Prospects for measuring the 229Th isomer energy using a metallic magnetic microcalorimeter
G. A. Kazakov, V. Schauer, J. Schwestka, S. P. Stellmer, J. H. Sterba, A. Fleischmann, L. Gastaldo, A. Pabinger, C. Enss, T. Schumm
aa r X i v : . [ phy s i c s . a t o m - ph ] A p r Prospects for measuring the
Th isomer energy usinga metallic magnetic microcalorimeter
G. A. Kazakov a,b, ∗ , V. Schauer b , J. Schwestka b , S. P. Stellmer b , J. H. Sterba b ,A. Fleischmann c , L. Gastaldo c , A. Pabinger c , C. Enss c , T. Schumm b , a Wolfgang Pauli Institute, Univ. Wien - UZA 4 Nordbergstrasse 15, A 1090, Vienna,Austria b Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien,Stadionallee 2, 1020, Vienna, Austria c Kirchhoff-Institute for Physics, Heidelberg University, INF 227, 69120 Heidelberg,Germany
Abstract
The Thorium-229 isotope features a nuclear isomer state with an extremely lowenergy. The currently most accepted energy value, 7 . ± . γ -spectrum following the α -decay of Uranium-233, corresponding to thedecay into the ground and isomer state, allows to measure the isomer transitionenergy without additional theoretical input parameters, and increase the energyaccuracy. We study the possibility of resolving the 29.18 keV line as a doubletand the dependence of the attainable precision of the energy measurement onthe signal and background count rates and the instrumental resolution. Keywords:
Thorium-229, isomer energy, gamma spectroscopy, design ofexperiment ✩ ∗ Corresponding author
Email address: [email protected] (G. A. Kazakov)
Preprint submitted to Nucl. Instr. and Meth. A October 29, 2018 . Introduction
The nuclear level scheme of the Thorium-229 isotope is expected to featurea long-lived isomer state, m Th, extremely close to the nuclear ground state.The most recent value for the isomer energy E is , 7 . ± . [1, 2], is withinthe reach of modern optical laser spectroscopy and could serve as a “nuclearfrequency standard” [3]. This standard could reach an uncertainty level of10 − [4], and provide a new powerful instrument for testing the stability offundamental constants [5, 6]. It has been shown that an ensemble of Thoriumnuclei doped into a transparent crystal may demonstrate superradiance with anon-trivial emission dynamics [7], and may be used for building an ultraviolet(UV) laser [8]. Finally, the frequency shifts and broadenings produced by such acrystal lattice environment might be used in studies of material properties, as iscommonly done in M¨ossbauer spectroscopy [3]. The necessary step towards allof these exciting applications is a direct observation and precise determinationof the isomer state energy.The existence of the low-energy state in the Th nucleus was first conjec-tured by Kroger and Reich based on studies of the γ -ray spectrum following the α -decay of Uranium-233 [9]. They concluded that this nucleus has a J π = 3 / + isomer level lying within 100 eV above the J π = 5 / + ground state level. The de-velopment of high quality germanium detectors (resolution from 300 to 900 eV)allowed Helmer and Reich to measure more precise γ -energies in 1989 – 1993and to predict the energy of the nuclear transition to be E is = 3 . ± . E is triggered a multitude of investigations, We will refer to these devices as “x-ray” spectrometers, corresponding to their primaryfield of application. In the measurements described here, they detect both, x-rays and γ -rays. igure 1: Partial level schemes of the Th nucleus with decay paths and energies (all inkeV). Boxes in each panel denote the energy combinations used to derive E is in the “indirect”methods discussed in the main text. (a): according to Helmer and Reich [10]; (b): accordingto Beck et al. [1, 2], the interband transitions (dashed arrows) are taken into account; (c):approach discussed here using a high-resolution (∆ inst ≃ indirect measurements, involving keV energy transitions whereas scheme (d) is direct , only measuringthe isomer transition of a few eV energy. J π = 3 / + excited state (suchas lifetime and magnetic moment). However, searches for direct photon emissionfrom the low-lying excited state performed in the late 90’s [11, 12] have failedto observe a signal [13, 14]. In 2005, Guim˜araes-Filho and Helene re-analysedthe data of Helmer and Reich, taking into account new information about thenuclear decay pattern [15]. They derived E is = 5 . ± . inst from 26 to 30 eV (FWHM) allowed Beck et al. [1] to perform a newindirect measurement of E is , involving lower energy nuclear states, as depictedin Figure 1 (b). In this measurement, the obtained transition energy (7 eV) wascorrected by accounting for the theoretical branching ratios 29 .
19 keV → g Thestimated as 1/13, and 42 .
