text: ACKNOWLEDGEMENTS Funding for this project was provided by a grant from la Re´gion Rhoˆne-Alpes. REFERENCES [1] T. Aldenberg and J. Jaworska. Uncertainty of the hazardous concentration and fraction affected for normal species sensitivity distributions. Ecotoxicology and Environmental Safety, 46(1):1–18, May 2000. [2] T. Aldenberg and E. Rorije. Species Sensitivity Distribution estimation from uncertain (QSAR-based) effects data. Alternatives to Laboratory Animals, 41(1):19–31, Mar. 2013. [3] T. Aldenberg and W. Slob. Conﬁdence limits for hazardous concentrations based on logistically distributed NOEC toxicity data. Ecotoxicology and Environmental Safety, 25(1):48–63, 1993. [4] F. A.M Verdonck, J. Jaworska, O. Thas, P. A. Vanrolleghem, and F. a.M Verdonck. Determining environ- mental standards using bootstrapping, bayesian and maximum likelihood techniques: a comparative study. Analytica Chimica Acta, 446(1-2):427–436, Nov. 2001. [5] F. Baty and M. L. Delignette-Muller. nlstools: tools for nonlinear regression diagnostics, 2013. [6] S. P. Brooks and A. Gelman. General methods for monitoring convergence of iterative simulations. Journal of computational and graphical statistics, 7(4):434–455, 1998. [7] F. De Laender, K. a. C. De Schamphelaere, P. A. Vanrolleghem, and C. R. Janssen. Is ecosystem structure the target of concern in ecological effect assessments? Water research, 42(10-11):2395–402, May 2008. [8] R. Dowse, D. Tang, C. G. Palmer, and B. J. Kefford. Risk assessment using the species sensitivity distribution method: data quality versus data quantity. Environmental Toxicology and Chemistry, 32(6):1360–1369, June 2013. [9] B. Efron and R. Tibshirani. An introduction to the bootstrap, volume 57. CRC press, 1993. [10] V. E. Forbes and P. Calow. Species sensitivity distributions revisited: a critical appraisal. Human and Ecological Risk Assessment, 8(3):473–492, 2002. [11] D. R. Fox and E. Billoir. Time-dependent species sensitivity distributions. Environmental toxicology and chemistry / SETAC, 32(2):378–83, Feb. 2013. [12] E. P. M. Grist and K. M. Y. Leung. Better bootstrap estimation of hazardous concentration thresholds for aquatic assemblages. Environmental Toxicology and Chemistry, 21(7):1515–1524, 2009. [13] R. H. Jagoe and M. C. Newman. Bootstrap estimation of community NOEC values. Ecotoxicology, 6(5):293– 306, 1997. [14] B. J. Kefford, R. B. Scha¨fer, and L. Metzeling. Risk assessment of salinity and turbidity in Victoria (Australia) to stream insects’ community structure does not always protect functional traits. The Science of the total environment, 415:61–8, Jan. 2012. [15] G. Kon Kam King, P. Veber, S. Charles, and M. L. Delignette-Muller. MOSAIC-SSD: A new web-tool for species sensitivity distribution to include censored data by maximum likelihood. Environmental Toxicology and Chemistry, pages n/a—-n/a, 2014. [16] F. Larras, A. Bouchez, F. Rimet, and B. Montuelle. Using Bioassays and Species Sensitivity Distributions to Assess Herbicide Toxicity towards Benthic Diatoms. PloS one, 7(8):e44458, Jan. 2012. [17] D. Moore, W. Warren-Hicks, S. Qian, A. Fairbrother, T. Aldenberg, T. Barry, R. Luttik, and H. Ratte. Uncertainty analysis using classical and bayesian hierarchical models. In W. Warren-Hicks and A. Hart, editors, Application of Uncertainty Analysis to Ecological Risk of Pesticides, pages 134–141. CRC press, Pensacola (Florida), US, 2010. [18] S. Pesce, I. Batisson, C. Bardot, C. Fajon, C. Portelli, B. Montuelle, and J. Bohatier. Response of spring and summer riverine microbial communities following glyphosate exposure. Ecotoxicology and environmental safety, 72(7):1905–1912, 2009. [19] M. Plummer. JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003). March, pages 20–22, 2003. [20] L. Posthuma, G. W. Suter II, and T. P. Traas. Species sensitivity distributions in ecotoxicology. CRC press, 2010. [21] M. Power and L. S. McCarty. Fallacies in Ecological Risk Assessment Practices. Environmental Science and Technology, 31(8):370A—-375A, 1997. [22] Q. Shao. Estimation for hazardous concentrations based on NOEC toxicity data: an alternative approach. Environmetrics, 11(5):583–595, Sept. 2000. [23] M. Singh, P. Lodha, and G. P. Singh. Seasonal diatom variations with reference to physico-chemical properties of water of Mansagar lake of Jaipur, Rajasthan. Research Journal of Agricultural Sciences, 1(4):451–457, 2010. [24] N. van der Hoeven. Estimating the 5-percentile of the species sensitivity distributions without any assumptions about the distribution. Ecotoxicology, 10(1):25–34, Feb. 2001. [25] C. Wagner and H. Lokke. Estimation of ecotoxicological protection levels from NOEC toxicity data. Water Research, 25(10):1237–1242, 1991. \ [26] S. Weisberg. Applied linear regression, volume 528. John Wiley & Sons, 2005. [27] J. R. Wheeler, E. P. M. Grist, K. M. Y. Leung, D. Morritt, and M. Crane. Species sensitivity distributions: data and model choice. Marine Pollution Bulletin, 45(1-12):192–202, Jan. 2002. 15

text: ACKNOWLEDGEMENTS

font: OACPDD+NimbusSanL-Bold

text: Funding

font: LDHZMW+URWPalladioL-Roma

text: for

text: this

text: project

text: was

text: provided

text: by

text: a

text: grant

text: from

text: la

text: Re´gion

text: Rhoˆne-Alpes.

text: REFERENCES

text: T.

text: Aldenberg

text: and

text: J.

text: Jaworska.

text: Uncertainty

text: of

text: the

text: hazardous

text: concentration

text: and

text: fraction

text: affected

text: for

text: normal

text: species

text: sensitivity

text: distributions.

text: Ecotoxicology

font: PXLDGA+URWPalladioL-Ital

text: and

text: Environmental

text: Safety,

text: May

text: T.

text: Aldenberg

text: and

text: E.

text: Rorije.

text: Species

text: Sensitivity

text: Distribution

text: estimation

text: from

text: uncertain

text: (QSAR-based)

text: effects

text: data.

text: Alternatives

text: to

text: Laboratory

text: Animals,

text: Mar.

text: T.

text: Aldenberg

text: and

text: W.

text: Slob.

text: Conﬁdence

text: limits

text: for

text: hazardous

text: concentrations

text: based

text: on

text: logistically

text: distributed

text: NOEC

text: toxicity

text: data.

text: Ecotoxicology

text: and

text: Environmental

text: Safety,

text: F.

text: A.M

text: Verdonck,

text: J.

text: Jaworska,

text: O.

text: Thas,

text: P.

text: A.

text: Vanrolleghem,

text: and

text: F.

text: a.M

text: Verdonck.

text: Determining

text: environ-

text: mental

text: standards

text: using

text: bootstrapping,

text: bayesian

text: and

text: maximum

text: likelihood

text: techniques:

text: a

text: comparative

text: study.

text: Analytica

text: Chimica

text: Acta,

text: Nov.

text: F.

text: Baty

text: and

text: M.

text: L.

text: Delignette-Muller.

text: nlstools:

text: tools

text: for

text: nonlinear

text: regression

text: diagnostics,

text: S.

text: P.

text: Brooks

text: and

text: A.

text: Gelman.

text: General

text: methods

text: for

text: monitoring

text: convergence

text: of

text: iterative

text: simulations.

text: Journal

text: of

text: computational

text: and

text: graphical

text: statistics,

text: F.

text: De

text: Laender,

text: K.

text: a.

text: C.

text: De

text: Schamphelaere,

text: P.

text: A.

text: Vanrolleghem,

text: and

text: C.

text: R.

text: Janssen.

text: Is

text: ecosystem

text: structure

text: the

text: target

text: of

text: concern

text: in

text: ecological

text: effect

text: assessments?

text: Water

text: research,

text: May

text: R.

text: Dowse,

text: D.

text: Tang,

text: C.

text: G.

text: Palmer,

text: and

text: B.

text: J.

text: Kefford.

text: Risk

text: assessment

text: using

text: the

text: species

text: sensitivity

text: distribution

text: method:

text: data

text: quality

text: versus

text: data

text: quantity.

text: Environmental

text: Toxicology

text: and

text: Chemistry,

text: June

text: B.

text: Efron

text: and

text: R.

text: Tibshirani.

text: An

text: introduction

text: to

text: the

text: bootstrap,

text: volume

text: CRC

text: press,

text: V.

text: E.

text: Forbes

text: and

text: P.

text: Calow.

text: Species

text: sensitivity

text: distributions

text: revisited:

text: a

text: critical

text: appraisal.

text: Human

text: and

text: Ecological

text: Risk

text: Assessment,

text: D.

text: R.

text: Fox

text: and

text: E.

text: Billoir.

text: Time-dependent

text: species

text: sensitivity

text: distributions.

text: Environmental

text: toxicology

text: and

text: chemistry

text: SETAC,

text: Feb.

text: E.

text: P.

text: M.

text: Grist

text: and

text: K.

text: M.

text: Y.

text: Leung.

