Marcello ; Trevisani
email: marcello.trevisani@unibo.it

Matilde Checchini

Roberto Rosmini

Lorenzo Vercellotti

Luca Taffetani

M Cecchini

QUANTIFICATION OF LEAD AND CADMIUM IN POULTRY AND BIRD GAME MEAT BY SQUARE WAVE ANODIC STRIPPING VOLTAMMETRY

Keywords: assurance, Metals analysis -stripping voltametry Additives/Contaminants: Heavy metals, Lead, Heavy metals -cadmium Food Types: Meat, Animal products -meat Metals analysis -stripping voltametry, meat, lead, cadmium

published or not. The documents may come

Introduction

The Joint FAO/WHO Expert Committee on Food Additives (JECFA) has established a provisional tolerable weekly intake (PTWI) for lead and cadmium (WHO, 2000;JECFA, 2005). These were set to 25 µg/kg body weight for lead and 7 µg/kg b.w. for cadmium. On this basis, and taking into account the food consumption patterns, maximum limits for these trace contaminants in food have been set (Berg and Licht, 2002) but many factors contribute to the uncertainty of analytical results and therefore characteristics of the analytical methods used for control and monitoring purpose have been defined (EC 2007). Although meat is not regarded as a major source of human exposure to lead and cadmium (EC, 2004a;EFSA, 2004aEFSA, 2004b) the results of monitoring programmes showed that some samples may exceed the maximum limits set and investigations are needed to trace the source to prevent contamination, and also to evaluate if they actually represent a hazard for target populations. Major concern is given indeed to offal consumption (i.e. liver) because these animal tissues accumulate toxic heavy metals, but only a few studies were addressed with respect to chicken liver, whose consumption is very popular in some regions like the Central Italy. The need for suitable methods that can produce valid data and give information on their full uncertainty has been noted in many reports especially in the context of a risk assessment. Atomic Adsorption Spectrometry (AAS) and Inductively Couple Plasma coupled with Mass Spectrometry (ICP/MS) are the most commonly used method for trace analysis and studies were already be done to evaluate their validity for analyses of many food matrices, including offal (D'Ilio et al, 2008;Forte and Bocca, 2007). Benefits and drawbacks of analytical methods for trace elements analysis has been discussed taking into account difficulties due to possible interferences, detection limits, throughput and cost [START_REF] Brown | Analytical techniques for trace element analysis: an overview[END_REF]. Undoubtedly spectrometric techniques like Atomic Adsorption Spectrometry (AAS) and Inductively Coupled Plasma coupled with Mass Spectrometry (ICP/MS) are the preferred methods for laboratories that have to process a large number of samples for their higher throughput, but for the determination of electro-active trace elements like heavy metals stripping voltammetry has been reported to provide comparable results in terms of precision and limits of detection. In addition it has the advantage of requiring relatively inexpensive instrumentation, having low initial and running costs and also being capable of determining trace level of heavy metals like lead and cadmium at the same time (Capar and Subjoc, 1982;Locatelli, 2000;Locatelli and Torsi, 2003;Omanovic and Branica, 2003;Vos et al., 1990;Wang, 2000). With the voltammetric stripping methods, the electro active metals are concentrated at the surface of a polarized modified electrode immersed in sample solution (i.e. by application of a plating potential) and then re-dissolved (stripped) from it by means of oxidoreductants or by progressively decreasing potential ramps. The stripping potential is specific, in controlled conditions (i.e. support electrolytes and pH) for the target analyte (Bard and [START_REF] Faulkner | Electrochemical Methods[END_REF]Wang, 1994). This study proposes a procedure for the simultaneous determination of lead and cadmium in poultry (muscle and liver) samples by Square Wave Anodic Stripping Voltammetry (SWASV). The method uses an optimized microwave acid digestion procedure and acetate buffer in the supporting electrolyte in order to improve the recovery of target analytes and reduce the interference from other electroactive compounds. One aim of this study was to identify the sources of uncertainty that could affect the results. The procedure recommended for in-lab validation studies in the EC report on the "relationship between analytical results, measurement uncertainty, recovery factors" was followed (EC, 2004b) and the laboratory control requirements for establishing the performance criteria of methods for lead and cadmium determination (EC, 2007) were tested. The method was validated using two certified reference meat and liver samples (with composition closest to poultry meat) as well as chicken meat and liver samples that were artificially spiked at different levels in order to define the quantification limit and the overall precision and accuracy (De Silva et al., 2003) in a range between 10 and 200 ng g -1 of lead in muscle (or 10 -100 ng g -1 of cadmium), and between 10 and 172 ng g -1 of lead in liver (or 10 -544 ng g -1 cadmium). Another aim was to investigate the level of these contaminants in chicken and game (pigeon) meat in regularly slaughtered animals.

Materials and methods

Instruments and labware

Voltammetric measurements were made using a potentiostat/galvanostat Autolab Pgstat12

Reagents

Ultrapure water (grade I, std CAP and ASTM) was used to prepare all solutions and wash the labware.

The supporting electrolyte in the electrochemical cell was HCl 0,1M, made with Suprapur HCl 30% (Merk, D). A mercury nitrate (Hg 2+ ) 1000 mg•L -1 solution, which is used to overlay a mercury film on the graphite working electrode, was made by dissolving metallic Hg, purified by distillation (Merk, D), in 5M HNO 3 .and water for dilution. Reagents used for mineralization of the meat and liver samples were nitric acid 65%, hydrogen peroxide 20% and perchloric acid 60-70% (Suprapur, Merck, D). An acetate buffer solution (pH 4.75) was made with acetic acid in HCl 0.1 M was used to set the pH of mineralized samples to pH 2 (±0.2). Working solutions 1 mg•L -1 of lead and cadmium in 0.1M HCl were made using standard 1000 mg•L -1 lead and cadmium (Titrisol, Merk, D). Certified Reference Materials BCR-184 (beef) and BCR-185R cattle liver, produced by the European Community Bureau of Reference, and spiked chicken meat and liver samples were used for the validation procedure.

These chicken meat samples at lead and cadmium concentration in a range between 10 and 100 ng•g - 1 were prepared using samples with very low concentration of lead and cadmium.

Sample collection and preparation

Muscle and liver samples from thirty different lots of regularly slaughtered broilers were analysed. In addition muscle samples nine lots of pigeons were examined. Each sample was obtained by combining aliquots (10 g each) of muscle or liver taken form 5 carcasses. Tight muscles were sampled from broilers whereas breast muscle were taken form pigeons. The samples that showed the lowest level of lead and cadmium were used for preparation of artificially spiked samples. The selected muscle sample was spiked with lead and cadmium (10, 50 and 100 ng•g -1 ) whereas different spike levels (50, 100 and 500 or 200 ng•g -1 ) was used for the liver sample. Due to the lack of certified reference samples from chicken meat, the Certified Reference samples BCR 184 Bovine Muscle (Pb 239 ± 11 ng•g -1 ; Cd 13 ± 2 ng•g -1 ) and BCR 185R Bovine Liver (Pb 172 ± 9 ng•g -1 ; Cd 544 ± 17 ng•g -1 )

were analysed and these muscle or liver certified samples were included in every batch of analysis, which also included a blank. The homogenised muscle or liver samples (0.5± 0.005 g) or the lyophilized certified samples (0.3± 0.005 g) were weighted directly in a Teflon mineralizing vessel. A mixture of HNO 3 :H 2 O 2 :HClO 4 in the ratio 6:2:1 was used for the microwave assisted acid digestion using an optimised program (250W x 2 min., 0W x 1 min, 250 W x 5 min, 450 W x 4 min, 600 W x 3 min, 250W x 10 min and an additional 60 min venting step). After cooling (overnight) the digested samples were brought in a Teflon beaker using ultra-pure water for washing the vessels and were dried (almost completely) on an hotplate set at 80°C. The dried digested samples were dissolved in the supporting electrolyte (HCl 0.1 M) and transferred in Teflon volumetric flasks. After the addition of small amount of acetate buffer for adjusting pH (approx. pH=2) they were diluted up to 50 mL with HCl 0.1M. A blank solution (reagents without sample) is also prepared for every batch of the analysis.

Every batch included one blank, one certified reference material, one un-spiked muscle or liver sample, the same sample spiked (two repetitions) the same sample with a different spike. Lead and cadmium concentrations were measured using a SWASV procedure and the standard addition method. A deposition potential equal to -1.1 V for the deposition of lead and cadmium and the anodic stripping was assessed in the range between -0.25 and -1.1 V. (Table 1). Peaks of cadmium and lead can be recorded at potentials of 0.65 and 0.45 V vs Ag/AgCl (0.1 MKCl). Lead and cadmium peaks were recorded in a blank solution before then in samples, in order to quantify the trace level in reagents or those originating from contamination of labware and environment. An aliquot of the diluted digested sample (10 mL) was put in the electrochemical cells and was purged with N 2 to remove dissolved oxygen. The peaks in the voltammogram were recorded (i vs E) and their area was measured. The relative concentration of lead and cadmium in the diluted samples was determined by the method of Standard Addition. All instrumental measures were repeated three times, using different aliquots of the diluted samples, and also the determination of the concentration in blank is done using the same procedure.

