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NUCLEAR RESONANCE ABSORPTION OF
GAMMA-RADIATION AND COHERENT DECAY
MODES
R. Mössbauer
To cite this version:
LA TTICE D YNA MICS
NUCLEAR RESONANCE ABSORPTION OF GAMMA-RADIATION
AND COHERENT DECAY MODES
Institut Max von Laue-Paul Langevin, Grenoble, France
R6sumb. - La section efficace pour l'absorption nucleaire rksonnante des radiations gamma est en gBneral calculee en negligeant I'influence des phenomknes de coherence interatomique. On examine la validit6 de cette hypothese. On en conclut que les mesures d'absorption dans les cristaux hors des orientations de Bragg et en general dans les poudres peuvent &tre interpretkes sans prendre en consideration les effets de coherence interatomique, moyennant certaines restrictions quant a la taille et la distribution des domaines cristallins parfaits. En particulier les Btudes par absorption de la dynamique des mouvements atomiques dans la matikre condensee peuvent dtre interprBtQs dans la plupart des cas en fonction des mouvements atomiques individuels, sans prendre en consi- deration les effets de correlation de paires.
Abstract. - The cross-section for nuclear resonance absorption of gamma-radiation is usually calculated under the assumption that interatomic coherence phenomena may be disregarded. The validity of this assumption is examined. The conclusion is drawn that absorption measurements on crystals in off-Bragg orientations and generally on powders can be interpreted without taking interatomic coherence effects into account, if proper precautions as to the size and distribution of individual perfect crystalline domains are taken. Absorption studies of the dynamics of atomic motions in condensed matter, in particular, can in most cases be interpreted in terms of individual atomic motions, without the necessity to include pair correlations effects.
1. Introduction.
-
The calculation of nuclear reso- nance absorption of gamma radiation by a system of nuclei is usually performed in an incoherent fashion, where one neglects correlations between motions of different nuclei. Such calculations basically regard absorption in a system of nuclei as the inverse of the emission process by a single nucleus, employing the principle of detailed balancing. In reality, this perfect reciprocity is not a priori applicable to actual experimental arrangements. Emission may in the kinematical approximation be regarded as a process occuring in an atom assigned to a well defined lattice site. The emitted photon is then directly detectable. Recoilless absorption, by contrast, is usually observed in transmission experiments, where an assembly of nuclei i s exposed to a plane wave, the attenuation of which is measured as a function of frequency. The cross-section for nuclear resonance absorption des- cribes the probability for the removal of energy from a primary incident beam by processes such as elastic or inelastic scattering or internal conversion. This cross-section will necessarily depend on details of the various decay modes, some of which may be of a coherent nature.2. Spin incoherence and isotope incoherence.
-
The main contributions to the total nuclear resonanceabsorption normally arise from incoherent phenome- na. For such events the intensity attenuation observed in an absorption experiment is due t o the sum of the intensity attenuations by individual nuclei, with interatomic interferences playing no role. The principle attenuation is due to genuine absorption, followed by the emission of internal conversion electrons. The influence of chemical bond effects is usually very small and it therefore suffices for most practical applications to evaluate the absorption for individual nuclei. Another contribution to the absorption cross-section arises from scattering events, both coherent and incoherent. Incoherent scattering contributions origi- nate from the spin dependence and the isotope dependence of the nuclear scattering. Let us first consider the spin dependence of the elastic scattering of photons by resonant nuclei. We regard for this purpose the scattering of a single isolated nucleus with initial magnetic substate m,,, intermediate substate
m, and final substate mi,, where J , and J are the spin
of the nucleus in the ground and the excited state respectively. The scattering process is associated with a nucIeus spin-flip, if in;, f m,,, even if the ground
state is degenerate. Such a spin-flip process destroyes the phase relation between incident and scattered photon waves and therefore gives rise to an incoherent scattering amplitude. Only scattering processes with
mi, = m,, contribute to the coherent scattering amplitude. By consequence, the cross-section o::~, for coherent elastic scattering becomes only a fraction of the total cross-section of1 for elastic scattering :
= <afl
.
The factor follows immediately from the argument, that for coherent scattering one has to evaluate the product of the transition amplitudes
V+(m,,, MJ) V-(m,, m,,),
i. e. the transition probability
]
V+(m,,, m,)12.