text: Better

text: bootstrap

text: estimation

text: of

text: hazardous

text: concentration

text: thresholds

text: for

text: aquatic

text: assemblages.

text: Environmental

text: Toxicology

text: and

text: Chemistry,

text: R.

text: H.

text: Jagoe

text: and

text: M.

text: C.

text: Newman.

text: Bootstrap

text: estimation

text: of

text: community

text: NOEC

text: values.

text: Ecotoxicology,

text: B.

text: J.

text: Kefford,

text: R.

text: B.

text: Scha¨fer,

text: and

text: L.

text: Metzeling.

text: Risk

text: assessment

text: of

text: salinity

text: and

text: turbidity

text: in

text: Victoria

text: (Australia)

text: to

text: stream

text: insects’

text: community

text: structure

text: does

text: not

text: always

text: protect

text: functional

text: traits.

text: The

text: Science

text: of

text: the

text: total

text: environment,

text: Jan.

text: G.

text: Kon

text: Kam

text: King,

text: P.

text: Veber,

text: S.

text: Charles,

text: and

text: M.

text: L.

text: Delignette-Muller.

text: MOSAIC-SSD:

text: A

text: new

text: web-tool

text: for

text: species

text: sensitivity

text: distribution

text: to

text: include

text: censored

text: data

text: by

text: maximum

text: likelihood.

text: Environmental

text: Toxicology

text: and

text: Chemistry,

text: pages

text: n/a—-n/a,

text: F.

text: Larras,

text: A.

text: Bouchez,

text: F.

text: Rimet,

text: and

text: B.

text: Montuelle.

text: Using

text: Bioassays

text: and

text: Species

text: Sensitivity

text: Distributions

text: to

text: Assess

text: Herbicide

text: Toxicity

text: towards

text: Benthic

text: Diatoms.

text: PloS

text: one,

text: Jan.

text: D.

text: Moore,

text: W.

text: Warren-Hicks,

text: S.

text: Qian,

text: A.

text: Fairbrother,

text: T.

text: Aldenberg,

text: T.

text: Barry,

text: R.

text: Luttik,

text: and

text: H.

text: Ratte.

text: Uncertainty

text: analysis

text: using

text: classical

text: and

text: bayesian

text: hierarchical

text: models.

text: In

text: W.

text: Warren-Hicks

text: and

text: A.

text: Hart,

text: editors,

text: Application

text: of

text: Uncertainty

text: Analysis

text: to

text: Ecological

text: Risk

text: of

text: Pesticides,

text: pages

text: CRC

text: press,

text: Pensacola

text: (Florida),

text: US,

text: S.

text: Pesce,

text: I.

text: Batisson,

text: C.

text: Bardot,

text: C.

text: Fajon,

text: C.

text: Portelli,

text: B.

text: Montuelle,

text: and

text: J.

text: Bohatier.

text: Response

text: of

text: spring

text: and

text: summer

text: riverine

text: microbial

text: communities

text: following

text: glyphosate

text: exposure.

text: Ecotoxicology

text: and

text: environmental

text: safety,

text: M.

text: Plummer.

text: JAGS:

text: A

text: program

text: for

text: analysis

text: of

text: Bayesian

text: graphical

text: models

text: using

text: Gibbs

text: sampling.

text: In

text: Proceedings

text: of

text: the

text: International

text: Workshop

text: on

text: Distributed

text: Statistical

text: Computing

text: (DSC

text: March,

text: pages

text: L.

text: Posthuma,

text: G.

text: W.

text: Suter

text: II,

text: and

text: T.

text: P.

text: Traas.

text: Species

text: sensitivity

text: distributions

text: in

text: ecotoxicology.

text: CRC

text: press,

text: M.

text: Power

text: and

text: L.

text: S.

text: McCarty.

text: Fallacies

text: in

text: Ecological

text: Risk

text: Assessment

text: Practices.

text: Environmental

text: Science

text: and

text: Technology,

text: Q.

text: Shao.

text: Estimation

text: for

text: hazardous

text: concentrations

text: based

text: on

text: NOEC

text: toxicity

text: data:

text: an

text: alternative

text: approach.

text: Environmetrics,

text: Sept.

text: M.

text: Singh,

text: P.

text: Lodha,

text: and

text: G.

text: P.

text: Singh.

text: Seasonal

text: diatom

text: variations

text: with

text: reference

text: to

text: physico-chemical

text: properties

text: of

text: water

text: of

text: Mansagar

text: lake

text: of

text: Jaipur,

text: Rajasthan.

text: Research

text: Journal

text: of

text: Agricultural

text: Sciences,

text: N.

text: van

text: der

text: Hoeven.

text: Estimating

text: the

text: of

text: the

text: species

text: sensitivity

text: distributions

text: without

text: any

text: assumptions

text: about

text: the

text: distribution.

text: Ecotoxicology,

text: Feb.

text: C.

text: Wagner

text: and

text: H.

text: Lokke.

text: Estimation

text: of

text: ecotoxicological

text: protection

text: levels

text: from

text: NOEC

text: toxicity

text: data.

text: Water

text: Research,

text: S.

text: Weisberg.

text: Applied

text: linear

text: regression,

text: volume

text: John

text: Wiley

text: Sons,

text: J.

text: R.

text: Wheeler,

text: E.

text: P.

text: M.

text: Grist,

text: K.

text: M.

text: Y.

text: Leung,

text: D.

text: Morritt,

text: and

text: M.

text: Crane.

text: Species

text: sensitivity

text: distributions:

text: data

text: and

text: model

text: choice.

text: Marine

text: Pollution

text: Bulletin,

text: Jan.


text: (a)C(5) (b)C(5)-O FIG. 6. Electron-density on a logarithmic scale for substitutional C and the CO dimer in the TiO 2 lattice, lighter colors denote larger electron densities. (a) Electron-density of the non-magnetic saddle point conﬁguration of C; (b) Electron-density of the stable conﬁguration of CO. (b) and (a) are diﬀerent planes of the crystal lattice, since the bond-axes of the CO dimer turns during formation out of plane (a). charge and spin density is shown in Fig.7(c), (d). For the case of carbon with oxygen on position (4) we ﬁnd a metastable magnetic solution with a magnetic moment of 2µ per supercell. Unlike the cases discussed earlier in this paper, B the magnetic moment is no longer located at carbon only, but is distributed between carbon, oxygen and 3 neighbouring Ti atoms. Figure 7(a), (b) shows the electron density and the spin density, respectively, where the polarization of the Ti d-states is easy to recognize. However, due to the unfavourable total energy of this magnetic state we expect that this case will not be present in real samples. VI. DISCUSSION Fig. 9 shows the calculated µ /µ (cid:48) and µ /µ (cid:48) phase diagrams as well as the correspond- C O N O ing defect formation energy Eform[X] versus oxygen chemical potential µ (cid:48). In Fig.9(a) µ O C vs. µ (cid:48) is plotted giving the thermodynamically stable phases of carbon doped TiO de- O 2 pendent on the chemical potentials of C and O. On the top x-axis the oxygen chemical potential is converted into equivalent oxygen pressure at a ﬁxed temperature (T=1000K). Highlighted in color are the thermodynamically diﬀerent phases for the single carbon sub- 14 ##LTFigure## ##LTFigure##

label: paragraph

text: FIG.

text: Electron-density

label: caption

text: on

text: a

text: logarithmic

text: scale

text: for

text: substitutional

text: C

text: and

text: the

text: CO

text: dimer

text: in

text: the

text: TiO

text: lattice,

text: lighter

text: colors

text: denote

text: larger

text: electron

text: densities.

text: (a)

text: Electron-density

text: of

text: the

text: non-magnetic

text: saddle

text: point

text: conﬁguration

text: of

text: C;

text: (b)

text: Electron-density

text: of

text: the

text: stable

text: conﬁguration

text: of

text: CO.

text: (b)

text: and

text: (a)

text: are

text: diﬀerent

text: planes

text: of

text: the

text: crystal

text: lattice,

text: since

text: the

text: bond-axes

text: of

text: the

text: CO

text: dimer

text: turns

text: during

text: formation

text: out

text: of

text: plane

text: (a).

text: charge

text: and

text: spin

text: density

text: is

text: shown

text: in

text: (d).

text: For

text: the

text: case

text: of

text: carbon

text: with

text: oxygen

text: on

text: position

text: we

text: ﬁnd

text: a

text: metastable

text: magnetic

text: solution

text: with

text: a

text: magnetic

text: moment

text: of

text: 2µ

text: per

text: supercell.

text: Unlike

text: the

text: cases

text: discussed

text: earlier

text: in

text: this

text: paper,

text: B

text: the

text: magnetic

text: moment

text: is

text: no

text: longer

text: located

text: at

text: carbon

text: only,

text: but

text: is

text: distributed

text: between

text: carbon,

text: oxygen

text: and

text: neighbouring

text: Ti

text: atoms.

text: Figure

text: (b)

text: shows

text: the

text: electron

text: density

text: and

text: the

text: spin

text: density,

text: respectively,

text: where

text: the

text: polarization

text: of

text: the

text: Ti

text: d-states

text: is

text: easy

text: to

text: recognize.

text: However,

text: due

text: to

text: the

text: unfavourable

text: total

text: energy

text: of

text: this

text: magnetic

text: state

text: we

text: expect

text: that

text: this

text: case

text: will

text: not

text: be

text: present

text: in

text: real

text: samples.

text: VI.

text: DISCUSSION

text: Fig.