Voltammetric determination

Quality controls

Analytical balance and micropipettes were regularly calibrated using approved procedures and checked for their precision (i.e. for the balance the precision ≤ 0.2 mg and for the micropipettes systematic error ≤ 1% in the range 2-10 µL or ≤ 0.4% in the range 20-50 µL or ≤ 0.8% at 100 µL). The stability of microwave oven was checked by applying specific power/time programmes to water samples (i.e. 1000 mL) and measuring the temperature increments. The measured delta value must be equal to the ratio between applied potency per time and a constant (e.g. ∆T = 1000W • 1 min/70).

The constant (70) is specified by the producers of the microwave oven and is related to leakage of energy and wavelength. The allowed tolerance (CV) must be ± 5%. The stability of the potential applied for the voltammentric method was also checked regularly, using a calibration cell and the procedures advised in the Autolab™ user's manual. Spiking, incomplete extraction, loses through evaporation, laboratory environmental contamination, standard addition calibration are covered in the precision study.

Detection and Quantification Limits

The detection and quantification limits (D.L. and Q.L.) were calculated from the average value (mean) and standard deviation (SD) of Lead and Cadmium level in the entire pool of analysed blank samples

Recovery and bias

The recovery and bias were calculated on the basis of at least five replicates of the digestion procedure for each artificially spiked sample or certified reference materials, and at least three repeated measurements on separate aliquots of the digested samples. In order to define the concentration of the original samples (muscle or liver) used to prepare the spiked samples, the analytical measurements were also replicated and repeated several times (i.e. 3 replicates and 7 repetitions). The mean bias is calculated as % relative error in ANOVA (interbatch) and its uncertainty u(bias) with the equations:

( ) ( ) ( ) ( ) ( ) [ ]

Repeatability, and HORRATr

Repeatability refers to degree of agreement of results in inter-batch (between-run) and intra-batch (within-run) precision within the laboratory. They were calculated through an Excel spreadsheet using formulae for an analysis of variance (ANOVA) based on the one-factor random effects model as indicated in the paper of DeSilva et al. (2003). In details, the inter-batch standard deviation (repeatability) is calculated as the square root of the ANOVA Mean Square intra-batch component MSw, if it is grater then the ANOVA Mean Square inter-batch (MSb) component (otherwise it is equal to the square root of the total ANOVA Mean Square). The inter-batch standard deviation (intermediate precision) is calculated as the square root of the total ANOVA variance components if MSb is grater then the MSw (otherwise the square root of the total ANOVA Mean Square). HORRATr = RSDr (found)/RSDr(calculated); where calculated RSDr depends on the level of analyte (typically 15% at concentration in the order of 10 µg kg -1 ). Acceptable values for this ratio are typically 0.5-2 and for EU Regulation 333/2007 must be below 2. In this study the intermediate precision was assessed within the same laboratory, by repeating the analyses on different days and using different batch of reagents and aliquots of the same samples.

Linearity

Reference materials (spiked samples at varying concentration and certified samples) were used in linearity study in order to validate a linear dynamic range for lead and cadmium in muscle and liver.

Specificity

Interference from natural constituents of samples was investigated. Occurrence and concentration of electro-active compound in the chicken muscle or liver producing voltammetric peaks that can overlap or drift those of lead or cadmium and their effect on measurements at different quantification levels were assessed.

Uncertainty components quantification

The up-down Eurachem/Citac method (error propagation) was used for calculating the overall uncertainty (Ellison et al, 2000).

Results and Discussion

Precision and accuracy

The results of the precision and accuracy study are reported in Tables 2a and2b. The analyses of certified reference materials (BCR 184 and BCR 185R) and of artificially spiked chicken tissues show that recovery of analytes (lead and cadmium) was almost complete. The mean bias for muscle was indeed in the range from -0.34 to -0.70 for lead and from 0.03 to -1.03 for cadmium. The expected tolerances for the mean recovery and inter-batch precision were within 5% for both lead and cadmium in muscle as well as for lead in liver, but reached approximately 11% for cadmium in liver.

Repeatability and reproducibility of analytical determination within the lab were always good with Horrat r and Horrat R values below 0.5 and fulfilled the EU regulation requirement (Horrat below 2).

Apart from uncertainty included in the precision and accuracy study other sources of uncertainty were identified. Uncertainty on producing the stock solutions and the calibration standards should be included in the overall uncertainty. The relative standard uncertainty related to the preparation of the stock and intermediate solution is reported in Table 3. Therefore, the nominal concentration and the expanded uncertainty (coverage factor k=2; 95%) of the standard solution used for calibration was 100 ± 0.24 ng•mL -1 . The uncertainty values for the spiked chicken meat samples at different level of lead and cadmium concentration and for the certified reference materials, which were used in the precision study, is reported in Table 4. The combined uncertainties of the lead and cadmium concentration in the artificially spiked were respectively 0.70% and 1.54%. The major contribution to these uncertainties was given by the repeatability of determinations, weight of samples and uncertainty of lead/cadmium concentration in the standard solution. It was showed that the combined uncertainty of artificially spiked samples was similar to those of certified reference materials (BCR184 and 185R). The relative overall uncertainty for samples with 57 ng g -1 of lead was 3.%, whereas for samples with 53 ng g -1 of cadmium was 4 % (data not shown). Therefore, the proposed analytical method fulfils the requirements specified by the EU regulation 333/2007 other than relative standard deviation calculated from results generated under reproducibility conditions (RSDR) which has not been assessed.

Detection, Quantification Limits and Linearity

The lower limit of detection and quantification (LLOD and LLOQ) calculated from the mean and standard deviation of the entire pool of analysed blank samples were equal to 1. and 2 ng g -1 for lead (n = 73) and 0.1 ng g -1 for cadmium (n= 85). The LLOQ calculated from the linearity study, i.e. ten times the intercept standard deviation at lead or cadmium level equal to zero (intercept) were instead 1.9 and 0.9 ng g -1 for lead and cadmium in muscles, 4.7 and 3.4 ng g -1 for lead and cadmium in liver.

The linearity study showed that relations between analytes concentrations (N = 4) analysed in six replicates (K = 6) and instrument responses were linear within the investigated ranges, that residual standard deviations increased with the concentration of lead or cadmium in the reference materials.

Therefore the model with proportional residual standard deviation, as defined in ISO 11095:1996, was used and the sum of squared weighted residuals (WSSE) and the sum of squared deviation due to pure error (WSSP) were calculated and the ratio between variability due to lack of fit and variability due to pure error proved to be not larger than the value of F (95%) calculated for N-2; N•K-N, showing that there was no evidence to reject the linear model (Table 5). The present method was validated in compliance with the requirements set in the EC Directive 2001/22 and it has detection and quantization limit that are at least one tenth and one fifth, respectively, of the acceptable maximum residual level specified in the Regulation EC 1881/2006 (i.e. in meat D.L. cadmium 5 ng g -1 and lead 10 ng g -1 , in liver lead and cadmium 50 ng g -1 ).

Lead and cadmium in chicken and pigeon meat

Higher levels of lead in chicken muscle were approximately 60 ng g -1 and most of the sample were in a range from 20 and 30 ng g -1 . The level of cadmium was below 10.5 ng g -1 in all the samples, with prevalent values in the range 2 -8 ng g -1 (Table 6). Therefore, chicken muscle proved to be relatively less contaminated than meat from other species commonly consumed. Level of these heavy metals was relatively low also in the chicken liver samples (8.0-84.4 ng g -1 for lead and 10.4 -90.6 ng g -1 for cadmium) and it was observed that the concentration in liver and muscle cannot be correlated (Figure 1). Probably the age of animals (below 60 days) does not allow a detectable accumulation in the liver and therefore consumption of chicken liver may be regarded favourably for this aspect. Level in farmed games, like pigeons, also show values similar to those observed in chicken, i.e. 7.9-63.6 ng g -1 for Pb and 2.2-8.0 ng g -1 for Cd in muscles.

Lead and cadmium food intake assessment studies have been undertaken with the aim of establishing the maximum permissible level in the different food on the basis of the Provisional Acceptable Weekly Intake (PWTI) and the risk for populations and the margin of exposure with different diet in Europe (EC. 2004). Meat, but also liver from chickens can only give a small contribution to consumer exposure to Pb and Cd. The management choice of Food Business Operators, i.e. feed producers, in controlling the sources of feedstuffs affect the level of contaminants in meat is however of great importance. Regular monitoring activities, which are facilitated by use of low cost analytical methods, can give insight in detecting contaminated sources of feedstuffs, and analysis of meat samples can provide a valuable tool for the risk management.

Conclusions

Efficient sample digestion and pH adjustment was recognized as an essential pre-treatment for the trace element measurement, however high qualitative standards are achievable using Square Wave Anodic Stripping Voltammertry for the assessment of lead and cadmium in chicken muscle and liver.

Although our results have to be confirmed by full collaborative studies to assess the reproducibility and robustness of this method, they showed that it can comply with the requirements set in the regulation of the European Union regarding the official food control, with good performance in term of detection and quantification limits, recovery, within-laboratory repeatability and measurement uncertainty for both lead and cadmium. Very low values of lead and cadmium were measured in chicken and pigeon meat and due to their very young age also chicken liver showed very low values.. Official Journal of the European Union. L77 of 16.3.200116.3. . pp. 14-21. EC (2004a)) Laboratories". In: Compendium of international analysis of methods -OIV.

