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A note on the time-dependence of Mössbauer scattered radiation
100
Nuclear
Instruments
and Methods
in Physics
Research
B16 (1986) lOOC105
North-Holland,
A NOTE Dimiter
ON THE
L. BALABANSKI,
OF MiiSSBAUER
Pave1 S. KAMENOV,
Veselin
of Physics, Uniuersity of Sofia“Clement Okhridsky”,
Depurtment
Received
TIME-DEPENDENCE
3 July 1985 and in revised form 19 November
SCATTERED DIMITROV
and
Amsterdam
RADIATION Emil
VAPIREV
1126~Sofia, Bulgarra
1985
Expressions for the amplitude and the intensity of resonantly scattered Mossbauer radiation are found within the classical dispersion theory. The influence of the electronic part of the complex index of refraction is taken into consideration in the calculations. It is shown that the interference between electronic and nuclear resonant scattering can be observed in time-differential M&.sbauer scattering experiments.
1. Introduction.
Thieberger et al. [l] were the first to study the time-dependence of MGssbauer scattered radiation. The results were interpreted with a theory of a nuclear complex index of refraction, developed by Lynch et al. [2] for the case of time-differential Mossbauer transmission. Later similar experiments were carried out by other groups [3-51 aimed at explaining a considerable discrepancy between theory and experiment. In a previous paper we have interpreted these results taking into account the properties of the detectors used. There we introduced in the formula for the intensity of the scattered radiation a term depending on the nonresonant efficiency of the detector. It was demonstrated that this approach fits quite well with our experimental data, and with those of Drost et al. [3,4]. Here we consider the problem for the time-dependence of resonantly scattered radiation entirely within the framework of a theory of a complex index of refraction. We show the importance and the consequences of the electronic part for scattering experiments in the time-domain. The paper is organized in the following way. In section 2 we derive the scattering amplitude. It has two terms corresponding to the electronical and nuclear resonant scattering. In section 3 we evaluate the intensity of scattered radiation and show the difference between this expession and the others used. In section 4 we discuss the conditions for an experimental verification of interference effects in the time-domain.
2. Amplitude
of the scattered
The model developed the general assumption
radiation.
“(w)=(l+~(o))1’2
B.V.
=1 +:x(w)
+ 0(x2)_
(I)
Here the complex susceptability x(o) contains both nuclear resonant and electronic contributions. A discussion on both parts of the index of refraction can be found in ref. [7]. x(w)
=x,,(w)
+Xnucl(W)
!I
=bx ,
_L_$!!E, w2 I
+w2-iiwy
2w
We consider a scatterer with a hyperfine structure. The transition energies of the components are w,, j = 1,2, . . and y,, j = 1,2, . . are their full widths at half maximum (fwhm). We assume A = c = 1. b is a constant depending on the resonant properties of the filter, Y depends on its electronical properties, p is a mass absorption coefficient and p is the material density. The incident radiation is described by its amplitude aa( or by its spectral function A,(w). These are connected by Fourier transformation. A,(w)
=FT[a,(r)l
and
a,(t)=FT-‘[A,(w)]. (3)
Each monochromatic component of the incident wave is altered in amplitude and phase when passing a distance 5 in the scatterer. The spectral function of the transmitted radiation is: A,,(w,
by Lynch et al. [2] is based on that any process of frequency
0168-583X/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
filtration alters the time-dependence of the filtered signal [6]. In the case of a time-differential Miissbauer experiment we face a process of resonant filtration. The physical properties of the filter are described by its complex index of refraction n (w ).
<)=&,(w).G(w,
t).
The corresponding amplitude sion is found by the inverse
(4) of Mijssbauer transmisFourier transformation.
D. L. Balabunski et al. / Time-dependence of Miissbauer scattered ruduzrion
Here G(w, 5) stands the resonant filter. G(w,
for the characteristic
function
of
(5)
[) = exp(-iwn(w)[).
The amplitude of the radiation scattered by an elementary layer d< of the filter is defined by the change of the amplitude of the transmitted radiation in this layer.
101
expressions the exponential function e-‘5K1 depends on the effective electron thickness D,(t) = PL~.z$.The total electron thickness De(x,,,) = ppxmax is small for the filters used in these experiments, so e-IEKI varies slightly with 5 and can be taken to be constant. Thus, for the amplitude of the scattered radiation we write: u,,(t,
6) = 9,a,,(t,
.C) + ?)2a,(t,
S).