text: shows

text: the

text: calculated

text: µ

text: /µ

text: and

text: µ

text: /µ

text: phase

text: diagrams

text: as

text: well

text: as

text: the

text: correspond-

text: C

text: O

text: N

text: O

text: ing

text: defect

text: formation

text: energy

text: Eform[X]

text: versus

text: oxygen

text: chemical

text: potential

text: µ

text: In

text: µ

text: O

text: C

text: vs.

text: µ

text: is

text: plotted

text: giving

text: the

text: thermodynamically

text: stable

text: phases

text: of

text: carbon

text: doped

text: TiO

text: de-

text: O

text: pendent

text: on

text: the

text: chemical

text: potentials

text: of

text: C

text: and

text: O.

text: On

text: the

text: top

text: x-axis

text: the

text: oxygen

text: chemical

text: potential

text: is

text: converted

text: into

text: equivalent

text: oxygen

text: pressure

text: at

text: a

text: ﬁxed

text: temperature

text: Highlighted

text: in

text: color

text: are

text: the

text: thermodynamically

text: diﬀerent

text: phases

text: for

text: the

text: single

text: carbon

text: sub-

text: ##LTFigure##

text: ##LTFigure##


text: N atoms, respectively, were occupied leading to a doping rate of 4.2%. This was done for two diﬀerent conﬁgurations investigating the eﬀect of increasing distance between the two impurity atoms. In the ﬁrst conﬁguration of this double substitutional state the distance between the two dopants was chosen to be minimal (position (1) and (2)). In the second one the two dopants were placed on arbitrary oxygen sites (positions (1) and (3)) increasing the distance. Moreover, two diﬀerent interstitial positions of C and N (positions (4) and (5)) in the TiO host lattice were examined. Finally, we performed a set of calculations exchanging 2 these interstitial dopants by their next nearest oxygen neighbour, so that O and C or N respectively switch position, with a doping concentration of again 2.1%. FIG. 1. (a) and (b) shows the 48-atom supercell of TiO . The large blue and the small red spheres 2 represent the Ti and the O atoms. The orange numbered spheres (1-3) denote the positions of O substituted by the C and N dopants and the violet numbered spheres (4-5) notify the interstitial doping positions. III. CALCULATIONAL DETAILS All calculations were performed employing the Vienna Ab-initio Simulation Package (VASP)20–25 which uses projector augmented wave (PAW) pseudopotentials26 to describe the potential between the ions. The semi-core s and p states for Ti and the 2s states 4 ##LTFigure## ##LTFigure##

text: N

text: atoms,

text: respectively,

text: were

text: occupied

text: leading

text: to

text: a

text: doping

text: rate

text: of

text: This

text: was

text: done

text: for

text: two

text: diﬀerent

text: conﬁgurations

text: investigating

text: the

text: eﬀect

text: of

text: increasing

text: distance

text: between

text: the

text: two

text: impurity

text: atoms.

text: In

text: the

text: ﬁrst

text: conﬁguration

text: of

text: this

text: double

text: substitutional

text: state

text: the

text: distance

text: between

text: the

text: two

text: dopants

text: was

text: chosen

text: to

text: be

text: minimal

text: (position

text: and

text: In

text: the

text: second

text: one

text: the

text: two

text: dopants

text: were

text: placed

text: on

text: arbitrary

text: oxygen

text: sites

text: (positions

text: and

text: increasing

text: the

text: distance.

text: Moreover,

text: two

text: diﬀerent

text: interstitial

text: positions

text: of

text: C

text: and

text: N

text: (positions

text: and

text: in

text: the

text: TiO

text: host

text: lattice

text: were

text: examined.

text: Finally,

text: we

text: performed

text: a

text: set

text: of

text: calculations

text: exchanging

text: these

text: interstitial

text: dopants

text: by

text: their

text: next

text: nearest

text: oxygen

text: neighbour,

text: so

text: that

text: O

text: and

text: C

text: or

text: N

text: respectively

text: switch

text: position,

text: with

text: a

text: doping

text: concentration

text: of

text: again

text: FIG.

text: (a)

text: and

text: (b)

text: shows

text: the

text: supercell

text: of

text: TiO

text: The

text: large

text: blue

text: and

text: the

text: small

text: red

text: spheres

text: represent

text: the

text: Ti

text: and

text: the

text: O

text: atoms.

text: The

text: orange

text: numbered

text: spheres

text: denote

text: the

text: positions

text: of

text: O

text: substituted

text: by

text: the

text: C

text: and

text: N

text: dopants

text: and

text: the

text: violet

text: numbered

text: spheres

text: notify

text: the

text: interstitial

text: doping

text: positions.

text: III.

text: CALCULATIONAL

text: DETAILS

text: All

text: calculations

text: were

text: performed

text: employing

text: the

text: Vienna

text: Ab-initio

text: Simulation

text: Package

text: which

text: uses

text: projector

text: augmented

text: wave

text: (PAW)

text: to

text: describe

text: the

text: potential

text: between

text: the

text: ions.

text: The

text: semi-core

text: s

text: and

text: p

text: states

text: for

text: Ti

text: and

text: the

text: states

text: ##LTFigure##

text: ##LTFigure##


text: case is lower in energy, while for the C substitution the ”close” C(1,2) conﬁguration is more stable, which is again a consequence of the stronger interaction and points to a tendency of the C atoms to form clusters. The multiple nitrogen substitution conﬁguration N(1,3) has a lower Eform per impurity atom than the single substitutional case N(1). TABLE II. Multiple carbon and nitrogen substitution: Cell volume, total magnetic Moment M tot of the supercell, defect formation energy per impurity atom Eform[X], energy diﬀerence of the FM − and AFM state ∆E[F M AF M] and direct band gap for spin-up and spin-down. X= C (1,2) N (1,2) C (1,3) N (1,3) Cell Volume (˚A3) 522.85(+2.5%) 519.51(+1.8%) 522.85(+2.5%) 517.87(+1.5%) M (µ ) 4 2 4 2 tot B Eform[X] (eV) 9.62 5.47 9.65 5.41 − × − × − ∆E[F M AF M] (eV) 0.160 0.018 1.14 10 3 -0.019 10 3 Spin up gap (eV) 1.40 2.45 1.90 3.16 Spin down gap (eV) 1.40 2.45 1.90 2.33 V. C/N-INTERSTITIAL The ﬁnal investigation deals with carbon and nitrogen placed on interstitial sites (sites (4),(5) in Fig.1) and their interaction with the respective neighbouring oxygen. Upon putting carbon on either of the 2 interstitial sites we ﬁnd that the electronic structure is very similar and no magnetic moment appears. We thus restrict ourselves to present only the density of states for interstitial site (4) (see ﬁgure 5), the respective electron density is given in ﬁgure 6(a). The interaction of the interstitial carbon with the host lattice is almost negligible and C forms atomic like ﬂat bands at the bottom of the TiO gap, which is also reﬂected 2 in the electron density. The non-magnetic state is easily explained from the even number of electrons. In contrast to free carbon, which has a total spin of 1, the small but present crystal ﬁeld from the host lattice leads to a double occupation of a single p-orbital and hence a total spin of zero. The same mechanism can be applied to nitrogen with the only 10 ##LTLine##

text: case

text: is

text: lower

text: in

text: energy,

text: while

text: for

text: the

text: C

text: substitution

text: the

text: ”close”

text: conﬁguration

text: is

text: more

text: stable,

text: which

text: is

text: again

text: a

text: consequence

text: of

text: the

text: stronger

text: interaction

text: and

text: points

text: to

text: a

text: tendency

text: of

text: the

text: C

text: atoms

text: to

text: form

text: clusters.

text: The

text: multiple

text: nitrogen

text: substitution

text: conﬁguration

text: has

text: a

text: lower

text: Eform

text: per

text: impurity

text: atom

text: than

text: the

text: single

text: substitutional

text: case

text: TABLE

text: II.

text: Multiple

text: carbon

text: and

text: nitrogen

text: substitution:

text: Cell

text: volume,

text: total

text: magnetic

text: Moment

text: M

text: tot

text: of

text: the

text: supercell,

text: defect

text: formation

text: energy

text: per

text: impurity

text: atom

text: Eform[X],

text: energy

text: diﬀerence

text: of

text: the

text: FM

text: and

text: AFM

text: state

text: ∆E[F

text: M

text: AF

text: M]

text: and

text: direct

text: band

text: gap

text: for

text: spin-up

text: and

text: spin-down.

text: X=

text: C

text: N

text: C

text: N

text: Cell

text: Volume

text: M

text: (µ

text: tot

text: B

text: Eform[X]

text: (eV)

text: ∆E[F

text: M

text: AF

text: M]

text: (eV)

text: Spin

text: up

text: gap

text: (eV)

text: Spin

text: down

text: gap

text: (eV)

text: V.

text: C/N-INTERSTITIAL

text: The

text: ﬁnal

text: investigation

text: deals

text: with

text: carbon

text: and

text: nitrogen

text: placed

text: on

text: interstitial

text: sites

text: (sites

text: in

text: and

text: their

text: interaction

text: with

text: the

text: respective

text: neighbouring

text: oxygen.

text: Upon

text: putting

text: carbon

text: on

text: either

text: of

text: the

text: interstitial

text: sites

text: we

text: ﬁnd

text: that

text: the

text: electronic

text: structure

text: is

text: very

text: similar

text: and

text: no

text: magnetic

text: moment

text: appears.

