Bremsstrahlung in alpha-Decay Reexamined
H. Boie, H. Scheit, U. D. Jentschura, F. Köck, M. Lauer, A. I. Milstein, I. S. Terekhov, D. Schwalm
aa r X i v : . [ nu c l - e x ] J u l Bremsstrahlung in α Decay Reexamined
H. Boie, H. Scheit, U. D. Jentschura, F. K¨ock, M. Lauer, A. I. Milstein, I. S. Terekhov, and D. Schwalm ∗ Max-Planck-Institut f¨ur Kernphysik, D-69117 Heidelberg, Germany Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
Abstract
A high-statistics measurement of bremsstrahlung emitted in the α decay of Po has beenperformed, which allows to follow the photon spectra up to energies of ∼
500 keV. The measureddifferential emission probability is in good agreement with our theoretical results obtained withinthe quasi classical approximation as well as with the exact quantum mechanical calculation. It isshown that due to the small effective electric dipole charge of the radiating system a significantinterference between the electric dipole and quadrupole contributions occurs, which is alteringsubstantially the angular correlation between the α particle and the emitted photon. PACS numbers: 23.60.+e, 27.80.+w, 41.60.-m α decay of an atomic nucleus is the archetypal quantum mechanical process, andso is the bremsstrahlung accompanied α decay. It is therefore somewhat surprising thatonly ten years ago the first fully quantum mechanical calculation of the latter process hasbeen performed, using first-order perturbation theory and the dipole approximation for thephoton field [1]. On the other hand, it is known that the α decay of a heavy nucleus and theradiation that accompanies this decay can be treated in the quasi-classical approximationas well, the applicability of this approximation being provided by the large value of theSommerfeld parameter η [2, 3], which amounts to e.g. η = 22 for the α decay of Po. Inall theoretical approaches the matrix element of bremsstrahlung incorporates contributionsof the classically allowed and classically forbidden (tunneling) region. The relative contribu-tion of the tunneling region is not small in general and can be interpreted as bremsstrahlungat tunneling. However, such an interpretation can only have a restricted meaning as thewavelength of the photon is much larger than the width of the tunneling region and evenlarger than the main classical acceleration region; it is therefore not possible to identify ex-perimentally the region where the photon was emitted. Nevertheless, the issue of tunnelingduring the emission process was widely discussed [1, 2, 3, 4, 5, 6, 7]. The authors used dif-ferent theoretical approaches leading to partly conflicting conclusions as to the contributionof the tunneling process to the bremsstrahlung, but - more seriously - also with regard tothe energy dependent emission probabilities.While the theoretical interest in the bremsstrahlung accompanied α decay was stirredup by an experiment published in 1994 [8], this and later experimental attempts [9, 10]to observe these rare decays produced conflicting results and did not reach the sensitivityto allow for a serious test of the various theoretical predictions concerning the emissionprobabilities for γ energies above E γ ∼
200 keV. In the present paper we report on thefirst high-statistics measurement of bremsstrahlung in the α decay of Po, where we havebeen able to observe the photon spectra up to E γ ∼
500 keV. Taking into account theinterference between the electric dipole and quadrupole amplitudes, which we derived withinthe framework of a refined quasi-classical approximation [11], we find good agreement of ourmeasured γ emission probabilities with those calculated in our quasi-classical approach aswell as with the quantum mechanical prediction of Ref. [1].The main experimental challenge is the very low emission rate for bremsstrahlung pho-tons. Even with a rather strong α source of ∼
100 kBq the emission rate is only of the order2 teel lamellae
Po sourcesSilicon stripdetectorsAluminumvacuum chamber Germanium clusterdetector5 cm copper platecooled to -20 o C FIG. 1: (color online) Cross section of the experimental setup. of one per day in the 300–400 keV energy range, i.e. in only one out of 10 α decays aphoton with an energy within that range will be emitted. Only by measuring the α parti-cles in coincidence with the bremsstrahlung photons and by identifying the bremsstrahlungphotons by requiring energy balance between the α energy and the photon one can thereforehope to sufficiently suppress randoms due to the copious room background. After a carefulevaluation of possible α emitters, Po, already used in the work of Ref. [9], was felt to be themost promising choice for such a measurement. It decays with a halflife of t / = 138 daysmainly to the ground state of the stable daughter nuclide Pb ( Q α = 5 .
