Resonant Scattering of $\gamma$-Rays by Nuclei

B. S. Dzhelepov

§ 1. Introduction

Resonant scattering of light quanta is a phenomenon well known in optics[^1]. It consists in the fact that an atom absorbs with particular readiness those quanta that correspond exactly to the excitation energy of one of the atomic states; after some time the atom becomes de-excited, emitting quanta that we call resonantly scattered.

Since nuclei also have excited states, the question naturally arose: is it possible to realize resonant scattering of $\gamma$-rays by nuclei? Experimental attempts to detect this phenomenon were undertaken beginning in 1929[^2–^5]. Up to 1951 they all ended unsuccessfully. This was due mainly to the fact that the special features associated with the large energy of the scattered quanta were not appreciated, and the experiments were therefore carried out under unsuitable conditions.

In optics, in order to observe resonant scattering, atoms are illuminated by quanta emitted by the same kind of atoms. A direct transfer of this method to the field of nuclear physics proves difficult.

Let there be a nucleus having an excited state with excitation energy $E^*$. In a coordinate system rigidly bound to the nucleus, the energy of the quanta that can be absorbed is equal to $E^*$; likewise, the energy of the quanta that will subsequently be emitted is equal to $E^*$.

However, in the laboratory coordinate system the primary and secondary quanta have different energies: upon absorption the primary quantum must impart its momentum to the nucleus and, consequently, spend part of its energy on giving kinetic energy to the nucleus. Upon emission of a quantum, part of the excitation energy likewise goes into the kinetic energy of the recoil nucleus. As a result, to excite the state $E^*$ a quantum with energy $h\nu_1$, somewhat greater than $E^*$, is required, while in the decay of this state a quantum with energy $h\nu_2$, somewhat smaller than $E^*$, appears. The difference $\Delta = h\nu_1 - h\nu_2$ is very small in comparison with $h\nu_1$ and $h\nu_2$ both in optics and in the scattering of $\gamma$-rays. In the latter case, however, it is considerably larger than in the former; in § 5 it will be shown that it is equal to $\dfrac{E^{*2}}{Mc^2}$, where $M$ is the mass of the atom or nucleus. In optics, where $E^*$ is very small, $\Delta$ does not exceed $10^{-8}$ eV, but in nuclear physics $\Delta$ may reach 1000 eV. Such a difference in the energies of the $\gamma$-quanta $h\nu_1$ and $h\nu_2$ is difficult to measure directly in experiment, since it amounts to less than $0.1\%$ of $h\nu$. Nevertheless, it will be seen from what follows that the magnitude $\Delta$ plays an important role in the process of resonant scattering. What is essential is the relation between the magnitude $\Delta$ and the natural width of the excited level $\Gamma$. In optics $\Delta \ll \Gamma$, and therefore the energy shift $\Delta$ plays no role. In nuclear physics, on the contrary, usually $\Delta > \Gamma$; the emission line $h\nu_2$ and the absorption line $h\nu_1$ almost do not overlap, and this is the principal obstacle to realizing resonant scattering.

§ 2. NATURAL ENERGY WIDTH OF EXCITED STATES OF NUCLEI

For resonant excitation of a nucleus to occur, it is not necessary that there be exact equality between the excitation energy of the nuclear level and the energy of the \(\gamma\)-quantum in the coordinate system of the nucleus. Since excited states of nuclei are unstable, they have an energy width. The shorter the mean lifetime of the excited state, the greater the energy width of the state. These quantities are connected by the uncertainty relation, which may be written in the form (see, for example, \(^{6,7}\))

\[ \bar T \Delta E \simeq \frac{h}{2\pi}. \tag{1} \]

For convenience of comparison, Table I gives the values of \(\Delta E\) and \(\bar T\) following from relation (1).

The energy width of the ground states of stable nuclei is zero. The ground states of all radioactive nuclei and isomeric states with a long half-life have so small a width that it does not affect any of the known phenomena and therefore has not yet been measured directly in experiment. However, the energy width of shorter-lived states is no longer so small, and several phenomena are known whose probability is directly connected with this width. Such phenomena include resonant scattering of \(\gamma\)-rays, excitation of nuclei by the Coulomb field of a passing particle\(^{8}\), competition between the \((p,p)\) and \((p,\gamma)\) processes in light nuclei\(^{9}\), competition between \(\alpha\)-decay and \(\gamma\)-emission in nuclei emitting long-range \(\alpha\)-particles\(^{10}\), the probability of formation of monochromatic positrons\(^{11}\), the probability of electron conversion in the presence of an unfilled atomic level, etc. The first four phenomena are used for experimental determinations of the magnitude \(\Gamma\).

Table I

Relation between the mean lifetime of a state and its energy width, following from the uncertainty relation

\(\bar T=\dfrac{6.6\cdot 10^{-16}}{\Gamma\ \text{eV}}\ \text{sec}\)	\(\Delta E \simeq \Gamma=\dfrac{6.6\cdot 10^{-16}}{T\ \text{sec}}\ \text{eV}\)
\(1\)	\(6.6\cdot 10^{-16}\)
\(10^{-6}\)	\(6.6\cdot 10^{-10}\)
\(10^{-10}\)	\(6.6\cdot 10^{-6}\)
\(10^{-12}\)	\(6.6\cdot 10^{-4}\)
\(10^{-14}\)	\(6.6\cdot 10^{-2}\)
\(10^{-16}\)	\(6.6\)
\(10^{-18}\)	\(660\)

The results of all measurements of \(\Gamma\), obtained by various methods up to May 1, 1956, as well as estimates of \(\Gamma\) following from lifetimes and from the uncertainty relation, are shown in Fig. 1. The points in the figure form two groups. For the points of the upper group, the quantity \(\Gamma\) has values from \(10^{-14}\) to \(10^{-24}\) eV, which corresponds to \(T_{1/2}\) from 0.05 sec to 15 years; this is the group of isomeric states. For the points of the lower group, \(\Gamma\) has values from \(10^{-2}\) to \(10^{-9}\) eV, which corresponds to \(T_{1/2}\) from \(5\cdot 10^{-14}\) to \(5\cdot 10^{-7}\) sec; this is the group of “ordinary” excited states. Of course, the division into isomeric and “ordinary” excited states is conventional, but it is justified by the good separation of the points into groups. The distribution of points shown in Fig. 1 ...

RESONANCE SCATTERING OF γ-RAYS BY NUCLEI

does not reflect the frequency with which nuclear states with one or another value of \(\Gamma\) occur in nature: the experimental conditions are not equally favorable for detecting points lying in different parts of the figure. Probably there exist very many highly excited nuclear states with lifetimes

[Figure: scatter plot. Vertical axis at left: “Energy width of the state”; vertical axis at right: \(T_{1/2}\), sec, “half-life”; horizontal axis: “Excitation energy,” \(E\) (keV).]

Fig. 1. Experimental data on the energy width of the lower excited states of atoms. ○ — results of direct determinations, half-life period (from observations of the decay of activity, from delayed \(\beta\)—\(\gamma\) and \(\gamma\)—\(\gamma\) coincidences, by the recoil method and by the Doppler effect); ■ — data obtained from resonance scattering; ● — data obtained from the \((p,\gamma)\) reaction; + — data obtained from spectra of long-range \(\alpha\)-particles; ▲ — data obtained from Coulomb excitation. An arrow means that only an upper or lower limit is known.

less than \(10^{-11}\) sec., but such times are very difficult to measure directly from decay, and few determinations of \(T\) by indirect methods have been made; for this reason the lower part of Fig. 1 is sparsely filled. In what follows we shall be interested in the excited states lying in the lower part of Fig. 1; it is precisely these that can be successfully excited by \(\gamma\)-rays, whereas no one has succeeded in directly exciting any state of the upper group by \(\gamma\)-rays. Since the resonant-scattering cross section is proportional to \(\Gamma_\gamma^2\), and \(\Gamma_\gamma < \Gamma\), it is clear that this task is incomparably more difficult for the upper group than for the lower.

Each excited state of a nucleus, like that of any other unstable quantum-mechanical system, is to some extent “smeared out.” Excited states of nuclei arise as a result of \(\alpha\)- or \(\beta\)-decay, \(\gamma\)-radiation, or a nuclear reaction (Fig. 2). In all these cases the emitted particles or quanta do not have exactly the same energy, and the excited nuclei obtained are not exactly identical: they have strictly identical characteristics, but slightly different excitation energies and, consequently, masses. This difference is quantum-mechanical in character. On the one hand, by studying, for example, the spectrum of \(\alpha\)-particles, we could in principle select \(\alpha\)-particles of as precisely defined an energy as desired; simultaneously with them recoil nuclei would be formed with an equally strictly defined mass, which subsequently, upon transforming into stable nuclei, would emit quanta of strictly defined energy, so that the sum of the energies of \(\alpha\) and \(\gamma\) would be exactly equal to the mass difference of the initial and final nuclei. For such nuclei the half-life could not be determined with arbitrary precision because of the uncertainty principle. On the other hand, if these nuclei were given the possibility of somehow interacting with the surrounding matter, this would lead to the actual appearance of states with different masses, and subsequently to the appearance of \(\gamma\)-quanta of different energy; but the half-life would then acquire a strictly definite value.

Fig. 2. The excited state \(E^*\) of a nucleus \(A\) may arise as a result of \(\alpha\)- or \(\beta\)-decay of other nuclei or of a \(\gamma\)-transition from a more excited state of the nucleus \(A\). In all these cases the state \(E^*\) arises with a natural distribution in mass, conventionally represented by the hatched area of a circle, in which the scale is greatly enlarged.

In principle, by studying the “microspectrum” of a \(\gamma\)-line, one could study the probability of formation of a nucleus with one mass or another. However, such an experiment lies far beyond the possibilities of experimental technique.

The probabilities of \(\alpha\)- and \(\beta\)-decay depend on the decay energy, but since the natural width of levels is usually very small, the probabilities of formation of nuclei with slightly different masses in \(\alpha\)- or \(\beta\)-decay are practically identical, and the excited state arises with a “natural” distribution in mass.

Quantum electrodynamics leads to the following expression for the “natural” distribution in energies (or masses) for any system having a state with excitation energy \(E^*\):

\[ W(E)=A\frac{\Gamma}{(E^*-E)^2+\left(\frac{\Gamma}{2}\right)^2}. \tag{2} \]

This expression is sometimes called the dispersion formula. In it \(A\) and \(\Gamma\) are two constants in which the individual properties of the excited state under consideration are manifested; they depend on the quantum characteristics

of the given state (spin, parity, etc.) and on the probabilities of all processes leading to the decay of this state. Formula (2) is represented by a curve of resonance type, shown in Fig. 3. The width of the peak at half-height is equal to \(\Gamma\), the cross section at the maximum is equal to \(\dfrac{4A}{\Gamma}\), and the area enclosed between the curve and the abscissa axis, i.e. the integral probability, is equal to

\[ \int_0^\infty W(E)\,dE = \]

\[ = 2A\left(\frac{\pi}{2}+\operatorname{arctg}\frac{2E^*}{\Gamma}\right)\simeq 2\pi A \]

(i.e. \(\dfrac{\pi}{2}\times\) height \(\times\) half-width).

Fig. 3. Dispersion curve \(W(E)\) according to (2).

The natural distribution over the masses or excitation energies is directly connected with the law of decay of the excited state. Fock and Krylov showed\(^6\) that the probability of finding the system in the initial state after a time \(t\) is expressed by the formula

\[ L(t)=\left|\int e^{-\frac{i}{\hbar}Et} W(E)\,dE\right|^2, \tag{3} \]

which, for the above form of \(W(E)\), takes the form

\[ L(t)=2\pi A e^{-\frac{\Gamma}{\hbar}|t|}. \tag{4} \]

Thus, it is precisely for the dispersion form of \(W(E)\) that an exponential law of decay of the excited state is obtained. In this case the mean lifetime

\[ \overline{T}=\frac{\hbar}{\Gamma}, \]

i.e. it coincides with formula (1).

In all experiments in which the law of decay of excited nuclear states was studied, an exponential law was always obtained. For all these cases the assumption of the “dispersion” form of the distribution \(W(E)\) is justified. Taking into account the theoretical justification of this distribution, one may suppose that it also occurs in unstudied cases of decay of excited states.

Fig. 4. Resonant excitation of a nucleus \(A\) by a \(\gamma\)-spectrum with a broad spectral composition (for example, a bremsstrahlung spectrum). For excitation of nucleus \(A\), only quanta lying inside the narrow hatched band of the spectrum are effective; the width of the band is approximately equal to \(\Gamma\) for the level being excited. The secondary rays \(\gamma_2\) have a spectrum corresponding to the distribution \(W(E)\) for the given level. The scale in the drawing is not preserved.

If the excited state of a nucleus arose as a result of some process whose probability does not depend on the exact value of the mass of the excited nucleus, then the \(\gamma\)-line obtained in the decay of this state will have a spectral composition (“microspectrum”) corresponding to the function \(W(E)\) (formula (2)). Similarly, if excited states arise in

when irradiated with $\gamma$ rays with a continuous spectrum or with a broad spectral composition (much broader than the microspectrum of the intrinsic $\gamma$ rays), the spectral composition of the radiation will correspond to the function $W(E)$ (Fig. 4).

But if the excited state arises as a result of the absorption of monochromatic $\gamma$ rays, then the $\gamma$ line of the decay must also be just as monochromatic. Of course, strictly monochromatic lines do not exist in nature, but there may be lines considerably narrower than the intrinsic width of the level under consideration, and in this case the secondary radiation will have as narrow a spectrum as the primary radiation$^{12}$ (Fig. 5).

Fig. 5. Resonant excitation of nucleus $A$ by $\gamma$ rays with a spectrum narrower than the natural width of the excited level $A$. The secondary rays $\gamma_2$ have almost as narrow a spectrum as the rays $\gamma_1$.

If, during the lifetime of the excited state, it is subjected to some external influences that can slightly change its energy, then the secondary $\gamma$ line will broaden and assume the form corresponding to the function $W(E)$ for the given state.

§ 3. PROBABILITY OF RESONANT EXCITATION OF NUCLEI AND RESONANT SCATTERING OF $\gamma$ RAYS

Let there be a nucleus with spin of the initial state $I_0$, having a state with excitation energy $E^*$ and spin $I^*$.

Quantum mechanics leads to the following expression for the cross section for excitation of a nucleus by strictly monochromatic $\gamma$ rays$^{12}$:

\[ \sigma_{\text{res. excit.}} = \frac{2I^*+1}{2I_0+1} \frac{\lambda^2}{8\pi} \frac{\Gamma_\gamma \Gamma}{(E^*-E)^2+\left(\frac{\Gamma}{2}\right)^2}, \tag{5} \]

where $\lambda$ is the wavelength of the incident rays in the resonance region; $E$ is the energy of the incident quanta in the coordinate system associated with the nucleus; $\Gamma$ is the total width of the level—the sum of the widths associated with all modes of decay of the nuclear state under consideration; $\Gamma_\gamma$ is the width of the level associated with absorption or emission of $\gamma$ rays.