text: We

text: thus

text: restrict

text: ourselves

text: to

text: present

text: only

text: the

text: density

text: of

text: states

text: for

text: interstitial

text: site

text: (see

text: ﬁgure

text: the

text: respective

text: electron

text: density

text: is

text: given

text: in

text: ﬁgure

text: The

text: interaction

text: of

text: the

text: interstitial

text: carbon

text: with

text: the

text: host

text: lattice

text: is

text: almost

text: negligible

text: and

text: C

text: forms

text: atomic

text: like

text: ﬂat

text: bands

text: at

text: the

text: bottom

text: of

text: the

text: TiO

text: gap,

text: which

text: is

text: also

text: reﬂected

text: in

text: the

text: electron

text: density.

text: The

text: non-magnetic

text: state

text: is

text: easily

text: explained

text: from

text: the

text: even

text: number

text: of

text: electrons.

text: In

text: contrast

text: to

text: free

text: carbon,

text: which

text: has

text: a

text: total

text: spin

text: of

text: the

text: small

text: but

text: present

text: crystal

text: ﬁeld

text: from

text: the

text: host

text: lattice

text: leads

text: to

text: a

text: double

text: occupation

text: of

text: a

text: single

text: p-orbital

text: and

text: hence

text: a

text: total

text: spin

text: of

text: zero.

text: The

text: same

text: mechanism

text: can

text: be

text: applied

text: to

text: nitrogen

text: with

text: the

text: only

text: ##LTLine##


text: ⊂ R For a ﬁnite conﬁguration Q 3, it is well-known that the energy per particle for the Lennard-Jones potential is bounded below. This condition is called “stability”. An introduction to this concept can be found in [5] section 3.2, jointly with useful criteria to determine stability, e.g. proposition 3.2.8. An important problem associated to these kind of potentials is to estimate the minimum possible energy p(cid:88)er particle. Thus, the “stability constant” is ≥ the minimum real B 0 such that 1 − ≥ − | | Φ(x y) B Q ∈ x,y(cid:54) Q x=y ⊂ R | | for every ﬁnite Q 3. Here Q denotes the cardinality of Q. In this paper ≤ we prove B 14.316 for the Lennard-Jones potential Φ (theorem 5.1). One good reason that motivates the quest for a good estimate for B is its direct implication on the convergence radius of the cluster expansion of the corresponding grandcanonical ensemble. This expansion shows that the system behaves like a gas in the convergence region, and allows to com- pute exactly the thermodynamic observables. In 1963, Penrose and Ruelle have independently shown, for certain class of stable potentials, the conver- | | ≤ − gence region λ (e2βB+1C(β)) 1, where λ is the activity, β the inverse temperature, and C a function that depends on Φ but can be computed in- dependently from B. This classical result was recently improved by Morais, Procacci, Scoppola [4] and de Lima, Procacci [1], so the convergence region can now be written as: | | ≤ 1 λ eβB+1C˜(β) ˜ where C is a function similar to C. So far, the best theoretical upper bound for B = B to their knowledge was 41.66, provided by Schachinger, Addis, LJ Bomze and Schoen [6], which is signiﬁcantly higher than our bound 14.316. Therefore, the present article enlarges drastically the proven convergence region for the Mayer series of the Lennard-Jones gas. As already mentioned in [6], there are not many articles establishing rig- orous results on the lowest energy conﬁgurations for the Lennard-Jones or other similar potentials. There has been, however, a lot of eﬀort put at com- putationally ﬁnding the optimal conﬁgurations for manageable amounts of particles; see for example [3], [8]. A reasonable lower bound for B can be obtained by considering a particular conﬁguration: a face centered cubic lat- ≥ tice. This gives B 8.61; see [6] for the details. Another important feature 2 ##LTLine## ##LTLine##

text: R

text: For

text: a

text: ﬁnite

text: conﬁguration

text: Q

text: it

text: is

text: well-known

text: that

text: the

text: energy

text: per

text: particle

text: for

text: the

text: Lennard-Jones

text: potential

text: is

text: bounded

text: below.

text: This

text: condition

text: is

text: called

text: “stability”.

text: An

text: introduction

text: to

text: this

text: concept

text: can

text: be

text: found

text: in

text: section

text: jointly

text: with

text: useful

text: criteria

text: to

text: determine

text: stability,

text: e.g.

text: proposition

text: An

text: important

text: problem

text: associated

text: to

text: these

text: kind

text: of

text: potentials

text: is

text: to

text: estimate

text: the

text: minimum

text: possible

text: energy

text: particle.

text: Thus,

text: the

text: “stability

text: constant”

text: is

text: the

text: minimum

text: real

text: B

text: such

text: that

text: Φ(x

text: y)

text: B

text: Q

text: Q

text: x=y

text: R

text: for

text: every

text: ﬁnite

text: Q

text: Here

text: Q

text: denotes

text: the

text: cardinality

text: of

text: Q.

text: In

text: this

text: paper

text: we

text: prove

text: B

text: for

text: the

text: Lennard-Jones

text: potential

text: Φ

text: (theorem

text: One

text: good

text: reason

text: that

text: motivates

text: the

text: quest

text: for

text: a

text: good

text: estimate

text: for

text: B

text: is

text: its

text: direct

text: implication

text: on

text: the

text: convergence

text: radius

text: of

text: the

text: cluster

text: expansion

text: of

text: the

text: corresponding

text: grandcanonical

text: ensemble.

text: This

text: expansion

text: shows

text: that

text: the

text: system

text: behaves

text: like

text: a

text: gas

text: in

text: the

text: convergence

text: region,

text: and

text: allows

text: to

text: com-

text: pute

text: exactly

text: the

text: thermodynamic

text: observables.

text: In

text: Penrose

text: and

text: Ruelle

text: have

text: independently

text: shown,

text: for

text: certain

text: class

text: of

text: stable

text: potentials,

text: the

text: conver-

text: gence

text: region

text: λ

text: where

text: λ

text: is

text: the

text: activity,

text: β

text: the

text: inverse

text: temperature,

text: and

text: C

text: a

text: function

text: that

text: depends

text: on

text: Φ

text: but

text: can

text: be

text: computed

text: in-

text: dependently

text: from

text: B.

text: This

text: classical

text: result

text: was

text: recently

text: improved

text: by

text: Morais,

text: Procacci,

text: Scoppola

text: and

text: de

text: Lima,

text: Procacci

text: so

text: the

text: convergence

text: region

text: can

text: now

text: be

text: written

text: as:

text: λ

text: where

text: C

text: is

text: a

text: function

text: similar

text: to

text: C.

text: So

text: far,

text: the

text: best

text: theoretical

text: upper

text: bound

text: for

text: B

text: B

text: to

text: their

text: knowledge

text: was

text: provided

text: by

text: Schachinger,

text: Addis,

text: LJ

text: Bomze

text: and

text: Schoen

text: which

text: is

text: signiﬁcantly

text: higher

text: than

text: our

text: bound

text: Therefore,

text: the

text: present

text: article

text: enlarges

text: drastically

text: the

text: proven

text: convergence

text: region

text: for

text: the

text: Mayer

text: series

text: of

text: the

text: Lennard-Jones

text: gas.

text: As

text: already

text: mentioned

text: in

text: there

text: are

text: not

text: many

text: articles

text: establishing

text: rig-

text: orous

text: results

text: on

text: the

text: lowest

text: energy

text: conﬁgurations

text: for

text: the

text: Lennard-Jones

text: or

text: other

text: similar

text: potentials.

text: There

text: has

text: been,

text: however,

text: a

text: lot

text: of

text: eﬀort

text: put

text: at

text: com-

text: putationally

text: ﬁnding

text: the

text: optimal

text: conﬁgurations

text: for

text: manageable

text: amounts

text: of

text: particles;

text: see

text: for

text: example

text: A

text: reasonable

text: lower

text: bound

text: for

text: B

text: can

text: be

text: obtained

text: by

text: considering

text: a

text: particular

text: conﬁguration:

text: a

text: face

text: centered

text: cubic

text: lat-

text: tice.

text: This

text: gives

text: B

text: see

text: for

text: the

text: details.

text: Another

text: important

text: feature

text: ##LTLine##

text: ##LTLine##


text: ≤ 5 we prove the main theorem, B 14.316. 2 Elementary facts As a convention, we will work with the minus-energy, instead of the en- ergy: R →−h R, h(r) = −r−12 + 2r−6 >0 The following function will play an important role: h˜ = χ + χ ∞ h (0,1] (1,+ ] where χ is the characteristic function of the set R, deﬁned as 1 on R and 0 R on its complement. (cid:39) The function h has a unique root at √1 0.8909. It is negative for 6 2 (cid:48) smaller values and positive for the rest. Its derivative h has a unique root at 1, it is positive before 1 and negative thereafter. R Formulas 2.1. Consider in 3 two balls B and B with radii r and r 1 2 ≤ 1 2 respectively, whose centers are separated by a distance d r + r . The 1 2 volume of the intersection is given by | ∩ | π − − − a) B B = (r + r d)2(d2 + 2d(r + r ) 3(r r )2) 1 2 1 2 1 2 1 2 12d If we call S and S the surfaces of B and B , the area of the portion of S 1 2 1 2 1 inside B is: 2 | ∩ | r − − 1 b) S B = π (r + r d)(r r + d) 1 2 1 2 2 1 d A quick way to prove these formulas is the following. Start showing that ∩ − − the height of the spherical cap S B is equal to 1 (r + r d)(r r + d). 1 2 2d 1 2 2 1 Multiplying this by 2πr we obtain the area. Then we can check the volume 1 | ∩ | formula: the derivative of the volume with respect to r must be S B − 1 1 2 and the volume at r = d r must be 0. 1 2 ⊂ R | | Notation. As it was already done, for a subset X 3, we denote by X the volume, area or cardinality of X according to the dimension of X. For a ∈ R (cid:107) (cid:107) point x 3, x denotes the Euclidean norm. (cid:107) (cid:107) Observation 2.2. Calling H(x) = h( x ), its Laplacian satisﬁes: ≥ (cid:107) (cid:107) ≥ ∆H(x) 0 for x (11)1 . 6 (cid:107) (cid:107) 5 ∆H(x) = 0 for x = (11)1 . 6 ≤ (cid:107) (cid:107) 5≤ ∆H(x) 0 for 0 < x (11)1 . 6 5 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: we

text: prove

text: the

text: main

text: theorem,

text: B

label: section

text: Elementary

text: facts

text: As

text: a

text: convention,

text: we

text: will

text: work

text: with

text: the

text: minus-energy,

text: instead

text: of

text: the

text: en-

text: ergy:

text: R

text: →−h

text: R,

text: h(r)

text: The

text: following

text: function

text: will

text: play

text: an

text: important

text: role:

text: h˜

text: χ

text: χ

text: h

text: where

text: χ

text: is

text: the

text: characteristic

text: function

text: of

text: the

text: set

text: R,

text: deﬁned

text: as

text: on

text: R

text: and

text: R

text: on

text: its

text: complement.

