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Spin fluctuation rates from Mössbauer spectra of high-spin ferrous rubredoxin
Volume 69A, number 5
PHYSICS LETTERS
8 January 1979
SPIN FLUCTUATION RATES FROM MOSSBAUER SPECTRA OF HIGH-SPIN FERROUS RUBREDOXIN H. WINKLER1, C. SCHULZ and P.G. DEBRUNNER
Department of Physics, University of illinois at Urbana-C7~ampaign,Urbana, IL 61801, USA Received 3 October 1978
Spin fluctuation rates deduced from 57Fe Mdssbauer spectra of the iron protein rubredoxin In applied fields between 10 K and 150 K are reproduced by a one-parameter model based on the direct process using Blume’s stochastic theory of line shape.
The spin—phonon interaction of paramagnetic centers has been studied extensively and, qualitatively, its basic mechanism is well understood [1,21 The lattice vibrations modulate the Stark splitting of the paramagnetic ions and induce, via spin orbit coupling, transitions among the spin states. Depending on the energy difference between spin levels and on temperature, one-phonon (direct) processes or two-phonon (Raman) processes dominate. Experimental data are available for most paramagnetic species from EPR measurements of spin lattice relaxation rates, but quantitative agreement withtheory is often poor [2]. We show that spin transition rates can be deduced from MOssbauer spectra of high-spin ferrous iron, Fe2~,S = 2, a system that cannot be studied by EPR because it lacks spin degeneracy. The method is applied to a magnetically dilute frozen solution of an iron protein, reduced rubredoxin, which contains a single high-spin ferrous ion per molecule in a distorted tetrahedral Fe(RS )4 complex [3,4] To our knowledge this is the first quantitative analysis of the effect of spin fluctuations on the Mössbauer spectra of highspin ferrous iron. We can simulate our data from 10 K to 150 K with a single adjustable parameter using Blume’s stochastic model of line shape [5] taking into .
account one-phonon transitions among all of the spin levels. Mössbauer spectroscopy measures the nuclear energy levels as they are perturbed by external fields and hyperfine interactions. The magnetic hyperfine term S A1, in particular, couples the nuclear spin Ito the electron spin S, and transitions among the electronic eigenstates lead to fluctuations in the nuclear hamiltonian. The resulting changes in the energy spectrum of the Mössbauer transition are readily observable if some nuclear Larmor frequency WL due to the SA1 term is comparable to the spin transition rate w but large compared to the nuclear decay rate l/r = 7 X 106 s 1 In the system studied here these criteria can be satisfied over a large range of temperatures. The electronic eigenstates and hyperfine parameters of reduced rubredoxin are known from Mössbauer measurements at 4.2 K in strong magnetic fields [4] The system is adequately described by the hamiltonian 2
~.
*1
Recipient of DAAD fellowship 430/402/580/7. On leave of absence from II. Institut für Experimentalphysik. Universität Hamburg. The samples were prepared by Dr. W. Lovenberg, N.I.H. Bethesda, Maryland.
360
~
=
~e ~
=D[S~ S(S+ 1)13 +X(S~
+j3SgH+~
S~)1
1
S=2.
The fine structure given by the first term of JC lifts the (2S + 1)-fold spin degeneracy, and in order to ob. tam sizable spin expectation values
Volume 69A, number 5
PHYSICS LETTERS
8 January 1979
states Ii) of energy E are obtained,and the spin operator Sin ~n can be replaced by (S)~,i = 1,..., 5, to ~
~
In
Diagonalizing ~CewithineQV~~ the spin quintet, eigen=(S) 1AI ~g~H1 + 41(21— 1) (2)
0.0
0.5
X [3I~ I(I+1)+7~(I~—I~)] .
