence of the relative intensity I for a transition involving a nonpolarized r-beam with polar angles 0, Q and direction cosines
ex=sinBcos@,
e,=sin@sinQ,
e,=cose,
takes the simple form (5)
where I pQ = 2(c, c_;+cy c;+c= cz‘) 6,,
- (cp c;icq c;> .
The quantities Ipp4, p. q = x,y, z can be interpreted as components of an intensity tensor which is real, symmetric and of second rank. Thus the intensity describes an ellipsoid in the three-dimensional space. The components of the intensity tensor, Ip4 are not iudependent of each other since the multiplication of cr, cy , c, by an arbitrary phase does not change their values. With Tp4 = Ip4 - UXX+1,,+I,,)6,, the interrelation
(6)
>
between the components
is expressed as (7)
If two transitions are degcntrate the total line intensity is characterized by an intensity tensor which is the sum of the intensity tensors for each transition. Since eq. (7) is, in general, valid only for !ines which result from a tnnsition between nondegenerate states, it is possible to differentiate between degenerate and nondegenerate lines. One of the advantages of the intensity tensor formulation arises from the close rzlaticnship between a particular symmetry at the nucleus and properties of the intensity tensor of every transition line between the rntdtiplets. In the presence of a nuclear electric field gradient (EFG), for instance, the intensity tensors have to be diagonal within the principal acres system of the EFG. Even if t!!ere is an effective magnetic field along one of the principal axes v XX v yy’ V,, of the EFG the components IXy, I xZ, I yc have to vanish. in both cases the principal axes system of the EFC can bc found by determination of that of the intensity tensor. The particu!ar values for the non-vanishing components
of the intensity tensor cannot be obtained by syrn-
metry arguments but have to be calculated by d.iagonahzing the hyperfine field hamiltonian and applying formula (3). A comp!ete discussion is beyond the scope of the present work and resu!ts are presented only for transitions withj, = &,jb = $ e7Fe, llgSn , ___)where the total intensity tensor of ezch of ihe two quadrupolar lines obeys 417
Volume 34, number 3 the
CHEMICAL
PHYSICS
LETTERS
1 August 1975
relation
I - 3,, = ypq . .pq In this case the electric anisotropy at the nucleus is directly reflected by the anisotropy of each line intensity. The asymmetry parameter T) of the traceless intensity tensor Ip4 - $8,, is equal to that of the EFG tensor. This equality can he successfully applied to the evaluation of paramagnetic single crystal 57Fe Mijssbauer spectra [lo] _ Further advantages of the utilization of an intensity tensor in nuclear solid state physics arise from its analogy to the susceptibfity tensor x, e.g.: (i) The powder mean value of the inter.sity is one third of the trace of the intensity tecsor Ip”‘v = (I,,+ly,,+1,,)/3
_
[ii) Formulae for .iextured material can be taken over from those of the x-tensor, (iii) For more than one equivalent lattice site the macroscopic intensity I can be described tensity
tenser
which
is the mean
value
of the local
intensity
tensors
of each
lattice.
Symmetries
by a macroscopic of the crystal
inare
reflected by the macroscopic intensity tensor in the same way as by the x-tensor. In a rhombic crystal, for instance, the principal The author
axes of the macroscopic thanks
Professor
intensity
tensor have to be parallel
to the twofold
axes.
H. Wegener, Dr. D. Nagy, Dr. H. Spiering and Dr. C. Ritter for helpful and enassistance of the Deutsche Forschungsgemeinschaft is much appreciated.
couraging discussions. The financial
R~fekxms [l]
H. Frauenfelder,
DE.
Naslr-, R.D. Taylor,
D.R.F.
Cochran
and W.M. Visschcr,
Phys. Rev. 126 (1962)
‘f?] J.R. Gabriel and S.L. Ruby, Irlucl. Instr. Methods 36 (1965) 23. [3] P. Zpry, Phys. Rev. 140A (1565) 1401. [4]
S.V. Karyagin,
Soviet Phys. Solid State 8 (1966)
1387. 961.
[5] G.R. Hoy and S. Chandra, J. (3hem. Phys. 47 (1967) [6] 171 [8 j [9] [IO]
W. Kiindig, Nucl. Instr. Methods 48 (1967) 219. A.J. Stone and W.L. Pitlinger. Phys. Rev. 165 (1968) 1319. S. De Bendetti, Nuclear inte.ractions (Wiley, New York, 1964). A. Messiah, Quantum mechanics, Vol. 2 (Not&-Holland, Amsterdam, R. Zimmermann, Nucl. Instr. hiethods, submitted for publication.
1970)
p. 1059.
1065.