text: The

text: function

text: h

text: has

text: a

text: unique

text: root

text: at

text: √1

text: It

text: is

text: negative

text: for

text: smaller

text: values

text: and

text: positive

text: for

text: the

text: rest.

text: Its

text: derivative

text: h

text: has

text: a

text: unique

text: root

text: at

text: it

text: is

text: positive

text: before

text: and

text: negative

text: thereafter.

text: R

text: Formulas

text: Consider

text: in

text: two

text: balls

text: B

text: and

text: B

text: with

text: radii

text: r

text: and

text: r

text: respectively,

text: whose

text: centers

text: are

text: separated

text: by

text: a

text: distance

text: d

text: r

text: r

text: The

text: volume

text: of

text: the

text: intersection

text: is

text: given

text: by

text: π

text: a)

text: B

text: B

text: (r

text: r

text: r

text: r

text: If

text: we

text: call

text: S

text: and

text: S

text: the

text: surfaces

text: of

text: B

text: and

text: B

text: the

text: area

text: of

text: the

text: portion

text: of

text: S

text: inside

text: B

text: is:

text: r

text: b)

text: S

text: B

text: π

text: (r

text: r

text: d)(r

text: r

text: d)

text: d

text: A

text: quick

text: way

text: to

text: prove

text: these

text: formulas

text: is

text: the

text: following.

text: Start

text: showing

text: that

text: the

text: height

text: of

text: the

text: spherical

text: cap

text: S

text: B

text: is

text: equal

text: to

text: (r

text: r

text: d)(r

text: r

text: d).

text: Multiplying

text: this

text: by

text: we

text: obtain

text: the

text: area.

text: Then

text: we

text: can

text: check

text: the

text: volume

text: formula:

text: the

text: derivative

text: of

text: the

text: volume

text: with

text: respect

text: to

text: r

text: must

text: be

text: S

text: B

text: and

text: the

text: volume

text: at

text: r

text: d

text: r

text: must

text: be

text: R

text: Notation.

text: As

text: it

text: was

text: already

text: done,

text: for

text: a

text: subset

text: X

text: we

text: denote

text: by

text: X

text: the

text: volume,

text: area

text: or

text: cardinality

text: of

text: X

text: according

text: to

text: the

text: dimension

text: of

text: X.

text: For

text: a

text: R

text: point

text: x

text: x

text: denotes

text: the

text: Euclidean

text: norm.

text: Observation

text: Calling

text: H(x)

text: h(

text: x

text: its

text: Laplacian

text: satisﬁes:

text: ∆H(x)

text: for

text: x

text: ∆H(x)

text: for

text: x

text: 5≤

text: ∆H(x)

text: for

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 Estimate for the energy of a particle and minimum interparticle distance (cid:88) It is useful to deﬁne, for every a > 0 { ⊂ R ≥ ∀ ∈ } µ(a) := sup h(x)/Q 3 ﬁnite, d(x, y) a x, y Q ∈ x Q It is the supremum of the possible minus-energies from a point when consid- ering conﬁgurations where every interparticle distance is at least a. In the following proposition, we show reasonable bounds for µ(a), where a < 0.7. The second one is slightly sharper and its proof demands more eﬀort. This improved bound is not essential for our main theorem. However, we included it because it allows to increase the lower bound for the minimum interparticle distance in an optimal conﬁguration, from approximately 0.679 [6] to 0.684. Proposition 3.1. (cid:90) ≤ ≤ I) For 0 a 0.7 we have: ∞ ≤ 24 26.95 µ(a) θ(w)w2dw < a3 a3 0.54 (cid:90) ≤ ≤ II) For 0.6 a 0.7 ∞ ≤ 24 24.05 µ(a) θ(w)w2dw < a3 a3 0.64 Proof. I) Take any conﬁguration Q such that every distance is at least a. ∈ For every x(cid:88)Q with no(cid:88)rm larger than(cid:88)0.89, con(cid:90)sider a ball Bx of radius a2 centered at x. These balls are disjoint, therefore, applying proposition 2.4: ≤ ≤ 1 (cid:107) (cid:107) ≤ h(x) h(x) | | θ( y )dy (cid:90) (cid:90)B x∈Q (cid:107) x(cid:107)≥∈Q (cid:107) x(cid:107)≥∈Q x Bx x 0.89 x 0.89 ∞ ≤ 1 (cid:107) (cid:107) 24 26.95 θ( y )dy = θ(r)r2dr < 4π.(a)3 (cid:107) (cid:107) a3 a3 3 2 y >0.54 0.54 where the integral can be solved analytically. The number 0.54 is equal to − 0.89 0.7. 2 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Estimate

text: for

text: the

text: energy

text: of

text: a

text: particle

text: and

text: minimum

text: interparticle

text: distance

text: It

text: is

text: useful

text: to

text: deﬁne,

text: for

text: every

text: a

text: R

text: µ(a)

text: sup

text: h(x)/Q

text: ﬁnite,

text: d(x,

text: y)

text: a

text: x,

text: y

text: Q

text: x

text: Q

text: It

text: is

text: the

text: supremum

text: of

text: the

text: possible

text: minus-energies

text: from

text: a

text: point

text: when

text: consid-

text: ering

text: conﬁgurations

text: where

text: every

text: interparticle

text: distance

text: is

text: at

text: least

text: a.

text: In

text: the

text: following

text: proposition,

text: we

text: show

text: reasonable

text: bounds

text: for

text: µ(a),

text: where

text: a

text: The

text: second

text: one

text: is

text: slightly

text: sharper

text: and

text: its

text: proof

text: demands

text: more

text: eﬀort.

text: This

text: improved

text: bound

text: is

text: not

text: essential

text: for

text: our

text: main

text: theorem.

text: However,

text: we

text: included

text: it

text: because

text: it

text: allows

text: to

text: increase

text: the

text: lower

text: bound

text: for

text: the

text: minimum

text: interparticle

text: distance

text: in

text: an

text: optimal

text: conﬁguration,

text: from

text: approximately

text: to

text: Proposition

text: I)

text: For

text: a

text: we

text: have:

text: µ(a)

text: II)

text: For

text: a

text: µ(a)

text: Proof.

text: I)

text: Take

text: any

text: conﬁguration

text: Q

text: such

text: that

text: every

text: distance

text: is

text: at

text: least

text: a.

text: For

text: every

text: with

text: larger

text: a

text: ball

text: Bx

text: of

text: radius

text: centered

text: at

text: x.

text: These

text: balls

text: are

text: disjoint,

text: therefore,

text: applying

text: proposition

text: h(x)

text: h(x)

text: θ(

text: y

text: )dy

text: x∈Q

text: x

text: Bx

text: x

text: x

text: θ(

text: y

text: )dy

text: y

text: where

text: the

text: integral

text: can

text: be

text: solved

text: analytically.

text: The

text: number

text: is

text: equal

text: to

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text: ##LTLine##


text: (cid:82) (cid:107) (cid:107) ≥ ≤ (cid:107) (cid:107) II) For r = x 0.64 + a we still can use h(x) | 1 | θ( y )dy but 2 (cid:90) ≤ (cid:107) (cid:107) ≤ Bx Bx to prove the assertion we need, for 0.89 x 0.64 + a 2 ≤ 1 (cid:107) (cid:107) h(x) | | θ0.64( y )dy B x Bx where θ0.64 = χ ∞ θ. We can integrate through spherical coordinates. (0.64,+ ) Let c = a. For a radius w between 0.64 and r+c we must consider the sphere 2 centered at 0 with radius w, and the area of its surface inside B . By the x formula 2.1 b), this area is equal to πw − − πw − − (c + w r)(c w + r) = ( w2 + 2rw + c2 r2) r r T(cid:90) hus we have, calling(cid:90)A = 360(11)16 121 5 (cid:107) (cid:107) ≥ π r+c − − 25 − − ≥ θ0.64( y )d(cid:90)y (Aw 1 )w( w2 + 2rw + c2 r2)dw r 11 Bx 0.64 ≥ π 1.19 − − 25 − − (Aw 1 )w( w2 + 2rw + c2 r2)dw r 11 0.64 We have used h(w) > t(w) if w > (11)1 (proposition 2.3) for the ﬁrst in- 6 5 equality, and that t(w) is positive if w < 1.49 for the second. Notice that 1.19 =(cid:90) 0.89 + 0.6 2 We compute the relevant primitives: (cid:90) − − −1 − w2 + 2rw + (c2 r2) dw = w3 + rw2 + (c2 r2)w = α(w) 3 − − −1 2 1 − w3 + 2rw(cid:90) 2 + (c2 r2)w dw = w4 + rw3 + (c2 r2)w2 = β(w) 4 3 2 1.19 − − 25 − − (Aw 1 )w( w2 + 2rw + c2 r2)dw = 11 0.64 − − 25 − ≥ = A(α(1.19) α(0.64)) (β(1.19) β(0.64)) 11 ≥ − − 0.7224c2 0.7225r2 + 1.2589r 0.5654 where the inequality is due to the truncation of decimal expressions. 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: II)

text: For

text: r

text: x

text: a

text: we

text: still

text: can

text: use

text: h(x)

text: θ(

text: y

text: )dy

text: but

text: Bx

text: Bx

text: to

text: prove

text: the

text: assertion

text: we

text: need,

text: for

text: x

text: a

text: h(x)

text: y

text: )dy

text: B

text: x

text: Bx

text: where

text: χ

text: θ.