43 keV → m Th estimated as 2% in [2]. This cor-rection yields the currently most accepted value E is = 7 . ± . ≈
160 nm).In the experiments decribed above ([1, 9, 10]) the isomer transition energy E is is not measured directly but is derived from the spectrum of higher-energy(keV) γ -radiation of a spontaneously decaying U source. We will refer tothese measurements as indirect passive . Possible alternatives are direct passive and active approaches.In the direct passive schemes (Figure 1 (d)), the aim is to perform spec-troscopy of the ultraviolet radiation emitted from the isomer appearing in the α -decay of U (2% of the nuclei decay is expected to lead into the isomerstate). This method has two main difficulties: a relatively high false count ratecaused by the Uranium sample radioactivity, and a high probability of non-radiative decay (quenching) of the isomer state in neutral Thorium atoms (upto 10 times higher than the radiative decay rate [16]). To overcome these prob-lems, it was proposed in [17] to extract α -recoil Thorium ions ejected from anUranium sample, and collect them in a small spot on a MgF coated surface tominimize the quenching rate. Vacuum ultraviolet spectroscopy of the emittedfluorescence radiation may then allow to measure the isomer transition energy.4n the contrary, in active approaches , Thorium nuclei (in the ground state)will be illuminated by tunable radiation to excite them to the isomer state. Inthe solid-state approach a macroscopic (10 − ) number of Thorium ionsdoped into UV transparent crystals can be excited, for example, by synchrotronradiation, and the emerging fluorescence signal can be studied [18–20]. Appar-ent advantage of this approach is the huge number of simultaneously excitednuclei. At the same time, crystal fluorescence can cause difficulties in identify-ing the Thorium isomer transition, and various crystal effects can hamper theprecise determination of E is . Another approach is the spectroscopy of trappedThorium ions . At PTB, Germany, work is under way to excite nuclei of Th + ions into the isomer state using a two-photon scheme, exploiting the electronicbridge mechanism [21, 22]. In Georgia University of Technology, USA, the lasermanipulation of Th ions is under investigation [4, 23]. Detection of the ex-citation of the Thorium into the isomer state may be based on a change of theelectronic hyperfine structure [3]. Studies of the hyperfine structure of Thoriumare also performed at the IGISOL facility in Jyv¨askyl¨a, Finland, in collaborationwith a group of the University of Mainz, Germany [24].We should also mention a number of studies aimed to measure the lifetime ofthe isomer state without a determination of E is . In [25], the half life of the isomerstate for a bare nucleus was derived theoretically based on the calculations of thematrix element of the nuclear magnetic moment and on the experimental dataconcerning transitions at higher energies. They predict a half-life of T / =(10 .
95 h) / (0 . E ) for the isomer transition, where E is given in eV, whichyields T / = 55 min for E = 7 . = 0 . ± σ )to find the transition with 95 % probability, or over 3 eV ( ± σ ) to find thetransition with 99.7 % probability. Sakharov re-estimated the influence of theuncertainty of the 29.39 keV peak on the isomer energy derivation in [1] andobtained an error of 1.3–1.5 eV [30]. Moreover, an analysis of more recent ex-perimental data led him to claim that the energy of the isomer state can beanywhere in the range 0–15 eV, if the isomer state exists at all.We believe it will be technically difficult, if not impossible, to cover such abroad energy range with a tunable narrow-band source of ultraviolet radiationin a reasonable time. We therefore propose to first increase the energy reso-lution on E is by an improved indirect measurement compared to [1]. As weshow below, it appears possible to resolve the 29.18 keV doublet [31] presentedin Figure 1 (c) with todays state-of-the-art x-ray spectrometers. Resolving thisdoublet would significantly increase confidence in the existence of the isomerstate. Moreover, the isomer energy would be measured without additional theo-retical input parameters like branching ratios etc. The aim of the present studyis to investigate the possibility of resolving the 29.18 keV line clearly as a doubletover a broad range of values for the isomer energy splitting and the branchingratio, and to analyze the precision that can be obtained on E is depending onthe relevant experimental parameters.