Formula (5) should be applied only near resonance. Far from resonance $\sigma$ is small, but may obey another law.

At exact resonance, i.e. for $E=E^*$, one obtains

\[ (\sigma_{\text{res. excit.}})_{\max} = \frac{2I^*+1}{2I_0+1} \frac{\lambda^2}{2\pi} \frac{\Gamma_\gamma}{\Gamma}. \]

Usually $I^*$ and $I_0$ are small, and the whole first factor differs little from 1. If the threshold of nuclear splitting has not been reached and not an isomeric

state, then \(\Gamma_\gamma \sim \Gamma\); in this case the magnitude of the cross section is determined by the factor \(\lambda^2/2\pi\), which can be rewritten as follows:

\[ \frac{\lambda^2}{2\pi} = \frac{1}{2\pi} \left(\frac{h}{m_0 c}\right)^2 \left(\frac{m_0 c^2}{E^*}\right)^2 = \frac{0.94\cdot 10^{-20}}{(E^*/m_0 c^2)^2}\ \text{cm}^2 . \tag{6} \]

In optics, in the visible part of the spectrum (\(\lambda \sim 5000\ \text{Å}\)), formula (6) leads to cross sections of the order of \(4\cdot 10^{-10}\ \text{cm}^2/\text{atom}\), i.e. considerably larger than the geometrical cross section of an atom. In nuclear physics formula (6) leads to large cross sections, considerably exceeding the geometrical cross sections of nuclei; thus, for example, at \(h\nu=1\ \text{MeV}\) we obtain \((\sigma_{\text{res. exc.}})_{\max}\cong 2\cdot 10^{-21}\ \text{cm}^2/\text{nucleus}\).

It should be remembered, however, that large cross sections are concentrated in an extremely narrow energy region around the resonance. Thus, for example, for those nuclear transitions (see the lower part of Fig. 1) for which

\[ \Gamma_\gamma \cong \Gamma \cong 10^{-6}\ \text{eV}, \]

already at a distance of \(1\ \text{eV}\) from exact resonance the cross section falls by \(4\cdot 10^{12}\) times. For this reason the integrated cross sections of resonant excitation of nuclei are by no means large. The integrated cross section \(\Sigma\) is obtained by integrating (5) over the resonance region; in view of the rapid decrease of the function, the integration may be carried out from 0 to \(\infty\), first taking \(\lambda^2\) outside the integral (its value at resonance):

\[ \Sigma_{\text{res. exc.}} = \frac{2I^*+1}{2I_0+1}\frac{\lambda^2}{8\pi} \int_0^\infty \frac{\Gamma_\gamma\Gamma}{(E^*-E)^2+\dfrac{\Gamma^2}{4}}\,dE = \frac{2I^*+1}{2I_0+1}\frac{\lambda^2}{8\pi}\,2\pi\Gamma_\gamma = \]

\[ = \frac{2I^*+1}{2I_0+1}\, 1.5\cdot 10^{-20}\, \frac{\Gamma_\gamma}{\left(\dfrac{E^*}{m_0c^2}\right)^2} \ \text{cm}^2\ \text{eV}. \]

With increasing energy \(E^*\) the denominator grows, but the numerator \(\Gamma_\gamma\) grows still faster: for single-particle dipole transitions it is proportional to \(E^{*3}\), and for quadrupole transitions to \(E^{*5}\). Therefore the integrated cross section of resonant excitation should increase with energy.

The cross section for resonant scattering of \(\gamma\)-rays has almost the same form as (5):

\[ \sigma_{\text{res. scat.}} = \frac{2I^*+1}{2I_0+1}\frac{\lambda^2}{8\pi} \frac{\Gamma_\gamma^2}{(E^*-E)^2+\left(\dfrac{\Gamma}{2}\right)^2}, \tag{7} \]

only instead of the total width \(\Gamma\), the numerator contains the radiative width \(\Gamma_\gamma\) (of all cases of resonant excitation, only the fraction equal to \(\Gamma_\gamma/\Gamma\) is associated with emission, i.e. with resonant scattering\({}^{13}\)).

The integrated cross section of resonant scattering is equal to

\[ \frac{2I^*+1}{2I_0+1}\frac{\lambda^2}{4}\frac{\Gamma_\gamma^2}{\Gamma}, \]

i.e. is proportional to

\[ \frac{\Gamma_\gamma}{E^{*2}}\frac{\Gamma_\gamma}{\Gamma}. \]

The first factor, as indicated above, grows with energy, but the second decreases as soon as new modes of decay of the excited state of the nucleus considered by us appear. When the excitation reaches such a value that emission of heavy particles becomes possible, \(\Gamma\) begins to grow rapidly. Therefore at \(h\nu \gg 10\ \text{MeV}\) the integrated cross section of resonant scattering should decrease.

To check the formulas of this section, independent determinations of \(\sigma_{\mathrm{res.\ scat}}\) and \(\Gamma\) (or \(\sigma_{\mathrm{res.\ scat}}\) and the half-life \(T_{1/2}\)) are needed. Such data are still few (see § 6), but those available confirm the correctness of formula (7).

§ 4. RESONANCE EXCITATION OF NUCLEI BY A CONTINUOUS \(\gamma\)-SPECTRUM

If nuclei are illuminated by a continuous \(\gamma\)-spectrum, for example the spectrum of bremsstrahlung, then only narrow spectral bands corresponding to the excited states of the irradiated nucleus prove effective for resonance excitation (see Fig. 4). The width of the bands is determined by the values of \(\Gamma\) for the nuclear levels under consideration.

Fig. 1 shows that the majority of low-lying isomeric nuclear levels have a width of \(10^{-5}\)—\(10^{-8}\) eV. As a consequence, excitation of nuclei by a bremsstrahlung spectrum occurs with small probability: from a continuous spectrum extending over hundreds of keV, narrow bands of width \(10^{-5}\)—\(10^{-8}\) eV are cut out, and only the quanta contained in these bands prove effective for resonance excitation of nuclei. For this reason, resonance phenomena practically do not alter the penetrating power of the primary beam.

Resonance excitation can be detected only by secondary phenomena:

a) by scattering, i.e. by emission of the same quantum that was absorbed (after subtracting \(\Delta\), see § 1),

b) by emission of other discrete lines of the \(\gamma\)-spectrum of the given nucleus,

c) by the discrete spectrum of particles emitted by the nucleus.

The first route proves the most difficult. Up to now no one has yet observed resonance scattering of \(\gamma\)-rays upon irradiation of nuclei by a continuous spectrum. It is very difficult to observe this phenomenon: in the total flux of bremsstrahlung the quanta of the required energy constitute \(10^{-10}\)—\(10^{-13}\) of the total number of quanta; since the resonance-scattering cross section for these quanta does not exceed \(10^{-20}\ \mathrm{cm}^2/\mathrm{atom}\), while the average cross section for nonresonance scattering of all bremsstrahlung quanta is \(\sim 10^{-25}\ \mathrm{cm}^2/\mathrm{atom}\), then in the flux of scattered quanta the resonance ones will constitute \(10^{-5}\)—\(10^{-8}\). The specific characteristics of the phenomenon (angular distribution, monochromaticity, time delay) can, of course, help to some extent in isolating it; however, up to now the isolation of resonance-scattered rays has not been accomplished.

Resonance excitation of nuclear levels by bremsstrahlung can be detected much better from processes not connected with scattering, for example from the appearance of long-lived isomeric states of the irradiated nucleus, from the discrete spectrum of any heavy particles emitted by excited states of the nucleus, etc.

In 1939 Pontecorvo and Lazar\(^{14}\) found that upon irradiation of \(\mathrm{In}^{115}\) by a bremsstrahlung spectrum with upper limit \(h\nu_{\max}=1.8\) MeV an isomer \(\mathrm{In}^{115*}\) arises, having a half-life of 4.5 hours; its presence in the target is readily detected after the tube that produced the bremsstrahlung has been switched off. This isomeric state \(\mathrm{In}^{115*}\) does not arise directly: although it has an excitation energy of 334 keV, it arises only in the case when \(h\nu_{\max}\) is greater than 1.02 MeV\(^{14-18}\). The reason for this is clear: being relatively long-lived, the isomeric state has a very small natural width: from the relation \(\overline{T}\Delta E \approx \dfrac{h}{2\pi}\) it follows that it must be of the order of \(4\cdot 10^{-20}\) eV. Naturally, a very small number of quanta in the bremsstrahlung spectrum fall within so narrow a band, and the state is practically not excited. However, above this level there are other states, considerably shorter-

living and, accordingly, broader, and if the energy is sufficient, they can be excited. In cascade transitions from these “activation levels” the isomeric state mentioned above, \( \mathrm{In}^{115*} \), sometimes arises.

It should be noted that \(h\nu_{\max}\) in the experiments of Pontecorvo and Lazar and of other authors was considerably below the threshold of the reactions \((\gamma,p)\) and \((\gamma,n)\); therefore there is no doubt that these authors observed precisely the resonance excitation of \( \mathrm{In}^{115} \) nuclei.

At present, another 10 cases analogous to the one described are known (see § 13).

Other clear examples of resonance excitation of nuclei can be seen in the fine structure of the yield curves of the reactions \((\gamma,n)\), \((\gamma,p)\), or \((\gamma,t)\) near threshold. At an excitation energy of the nucleus only slightly exceeding the separation threshold, the nuclear levels are still sufficiently far apart and overlap only weakly; in this case it is still possible to distinguish, by energy, protons, neutrons, or other particles arising in the decay of individual states. For example, in Fig. 6, a and b, the dependences of the cross sections of the reactions \( \mathrm{Li}^{7}(\gamma,t)\alpha \) (see \(^{19}\)) and \( \mathrm{O}^{16}(\gamma,p)\mathrm{N}^{15} \) (see \(^{20,21}\)) on energy are shown. The maxima located near threshold (up to the giant resonance) correspond to the excitation of individual levels or groups of levels of \( \mathrm{Li}^{7*} \) and \( \mathrm{O}^{16*} \). At higher \(\gamma\)-quantum energy, the resonance excitation of individual states is no longer manifested so clearly: on the one hand, the distances between levels decrease and the resolving power of the apparatus proves insufficient, while on the other hand, qualitatively new phenomena arise—the giant resonance, emission of several particles, etc.

Fig. 6. Examples showing resonance excitation by \(\gamma\)-rays of nuclear states having an excitation energy slightly above the splitting threshold. a) Cross section for the photodisintegration process \( \mathrm{Li}^{7}(\gamma,t)\alpha \) \(^{19}\). b) Cross section for the reaction \( \mathrm{O}^{16}(\gamma,p)\mathrm{N}^{15} \) \(^{20,21}\).

The examples given above, strictly speaking, show only that, when nuclei are irradiated with a bremsstrahlung spectrum, discrete nuclear levels are excited. The mechanism of excitation remains undisclosed and, by calling it resonance from the very beginning of this paragraph, we have only expressed thereby our idea of the mechanism. A justification of such an idea can be found in the following five paragraphs.

§ 5. EXCITATION OF NUCLEI BY \(\gamma\)-RADIATION OF THE SAME NUCLEI; GENERAL QUESTIONS

Most lower excited states of nuclei have an energy width of \(10^{-8}\)—\(10^{-3}\) eV; such is the width of the natural micro-spectrum for most \(\gamma\)-lines. The thermal motion of the emitting atoms broadens this band to \(\sim 0.1\) eV (see § 1). Since the range of excitation energies

amounts to hundreds of keV, there is no hope that a case will be found in which the energy of the γ-rays of one radioactive substance would by chance be exactly that needed to excite the nuclei of another substance.

At first glance it seems that the process of resonant excitation of a nucleus can be easily realized if nuclei are irradiated with γ-rays arising in the decay of other nuclei of the same kind. In reality this is not so. Let \(A\) and \(B\) (Fig. 7) be identical nuclei with first-level excitation energy \(E^*\).

Fig. 7. Excitation of a nucleus by γ-radiation from a nucleus of the same kind.

The quantum emitted by nucleus \(A\) has an energy somewhat smaller than \(E^*\), owing to the recoil of nucleus \(A\) in the process of radiation. The laws of conservation of momentum and energy give the equations:

\[ \frac{h\nu_1}{c}=Mv^2, \]

\[ h\nu_1+\frac{1}{2}Mv^2=E^*, \]

whence, approximately,

\[ h\nu_1=E^*-\frac{E^{*2}}{2Mc^2}. \]

On the other hand, the quantum \(h\nu_1\), upon reaching nucleus \(B\), cannot transfer all its energy to the excitation of this nucleus, since part of it goes into imparting to nucleus \(B\) the velocity that follows from the law of conservation of momentum. This part is again \(\dfrac{(h\nu_1)^2}{2Mc^2}\), and, consequently, only the energy

\[ h\nu_1-\frac{(h\nu_1)^2}{2Mc^2}\simeq E^*-\frac{E^{*2}}{Mc^2}. \]

can go into exciting nucleus \(B\).

The deficit \(\Delta\) is therefore \(\dfrac{E^{*2}}{Mc^2}\). Usually \(\Delta\) is a small quantity: in a heavy nucleus (\(A=200\)) at \(E^*=50\) keV it is only \(0.013\) eV, but in \(\mathrm{Li}^6\) at \(E^*=2189\) keV this difference reaches \(850\) eV. Since the overwhelming majority of the lower excited states of nuclei have a considerably smaller width, it is clear that, as a rule, resonant scattering of hard γ-rays according to the scheme of Fig. 7 cannot occur unless special measures are taken to compensate the energy \(\dfrac{E^{*2}}{Mc^2}\) that passes into kinetic energy in absorption and emission\(^{22}\). High excited states of nuclei, capable of decaying by emitting nucleons, have a large energy width, which in individual cases may exceed \(\Delta\) (see in § 9 the example with the excited state \(8.06\) MeV of \(\mathrm{N}^{14}\)). However, it is difficult to observe resonant scattering in these cases because γ-radiation appears only in a very small fraction of all decay events of the excited state.

Moderate excitation energies, \(h\nu < 6\) MeV. Let us first consider the situation existing when the \(\gamma\)-ray energy is less than the threshold for knocking out nucleons; it is schematically shown in Fig. 8.