text: We

text: can

text: integrate

text: through

text: spherical

text: coordinates.

text: Let

text: c

text: a.

text: For

text: a

text: radius

text: w

text: between

text: and

text: r+c

text: we

text: must

text: consider

text: the

text: sphere

text: centered

text: at

text: with

text: radius

text: w,

text: and

text: the

text: area

text: of

text: its

text: surface

text: inside

text: B

text: By

text: the

text: x

text: formula

text: b),

text: this

text: area

text: is

text: equal

text: to

text: πw

text: πw

text: (c

text: w

text: r)(c

text: w

text: r)

text: r

text: r

text: hus

text: we

text: have,

text: π

text: r+c

text: y

text: (Aw

text: )w(

text: r

text: Bx

text: π

text: (Aw

text: )w(

text: r

text: We

text: have

text: used

text: h(w)

text: t(w)

text: if

text: w

text: (proposition

text: for

text: the

text: ﬁrst

text: in-

text: equality,

text: and

text: that

text: t(w)

text: is

text: positive

text: if

text: w

text: for

text: the

text: second.

text: Notice

text: that

text: We

text: compute

text: the

text: relevant

text: primitives:

text: −1

text: dw

text: α(w)

text: −1

text: dw

text: β(w)

text: (Aw

text: )w(

text: where

text: the

text: inequality

text: is

text: due

text: to

text: the

text: truncation

text: of

text: decimal

text: expressions.

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text: 5. INDIVIDUAL BURSTS The number of components (spikes) per burst (Figure 4, right panel) has a posterior distribution that is bounded, and The generative model described above considers each burst individually, and tries to ﬁt the burst light curve with a su- all samples drawn from this distribution contain between 10 perposition of spikes with unknown parameters, where the and 20 components, with most of the posterior mass between number of spikes in a burst is not ﬁxed a priori. The results 11 and 16 components. This range is slightly higher than the features that are immediately obvious to the eye in the burst are presented in the form of samples from the posterior distri- itself, but may in part be due to some visible features being bution of all model parameters for all spikes, the number of superpositions of spikes. spikes, as well as the hyperparameters listed in Table 1. We In Figure 5, we show a grid of light curves with random can make inferences of individual parameters by marginalis- draws from the posterior distribution of the model for each ing (either integrating for continuous variables, or summing burst displayed. We choose a wide range of peak count rates for discrete variables) over all nuisance parameters (e.g. the from very weak bursts to the brightest examples in our sample, hyperparameters). as well as light curves that vary strongly in complexity, from In Figure 4, we show an example of a burst light curve, to- light curves with only one or two peaks to examples with gether with 10 random draws from the posterior distribution many distinct features. In general, the model will pick out the (left panel). The burst was chosen speciﬁcally for its multi- brightest features and model each of them with a single spike, peaked structure such that we can investigate how well the or a superposition of spikes if the features shows complex method does in inferring the properties of a single burst. Over- structure or a broad peak (the latter are not well matched to our all, the posterior distribution is narrow and peaked for bright simple exponential model shape). While brighter bursts pref- features in the data; the presence of Poisson noise leads to un- erentially have models with more model spikes than weaker certainty in the weaker features, leading to a broader posterior ones (Figure 5, compare left and right columns), the number distribution in those dimensions. of spikes for a given burst as well as spike parameters are very There is some ambiguity for some features on whether there well constrained for high signal-to-noise ratios. For the large should be a component, or whether perhaps a particular feature majority of low-count rate bursts (Figure 5, keft column), the should be modelled as a superposition of two components, but model is nevertheless well constrained to few model compo- this ambiguity is generally small. nents per light curve. The sole exception are the very weakest If we were interested in deciding whether a feature should bursts, with very little emission above the sky background. In be modelled with a spike component, we could marginalise out this case, the model has no obvious features to pick out, and all other parameters except for the position parameter and infer instead ﬁts a large number of very-low amplitude spikes at the presence of a feature based on its position and the number random positions in the light curve (see Figure 5, last row, left of samples in which a component appears at that position. For panel). the quantities we are interested in below, this is not necessary, altDhoRAuFgThVtEhReSIuOnNcMerAtYai2n7t,y20in20spike position may enter as an Preprint typeset using LATEX style emulateapj v. 12/16/11 uncertainty in the distribution of waiting times between spikes. DISSECTING MAGNETAR VARIABILITY WITH BAYESIAN HIERARCHICAL MODELS DANIELA HUPPENKOTHEN1, 2, 3, 4, BRENDON J. BREWER5, DAVID W. HOGG3,2, IAIN MURRAY6, MARCUS FREAN7, CHRIS ELENBAAS1, ANNA L. WATTS1, YURI LEVIN8, ALEXANDER J. VAN DER HORST1, CHRYSSA KOUVELIOTOU9,10 Subject headings: pulsars: individual (SGR J1550-5418), stars: magnetic ﬁelds, stars: neutron, X-rays: bursts, methods:statistics 6. RESULTS the amplitude A, and derived quantities such as the ﬂuence (as a proxy for total dissipated energy) and duration for model We explored statistical distributions of model parameters, spikes for the whole sample of unsaturated Fermi/GBM bursts focussing predominantly on the exponential rise time scale τ, from SGR J1550-5418. We sampled from the posterior dis- tribution for each individual burst light curve, then combined 1 Anton Pannekoek Institute for Astronomy, University of Amsterdam, samples from many bursts to make inferences across the popu- Postbus 94249, 1090 GE Amsterdam, the Netherlands lation. In a naive way, this can be done by picking a parameter 2 Center for Data Science, New York University, 726 Broadway, 7th set from the posterior distribution run for each burst, and com- Floor, New York, NY 10003 3 Center for Cosmology and Particle Physics, Department of Physics, bining these parameter sets from all bursts to a single ensemble New York University, 4 Washington Place, New York, NY 10003, USA to compute the quantity of interest (say, a correlation between 4 E-mail: daniela.huppenkothen@nyu.edu two model parameters). This procedure can be repeated, such 5 Department of Statistics, The University of Auckland, Private Bag that we build up a sample for the quantity in question. 92019, Auckland 1142, New Zealand There are some important limitations to this approach. The 6 School of Informatics, University of Edinburgh most important one is the inherent assumption of independence 7 School of Engineering and Computer Science, Victoria University of Wellington, New Zealand between bursts. By computing and sampling from the posterior 8 Monash Center for Astrophysics and School of Physics, Monash Uni- for each burst individually, we have effectively assumed that versity, Clayton, Victoria 3800, Australia each burst is independent of all other bursts; thus each burst has 9 Astrophysics Ofﬁce, ZP 12, NASA/Marshall Space Flight Center, its own posterior distribution for model shape parameters as Huntsville, AL 35812, USA well as hyperparameters. In practice, however, it is more likely 10 NSSTC, 320 Sparkman Drive, Huntsville, AL 35805, USA

text: INDIVIDUAL

text: BURSTS

text: The

text: number

text: of

text: components

text: (spikes)

text: per

text: burst

text: (Figure

text: right

text: panel)

text: has

text: a

text: posterior

text: distribution

text: that

text: is

text: bounded,

text: and

text: The

text: generative

text: model

text: described

text: above

text: considers

text: each

text: burst

text: individually,

text: and

text: tries

text: to

text: ﬁt

text: the

text: burst

text: light

text: curve

text: with

text: a

text: su-

text: all

text: samples

text: drawn

text: from

text: this

text: distribution

text: contain

text: between

text: perposition

text: of

text: spikes

text: with

text: unknown

text: parameters,

text: where

text: the

text: and

text: components,

text: with

text: most

text: of

text: the

text: posterior

text: mass

text: between

text: number

text: of

text: spikes

text: in

text: a

text: burst

text: is

text: not

text: ﬁxed

text: a

text: priori.

text: The

text: results

text: and

text: components.

text: This

text: range

text: is

text: slightly

text: higher

text: than

text: the

text: features

text: that

text: are

text: immediately

text: obvious

text: to

text: the

text: eye

text: in

text: the

text: burst

text: are

text: presented

text: in

text: the

text: form

text: of

text: samples

text: from

text: the

text: posterior

text: distri-

text: itself,

text: but

text: may

text: in

text: part

text: be

text: due

text: to

text: some

text: visible

text: features

text: being

text: bution

text: of

text: all

text: model

text: parameters

text: for

text: all

text: spikes,

text: the

text: number

text: of

text: superpositions

text: of

text: spikes.