(

  EcoChemie,NL) controlled by the software GPES 4.7 for Window. The potentiostat was linked to a three electrode electrochemical cell (C3 cell stand, BAS, USA) with glassy carbon working electrode (3 mm diameter), Ag/AgCl reference electrode and platinum counter electrode. The working electrode was cleaned using aluminium oxide powder (1 µm and 0.3 µm ) and an electrochemical cleaning step before measurements. The samples were digested by acid oxidizing reagents in Teflon closed vessels with a microwave controlled heating program (Ethos D, Milestone, USA). The labware used for dilution and preparation of reagents were certified Teflon volumetric class A flasks, Teflon beakers and polyethylene bottles carefully cleaned with HNO 3 10% and washed with Ultrapure water.

  (i.e. D.L. = mean + 3 SD; Q.L. = mean + 10 SD) but also calculated by extrapolating the concentration of lead or cadmium in blank using the linear regression curves interpolating data from analysis of artificially spiked and the certified reference samples (i.e D.L. = (3•Sa)/b; Q.L. = (10•Sa)/b, where Sa is the calculated standard deviation of intercept in the linearity study).

  where C ref (or C spike ) is the concentration of lead or cadmium in the certified reference material (or in the spiked samples) and RSDc ref (or RSDc spike ) is the Relative Standard Deviation of this value. RSD wgt is the Relative Standard Deviation of the sample weight.

  Materials at four concentration level for lead and cadmium.The ISO 11095:1996 and ISO 8466- 2:2001 type approaches were used. The Fisher-Snedecor test was used to verify the linearity within the 17.4-239.0 range for lead and within the 13.0-103.2 range for cadmium in muscle, within the 59.2-509.2 range for lead and the 60.5-544.0 range for cadmium in liver.

  G. and Subjoc G.A. (1982). Defining a lowest level of reliable measurement for lead in foods.J AssocOff Anal Chem. 65.1025-1029. Danzer K. and Currie L.D. (1998). Guideline for calibration in analytical chemistry. Part 1: Fundamentals and single component calibration. Pure & Appl. Chem. 70(4). 993-1014. De Silva B.. Smith W.. Weiner R.. Kelley M.. Smolec J.. Lee B.. Khan M.. Tacey R.. Hill H.. Celniker A. (2003). Recommendations for the bioanalytical method validation of ligand-binding assay to support pharmacokinetic assessment of macromolecules. Pharmaceutical Research 20(11) 1885-1900. D'ILio S., Petrucci F., D'Amato M., Di Gregorio M. Senofonte O. and Violante N. (2008). Method Validation for determination of cadmium and lead in offal by means of quadruple inductively couplet plasma mass spectrometry. J. Agric. Food Chem. 56,11584-11588. EC (2001) Commission Directive 2001/22/EC of 8 March 2001 laying down sampling methodsof analysis for the official control of the levels of lead, cadmium, mercury and 3-MCPD in foodstuffs.

  . Commission regulation (EC) No 1881/2006 of 19 December 2006 setting maximum levels for certain contaminants in foodstuffs. Official Journal of the European Union. L364 of 20.12.2006. pp. 5-24. EC (2007). Commission regulation (EC) No 333/2007 of 28 March 2007 laying down the methods of sampling and analysis for the official control of the levels of lead. cadmium. mercury. inorganic tin. 3-MCPD and benzo(a)pyrene in foodstuffs. Official Journal of the European Union. L88 of 29.3.2007. pp. 29-38. EFSA (2004a). Opinion of the Scientific Panel on Contaminants in the Food Chain on a request from the Commission related to lead as undesirable substance in animal feed. The EFSA Journal (2004) 71. 1-20. EFSA (2004b). Opinion of the Scientific Panel on Contaminants in the Food Chain on a request from the Commission related to cadmium as undesirable substance in animal feed. The EFSA Journal (2004) 72. 1-24. Ellison S.R.M. Rosslein M.. Williams A. (2000). Quantifying Uncertainty in Analytical Measurement. 2 nd ed. Eurachem/Citac. ISBN 0 948926 15 5. Forte G. and Bocca B. (2007). Quantification of cadmium and lead in offal by SF-ICP-MS: method development and uncertainty estimate. Food Chemistry 105, 1591-1598. ISO 8466-1 1990. "Water quality -Calibration and evaluation of analytical methods and estimation of performance characteristics -Part 1: Statistical evaluation of the linear calibration function. ISO 8466-2 2001. "Water quality -Calibration and evaluation of analytical methods and estimation of performance characteristics -Part 2: calibration strategy for non-linear second-order calibration functions. ISO 17025 2005. "General Requirements for the Competence of Testing and Calibration

Figure 1 -

 1 Figure 1 -Correlation between lead or cadmium concentration in broilers' muscle and liver Lead y = 0.6173x + 22.933 R 2 = 0.0639

  Faulkner,

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

  . European Commission SCOOP 2004. task 3.2.11. Assessment of the dietary exposure to arsenic. cadmium. lead and mercury of the population of the EU Member States. European Commission. Directorate-General Health and Consumer Protection. Reports on tasks for scientific

	co-operation				March			2004.
	http://europa.eu.int/comm/food/food/chemicalsafety/contaminants/scoop_3-2-
	11_heavy_metals_report_en.pdf
	EC (2004b). Report on the relationship between analytical results. Measurement uncertainty. recovery
	factors and the provisions of EU food and feed legislation. with particular reference to Community
	legislation concerning: -contaminants in food (Council Regulation (EEC) no 315/93 of 8 February
	1993 laying down community procedures for Contaminants in food);-undesirable substances in feed
	(directive 2002/32/ec of the European parliament and of the Council of 7 may 2002 on Undesirable
	substances	in	animal	feed).	Available	at	web	address:

ec.europa.eu/food/food/chemicalsafety/contaminants/report-ampling_analysis_2004_en.pdf -Accessed on April 2008.

Table 1 -

 1 Locatelli C.(2000). Proposal of new analytical procedures for heavy metal determinations in mussels. clams and fishes. Food Additives and Contaminants 17. 769-774. Parameters used for Lead and Cadmium Anodic Stripping Voltammetry

	Guide for the validation -

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Table 2a -

 2a Results of precision and accuracy study relative to chicken muscle

	Lead concentrations		Spiked chicken muscle		BCR-184
	(ng•g -1 )		10+7,41 50+7,41	100+7,41	239
	Results	n	19	19		18	26
	Accuracy	Mean bias (%RE)	-0.34	-0.70		-0.55	-0.50
		LCL	-0.34	-0.72		-0.58	-0.57
		UCL	-0.33	-0.68		-0.53	-0.43
	Precision	Intra-batch(%CV)	1.11	2.22		0.44	3.62
		Inter-batch(%CV)	0.94	1.89		0.85	3.07
	Accuracy + Precision 90% expected tolerance level F |mean| + Inter-batch Lower Limit (%RE) Upper Limit (%RE) Horrat r o Horrat R r	1.28 -0.36 -0.31 0.05 0.04	2.59 -0.86 -0.54 0.12 0.11		1.40 -0.68 -0.42 0.03 0.05	3.57 -1.49 0.49 0.26 0.22
	Cadmium concentrations	P	Spiked chicken muscle		BCR-184
	(ng•g -1 ) Results Accuracy	10+3.2 19 -1.03 -1.04 r e Mean bias (%RE) n LCL e	50+3.2 19 -0.97 -0.99	100+3.2 18 -0.88 -0.89	13 28	0.03 0.02
	UCL Intra-batch(%CV) Inter-batch(%CV) |media| + Inter-batch Lower Limit (%RE) Upper Limit (%RE) Horrat r Horrat R a a repeatability within the same laboratory Precision Accuracy + Precision 90% expected tolerance level Notes:	-1.03 0.95 1.71 2.74 -1.06 -1.00 0.06 0.11 R e v -0.96 0.55 1.10 2.07 0.55 1.10 0.03 0.07 i e w	-0.86 0.64 0.86 1.74 -1.00 -0.75 0.04 0.05	0.03 2.03 1.75 1.77 0.00 9.90 0.14 0.12
					O n
						l y