For resonant excitation a frequency band \(B\) is needed, while the scatterer receives a band of lower frequencies \(A\). The shift \(\Delta\) is much larger than \(\Gamma\). Although each of the bands \(A\) and \(B\) has, according to (2), infinitely long tails,

Fig. 8. For resonant scattering a frequency band \(B\) is needed, but because of recoil a band of lower frequencies \(A\) arrives, shifted by \(\Delta\); the bands \(A\) and \(B\) nevertheless overlap slightly.

extending in both directions, their ordinates far from the bands are very small, and the probability of resonant excitation, although different from zero, is negligible. The mean cross section for resonant excitation is expressed by the formula

\[ \bar{\sigma} = \frac{ \displaystyle \int_{0}^{\infty} W(E)\,\sigma_{\text{res. exc}}\,dE }{ \displaystyle \int_{0}^{\infty} W(E)\,dE }, \]

where \(W\) is given by formula (2), and \(\sigma_{\text{res. exc}}\) by formula (7). Substituting \(W\) and \(\sigma_{\text{res. exc}}\), we obtain:

\[ \bar{\sigma} = \frac{2I^{*}+1}{2I_{0}+1}\, \frac{\lambda^{2}}{16\pi^{2}}\, \Gamma^{2}\Gamma_{\gamma} \int_{0}^{\infty} \frac{dE}{ \left\{(E^{*}-E+\Delta)^{2}+\left(\frac{\Gamma}{2}\right)^{2}\right\} \left\{(E^{*}-E)^{2}+\left(\frac{\Gamma}{2}\right)^{2}\right\} }. \]

For an approximate evaluation of the integral one may use the circumstance that the integrand has two very sharp maxima, which arise when \(E^{*}-E+\Delta \simeq 0\) and when \(E^{*}-E \simeq 0\); at all other energies the integrand is negligibly small (see p. 9). If \(\Delta \gg \Gamma\), then the maxima are well separated; over the extent of the narrow maximum caused by the first curly bracket, the value of the second bracket does not have time to change appreciably and may be regarded as constant; the same applies to the maximum caused by the second bracket. Therefore the integral is equal to

\[ 2 \times \frac{\pi}{2} \times \text{half-width} \times \text{height} = \frac{\pi\Gamma}{ \displaystyle \frac{\Gamma^{2}}{4} \left(\Delta^{2}+\frac{\Gamma^{2}}{4}\right) } \simeq \frac{4\pi}{\Gamma\Delta^{2}}. \tag{7a} \]

Consequently, the cross section for resonant excitation is

\[ \bar{\sigma}_{\text{res. exc}} = \frac{2I^{*}+1}{2I_{0}+1}\, \frac{\lambda^{2}}{4\pi}\, \frac{\Gamma\Gamma_{\gamma}}{\Delta^{2}}. \tag{7b} \]

The cross section for resonance scattering will be equal to

\[ \bar{\sigma}_{\text{res. scat.}}= \frac{2I^{*}+1}{2I_{0}+1}\, \frac{\lambda^{2}}{4\pi}\, \frac{\Gamma_{\gamma}^{2}}{\Delta^{2}} . \tag{7в} \]

Let us consider a specific case. The excited nuclei \(Hg^{198*}\), produced in the \(\beta\)-decay of \(Au^{198}\), emit quanta \(h\nu=411\) keV, which can be resonantly scattered by \(Hg^{198}\) nuclei; the width of the 411-keV level of \(Hg^{198*}\) is \(\Gamma=3.0\cdot10^{-5}\) eV (see below); \(\frac{\Gamma_{\gamma}}{\Gamma}=0.97\) (three percent is internal conversion). The distance

\[ \Delta=\frac{(h\nu)^{2}}{Mc^{2}}=0.91\ \text{eV}; \]

in the present case \(\Delta\gg\Gamma\), and formulas (7a) and (7в) should be applicable; calculation of the average cross section gives \(\sigma=3.9\cdot10^{-29}\ \text{cm}^{2}/\text{atom}\). Thus, if the nuclei \(Hg^{198*}\) and \(Hg^{198}\) were at rest at the moment of emission and absorption, the resonance excitation would be very small. Thermal motion substantially increases the excitation cross section. If the emitting nuclei are moving at the moment of emission, then the energy of the quanta they emit changes because of the Doppler effect; in exactly the same way, moving absorbing nuclei can absorb quanta which they could not absorb while at rest. If, at the moment of flight of the quantum, the nuclei are moving toward one another, this should bring bands \(A\) and \(B\) closer together and in any case increase their overlap. The importance of this phenomenon is shown even by a rough estimate. At room temperature the atoms \(Hg^{198}\) have a root-mean-square velocity following from the equality

\[ \frac{1}{2}MV^{2}=\frac{3}{2}kT=0.025\ \text{eV},\quad \text{whence } V\simeq1.6\cdot10^{4}\ \text{cm/sec}. \]

At such a velocity the Doppler displacement of the frequency is

\[ D_{T}=h\nu\frac{V}{c}=4.11\cdot10^{5}\cdot \frac{1.6\cdot10^{4}}{3\cdot10^{10}} =0.21\ \text{eV}. \]

Although \(D_T\) is smaller than \(\Delta\), they are comparable. It should be taken into account that both the atoms \(Hg^{198*}\) and \(Hg^{198}\) are moving, and also that we have calculated \(D_T\) for the root-mean-square velocity, while in a Maxwellian distribution there are larger velocities as well. Therefore one may expect that the overlap of the bands \(A\) and \(B\) will not be very small. Calculations lead to the value \(7.7\cdot10^{-28}\ \text{cm}^{2}/\text{atom}\). This means that thermal motion, even at room temperature, increases the cross section by a factor of 20. Nevertheless, the cross section remains small—approximately one hundred million times smaller than predicted by (7) for the case \(E=E^{*}\),

\[ \sigma_{E=E^{*}}= \frac{2I^{*}+1}{2I_{0}+1}\, \frac{\lambda^{2}}{2\pi} \left(\frac{\Gamma_{\gamma}}{\Gamma}\right)^{2} =6.9\cdot10^{-20}\ \text{cm}^{2}/\text{atom} \]

or, by the formula pertaining to the case where the bands \(A\) and \(B\) are not displaced at all:

\[ \bar{\sigma}=\frac{1}{2}\sigma_{E=E^{*}}=3.5\cdot10^{-20}\ \text{cm}^{2}/\text{atom}. \]

In this respect the situation in resonance scattering of \(\gamma\)-rays differs sharply from the situation existing in optics.

In the visible part of the optical spectrum the energy of the quanta does not exceed 3 eV and, correspondingly, the quantity \(\Delta\) does not exceed \(10^{-8}\) eV. The Doppler broadening of optical lines associated with the thermal motion of the emitting atoms or molecules must, even at room temperature, exceed the quantity \(\Delta\) by hundreds of times; therefore the displacement \(\Delta\) practically does not interfere with resonance scattering.

The overlap of bands \(A\) and \(B\) can be increased in four ways:

1) by heating the source or scatterer (§ 6);
2) by mechanical motion of one relative to the other (§ 7);
3) by using the recoil arising in the act of decay (\(\beta\) or \(\gamma\)) of the nucleus \(A\) that preceded the \(\gamma\)-radiation (§ 8);
4) by using the velocity acquired by the emitting nucleus when it is created as a result of a nuclear reaction (§ 9).

At present all these methods have been successfully applied experimentally. The corresponding experiments are described in the following sections.

Large excitation energies, \(h\nu > 8\) nucleon-emission threshold. If the excitation energy of the nucleus considerably exceeds the reaction threshold for the emission of neutrons or protons, then \(\Gamma\), the energy width of the excited state, may be very large. Thus, for example, excited states arising from direct capture of protons or neutrons have widths sometimes reaching many keV (for example, the reaction \(\mathrm{C}^{13}(p,\gamma)\mathrm{N}^{14}\) has a resonance at \(E_p = 554\) keV with a half-width of 32 keV). In many cases \(\Gamma \gg \Delta\), and consequently the existence of the deficit \(\Delta\) plays only a small role. The formula for resonance excitation in this case has the form

\[ \bar{\sigma}_{\mathrm{exc}} = \frac{2I^*+1}{2I_0+1}\, \frac{\lambda^2}{4\pi}\, \frac{\Gamma_\gamma}{\Gamma} \left( 1-\frac{\Delta^2}{\Gamma^2} \right), \]

and for resonance scattering

\[ \bar{\sigma}_{\mathrm{scatt}} = \frac{2I^*+1}{2I_0+1}\, \frac{\lambda^2}{4\pi} \left( \frac{\Gamma_\gamma}{\Gamma} \right)^2 \left( 1-\frac{\Delta^2}{\Gamma^2} \right). \]

The large value of the first factors (\(10^{-23}\,\mathrm{cm}^2/\mathrm{atom}\) at 10 MeV) is partly compensated by the smallness of the quantity \(\Gamma_\gamma/\Gamma\), which may be \(10^{-4}\)—\(10^{-6}\). This is especially pronounced in resonance scattering, whose cross section contains the factor \((\Gamma_\gamma/\Gamma)^2\). Resonance excitation of nuclei by hard \(\gamma\)-rays has been observed experimentally, but resonance scattering never has.

§ 6. INCREASING RESONANCE SCATTERING BY HEATING THE SOURCE

In the preceding section it was pointed out that the thermal motion of the emitting and absorbing atoms substantially increases the magnitude of resonance scattering. The question arises: is it possible, by heating the \(\gamma\)-ray source and the scatterer to high temperatures, to obtain conditions under which the scattering cross section will reach \(10^{-25}\,\mathrm{cm}^2/\mathrm{atom}\) and will be easily detectable experimentally? In order to answer this question, one must solve the problem of the overlap of bands \(A\) and \(B\) (Fig. 8), taking exact account of the Maxwell distribution and of all possible directions of the velocities of the nuclei of the source and scatterer, and using formula (2) for the “microspectrum” of the radiation and formula (7) for the cross section of resonance scattering.

Such calculations lead to the following formula for the mean (or effective) cross section of resonance scattering:

\[ \bar{\sigma} = \frac{2I^*+1}{2I_0+1} \frac{\lambda^2}{4\pi} \left\{ \frac{\Gamma^2}{\Delta^2} + \sqrt{\pi}\, \frac{\Gamma \cdot Mc^2}{E^{*2}}\, g e^{-g^2} \right\}, \tag{8} \]

where all notation is as before, except for the dimensionless quantity \(g\), defined by the equality

\[ g^2=\frac{E^{*2}}{2k(T_1+T_2)Mc^2}; \]

\(k\) is Boltzmann’s constant, \(T_1\) and \(T_2\) are the absolute temperatures of the source and the scatterer. The first term does not depend on the temperature and represents the value of the cross section for initially stationary atoms of the emitter and scatterer (the limit as \(T_1\) and \(T_2 \to 0\)). The second term, first obtained by Moon, gives the temperature dependence. The temperature enters through the function \(g e^{-g^2}\). The form of this function is shown in Fig. 9; it has a maximum equal to

\[ \frac{1}{\sqrt{2e}} \]

at

\[ g=\frac{1}{\sqrt{2}} . \]

Thus, unlimited heating should not lead to an unlimited increase of the cross section—after reaching a maximum, the cross section will begin to decrease. The physical causes of this decrease are clear: heating broadens bands \(A\) and \(B\), and at the same time they decrease strongly in height; when the bands overlap strongly, their further broadening will only decrease the cross section.

Fig. 9. The function \(g e^{-g^2}\), determining the temperature dependence of resonant scattering. In the figure

\[ g^2=\frac{E^{*2}}{2k(T_1+T_2)Mc^2}. \]

Fig. 10. Dependence of the cross section of resonant scattering of \(\gamma\)-rays of \(Au^{198}\) in \(Hg^{198}\) on the temperature of the source. The scatterer is throughout at \(T_2 = 20^\circ C\).

However, to attain the maximum value of the function \(g e^{-g^2}\) by direct heating of the source is not easy: for this it must be heated to the temperature

\[ T_M=\frac{E^{*2}}{kMc^2}-T_2, \]

and \(T_M\), as a rule, is very large: \(9.7\cdot 10^5\,^\circ K\) for \(Na^{24}\), \(10\,200^\circ K\) for \(Au^{198}\). It is easier to attain the optimum temperature in substances having small \(E\) and large \(M\). Of course, in order to observe resonant scattering, there is no need to heat the sample so strongly as to attain the maximum cross section. It is sufficient to heat it enough that the resonant scattering becomes clearly noticeable against the background of Rayleigh scattering, which is especially large, as is the resonant scattering, for small \(h\nu\) and large \(M\). The magnitude of the second term of formula (8) depends not only on the constant \(g\), but also on the energy width \(\Gamma\) of the excited state. In practice, a noticeable effect can be achieved only for quadrupole electric transitions in heavy nuclei at an excitation energy not exceeding \(500\) kev. For illustration, in Fig. 10 we give the dependence of \(\overline{\sigma}\) on temperature for a particular case: \(\gamma\)-rays of \(Au^{198}\), having \(h\nu=411\) kev, resonantly excite a level of \(Hg^{198}\) of type \(2^+\), having \(\Gamma=3.0\cdot 10^{-5}\) ev. The temperature of the source is varied, while the scatterer, consisting entirely of \(Hg^{198}\), is throughout at a temperature of \(20^\circ C\). The large cross sections of resonant scattering at high temperatures deserve attention. The Rayleigh-scattered radiation, which

resonant radiation has the same wavelength as the resonantly scattered radiation, and therefore cannot be separated from it by filtration and discrimination; at scattering through \(140^\circ\) it has a cross section of about \(1.2\cdot 10^{-25}\ \mathrm{cm}^2/\mathrm{atom}\).

Thus, using a source heated above \(1000^\circ\mathrm{C}\), a scattering angle \(>140^\circ\), and isotopically separated \(Hg^{198}\) as the scatterer, one can obtain an advantage of resonant scattering over Rayleigh scattering. At the same time, it should be noted that it is almost impossible to observe resonant scattering at room temperature.

Malmfors\(^{23}\) experimentally investigated the resonant scattering of \(\gamma\)-rays of \(Au^{198}\) on Hg, heating the source to \(1100^\circ\mathrm{C}\).

The arrangement of the experiments is shown in Fig. 11. A small gold sphere was heated in a furnace. The \(\gamma\)-rays were scattered by ordinary mercury \((10\%\, Hg^{198})\) and by lead. The detector was a NaI(Tl) crystal. Pulses from it passed through a discriminator that selected the band of large pulses. Owing to this it was possible to get rid of soft Compton-scattered \(\gamma\)-rays. By studying the scattering alternately in Hg and in Pb, it was possible to exclude Rayleigh scattering.

The experiments showed that heating \(Au^{198}\) to \(1100^\circ\) increases the counting rate both with a mercury scatterer and with a lead scatterer.

The latter result was unexpected, since Rayleigh scattering in lead should not depend on temperature. It turned out to be connected with a secondary effect: the gold sphere expands upon heating; moreover, at the instant of melting of gold \((1065^\circ\mathrm{C})\) the volume of the sphere increases abruptly by \(5.2\%\). Because of the expansion, there is less matter in the path of the \(\gamma\)-quanta and they are less absorbed in the source itself (the density decreases as \((1+\beta t)^3\)), while the path increases as \(1+\beta t\), where \(\beta\) is the coefficient of linear expansion.