2. Statistical aspects of the envisioned experiment
The operation principle of high-resolution x-ray microcalorimeters is to de-tect the heat deposited by an x- or γ -ray interacting with an absorber, usinga very sensitive thermometer. Interaction with the absorber material mainlyproceeds through the photoeffect. The energy of the produced photoelectronas well as Auger electrons together with their thermalization cascade should beeffectively deposited within the volume of the absorber [32, 33]. On the otherhand, the absorber should have a small heat capacity C a for good instrumental6nergy resolution ∆ inst . Various microcalorimeters differ in geometry, absorbermaterial, sensor, etc., which leads to different energy resolutions, stopping pow-ers, total detector surfaces etc. Many of these parameters are connected and cannot be optimized independently. For example, increasing the size of absorberincreases the solid angle and/or stopping power but degrades the instrumen-tal energy resolution. Finally, we note that after a detection event, dissipationof the deposited heat leads to a detector-specific dead time, during which theenergy of a successive photon can not be measured correctly. Therefore it isimpossible to improve the precision of the measurements infinitely simply byusing a more active sample, or by placing the sample very close to the detector.The total count rate can be reduced using a designed filter which will primarilyabsorb photons outside the 29.18 keV region of interest.The present study aims to answer two questions: how does the possibility toresolve the 29.18 keV peak as a doublet depend on the experimental parameters,and how to attain the most precise determination of the isomer transition energy E is . As outlined above, parameters of the experimental setup can be controlledto some extent only. For a proper design of suitable detectors and experimentalconfigurations, we analyze how the key parameters affect these two points. The model employed for the statistical study should not contain too manyparameters to make it accessible to a multi-factor analysis. On the other hand,it should be sufficiently comprehensive for a realistic feasibility study. For thesake of convenience, we assume a fixed total measurement time of t = 10 s,approximately 11 days, which corresponds to the total time of the successfullmeasurement in [1]. Also we suppose that the background count rate near the29.18 keV doublet is flat and symmetric, and that the monoenergetic line hasGaussian shape with full width at half maximum equal to ∆ inst (see section 4for a discussion of these approximations).The considered total energy interval (0-70 eV) is subdivided to a set of 0.4 eVbins (approximately a factor 10 below the expected instrumental energy reso-7ution). The number of counts in the i th energy bin is a Poissonian randomnumber n i with a mean value λ i equal to λ i = d · R · tσ √ π (cid:20) (1 − b ) exp (cid:18) − ( E i − E ) σ (cid:19) + b exp (cid:18) − ( E i − E ) σ (cid:19)(cid:21) + d · r bg · t, (1)where R is the signal count rate, r bg is a specific rate of background countsper 1 eV energy interval, E , E are the centers of lines of the components ofthe 29.18 keV doublet, σ = ∆ inst √ , d = 0 . E i is themean energy of the i th bin, and b is the branching ratio. The set { n , ..., n N } ofexperimental data can be represented as a position vector n in an N -dimensional“sample space” ( N = 175).For a given sample n , we perform a nonlinear regression fit by a vectorfunction f with N components f i = d · ts √ π " J exp − ( E i − ˜ E ) s ! + J exp − ( E i − ˜ E − ˜ E is ) σ ! + d · ˜ r bg · t. (2)This fit has 6 free parameters { J , J , ˜ E , ˜ E is , ˜ r bg , ˜ σ } = { θ , ..., θ } ≡ θ . Forthe estimation of these parameters, we use the maximum likelihood method.The likelihood function L ( n | θ ) is the probability for realizing the set n , if truemean values λ i are equal to f i ( θ ). We also introduce the logarithmic likelihoodfunction ℓ ( n | θ ) = log L ( n | θ ) = N X i =1 [ n i log f i ( θ ) − log( n i !) − f i ( θ )] . (3) As it was outlined above, resolving the 29.18 keV line as a doublet wouldsignificantly reduce the doubts [30] in the existence of the isomer state in the
Th nucleus. Our first aim is to discuss the feasability of such an identificationdepending on experimental parametes and the (yet unknown) values of theisomer energy splitting E is = E − E and the branching ratio b .8 igure 2: Curves of constant levels of signal count rate R (in mHz) required to resolvethe 29.18 keV line as a doublet at 1 % significance level for different values of the detectorresolution ∆ inst and the signal-to-noise ratio R /r bg at 10 seconds of measurement time.The red spot corresponds to the area of the branching ratio b and isomer transition energy E is according to [2]. igure 3: Curves of constant levels of signal count rate R (in mHz) required to resolvethe 29.18 keV line as a doublet at 1 % significance level for different values of the detectorresolution ∆ inst and the signal-to-noise ratio R /r bg at 10 seconds of measurement time.The red spot corresponds to the area of the branching ratio b and isomer transition energy E is according to [2] (continued).
10o check whether a given set of experimental data corresponds to a singleline or to a doublet, one can apply the likelihood ratio test [34–36]. The essenceof this test and the method to estimate the significance level is described inAppendix A. Here we define the significance level as the probability to identifyincorrectly the single peak as a doublet or the doublet as a single peak usingthe likelihood ratio test in the situation when we have either a single peak ora doublet whose parameters are specified. Figures 2, 3 show level curves forthe signal count rates R which are necessary to attain a significance levelof 1 % for various values of the instrumental resolution ∆ inst and the R /r bg ratio, in the ( b, E is ) plane. It is interesting to note that the optimal energyresolution is attained when b ≃ .