If (S)1 is stationary for periods comparable to the nuclear lifetime r, the MOssbauer spectrum calculated from (2) is a superposition of five-component spectra arising from the spin states Ii) weighted with the Boltzmann factors p~, p1
=
exp(—E/kT)/ ~exp(—E~/kT). /
(3)
HZ
0 5
0
~00
LU
3-
~0.5
~0.0
UUJ 0.5
In the opposite, fast fluctuation limit, the transition rates between electronic eigenstates are larger than the nuclear Larmor frequencies WL, and the expectation values (S)~in (2) can be replaced by their thermal average
1.5
2.0 I
4.0
3.0
2.0
I
1.0
0.0
~
1.0
2.0
3.0
4.0
VELOCITY (mm/sec) Fig. 2. Mdssbauer spectra of reduced rubredoxin at (a) 10 K, ~
while (b) 25 curve K in a(e)field is a of theoretical 3.6 kG and spectrum (c) 100 in K,the (d)fast-fluctuation 150 K in a field of 6.6 kG perpendicular to the transmitted gamma rays. The solid lines are simulations based on eqs. (5), (6), (8), (9)
F—
limit at 150 K. The same parameters were used as in fig. 1. Zero velocity corresponds to the isomer shift of metallic Fe at 300 K.
I-) LU UU-
;~0L__ 4.0
3.0
~..n
.~
5.0
_ 1.0
.0
3.0
4.0
VELOCITY (mm/sec)
Fig. 1. Mössbauer spectra of reduced rubredoxin at 4.2 K in a magnetic field of (a) 3.6 kG and (b) 6.6 kG perpendicular to the transmitted gamma rays. The solid lines are simulations based on eqs. (1) (3) (slow-fluctuation limit) using the parameters [4J D = 10.9 K, ~. = 0.28, g = (2.11,2.19,2.00), eQVzzI 2 = 3.04 mm/s. s~= 0.65, A (99, 41, 186) X 106 rad/s, FWHM = 0.3 mm/s. The zero-field energies kT, of the electronic states Ii>are 0,4.1,22.7,46.1,48.6K. Zero velocity corresponds to the isomer shift of metallic Fe at 300 K.
Z~
= p1(S)j. (4) Figs. 1 and 2 show medium-field MOssbauer spectra of reduced rubredoxin. The limiting cases of slow and fast spin fluctuation rates, respectively, are inadequate even at the lowest and highest temperatures studied (4.2 K, fig. 1, and 150 K, figs. 2d, e, respectively) and we thus have to resort to a dynamical theory. Since (2) is diagonal in the electronic states, Blume’s stochastic model of line shape [5] applies. This model has been used successfully to describe similar dynamic effects in angular correlation and Mössbauer experi-
361
Volume 69A, number 5
PHYSICS LETTERS
ments [6 9]. The expression for the Mössbauer absorption spectrum written in the Liouville formalism
takes on the concise form
[10]
speed of sound of the medium. The matrix element (iIV(’)If) is discussed in ref. [11]. For the transition If) li) with emission of a phonon one has —~
(i/h)~C~ ~‘] 1l~c()),
A(p)oxRe(~C( )pi[pE
~w~ 1(N+l)/N—w~yexp(71/T), T11=1E1
(5)
with p = [‘1(2/I) ice, where F is the combined linewidth of source and absorber, and p the density matrix of the initial state, i.e. the thermal equilibrium distribution of the electronic states and the nuclear ground state. ~ic( is the hamiltonian for photon absorption, E is the unity (super-)operator and ~IC~and ~ are de—
fined as (~(1)x
and the principle of detailed balance is satisfied, P1Wj/ p1w11, with the a priori probabilities p1 (3). For sim-
plicity1)lj)I we in further all matrix elements eq. (7)assume are thethat same [11]. Thus w~ I(iIV( 1only depends on temperature and transition energy kT~1,
‘
(~~)
wpI~/(exp(7~/T)1), 3 IVU)12/(2irh4vSd). 3k
(9a) (9b)
w0 The model defined by eqs. (5), (6) and (9) leads to the —
.
(8)
E~I/k,
=
~‘ij =
(iI~C~Ij) =
8 January 1979
spectral simulations shown as solid lines (a) (d) in fig. 2. The agreement with the experimental data is remarkable in view of the fact that w
/
I
(ii ~If)
=
0 is the only adjustable parameter. Several conclusions can be drawn from our ~ Wlk k+ 1
~12
21
\ ...