(13)
where
&++G(w, and the spectral
function
1
S)
of the scattered
radiation
a,,(t,
exp(ib[[
4) = FT-’
(14)
w - wO- iy/2
(6) is:
w - wI - iy/2]-‘) I
and exp(ib[[w
- w1 - iy/2]-‘)
wO-iy/2)(w-wl--y/2) For a filter with a single-line Miissbauer spectrum t with transition energy w, and fwhm y) we are able to present all the results in terms of analytical functions. We suppose that the incident radiation is emitted from J Miissbauer source (with transition energy wO and I‘whm y). Its amplitude is a damped plane wave.
do(t) =
r
t i 0,
0,
exp(iw,t
and its spectral
zlo(w)
=A
- yt/2),
function
t > 0.
is a Lorentzian
one
1 IPi
(271)
w-w,-iy/2’
The quantity K(o) the wave propagating written in the form:
= an(w) is the wave number of in the resonant medium. It is
(10)
ti: + w* - iwy
In further considerations we omit the first two terms, because they do not vary considerably over the range of the resonant frequencies and lead to an unmeasurable phase shift. For Massbauer lines w1 > y and the poles of G( w, y) are at the points u = +wl + iy/2. As usual, the contribution of w = -wl + iy/2 is neglected (see the appendix of ref. [2] and finally, accepting w, + w = 2 w, for K(w) we find:
K(w)=K,+K,(w)= The spectral comes: A,,(w,
-iy-
function
.$) = -i(K,
These two terms correspond to the amplitudes of the electronic and the nuclear resonant scattering. a,(t, 5) is the amplitude considered by Thieberger et al. [l] and for a single resonant scatterer u,,(t, 5) is evaluated by Lynch et al. [2]. The advantage of the approach of this paper is that we manage to derive the relations defining both amplitudes within a theory of a complex index of reflection. Using the Lommel functions of two arguments U,,(v, z) [8] for a,(f, z) and a,,(t, t) we write: a,(T, at,(T,
b
of the scattered
+K,(w))A,(o)
xew(iSK2(o)).
w-w,
b - iy/2. radiation
1
(15)
P) = F(-c;Cv,
z)+iU,(u,
z)),
P) = eiW1’~T’2(LiO(v, z) + iU,(u,
z)).
(16) (17)
In these formulae (Y= (wl - wO)/y is the isomer shift; T= yt is time in units of the mean lifetime; p = 46(/y is the effective thickness, corresponding to a distance 5; v = 2aT and z = (pT)‘/*. Thus, in this case for the amplitude of the scattered radiation usc(T, /3) we find: usc(T,
/?) =q,
e’w1’-T’2(Uo(u,
z) +iC;(v,
z))
e’l.J,‘-T/2 +7J2
z)+iU,(v,
-----(-U2(u, (y
z)). (18)
(II) be-
em’5Kl (12)
It is most convenient to introduce q, = iK,e-lSK1 and qZ = (ib/y)e-‘E”l which depend on the electronic and the nuclear resonant properties of the filter. In these
3. Time-dependence
of the intensity
of the scattered
radiation
The integral over the filter thickness of the square magnitude of the scattering amplitude is the intensity of the scattered radiation. I,,(T)
=Irm”
For lalc(T,
d5la,,(T,
P)l2 we
P) I*.
find, introducing
(19) K =
q/q2
and
D.L. Balabanski et al. / Time-dependence of Mijssbauer scattered radiation
102
using the relation Ue(u, z)=JO(z)J,(z) is the first order Bessel function.
Q(u,
2@(%
z).
Here
z)+q2(4
9 Tb and .Fvn stand for the intergals evaluated in ref. [l] and ref. [5], respectively. D is the effective thickness of the filter and I = (DT)‘/2.
2)
o V,‘(U> z) + u:cu>
2)
a2 The intensity
of the scattered
radiation
is:
9
(K+cI)~.F~+~(K+LY)S~~ +$(4%)
+W)].
DJO(Z)U,(U3 VD=
I0
:
1. 1.‘\\
\
1u*- ! I
U.
2.0
’
I
L
I
1.0
z)
a
Eq. (21) differs from the expressions considered for the intensity of the scattered radiation up to now. Thieberger et al. [l] studied the nuclear resonant scatter-
(21)
.I
dp
3.0
LO T/r
bl
\ \\ -p, =“\ ‘\
\
‘\
:
‘\ \, ,/.-\
ti -I I 1o-3
Ii
\\
1.0
I
2.0
‘\
\
\\
?