Table 2b -

 2b Results of precision and accuracy study relative to chicken liver

	Lead concentrations		Spiked chicken liver	BCR-185R
	(ng•g -1 )		50+9.2	100+9.2	500+9.2	172
	Results	n	18	18		18	18
	Accuracy	Mean bias (%RE)	-0.29	-0.75		-0.41	0.46
		LCL	-0.31	-0.77		-0.43	0.37
		UCL	-0.27	-0.73		-0.39	0.56
	Precision	Intra-batch(%CV)	0.26	0.58		1.53	4.35
		Inter-batch(%CV)	0.64	1.93		3.16	3.81
	Accuracy + Precision 90% expected tolerance level F |mean| + Inter-batch Lower Limit (%RE) Upper Limit (%RE) Horrat r o Horrat R r	0.93 -0.38 -0.19 0.02 0.04	2.68 -1.02 -0.48 0.04 0.12		3.57 -0.85 0.02 0.10 0.20	4.27 -0.49 1.41 0.28 0.24
	Cadmium concentrations	P	Spiked chicken liver	BCR-185R
	(ng•g -1 ) Results Accuracy	50+10.4 100+10.4 18 18 -0.33 -0.81 -0.35 -0.83 r e Mean bias (%RE) n LCL e	200+10.4 18 -0.32 -0.34	544 21 -3.84 -4.68
	UCL Intra-batch(%CV) Inter-batch(%CV) |media| + Inter-batch Lower Limit (%RE) Upper Limit (%RE) Horrat r Horrat R a a repeatability within the same laboratory Precision Accuracy + Precision 90% expected tolerance level Notes:	-0.31 0.19 0.63 0.96 -0.43 -0.24 0.01 0.04 R e v -0.79 0.51 1.91 2.72 -1.08 -0.55 0.03 0.12 i e w	-0.30 0.52 1.18 1.50 -0.49 -0.16 0.03 0.07	-3.01 4.35 6.30 10.14 -8.81 1.13 0.28 0.40
					O n
						l y

Table 3 -

 3 Relative standard uncertainty related to the preparation of stock and intermediate solutions

		Food Additives and Contaminants	Page 16 of 20
	Source	Tolerance	Uncertainty	Relative uncertainty
	Standard weight and purity 1000 mg	± 2	1.02000	0.1020%
	Volumetric flask. 1000 mL	± 0.2	0.08165	0.0082%
	Pipette. 500 µL	± 1.6%	0.00656	0.0131%
	Volumetric flask. 50 mL	± 0.06	0.02450	0.0490%
	Pipette. 1000 µL	± 1%	0.04127	0.0041%
	Volumetric flask. 100 mL	± 0.1	0.04083	0.0408%
	Combined uncertainty			0.1206%

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Table 4 -

 4 Relative standard uncertainty related to the preparation of the Certified Reference materials and the artificially spiked samples used for precision analysis St.Dev. result of 7 independently repeated determinations; b calculation reported in Table3; c for standards at 10. 50 and 100 ng g -1

	Source	Tolerance	Uncertainty	Relative
				uncertainty
	BCR184 purity:
	Lead 239 ng•g-1	± 11	5.61	2.35%
	Cadmium 13 ng•g-1	± 2	1.02	7.85%
	weight 0.3 g	0.002 g linearity;	± 0.0014	0.47%
		0.001 g readability
	combined uncertainty
	lead			2.39%
	cadmium			7.86%
	BCR185R purity
	Lead 172 ng•g-1	± 9	4.59	2.67%
	Cadmium 544 ng•g-1	± 17	8.67	1.59%
	weight 0.3 g	0.002 g linearity;	± 0.0014	0.47%
		0.001 g readability
	Combined uncertainty
	lead			2.71%
	cadmium			1.66%
	Chicken muscle:
	Lead 7.41 ng•g-1	series of measures	± 0.044 a	0.59%
	Cadmium 3.21 ng•g-1	series of measures	± 0.048 a	1.49%
	weight 0.5 g	0.002 g linearity;	± 0.0018	0.36%
		0.001 readability
	Standard solution 100 ng•g-1	0.103% b	0.01206	0.12%
	(lead and cadmium)
	Pipette. 50 µL (10 ng•g-1)	± 1%	0.006532	0.0131%
	Pipette. 250 µL (50 ng•g-1)	± 2%	0.008187	0.0033%
	Pipette. 500 µL (100 ng•g-1)	± 1%	0.006560	0.0013%
	Combined uncertainty
	Lead c			0.70%
	Cadmium c			1.54%
	Notes:

a

Table 5 -

 5 Results of the linearity study

	Muscle		Liver
	Pb	Cd	Pb	Cd

Table 6 -

 6 Concentration (in ng g -1 ) of lead and cadmium in broilers and pigeons

		broilers' muscle	broilers' liver	pigeons' muscle
		Lead	Cadmium Lead	Cadmium Lead	Cadmium
	mean	21.

http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants


A Nucciotti
email: angelo.nucciotti@mib.infn.it

E Ferri

O Cremonesi

Expectations for a new calorimetric neutrino mass experiment

Keywords: Neutrino mass, Beta decay, Low-temperature detectors, 187 Re, Montecarlo simulations, Systematic errors

A large calorimetric neutrino mass experiment using thermal detectors is expected to play a crucial role in the challenge for directly assessing the neutrino mass. We discuss and compare here two approaches to the estimation of the experimental sensitivity of such an experiment. The first method uses an analytic formulation and allows to readily obtain a sensible estimate over a wide range of experimental configurations. The second method is based on a frequentist Montecarlo technique and is more precise and reliable. The Montecarlo approach is then exploited to study the main sources of systematic uncertainties peculiar to calorimetric experiments. Finally, the tools are applied to investigate the optimal experimental configuration for a calorimetric experiment with Rhenium based thermal detectors.

Introduction

Assessing the neutrino mass scale is one of the major challenges in today particle physics and astrophysics. This requires to measure the mass of one of the three neutrinos and the kinematical neutrino mass measurement is the only model independent method. In particular, the electron anti-neutrino mass can be measured by precisely analyzing the kinematics of electrons emitted in beta decays. In practice this means measuring the minimum energy carried away by the anti-neutrino, i.e. its rest mass, by observing the highest energy electrons emitted in the decay. To date, the study of the 3 H beta decay endpoint by means of electrostatic spectrometers has proved to be the most sensitive approach, yielding an upper limit on the electron anti-neutrino mass of 2.2 eV [1]. Starting from 2012 the new experiment KATRIN will analyze the 3 H beta decay end-point with a much more sensitive electrostatic spectrometer and with an expected statistical sensitivity of about 0.2 eV [START_REF]KATRIN LoI[END_REF] However, these spectrometric experiments suffer from many systematic uncertainties because the measured electron energy has to be corrected for the energy lost in exciting atomic and molecular states, in crossing the source, in scattering through the spectrometer, and more. To avoid these uncertainties it was proposed to embed the beta source in a detector and to perform a so called calorimetric measurement. Ideally, in such a configuration for each decay the detector measures all the energy released except for the energy carried away by the neutrino. 1A drawback of calorimetry is that a calorimeter is forced to detect all the beta decays while only the ones very close to the end-point E 0 are useful for measuring the neutrino mass. The fraction of useful decays in a small interval ΔE below E 0 is approximately given by (ΔE/E 0 ) 3 , therefore it pays to select a beta decaying isotope with the lowest E 0 value. In the past, calorimetric neutrino mass experiments have been performed implanting 3 H in Silicon diode detectors [START_REF] Simpson | [END_REF]. The use of 187 Re as beta source seems more promising since it is the betaactive nuclide with the second lowest known transition energy (E 0 ∼ 2.5 keV). In the '80s S. Vitale proposed to realize calorimetric neutrino mass measurements using 187 Re as beta source and exploiting the thermal detection technique [4]. Since then two experiments of this kind have been carried out: the MANU [8,7] and MIBETA [5,6] experiments. MANU used one detector with a NTD thermistor glued to a 1.6 mg metallic rhenium single crystal, while MIBETA used an array of ten silicon implanted thermistors with AgReO 4 crystals for a total mass of about 2.2 mg. The two experiments collected statistics corresponding to 10 7 and 1.7×10 7 decays respectively, yielding limits on m ν of about 26 eV at 95% CL and 15 eV at 90% CL respectively.

Thanks to the recent developments in the thermal detection technique, a new very large calorimetric experiment aiming at a sub-eV sensitivity may be feasible: this is what the MARE project is about [START_REF]the MARE proposal[END_REF][START_REF] Nucciotti | [END_REF].

In this paper we present a comprehensive discussion of the potential sensitivity to the neutrino mass for a calorimetric experiment. We define the sensitivity on the neutrino mass as the statistical and systematic uncertainties on m 2 ν [START_REF]KATRIN LoI[END_REF]. First, through an analytical approach, we derive an algorithm to assess the statistical sensitivity for a given experimental configuration. Then a Montecarlo method is described which allows to get more precise statistical sensitivity estimates. The results of the analytic approach are then validated through the comparison with the Montecarlo results over a wide range of experimental parameters. The second part of this paper focuses on the systematic uncertainties peculiar to the calorimetric technique by applying extensively the Montecarlo approach to their investigation. We conclude with a discussion of the possible experimental configurations for future large scale calorimetric experiments.

Before moving to the next sections, it is worth noting that, because of neutrino mixing, the electron (anti-)neutrino emitted in beta decays is a superposition of neutrino mass eigenstates. It can be shown that in the mass range of interest for future kinematical experiments -m ν > 0.1 eV -the experimental observable m ν could be written more correctly as

m ν = i m 2 i |U ei | 2 ≈ m 1 ≈ m 2 ≈ m 3
where m i are the masses of the three neutrino mass eigenstates and U ei are the elements of the electron sector of the neutrino mixing matrix [11].