25 and not for equally strong components ofthe doublet. This is explained by the fact that such a branching ratio leadsto a noticeable asymmetry of the peak which facilitates the identification of asecond component, increasing b leads to a reduction of the main peak at fixedtotal signal count rate R . E is : Monte-Carlo simulations The aim of the proposed spectroscopy study is not only to resolve the29.18 keV line in the γ -spectrum of U as a doublet but to determine theenergy splitting with maximum precision.We study the standard deviation δE is = q h ( ˆ E is − E is ) i (4)of the isomer energy as a characteristic measure of precision (in the followingwe call δE is the uncertainty of E is ). Here and below, angular brackets denoteexpectation values, E is and ˆ E is denote “true” and measured values of the isomertransition energy respectively. For the sake of brevity, we suppose that thetrue values of the energy splitting E is and of the branching ratio b are equal to7.8 eV [2] and 1/14 [10] respectively . As before, we assume a total measurement According to [1], b = 1 /
13 has 8% error, therefore the value b = 1 /
14 can be considered asa consistent but slightly pessimistic (from the point of view of spectral resolution) estimation. igure 4: Examples of Monte-Carlo simulated “experimental data” (black dots) and fit (redcurves) for ∆ inst = 3 eV (a), 6 eV (b) and 9 eV (c). Other parameters are: R = 7 .
74 mHz, r bg = 3 . µ Hz/eV, t = 10 s. Plots are depicted in a “square-root scale” where the Poissoniannoise is mapped onto signal-independent deviations. The origin of the energy axis is chosenarbitrarily. time t = 10 s. Scaling the results to other values of E is , b , or t is straightforward.To investigate the dependence of δE is on the instrumental energy resolu-tion ∆ inst , the signal count rate R , and the specific background count rate r bg , we perform a Monte-Carlo study of δE is . For any set of parameters,we simulate the sample n as shown in Figure 4, and estimate the parameters θ = { J , J , ˜ E , E is , ˜ r bg , s } maximizing the sum (3). We repeat this procedure10 times and calculate δE is according to (4).In Figure 5 we present curves of constant level of δE is in the plane ( R , r bg ) obtained from the Monte-Carlo simulation. Finally, in Figure 6 we presentthe curves of constant δE is in the plane ( R , ∆ inst ) for fixed ratios R /r bg .One can see that improving the instrumental resolution by 1 eV increases theprecision on the determination of δE is by the same amount as doubling the29.18 keV signal count rate, or doubling the measurement time t .12 igure 5: Curves of constant δE is labeled in eV in the ( R , r bg ) plane for different values of∆ inst and 10 s of total measurement time. igure 6: Curves of constant δE is labeled in eV in the ( R , ∆ inst ) plane for different valuesof R /r bg and 10 s of total measurement time. . Experimentally attainable count rates and expected precision In this section we estimate the attainable count rates and the resulting pre-cision in a measurement of the isomer energy E is that can be achieved with astate-of-the-art high-resolution microcalorimeter. In particular we consider themetallic magnetic microcalorimeter maXs-20 as described in ref. [33]. The aimis to demonstrate, that valuable results can be obtained with currently avail-able technology. In section 5 we describe ongoing work towards a more refined,dedicated detector setup.The maXs-20 microcalorimeter consists of 8 detector elements (pixels) eachof which has an absorber plate for incoming radiation (250 × × µ m Auplate) connected to a 160 × × . µ m paramagnetic temperature sensor(Er-doped Au) through 24 gold stems (10 µ m diameter and 5 µ m height each).Each sensor is connected to a thermal bath, the system is installed in a cryostatoperating at a temperature of about 30 mK. Energy deposited into the absorberplate heats the paramagnetic sensor and causes a change of its magnetizationin an external magnetic field. Measuring this change in magnetization usingSQUIDS, it is possible to determine the amount of absorbed energy and hencethe energy of the incoming x- or γ -rays. Note that after the detection of an x-or γ -ray, the individual pixel can only detect again after a certain relaxationtime of the order of 100 ms. Therefore the total count rate R T should not betoo high.As a sample, we consider 1 mCi of U electrodeposited as a film onto ametal planchet with a radius of R = 10 mm. We assume the sample to besituated 40 mm from the detector (outside the cryostat) with a total surface s = 0 . . Also we suppose the presence of additional material related tothe cryostat vacuum system (sealing, input window of the cryostat, some otherintentionally positioned shielding etc.), which we refer to as filters .According to the NuDat 2.6 database [37], each single decay of U is accom-panied (on average) by one α particle with an energy from 4.309 to 4.824 MeV,0.213 conversion electrons with energies from 2.3 to 600 keV (97.7 % of the elec-15rons have energies below 50 keV), and 0.0544 photons most of which (0.052per decay) are L -shell x-rays with a mean energy of 13 keV. As no individual L x-rays are listed in NuDat 2.6 (only average energy and total intensity), wehave taken the lacking data from the X-Ray Data Booklet [38].To estimate the detector count rates, we suppose that all α particles andelectrons emitted from the Uranium sample are stopped by the sample itselfor by the filter materials, therefore, we consider only x- and γ -rays. Also it issupposed that all secondary electrons and photons generated in the filters areabsorbed in the material locally. This assumption is correct for relatively thickfilters made from light materials like Aluminium.We take into account absorption of the photons within the sample itself, thefilters, and the detector. The total count rate R T is: R T = i max X i =1 A · Ω4 π · I i − e − ℓ U a U ( E i ) ℓ U a U ( E i ) × e − ℓ f a f ( E i ) · (cid:16) − e − ℓ Au a Au ( E i ) (cid:17) . (5)Here the sum is taken over all photon energies E i , A is the activity of theUranium sample, I i is the relative intensity (quantum output) of photons withthe energy E i per single decay event, ℓ Au and ℓ f are the thicknessess of the goldabsorber and filters respectively. Linear absorption coefficients a κ ( E i ) ( κ =U , Au , f) were taken from the XCOM Photon Cross Sections Database [39].The count rate R of signal photons is: R = X i =1 , A · Ω4 π I i − e − ℓ U a U ( E i ) ℓ U a U ( E i ) e − ℓ f a f ( E i ) × " − e − ℓ Au a Au ( E i ) − ̟ X X I X P es ( E i , E X , ℓ Au ) , (6)where i = 1 , ̟ = 0 .