‘\I I
\
\. I
30
\,
LO
T/Z
1.0
2.0
3.0
LKI
VT
Fig. 1. Time-dependence of the intensity of scattered radiation for a filter with an effective thickness D = 2 for different isomer shifts: a) (r=l.O; b) (Y= 3.0; c) a= 5.0. intensity of the scattered radiation within the theory of this paper; - - - - - - intensity of the scattered radiation when interference effects are neglected; -. -. pm-e resonance scattering.
D.L. Balabanski
et al. / Time-dependence
103
ofMiissbauer scattered radiation
30
LO
TX+
InIlTlI
!I
II
1.0
I
I
L
II
20
\ 3.0
/ L.0
T/T
Fig. 2. Time-dependence of the intensity of scattered radiation for a filter with an effective thickness D = 0.5 for different isomer shifts: a) a = 1.0; b) a = 3.0; c) (Y= 5.0. The lines have the same meaning as in fig. 1.
ing amplitude and for the intensity radiation they found the expression:
of the scattered
(22) Drost et al. [3,4] used this expression and found significant deviations from the experimental results. In our previous work [5], we explained this discrepancy by considering the properties of the resonant detectors used. We took into account the possibility for nonresonant registration of the radiation. There for the intensity of scattered radiation we used the expression: (K2+C12)9&+2&FvD ZvI,(T) - emT [ (23)
This corresponds to an expression in which the interference between the electronic and nuclear resonant scattering is neglected. The influence of interference on the time-dependence of the scattered radiation is: lint(T)
-
eeT [2lc(Y.9& + 2KSVD].
(24)
In figs. 1 and 2 we show the time-dependence of scattered radiation for different isomer shifts and effective filter thicknesses. Results with the three different expressions are shown. The calculations are done for the value of K = 1. The shape of the curves do not vary appreciably when interference terms are taken into account. Therefore, since experimental results are compared to the theoretical curves by normalization to the first maximum, the approach in ref. [5] appears to be in a reasonable agreement with experiment.
104
D.L. Eu~abans~i et al. / Time~de~ende~ce o~~~s~bauer
4. Discussion There are two possibilities for carrying out Mijssbauer scattering experiments. In the first case some type of resonance detector is used and electrons following the scattering process are detected. In this case the competitive processes are nuclear resonant scattering, resulting in the emission of conversion electrons and Compton and photoelectron scattering. In this case interference of photoelectrons and conversion electrons can be observed in the time-dependence of the scattered radiation. The necessary condition is that as a result of these two inelastic processes the atom should have an identical final state. In this case both amplitudes in eq. (13) add. The other possibility is to detect scattered gamma-radiation. In this case interference between resonant and Releigh scattering is observed in the time-domain. Interference effects depend on the value of K,
1
scattered rad~ot~on
which depends both on the electronic and the nuclear resonant properties of the scatterer. Interference phenomena have been studied in different Mossbauer scattering experiments using the method of Dopplervelocity spectroscopy. These result in asymetric velocity spectra [9]. For these experiments see also ref. [lo] and the papers cited therein. The theory fll-131 describes well the results. It must be pointed out that the interference effect is large for El Mossbatter transitions, which is connected with the predominantly dipole character of the photoeffect for low energy y-rays. Interference is also established in some E2 and Ml transitions. Aleksandrov [14] considered the influence of interference effects on the time-dependence of Mossbauer gamma-transition. He concluded that an experimental study would be worthw~le. In this paper we have shown that precise time-differential Mossbatter experiments can also be used for the experimental verification of these effects. In fig. 3 the contributions from both scattering processes and their interference to the intensity of the scattered radiation are shown. The resonant scattering amplitude dominates in the interference term and it follows the minima and maxima of the nuclear resonant scattering term. The electron scattering contribution is a nearly exponential function. Its points of inflection are located at the minima and maxima of the other terms. Therefore, electron scattering smooths the resultant scattering curve. In our opinion, most suitable for an experimental investigation of the interference between nuclear resonant and electronic scattering in the time-domain are resonance detectors having an effective thickness D = 0.5-1.0 and a low value of the parameter I( < 1. Precise experiments have to be performed, because in the background of the other processes in the scattered radiation, interference effects have only a small contribution. The authors are thankful to Dr. Ana Proykova her interest in this work and for useful discussions.
for
References [I] P. Thieberger, [2] [3] 0.0
1.0
2.0
3.0
1.0
T/r
Fig. 3. Contributions from different process to the timodependence of scattered radiation: @ intensity of the scattered radiation; @ contribution from electron scattering; @ contribution from interference between resonance and electron scattering; @ contribution from pure resonance scattering. Calculations are carried out for a filter with an effective thickness D = 2 for an isomer shift (Y= 3.