Statistical sensitivity

The analytic approach

In the following we derive an approximate analytic expression for the statistical sensitivity of a calorimetric neutrino mass experiment. The primary effect of a finite mass m ν on the beta spectrum is to cause the spectrum to turn more sharply down to zero an energy distance m ν below the end-point E 0 (lower panel of Figure 1). To rule out a finite mass, we must be sensitive to the number of counts expected in this interval. The fraction of the total spectrum within an interval ΔE below the end-point E 0 is given by

F ΔE (m ν ) = E0 E0-ΔE N β (E, m ν )dE (1)
where N β (E, m ν ) is the beta energy spectrum for a neutrino mass m ν and normalized to unity. The signal to detect in counts is therefore

signal = A β N det |F ΔE (m ν ) -F ΔE (0)|t M (2)
where A β is the single detector source activity, N det is the number of identical detectors and t M is the measuring time (see lower panel of Figure 1). The detection of this signal is impaired by the noise caused by the statistical fluctuations of the total measured spectrum in the interval ΔE. For a calorimeter the total measured spectrum is obtained summing up -from all detectors -the beta decay events, the counts due to unresolved pile-up of two or more decays, and any additional background counts.

As a first approximation we can neglect the pile-up of more than two decays. We can then crudely approximate the pile-up spectrum by assuming a constant pulse-pair resolving time, τ R , such that events with greater separation are always detected as being doubles, while those at smaller separations are always interpreted as singles with an apparent energy equal to the sum of the two events. In fact, the resolving time will depend on the amplitude of both events, and the sum amplitude will depend on the separation time and the filter used, so a proper calculation would have to be done as a Monte Carlo with the actual filters and pulse-pair detection algorithm being used. However, this approximation is good enough to get the correct scaling and an approximate answer. The parameter τ R is related to the detector signal bandwith and high frequency signal-to-noise ratio: in practice τ R is of the order of the detector rise time.

N β (E,m ν ) τ R A β N β (E,0)⊗N β (E,0) ΔE N β (E,m ν =15eV) N β (E,m ν =0) N β (E,m ν = 0) -N β (E,m ν =15 eV)
The two event pile spectrum is given by

N pp (E) = (1 -e -A β τR )N β (E, 0) ⊗ N β (E, 0) (3)
and the fraction of this spectrum within the interval ΔE below the end-point E 0 is obtained by

F pp ΔE = E0 E0-ΔE N pp (E)dE ≈ τ R A β E0 E0-ΔE N β (E, 0) ⊗ N β (E, 0)dE (4)
where, at first order, τ R A β is the probability for the two event pile-up to occur, i.e. the fraction of unresolved pile-up events, f pp . From (1) and (4) one can write the noise in counts as

noise = A β N det (F ΔE (0) + F pp ΔE )t M + N det bΔEt M ( 5
)
where b is the average background counting rate for unit energy and for a single detector. We can then write the signal to noise ratio in a region within ΔE of the end-point E 0 as

signal noise = A β N det t M |F ΔE (m ν ) -F ΔE (0)| F ΔE (0) + F pp ΔE + bΔE/A β (6)
It is now useful to introduce the exposure T = N det t M and total number of events or total statistics of the experiment

N ev = A β N det t M .
The value of m ν which makes this ratio equal to 1.7 is the sensitivity at 90% confidence level, Σ 90 (m ν ). Therefore one has to solve for m ν the following equation

N ev |F ΔE (m ν ) -F ΔE (0)| F ΔE (0) + F pp ΔE + bΔE/A β = 1.7 ( 7 )
To evaluate (7), we can consider approximate expressions for F ΔE (m ν ) and F pp ΔE . In particular if we restrict ourselves to 187 Re, which has a first forbidden unique beta transition, we can make use of the following empirical spectrum

N β (E, m ν ) ≈ 3 E 3 0 (E 0 -E) 2 1 - m 2 ν (E 0 -E) 2 (8)
which is an extremely good approximation of the expected theoretical shape [START_REF] Dvornický | Measuring mass of neutrinos with beta decays of tritium and rhenium[END_REF] as well as a perfect description of the experimental observations [6,7]. For a null mass, from (8) we can derive

F ΔE (0) = ΔE E 0 3 (9)
while, for a small but finite mass m ν , using a second order expansion in m ν /ΔE we have approximately

F ΔE (m ν ) ≈ F ΔE (0) 1 - 3m 2 ν 2ΔE 2 + 3m 4 ν 8ΔE 4 (10)
For the pile-up spectrum (3) using (8) we can calculate (between 0 and E 0 )

N pp (E) = (1 -e -A β τR ) 1 E 0 9 E E 0 -18 E 2 E 2 0 + 12 E 3 E 3 0 -3 E 4 E 4 0 + 3 10
E 5 E 5 0 (11)
Between E 0 and 2E 0 the expression for N pp (E) is more complicated and it is of no use in this context. Substituting (11) in (4) and carrying out the integration, we obtain

F pp ΔE = f pp 1 20 6 ΔE E 0 + 15 ΔE 2 E 2 0 + 20 ΔE 3 E 3 0 -15 ΔE 4 E 4 0 -6 ΔE 5 E 5 0 - ΔE 6 E 6 0 ( 12
)
where we have used the approximation (1e -A β τR ) ≈ A β τ R = f pp because in all interesting experimental configurations f pp 0.01. Substituting ( 9), ( 10) and ( 12) in (7), keeping only the terms up to (ΔE/E 0 ) 3 and considering that (1 + f pp ) ≈ 1, we obtain

m 2 ν E 3 0 3 2 ΔE - 3m 2 ν 8ΔE N ev = 1.7 ΔE 3 E 3 0 + f pp 3ΔE 10E 0 + 3ΔE 2 4E 2 0 + bΔE/A β ( 13
)
which can be solved for m ν to give the sensitivity at 90% confidence level, Σ 90 (m ν ). Considering only the leading terms in [START_REF] Nucciotti | [END_REF] and ( 12) then the solution is

Σ 90 (m ν ) = 1.13 E 0 4 √ N ev ΔE E 0 + E 0 ΔE 3 10 f pp + b E 0 A β 1 4 (14)
In order to make meaningful use of ( 14) one has to interpret correctly the energy interval ΔE. The same applies to the solution of [START_REF] Gatti | [END_REF].

The two terms in the square bracket in (14) represent the contributions to the noise from the statistical fluctuations of the beta and pile-up spectra respectively (here we neglect the background term for sake of clarity). When the pile-up term is negligible (because of the low rate A β or of the short resolving time τ R ) the left term dominates and it is worth to keep ΔE as small as possible: the limit is of course the detector energy resolution. On the other extreme, when the end-point of the beta spectrum is buried in the pile-up spectrum, the noise is dominated by the right term. In this case the signal-to-noise ratio improves by enlarging the energy interval ΔE.

It is then clear that there is no defined value of ΔE to plug in (14): the solution we have found is to choose ΔE as the value that minimizes Σ 90 (m ν ) for a given set of experimental parameters, with the boundary condition that ΔE cannot be smaller than the detector energy resolution ΔE FWHM .

In particular, for the simpler case of ( 14), by searching the positive zero of the derivative with respect to ΔE, we can obtain the following

ΔE = max E 0 3 10 f pp + b E 0 A β , ΔE FWHM (15)
This approach for defining ΔE can be applied to the solution of ( 13) as well. In this case ΔE can be evaluated numerically following the above prescriptions. In particular to obtain the results presented in this paper, after making the substitution ΔE → |ΔE opt | + ΔE FWHM , we have found numerically the ΔE opt which minimizes the solution of ( 13)

Σ 90 (m ν ) = f (ΔE opt , ΔE FWHM , τ R , A β , N det , t M , b) (16)
It is worth noting that, in this analysis, equations ( 15) and ( 16) are the only places where the detector energy resolution ΔE FWHM shows up.

Montecarlo approach

In this section we describe a frequentist Montecarlo code developed to estimate the statistical sensitivity of a neutrino mass experiment performed with thermal calorimeters. The approach is to simulate the beta spectra that would be measured by a large number of experiments carried out in a given configuration: the spectra are then fit as the real ones [6] and the statistical sensitivity is deduced from the distribution of the obtained m 2 ν parameters. The Montecarlo parameters describing the experimental configuration are the total statistics N ev , the FWHM of the Gaussian energy resolution ΔE FWHM , the fraction of unresolved pile-up events f pp and the background B(E). These input parameters can be derived from the ones actually characterizing a real experiment: N ev = N det A β t M and, by recalling (4), f pp = F pp ΔE=E0 ≈ A β τ R , where again N det is the number of detectors, A β is the beta decay activity of a single detector, t M is the measuring time and τ R is the pile-up resolving time.

The procedure to estimate the statistical sensitivity goes through the following steps:

• The theoretical spectrum S(E) which is expected to be measured by the virtual experiments is evaluated:

S(E) = [N ev (N β (E, m ν ) + f pp N β (E, 0) ⊗ N β (E, 0)) + B(E)] ⊗ R(E) ( 17
)
where N β (E, m ν ) is again the 187 Re beta spectrum normalized to unity (8), B(E) the background energy spectrum and R(E) is the detector energy response function. The B(E) function is usually taken as a constant

B(E) = bT . The response function R(E) is assumed to be a symmetric Gaussian G(E) = 1 σ √ 2π e -E 2 2σ 2 (18)
with standard deviation σ = ΔE FWHM /2.35.

