Corrections of alpha- and proton-decay energies in implantation experiments

Energies from alpha- and proton-decay experiments yield information of capital importance for deriving the atomic masses of superheavy and exotic nuclides. We present a procedure to correct the published decay energies in case the recoiling daughter nuclides were not considered properly in implantation experiments. A program has been developed based on Lindhard's integral theory, which can accurately predict the energy deposition of heavy atomic projectiles in matter.


Introduction
The study of different decay modes reveals important nuclear structure information. In particularly, α decay and proton decay are two unique tools to explore the most proton-rich atomic nuclei [1,2]. According to the latest Atomic Mass Evaluation (AME) [3], around 65% of the input data in the mass range A > 200 result from αdecay experiments. In lighter mass regions there are a large number of proton-decay data which share many similarities with α-decay data. Energies from α and proton decay yield information of capital importance for deriving mass values. There are four major experimental approaches for α-decay measurements: The first one uses a magnetic spectrograph [4], from which α-kinetic energies are determined by direct measurements of the orbit diameters and the magnetic induction field. All α-energy standards use this method. The second one uses the scintillating bolometer technique which detects the total α-decay energy at temperatures below 100 mK [5]. In the third method the nuclide of interest is implanted into a foil and the α particle is detected by surrounding Si detectors [6]. Last but not least the radioactive species, which are produced in a nuclear reaction are directly implanted into a Si detector: e.g. a double-sided silicon-strip detector (DSSD) or a resistive-strip detector [7]. The first three methods measure either the pure α-particle energy or the total αdecay energy, while the implantation method detects the α (or proton) particle and the heavy recoil daughter nuclide in coincidence. The knowledge of the behaviour of the recoil nuclide is crucial for obtaining the accurate decayenergy value.

Energy calibration
In the α-decay implantation in detector experiments, authors often make the simple assumption that only the αparticle energy is measured in the detector while in the e-mail: huang@csnsm.in2p3.fr proton decay, it is often considered that both the proton and the heavy recoil are detected at the same time but neither of these statements is correct: α-particles and protons with energies of a few MeV have almost 100% detection efficiency, which is not the case for the heavy species.
Suppose there are three equidistant lines in an α-decay spectrum (see Fig. 1). Two well-known α-energy activities line-1 (with E(α 1 ) = 5000 keV) and line-2 (with E(α 2 ) = 5200 keV) are used as calibrants and line-3 is assigned to the unknown nuclide. If the detector does not detect the recoiling nuclide as in Fig. 1 (a), then what is measured would be the α-particle energy and E(α 3 ) = 5400 keV is easily obtained. In the other extreme case, when the detector measures all the energy of the recoiling ion, then the energy scale will change as in Fig. 1 (b). If line-1 and line-2 correspond to a nuclide of mass number A = 150, the new scales will change to Q α (line-1) = 5137 keV and Q α (line-2) = 5342 keV based on the simple relation: where M is the mass number of the parent nuclide and M 4He is the mass number of helium-4. In this case we measure the α-decay energy Q α and obtain Q α (line-3) = 5547 keV. If line-3 corresponds to a nuclide of mass number A = 150, its energy E α is deduced to be 5399 keV according to the transformation of Eq. 1, which is 1 keV smaller than the value obtained from Fig. 1 (a). However, if line-3 corresponds to a nuclide with a different mass number for example, A = 200, E α will increase from 5400 keV to 5436 keV, which is already off by 36 keV. Moreover the detector is not 100% sensitive to the recoiling nuclide and this more relativistic case will be developed in the next section.