331 isthe Au L shell fluorescence yield [40], I X is the probability that an energy of afluorescence photon emitted by a Au atom is equal to E X , and P es ( E γ , E X , ℓ Au )is a probability that an incoming γ -quant with energy E i will be absorbed, andan x -ray photon following this absorption leaves the absorber (escape line). Sup-16osing an isotropic spatial distribution of these secondary photons, we obtain P es = 12 ℓ Z a γ e − a γ x π/ Z exp (cid:20) − a X ( ℓ − x )cos θ (cid:21) sin θdθ + π Z π/ exp h a X x cos θ i sin θdθ dx, (7)where ℓ = ℓ Au , a γ = a Au ( E γ ), a X = a Au ( E X ). Also we suppose that the x -rayphoton is emitted from the L shell, i.e. the deepest shell that is accessible byenergy conservation, and the probability I X for emission of the photon is therelative intensity tabulated in [39] normalized to the sum of relative intensitiesfrom the L shell.Evaluating the expressions (5) – (7) yields a total detector count rate R T =1 Hz and a signal count rate of R = 7 .
74 mHz for a 1.3 mm thick Aluminiumfilter. Without any filter, the count rates for the same parameters are: R T =13 . R = 11 .
78 mHz. We see that the Aluminium filter absorbs ap-proximately 92 % of all photons emitted from the sample, but only about 34 %of the signal photons. We conclude that filtering is an effective method to de-crease the total count rate R T , caused mainly by low-energy Thorium L shellx-ray.The background count rate r bg is caused by the escape of some fraction ofthe dissipated energy of γ -quants absorbed in the detector. In [1], the number ofbackground counts close to the 29.18 keV doublet was about 30 – 40 events per3 eV bin whereas the total number of counts in the 29.18 keV peak was about2 . · events. This yields the ratio R /r bg = 2000 eV. Assuming that a similarratio can be realized with the maXs-20 detector, we find that the uncertainty δE is on the measured isomer transition energy E is will be equal to 0.06 eV foran instrumental resolution of ∆ inst = 3 eV, signal count rate R = 7 .
74 mHzand total measurement time t = 10 s. Therefore the proposed experiment todetermine the isomer energy E is is expected to be almost one order of magnitudemore precise than the results obtained in the previous experiment [1]. Reducing17he experimental resolution to 6 eV and 9 eV yields δE is = 0 .
19 eV and δE is =0 .
56 eV respectively. Increasing the total measurement time to t = 2 . · s, 1month, we can measure the isomer transition energy with an uncertainty δE is =0 .
037 eV for ∆ inst = 3 eV, δE is = 0 .
12 eV for ∆ inst = 6 eV, or δE is = 0 .
33 eV for∆ inst = 9 eV.