• The virtual outcome of a large number (between 100 and 1000) of experiments is numerically generated by letting the spectrum S(E) fluctuate according to a Poisson statistics. The simulated experimental spectra are generated on an energy interval which is smaller than the full 0 -2E 0 interval.

• Each simulated spectrum is fitted using (17) and leaving m 2 ν , E 0 , N ev , f pp and b as free paramaters. The fit is restricted to an energy interval smaller than the one used for the simulated spectrum generation.

• The 90% C.L. m ν statistical sensitivity Σ 90 (m ν ) of the simulated experimental configuration is given by Σ 90 (m ν ) = 1.7σ m 2 ν , where σ m 2 ν is the standard deviation of the distribution of the m 2 ν found by fitting the spectra.

σ 2

m 2 ν = 1 N -1 i (m 2 νi -m 2 ν ) 2 = N N -1 (m 4 ν -m 2 ν 2 ) ( 19
)
where N is the number of generated spectra and m 2 νi are the values found in each fit for m 2 ν fit parameter. • The statistical error on the 90% C.L. m ν statistical sensitivity is estimated as follows. By defining

y i = (m 2 νi -m 2 ν ) 2 , we have y ≈ σ 2 m 2 ν
and we can write

σ 2 y = N N -1 (y 2 -y 2 ) ≈ N N -1 1 N i (m 2 νi -m 2 ν ) 4 -σ 4 m 2 ν ( 20
)
The error on √ y = σ m 2 ν is given by

√ y = 1 2 σ 2 y N σ m 2 ν (21)
and therefore the error on Σ 90 (m ν ) is obtained

Σ90(mν ) = 1.7 2 √ y Σ 90 (m ν ) (22)
Using equation ( 22) one finds that the statistical error on the Montecarlo results is around 3% and 1% for about 100 and 1000 simulated experiments respectively.

Analytic vs. Montecarlo

We have compared the predictions of the two approaches described in the previous sections for a wide range of experimental configurations suitable for obtaining a sub-eV neutrino mass sensitivity (Figure 2, 3, 4 and 5). In all plots the continuous lines are obtained through equations ( 14) and (15), while the dashed lines are obtained from ( 13) as described at the end of the section on the statistical sensitivity. The symbols are the results of the Montecarlos, which have negligible errors on these scales (see equation ( 22)).

Figure 2 shows how the sensitivity improves for increasing single detectors activity A β , with the other experimental parameters fixed and for an exposure T of 10000 det×year. From this plot one can deduce that it pays to increase the single detector activity at the expense of an increased fraction of pile-up events, f pp . The upper panel of Figure 2 shows the value of the optimal energy interval ΔE opt defined as discussed above.

Figure 3 demonstrates how the impact of the detector FWHM energy resolution ΔE on the sensitivity is reduced by a high pile-up fraction f pp . This Figure shows also one of the limits of the analytic approach described above, i.e. the poor consideration of the detector energy resolution which translates in a too weak dependence of the sensitivity on this parameter.

From Figure 4 it is possible to appreciate the importance of the total statistics N ev to reach a neutrino mass sensitivity of the order of 0.1 eV. In particular energy resolutions ΔE of about 1 eV and pile-up fractions of the order of 10 -6 are required. The fine dashed line on the plot corresponds to a N -1/4 ev functional dependence of the sensitivity (recall equation ( 14)): this dependence may be exploited to scale the Montecarlo results.

In Figure 5 the impact of the continuous background below the beta spectrum is shown. The abscissa is the ratio between N ev and the total number of background counts N bkg between 0 and E 0 , i.e. N bkg = bE 0 T . Clearly the impact is lower for higher pile-up fractions f pp .

For more details on each plots the reader can refer to their captions. In the final section of this paper these results will be used to assess the potential of a calorimetric neutrino mass experiment using 187 Re. 14) with ( 15) and ( 16) respectively. The statistical sensitivity is evaluated for an exposure T = 10000 detector×year. The rightmost point corresponds to a total statistics Nev = 3 × 10 12 . The upper panel shows how ΔEopt gets larger as the pile-up spectrum increase its weight.

ΔE FWHM =1 eV; τ R =1 μs ΔE FWHM =5 eV; τ R =10 μs
Although the agreement is only partial, the comparison confirms that the analytic formulation goes in the right direction to be used to make useful predictions. Nevertheless we believe that the most accurate estimate of the sensitivity is the one obtained through the Montecarlo frequentist approach. From the Figures it is apparent that the formula tends to overestimate the sensitivity for increasing ΔE opt , i.e. when the sensitivity is limited by the pile-up or the continuous background. To improve the predictive power of the analytical approach it is possible to introduce free parameters in the formulas and to adjust them to better reproduce the Montecarlo predictions, but this is out of the scope of this paper.

As a general check, both approaches can be applied taking as input the experimental parameters of the Milano experiment with an array of AgReO 4 crystals whose results are presented in [6]. For N ev = 1.7 × 10 7 , ΔE FWHM = 28.5 eV, f pp = 2.3×10 -4 and N ev /N bkg = 3.28×10 4 one obtains a sensitivity at 90% C.L. of about 15 and 17 eV, using equations ( 14) and the Montecarlo respectively, while the limit on the neutrino mass reported in [6] is about 16 eV at 90% C.L.

ΔE FWHM =1 eV; f pp =10 -6 ΔE FWHM =5 eV; f pp =10 -5 ΔE FWHM =30 eV; f pp =10 -4
ΔE FWHM =5 eV; f pp =10 -4 ΔE FWHM =1 eV; f pp =10 -4 ΔE FWHM =5 eV; f pp =10 -5 ΔE FWHM =1 eV; f pp =10 -5
Figure 5: Comparison between the statistical sensitivity as estimated by a Montecarlo approach (symbols) and by the analytic formulation (lines). The integrated signal-to-background ratio is given by Nev /N bkg where N bkg = bE 0 T (see text) and Nev = 10 14 . An integrated ratio of about 3×10 4 corresponds to the background level measured in the Milano experiment.

Systematic uncertainties

Although, as mentioned in the introduction, a calorimetric neutrino mass experiment is considered free from systematics related to the external source effects, still it may be affected by other uncertainties.

As it will be discussed in the following, the origin of some of these effects (electron escape, beta decay spectral shape and beta environmental fine structure) is indeed related to the beta source and may be unavoidable in spite of the calorimetric configuration. In order to minimize the related uncertainties, this kind of effects must be precisely modelled with the help of theoretical investigations, independent experiments and Montecarlo simulations.

Other systematic uncertainties arise from instrumental effects and can be mitigated through an improved detector design and characterization as well as a careful off-line data analysis.

The frequentist Montecarlo code described above can be readily adapted to estimate the many systematic effects which in general fall under two categories. To the first belong the uncertainties due to lack of accuracy with which the experimental parameters are determined. To assess the corresponding systematic uncertainties in the generated spectra, the parameters are randomly fluctuated -according to the given accuracy -while they are kept fixed to their average value in the fitting function S(E) (see (17)). The second category consists in the effects caused by an incomplete or incorrect modelling of the data. In this case the adopted approach is to include the effects in the generated spectra, but not in the fitting function S(E).

In general these procedures result in a shift of m 2 ν away from zero and, in some cases, in a sensible deterioration of the sensitivity as shown by the wider error bars in the plots. The purpose of this analysis is to identify the size of the inaccuracy or of the neglected effect for which the shift of m 2 ν remains within a given limit. Table 1 reports the results in terms of the systematic uncertainty giving a shift of |m 2 ν | less than 0.01 eV 2 -value for which the systematic uncertainties remain smaller than the statistical error of an experiment aiming at a sub-eV sensitivity.

In the following we analyze in more details the effects that are more likely to introduce systematic uncertainties in calorimetric neutrino mass experiments. Except where differently stated, the plots in this section are obtained for N ev = 10 14 , ΔE FWHM = 1.5 eV and f pp = 10 -6 .

Source related uncertainties

Excited final states. Beta decays to any kind of excited final state are not going to perturb the beta spectrum end-point as long as the state lifetimes are shorter than the detector integration time -which is always more than about 1 μs. In fact, while an excitation energy E exc lost in the beta decay shifts the beta spectrum end-point to E 0 = E 0 -E exc < E 0 , at the same time the coincident detection of the energy E exc released in the state de-excitation adds an energy offset to the beta spectrum. The final outcome is that to each excited state corresponds a beta spectrum starting at E exc and with end-point E 0 = E 0 + E exc . Thanks to its simple functional energy dependency (8) the 187 Re beta decay spectral shape above E exc remains unperturbed. 2 Electron escape. A fraction of electrons emitted in the decays of 187 Re nuclei close to the detector surface will not be contained in the calorimeter. We have used the Geant4 toolkit [15] to estimate the type and magnitude of this effect on the measured spectrum. Figure 6 shows the results for a 1 mg cubic Rhenium detector -i.e. with linear dimensions of about 0.362 mm -in terms of relative deviation with respect to the spectrum given by (8). The simulation has been repeated for the two available Geant4 low energy extensions (the results in Figure 6 are the ones obtained using the Penelope extension) and for different low energy cuts applied in the electron transport. These tests has confirmed the shape of the effect and its magnitude, while giving slightly different results. In conclusion the Montecarlo simulation cannot be considered reliable to precisely calculate the effect, also considering the uncertainties in the shape and size of the detector rhenium absorber. The effect on the measured spectrum can be 2 This is not true for a more general beta spectrum shape as for example in the case of allowed transitions with N (E) ∝ pE(E 0 -E) 2 F (Z, E), where p is the electron momentum and F (Z, E) is the Fermi factor. For calorimeters, it is more generally true that at the end-point the correction for the presence of excited final states vanishes approximately as È i (1 + V i /E), where V i are the final state energies. parametrized as a multiplicative factor to include in (8) given by

f esc (E) = 1 -a esc E E 0 (23)
where the dimensionless a esc parameter will have to be left free in the data analysis. The solid line in Figure 6 corresponds to a esc = 1.9 × 10 -5 . We estimated the systematic error arising when this effect is not included in the data analysis for various values of a esc . The results are plotted in Figure 7.