W51
W52
W25
w5~
...
k15
)
I
® in ® i~.
(6b) Here ~ denotes the Liouville operator associated with the nuclear hamiltonian (2), and in and 1~are the unit operators in nuclear ground and nuclear excited state spin space, respectively. All information about the spin phonon interaction is contained in the transition probabilities wq between eigenstates Ii) and Ij) Of~JCe.For a direct process with absorption of a phonon of energy — E,I = kT 11 the transition ability is given, in the long-wavelength limit, byprob[1,2] =
3k3(2rrh4v5d)
N = [exp(7~1/7)
(il J/(l)lj)12T3N
1
(7)
1] —l
Here the phonon spectrum is described in the Debye
approximation,d is the density and v is an average 362
analysis.
(i) The functional dependence of the transition rate w~1on temperature and transition energy is well approximated by eq. (9), i.e., it is essentially given by the energy density of the phonons “on speaking terms” with the spin system [121 . (ii) For the closely spaced electronic states Ii), 12) and 14), IS), respectively, Orbach processes dominate at all but the lowest temperatures. These processes involve two subsequent one-phonon transitions and are implicitly accounted for by eq. (6b). 3 tad sThe 1Kexperimental 3, implies anvalue effective element (iii) of w0,matrix w0 = (4.4 ±1.1) X i0 Iv(l)I/k = (w 4v5d/3k5)1/2 4.2 K if the average 027rh speed of sound is taken to be u 1500 ill/s and the density d is 1 g cm 3. The assumption that IVU)l ap~ proximately equals the average splitting ~ 28 K leads to a speed of sound v 3200 m/s. (iv) Raman processes are estimated to be negligible even at 150 K,
2.(v) Reduced rubrein agreement with experiment~ doxin offers particularly favorable conditions for the *2
Assuming
iv(2)i iv(1)i —
—
—
KT~pj>z~28K the Raman
transition rate [1,21 estimated to be WR2flW~T7I 2). With a is Debye temperature 0 150 K this leads 6(O/ fl/(irk(Tjj) to a value of wR ~ x l0~rad/s 1 at 150 K, while the smallest diagonal element ofeq. (6b)is 1w 331 1.3 X iO~rad/ 1.
s~
Volume 69A, number 5
PHYSICS LETTERS
study of spin fluctuation rates by MOssbauer spectro-
scopy, but the method developed can be applied to other systems as well.
8 January 1979
[3] W.A. Eaton and W. Lovenberg, Iron—sulfur proteins II (Academic, New York, London, 1973) Ch. 3, p. 131. [4] C. Schulz and P.G. Debrunner, J. de Phys. Colloq. 37 (1976) C6-153. [5] M. Blume, Phys. Rev. 174 (1968) 351.
This research was supported in part by USPH GM 16406 and NSF PCM76-8 1025.
[6] K. Spartalian, G. Lang, J.P. Coilman, R.R. Gagne and C.A. Reed, J. Chem. Phys. 63 (1975) 5375.
[7] D. Niarchos and V. Petrouleas, J. de Phys. Colloq. 37
References [1] A. Abragam and B. Bleaney, Electron paramagnetic resonance of transition ions (Clarendon, Oxford, 1970) Ch. 10,p. 541.
[2] R. Orbach and H.J. Stapleton, Electron paramagnetic resonance, ed. S. Geschwind (Plenum, New York,
C6-729. [8] (1976) H. Winkler and E. Gerdau, Z. Phys. 262 (1973) 363. [9] E. Gerdau, J. Birke, H. Winkler, M. Forker and G. Netz, Z. Phys. 263 (1973) 5. [10] M.J. Clauser, Phys. Rev. B3 (1971) 3748. [111 R. Orbach, Proc. Roy. Soc. (London) A264 (1961)458.
[12] J.H. Van Vieck, Phys. Rev. 59 (1941) 724.