The problem then reduces to the question, in how many ways one may combine the magnifold of initial states of the nucleus and of the photon, i. e. (2 J,+
1) (2 L+
I), to arrive at the magnifold of final states of the nucleus, i. e. (2 J+ I), where L is the multipol
order of the photon. The answer, apparently, is given byIsotope incoherence likewise leads to an incoherent contribution to the total elastic scattering and there- fore to the total absorption cross-section. Such an incoherence arises in a single crystal even if only one isotopic species gives rise to nuclear resonance scatter- ing, if the resonant isotope is not occupying all equivalent positions in the elementary cells, due to isotopic dilution or due to the presence of vacancies.
3. Coherent contributions.
-
Coherent contributions to the total cross-section for nuclear resonance absorption have essentially two sources :3.1 Intraatomic interferences between the transition
amplitudes of internal conversion electrons and of
photo-electrons. This effect is particularly pronounced for nuclear transitions of El character [I, 21. The cross-section for nuclear resonance absorption in the presence of such coherent decay modes no longer exhibits the familiar Lorentzian shape, as has first been experimentally demonstrated in the case of the 6 keV resonance in lS1Ta [3]. Another intraatomic coherent contribution to the absorption cross-section should arise from interferences between the amplitudes for coherent nuclear resonant scattering and for electronic Rayleigh scattering. The corresponding interference term, being prohibitively small, has not yet been experimentally studied.
3 .2 Interatomic interferences due to coherent scattering events. Measurements of the radiation intensity behind a single crystal containing resonant nuclei will show a dramatic reduction whenever the crystal is brought into a position which causes a Bragg reflection of the incident radiation. The corresponding increase in the absorption cross-section, which is both of electronic and of nuclear origin, demonstrates the dependence of the absorption phenomena on coherent interatomic decay modes.
The question arises, under which circumstances such coherent decay modes must be taken into account in calculations of the cross-section for nuclear resonance absorption. We begin the discussion of this question by considering first the differential cross- section for nuclear resonance scattering, which when summed over a11 final states and when integrated over the entire solid angle gives the scattering contribution to the total cross-section for nuclear resonance absorption. Employing the Born approximation, we obtain by a generalization of the usual dispersion theory
Here the energy of the incident photon with wave- vector k,, the nuclear excitation energy and the total
level width of the excited nuclear state are specified by
E,, E, and F , respectively. The energy of the lattice in the intermediate state ( il
>
is given by E,, while the initial lattice states of energy ei are averaged with weighing factors gi. The summation over the final lattice states1
f>
implies that the energy of the scattered photon of wavevector kf is not measured.The &function provides for energy conservation within the entire system of nucleus, lattice and radia tion field. We emphasize that the summation over the
scattering amplitudes originating from the different nuclei, which are specified by their position coordinates
r,, is performed in equation (1) in a coherent fashion. We emphasize furthermore, that the summation is not performed invicidually for each matrix element, thus excluding direct transfers of excitation energy from one nucleus to another (nuclear excitons).
NUCLEAR RESONANCE ABSORPTION OF GAMMA. -RADIATION AND COHERENT DECAY MODES C6-7
Let us now consider the special case n = m, which amounts to performing the summation over the different nuclei in equations (1) and (2) in an incoherent rather than in a coherent fashion. We then obtain
The second-order perturbation treatment apparently reduces to the usual first-order expression for the absorption cross-section, if we are only dealing with a single specified nucleus. In the general case, however, coherence effects, i. e. spatial correlations involving different nuclei., must be taken into account.
4. Perfect single crystals and assemblies of small perfect crystal units.
-
Coherent phenomena are parti- cularly pronounced in the case of perfect single crystals. Resonance emission, absorption and scattering in such crystals has been dealt with in a number of theoretical and experimental papers. The dynamical theory of nuclear resonant diffraction of gamma-rays has been worked out for both Laue-diffraction and Bragg- diffraction [4-4-12]. The dynamical theory for nuclear resonance emission [13-171 predicts, in particular, the appearence of Kossel cones. The anomalous atte- nuation behaviour of the resonance radiation in perfect crystals has been experimentally studied in several cases [IS-231. We shall especially be interested in the absorption properties of a small mosaic unit consisting of a perfect single crystal oriented such as to give a Bragg-reflection. For convenience we shall consider the case of a symmetric Laue-reflection for M 1 radiation polarized in the scattering plane and shall assume a crystal of cubic symmetry with only one resonant isotope in the elementary cell. The situation is illustrated in figure 1. Applying the dynamic diffraction theory of Afanase'ev and Kagan [12]FIG. 1. - Schematic representation of the passage of an incident beam lo through a perfect crystal of thickness I for the symmetric Laue cause. ID and h specify the diffracted and the transmitted beams, respectively. The effective pathlength L is defined by
1 = ~ ~ 0 s e .