Beta spectrum uncertainties

Spectral shape. Although the use of equation ( 8) for the 187 Re beta decay spectrum is up to the purpose of the present work, future high statistic experiments will need a more precise description of the spectrum. In order to estimate the sensitivity to deviations from the simple equation ( 8) we have considered the corrective factor

f corr (E) = 1 + a 1 E + a 2 E 2 (24)
which is actually an extension of the correction for the escape of beta electrons (23). The effect of deviations of this kind has been investigated by generating the experimental spectra according to a modified beta spectrum N (E) = N (E)f corr (E) and fitting them using the regular beta spectrum N (E) in S(E) (17). Figure 8 shows the effect on m 2 ν varying the a i coefficients (see the caption for more details). Figure 9 displays the results for positive values of the a i coefficients. For sake of comparison, one can consider that the deviation of the approximate beta spectrum shape described by equation ( 8) from the theoretical one given in [START_REF] Dvornický | Measuring mass of neutrinos with beta decays of tritium and rhenium[END_REF] can be parametrized as

f corr (E) ≈ 1.0 -1.8 × 10 -5 E + 2.8 × 10 -10 E 2 -3.5 × 10 -15 E 3 + ... (25)
Neglecting such a correction in the data analysis would systematically shift m 2 ν by about -280 eV 2 .

Beta Environmental Fine Structure. The Beta Environmental Fine Structure (BEFS) is a modulation of the beta emission probability due to the atomic and molecular surrounding of decaying nuclei: it is the analogous of the oscillation observed in the Extended X-ray Absorption Analysis (EXAFS) and it is explained by the electron wave structure in terms of reflection and interference. Although the phenomenon is completely understood, its description is quite complex and the parameters involved are not known a-priori. Because of its faintness, so far the BEFS has been observed in metallic Rhenium [START_REF] Gatti | [END_REF] and in AgReO 4 [14] only below 1.5 keV where it is larger. It is clear that future neutrino mass experiments will cope with the need of a very accurate description of the BEFS modulation up to the beta spectrum end-point. The parameters presently available are still affected by large statistical errors: for a safe extrapolation up to the end-point the BEFS must be characterized using much higher statistics spectra. Meanwhile the Montecarlo approach can be used to show the shift on m 2 ν when data with BEFS included are fitted to a model without BEFS. For the BEFS function it is assumed that the one used to interpolate the data up to 1.5 keV can be used up to the end-point without modifications. A Montecarlo simulation of the Rhenium BEFS in a measurement with a statistics of about 10 10 events is shown in Figure 10 in terms of residuals of the fit. Figure 11 shows what happens to m 2 ν when fitting spectra like the one in Figure 10 with different left boundaries of the fitting energy interval. The effect worsens when the left boundary is moved to lower energies where the BEFS gets larger. The plot confirms that the inclusion of the BEFS in the end-point analysis is mandatory.

Pile-up spectrum. As discussed in § 2.1, formula (3) holds only under the assumption of a constant resolving time τ R . For real detectors τ R depends on the pulse shape and on the noise level: in practice τ R tends to increase for smaller pulses. The detailed behavior of the resolving time is difficult to predict and must be modelled by Montecarlo methods taking in account both the actual pulse and noise frequency spectra and the algorithm used to identify the double pulses. Examples of such analysis can be found in [16,17]. In order to get a sensible idea of the systematics related to this effect, we have used the results presented in [16]. We have numerically evaluated the pile-up spectrum introducing a variable pile-up rejection efficiency described by an effective resolving time

τ eff R τ eff R = τ R 1 + (n τR -1)e -rAλr ( 26
)
where r A < 1 is the ratio between the amplitudes of the two events to discriminate, n τR is 2 and λ r is 4.0. The function (26) roughly approximates the one described in [16], even though it neglects the time ordering of the two events.

without BEFS; ΔE=5 eV; f pp =10 -6

; E min =2 keV with BEFS; ΔE=5 eV; f pp =10 -6

; E min =2 keV with BEFS; ΔE=5 eV; f pp =10 -6

; E min =2.15 keV with BEFS; ΔE=5 eV; f pp =10 -6

; E min =2.3 keV Figure 11: Systematic effect caused on m 2 ν by neglecting the BEFS when fitting the Montecarlo generated spectra with different left energy boundaries (see Figure 10).

With this approximation we find about 30% more unresolved pile-up events and a good description of the resulting pile-up spectrum for A β τ R 0.1 is given by

N pp (E) = (1 -e -A β τR )N β (E, 0) ⊗ N β (E, 0) 1 + 0.35 e (E-E0)/(480.0 eV) + 1 (27)
Figure 12 shows that the systematic shift caused by neglecting this deviation increases with the pile-up probability τ R A β . In particular it is apparent as a proper modelling of the pile-up spectrum is crucial for a pile-up rate as low as 10 -6 .

Instrumental uncertainties

Response function uncertainty. In (17) the simplest response function R(E) used to model the data is a Gaussian G(E) (18) which is completely determined by its standard deviation σ = ΔE/2.35. The detector FWHM energy resolution ΔE is usually determined by means of a calibration procedure using radioactive sources. The accuracy with which ΔE, and therefore R(E), is known is mainly limited by statistics. Assuming a purely Gaussian R(E), the systematics due to the finite accuracy with which the detector FWHM energy resolution ΔE is known have been evaluated by letting fluctuate the detector energy resolution in the simulated spectra around a central value ΔE which is the fixed resolution used to fit the spectra. The detector energy resolution fluctuates according to a Gaussian distribution centered in ΔE with standard deviation σ err (ΔE). The resulting shift of m 2 ν is shown in the left panel of Figure 13. With the number of events N ev and the pile-up fraction f pp considered in the present analysis, a calibration peak at an energy just above the beta decay end-point E 0 would have only the pile-up spectrum as background. In these conditions, a perfectly Gaussian peak with 10 4 counts would allow an estimation of the FWHM energy resolution ΔE with an accuracy of about 1%.

Response function tails. The actual response function R(E) may be as simple as a Gaussian, though presenting additional extra features which are difficult to identify in the calibration peaks. One example are small tails on the left side of the main Gaussian peak. For a Gaussian with variance σ, the function

T (E) = A tail λ 2 exp (E -E 0 )λ + σλ √ 2 2 1 -erf E -E 0 σ √ 2 + σλ √ 2 (28)
represents an exponential tail with area A tail and decay constant λ. The effect of such an exponential tail has been studied using a response function R(E) = G(E) + T (E) in S(E) when generating the experimental spectra, while keeping the standard Gaussian response function in the fit: Figure 14 shows the effect for various values of λ and A tail < 1.

Of course, identifying an exponential tail with a relative area A tail as small as 10 -4 requires a main Gaussian peak with statistics much larger than 10 4 counts. Calibration error in array. Future calorimetric experiments will be carried out with large arrays of thermal detectors (> 10 4 channels). Since each individual channel has to be energy calibrated prior to be summed up, a systematic effect may arise because of the finite calibration accuracy. For this analysis each simulated experimental spectrum is the sum of 10000 slightly mis-calibrated spectra. The mis-calibration is simulated shifting the energy calibration according to a Gaussian distribution centered in K with standard deviation σ(K), where K is the correct calibration factor. The center panel of Figure 13 displays the effect on m 2 ν . In the simplest hypothesis of a linear energy calibration without pedestal, a couple of calibration peaks close to beta end-point with about 10 4 counts each and with a background arising solely from the pile-up spectrum would in principle allow to determine the calibration factor with an accuracy better than 0.1%.

Response function dispersion in array.

A second effect that can be observed when summing up many channels is due to the spread in the Gaussian response functions of the single detectors: the response function of the sum spectrum will not be Gaussian. In this case the simulated spectrum is the sum of ones whose Gaussian response functions have FWHMs varying according to a Gaussian distribution centered in ΔE with standard deviation σ spread (ΔE). The sum spectrum is analyzed assuming a response function with FWHM equal to ΔE. The results is shown in the left panel of Figure 13. 3The dispersion of the energy resolution in an array strictly depends on the detector technology and it is therefore difficult to predict. Nevertheless, based on the experience with running arrays of thermal detectors, a σ spread better than 10% should be realistic.