Fig. 11. Scheme of the experiments of Malmfors\(^{23}\).

The expansion of the gold completely explained the increase of scattering in lead; but after allowance for this phenomenon, the scattering in mercury upon heating nevertheless increased by \(3\%\).

In order to determine \(\bar{\sigma}\) from this number, it is necessary: a) to adopt a value of the cross section for Rayleigh scattering on \(Hg^{198}\) at \(140^\circ\). Malmfors takes the calculated value \(1.2\cdot 10^{-25}\ \mathrm{cm}^2/\mathrm{atom}\), which may prove inaccurate.

b) To take into account the angular distribution of the resonantly scattered radiation, which can be done quite unambiguously (see § 10).

Having determined \(\bar{\sigma}\), one can then compute from formula (8) the quantity \(\Gamma\). (\(\Gamma\) and \(\Gamma_\gamma\) for the 411-keV level of \(Hg^{198}\) practically coincide, since, besides emission of the \(\gamma\)-line with \(h\nu=411\ \mathrm{keV}\), the nucleus can undergo only internal conversion, whose coefficient is small: \(0.03\), according to\(^{24}\).) The last value, given by Malmfors\(^{25}\),

\[ \Gamma = 1.3\cdot 10^{-5}\ \mathrm{eV}, \]

corresponds to the half-life of the excited state \(Hg^{198*}\)

\[ T_{1/2}\sim 3.5\cdot 10^{-11}\ \mathrm{sec}. \]

The results of Malmfors’s experiments were confirmed in the works of Metzger and Todd\(^{26}\) and of Metzger\(^{27}\). The latter succeeded in considerably increasing the relative role of resonant scattering and bringing it up to \(8\%\). Metzger and Todd obtained for \(Hg^{198*}\)

\[ \Gamma=(3.0\pm 0.3)\cdot 10^{-5}\ \mathrm{eV}, \qquad T_{1/2}=(2.2\pm 0.2)\cdot 10^{-11}\ \mathrm{sec}. \]

This value of \(T_{1/2}\) is in satisfactory agreement with values obtained by other methods:

\[ T_{1/2}=(2.2\pm0.5)\cdot10^{-11}\ \text{sec.} \quad \text{(Moon and Davey}^{28}\text{, mechanical motion of the source),} \]

\[ T_{1/2}=(1.0\pm1.7)\cdot10^{-11}\ \text{sec.} \quad \text{(Graham and Bell}^{29}\text{, delayed in }\beta-\gamma\text{ coincidences).} \]

In 1954–1955 the method of heating the source was successfully applied by Metzger and co-workers to determine the lifetimes of the excited states of \(\mathrm{Hg}^{199}\), \(\mathrm{Tl}^{202}\), and \(\mathrm{Tl}^{203}\). The results they obtained are as follows:

\[ \begin{aligned} \mathrm{Hg}^{199}\ E^* &= 209\ \text{kev}, & T_{1/2} &= (3.1\pm0.9)\cdot10^{-10}\ \text{sec.}^{30,31},\\ \mathrm{Hg}^{202}\ E^* &= 439\ \text{kev}, & T_{1/2} &= (3.4\pm0.7)\cdot10^{-11}\ \text{sec.}^{32},\\ \mathrm{Tl}^{203}\ E^* &= 280\ \text{kev}, & T_{1/2} &= (9\pm4)\cdot10^{-10}\ \text{sec.}^{30}. \end{aligned} \]

The first result was confirmed by the method of delayed coincidences \(^{29,33}\), and the last by the methods of \(\beta\)-recoil and mechanical motion \(^{30}\).

The method of delayed coincidences has no relation whatever to the formulas for resonance scattering (5) and (7). Therefore the closeness of the values of \(T_{1/2}\) obtained for \(\mathrm{Hg}^{198}\) and \(\mathrm{Hg}^{199}\) by the coincidence method and by the scattering method indirectly confirms the correctness of the formulas of § 3.

It should be noted that in the case of \(\mathrm{Hg}^{199}\) resonance scattering occurs not at the first, but at the second excited level: the first level is metastable, has a small \(\Gamma\), and is not excited by \(\gamma\)-rays.

In summary, it should be emphasized that, although the method of heating the source leads only to a small resonance scattering, nevertheless the width \(\Gamma\) is determined quite unambiguously from the value of \(\sigma\) found. We shall encounter later, in § 8, the method of gaseous sources, which is considerably more effective for demonstrating and studying resonance scattering, but which does not allow \(\Gamma\) to be determined so clearly and simply.

§ 7. EXCITATION OF RESONANCE SCATTERING BY MECHANICAL MOTION OF THE SOURCE

In § 1 it was indicated that the cause which prevents resonance excitation of nuclei by \(\gamma\)-radiation from identical nuclei is the recoil of the nucleus that occurs in the emission and absorption of a quantum. The velocity of the recoil nucleus is then small:

\[ v=\frac{h\nu}{Mc},\qquad \beta=\frac{v}{c}=\frac{h\nu}{Mc^2} =\frac{h\nu}{m_0c^2}\frac{m_0}{M}. \]

Thus, for example, when a quantum \(h\nu=411\ \text{kev}\) is emitted by a nucleus \(\mathrm{Hg}^{198*}\),

\[ v=\frac{3\cdot10^{10}\cdot411}{1836\cdot198\cdot511} =6.6\cdot10^4\ \text{cm/sec}. \]

A bullet flies out of a rifle at approximately this speed. By firing into mercury with an activated gold bullet at twice the speed, it is possible to achieve resonance excitation of \(\mathrm{Hg}^{198}\) nuclei. Of course, the experiment can be arranged more simply. After testing several unsuccessful designs \(^{34,35}\), Moon and Storruste built the apparatus shown in Fig. 12.

\(\mathrm{Au}^{198}\) was deposited electrolytically on the edge of a steel disk. The disk could rotate very rapidly, making up to \(2000\ \text{rev/sec}\); in this case, at its periphery a linear velocity of up to \(8\cdot10^4\ \text{cm/sec}\) was produced. Between the disk and the scatterer (\(\mathrm{Hg}\) or \(\mathrm{Pb}\)), lead diaphragms were arranged like the blades of a water wheel: they were to select preferentially the \(\gamma\)-quanta emitted along the direction of the mechanical velocity of motion of the radiating atom. The scattered quanta were recorded with the aid of a scintillation counter and a discriminator, which selected only elastically scattered quanta. The scatterers

from mercury and lead were chosen so that the Rayleigh scattering produced by them was the same. The ratio of the counting rates of γ-quanta scattered elastically in Hg and Pb was measured at different disk rotation speeds. It began to increase because of resonant scattering at a linear speed of

Fig. 12. Diagram of the apparatus of Moon and Storruste^36. Vertical and horizontal sections.
\(R\)—rotating disk, \(S\)—sources, \(A\)—lead collimators, Hg, Pb—scatterers: mercury or lead, S. c.—scintillation counter.

\(2\cdot 10^4\ \text{cm}\); at \(v=7\cdot 10^4\ \text{cm/sec}\) mercury scattered 2.5 times more strongly than lead (Fig. 13). Thus, the resonant scattering was 1.5 times greater than the Rayleigh scattering. Let us recall that in the experiments of Malmfors et al. with heating of the source, resonant scattering could be brought only up to a few percent of the Rayleigh scattering.

The theory of the experiments of Moon and Storruste is very simple; generalization of formula (8) to the case of motion of the source relative to the scatterer leads to the expression

\[ \bar{\sigma}_v= \frac{2I^*+1}{2I_0+1}\, \frac{\lambda^2}{4\pi} \left\{ \frac{\Gamma^2}{E^{*2}} + \sqrt{\frac{\pi Mc^2}{kT}}\, \frac{\Gamma}{E^*}\, e^{-\frac{Mc^2}{4kT}\left(\frac{E^*}{Mc^2}-\frac{v}{c}\right)^2} \right\}. \tag{9} \]

In the case of Au\(^{198}\), the first term in the braces is always much smaller than the second, and it may be neglected. The second term has a maximum value, as was to be expected, at

\[ \frac{v}{c}=\frac{E^*}{Mc^2}. \]

On both sides of this value the decrease is symmetric.

It is interesting to note that the maximum value of \(\bar{\sigma}_v\) is proportional to \(1/\sqrt{T}\): thermal motion increases \(\bar{\sigma}\) for a stationary emitter and scatterer, but in the experiments of Moon and Storruste it only uselessly broadens the γ-spectrum and reduces the cross section.

Fig. 13. Results of the experiments of Moon and Storruste.

In Fig. 13 the solid curve is drawn according to (9); the agreement with experiment is sufficiently convincing. The scattering observed from lead and the scattering from mercury at \(v=0\) are almost entirely Rayleigh scattering. The calculation gives

for its cross section (at the given scattering angle) \(1.32\cdot 10^{-26}\ \dfrac{\mathrm{cm}^{2}}{\mathrm{atom\ ster}}\). On the basis of this number and Fig. 13, one can find \(\bar{\sigma}_{v}\) and then \(\Gamma\); here, too, one has to introduce a correction for the angular distribution of the resonantly scattered radiation (see § 9).

The results of Moon and Storruste have already been cited:

\[ \Gamma=(3.0\pm0.7)\cdot 10^{-5}\ \mathrm{eV},\qquad T_{1/2}=(2.2\pm0.5)\cdot 10^{-11}\ \mathrm{sec}. \]

§ 8. USE OF NUCLEAR RECOIL IN PRECEDING β-DECAY, K-CAPTURE, OR γ-RADIATION

Suppose that an excited state \(E^*\) of a nucleus \(A\) is produced as a result of \(\beta\)-decay of a nucleus \(C\) (Fig. 14, a), as a result of capture of an orbital electron, or as a result of a \(\gamma\)-transition from a more highly excited state of the nucleus \(A\) (Fig. 14, b). In all these cases the nucleus undergoes recoil. If the time of emission of the subsequent \(\gamma\)-quantum is much less than the braking time of the nucleus, then the radiation must occur from a moving nucleus. Since various velocities of the recoil nuclei and various angles between the directions of recoil and of the \(\gamma\)-quantum are possible, the Doppler effect must lead to the fact that, in any direction, not strictly monochromatic \(\gamma\)-rays will travel, but rather a certain spectral band.

Fig. 14. The excited state \(E^*\) of nucleus \(A\) is produced after \(\beta\)-decay of nucleus \(C\) or after \(\gamma\)-de-excitation of a higher excited state of nucleus \(A\).

Let us determine the width of this band. Let a nucleus having mass \(M\) and excitation energy \(E^*\), before emitting the \(\gamma\)-quantum, have velocity \(V\), directed at an angle to the direction of flight of the \(\gamma\)-quantum (Fig. 15). In this case the energy of the quantum will be approximately equal to:

\[ h\nu=E^*-\frac{E^{*2}}{2Mc^2}+E^*\frac{V}{c}\cos\alpha . \tag{10} \]

Since \(V/c\) and \(E^*/Mc^2\) are very small, terms with higher powers of these quantities have been dropped here. The second term on the right represents the recoil energy in \(\gamma\)-emission (see § 4), and the third is the Doppler change of energy due to the velocity of the nucleus at the moment of recoil.

For determining the width of the spectral band it is sufficient for us to assume that any \(\alpha\) are possible and to determine the doubled maximum value of the third term, i.e., to find \(2E^*V_{\max}/c\), where \(V_{\max}\) is the maximum value of the recoil-nucleus velocity in the preceding \(\beta\)-decay, \(K\)-capture, or \(\gamma\)-radiation. In \(\beta\)-decay the recoil nucleus receives the greatest velocity in the case when the neutrino is emitted with zero energy,

Fig. 15. Emission of a \(\gamma\)-quantum by a nucleus moving with velocity \(V\).

\[ V_{\max}=\frac{m_0c}{M}\sqrt{\left(\frac{\varepsilon_0}{m_0c^2}\right)^2+2\frac{\varepsilon_0}{m_0c^2}}, \tag{11} \]

where \(\varepsilon_0\) is the endpoint energy of the \(\beta\)-spectrum.

Thus, the width of the band in the absence of braking of the recoil nuclei is

\[ D_{\beta}=2E^*\frac{m_0}{M}\sqrt{\left(\frac{\varepsilon_0}{m_0c^2}\right)^2+2\frac{\varepsilon_0}{m_0c^2}} . \tag{12} \]

Let us give, as an example, the values of \(D_\beta\) for some emitters.

Table II

Values of \(D_\beta\) for some emitters

Substance	\(\varepsilon_0,\ \mathrm{MeV}\)	\(E^*,\ \mathrm{MeV}\)	\(T_{\mathrm{eff}},\ \mathrm{deg.}\)	\(D_\beta,\ \mathrm{eV}\)	\(\Delta,\ \mathrm{eV}\)
\(\mathrm{Na}^{24}\)	1.390	1.368	\(8.6\cdot10^5\)	222	83
\(\mathrm{Zn}^{63}\)	1.40	0.96	\(3.3\cdot10^5\)	60	15.5
\(\mathrm{Au}^{198}\)	0.957	0.411	\(0.59\cdot10^5\)	6.1	0.91

In order for resonant excitation to be possible, it is necessary that \(D_\beta/2>\Delta\). In all the examples given this is realized (this always occurs when \(E^*<\varepsilon_0\sqrt{1+2m_0c^3/\varepsilon_0}\)).

The same situation arises in those cases when the \(\gamma\)-transition under consideration is preceded not by \(\beta\)-decay, but by \(K\)-capture or a \(\gamma\)-transition. The recoil of the nuclei then creates in the spectrum of \(\gamma\)-rays a band \(D_\gamma\):

\[ D_\gamma \simeq 2E^*\frac{h\nu_1}{Mc^2}, \tag{13} \]

where \(h\nu_1\) is the energy of the first quanta (or the energy of the neutrino in \(K\)-capture). For a \(\beta\)-particle energy equal to \(\varepsilon_0\), and \(\gamma\)-quanta \(h\nu\), \(D_\gamma\) is somewhat less than \(D_\beta\) (from formula (12) it is seen that \(D_\beta\) goes over into \(D_\gamma\) if the second term under the radical is neglected).

Table III gives, as an example, the values of \(D_\gamma\) for some radioactive substances.

