Two-photon decay in gold atoms

We have measured the energy differential transition probabilities for the two-photon decay of $K$ vacancies in gold atoms (nuclear charge $Z=79$). This is the heaviest atom for which this information has been obtained, and so is most sensitive to relativistic effects. The experiment determined the shape of the continuum radiation for the transitions $2s\to 1s$, $3s\to 1s$, $3d\to 1s$, and $(4s+4d)\to 1s$ at an emission pair opening angle $\theta =\pi ∕2$. Our results for $3d\to 1s$ and $(4s+4d)\to 1s$ extend to energies above and below the region of the intermediate state resonances. No relativistic calculations exist for Au, so we compare with calculations by Mu and Crasemann and Tong et al. for $\mathrm{Ag}(Z=47)$ and $\mathrm{Xe}(Z=54)$. For equal-energy, back-to-back two-photon decay, the calculations show an increase in transition probability with $Z$ for the $2s\to 1s$ and $3d\to 1s$ transitions. In contrast, our data, at $Z=79$, corrected for the angular distribution, give a smaller transition probability than the lower-$Z$ experimental results of Ilakovac et al. and Mokler et al. for Ag and Xe. The shapes of the two-photon continua in our data are in general agreement with theory except that we find anomalously high values for the differential two-photon transition probability for the $3s\to 1s$ transition near $y=0.35$, where $y$ is the fraction of the transition energy carried by the lower-energy photon.

Figure 1

Schematic diagram of the interaction region. Two germanium solid-state detectors measure two-photon decays with opening angles near ${90}^{\circ }$. Cross-talk shields prevent coincidences caused by photons scattering between the detectors.

Figure 2

Log plot of the spectrum recorded in detector 1. The feature near $90\mathrm{keV}$ is due to Rayleigh and Compton scattering of the incident x-ray beam. The four peaks below $20\mathrm{keV}$ are $\mathrm{Au}L\mathrm{x}$ rays, and the four prominent peaks between 60 and $80\mathrm{keV}$ are the $K{\alpha }_2$, $K{\alpha }_1$, $K{\beta }_{1,3}$, and $K{\beta }_2$ x rays, respectively.

Figure 3

Coincidence intensity after subtraction of random coincidences. The data are plotted as a function of the energies $E_1$ and $E_2$ deposited in detectors 1 and 2 with $90\mathrm{keV}$ x rays incident on a $1\mathrm{mg}∕{\mathrm{cm}}^2$ Au target. The contrast was adjusted to emphasize two-photon decays, which appear as diagonal lines near the center of the figure. The double arrow labeled $\mathrm{\Delta }E$ shows the energy cut for Fig. 4. The arrows on the right indicate the centers of the windows corresponding to panels (a), (b), (c), and (d) of Fig. 5.

Figure 4

The histogram is the sum-energy spectrum of coincidences between the two detectors with the condition that the lower-energy photon $E_{\mathrm{low}}$ lie between 22 and $33\mathrm{keV}$ (see double arrow in the upper part of Fig. 3). Random coincidences have been subtracted. The dashed curves show fits to the background and four peaks corresponding to the $2s\to 1s$, $3s\to 1s$, $3d\to 1s$, and $(4s+4d)\to 1s$, two-photon transitions. The heavy solid line is the sum of the components of the fit. The arrow shows the energy where a $K{\alpha }_1$ line caused by cross-talk would occur.

Figure 5

Sum-energy spectra for coincidences between the two detectors with a condition that the lower energy photon be in a window centered at (a) $35.7\mathrm{keV}$, (b) $27.7\mathrm{keV}$, (c) $19.7\mathrm{keV}$, and (d) $7.5\mathrm{keV}$ (see arrows in Fig. 3). Random coincidences have been subtracted. The window widths for (a), (b), and (c) are $4\mathrm{keV}$, while that for (d) is $0.4\mathrm{keV}$. The arrows in (b) show the positions of the $2s\to 1s$, $3s\to 1s$, $3d\to 1s$, and $(4s+4d)\to 1s$ two-photon transitions.

Figure 6

Solid circles with error bars are the experimental results for the relative differential transition probability $P$ [Eq. (5)] as a function of $E∕E_0$ for (a) $3s\to 1s$ and (b) $2s\to 1s$ two-photon decays at ${90}^{\circ }$ opening angle. $E$ is the energy of the lower-energy photon of the coincident pair. The solid lines are the Xe theoretical results of Mu and Crasemann [18] for ${180}^{\circ }$ divided by 2 to correct for angular distribution (see text). The open diamonds are the Ag theory of Mu and Crasemann [16] and the open boxes that of Tong et al. [15], both for Ag atoms at ${90}^{\circ }$ opening angle. The dashed lines give the positions of the major $L$ x-ray groups ($L\ell$, $L\alpha$, $L\beta$, $L\gamma$).

Figure 7

Solid circles with error bars are the experimental results for the relative differential transition probability $P$ [Eq. (5)] at $90°$ opening angle as a function of $y=E∕E_0$ for (a) the sum of $4s\to 1s$ and $4d\to 1s$ two-photon transitions, and (b) $3d\to 1s$ two-photon transition. $E$ is the energy of the lower-energy photon of the coincident pair. The solid lines in both panels are the theoretical results of Mu and Crasemann [18] and the open triangles at $y=0.5$ are those of Tong et al. [15] both for Xe atoms at $180°$ but corrected here to apply to $90°$ for comparison to our data (see text). In (a), the dashed line and dashed open triangle (at $y=0.5$) are, respectively, Mu and Crasemann and Tong et al.’s results for the Xe $4s\to 1s$ transition also corrected for to apply to $90°$ opening angle. The $4s\to 1s$ calculations have been multiplied by 5 for better visualization. In (b), the open diamonds are the theory of Mu and Crasemann [16] and the open boxes that of Tong et al. [15], both for Ag atoms at $90°$ opening angle.

Figure 8

Solid circles with error bars are the experimental results for the relative differential transition probability $P$ [Eq. (5)] at $90°$ opening angle as a function of $y=E∕E_0$ in the region of the cascade resonances, for (a) the sum of $4s\to 1s$ and $4d\to 1s$ two-photon transitions, and (b) the $3d\to 1s$ two-photon transition. In (b), the open diamonds are the theory of Mu and Crasemann [16] and the open boxes that of Tong et al. [15], both for Ag atoms at $90°$ opening angle. The dashed lines show the positions of the cascade resonances through the $2{}^2{P}_{1∕2}$ and $2{}^2{P}_{3∕2}$ levels. Note the different vertical scales.

Figure 9

Theoretical and experimental results for $P$ [Eq. (5)] as a function of the nuclear charge $Z$. Experimental results for $E_1=E_2$ and $\theta =\pi$ are shown as points with error bars for (a) $2s\to 1s$ and (b) $3d\to 1s$. Solid triangles are results by Ilakovac et al. [25, 26]. The solid circle at $Z=47$ in (a) is the silver result of Mokler et al. [28, 38]. The solid circles at $Z=79$ are the results of the present experiment. Our data have been extrapolated to $180°$ [in (a) we multiplied by 2 and in (b) we multiplied by $14∕13$, as described in the text]. The solid curves show the theoretical calculations by Tong et al. [15] and the dashed curves are the theoretical calculations by Mu and Crasemann [16, 18] for $E_1=E_2$ and $\theta =\pi$.