4. Further statistical aspects
We are aware of certain simplifications and assumptions in the above anal-ysis. Here we briefly resume some additional issues that could arrise, a detaileddiscussion of these points is beyond the scope of this work.First, the shape of signal peaks can deviate from Gaussian. For example,a long low energy tail on the spectral lines may lead to the appearance of anoticeable step in the backgound count rate (see, for example, Figure 2 (a) in[1]). We believe that in the work of ref. [1] this effect is caused mainly by theescape of energy from the absorber material, for example in the form of athermalphonons [33]. The yield of Compton scattering is not sufficient to explain thisstep, see Appendix B. To take this effect into account correctly, we will haveto modify our model of the background. A more difficult situation arises whenthe escaping energy is relatively small, which would lead to an asymmetry ofthe line rather than the appearance of a tail. In this case, it would be useful tostudy an isolated single peak separated from the doublet of interest but intenseenough to give good statistics, and/or to perform an independent study withanother γ source, e.g. Am.Another issue that may appear is a slow time-dependent fluctuation of theresponse function caused by an uncontrollable drift of ambient magnetic fieldsand/or cryostat temperature over the duration of the measurement. We believewe can suppress such drifts below 10 eV by temperature stabilization and mu-metal shielding of the setup. Additionally, we will monitor the position of aseries of reference x- and γ -ray lines for a correct tracing of this drift, realizing atime-dependent calibration of the detector. Note that x-ray lines generally have18 much broader linewidth than γ lines [42] which simplifies the identification.An auxiliary calibration source, for example Am, can help to enhance thequality of this callibration.Also we should mention possible interference of the 29.18 keV doublet withcoincidence and escape lines of x- and γ -rays of U and other elements presentin the sample. We plan to study the composition of the sample using “ordinary”low-precision γ -spectrometry.
5. Planned experimental implementation
We are currently developing a dedicated new detector for the measurementdescribed above, to some extend interpolating between the maXs-20 (0-20 keV)and the maXs-200 (0-200 keV) series [33]. It will feature a linear array of mag-netic calorimeters, each with an active area of 250 × µ m . We will increasethe absorber thickness by a factor of 2 to 3 in comparison with 5 µ m in themaXs-20, leading to a stopping power of about 50 % at 30 keV. We will operatethe detectors in a dry 3He/4He-dilution refrigerator at about 20 mK. In this sit-uation, the intrinsic energy resolution of the detector caused by thermal noisesof all kinds is expected to be below 2 eV (FWHM). According to the calorimetricdetection principle of metallic magnetic calorimeters [41], this resolution is inde-pendent of energy as long as the total gain (including operational temperatureand external magnetic fields) is stable, the dependence of the detector responseon the event position in the absorber is negligible, and the photon energy E isstill small enough to be within the range of linear detector response Φ( E ). Alsowe expect that the minimal time between two correctly measurable counts in asingle detector element (pixel) will be about 100 ms.So far, we have achieved resolving powers up to about E/ ∆ inst = 3700(corresponding to 1.6 eV (FWHM) at 5.9 keV) with our maXs-20 devices (un-published), being limited by a combination of both, instabilities of the operatingtemperature and a position dependence. We believe that we can improve the19 igure 7: Detector response versus the energy of absorbed photons (a) and deviation fromthe linear behaviour (b) based on the measurement of 3 characteristic lines of an Am γ -spectrum (c). short-term stability of the total gain and keep the position dependence of absorb-tion events small enough to allow resolving powers beyond 10 in the plannedexperiment. Also, the response of the present maXs-20 detector to photon en-ergies below 60 keV has a small quadratic deviation from a linear behavior, seeFigure 7. At an energy of 30 keV this deviation is only about 3 %, which yieldsa 6 % degradation of the intrinsic energy resolution compared to the low-energysignals, i.e. below 2.12 eV on an absolute scale. The sample should ideally consist of isotopically pure
U to avoid a toohigh count rate not carrying relevant information and possible interference withthe 29.18 keV Thorium doublet signal. For this project, we have 560 mg (about20 igure 8: Left: Inductively coupled plasma mass spectrometer (ICP-MS) data of the raw
U material composition. Mass signals above 240 amu are molecular fragments and can beignored. Right: Photo of the electroplated UO test sample (with U). γ -spectrum wesuspect that originally, Th has been activated in a high-flux neutron reactorand the
U has been separated chemically. A mass spectrum of the rawmaterial, produced by an in-house ICP-MS can be seen in Figure 8. The rawmaterial contains >
90 %
U, together with traces of U, U, U, U,and the decay product
Th. Further daughter product of the
U chain havenot been detected.To further purify the sample, we will perform a PUREX Uranium extractionprocedure. We have also observed an efficient additional element separation inthe electrodeposition process. In-house analysis using γ - and α -spectroscopy,ICP-MS, and neutron activation analysis will allow us to quantify the success ofthis procedures and finally know the exact composition of the final measurementsample.The sample will be produced by electroplating U from a liquid solutiononto a stainless steel or aluminium planchets. The target activity of 1 mCi cor-21esponds to 104 mg of pure
U or 118 mg of UO . Producing correspondinglythick films (15-20 µ m) turned out to be difficult in electroplating [43, 44]. Wehave therefore developed a process to deposit up to 20 mg Uranium onto stainlesssteel or aluminium foils of only 10-50 µ m thickness. These samples can easilybe stacked to realize the target activity without the carrier foils significantlyreducing the count rate in the 29.18 keV peak.