407 MeV), withonly a small fraction of 1 . × − proceeding through the first exited J π = 2 + stateat an excitation energy of 803 keV [12]. While no other γ rays are emitted by the source,the weak 803 keV branch constitutes a convenient calibration point for the overall detectionefficiency reached in the experiment.The experimental setup used in the present work is shown in Fig. 1. Two Po α sourcesare placed at the bottom of a common vacuum chamber and are viewed by two segmentedSilicon detectors, each placed about 30 mm above the source to measure the energy ofthe α particles. Directly below the center of the vacuum chamber an efficient high-purityGermanium triple cluster detector of the MINIBALL design [13] was placed to record theemitted bremsstrahlung photons.The source material was evenly spread on two 0.2 mm thick circular Ni foils with adiameter of 16 mm, which were mounted on Aluminum disks of 0.5 mm thickness each. Bydistributing the source material the α energy loss in the material was minimized; moreover,sputtering of source material due to the recoil of a nearby α decay was avoided. The arealuniformity of the activity (100 kBq per source) within the active area of the source was3ested by autoradiography; no intensity variations could be discerned.The α particles were detected by two 5 × Silicon detectors, which were electricallysegmented into 16 strips each. The α particles were incident on the unsegmented side of thedetector in order to avoid events with incomplete charge collection occurring in the inter-strip region. Both detectors were mounted on a copper plate cooled to − ◦ C to improve theenergy resolution and to reduce damage of the Si detectors due to the implanted α particles(about 10 /cm at the end of experiment). The energy resolution was on the order of30–35 keV (FWHM) and deteriorated only for a few strips up to 45 keV at the end of theproduction run. The direct path between the left source and the right Si detector (and viceversa) was mechanically blocked to avoid large angles of emission and incidence. Moreover,events with two responding strips were rejected in the off-line analysis, which considerablyimproved the low energy tail of the α peaks. Typical counting rates were ∼ . γ - α timeresolution (FWHM) to 29 ns at E γ = 100 keV and 15 ns at E γ = 500 keV. The γ energywas determined by adding the measured core energies of all three crystals. After carefullyshielding the setup with copper and lead, the counting rate was as low as ∼
20 Hz for athreshold at about 40 keV.In view of the spreaded source and the close source-detector geometry, detailed simula-tions of the experimental setup were performed to determine the response function of thecluster detector as well as the detection efficiency of the setup as a function of E γ and of theangle ϑ between the direction of the α particle and the bremsstrahlung quanta [14]. Thesimulated absolute γ full-energy peak efficiencies were compared to measurements performedwith radioactive sources and to the result deduced from the 803 keV branch of the Podecay; their accuracies were found to be better than 4% for γ energies above 200 keV butto deteriorate slightly up to 9% at 100 keV. The absolute γ full-energy peak-efficiency at4
100 200 300 400 500 600 700 800 900440046004800500052005400 110 E a ( k e V ) C oun t s / ( k e V · k e V ) E g (keV) FIG. 2: (Color online) Scatter plot showing the energy of the α particle versus the energy of the γ ray, detected coincident within a time window of ±
50 ns. The bremsstrahlung events can beclearly discerned along the diagonal line starting at E α = E α = 5304 keV.