text: spikes,

text: as

text: well

text: as

text: the

text: hyperparameters

text: listed

text: in

text: Table

text: We

text: In

text: Figure

text: we

text: show

text: a

text: grid

text: of

text: light

text: curves

text: with

text: random

text: can

text: make

text: inferences

text: of

text: individual

text: parameters

text: by

text: marginalis-

text: draws

text: from

text: the

text: posterior

text: distribution

text: of

text: the

text: model

text: for

text: each

text: ing

text: (either

text: integrating

text: for

text: continuous

text: variables,

text: or

text: summing

text: burst

text: displayed.

text: We

text: choose

text: a

text: wide

text: range

text: of

text: peak

text: count

text: rates

text: for

text: discrete

text: variables)

text: over

text: all

text: nuisance

text: parameters

text: (e.g.

text: the

text: from

text: very

text: weak

text: bursts

text: to

text: the

text: brightest

text: examples

text: in

text: our

text: sample,

text: hyperparameters).

text: as

text: well

text: as

text: light

text: curves

text: that

text: vary

text: strongly

text: in

text: complexity,

text: from

text: In

text: Figure

text: we

text: show

text: an

text: example

text: of

text: a

text: burst

text: light

text: curve,

text: to-

text: light

text: curves

text: with

text: only

text: one

text: or

text: two

text: peaks

text: to

text: examples

text: with

text: gether

text: with

text: random

text: draws

text: from

text: the

text: posterior

text: distribution

text: many

text: distinct

text: features.

text: In

text: general,

text: the

text: model

text: will

text: pick

text: out

text: the

text: (left

text: panel).

text: The

text: burst

text: was

text: chosen

text: speciﬁcally

text: for

text: its

text: multi-

text: brightest

text: features

text: and

text: model

text: each

text: of

text: them

text: with

text: a

text: single

text: spike,

text: peaked

text: structure

text: such

text: that

text: we

text: can

text: investigate

text: how

text: well

text: the

text: or

text: a

text: superposition

text: of

text: spikes

text: if

text: the

text: features

text: shows

text: complex

text: method

text: does

text: in

text: inferring

text: the

text: properties

text: of

text: a

text: single

text: burst.

text: Over-

text: structure

text: or

text: a

text: broad

text: peak

text: (the

text: latter

text: are

text: not

text: well

text: matched

text: to

text: our

text: all,

text: the

text: posterior

text: distribution

text: is

text: narrow

text: and

text: peaked

text: for

text: bright

text: simple

text: exponential

text: model

text: shape).

text: While

text: brighter

text: bursts

text: pref-

text: features

text: in

text: the

text: data;

text: the

text: presence

text: of

text: Poisson

text: noise

text: leads

text: to

text: un-

text: erentially

text: have

text: models

text: with

text: more

text: model

text: spikes

text: than

text: weaker

text: certainty

text: in

text: the

text: weaker

text: features,

text: leading

text: to

text: a

text: broader

text: posterior

text: ones

text: (Figure

text: compare

text: left

text: and

text: right

text: columns),

text: the

text: number

text: distribution

text: in

text: those

text: dimensions.

text: of

text: spikes

text: for

text: a

text: given

text: burst

text: as

text: well

text: as

text: spike

text: parameters

text: are

text: very

text: There

text: is

text: some

text: ambiguity

text: for

text: some

text: features

text: on

text: whether

text: there

text: well

text: constrained

text: for

text: high

text: signal-to-noise

text: ratios.

text: For

text: the

text: large

text: should

text: be

text: a

text: component,

text: or

text: whether

text: perhaps

text: a

text: particular

text: feature

text: majority

text: of

text: low-count

text: rate

text: bursts

text: (Figure

text: keft

text: column),

text: the

text: should

text: be

text: modelled

text: as

text: a

text: superposition

text: of

text: two

text: components,

text: but

text: model

text: is

text: nevertheless

text: well

text: constrained

text: to

text: few

text: model

text: compo-

text: this

text: ambiguity

text: is

text: generally

text: small.

text: nents

text: per

text: light

text: curve.

text: The

text: sole

text: exception

text: are

text: the

text: very

text: weakest

text: If

text: we

text: were

text: interested

text: in

text: deciding

text: whether

text: a

text: feature

text: should

text: bursts,

text: with

text: very

text: little

text: emission

text: above

text: the

text: sky

text: background.

text: In

text: be

text: modelled

text: with

text: a

text: spike

text: component,

text: we

text: could

text: marginalise

text: out

text: this

text: case,

text: the

text: model

text: has

text: no

text: obvious

text: features

text: to

text: pick

text: out,

text: and

text: all

text: other

text: parameters

text: except

text: for

text: the

text: position

text: parameter

text: and

text: infer

text: instead

text: ﬁts

text: a

text: large

text: number

text: of

text: very-low

text: amplitude

text: spikes

text: at

text: the

text: presence

text: of

text: a

text: feature

text: based

text: on

text: its

text: position

text: and

text: the

text: number

text: random

text: positions

text: in

text: the

text: light

text: curve

text: (see

text: Figure

text: last

text: row,

text: left

text: of

text: samples

text: in

text: which

text: a

text: component

text: appears

text: at

text: that

text: position.

text: For

text: panel).

text: the

text: quantities

text: we

text: are

text: interested

text: in

text: below,

text: this

text: is

text: not

text: necessary,

text: position

text: may

text: enter

text: as

text: an

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: uncertainty

text: in

text: the

text: distribution

text: of

text: waiting

text: times

text: between

text: spikes.

text: DISSECTING

text: MAGNETAR

text: VARIABILITY

text: WITH

text: BAYESIAN

text: HIERARCHICAL

text: MODELS

text: DANIELA

text: BRENDON

text: J.

text: DAVID

text: W.

text: IAIN

text: MARCUS

text: CHRIS

text: ANNA

text: L.

text: YURI

text: ALEXANDER

text: J.

text: VAN

text: DER

text: CHRYSSA

text: Subject

text: headings:

text: pulsars:

text: individual

text: (SGR

text: stars:

text: magnetic

text: ﬁelds,

text: stars:

text: neutron,

text: X-rays:

text: bursts,

text: methods:statistics

text: RESULTS

text: the

text: amplitude

text: A,

text: and

text: derived

text: quantities

text: such

text: as

text: the

text: ﬂuence

text: (as

text: a

text: proxy

text: for

text: total

text: dissipated

text: energy)

text: and

text: duration

text: for

text: model

text: We

text: explored

text: statistical

text: distributions

text: of

text: model

text: parameters,

text: spikes

text: for

text: the

text: whole

text: sample

text: of

text: unsaturated

text: Fermi/GBM

text: bursts

text: focussing

text: predominantly

text: on

text: the

text: exponential

text: rise

text: time

text: scale

text: τ,

text: from

text: SGR

text: We

text: sampled

text: from

text: the

text: posterior

text: dis-

text: tribution

text: for

text: each

text: individual

text: burst

text: light

text: curve,

text: then

text: combined

text: Anton

text: Pannekoek

text: Institute

text: for

text: Astronomy,

text: University

text: of

text: Amsterdam,

text: samples

text: from

text: many

text: bursts

text: to

text: make

text: inferences

text: across

text: the

text: popu-

text: Postbus

text: GE

text: Amsterdam,

text: the

text: Netherlands

text: lation.

text: In

text: a

text: naive

text: way,

text: this

text: can

text: be

text: done

text: by

text: picking

text: a

text: parameter

text: Center

text: for

text: Data

text: Science,

text: New

text: York

text: University,

text: Broadway,

text: set

text: from

text: the

text: posterior

text: distribution

text: run

text: for

text: each

text: burst,

text: and

text: com-

text: Floor,

text: New

text: York,

text: NY

text: Center

text: for

text: Cosmology

text: and

text: Particle

text: Physics,

text: Department

text: of

text: Physics,

text: bining

text: these

text: parameter

text: sets

text: from

text: all

text: bursts

text: to

text: a

text: single

text: ensemble

text: New

text: York

text: University,

text: Washington

text: Place,

text: New

text: York,

text: NY

text: USA

text: to

text: compute

text: the

text: quantity

text: of

text: interest

text: (say,

text: a

text: correlation

text: between

text: E-mail:

text: daniela.huppenkothen@nyu.edu

text: two

text: model

text: parameters).

text: This

text: procedure

text: can

text: be

text: repeated,

text: such

text: Department

text: of

text: Statistics,

text: The

text: University

text: of

text: Auckland,

text: Private

text: Bag

text: that

text: we

text: build

text: up

text: a

text: sample

text: for

text: the

text: quantity

text: in

text: question.

text: Auckland

text: New

text: Zealand

text: There

text: are

text: some

text: important

text: limitations

text: to

text: this

text: approach.

text: The

text: School

text: of

text: Informatics,

text: University

text: of

text: Edinburgh

text: most

text: important

text: one

text: is

text: the

text: inherent

text: assumption

text: of

text: independence

text: School

text: of

text: Engineering

text: and

text: Computer

text: Science,

text: Victoria

text: University

text: of

text: Wellington,

text: New

text: Zealand

text: between

text: bursts.

text: By

text: computing

text: and

text: sampling

text: from

text: the

text: posterior

text: Monash

text: Center

text: for

text: Astrophysics

text: and

text: School

text: of

text: Physics,

text: Monash

text: Uni-

text: for

text: each

text: burst

text: individually,

text: we

text: have

text: effectively

text: assumed

text: that

text: versity,

text: Clayton,

text: Victoria

text: Australia

text: each

text: burst

text: is

text: independent

text: of

text: all

text: other

text: bursts;

text: thus

text: each

text: burst

text: has

text: Astrophysics

text: Ofﬁce,

text: ZP

text: NASA/Marshall

text: Space

text: Flight

text: Center,

text: its

text: own

text: posterior

text: distribution

text: for

text: model

text: shape

text: parameters

text: as

text: Huntsville,

text: AL

text: USA

text: well

text: as

text: hyperparameters.