[4] [5] [6]
J.A. Moragues and A.W. Sunyar, Phys. Rev. 171 (1968) 425. F.G. Lynch, R.E. Holland and M. Hammermesh, Phys. Rev. 120 (1960) 513. H. Drost, H. v. Lojevsky, K. Palow, R. Wallenstein and G. Weyer, Proc. 5th. Int. Conf. on Miissbauer Spectroscopy, Bratisiava (1973) p. 713. H. Drost, K. Paiow and G. Weyer, J. Phys. (Paris) C6 35 (1974) 679. E.I. Vapirev, P.S. Kamenov, V. Dimitrov and D.L. Balabanski, Nucl. Instr. and Meth. 219 (1984) 376. J. Max, Methodes et techniques de traitement du signal et applications aux mesures physique, vol. 1 (Masson, Paris, 1983).
D. L. Balubanskr et al. / Tme-dependewe [7] G.C. Baldwin, Nucl. Instr. and Meth. 159 (1979) 309. [8] I.S. Gradsteyn and 1.M. Ryzhik, Tablitzi integralov, summ. rjadov i proizvedenij (Fizmatgiz, Moscow, 1963). p. 1001. [9] C. Sauer, E. Matthias and R.L. Miissbauer, Phys. Rev. Lett. 21 (1968) 961. [lo] D.C. Shampeney, Rep. Prog. Phys. 42 (1979) 1017. [II] G.T. Trammel1 and J.P. Hannon. Phys. Rev. 180 (1969) 337.
ofMijsshauer
scattered radiution
105
[12] Yu. Kagan, A.M. Afanas’ev and V.K. Voitovetskii, Pis’ma Zh. Eksp. Teor. Fiz. [JETP Lett.] 9 (1969) 155. [13] A.M. Afanas’ev and Yu. Kagan, Phys. Lett. 31A (1970) 38. [14] P.A. Aleksandrov. Zh. Eksp. Teor. Fiz. [Sov. Phys. JETP] 65 (1973) 2047.
Nuclear
Instruments
and Methods
in Physics
Research
B16 (1986) lOOC105
North-Holland,
A NOTE Dimiter
ON THE
L. BALABANSKI,
OF MiiSSBAUER
Pave1 S. KAMENOV,
Veselin
of Physics, Uniuersity of Sofia“Clement Okhridsky”,
Depurtment
Received
TIME-DEPENDENCE
3 July 1985 and in revised form 19 November
SCATTERED DIMITROV
and
Amsterdam
RADIATION Emil
VAPIREV
1126~Sofia, Bulgarra
1985
Expressions for the amplitude and the intensity of resonantly scattered Mossbauer radiation are found within the classical dispersion theory. The influence of the electronic part of the complex index of refraction is taken into consideration in the calculations. It is shown that the interference between electronic and nuclear resonant scattering can be observed in time-differential M&.sbauer scattering experiments.
1. Introduction.
Thieberger et al. [l] were the first to study the time-dependence of MGssbauer scattered radiation. The results were interpreted with a theory of a nuclear complex index of refraction, developed by Lynch et al. [2] for the case of time-differential Mossbauer transmission. Later similar experiments were carried out by other groups [3-51 aimed at explaining a considerable discrepancy between theory and experiment. In a previous paper we have interpreted these results taking into account the properties of the detectors used. There we introduced in the formula for the intensity of the scattered radiation a term depending on the nonresonant efficiency of the detector. It was demonstrated that this approach fits quite well with our experimental data, and with those of Drost et al. [3,4]. Here we consider the problem for the time-dependence of resonantly scattered radiation entirely within the framework of a theory of a complex index of refraction. We show the importance and the consequences of the electronic part for scattering experiments in the time-domain. The paper is organized in the following way. In section 2 we derive the scattering amplitude. It has two terms corresponding to the electronical and nuclear resonant scattering. In section 3 we evaluate the intensity of scattered radiation and show the difference between this expession and the others used. In section 4 we discuss the conditions for an experimental verification of interference effects in the time-domain.