Table III

Values of \(D_\gamma\) for some radioactive substances

Substance	\(h\nu_1,\ \mathrm{MeV}\)	\(E^*,\ \mathrm{MeV}\)	\(T_{\mathrm{eff}},\ \mathrm{deg.}\)	\(D_\gamma,\ \mathrm{eV}\)	\(\Delta,\ \mathrm{eV}\)
\(\mathrm{Na}^{24}\)	2.755	1.368	\(19\cdot10^5\)	334	83
\(\mathrm{Co}^{60}\)	1.1715	1.3316	\(1.7\cdot10^5\)	55	32
\(\mathrm{Sb}^{124}\)	1.692	0.603	\(1.4\cdot10^5\)	17.5	3.2

In order for resonant excitation to be possible, it is necessary that \(h\nu_1>E^*\) and, correspondingly, \(\frac{1}{2}D_\gamma>\Delta\). In the examples given this is realized in the decay of \(\mathrm{Na}^{24}\) and \(\mathrm{Sb}^{124}\), but is not realized in the decay of \(\mathrm{Co}^{60}\).

If the decay scheme of nucleus \(C\) in Fig. 14 is still more complicated and the \(\beta\)-decay is accompanied by several acts of \(\gamma\)-radiation, which occur so rapidly that the braking has no time to manifest itself, then in the most favorable

B. S. DZHELEPOV

case the recoil velocities are added, and since the quantities \(D\) are proportional to them, for the full width of the band one obtains

\[ D = D_{\beta} + D_{\gamma_1} + D_{\gamma_2} + \ldots \tag{14} \]

Thus, for example, in \(\mathrm{Co}^{60}\) \(D\) turns out to be equal to \(85\ \mathrm{eV}\), and consequently in this case, which was the only exception in Table III, nuclear recoil makes resonance scattering possible.

Tables II and III and formula (14) show us that the recoil of nuclei is sufficiently large to compensate \(\Delta\). Figuratively speaking, we may say that in \(\beta\)-decay and \(\gamma\)-emission the recoil nuclei are obtained sufficiently “hot.” We can determine their effective temperature: the temperature of a gas at which the most probable velocity of an atom is equal to the maximum recoil velocity

\[ \left. \begin{aligned} (T_{\mathrm{eff}})_{\beta} &= \frac{m_0}{M} \left\{ \left(\frac{\varepsilon_0}{m_0 c^2}\right)^2 + 2\frac{\varepsilon_0}{m_0 c^2} \right\} \frac{m_0 c^2}{2k}, \\[6pt] (T_{\mathrm{eff}})_{\gamma} &= \left(\frac{h\nu_1}{m_0 c^2}\right)^2 \frac{m_0}{M}\frac{m_0 c^2}{2k}. \end{aligned} \right\} \tag{15} \]

The corresponding values are given in Tables II and III; they are all of the order of \(10^5\) degrees—just what is needed to reach the maximum of resonance scattering (see § 5).

This directly shows that the use of recoil energy for detecting resonance scattering is undoubtedly more promising than direct heating of the specimen. It is only necessary to be able to make use of the high “temperature” of the recoil nuclei before they “cool,” i.e., are slowed down.

It will be seen below that resonance scattering of \(\gamma\)-rays can be used to determine the spin and the energy width \(\Gamma\) of the excited state arising in scattering. The first problem is solved by studying the angular distribution of the scattered radiation, and for its solution only a sufficient scattering intensity is necessary. With the second problem the matter is much more complicated. To determine \(\Gamma\) it is necessary to use formula (7) for the cross section, and it contains \(h\nu\); therefore, for the correct application of this formula it is necessary to know the “microspectrum” of the incident radiation. In the preceding sections we determined the boundaries of this spectrum (the quantities \(D\)), and now it is necessary to know its shape. However, the shape of this spectrum is affected by many factors.

1) To calculate it, one must know the initial energy distribution of the recoil nuclei. In those cases where the preceding radiation consists of \(\gamma\)-quanta or neutrinos (electron capture), the spectrum of recoil nuclei is simplest: all nuclei have identical velocities. However, in \(\beta^\pm\)-decay, even allowed decay, the spectrum of recoil nuclei must depend on the angular correlation between the electron and the neutrino; this correlation has the form \(1 + k\cos\theta\); the constant \(k\) depends on the type of interaction leading to \(\beta\)-decay, and this type, as is known, is still not sufficiently clear.

2) It is necessary to know the angular distribution of the recoil nuclei with respect to the direction of flight of the \(\gamma\)-quanta. Sometimes this distribution is isotropic, sometimes it is not. In the general case, knowledge is required of the quantum characteristics of all the participating states.

3) It is necessary to know the influence of molecular bonds on the first two factors; only a few substances can be used in the form of monatomic vapors. In a complex molecule the recoil energy experienced by one of the atoms may be partly or wholly expended on the destruction of the molecule or on bringing it into rotational or vibrational motion. It should be remembered that after \(\beta\)-decay or \(K\)-capture the atom has a different \(Z\), and consequently all chemical bonds change.

4) It is necessary to know the law of slowing down of recoil nuclei. The question concerns relatively slow atoms, molecules, or ions:

\[ v \simeq 10^5—10^6\ \text{cm/sec}; \qquad E_{\text{kin}} \simeq 1—100\ \text{eV}. \]

The slowing down of such atoms or ions in gases can be calculated from the formulas of the kinetic theory of gases, but the laws of slowing down in liquids or solids are practically unknown.

The quantities \(D\) derived by us are applicable to the case when all transitions occur so rapidly that the recoil nuclei, in the interval between events, do not have time to change their velocity. In reality, however, the \(\gamma\)-quantum is not emitted immediately after \(\beta\)-decay. If slowing down takes place, then the effective value of \(D\) decreases (the microspectrum narrows). If the state \(E^*\) lives so long that all nuclei \(A\), before emission, have time to come to rest, then \(D_{\text{eff}}=0\).

The totality of the available data indicates that, in gases at atmospheric pressures, the slowing-down time is of the order of \(10^{-10}—10^{-9}\) sec.; correspondingly, in a medium with density 1 it is about \(10^{-13}—10^{-12}\) sec.

As a consequence, the use of recoil in a dense source for observing resonance scattering is possible only if the time of \(\gamma\)-emission \(T_\gamma \ll 10^{-12}\) sec. The absence of scattering indicates that \(T_\gamma > 10^{-12}\) sec. From this point of view it is clear why, at ordinary temperature, resonance scattering of the \(\gamma\)-rays of Au\(^{198}\) in Hg\(^{198}\), caused by \(\beta\)-recoil, is practically not observed: the lifetime of Hg\(^{198*}\), \(T_\gamma = 2.2\cdot10^{-11}\) sec.,\(^{23,36}\) is too large for this. Metzger\(^{27}\), it is true, noted a small scattering at room temperature, but himself pointed out that further verification of these experiments was needed.

A number of attempts to detect resonance scattering stimulated by \(\beta\)- or \(\gamma\)-recoil of nuclei ended unsuccessfully: no resonance scattering was found for

\[ \begin{aligned} &\gamma\text{-rays of Na}^{24}, &&(h\nu=1.368\ \text{MeV})\ \text{in Mg}^{24}\quad (\text{see }^{37}),\\ &\gamma\text{-rays of Tc}^{96}, &&(h\nu=0.8\ \text{MeV})\ \text{in Mo}^{96}\quad (\text{see }^{38}),\\ &\gamma\text{-rays of Nb}^{97}, &&(h\nu=0.665\ \text{MeV})\ \text{in Mo}^{97}\quad (\text{see }^{39}),\\ &\gamma\text{-rays of Mn}^{56}, &&(h\nu=0.84\ \text{MeV})\ \text{in Fe}^{56}\quad (\text{see }^{37,40}). \end{aligned} \]

The reason for the failure in all these experiments was that the recoil nuclei were slowed down in solid targets. Only Deljagin and Shpinel\(^{69}\) observed an effect with a solid source containing Na\(^{24}\).

The experiments began to give positive results when liquid (Ilakovac\(^{40,41}\), Burgov and Terekhov\(^{70}\)) or gaseous sources (Metzger\(^{42,68}\)) began to be used.

From what has been set forth in this section the following conclusions may be drawn:

1. For determining the energy widths of excited states of atomic nuclei, it is desirable to use gaseous radioactive sources containing monatomic vapors of the radioactive substance at sufficiently low total pressure (the time between collisions must be much greater than \(T_{1/2}\)).

2. For the chosen radioactive substance, the decay scheme must be carefully studied and the quantum characteristics of all participating states determined, since calculation of the microspectrum requires knowledge of the energies and relative intensities of all decay branches, the degree of resolution of all transitions, and all angular correlations.

3. If it is impossible to prepare a source satisfying item 1, then a molecular gaseous compound may be used. In this case it is desirable that the masses of all other atoms be much smaller than that of the radioactive one,

and so that the binding energy would be smaller. Calculations of the influence of molecular bonds on the spectrum of recoil nuclei are necessary if it is desired to determine $\Gamma$ with sufficient accuracy.

1. If it is not possible to prepare a gaseous source, then a liquid one may be used. The lower the density of the liquid, the greater the chances of success. So far no one has used the transition of a liquid into the supercritical state; meanwhile, by this method one can obtain densities of the order of 0.01–0.10, and precisely such densities are desirable in the study of many cases of resonance scattering.

Calculations of $\Gamma$ and $T_{1/2}$ when a liquid source is used can be regarded only as approximate; for studying the angular distribution such sources are quite applicable.

1. For the investigation of resonance scattering, especially good are short-lived states: first, because the “acceptance band” $B$ in Fig. 8 is broad, and second, because the nuclei do not have time to stop and, consequently, the “emission band” $A$ is sufficiently broad.

Ilakovac’s experiments (liquid source). In 1954 Ilakovac$^{40,41}$ studied resonance scattering of $\gamma$-rays of Zn$^{63}$ in the usual arrangement of Fig. 11, with a somewhat modified geometry. The scatterers were, in turn, Cu and Fe, taken in approximately equal weights. The scattering occurred at $90^\circ$. The detector was a luminescent NaJ(Tl) crystal; the light pulses were received by a photomultiplier with a discriminator selecting the $\gamma$-line 0.96 MeV.

The source was Zn$^{63}$, dissolved in nitric acid (several hundred millicuries). The decay scheme of Zn$^{63}$ is shown in Fig. 16. Only 8% of the decays pass through the 0.96 MeV state of Cu$^{63}$; in the main part of these 8% the $\gamma$-rays $h\nu - 0.96$ MeV follow the $\beta^+$-decay with $\varepsilon_0 = 1.40$ MeV$^{41}$.

Fig. 16. Decay scheme of Zn$^{63}$.

Ilakovac found that, when a liquid source was used, the copper scatterer gave $3.6 \pm 0.9$ times more additional pulses than the iron one. However, when the source was crystalline copper containing inclusions of Zn$^{63}$, the additional scattering in copper exceeded the additional scattering in iron by only $1.6 \pm 0.7$ times. This fact is explained by the density of the stopping substance being, in the case of copper, about 8 times greater than in the case of the liquid; probably, the stopping time of the recoil nuclei in copper is correspondingly shorter than in the liquid. The cross section for the additional scattering in copper is $\sigma = (8.4 \pm 1.4)\,10^{-27}\ \mathrm{cm}^2/\mathrm{atom}$. Attributing the observed effect to resonance scattering, Ilakovac obtained for the Cu$^{63}$ level under consideration

\[ T_\gamma = 6 \cdot 10^{-13}\ \mathrm{sec}. \]

During this time the Cu$^{63}$ nuclei in copper have time to stop, whereas in water they do not.

The presence of resonance scattering of $\gamma$-rays emitted by a liquid Zn$^{63}$ preparation means only that the stopping and emission times in this case are comparable. It is impossible to determine the half-life at all reliably without knowing the laws of stopping. Only rough estimates can be involved.

For example, we give the following estimate for Ilakovac’s experiments for \( \mathrm{Zn}^{63} \). The mean distance of a \( \mathrm{Cu}^{63} \) atom from its neighbors in the solution is \(\sim 3\,\text{\AA}\); the initial velocity of the recoil atom is \(5\cdot 10^5\ \mathrm{cm/sec}\). Consequently, the first collision will occur after \(t_0=6\cdot 10^{-14}\) sec. The mean lifetime of the excited state \(\bar T\) is greater than this quantity—otherwise scattering would also have been observed with a solid source. We shall assume that during the time \(t_0\), \(t_0/T\) of the nuclei emit, and that emission after a collision does not lead to scattering. In the initial formula for the cross section

\[ \sigma=\frac{2I^*+1}{2I_0+1}\frac{\lambda^2}{8\pi} \frac{\Gamma_\gamma \Gamma}{(E^*-E_0)^2+\frac{\Gamma^2}{4}} \]

we set, for simplicity, \(\Gamma_\gamma=\Gamma\) and the first factor equal to 1. Then

\[ \sigma_{\max}=\frac{\lambda^2}{2\pi} =\frac{1}{\pi}\left(\frac{h}{mc}\frac{mc^2}{h\nu}\right)^2 =\frac{1}{2\pi}\left(2.426\cdot 10^{-10}\frac{511}{960}\right)^2 =2700\ \text{barns}. \]

The effective \(\sigma\) is smaller, since 1) the spectrum of the \(\gamma\)-rays is smeared over a band of width \(D\), and 2) only \(t_0/\bar T\) of all nuclei emit. Let us assume, for simplicity, that the smearing is uniform (a refinement does not strongly change the result). Then

\[ \sigma_{\mathrm{eff}}=\sigma_{\max}\frac{\bar\Gamma}{2}\frac{\Gamma}{2}\frac{t_0}{\bar T}. \]

Replacing \(\Gamma\) by \(h/2\pi T\), we obtain

\[ \bar T=\left(\frac{h}{4}\frac{\sigma_{\max}}{\sigma_{\mathrm{eff}}}\frac{t_0}{D}\right)^{1/3}. \]

Substituting here \(\sigma_{\max}=2700\) barns, \(\sigma_{\mathrm{eff}}=\sigma_{\mathrm{exp}}=(8.4\pm 1.4)\) mbarns, \(t_0=6\cdot 10^{-14}\) sec.,

\[ D=2E^*\frac{V}{c}=62\ \mathrm{eV}, \]

we obtain

\[ \bar T=6\cdot 10^{-13}\ \text{sec}. \]

It is quite clear that in these estimates the crudest assumption is that concerning the number of emitting nuclei; the method of determining \(\bar T\) can become quantitative only after the process of slowing down of recoil nuclei has been studied.

Metzger’s experiments (gaseous sources). A few elements can be obtained in the form of gaseous compounds or vapors of sufficiently high density at easily attainable temperatures. These include mercury, thallium, arsenic, and cobalt, with which experiments using gaseous sources have now been carried out. Metzger’s experiments with \( \mathrm{Hg}^{203} \) and \( \mathrm{Tl}^{202} \) were exploratory

Fig. 17. Scheme of Metzger’s experiments \(^{23}\) on the study of the angular distribution of resonance-scattered \(\gamma\)-rays.