803 keV was determined to be 8 . . × α particles being recorded. At the same time about6 × γ rays have been detected out of which e.g. only about 150 are expected to be dueto bremsstrahlung events in the γ energy region above 300 keV.The α - γ coincidence matrix displaying the measured α particle energy versus the γ -rayenergy is shown in Fig. 2. The upper horizontal band corresponds to α particles, which weredetected with their full energy of E α = 5304 keV in random coincidence with a backgroundphoton. The lower horizontal band terminating at 803 keV γ -ray energy is caused by theresponse of the Ge detector to the 803 keV γ -rays, emitted in coincidence with α particlesof 4517 keV leading to the first excited state of Pb. The events at γ energies aroundbetween 70 keV and 85 keV for α energies below 5304 keV correspond predominantly to PbX-rays emitted after the knockout of an electron by the escaping α particle [15]. Finally,on the diagonal line with E α + (206 / E γ = const as required by energy and momentumconservation, bremsstrahlung events can be clearly observed up to γ energies in excess of400 keV.To determine the differential bremsstrahlung emission probabilities dP/dE γ coincident α - γ events with γ energies within 20 keV up to 100 keV broad gates were projected alongthe diagonal line by plotting them as a function of E p = E α + (206 / E γ . For each γ gate the coincidence time window ∆ t was individually adjusted to include all counts5 p = E a + / E g (keV) C oun t s / k e V
190 keV £ E g < 210 keV FIG. 3: Projected data and the result of a least-square fit (solid curve) for the γ energy bin of190 keV ≤ E γ <
210 keV. The peak centered at 5304 keV corresponds to the full-energy peak ofphotons from the bremsstrahlung accompanied α decay, while the right structures is due to random α - γ coincidences. within ± . σ t ( E γ ) around the centroid of the time peak with σ t denoting its variance.In the projected spectra the bremsstrahlung events are expected to show up in a sharppeak around E p = E α = 5304 keV independent of the width of the γ energy gate. As anexample, the projected energy spectrum for the 190 keV ≤ E γ <
210 keV bin is shown inFig. 3: Riding on the low energy tail of the random α - γ line the bremsstrahlung eventsare clearly born out. The solid curve corresponds to the result of a least-squares fit of thedata, where only the intensity and central energy of the bremsstrahlung peak was varied.The line-shape and intensity of the random α - γ peak (dashed line) was determined fromthe corresponding projected energy spectrum of the random α - γ matrix, scaled by the ratioof the time windows. As the random matrix has ∼
10 times more statistics, the tail underthe bremstrahlung line could be determined to better than 5% for all γ energy gates. Theshaded area reflects the Compton distribution caused by bremsstrahlung events with higherenergies. Its intensity relative to the full energy events and its shape was determined fromthe simulation; the accuracy of this contribution is mainly determined by the intensities ofthe bremsstrahlungs peaks of the next higher γ gates and is estimated to be ∼
10% for thebin shown in Fig. 3.From the intensity of the bremsstrahlung line the (solid-angle integrated) differential emis-sion probability dP ( E γ ) /dE γ can be determined if the total number of α particles detected inthe Si detectors, N α , and the probability to detect a bremsstrahlung photon of energy E γ isknown. While N α can be readily determined from the down-scaled single spectrum recorded6uring the production run to be N α = 4 . × , the detection efficiency requires someextra considerations as the photons are preferentially detected at backward angles with re-spect to the direction of the α particle. Hence the α - γ angular correlation must be takeninto account to extract the differential emission probability. Usually bremsstrahlung is as-sumed to be pure E α particle and the residual daughter nucleus Pb, which have rather similar charge to mass ratios suchthat the effective dipole charge amounts only to Z eff E = µ ( z/m − Z/M ) = 0 .