London, 1972) Ch. 2, p. 121.
363
PHYSICS LETTERS
8 January 1979
SPIN FLUCTUATION RATES FROM MOSSBAUER SPECTRA OF HIGH-SPIN FERROUS RUBREDOXIN H. WINKLER1, C. SCHULZ and P.G. DEBRUNNER
Department of Physics, University of illinois at Urbana-C7~ampaign,Urbana, IL 61801, USA Received 3 October 1978
Spin fluctuation rates deduced from 57Fe Mdssbauer spectra of the iron protein rubredoxin In applied fields between 10 K and 150 K are reproduced by a one-parameter model based on the direct process using Blume’s stochastic theory of line shape.
The spin—phonon interaction of paramagnetic centers has been studied extensively and, qualitatively, its basic mechanism is well understood [1,21 The lattice vibrations modulate the Stark splitting of the paramagnetic ions and induce, via spin orbit coupling, transitions among the spin states. Depending on the energy difference between spin levels and on temperature, one-phonon (direct) processes or two-phonon (Raman) processes dominate. Experimental data are available for most paramagnetic species from EPR measurements of spin lattice relaxation rates, but quantitative agreement withtheory is often poor [2]. We show that spin transition rates can be deduced from MOssbauer spectra of high-spin ferrous iron, Fe2~,S = 2, a system that cannot be studied by EPR because it lacks spin degeneracy. The method is applied to a magnetically dilute frozen solution of an iron protein, reduced rubredoxin, which contains a single high-spin ferrous ion per molecule in a distorted tetrahedral Fe(RS )4 complex [3,4] To our knowledge this is the first quantitative analysis of the effect of spin fluctuations on the Mössbauer spectra of highspin ferrous iron. We can simulate our data from 10 K to 150 K with a single adjustable parameter using Blume’s stochastic model of line shape [5] taking into .
account one-phonon transitions among all of the spin levels. Mössbauer spectroscopy measures the nuclear energy levels as they are perturbed by external fields and hyperfine interactions. The magnetic hyperfine term S A1, in particular, couples the nuclear spin Ito the electron spin S, and transitions among the electronic eigenstates lead to fluctuations in the nuclear hamiltonian. The resulting changes in the energy spectrum of the Mössbauer transition are readily observable if some nuclear Larmor frequency WL due to the SA1 term is comparable to the spin transition rate w but large compared to the nuclear decay rate l/r = 7 X 106 s 1 In the system studied here these criteria can be satisfied over a large range of temperatures. The electronic eigenstates and hyperfine parameters of reduced rubredoxin are known from Mössbauer measurements at 4.2 K in strong magnetic fields [4] The system is adequately described by the hamiltonian 2
~.
*1
Recipient of DAAD fellowship 430/402/580/7. On leave of absence from II. Institut für Experimentalphysik. Universität Hamburg. The samples were prepared by Dr. W. Lovenberg, N.I.H. Bethesda, Maryland.
360
~
=
~e ~
=D[S~ S(S+ 1)13 +X(S~
+j3SgH+~
S~)1
1
S=2.
The fine structure given by the first term of JC lifts the (2S + 1)-fold spin degeneracy, and in order to ob. tam sizable spin expectation values
Volume 69A, number 5
PHYSICS LETTERS
8 January 1979
states Ii) of energy E are obtained,and the spin operator Sin ~n can be replaced by (S)~,i = 1,..., 5, to ~
~
In
Diagonalizing ~CewithineQV~~ the spin quintet, eigen=(S) 1AI ~g~H1 + 41(21— 1) (2)
0.0
0.5
X [3I~ I(I+1)+7~(I~—I~)] .