…character^30, 32, but nevertheless proved the effectiveness of gaseous sources and their applicability for studying the angular distribution and other aspects of the phenomenon. The experiments described in 1956 with As^72 and As^74 were carried out much more carefully^42; we shall dwell on them in greater detail.

Metzger’s experiments were performed in the usual arrangement of Fig. 17. The main attention was devoted to the source. As^72 ($\tau = 27$ hr) and As^74 ($\tau = 19$ days) were produced in a cyclotron by bombarding germanium with deuterons. They were chemically separated with a minimal addition of carrier ($50\,\mu\text{g}$) and placed in a quartz ampoule of volume $0.6\,\text{cm}^3$. The ampoule was enclosed in a stainless-steel container; it could be heated to $700^\circ\text{C}$. The arsenic was present either in the form of molecular vapors As$_4$, or in the form of arsine molecules AsH$_3$. The decay schemes of As^72 and As^74 are shown in Fig. 18; owing to the difference in half-lives, the effects belonging to the first and the second are easily separated.

Fig. 18. Decay schemes of As^72 and As^74.

Germanium and zinc large rings were used in turn as scatterers; their number was selected so that under nonresonance conditions they scattered equally to within 3%. The detector was a large NaI(Tl) crystal ($\varnothing = 35\,\text{mm}$, $l = 40\,\text{mm}$), surrounded by a small shield against soft Compton quanta; the single-channel analyzer was adjusted to elastically scattered quanta. Under resonance conditions the apparatus counted $\sim 100$ pulses/min from a source of activity $1\,\mu\text{Ci}$.

In experiments with As^74, when the source was heated, scattering in zinc remained unchanged, while scattering in germanium increased by a factor of $1\frac{1}{2}$. This is illustrated by Fig. 19. The points correspond to the counting rate of resonance scattering, and the solid curve to the dependence of the vapor density on temperature (shifted at $600^\circ$). When all the arsenic passes into the gaseous state,

RESONANCE SCATTERING OF γ-RAYS BY NUCLEI

a horizontal section of the curve is reached; the slight rise above \(800^\circ\mathrm{C}\) is associated with partial dissociation of \(\mathrm{As}_4\).

Fig. 19 demonstrates very clearly the advantages of a gaseous source. The fact that it is precisely the gaseous state, and not the high temperature, that causes the considerable resonance scattering became still clearer when a heated quartz ampoule with As burst inside a steel container: the resonance scattering immediately fell by more than a factor of 50. All the arsenic remained in the container and retained its temperature, but, being chemically active, it reacted with the walls of the container and thus passed into the solid phase.

To prove that the increase in scattering by germanium upon heating the source is connected precisely with resonance scattering, Metzger used his pulse analyzer to record the spectrum of the scattered \(\gamma\)-rays.

Fig. 19. Dependence of resonance scattering on the temperature of the source. The points correspond to the counting rate of resonance scattering, and the solid curve to the dependence of the vapor density on temperature (shifted at \(600^\circ\)).

Fig. 20. Spectrum of the scattered \(\gamma\)-radiation of \(\mathrm{As}^{72}\).

He showed that, when the source is heated, a distinct \(\gamma\)-line \(h\nu = 596\ \mathrm{keV}\) appears in the spectrum in the case of \(\mathrm{As}^{74}\), and \(h\nu = 835\ \mathrm{keV}\) in the case of \(\mathrm{As}^{72}\); these are completely invisible against the background of ordinary scattering of the harder \(\gamma\)-lines of \(\mathrm{As}^{72}\) and \(\mathrm{As}^{74}\) with a cold source or a zinc scatterer. Fig. 20 shows the spectrum of the scattered \(\gamma\)-radiation for \(\mathrm{As}^{72}\). Metzger carried out measurements of \(\sigma\) and \(I^*\) for \(\mathrm{As}^{72}\) and \(\mathrm{As}^{74}\). He found the former quantities by comparing the intensities of the scattered and primary radiation; in this way he obtained:

\[ \sigma(835\ \mathrm{keV}\ \mathrm{Ge}^{72}) = (7.8 \pm 1.1)\cdot 10^{-26}\ \mathrm{cm}^2/\mathrm{atom}, \]

\[ \sigma(596\ \mathrm{keV}\ \mathrm{Ge}^{74}) = (7.5 \pm 1.2)\cdot 10^{-26}\ \mathrm{cm}^2/\mathrm{atom}. \]

Metzger determined the spins of the excited states from the angular distribution of the scattered radiation (see § 10); they proved to be equal to 2 in both cases.

From these data it is necessary to calculate \(\Gamma\) for both levels. For this one must first calculate the microspectrum. In order to reduce the influence on the microspectrum of molecular bonds, which are difficult to take into account, the source in these experiments was taken in the form of arsine \(\mathrm{AsH}_3\). A molecule of this type, when excited, cannot take up a large fraction of the recoil energy; the two limiting cases—the breakup of the molecule upon the transformation of As into Ge and the recoil of the molecule as a whole—differ only slightly because of the mass (for example, 74 and 77 in the case of the decay of \(\mathrm{As}^{74}\)). If molecular bonds and collisions in the gas are neglected, then it remains to construct a microspectrum based on the scheme

decay. In view of the small width of the absorption line (line B in Fig. 8), a narrow band is cut out of the microspectrum, over the extent of which the ordinate of the microspectrum practically does not change. Let us denote, following Metzger, by \(\dfrac{N_{\mathrm{res}}}{N}\) the relative fraction of the microspectrum contained in a band of \(1\) eV about the resonance energy. The problem consists precisely in calculating this quantity; if it is found, then \(\Gamma\) is calculated simply:

\[ \bar{\sigma} = \frac{\displaystyle \int_{0}^{\infty} S(E)\sigma_{\mathrm{res,scat}}\,dE} {\displaystyle \int S(E)\,dE} = \frac{N_{\mathrm{res}}}{N}\, \frac{\pi}{2}\, \Gamma\, \frac{2I^{*}+1}{2I_{0}+1}\, \frac{\lambda^{2}}{2\pi} \left(\frac{\Gamma_{\gamma}}{\Gamma}\right)^{2}, \tag{16} \]

where \(S(E)\) is the microspectrum of the \(\gamma\)-rays. Taking \(\Gamma_{\gamma}=\Gamma\), which is close to the truth for low-lying and nonisomeric levels, we find:

\[ \Gamma= \frac{4\bar{\sigma}} {\left(\dfrac{N_{\mathrm{res}}}{N}\right) \dfrac{2I^{*}+1}{2I_{0}+1}\lambda^{2}}. \tag{17} \]

Fig. 21. Microspectrum (line shape) of the \(\gamma\)-line 596 keV of \(Ge^{74}\). The rectangular part is due to \(K\)-capture (59%), the bell-shaped part to \(\beta^{+}\)-decay of \(As^{74}\) (41%) to the indicated level. For \(\bar{\sigma}\), the value of the ordinate of the microspectrum at \(E^{*}\) is essential.

Figure 21 shows the microspectrum of the \(\gamma\)-rays \(h\nu=596\) keV of \(Ge^{74}\), calculated by Metzger under the following assumptions:

1) 59% of the transitions to the 596 keV state of \(Ge^{74*}\) occur by \(K\)-capture, \(\Delta E=1.94\) MeV; the \(\gamma\)-radiation is considered isotropic with respect to the direction of neutrino emission;

2) 41% of the transitions occur by \(\beta^{+}\)-decay with \(E_{\mathrm{gr}}=0.92\) MeV; \(\beta^{+}-\nu\) and \(\beta^{+}-\gamma\) correlations are not taken into account; the role of thermal motion and the Auger effect is likewise not taken into account.

These assumptions probably do not correspond exactly to reality, but they greatly simplify the calculations.

It is seen from the figure that for \(Ge^{74}\)

\[ \frac{N_{\mathrm{res}}}{N}=4.00\%; \]

the error is 5%. For \(Ge^{72}\)

\[ \frac{N_{\mathrm{res}}}{N}=(2.0\pm0.4)\%; \]

the increased error is connected with the inaccurate knowledge of the decay scheme. Using these \(\dfrac{N_{\mathrm{res}}}{N}\), Metzger obtained:

\[ Ge^{72}\quad E^{*}=835\ \text{keV},\qquad T=(4.6\pm1.2)\cdot10^{-12}\ \text{sec.}, \]

\[ Ge^{74}\quad E^{*}=596\ \text{keV},\qquad T=(1.9\pm0.3)\cdot10^{-11}\ \text{sec.} \]

Recently these same quantities \(\bar{T}\) were determined by Heidenburg and Temmer\(^{43}\) from Coulomb excitation of germanium; they were found to be \(1.9\cdot10^{-12}\) and \(1.3\cdot10^{-11}\) sec. The agreement in the case of \(Ge^{74}\) is satisfactory, but in \(Ge^{72}\) \(T\) differs by \(2^{1}/_{2}\) times; the reasons are as yet unknown.

In Metzger’s experiments\(^{68}\) with \(Co^{60}\), a \(Co^{60}Cl_{2}\) preparation (\(\sim 5\) microcuries) was used, heated to \(1010^{\circ}C\); at this temperature all the \(CoCl_{2}\) had to be in the gaseous state at a pressure of about 550 mm Hg. The resonance scattering was very strong, and this enabled Metzger to change the method for determining the lifetime. As was indicated above, in the description of the experiments with \(Ge^{72}\) and \(Ge^{74}\), in determining \(\Gamma\) one must know

...the shape of the microspectrum of the γ-rays emitted by the source. The shape of the microspectrum is difficult to calculate for sources consisting of complex molecules, in particular for CoCl\(_2\) molecules. However, the need to know the microspectrum is eliminated if one makes use of resonance absorption of γ-rays on their path to the resonant scatterer. In experiments with Co\(^{60}\), before reaching the nickel scatterer the γ-rays passed through a nickel or a steel absorber, chosen so that the total absorption in them was the same to within 1%. Of course, the resonance absorption constitutes a negligible fraction of the total; but the apparatus placed behind the absorber—the resonant scatterer and the luminescent crystal tuned to the resonance radiation—is a selective detector, and thanks to this the selective absorption in \(N_1\) becomes noticeable.

Metzger obtained for Co\(^{60}\)

\[ \Gamma = 5.97 \cdot 10^{-4}\ \text{eV}, \]

\[ T_\gamma = (1.1 \pm 0.2)\cdot 10^{-12}\ \text{sec}. \]

This value of \(T_\gamma\) is approximately 5 times smaller than follows from the Weisskopf formula for single-particle transitions.

On the use of \(\beta\)—γ or γ—γ coincidences. As early as 1948, Pollard and Alburger raised the question of using \(\beta\)—γ or γ—γ coincidences to isolate resonance scattering (Fig. 22).

Let us suppose that the initial β-active nucleus (for example, Na\(^{24}\)) emitted a β-particle, recoiled, and after this emitted a quantum \(h\nu_1\), which entered counter \(C_1\). After that, without stopping, the nucleus emitted a quantum \(h\nu_2\), which subsequently underwent resonance scattering through a large angle and entered counter \(C_2\). The apparatus registered a coincidence of pulses in \(C_1\) and \(C_2\).

Fig. 22. Diagram of experiments in which γ—γ coincidences are used to isolate resonance scattering.

In order for the Doppler effect to compensate exactly the magnitude \(\Delta\), it is necessary that

\[ h\nu_2 \frac{h\nu_1}{Mc^2}\cos\theta = \Delta \simeq \frac{(h\nu_2)^2}{2Mc^2}, \]

whence

\[ \cos\theta = \frac{h\nu_2}{h\nu_1}. \]

Thus, for example, for the case of Na\(^{24}\) we obtain:

\[ \cos\theta = \frac{1.368}{2.758}=0.496 \]

and

\[ \theta = 29^\circ 30'. \]

The main obstacle to carrying out such an experiment is the background of random coincidences. The counters \(C_1\) and \(C_2\) turn out to be heavily loaded, and the expected effect is several orders of magnitude smaller than the background if Geiger–Müller counters are used.

However, if luminescent counters and a coincidence amplifier with a time resolution \(\tau \sim 10^{-9}\) sec are used, observation of resonance scattering becomes feasible. This was first shown by Burgo and Terekhov\(^{70}\). They used a solution of \(\mathrm{Na}^{24}\mathrm{OH}\) and, with the aid of two tolane crystals and a circuit with \(\tau \sim 5\cdot 10^{-9}\) sec, recorded two \(\gamma\)-quanta (\(h\nu = 1.38\) and \(2.76\) MeV) flying apart at an angle \(\sim 120^\circ\). Placing a resonant magnesium absorber in the path of the quantum \(h\nu = 1.38\) MeV, they observed selective absorption.

Determination of \(\Gamma\) is difficult, since a liquid source was used; a preliminary estimate is \(\Gamma > 3\cdot 10^{-4}\) eV.