Hidden background. In calorimetric experiments, since the beta source cannot be switched off, the background in the energy range of the beta spectrum cannot directly assessed. Therefore a costant background is usually included in the fit model S(E) as the safest hypothesis. Nevertheless we have analyzed the effect of neglecting this term. Figure 15 shows the effect for various levels of constant background bT and confirms the importance of including the background term in S(E).

If no specific measure will be taken to reduce the background, future experiments are expected to have a signal-to-noise ratio similar to the one of the Milano experiment, i.e. about 10 4 -10 5 . In fact increasing the size of the single detector is unlikely to improve much the ratio since the background is expected to scale approximately as the detector mass.

Eventually the background hidden below the beta spectrum could be not flat. We have explored this more critical situation making the simple hypothesis of a linear deviation from flatness starting just on the left of the pile-up spectrum end-point at 2E 0 , expressed as

B(E) = bT 1 + b 1 E 0 (2E 0 -E) (29)
The experimental spectra generated with the above linear background were fit with only the constant term in S(E): the results are shown in Figure 16 for various values of constant background bT and various values for the deviation from flatness b 1 .

Future calorimetric experiments

Given that the single channel activity A β is limited by technical considerations concerning the performance of the thermal detector (heat capacity, quasiparticle diffusion length, ...) the question is whether it is desirable to keep the pile-up negligible or not. There is no unique answer, although increasing the pile-up by increasing A β allows to accumulate more quickly large statistics, and, when pile-up dominates, the dependence on the energy resolution -which tends to degrade when A β increases -is attenuated (see Figure 2 and3). On the other hand the background caused at the end-point by the pile-up, together with a degraded energy resolution, may impair the ability to recognize and understand systematic effects. As a conclusion the optimal design of a neutrino mass experiment depends on the detection technique and, in particular, it depends on the effect of large absorbers on the detector performance. Nevertheless it may be worth to increase the single channel activity as much as possible, therefore relaxing the need of a high energy resolution.

As an example, Table 2 and 3 report the scaled Montecarlo results for a target neutrino mass sensitivity Σ 90 (m ν ) equal to 0.2 eV and 0.1 eV respectively. Results are obtained in absence of background. The first line is a sort of baseline experimental configuration characterized by very demanding energy and time resolution and by very limited pile-up fraction f pp obtained by keeping the single detector activity A β at 1 Hz: in this conditions the target sensitivity is achieved with a relatively low statistics N ev at the expenses of a large required exposure T . In the other lines of the tables a larger activity of 10 Hz is considered together with a progressive degradation of energy and time resolution. While a larger activity implies a lower required exposure, the poorer performances are compensated by the need of a larger statistics. From the tables it is clear that one can find a compromise between performances and exposure which is more convenient than the baseline high performance experimental configuration.

For example a target neutrino mass sensitivity of 0.1 eV be expected running for 10 years 3 × 10 5 rhenium detectors, each with a mass of 10 mggiving an activity of about 10 Hz -and with energy and time resolutions of about 1 eV and 1 μs respectively. The total required mass of rhenium is about 3 kg.

Conclusions

In this paper we have thoroughly discussed the statistical sensitivity of calorimetric Rhenium based neutrino mass experiments.

To estimate the statistical sensitivity, two methods have been developed. They are based respectively on an analytic and a Montecarlo approach: the results presented and compared in § 2.3 show the prominent importance of the total statistics collected by such an experiment in order to reach a sub-eV sensitivity.

Extending the application of the Montecarlo approach, we have then analyzed the expected sources of systematic uncertainties peculiar to this kind of experiments. In particular, in § 3.1 we have shown how crucial is for future experiments the understanding of the theoretical 187 Re beta decay spectrum and of BEFS. On the other hand, in § 3.3 we have shown how instrumental systematic uncertainties may be kept under control by a proper characterization of the response function R(E) and by an accurate detector calibration: tasks that may be accomplished by controlling the calibration peak statistic.

Finally, in § 4, we have exploited the statistical analysis to devise a plausible experimental configuration capable to achieve a sensitivity of about 0.1 eV on the neutrino mass.

As a concluding remarks, we believe we have demonstrated that calorimetric neutrino mass experiments with Rhenium based detectors offer a realistic chance to reach sensitivities comparable, or even beyond, the KATRIN goal. Moreover we have shown that, although systematics related to 187 Re beta decay theory and to BEFS require further investigations, these experiments should not be plagued by large systematic uncertainties.

Figure 1 :

 1 Figure 1: Higher panel: beta spectrum as in (8) compared with pile-up spectrum (3). Lower panel: zoom around the end point, with a comparison between 0 and finite-neutrino-mass beta spectra

Figure 2 :

 2 Figure2: Comparison between the statistical sensitivity as estimated by a Montecarlo approach (symbols) and by the analytic formulation (lines). The continuous and dashed lines are obtained using (14) with (15) and (16) respectively. The statistical sensitivity is evaluated for an exposure T = 10000 detector×year. The rightmost point corresponds to a total statistics Nev = 3 × 10 12 . The upper panel shows how ΔEopt gets larger as the pile-up spectrum increase its weight.

6 Figure 3 :

 63 Figure 3: Comparison between the statistical sensitivity as estimated by a Montecarlo approach (symbols) and by the analytic formulation (lines) for a total statistics Nev of 10 14 events. Continuous and dashed lines are as in Figure 2.

Figure 4 : 4 eV

 44 Figure 4: Comparison between the statistical sensitivity as estimated by a Montecarlo approach (symbols) and by the analytic formulation (lines). Continuous and dashed lines are as in Figure 2. The fine dashed line shows a N 1/4 eV functional dependence for sake of comparison.

Figure 6 :

 6 Figure 6: Spectral distortion caused by surface electron escape as calculated by a Montecarlo simulation for a 1 mg cubic Rhenium detector. The effect plotted in the graph is (N (E) -N (E))/N (E), where N (E) is given by equation (8) and N (E) = N (E)fesc(E)

Figure 7 :Figure 8 : 2 Figure 9 :

 7829 Figure 7: Systematic m 2 ν shift caused by ignoring the for the surface electron escape. Points obtained for Nev = 10 14 , ΔE FWHM = 1.5 eV, fpp = 10 -6 and bT = 0 c/eV.

Figure 10 :

 10 Figure 10: Residuals from the fit of a Montecarlo generated beta spectrum with BEFS using a fit function without BEFS. The Montecarlo is for Nev = 10 10 , ΔE FWHM = 5 eV, fpp = 10 -5 and bT = 0 c/eV.

Figure 12 :Figure 13 :

 1213 Figure 12: Systematic m 2 ν shift caused by ignoring the pile-up spectrum correction for an energy dependent pile-up rejection efficiency. Points obtained for Nev = 10 14 , ΔE FWHM = 1.5 eV, and bT = 0 c/eV.

Figure 14 :

 14 Figure 14: Systematic effect caused on m 2 ν by an undetected exponential in a Gaussian response function R(E). The Montecarlo parameters are Nev = 10 14 , ΔE FWHM = 1.5 eV, fpp = 10 -6 and bT = 0 c/eV.

Figure 15 :

 15 Figure 15: Systematic shift caused on m 2ν by an undetected constant background bT under the beta spectrum. The Montecarlo parameters are Nev = 10 14 , ΔE FWHM = 1.5 eV and fpp = 10 -6 . The Milano experiment signal-to-background ratio of about 3 × 10 4 would translates in to a constant background bT of about 10 6 c/eV on this plot.

Table 2 :

 2 Experimental exposure required for a 0.2 eV mν statistical sensitivity.

	A β	τ R	ΔE	N ev	exposure T
	[Hz] [μs] [eV]	[counts]	[detector×year]
	1	1	1	0.2 × 10 14	7.6 × 10 5
	10	1	1	0.7 × 10 14	2.1 × 10 5
	10	3	3	1.3 × 10 14	4.1 × 10 5
	10	5	5	1.9 × 10 14	6.1 × 10 5
	10	10	10	3.3 × 10 14	10.5 × 10 5

Table 3 :

 3 Experimental exposure required for a 0.1 eV mν statistical sensitivity.

	A β	τ R	ΔE	N ev	exposure T
	[Hz] [μs] [eV]	[counts]	[detector×year]
	1	0.1	0.1	1.7 × 10 14	5.4 × 10 6
	10	0.1	0.1	5.3 × 10 14	1.7 × 10 6
	10	1	1	10.3 × 10 14	3.3 × 10 6
	10	3	3	21.4 × 10 14	6.8 × 10 6
	10	5	5	43.6 × 10 14	13.9 × 10 6

In practice particles emitted at the detector surface or with enough energy may also escape detection. In most cases, however, only a small fraction of the decays are affected by such an effect.

A similar effect due the spread in the resolving time τ R is negligible since it affects only the total pile-up spectrum normalization.

The authors wish to thank Prof. Dan McCammon for the many stimulating discussions on the topic of this paper.

ν by an undetected deviation from flatness of the background under the beta spectrum. The Montecarlo parameters are Nev = 10 14 , ΔE FWHM = 1.5 eV and fpp = 10 -6 . The Milano experiment signal-to-background ratio corresponds to a constant background bT of about 10 6 c/eV, therefore one order of magnitude higher than the worst constant background considered in this analysis.