6. Conclusion
We have analyzed the feasability of an indirect measurement of the low-energy isomer state in
Th using a high-resolution magnetic microcalorimeter.We propose to resolve the 29.18 keV doublet in the γ -radiation spectrum follow-ing the α -decay of U. Such a measurement would provide a strong indicationfor the existence of the isomer state and improve the accuracy on the energymeasurement significantly. The measurement appears feasable with currentlyavailable detector technology and samples.
7. Acknowledgements
We thank C. Streli, P. Wobrauschek, and J. Seres for discussions andcalculations concerning γ -ray filters. We thank S. Smolek and M. Gollow-itzer for preliminary tests and characterizations of U samples. This workwas supported by the ERC project 258604-NAC, the FWF Project Y481, theWPI Thematic Program “Tailored Quantum Materials”, and the FWF projectM1272-N16 “TheoNAC”. 22 ppendix A. Likelihood ratio test and estimation of the significancelevel a Here we describe the essence of likelihood ratio tests for regression modelsin the simple case of normally distributed observables with known dispersions,and the method which we actually used for the estimation of the significancelevel attainable in the experiment.Let us have N experimental observables y i normally distributed around their(unknown) expectation values λ i with known dispersions σ i . Without lost ofgenerality, we can set σ i = 1 for all i . Also, we have 2 regression models, oneof which ( short model ) is a particular case of another one ( long model ). In thelong model, it is supposed that the expectation values of observations y i aresome known functions f i ( θ ) of the l -dimensional parameter θ = { θ , ..., θ l } . Inthe short model, it is supposed also that the parameters θ has some additionalrestrictions, and the short model has s = l − r degrees of freedom. It is supposedthat the long hypothesis anyhow is correct. The question we want to answeris whether the short hypothesis is correct? Or, more precisely: how plausible(or unplausible) is it to obtain the set of observables y i , if the short model iscorrect?The likelihood ratio test is a powerful method to answer this question. Toillustrate the essence of this test, let us represent N observations y i as a point y in the N -dimensional Euclidean sample space. N functions f i ( θ , ..., θ l ) forman l -dimensional surface C corresponding the long hypothesis. In turn, thesefunctions with additional conditions corresponding the short hypothesis formthe s -dimensional surface SS within C , see Figure 9. We suppose that thesesurfaces are sufficiently smooth. In the case of linear regression models, thefunctions f i ( θ ) are linear, and the surfaces C and SS are just hyperplanes.For normally distributed observables y i with zero mean and unit dispersionand for some set of parametes θ , the logarithmic likelihood function is ℓ ( y | θ ) = const − N X i =1 ( f i ( θ ) − y i ) . (8)23 igure 9: Sketch of the sample space: C and SS are the long- and short hypotheses sur-faces respectively. Point y ∗ = f ( θ ∗ ) corresponds to the true value of parameters, SS ′ is an s -dimensional surface passing through y ∗ whose points are equidistant to SS . y is an experi-mental point, L = f ( θ L ), and S = f ( θ S ) corresponds the best fits within the long and shorthypotheses respectively. The non-centrality parameter is the square of the distance a between SS and SS ′ . Therefore the maximization of ℓ ( θ ) is equivalent to a minimization of the dis-tance between the points f ( θ ) and y in the sample space. The square of thisdistance we denote as | f θ − y | . Let L = f ( θ c ) and S = f ( θ s ), where θ c and θ s are the best likelihood estimations of parameters θ within the long and shorthypotheses respectively. It is easy to see that | y − S | ≃ | y − L | + | L − S | which yields | L − S | = 2 (cid:0) ℓ ( y | θ c ) − ℓ ( y | θ s ) (cid:1) . In turn, | L − S | is a sum ofsquares of r normally distributed random values κ α with unit dispersion anddifferent means µ α . Therefore, | L − S | is a non-central χ random value with r degrees of freedom and non-centrality parameter a = P rα =1 µ α . For the sakeof brevity, we denote this random value as χ r ( a ). It is easy to see that a is justa distance between the short hypothesis surface SS and the point y ∗ = f ( θ ∗ )corresponing to the true value θ ∗ of parameters θ , see Figure 9. If the shorthypothesis is true (it corresponds the situation when the surface SS coincidedwith SS ′ in Figure 9), then a = 0 and | L − S | is just a “usual” χ r randomvalue with r degrees of freedom.To test whether the short hypothesis is true or not, one should choose some24esirable significance level α , and compare the value D = | L − S | = 2( ℓ ( n | θ c ) − ℓ ( n | θ s )) (9)with some critical value λ α ( χ r ) such, that the probability P (cid:0) χ r > λ α ( χ r ) (cid:1) = α. (10)If D > λ α ( χ r ), the short hypothesis is rejected on significance level α , otherwiseit is accepted. The probability to reject the short hypothesis incorrectly isequal to α . On the other hand, the probability to accept the short hypothesisincorrectly is P ( χ r ( a ) < λ α ( χ r )). Therefore if the non-centrality parameter a will be larger than a r ( α ) such that P (cid:2) χ r ( a r ( α )) < λ α ( χ r ) (cid:3) = α, (11)the probability to accept the short hypothesis falsely will be less than α .In our case, the observables n i are not normal random values with knowndispersion but Poissonian random values. To estimate the possibility to identifythe 29.18 keV line as a doublet, we consider √ n i as observables and approximatetheir distribution function by a normal distribution with mean √ λ i and disper-sion 1 /
4. This approximation is not very precise (the bias is about 20 % for λ = 1), but it seems to be applicable for a coarse estimation. Then, for specificvalues of b , E is t , ∆ inst , R , and r bg we calculate the set √ λ i according (1),and fit it by a vector function whose components are given by the square rootof (2) with additional restriction J = 0. Then we calculate the non-centralityparameter a = 4 N X i =1 (cid:16)p f i ( θ s ) − p λ i (cid:17) . (12)Our short hypothesis (single peak fit) has 4 degrees of freedom whereasthe long hypothesis (double peak fit) has 6 degrees of freedom which yields r = 2. For significance level α = 0 .