§ 9. RESONANCE EXCITATION OF NUCLEI BY HARD \(\gamma\)-RADIATION OF THE SAME NUCLEI

In the region of relatively soft \(\gamma\)-rays—up to \(h\nu = 1\) MeV—resonance excitation of atomic nuclei by \(\gamma\)-rays was clearly demonstrated in the experiments on resonance scattering described in the preceding three paragraphs. In the region of harder \(\gamma\)-rays, successful experiments of this type have not yet been obtained. The reason for this is clear: for harder \(\gamma\)-rays the displacement \(\Delta\) is larger; if \(\Delta\) is compensated by heating or by mechanical motion of the source, then temperatures that are too high or velocities that are too large are required; recoil in the preceding transformation cannot be used, since cases are very rare in which emission of hard \(\gamma\)-rays is preceded by a decay with no smaller energy (see § 6). It is still more difficult to observe resonance scattering of very hard rays; the reasons were set forth in § 5. The situation is somewhat different with resonance excitation. In § 4 experiments were described in which the formation of nuclear isomers and the appearance of line spectra of secondary particles under irradiation of nuclei by hard bremsstrahlung \(\gamma\)-spectra were demonstrated. These experiments undoubtedly show that discrete nuclear states are excited in this process, but because of the continuity of the primary spectrum they cannot prove the resonant character of the excitation process taking place, and the complexity of the analysis does not make it possible to verify how correctly the phenomena occurring are described by the formulas of § 3. The resonant character of the excitation of a high-lying nuclear level was demonstrated very clearly in 1956 by Griffiths\(^{44}\). Griffiths bombarded \(\mathrm{C}^{13}\) with protons of energy up to 700 keV. The cross section of the capture reaction \(\mathrm{C}^{13}(p,\gamma)\mathrm{N}^{14*}\) has a resonance maximum at \(E_p = 554\) keV with a half-width \(\Gamma = 32\) keV\(^{45}\). The excited state of \(\mathrm{N}^{14*}\) that arises in this process has such a large energy width because it can decay both by \(\gamma\)-emission and by proton emission. It has an excitation energy \(E^* = 8.06\) MeV and belongs to the type \(1^-\), \(T = 1^{46}\). In 90% of all cases of \(\gamma\)-de-excitation of this state a \(\gamma\)-quantum \(h\nu = 8.06\) MeV appears, and the ground state of \(\mathrm{N}^{14}\) is formed\(^{47}\). These \(\gamma\)-rays can resonantly excite \(\mathrm{N}^{14}\) nuclei. The deficit \(\Delta = \dfrac{E^{*2}}{Mc^2}\) in this case is equal to 5 keV, i.e. much smaller than the energy width of the emitting state. Gamma rays traveling in the direction of the proton beam must have a somewhat increased energy because they are emitted by nuclei that have not yet had time to stop after proton capture. To the width \(\Gamma = 32\) keV there corresponds a mean lifetime of the state \(T = 2\cdot 10^{-20}\) sec. Having captured a proton, the \(\mathrm{C}^{13}\) nuclei move with the velocity

\[ V = \sqrt{\frac{m_0}{m_p}}\,\frac{c}{A}\sqrt{2\frac{E_p}{m_0c^2}} \simeq 8\cdot 10^7 \text{ cm/sec} \]

and before de-excitation travel only \(1.6\cdot 10^{-12}\) cm; this distance is much smaller than the average distance between neighboring atoms in a solid body (\(\sim 10^{-8}\) cm). As a result, the majority of excited nuclei will de-excite before the first collision. In this case the Doppler change in energy is

\[ \Delta E_{\text{Doppl}}=h\nu\cdot \frac{V}{c} =8.06\cdot 10^6\frac{8\cdot 10^7}{3\cdot 10^{10}} =21\ \text{keV} \]

and thus more than covers \(\Delta\). Were it not for the large width \(\Gamma\), resonance would be practically excluded: band \(A\) in Fig. 8 would turn out to be to the right of band \(B\) by a fourfold width. To avoid this, it is necessary to use \(\gamma\)-rays traveling at large angles to the proton beam. Griffiths did exactly this. A proportional counter filled with \(N_2\) was placed to the side of the target. The magnitude of the pulses in this counter was measured. In addition to pulses from Compton electrons and pairs, a well-defined maximum was observed, 30 times exceeding the background, at an energy of \(\sim 550\) keV, which it was natural to ascribe to the reaction \(N^{14}(\gamma,p)C^{13}\), the “reverse” of that in which the \(\gamma\)-rays arose. This maximum was not observed when the counter was irradiated with other \(\gamma\)-rays (\(h\nu=6;\ 12\) or \(17\) MeV); this proved the resonant character of the excitation of the 8.06 MeV state of \(N^{14}\). The success of Griffiths’ experiments is due to the high selectivity of the detector he chose, for the cross section of the process is very small. It may be expected that the direction chosen by Griffiths will be used in many works. (See the annotation to work \(^{51}\) on p. 40.)

It is curious that as early as 1943 Zuber \(^{4}\) attempted to find resonant scattering of \(\gamma\)-rays from the reaction \(B^{11}(p,\gamma)C^{12}\) (\(h\nu=4.3;\ 11.8\) and \(16.6\) MeV) in paraffin (on \(C^{12}\) nuclei). However, the method he chose for detecting the resonance state—measuring the total absorption coefficient of the scattered \(\gamma\)-rays—was not sufficiently sensitive for solving the problem posed.

§ 10. ANGULAR DISTRIBUTION OF RESONANTLY SCATTERED \(\gamma\)-RAYS

If the absorption of a quantum by a nucleus is accompanied by a change in orbital angular momentum (\(\Delta l\ne 0\)), then the excited nucleus that arises will be oriented in space with respect to the direction of incidence of the \(\gamma\)-ray. If emission of the \(\gamma\)-quantum follows quickly after absorption, the radiation will be spatially anisotropic. If, however, the excited state that has arisen lives a long time, external influences will have time to disorient the nucleus, and after this the secondary radiation will already be isotropic.

Let \(A\) be the angle between the directions of the primary and secondary quanta. The correlation function \(\sigma(\theta)\) for instantaneous radiation depends only on the spins of the participating states; it can be calculated completely unambiguously according to the rules of quantum mechanics.

In 1940 Hamilton \(^{48}\) pointed out that the angular distribution of the directions of two successive quanta does not depend on whether both quanta are emitted or the first is absorbed and the second emitted. Therefore the angular distribution of resonantly scattered quanta relative to the primary quanta must be the same as in the successive emission of two quanta between levels with the same quantum characteristics. Thus one can use the functions \(\sigma(\theta)\) calculated for the successive instantaneous emission of two quanta; they can be found in the summary \(^{49}\). They depend only on the spins of the three states connected by the \(\gamma\)-transitions.

In resonant scattering there is a circumstance that substantially simplifies the analysis: the initial and final states are identical. If the spin of this state is known, then \(\sigma(\theta)\) depends only on one quantity—the spin of the intermediate state.

intermediate state. Thus, the study of the curve \(\sigma(\theta)\) is a direct means of determining this spin.

Metzger\(^{28,32,42}\) is so far responsible for the only series of works in this direction. The scheme of his experiments is almost the same as in the experiments of Malmfors; it is shown in Fig. 17. The radiation sources were preparations of \(\mathrm{Au}^{198}\) and \(\mathrm{Tl}^{202}\), heated to a high temperature, and gaseous preparations of \(\mathrm{As}^{72}\) and \(\mathrm{As}^{74}\). The scatterers were alternately rings made of the substance in which resonance scattering was to occur, and of a substance with neighboring \(Z\). The scattering angles could be varied from \(90\) to \(165^\circ\); at smaller angles Rayleigh scattering was too strong, while at larger angles the aperture ratio of the apparatus was considerably reduced. However, even this restricted interval of angles is quite sufficient for an unambiguous determination of the spin of the excited state; in the study of the short-lived \(\mathrm{As}^{72}\), the spin could be determined on the basis of measuring the ratio \(\sigma(90^\circ)/\sigma(121^\circ)\).

Figure 23 shows the results of measurements of angular distributions in the scattering of \(\gamma\)-rays of \(\mathrm{Au}^{198}\) and \(\mathrm{Tl}^{202}\). The decay products of these nuclei are even-even nuclei, which, according to the general empirical rule, have spin 0 in the ground state.

Fig. 23a. Angular distribution of \(\gamma\)-rays of \(\mathrm{Au}^{198}\), resonantly scattered in \(\mathrm{Hg}^{198}\). Solid curves are theoretical for the sequence of spins indicated near the curve.

Fig. 23b. Angular distribution of \(\gamma\)-rays of \(\mathrm{As}^{74}\), resonantly scattered in \(\mathrm{Ge}^{74}\). Solid curves are theoretical for the sequence of spins indicated near the curve.

Therefore the theoretical curves needed for comparison were calculated for the spin sequences \(0-1-0\), \(0-2-0\), and \(0-3-0\). The theoretical curves shown in the figures have already been prepared for comparison with experiment: they already take into account the finite interval of scattering angles used in each experiment.

The experimental points quite unambiguously indicate that in both cases states with spin 2 arise. The same result was obtained for the states 835 keV of \(\mathrm{Ge}^{72}\) and 596 keV of \(\mathrm{Ge}^{74}\).\(^{51}\) In all these cases the result obtained is not new: it was already known from measurements of internal-conversion coefficients and angular correlations\(^{43,50,51}\), and also followed from empirical regularities in the energy and spin of excited states of even-even nuclei.\(^{52,53,54}\)

It should be noted one further conclusion following from Fig. 22: the angular distribution is no more isotropic than it should be for the proces-

0–2. This means that during the lifetime of the 411-keV state of Hg\(^{198}\) (about \(2.2\cdot 10^{-11}\) sec.\(^{23,36}\)) and the 439-keV state of Hg\(^{202}\) (about \(3.4\cdot 10^{-11}\) sec.), external forces do not change the orientation of the nucleus that arose upon absorption of the \(\gamma\)-quantum. The question of how rapidly nuclei lose their orientation has as yet been studied very little\(^{55,56,57}\); cases are known, however, in which the “correlation memory” is lost relatively quickly and the subsequent radiations occur independently of one another\(^{58}\).

The study of deviations from the theoretical angular distribution in resonant scattering, as well as deviations from angular correlation in \(\gamma-\gamma\), \(\beta-\gamma\), \(e^{-}-e^{-}\), and other similar cascades, is the most direct method for investigating the process of “loss of memory.”

§ 11. ON THE POSSIBILITY OF USING THE COMPTON EFFECT FOR OBSERVING RESONANT SCATTERING

In Compton scattering of monochromatic \(\gamma\)-rays, the spectrum of the radiation scattered at a definite angle consists of a relatively narrow line. By selecting the energy of the primary quanta and the scattering angle, one can obtain a line of any energy and, by illuminating a scatterer with it, achieve resonant scattering\(^{64}\). The width of the line is determined mainly by the velocity and binding of the electrons on which the Compton effect occurs. In the scattering of \(\gamma\)-rays with \(h\nu = 1\) MeV by light elements, the half-width of the line is \(\sim 1\) keV\(^{59}\). In real experiments there will be added the width associated with the finiteness of the interval of selected angles. The line will thus turn out to be considerably wider than the absorption line, and this will lead to a decrease in \(\sigma\). Another obstacle is the low intensity of double scattering; it necessitates special detectors with low background.

§ 12. ON THE PROSPECTS FOR USING RESONANT SCATTERING OF \(\gamma\)-RAYS

The phenomenon of resonant scattering of \(\gamma\)-rays is of interest not only in itself as a process occurring with atomic nuclei. It provides experimenters with a powerful means for studying various properties of nuclear states. We shall briefly dwell on some of these possibilities.

a) Determination of the lifetime of rapidly decaying excited states of nuclei. Modern radio engineering makes it possible, in \(\beta-\gamma\) and \(\gamma-\gamma\) coincidence schemes, to measure half-lives down to \(10^{-11}\) sec.\(^{60,61}\); the difficulties increase very rapidly as this limit is approached, and it is hardly likely that, without changing the method, it will be possible to go much farther than \(10^{-11}\) sec. Meanwhile, many nuclear states decay much more rapidly: thus, for example, the 411-keV state of Hg\(^{198}\) has a period of \(2\cdot 10^{-11}\) sec., and all states with higher excitation energy should, as a rule, live still less. Resonant scattering can provide direct assistance here: the shorter the lifetime, the larger the cross section of resonant scattering.

In § 8 an example was given of measuring the lifetime of Cu\(^{63}\), \(T\sim 6\cdot 10^{-13}\) sec.; probably the half-lives of still shorter-lived states will soon be measured.

In this respect resonant scattering has features in common with Coulomb excitation of nuclei. The theoretical description of Coulomb excitation reduces to replacing the variable electric field of a particle flying past the nucleus by an equivalent continuous \(\gamma\)-spectrum. The smaller \(T\) and the larger \(\Gamma\) of the level, the more easily it is excited.

Coulomb excitation of nuclei plays a very large role in the search for and study of collective excited states of nuclei. These states decay tens and hundreds of times faster than single-particle states at the same energy; they therefore have a large natural width and are consequently more readily excited by the Coulomb field of a passing particle. One may expect that resonant scattering of nuclei will also be considerably greater on collective levels than on single-particle ones.

Work on measuring the half-lives of excited states of nuclei began to be carried out on a broad front after the importance of measurements of the lifetimes of excited states for solving primary problems of nuclear physics had been recognized.

Five years ago it was generally accepted that the lifetime of an excited state of a nucleus is determined by the excitation energy, the decay channels, the spin and parity of the initial and final states of the nucleus in each of the channels. It seemed that, having only these data and improved tables for processing them, one could calculate without error the lifetimes of excited states. It turned out that this is not so. It became clear that the structure of the nuclear states participating in the decay plays an essential role. The filling of nuclear shells plays the most important role in this and can change the lifetime by a factor of hundreds.

Fig. 24. Experimental values of \(\lg (T_{\gamma^{2}} A^{4/3} E_{\gamma}^{5})\) for the first excited states of even-even nuclei; sharp deviations from the predictions of the single-particle model of the nucleus are visible—the horizontal line “\(E2\)—single-particle.” Far from filled shells the decay proceeds 100 times faster than according to the theory.

As an example we shall present data relating to a narrow class of nuclear transitions, collected in the work of Sunyar\({}^{60}\). The decay of the first excited state of nuclei with even \(A\) and \(Z\) in the region \(69 < Z < 92\) is considered. It has been established experimentally that in such nuclei the first excited state belongs to the type \(2+\) (spin equal to 2, wave function even), while the ground state belongs to the type \(0+\). The transition between these states is always electric quadrupole; Weisskopf’s theory\({}^{62}\), which does not take into account the internal structure of the state, leads to the following formula for the lifetime of such states:

\[ T = 4.3 \cdot 10^{-9} \cdot A^{-4/3} E^{-5}\ \text{sec.}, \tag{18} \]

where \(A\) is the mass number of the nucleus in which the decay occurs, and \(E\) is the decay energy in MeV; the conversion probability is small and is not taken into account.

Weisskopf’s theory is very approximate, and one should not attach great significance to the numerical coefficient in formula (18). What is essential, however, is that the product \(T \cdot A^{4/3} \cdot E^{5}\), according to this theory, should be the same for all radiators of the type under consideration. Figure 24 shows the values of this product for a number of \(\gamma\)-emitters in the interval

\(A\) from 150 to 200 (only even-even nuclei). The points corresponding to \( \mathrm{Hg}^{198} \) and \( \mathrm{Hg}^{202} \) are based on measurements of resonance scattering in works\(^{27,32}\); the remaining points are based on materials collected by Sanyer. Instead of lying on a single horizontal straight line, as formula (18) requires, the points are arranged along a smooth curve, which indicates that deviations from the value of \(T\) according to (18) reach as much as a hundredfold. Large deviations are also observed for light nuclei. Thus, for example, the half-lives found by Metzger for the 835-keV states of \( \mathrm{Ge}^{72} \) and 596-keV states of \( \mathrm{Ge}^{74} \) are 11 and 14 times smaller than those following from Weisskopf’s formula (18).

The deviations are greatest in nuclei with a strongly unfilled neutron shell; the theory of collective motions in nuclei indicates that these are strongly deformed nuclei with a large quadrupole moment. The excited states of such nuclei decay especially rapidly.