40, where z, m and
Z, M denote the charge and mass of the α and the daughter nucleus, respectively, and µ the reduced mass, while the effective quadrupole charge Z eff E = µ ( z/m + Z/M ) = 1 . E E E α - γ angular correlation and thus thedetection sensitivity of our setup.We used a refined version of the quasi-classical approach to the bremsstrahlung emis-sion process [11] to study this question in more detail, and find indeed a substantial E /E E γ energies. Including only the interference term the angular correlation can be expressedby dP ( ϑ ) /d Ω ∝ sin ϑ (1 + 2 χ ( E γ ) cos ϑ ), where χ ( E γ ) is proportional to the ratio of thequadrupole to the dipole matrix element. In leading order in 1 /η we find that χ ( E γ ) ap-proaches zero for E γ → E γ to take e.g. values of +0.09 at 100 keVup to +0.22 at 500 keV. The influence of the E ϑ for bremsstrahlung photons of 100 keV and 500 keV in comparisonto a pure dipole emission characteristic ( χ = 0). Multiplied with the acceptance of our setupand integrated over the solid angle, the E χ we can estimate the uncertainty to thedetection efficiency caused by using only the leading order term in the angular correlation7
0 20 40 60 80 100 120 140 160 180 e ff i c i e n c y ( a r b . un it s ) d P ( J ) / d W emission angle J ( ) ·
100 keV500 keVdipole500 keV100 keV
FIG. 4: Calculated, normalized α - γ correlations dP/d Ω for bremsstrahlung photons of 100 and500 keV in comparison to a pure dipole correlation. The increasing deviation from the dipolecharacteristic is due to the E to be less than 3%.The resulting solid-angle integrated and efficiency corrected differential emission proba-bilities dP ( E γ ) /dE γ are displayed in Fig. 5 by the solid points. The 1 σ errors shown comprisethe statistical and systematic uncertainties and are smaller than the point size for γ energiesbelow 250 keV; they amount to e.g. σ sta = 2% and σ sys = 8% at < E γ > = 139 keV, and σ sta = 19% and σ sys = 5% at < E γ > = 373 keV. Also shown are the earlier results obtainedby Kasagi et al. [9]. Note, that external bremsstrahlung contributions, which stem from theslowing down of the α particles in the Si detector material, are several orders of magnitudesmaller than the measured probabilities.In Fig. 5 our data are also compared with the predictions of Papenbrock and Bertsch [1]and of our quasi-classical approach [11]; the two theoretical approaches actually agree witheach other to better than 2% as shown in [11] and are thus indistinguishable on the scale ofFig. 5. Overall, very good agreement between theory and experiment is observed, however,small deviations of up to 20% are encountered at energies below 200 keV, which are alsosupported by the data of Kasagi et al. [9]. It remains to be seen if these deviations can betraced back to the use of the potential model so far employed in all theoretical calculations todescribe the interaction of the α particle and the daughter nucleus at distances of the orderof the nuclear radius. The present high precision data clearly demonstrates the failure of aclassical Coulomb acceleration calculation (see e.g. [9, 14]) to describe the bremsstrahlung8
0 100 200 300 400 500 600 Po present experimentKasagi et al. E g (keV) d P ( E g ) / d E g ( k e V - ) -12 -11 -10 - - FIG. 5: Differential Bremsstrahlung emission probability dP ( E γ ) /dE γ for the decay of Po (solidpoints: present work, open symbols: Kasagi et al. [9]). The solid curve reflects the calculation ofPapenbrock and Bertsch [1] as well as the result of our quasi-classical calculation [11]. The resultof a classical Coulomb acceleration calculation is shown by the dashed line. emission in α decay, and rules out theoretical suggestions put forward by the authors ofRefs. [6, 7, 9].U.D.J. acknowledges support from the Deutsche Forschungsgemeinschaft (Heisenbergprogram) and D.S. support by a Joseph Meyerhoff Visiting Professorship granted by theWeizmann Institute of Science. A.I.M. and I.S.T. gratefully acknowledge the Max-Planck-Institute for Nuclear Physics, Heidelberg, for warm hospitality and support. The work wasalso supported by RFBR Grant No. 03-02-16510. ∗ Present affiliation: Department of Particle Physics, Weizmann Institute of Science, Rehovot,Israel[1] T. Papenbrock and G. F. Bertsch, Phys. Rev. Lett. , 4141 (1998).[2] M. I. Dyakonov and I. V. Gornyi, Phys. Rev. Lett. , 3542 (1996).[3] M. I. Dyakonov, Phys. Rev. C , 037602 (1999).[4] N. Takigawa et al. , Phys. Rev. C , R593 (1999).[5] E. V. Tkalya, Phys. Rev. C , 054612 (1999).[6] C. A. Bertulani, D. T. de Paula, and V. G. Zelevinsky, Phys. Rev. C , 031602(R) (1999).[7] S. P. Maydanyuk and V. S. Olkhovsky, Prog. Theor. Phys. , 203 (2003), and Eur. Phys.J. A , 283 (2006).
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