If (S)1 is stationary for periods comparable to the nuclear lifetime r, the MOssbauer spectrum calculated from (2) is a superposition of five-component spectra arising from the spin states Ii) weighted with the Boltzmann factors p~, p1
=
exp(—E/kT)/ ~exp(—E~/kT). /
(3)
HZ
0 5
0
~00
LU
3-
~0.5
~0.0
UUJ 0.5
In the opposite, fast fluctuation limit, the transition rates between electronic eigenstates are larger than the nuclear Larmor frequencies WL, and the expectation values (S)~in (2) can be replaced by their thermal average
1.5
2.0 I
4.0
3.0
2.0
I
1.0
0.0
~
1.0
2.0
3.0
4.0
VELOCITY (mm/sec) Fig. 2. Mdssbauer spectra of reduced rubredoxin at (a) 10 K, ~
while (b) 25 curve K in a(e)field is a of theoretical 3.6 kG and spectrum (c) 100 in K,the (d)fast-fluctuation 150 K in a field of 6.6 kG perpendicular to the transmitted gamma rays. The solid lines are simulations based on eqs. (5), (6), (8), (9)
F—
limit at 150 K. The same parameters were used as in fig. 1. Zero velocity corresponds to the isomer shift of metallic Fe at 300 K.
I-) LU UU-
;~0L__ 4.0
3.0
~..n
.~
5.0
_ 1.0
.0
3.0
4.0
VELOCITY (mm/sec)
Fig. 1. Mössbauer spectra of reduced rubredoxin at 4.2 K in a magnetic field of (a) 3.6 kG and (b) 6.6 kG perpendicular to the transmitted gamma rays. The solid lines are simulations based on eqs. (1) (3) (slow-fluctuation limit) using the parameters [4J D = 10.9 K, ~. = 0.28, g = (2.11,2.19,2.00), eQVzzI 2 = 3.04 mm/s. s~= 0.65, A (99, 41, 186) X 106 rad/s, FWHM = 0.3 mm/s. The zero-field energies kT, of the electronic states Ii>are 0,4.1,22.7,46.1,48.6K. Zero velocity corresponds to the isomer shift of metallic Fe at 300 K.
Z~
= p1(S)j. (4) Figs. 1 and 2 show medium-field MOssbauer spectra of reduced rubredoxin. The limiting cases of slow and fast spin fluctuation rates, respectively, are inadequate even at the lowest and highest temperatures studied (4.2 K, fig. 1, and 150 K, figs. 2d, e, respectively) and we thus have to resort to a dynamical theory. Since (2) is diagonal in the electronic states, Blume’s stochastic model of line shape [5] applies. This model has been used successfully to describe similar dynamic effects in angular correlation and Mössbauer experi-
361
Volume 69A, number 5
PHYSICS LETTERS
ments [6 9]. The expression for the Mössbauer absorption spectrum written in the Liouville formalism
takes on the concise form
[10]
speed of sound of the medium. The matrix element (iIV(’)If) is discussed in ref. [11]. For the transition If) li) with emission of a phonon one has —~
(i/h)~C~ ~‘] 1l~c()),
A(p)oxRe(~C( )pi[pE
~w~ 1(N+l)/N—w~yexp(71/T), T11=1E1
(5)
with p = [‘1(2/I) ice, where F is the combined linewidth of source and absorber, and p the density matrix of the initial state, i.e. the thermal equilibrium distribution of the electronic states and the nuclear ground state. ~ic( is the hamiltonian for photon absorption, E is the unity (super-)operator and ~IC~and ~ are de—
fined as (~(1)x
and the principle of detailed balance is satisfied, P1Wj/ p1w11, with the a priori probabilities p1 (3). For sim-
plicity1)lj)I we in further all matrix elements eq. (7)assume are thethat same [11]. Thus w~ I(iIV( 1only depends on temperature and transition energy kT~1,
‘
(~~)
wpI~/(exp(7~/T)1), 3 IVU)12/(2irh4vSd). 3k
(9a) (9b)
w0 The model defined by eqs. (5), (6) and (9) leads to the —
.
(8)
E~I/k,
=
~‘ij =
(iI~C~Ij) =
8 January 1979
spectral simulations shown as solid lines (a) (d) in fig. 2. The agreement with the experimental data is remarkable in view of the fact that w
/
I
(ii ~If)
=
0 is the only adjustable parameter. Several conclusions can be drawn from our ~ Wlk k+ 1
~12
21
\ ...