2. Amplitude
of the scattered
The model developed the general assumption
radiation.
“(w)=(l+~(o))1’2
B.V.
=1 +:x(w)
+ 0(x2)_
(I)
Here the complex susceptability x(o) contains both nuclear resonant and electronic contributions. A discussion on both parts of the index of refraction can be found in ref. [7]. x(w)
=x,,(w)
+Xnucl(W)
!I
=bx ,
_L_$!!E, w2 I
+w2-iiwy
2w
We consider a scatterer with a hyperfine structure. The transition energies of the components are w,, j = 1,2, . . and y,, j = 1,2, . . are their full widths at half maximum (fwhm). We assume A = c = 1. b is a constant depending on the resonant properties of the filter, Y depends on its electronical properties, p is a mass absorption coefficient and p is the material density. The incident radiation is described by its amplitude aa( or by its spectral function A,(w). These are connected by Fourier transformation. A,(w)
=FT[a,(r)l
and
a,(t)=FT-‘[A,(w)]. (3)
Each monochromatic component of the incident wave is altered in amplitude and phase when passing a distance 5 in the scatterer. The spectral function of the transmitted radiation is: A,,(w,
by Lynch et al. [2] is based on that any process of frequency
0168-583X/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
filtration alters the time-dependence of the filtered signal [6]. In the case of a time-differential Miissbauer experiment we face a process of resonant filtration. The physical properties of the filter are described by its complex index of refraction n (w ).
<)=&,(w).G(w,
t).
The corresponding amplitude sion is found by the inverse
(4) of Mijssbauer transmisFourier transformation.
D. L. Balabunski et al. / Time-dependence of Miissbauer scattered ruduzrion
Here G(w, 5) stands the resonant filter. G(w,
for the characteristic
function
of
(5)
[) = exp(-iwn(w)[).
The amplitude of the radiation scattered by an elementary layer d< of the filter is defined by the change of the amplitude of the transmitted radiation in this layer.
101
expressions the exponential function e-‘5K1 depends on the effective electron thickness D,(t) = PL~.z$.The total electron thickness De(x,,,) = ppxmax is small for the filters used in these experiments, so e-IEKI varies slightly with 5 and can be taken to be constant. Thus, for the amplitude of the scattered radiation we write: u,,(t,
6) = 9,a,,(t,
.C) + ?)2a,(t,
S).
(13)
where
&++G(w, and the spectral
function
1
S)
of the scattered
radiation
a,,(t,
exp(ib[[
4) = FT-’
(14)
w - wO- iy/2
(6) is:
w - wI - iy/2]-‘) I
and exp(ib[[w
- w1 - iy/2]-‘)
wO-iy/2)(w-wl--y/2) For a filter with a single-line Miissbauer spectrum t with transition energy w, and fwhm y) we are able to present all the results in terms of analytical functions. We suppose that the incident radiation is emitted from J Miissbauer source (with transition energy wO and I‘whm y). Its amplitude is a damped plane wave.
do(t) =
r
t i 0,
0,
exp(iw,t
and its spectral
zlo(w)
=A
- yt/2),
function
t > 0.
is a Lorentzian
one
1 IPi
(271)
w-w,-iy/2’
The quantity K(o) the wave propagating written in the form:
= an(w) is the wave number of in the resonant medium. It is
(10)
ti: + w* - iwy
In further considerations we omit the first two terms, because they do not vary considerably over the range of the resonant frequencies and lead to an unmeasurable phase shift. For Massbauer lines w1 > y and the poles of G( w, y) are at the points u = +wl + iy/2. As usual, the contribution of w = -wl + iy/2 is neglected (see the appendix of ref. [2] and finally, accepting w, + w = 2 w, for K(w) we find:
K(w)=K,+K,(w)= The spectral comes: A,,(w,
-iy-
function
.$) = -i(K,
These two terms correspond to the amplitudes of the electronic and the nuclear resonant scattering. a,(t, 5) is the amplitude considered by Thieberger et al. [l] and for a single resonant scatterer u,,(t, 5) is evaluated by Lynch et al. [2]. The advantage of the approach of this paper is that we manage to derive the relations defining both amplitudes within a theory of a complex index of reflection. Using the Lommel functions of two arguments U,,(v, z) [8] for a,(f, z) and a,,(t, t) we write: a,(T, at,(T,
b
of the scattered
+K,(w))A,(o)
xew(iSK2(o)).
w-w,
b - iy/2. radiation
1
(15)
P) = F(-c;Cv,
z)+iU,(u,
z)),
P) = eiW1’~T’2(LiO(v, z) + iU,(u,
z)).