Detection efficiency
The recoiling ions lose their energies in the Si detector in two ways: excitation and ionization of the electrons of the atoms (electronic process), or collision with nuclei of the atoms (nuclear process). The electronic process produces a signal in the detector, while the nuclear process does not. Knowledge of both processes is important for implantation α-decay and proton-decay experiments where the heavy recoil is detected simultaneously with the light particle. In 1963 Lindhard et al. [8] derived a theory to describe these processes, from which the detection efficiency K was defined as: whereη R is the part of the recoiling energy that is effectively detected in the detector, E R is the total recoiling energy, is called the "dimensionless reduced energy" related to E R , k is a coefficient related to the mass number and the atomic number of the recoil nuclide and the target nuclide, g( ) is a semi-empirical function (for more details please refer to Ref [8]). This theory was derived to predict the detected energy of heavy atomic projectiles in matter and the agreement between calculations and experiments data is remarkable [9,10]. Fig. 2 shows the calculations of the detection efficiency K for different nuclides based on Lindhard's theory. For light nuclides (e.g. 20 Ne and 40 Ca), the detection efficiencies increase rapidly as their energies increase. For intermediate (e.g. 60 Zn and 100 Sn) and heavy nuclides (e.g. 150 Yb and 210 Th), the detection efficiencies increase much more slowly than those of the light nuclides. For α particles and protons with energies larger than 1 MeV, both detection efficiencies can be considered to be 100%. For the implantation method where both the energies of the emitted particles and a part of the heavy recoil are detected, one needs to consider properly the energy loss of the heavy recoil in the detector. Some experimentalists have already noticed this effect and made the correction for their results [11][12][13]. In the following we come up with a concept about how to treat the calibration line and make a correction to the published experimental result, when the partial recoiling effect was not taken into account.
Here we take α decay as an example. If we consider the recoiling energy, the new scale should be adjusted to: where E d is the total detected energy, E α is the kinetic energy of the α particle, E R is the recoiling energy and K is the detection efficiency for the recoil nuclide at energy E R . It is E d that should be used in the energy calibration rather than E α . Also the recoiling energy can be expressed as: where M is the mass number of the mother nuclide. Combining Eq. 3 and Eq. 4, the pure α-particle energy can be obtained: For proton-decay experiments where Q p is often used in the calibration (as one considers erroneously that the energies of the proton and of the heavy recoil nuclide are fully detected at the same time), one can obtain a similar relation as Eq. 3: where E p is the proton energy and for the proton decay. Combining Eq. 6, 7 and 8, one can obtain: In the next section, we will illustrate how to make the correction for some experimental results.

255 Lr m (α)
In Ref [14], the detector was calibrated using the wellknown α-particle energy 7923(4) keV of 216 Th [16]. The recoiling energy of the daughter nuclide 212 Ra is calculated as 7923 * 4/212 ≈ 150 keV and at this energy the detection efficiency K is 29.12%. The calibration line of 216 Th should be adjusted to E d ( 216 Th) = 7923 + 150 * 0.2912 = 7967 keV. In the α-decay spectrum, the αparticle energy of 255 Lr m is 8371 keV, from which the detected energy of 255 Lr m can be deduced as E d ( 255 Lr) = 7967 * 8371/7923 = 8417 keV. The recoiling energy of the α-decay daughter nuclide 251 Md can be calculated approximately as 8417 * 4/255 ≈ 131 keV and at this energy, its detection efficiency is 29.08%. According to the Eq. 5, the pure α-particle energy of 255 Lr m is calculated to be 8378 keV. The difference between the published value and the corrected value is 7(10) keV. The same routine can be applied to the α-decay energy of the 255 Lr ground state.

69 Kr(βp)
In Ref [15], the β-delayed proton-decay energy of 69 Kr was determined to be 2939(22) keV using known βdelayed proton decay energies of 806, 1679, 2692 keV for 20 Mg and 1320, 2400, 2830, 3020, 3650 keV for 23 Si. The authors assumed (erroneously) that the recoil energy would be fully recorded at the same time [17]. As one can see from Fig. 2 the detection efficiency for the intermediate nuclide e.g. 60 Zn, is between 30%∼40% and its neighbouring nuclides show similar behaviour. The recoiling energy of the β-delayed proton-decay 23 Si at 3020 keV is 3020/23 ≈ 131 keV and the detection efficiency for the decay daughter nuclide 22 Mg is 59.75%. The effectively detected energy of this calibration line is 2967 keV according to Eq. 6. The detected energy of β-delayed proton-decay nuclide 69 Kr is deduced to be 2967 * 2939/3020 ≈ 2887 keV. The detection efficiency of the daughter nuclide 68 Se is 30.79% at the corresponding recoiling energy. Applying Eq. 9, the β-delayed proton decay energy of 69 Kr is calculated to be 2916 keV. The difference between the corrected value and the published one is 23(22) keV, which exceeds 1σ.
From the two examples discussed above, we demonstrated that the recoiling effect should not be ignored. In Ref [13], the detection efficiency K was assumed to be 0.28 and was applied to all the calibration lines and the nuclide of interest. It is reasonable to use K = 0.28 universally in this case as one can see from Fig. 2 that K becomes almost constant for heavy nuclides. For light nuclides, K differs quite a lot (59.75% for 22 Mg and 30.79% for 68 Se) and should be treated differently.

Conclusion
As the implantation method is widely used for decay experiments, the effect of the recoil nuclide should be carefully taken into account. Lindhard's theory predicts quite well the energy deposition of heavy nuclides in matter and it bas been proven to be reliable by Ref [9,10]. We propose a way to correct the result if the recoiling effect was not considered in the energy calibration. Here we strongly recommend that the authors specify in the publication how they treat the recoil nuclide in the experiment. Our next step will be to reexamine all the precise alpha-and protondecay energy data and make the required corrections when necessary.