to this case, we obtain for the transmitted and diffract- ed intensities, respectively :
Here I,, ID and I, respectively are the intensities of the transmitted, diffracted and incident radiation. lc = 2 n/A
is the wavevector of the resonance radiation of wave- length A, L = Zlcos 0 is the effective length of passage of the radiation in the crystal plate of thickness 1, compare figure 1. The quantities E,,, are given by
where a specifies the angular deviation from the Bragg angle, while I C ~ is simply related to the complex coherent
forward elastic scattering amplitude f (0) : ~g = ~ ( g '
+
ig") = - qQ(4 nlrc) f (0)Here 11 is the relative abundance of the resonant isotope,
D
is the number of atoms per unit volume,r,
andr
are the radiation width and the total width of the nuclear excited state, f (k,) is the Lamb (amplitude)factor while the other quantities have been introduced previously. We note, in particular, the relation
where
where o,(E) is the total (absorption) cross-section, while p, is the effective linear absorption coefficient. Equation (5) represents the optical theorem.
Let us now consider the intensities in the transmitted and in the diffracted beams, in the immediate vicinity of a Bragg angle, assumming a
4
g. We furthermore assume crystals up to moderate thicknessess, ICL<
1. Expansion of equation (4) yields, if we keep the angular dependence only in the terms depending on L :transmission and the diffraction case. One wave is given by
i r c ~ a ~ i ~ ~ a ' ( ~ '
-
ig") exp[-
-1
= exp[-
16 g 18
l2
I
The intensity associated with this wave apparently exists only in a narrow angular range a, around the Bragg angle, which is defined by
7CLg" a2
exp
[-
-1
= exp[-
(aiE0)'] (70)l g l 2
where the factor 112 in equation (7a) was introduced to account for the distinction between amplitude and intensity. Equation (7) is the result obtained by Afanas'ev and Kagan [12]. We cite as an example the case of the 14.4 keV radiation in pure 57Fe at room temperature, where we obtain a,
--
32" for a path length L of 1 y. Outside the narrow range a, exists only the second wave in equations (6), i. e.This wave shows the normal exponential attenuation with distance.
An ideal measurement of the absorption coefficient would involve an incident beam which is a perfect plane wave without showing any divergence. The transmitted and diffracted intensities for this idealistic situation follow in the case of a Bragg orientation from equations (6) by putting a = 0. We then obtain (IT/Io),,o =
+
[l+
exp(- 2 rcLgrr)+
+
2 exp(- K L ~ " ) cos (rcLgl)] (8a)(&/&)n-O =
t
[ l t exp(-- 2 xLg")-
2 exp(- rcLg") cos (xLgr)].
(8b) Equation (8a) can be rewritten in the following form (&/I,),-0 = expr- Po Ll C(L) ==exp[-po L]
[+
cosh (p, L)++ cos (rcLgr)].
(9) Expansion for sufficiently thin crystals, whereI
g1
L<
1, yieldsWe can now on the basis of these relations draw the following conclusions :
4.1 The transmitted and diffracted intensities show an oscillatory behaviour, where with increasing thickness I = L cos 9 of the crystal plate energy is alternatively transferred between the transmitted and the diffracted beams. This is the phenomena of Pen- dellosung, which was first noted by Ewald. The
period of the Pendellosung is given by Lp = 2 z/(rcgr). We cite as example the case of the 14.4 keV radiation in pure 57Fe at room temperature, where we obtain at E,
-
E, = r / 2 the value Lp--
cm.4.2 For sufficiently thin crystal plates, where rcLgr < n/2, the diffracted intensity increases with increasing scattering amplitude g' at the expense of the transmitted intensity.
4 . 3 The combined intensity in the diffracted and the transmitted beams adds up to 1, in case absorption can be neglected (g" = 0). This manifests energy conser- vation.
4.4 The intensity transmitted through a sufficiently thin absorber, according to equation (10) exhibits interesting coherence features : the transmission increases with g", a manifestation of the nuclear Borrman effect, the gradual suppression of inelastic decay channels. Under the assumptions made above, this suppression becomes complete, as has been shown by Afanas'ev and Kagan [12]. By contrast the transmission diminishes with increasing value of g', a manifestation of primary extinction.