(16) (17)
In these formulae (Y= (wl - wO)/y is the isomer shift; T= yt is time in units of the mean lifetime; p = 46(/y is the effective thickness, corresponding to a distance 5; v = 2aT and z = (pT)‘/*. Thus, in this case for the amplitude of the scattered radiation usc(T, /3) we find: usc(T,
/?) =q,
e’w1’-T’2(Uo(u,
z) +iC;(v,
z))
e’l.J,‘-T/2 +7J2
z)+iU,(v,
-----(-U2(u, (y
z)). (18)
(II) be-
em’5Kl (12)
It is most convenient to introduce q, = iK,e-lSK1 and qZ = (ib/y)e-‘E”l which depend on the electronic and the nuclear resonant properties of the filter. In these
3. Time-dependence
of the intensity
of the scattered
radiation
The integral over the filter thickness of the square magnitude of the scattering amplitude is the intensity of the scattered radiation. I,,(T)
=Irm”
For lalc(T,
d5la,,(T,
P)l2 we
P) I*.
find, introducing
(19) K =
q/q2
and
D.L. Balabanski et al. / Time-dependence of Mijssbauer scattered radiation
102
using the relation Ue(u, z)=JO(z)J,(z) is the first order Bessel function.
Q(u,
2@(%
z).
Here
z)+q2(4
9 Tb and .Fvn stand for the intergals evaluated in ref. [l] and ref. [5], respectively. D is the effective thickness of the filter and I = (DT)‘/2.
2)
o V,‘(U> z) + u:cu>
2)
a2 The intensity
of the scattered
radiation
is:
9
(K+cI)~.F~+~(K+LY)S~~ +$(4%)
+W)].
DJO(Z)U,(U3 VD=
I0
:
1. 1.‘\\
\
1u*- ! I
U.
2.0
’
I
L
I
1.0
z)
a
Eq. (21) differs from the expressions considered for the intensity of the scattered radiation up to now. Thieberger et al. [l] studied the nuclear resonant scatter-
(21)
.I
dp
3.0
LO T/r
bl
\ \\ -p, =“\ ‘\
\
‘\
:
‘\ \, ,/.-\
ti -I I 1o-3
Ii
\\
1.0
I
2.0
‘\
\
\\
?
‘\I I
\
\. I
30
\,
LO
T/Z
1.0
2.0
3.0
LKI
VT
Fig. 1. Time-dependence of the intensity of scattered radiation for a filter with an effective thickness D = 2 for different isomer shifts: a) (r=l.O; b) (Y= 3.0; c) a= 5.0. intensity of the scattered radiation within the theory of this paper; - - - - - - intensity of the scattered radiation when interference effects are neglected; -. -. pm-e resonance scattering.
D.L. Balabanski
et al. / Time-dependence
103
ofMiissbauer scattered radiation
30
LO
TX+
InIlTlI
!I
II
1.0
I
I
L
II
20
\ 3.0
/ L.0
T/T
Fig. 2. Time-dependence of the intensity of scattered radiation for a filter with an effective thickness D = 0.5 for different isomer shifts: a) a = 1.0; b) a = 3.0; c) (Y= 5.0. The lines have the same meaning as in fig. 1.
ing amplitude and for the intensity radiation they found the expression:
of the scattered
(22) Drost et al. [3,4] used this expression and found significant deviations from the experimental results. In our previous work [5], we explained this discrepancy by considering the properties of the resonant detectors used. We took into account the possibility for nonresonant registration of the radiation. There for the intensity of scattered radiation we used the expression: (K2+C12)9&+2&FvD ZvI,(T) - emT [ (23)
This corresponds to an expression in which the interference between the electronic and nuclear resonant scattering is neglected. The influence of interference on the time-dependence of the scattered radiation is: lint(T)
-
eeT [2lc(Y.9& + 2KSVD].