The examples given clearly show the influence of structure on the lifetime of the excited state of a nucleus. One may think that this influence is not limited to the degree of filling of shells. More accurate measurement of lifetimes will probably make it possible to learn new details of nuclear structure.

b) Determination of the sequence of emission of two \(\gamma\)-quanta\(^{59}\). Consider the well-known decay of \( \mathrm{Co}^{60} \): after the emission of \(\beta\)-particles with \(E_{\Gamma}=0.306\ \mathrm{MeV}\), \( \mathrm{Ni}^{60} \) is formed, emitting in succession two quanta \(h\nu_1=1.33\ \mathrm{MeV}\) and \(h\nu_2=1.17\ \mathrm{MeV}\). For the moment let us neglect recoil in \(\beta\)-decay and the thermal motion of the atoms. If the emission of the \(\gamma\)-quanta takes place in the indicated sequence in a gaseous source of sufficiently low density, then the recoil upon emission of the first quantum is sufficient to compensate \(\Delta\) in the resonance scattering of the second quantum by a \( \mathrm{Ni}^{60} \) nucleus; the necessary condition is \(h\nu_1>h\nu_2\). If the emission occurs in the reverse sequence, then the recoil will be insufficient and resonance scattering of the second quantum will not occur.

Analogous cases are encountered in the decay of other isotopes. Schemes with a larger number of successive transformations can also be subjected to analysis.

c) Measurement of the transformation energy in electron capture. Suppose that the emission of a quantum \(h\nu\) occurs after electron capture and emission of a neutrino with energy \(E\). If \(E<h\nu\), then the recoil energy upon emission of the neutrino will be insufficient to compensate \(\Delta\), and resonance scattering will practically not occur. If \(E>h\nu\), then scattering is possible.

The width of the microspectrum of \(\gamma\)-rays following \(K\)-capture is directly proportional to the decay energy; since the area bounded by the microspectrum is equal to 1, its ordinate, and consequently also \(\sigma\), are inversely proportional to the decay energy\(^{32}\) (provided only that the absorption lines are not at the very boundary of the microspectrum; in that case the dependence is more complicated).

d) Study of the laws of slowing down of slow atoms in gases, in solids and liquids. The quantity \(\bar{\sigma}\) is very sensitive to the law of slowing down. Even in gases \(\bar{\sigma}\) should decrease as the pressure is increased. Thus, for example, for a gaseous source \( \mathrm{As}^{74}_{4} \) the quantity \(\bar{\sigma}\) decreases when the pressure becomes equal to atmospheric\(^{42}\). Practically nothing is known about slowing down in solids and liquids.

e) Study of the process of disorientation of nuclei during motion through matter. The angular distribution of resonantly scattered radiation, determined by quantum mechanics, should be observed only if the nuclei do not become disoriented before luminescence. As ...

they collide with other atoms, the angular distribution should gradually become isotropic.

e) Study of the behavior of molecules under the $\beta$-transformation of one of the atoms entering into the molecule. In a $\beta$-transformation 1) the charge of the nucleus changes, 2) recoil arises. Whether these causes will lead to complete or partial breakup of the molecule, whether they will convert it into an ion, whether rotational and vibrational excited states will arise in this process—to all these questions at present one can obtain only approximate answers. Resonance scattering is very sensitive to the velocity acquired by the atom as a result of all the processes; it can therefore be used for their study.

§ 13. INELASTIC RESONANCE SCATTERING OF $\gamma$-RAYS

In all the preceding sections we considered the case when, following excitation of the nucleus, the $\gamma$-quantum carried away almost all the excitation energy $\left(E^*-\dfrac{\Delta}{2}\right)$, and the nucleus remained in the unexcited state. This process is sometimes called elastic resonance scattering.

If the first nuclear level is excited, then there can be no other scattering. But if the second or higher nuclear levels are excited, then during de-excitation cascades may arise and $\gamma$-quanta of considerably lower energy may appear (Fig. 25), leaving the nucleus in an excited state. Such a process is sometimes called inelastic resonance scattering of $\gamma$-rays.

Fig. 25. Occurrence of inelastic resonance scattering of $\gamma$-rays.

The scattering process may be regarded in two stages: first there is resonance excitation of the state $E_2^*$, and then emission of quanta $\gamma_2$. The latter occurs, of course, in competition with the emission of elastically scattered quanta $\gamma_1$.

The formula for the cross section may be written by analogy with (7):

\[ \sigma_{\text{inel. scat.}} = \frac{2I^*+1}{2I_0+1}\, \frac{\lambda^2}{8\pi}\, \frac{\Gamma_{\gamma_1}\Gamma_{\gamma_2}} {(E^*-E)^2+\left(\dfrac{\Gamma}{2}\right)^2}, \]

where $\Gamma_{\gamma_1}$, $\Gamma_{\gamma_2}$ are the radiative widths of the channels $\gamma_1$ and $\gamma_2$ of the level $E_2^*$, and $\Gamma=\Gamma_{\gamma_1}+\Gamma_{\gamma_2}+\Gamma_{\mathrm{dr}}$, where $\Gamma_{\mathrm{dr}}$ denotes the width corresponding to all other nonradiative modes of decay of the level $E_2$.

The angular distribution of inelastically scattered quanta with respect to the primary ones should be described by the usual Hamilton formulas$^{48}$ and others$^{49,64}$; it is determined uniquely by the spins of the three participating states (if there is no disorientation of the nucleus during the emission of the state).

It is not possible in general form to foresee the ratio between elastic and inelastic resonance scattering, expressed by the ratio $\dfrac{\Gamma_{\gamma_1}}{\Gamma_{\gamma_2}}$, since this ratio, first, depends on the spins and parities of all three participating states, and, second, may depend on structural factors, still unknown, that affect the probability of emission of $\gamma$-rays.

Experimental investigations of inelastic resonance scattering are very difficult. In addition to the general difficulties associated with the observation

of relatively weak nuclear scattering against the background of strong electron scattering, a specific complication is added: the \(\gamma\)-rays \(\gamma_2\) have lower energy than \(\gamma_1\) and, consequently, they must be separated from the strong background of \(\gamma\)-quanta arising in the Compton effect. Therefore there are almost no works in which the phenomenon itself has been studied. The phenomenon of inelastic scattering can be observed relatively easily from \(\gamma\)-rays \(\gamma_3\) in those cases where the lifetime of the level \(E_1^*\) is so large that there is time to switch off the flux of primary quanta. This is precisely what occurs when isomeric states are excited by \(\gamma\)-rays, for example, in the above-described experiments of Pontecorvo and Lazar\({}^{14}\) and others,\({}^{15,16,17,18}\) in which an isomeric state of \(\mathrm{In}^{115}\) was excited by \(\gamma\)-rays. Up to the present time 10 more cases of excitation of isomers by \(\gamma\)-rays have been found:

\[ \mathrm{Se}^{77},\ \mathrm{Kr}^{83},\ \mathrm{Sr}^{87},\ \mathrm{Rh}^{103},\ \mathrm{Ag}^{107},\ \mathrm{Ag}^{109},\ \mathrm{In}^{113},\ \mathrm{Lu}^{176},\ \mathrm{Au}^{197},\ \mathrm{Hg}^{199}. \]

However, it is difficult to use isomeric states for investigating the very process of inelastic scattering, since the cascade \(\gamma\)-transitions that lead to the formation of an isomeric state have not been studied. The relative probabilities of cascade transitions depend on the spins, parities\({}^{11}\), and possibly other characteristics of all the participating states; and since many states with high excitation energy (“activation levels”) usually participate in the formation of isomers, calculation of the cascades is at present impracticable.

§ 14. RESONANCE SCATTERING AT HIGH \(\gamma\)-RAY ENERGY

Suppose that a nucleus is irradiated with monochromatic \(\gamma\)-rays of gradually increasing energy. At first, the lower levels of the nucleus will be resonantly excited in succession. As is known, most \(\gamma\)-transitions between the lower levels of nuclei are quadrupole or octupole transitions; dipole transitions are rare. The resonance-scattering cross section is proportional to \(\Gamma_\gamma^2\); this quantity is much larger for dipole transitions than for quadrupole and octupole transitions. Therefore, when the first dipole level is reached (a level from which at least one dipole transition occurs), the resonance scattering will become large. It will be especially large if a dipole transition connects this level with the ground state: here dipole absorption and dipole emission will occur.

So far there have been no experiments in which this phenomenon has been clearly demonstrated. There are several reasons: dipole levels lie high—usually above \(10\) MeV; in this region there are no sources of monochromatic radiation; at such high energies, exceeding the thresholds for nucleon emission, the levels begin to broaden and overlap.

With a further increase in the energy of the \(\gamma\)-quanta, the scattering picture should be complicated by the superposition of effects from different levels.

However, soon, at \(h\nu = 12\)–\(18\) MeV, we enter the region of the “giant resonance.” In this region, under the action of a \(\gamma\)-quantum, collective dipole oscillations are excited in the nucleus: protons oscillate in phase relative to neutrons. The collective character of the giant resonance is evident from the fact that, in nuclei close in mass but distant in structure, it occurs approximately in the same energy region and always has a width of \(4\)–\(6\) MeV.

Whatever its mechanism, the giant resonance in nuclear excitation exists, and it should be expected that it will appear in all processes of de-excitation of the resulting states, including strong resonance scattering of \(\gamma\)-rays, both elastic and inelastic.

Among the works devoted to the search for this scattering\(^{64,65,66}\), the work of Stearns\(^{64}\) deserves special attention.

Stearns observed the scattering of γ-rays (\(h\nu = 14.8\) and \(17.6\) MeV, γ-rays \(\mathrm{Li}+p\)) on a series of elements—Cu, Sn, Pb, Bi. The arrangement of her apparatus is shown in Fig. 26. γ-rays scattered through \(116^\circ \pm 17^\circ\) were detected by a large NaJ(Tl) crystal, shielded by a ring of Geiger counters \(C\), connected in an anticoincidence circuit. A brass screen \(P\) in front of the counters reduced the load from soft γ-rays. Only those pulses were recorded which corresponded to absorption in the crystal of more than \(12.9\) MeV (this threshold could be shifted).

Fig. 26. Scheme of Stearns’ experiments\(^{64}\) on the study of inelastic scattering of hard γ-rays

Stearns found that the differential cross section

\[ \left(\frac{d\sigma}{d\Omega}\right)^{116} \]

1) varies proportionally to \(Z^{(2.5\pm0.5)}\), if the counter is strongly gated (\(90\%\, h\nu > 14.4\) MeV), and proportionally to \(Z^{(2.9\pm0.3)}\), if the counter is gated less strongly (\(90\%\, h\nu > 12.3\) MeV);

2) in absolute magnitude it is \(0.066\ \mathrm{mb}/\mathrm{ster}\) for Cu and \(1.2\ \mathrm{mb}/\mathrm{ster}\) for Pb.

Under the experimental conditions Stearns measured the sum of elastic and inelastic scattering: including inelastic scattering with an energy loss not exceeding \(3.2\) MeV if the counter is strongly gated, and not exceeding \(5.3\) MeV if it is gated less strongly.

Thus, with the less strongly gated counter more inelastically scattered rays are registered, and this increases the power of \(Z\). Therefore the different dependence on \(Z\) found by Stearns indicates a significant role of inelastic scattering at \(h\nu = 17.6\) MeV.

If it is assumed that with a strongly closed counter only elastically scattered radiation is registered, then, adopting\(^{68}\) for it an angular distribution of the type \(P(\varphi)\sim \dfrac{1}{2}(1+\cos^2\varphi)\), one can determine the total cross section of elastic resonance scattering. From Stearns’ data one obtains:

Cu	Sn	Pb
\(\sigma = 0.62\)	\(2.8\)	\(7.4\ \mathrm{mb}\)

In order to estimate these numbers, let us imagine that the giant resonance is simply a broad level. We shall make the calculation for copper. Taking for \(E^*\) and \(\Gamma\) the experimental characteristics of the giant resonance (\(E^* = 17.5\) MeV, \(\Gamma = 6.0\) MeV), and for \(\sigma_{\mathrm{res.\,scatt}}\) the value obtained by Stearns, we find for copper from formula (7) (omitting the factor \(\dfrac{2I^*+1}{2I_0+1}\)):

\[ \frac{\Gamma_\gamma}{\Gamma}=8\cdot 10^{-3}, \]

whence \(\Gamma_\gamma = 48\) keV (the main width of the “level” is particle, i.e. neutron and proton). This value is not in contradiction with generally accepted assumptions concerning the radiative width of dipole transitions. From the Weisskopf formula \(T = 6\cdot 10^{-15} A^{-2/3} E^{-3}\) sec. it follows for our

of the case \(\Gamma_\gamma=90\) keV, i.e. a close value. Approximately the same agreement is also obtained for Sn and Pb. We can approach the estimate of \(\Gamma_\gamma/\Gamma\) in another way.

Calculating the integral cross section of the resonance section

\[ \Sigma \simeq \frac{\pi}{2}\times \text{height}\times \text{half-width} \quad (§5), \]

we obtain for copper about \(6\) mb MeV, which is approximately 120 times smaller than the integral cross section observed experimentally for the reaction \((\gamma,n)\) on the same copper. This gives

\[ \Gamma_\gamma/\Gamma=\frac{1}{120}\simeq 8\cdot 10^{-3}. \]

Since \(\Gamma_n\sim \Gamma\), the former value is obtained for \(\Gamma_\gamma\). Thus, no internal contradictions arise in describing the giant resonance as a resonance caused by a very broad, single dipole level.

The question suggests itself: can the broad giant resonance in the region \(12\)–\(18\) MeV cause additional resonance scattering “in the tail,” in the region of small energies?

We cannot directly calculate this scattering from (7), since this formula is applicable only near the resonance. Levinger\(^{68}\) carried out calculations using more exact formulas and came to the conclusion that the cross section of such resonance scattering is described by the formula

\[ \sigma\sim \frac{8\pi}{3}\,\sigma_0\left(\frac{m_0}{M}\right)^2 Z^4\left(\frac{h\nu}{E^*}\right)^4 . \]

(\(E\) is the energy of the giant resonance).

All factors except the last are the cross section of Thomson scattering by a nucleus, amounting for copper to \(\sim 3.4\cdot 10^{-29}\ \text{cm}^2/\text{nucleus}\). The last factor is considerably less than 1, and consequently the resonance scattering “in the tail” is still smaller; true, it is coherent with the Thomson scattering, and therefore amplitudes, not intensities, must be added. This somewhat increases its role, but nevertheless it remains for the present beyond the limits of observational possibility.

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46. N. Burgov, Yu. Terekhov, ibid., p. 41.
47. C. Swann, F. Metzger, Physica XXII, 1138 (1956). The $\gamma$ rays of $O^{16*}$, arising in the reaction $F^{19}(P,\alpha)O^{16*}$, were used for resonant scattering on $O^{16}$ nuclei (in ordinary water). Experiments carried out by the resonant-scattering method (see the article on experiments with $\mathrm{CoCl}_2$) showed that the mean lifetime of the 6.9 and 7.1 MeV states of $O^{16}$ lies within the range $2 \cdot 10^{-15}$—$2 \cdot 10^{-14}$ sec.