W51
W52
W25
w5~
...
k15
)
I
® in ® i~.
(6b) Here ~ denotes the Liouville operator associated with the nuclear hamiltonian (2), and in and 1~are the unit operators in nuclear ground and nuclear excited state spin space, respectively. All information about the spin phonon interaction is contained in the transition probabilities wq between eigenstates Ii) and Ij) Of~JCe.For a direct process with absorption of a phonon of energy — E,I = kT 11 the transition ability is given, in the long-wavelength limit, byprob[1,2] =
3k3(2rrh4v5d)
N = [exp(7~1/7)
(il J/(l)lj)12T3N
1
(7)
1] —l
Here the phonon spectrum is described in the Debye
approximation,d is the density and v is an average 362
analysis.
(i) The functional dependence of the transition rate w~1on temperature and transition energy is well approximated by eq. (9), i.e., it is essentially given by the energy density of the phonons “on speaking terms” with the spin system [121 . (ii) For the closely spaced electronic states Ii), 12) and 14), IS), respectively, Orbach processes dominate at all but the lowest temperatures. These processes involve two subsequent one-phonon transitions and are implicitly accounted for by eq. (6b). 3 tad sThe 1Kexperimental 3, implies anvalue effective element (iii) of w0,matrix w0 = (4.4 ±1.1) X i0 Iv(l)I/k = (w 4v5d/3k5)1/2 4.2 K if the average 027rh speed of sound is taken to be u 1500 ill/s and the density d is 1 g cm 3. The assumption that IVU)l ap~ proximately equals the average splitting ~ 28 K leads to a speed of sound v 3200 m/s. (iv) Raman processes are estimated to be negligible even at 150 K,
2.(v) Reduced rubrein agreement with experiment~ doxin offers particularly favorable conditions for the *2
Assuming
iv(2)i iv(1)i —
—
—
KT~pj>z~28K the Raman
transition rate [1,21 estimated to be WR2flW~T7I 2). With a is Debye temperature 0 150 K this leads 6(O/ fl/(irk(Tjj) to a value of wR ~ x l0~rad/s 1 at 150 K, while the smallest diagonal element ofeq. (6b)is 1w 331 1.3 X iO~rad/ 1.
s~
Volume 69A, number 5
PHYSICS LETTERS
study of spin fluctuation rates by MOssbauer spectro-
scopy, but the method developed can be applied to other systems as well.
8 January 1979
[3] W.A. Eaton and W. Lovenberg, Iron—sulfur proteins II (Academic, New York, London, 1973) Ch. 3, p. 131. [4] C. Schulz and P.G. Debrunner, J. de Phys. Colloq. 37 (1976) C6-153. [5] M. Blume, Phys. Rev. 174 (1968) 351.
This research was supported in part by USPH GM 16406 and NSF PCM76-8 1025.
[6] K. Spartalian, G. Lang, J.P. Coilman, R.R. Gagne and C.A. Reed, J. Chem. Phys. 63 (1975) 5375.
[7] D. Niarchos and V. Petrouleas, J. de Phys. Colloq. 37
References [1] A. Abragam and B. Bleaney, Electron paramagnetic resonance of transition ions (Clarendon, Oxford, 1970) Ch. 10,p. 541.
[2] R. Orbach and H.J. Stapleton, Electron paramagnetic resonance, ed. S. Geschwind (Plenum, New York,
C6-729. [8] (1976) H. Winkler and E. Gerdau, Z. Phys. 262 (1973) 363. [9] E. Gerdau, J. Birke, H. Winkler, M. Forker and G. Netz, Z. Phys. 263 (1973) 5. [10] M.J. Clauser, Phys. Rev. B3 (1971) 3748. [111 R. Orbach, Proc. Roy. Soc. (London) A264 (1961)458.
[12] J.H. Van Vieck, Phys. Rev. 59 (1941) 724.
London, 1972) Ch. 2, p. 121.
363