(24)
In figs. 1 and 2 we show the time-dependence of scattered radiation for different isomer shifts and effective filter thicknesses. Results with the three different expressions are shown. The calculations are done for the value of K = 1. The shape of the curves do not vary appreciably when interference terms are taken into account. Therefore, since experimental results are compared to the theoretical curves by normalization to the first maximum, the approach in ref. [5] appears to be in a reasonable agreement with experiment.
104
D.L. Eu~abans~i et al. / Time~de~ende~ce o~~~s~bauer
4. Discussion There are two possibilities for carrying out Mijssbauer scattering experiments. In the first case some type of resonance detector is used and electrons following the scattering process are detected. In this case the competitive processes are nuclear resonant scattering, resulting in the emission of conversion electrons and Compton and photoelectron scattering. In this case interference of photoelectrons and conversion electrons can be observed in the time-dependence of the scattered radiation. The necessary condition is that as a result of these two inelastic processes the atom should have an identical final state. In this case both amplitudes in eq. (13) add. The other possibility is to detect scattered gamma-radiation. In this case interference between resonant and Releigh scattering is observed in the time-domain. Interference effects depend on the value of K,
1
scattered rad~ot~on
which depends both on the electronic and the nuclear resonant properties of the scatterer. Interference phenomena have been studied in different Mossbauer scattering experiments using the method of Dopplervelocity spectroscopy. These result in asymetric velocity spectra [9]. For these experiments see also ref. [lo] and the papers cited therein. The theory fll-131 describes well the results. It must be pointed out that the interference effect is large for El Mossbatter transitions, which is connected with the predominantly dipole character of the photoeffect for low energy y-rays. Interference is also established in some E2 and Ml transitions. Aleksandrov [14] considered the influence of interference effects on the time-dependence of Mossbauer gamma-transition. He concluded that an experimental study would be worthw~le. In this paper we have shown that precise time-differential Mossbatter experiments can also be used for the experimental verification of these effects. In fig. 3 the contributions from both scattering processes and their interference to the intensity of the scattered radiation are shown. The resonant scattering amplitude dominates in the interference term and it follows the minima and maxima of the nuclear resonant scattering term. The electron scattering contribution is a nearly exponential function. Its points of inflection are located at the minima and maxima of the other terms. Therefore, electron scattering smooths the resultant scattering curve. In our opinion, most suitable for an experimental investigation of the interference between nuclear resonant and electronic scattering in the time-domain are resonance detectors having an effective thickness D = 0.5-1.0 and a low value of the parameter I( < 1. Precise experiments have to be performed, because in the background of the other processes in the scattered radiation, interference effects have only a small contribution. The authors are thankful to Dr. Ana Proykova her interest in this work and for useful discussions.
for
References [I] P. Thieberger, [2] [3] 0.0
1.0
2.0
3.0
1.0
T/r
Fig. 3. Contributions from different process to the timodependence of scattered radiation: @ intensity of the scattered radiation; @ contribution from electron scattering; @ contribution from interference between resonance and electron scattering; @ contribution from pure resonance scattering. Calculations are carried out for a filter with an effective thickness D = 2 for an isomer shift (Y= 3.
[4] [5] [6]
J.A. Moragues and A.W. Sunyar, Phys. Rev. 171 (1968) 425. F.G. Lynch, R.E. Holland and M. Hammermesh, Phys. Rev. 120 (1960) 513. H. Drost, H. v. Lojevsky, K. Palow, R. Wallenstein and G. Weyer, Proc. 5th. Int. Conf. on Miissbauer Spectroscopy, Bratisiava (1973) p. 713. H. Drost, K. Paiow and G. Weyer, J. Phys. (Paris) C6 35 (1974) 679. E.I. Vapirev, P.S. Kamenov, V. Dimitrov and D.L. Balabanski, Nucl. Instr. and Meth. 219 (1984) 376. J. Max, Methodes et techniques de traitement du signal et applications aux mesures physique, vol. 1 (Masson, Paris, 1983).
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ofMijsshauer
scattered radiution
105
[12] Yu. Kagan, A.M. Afanas’ev and V.K. Voitovetskii, Pis’ma Zh. Eksp. Teor. Fiz. [JETP Lett.] 9 (1969) 155. [13] A.M. Afanas’ev and Yu. Kagan, Phys. Lett. 31A (1970) 38. [14] P.A. Aleksandrov. Zh. Eksp. Teor. Fiz. [Sov. Phys. JETP] 65 (1973) 2047.