01, conditions (10) and (11) are fulfilled at λ . ( χ ) = 2 log(100) ≃ .
21, and a (0 . ≃ .
4. The values of R necessaryto attain a = 27 . ppendix B. The role of Compton scattering in the formation of anasymmetry or low-energy tail in absorption lines In ref. [1] the 29.18 keV line shows a low-energy tail, causing a noticeablestep of about 10 counts per 3 eV bin (see Figure 2 (a) in [1]). This step can becaused either by the absorbtion of photons which had 29.18 keV energy initially,but lost some fraction of it in the source or filter material, or by the escape ofsome energy fraction from the absorber. The first scenario could be explainedby the Compton “almost forward” scattering, when a photon loses tiny parts ofits energy, about a few eV. The Klein-Nishina differential cross-section of theCompton scattering into the elementary solid angle is dσd
Ω = r e (cid:18) E ′ γ E γ (cid:19) (cid:20) E γ ′ E γ + E γ E ′ γ − sin θ (cid:21) , (13)where E γ and E ′ γ are the energies of the photon before and after the Comptonscattering respectively, r e = e / ( m e c ) is the classical electronic radius, and θ isthe scattering angle. Using a well-known relation between E ′ γ and θ , it is easyto express the differential cross-section in units of E ′ γ : dσdE ′ γ = πr e m e c E γ (cid:18) E ′ γ E γ (cid:19) (cid:20) E γ ′ E γ + E γ E ′ γ − sin θ (cid:21) . (14)Note that E ′ γ = E γ −
20 eV corresponds to θ ≃ . ◦ . For such small θ , (14)yields approximately dσdE ′ γ ≃ πE γ r e m e c ≃ · − cm / eV . (15)The differential probability for the 29.18 keV photon to be scattered into the1 eV range can be coarsely estimated as dPdE ′ γ = dσdE ′ γ (cid:20) z U n U ℓ U z f n f ℓ f (cid:21) , (16)where z i , n i , and ℓ i are the nuclear charge number, the atomic density, and thethickness of the Uranium ( i = U) or filter ( i = f) material respectively.In the experiment [1], the Uranium activity of one 19 mm diameter planchetwas about 0.02 mCi which corresponds to ℓ U ≃ . µ m. It was covered by26 Titanium foil (“filter”) with ℓ f = 50 . µ m. An estimation following (16)yields dP/dE ′ γ ≃ . · − eV − . The peak corresponding to the absorption of29.18 keV photons has Gaussian shape with about 3 · events per 3 eV binheight in the maximum, and 26 eV FWHM, see Figure 2(a) in [1]. This corre-sponds to a total number of counts forming this peak of about 3 · . Therefore,Compton scattering of 29.18 keV photons in the source or filter material pro-duces only 0.053 events per 1 eV interval, or 0.16 events per 3 eV bin. This valueis significantly lower than the observed step of 10 counts per 3 eV bin. A similarestimation for a ℓ U = 15 µ m layer of Uranium and ℓ f = 1 . dP/dE ′ γ ≃ . · − eV − .Note that in (16) we treat all electrons as free. A more accurate estimationrequires the substitution of the incoherent scattering function instead of z butthis function does not exceed z , see [45] for details. Therefore, a more accuratecalculations can only decrease the contribution of Compton scattering to thestep of the background count rate. ReferencesReferences [1] B. R. Beck, J. A. Becker, P. Beiersdorfer, G. V. Brown, K. J. Moody,J. B. Wilhelmy, F. S. Porter, C. A. Kilbourne, and R. L. Kelley,
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