4 . 5 Corrections to the absorption coefficient due to coherent effects play a role only within angular ranges a, around a Bragg angle.
4.6 The factor C(L) gives the correction to the absorption factor due to coherent phenomena, in case of a perfect collimation of the incident beam. Coherent corrections can be neglected if rc ( g
I
L
<
1. This condition is equivalent to the condition for vanishing primary extinction as obtained by Zacha- riasen [24], if one replaces the electronic scattering amplitude by the absolute value of the nuclear scatter- ing amplitude. We cite as an example again the case of the 14.4 keV radiation in pure 5 7 ~ e at room tempe- rature, where we obtain as condition for vanishing primary extinction L<
1 200A
for E, - E, = r/2.5. Absorption and coherence. - The idealistic assumption of an incident wave with perfect collimation is hardly realized in actual resonance experiments. Instead, one usually works under condi- tions, where the intensities given in equations (6) are integrated over angles a extending well outside the characteristic range a,. In the case of the diffracted beam one thereby obtains a strong dependence on coherent phenomena such as nuclear Borrman effect and primary extinction. These phenomena show usually a much less pronounced effect in absorption experiments. Here, they naturally play a role in the case of a perfect crystal set at a Bragg angle, if the collimation angle of the incident radiation is of order a, or less. The single atom absorption coefficient p,
applies to all situations, were the crystal orientations differs by more than a, from the Bragg angle.
NUCLEAR RESONANCE ABSORPTION OF GAMMA-RADIATION AND COHERENT DECAY MODES C6-9
in the case of real crystals, which are oriented at a Bragg angle. A general expression of the scattering
cross-section has been derived in Born approximation by Boyle and Hall [25]. Integration of the scattered radiation over the full solid angle in principle yields the scattering contribution to the absorption cross-section. In actual calculations complications arise due to insufficient knowledge of details of the crystal compo- sition, in particular of its mosaic structure. The normal single atom absorption cross-section applies again to all absorption measurements, were the crystal orientation differs by more than ah from the Bragg angle, with ab now representing an angle of the order of the characteristic angle of the mosaic distribution. Of particular interest is the case of a powder or of a solid foil with random distributions of the individual perfect crystal blocks. A single grain or domain may be made up of a number of such blocks. Coherent interatomic contributions to the absorption cross- section can be neglected, if the incident beam covers a sufficiently large number of individual blocks with different orientations. This situation will usually prevail, taking into account the size of the beam cross- section which is normally very large compared to the geometrical cross-section of individual mosaic units. If really necessary, one could always employ grinding procedures to reduce individual particle sizes in a powdered material in an effort to reduce primary extinction.
In summarizing we conclude, that a calculation of the absorption cross-section where one neglects coherent interatomic contributions is usually appli- cable to most experimental situations, with the excep- tion of single crystals with orientations in the vicinity of Bragg reflections. The approximation, of course, is particularly good in the case of amorphous systems and liquids.
6. Resonance spectroscopy and the dynamics of atomic
motions. - The remarks presented above bear upon applications of gamma-resonance spectroscopy to studies of the dynamics of atomic motions in condensed matter. Gamma-resonance spectroscopy due to its intrinsic high energy resolution is in principle suitable for such studies [25-281. Pertinent experiments can be performed in three variants :
6.1 Studies of the dynamics of atomic motion in non-resonant scatterers. A resonance absorber is employed to analyze the energy distribution of the radiation which is emitted by a resonant source and Rayleigh scattered by the non-resonant scattering system under study. Such measurements when carried
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6 . 2 Studies of the dynamics of atomic motion in resonant scatterers. Such measurements, when carried out as function of energy and momentum transfer yield the dynamical information in a very complex form. Considering that the scattering cross-section given in equation .(2) contains 4 different resonant denominators which are linear in energy, we realize that the conventional transfer to time dependent coor- dinates will lead to space-time correlations involving 4 different times. This most general case has been discussed in some detail by Kazarnovskii and Stepa- nov [30]. Taking into account the narrowness of the resonance line width as compared to typical phonon transition energies, and employing procedures similar to the ones used by Boyle and Hall [25], one arrives at a result, where the scattering cross-section becomes proportional to the product of Fourier-transforms of a general pair correlation function and a static pair correlation function. This complex situation renders this type of measurement rather douteful for studies of the dynamics of atomic motions.
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