text: In

text: practice,

text: however,

text: it

text: is

text: more

text: likely

text: NSSTC,

text: Sparkman

text: Drive,

text: Huntsville,

text: AL

text: USA


text: stsrub fo rebmuN time scale to fall time scale deviates signiﬁcantly from 1. We conclusions from the data set comes from the priors for am- parametrize the ﬂux as the peak amplitude A of a compo- plitudes and rise time. An exponential distribution favours n nent n divided by the rise time τ of that component; this low-amplitude spikes with short rise time, which, in principle, n quantity is effectively a measure of the energy injected into could give rise to a population of very short, very small spikes the emission region. If a ﬁreball is formed during the energy modelling individual photons in high-resolution light curves. release in brighter events, one might expect to see longer decay We have already shown in Section that the generally low num- times than otherwise detected, as the ﬁreball would radiate ber of components per burst disfavours this interpretation, but away energy over a longer time interval than if no ﬁreball was that there is a population of spikes with very small amplitudes. formed. On the other hand, ﬁreball formation has no direct In order to improve our understanding of how this prior could impact on the rise time, which therefore should remain unaf- inﬂuence our results, we implemented a log-normal prior for fected. We would therefore expect peaks of larger luminosity both rise time and amplitude, with the mean and standard devi- to be skewed towards preferentially longer decay times. Over- ations of each distribution free hyperparameters with truncated all, the spikes modelling magnetar bursts are skewed (see also Cauchy priors on the location parameter of the log-normal dis- Go¨gˇu¨s¸ et al. 1999; van der Horst et al. 2012): the population tribution, and a uniform prior on the standard deviations (see of skewness parameters resides largely above 1 (corresponding also Table 1). We do not perform formal model comparison to log (s) = 0), indicating that most spikes have faster rise (e.g. using the marginal likelihood introduced in Equation 9), 10 time scales than decays. Moreover, we ﬁnd that there is a but simply explore how the use of different priors changes the positive correlation between our ﬂux measure and the skew- results reported in the previous section. − ness (p < 3.4 × 10 4): brighter bursts have a higher ratio In Figures 11 and 12 we present a comparison between the between rise and fall time scales. However, due to the large distributions for the number of spikes per burst for each set scatter in the distribution, it is not possible to say whether this of priors, and the differential distributions for the most impor- correlation deviates from a linear relationship (as expected for tant quantities for both priors. Overall, there is a signiﬁcant a ﬁreball formation) in the limit of large peak ﬂux. increase in the number of components per burst when using a log-normal prior, related to an excess in short-duration spikes for a log-normal prior in the differential distribution for dura- 60 distribution of the number of components per burst tion, while the distributions for amplitude and ﬂuence remain exponential prior qualitatively the same (Figure 12) and have an overall excess lognormal prior in peaks at the mode of both distributions compared to what 50 is observed in the exponential prior. Much like the differen- tial distributions shown in Figure 12, both the waiting time 40 distribution presented in Figure 8 as well as the correlations between various parameters presented in Figures 9 and 10 re- main statistically compatible within bounds of the intra-model 30 variance (comparisons of priors for these choices not shown here), allowing us to remain conﬁdent in our results in the 20 face of different prior choices. Why the log-normal prior leads to a larger number of model components is not immediately obvious and merits further exploration. However, given that 10 the minor changes in distributions do little to change our con- clusions from the data, we defer this in-depth exploration of 0 priors to future work. 5 10 15 20 25 30 Number of components We also tested whether using binned data rather than the full unbinned Poisson likelihood would have any effect on the FIG. 11.— The number of components (spikes) per burst for 332 bursts for derived results. For 50 randomly chosen bursts, we used the both exponential priors (blue) for amplitude and rise time scale, the standard same prior assumptions and the same generative model, but an prior assumed in deriving the results of this work, and log-normal priors (green) as an alternative hypothesis for the prior distributions on amplitude unbinned Poisson likelihood of the form and rise time scale. Changing the prior to a log-normal distribution leads to an increase in the number of components per burst. (cid:90) (cid:89) L(N, α, {θ })=p(t | N, α, {θ }, H) (12) n n n 6.2. Testing the Prior Assumptions =exp(− λ(t)dt) λ(tn) , Because we have little prior information about the nature n of magnetar bursts and the details of the underlying physical processes producing them, we choose mostly uninformative where t is the nth photon arrival time and λ(t ) is the n n priors for all model parameters. There remains some leeway model ﬂux as deﬁned in Equation 1. The posterior distributions in the choice of these prior distributions, and thus it merits derived with either the binned or the unbinned likelihood are investigation whether changing any priors has an appreciable very similar; there is no qualitative difference in the results DRAFT VERSION MAY 27, 2020 imPpreapcritntotnyptehseet ucsionngcLAlTuEXsisotynlesewmueladteeaprjivv.e1.2/16/11 between the full unbinned likelihood and the binned likelihood An important source of uncertainty in terms of deriving implemented in Equation 4 . ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: stsrub

font: default

text: fo

text: rebmuN

text: time

text: scale

text: to

text: fall

text: time

text: scale

text: deviates

text: signiﬁcantly

text: from

text: We

text: conclusions

text: from

text: the

text: data

text: set

text: comes

text: from

text: the

text: priors

text: for

text: am-

text: parametrize

text: the

text: ﬂux

text: as

text: the

text: peak

text: amplitude

text: A

text: of

text: a

text: compo-

text: plitudes

text: and

text: rise

text: time.

text: An

text: exponential

text: distribution

text: favours

text: n

text: nent

text: n

text: divided

text: by

text: the

text: rise

text: time

text: τ

text: of

text: that

text: component;

text: this

text: low-amplitude

text: spikes

text: with

text: short

text: rise

text: time,

text: which,

text: in

text: principle,

text: n

text: quantity

text: is

text: effectively

text: a

text: measure

text: of

text: the

text: energy

text: injected

text: into

text: could

text: give

text: rise

text: to

text: a

text: population

text: of

text: very

text: short,

text: very

text: small

text: spikes

text: the

text: emission

text: region.

text: If

text: a

text: ﬁreball

text: is

text: formed

text: during

text: the

text: energy

text: modelling

text: individual

text: photons

text: in

text: high-resolution

text: light

text: curves.

text: release

text: in

text: brighter

text: events,

text: one

text: might

text: expect

text: to

text: see

text: longer

text: decay

text: We

text: have

text: already

text: shown

text: in

text: Section

text: that

text: the

text: generally

text: low

text: num-

text: times

text: than

text: otherwise

text: detected,

text: as

text: the

text: ﬁreball

text: would

text: radiate

text: ber

text: of

text: components

text: per

text: burst

text: disfavours

text: this

text: interpretation,

text: but

text: away

text: energy

text: over

text: a

text: longer

text: time

text: interval

text: than

text: if

text: no

text: ﬁreball

text: was

text: that

text: there

text: is

text: a

text: population

text: of

text: spikes

text: with

text: very

text: small

text: amplitudes.

text: formed.

text: On

text: the

text: other

text: hand,

text: ﬁreball

text: formation

text: has

text: no

text: direct

text: In

text: order

text: to

text: improve

text: our

text: understanding

text: of

text: how

text: this

text: prior

text: could

text: impact

text: on

text: the

text: rise

text: time,

text: which

text: therefore

text: should

text: remain

text: unaf-

text: inﬂuence

text: our

text: results,

text: we

text: implemented

text: a

text: log-normal

text: prior

text: for

text: fected.

text: We

text: would

text: therefore

text: expect

text: peaks

text: of

text: larger

text: luminosity

text: both

text: rise

text: time

text: and

text: amplitude,

text: with

text: the

text: mean

text: and

text: standard

text: devi-

text: to

text: be

text: skewed

text: towards

text: preferentially

text: longer

text: decay

text: times.

text: Over-

text: ations

text: of

text: each

text: distribution

text: free

text: hyperparameters

text: with

text: truncated

text: all,

text: the

text: spikes

text: modelling

text: magnetar

text: bursts

text: are

text: skewed

text: (see

text: also

text: Cauchy

text: priors

text: on

text: the

text: location

text: parameter

text: of

text: the

text: log-normal

text: dis-

text: Go¨gˇu¨s¸

text: et

text: al.

text: van

text: der

text: Horst

text: et

text: al.

text: the

text: population

text: tribution,

text: and

text: a

text: uniform

text: prior

text: on

text: the

text: standard

text: deviations

text: (see

text: of

text: skewness

text: parameters

text: resides

text: largely

text: above

text: (corresponding

text: also

text: Table

text: We

text: do

text: not

text: perform

text: formal

text: model

text: comparison

text: to

text: log

text: (s)

text: indicating

text: that

text: most

text: spikes

text: have

text: faster

text: rise

text: (e.g.

text: using

text: the

text: marginal

text: likelihood

text: introduced

text: in

text: Equation

text: time

text: scales

text: than

text: decays.

text: Moreover,

text: we

text: ﬁnd

text: that

text: there

text: is

text: a

text: but

text: simply

text: explore

text: how

text: the

text: use

text: of

text: different

text: priors

text: changes

text: the

text: positive

text: correlation

text: between

text: our

text: ﬂux

text: measure

text: and

text: the

text: skew-

text: results

text: reported

text: in

text: the

text: previous

text: section.

text: ness

text: (p

text: brighter

text: bursts

text: have

text: a

text: higher

text: ratio

text: In

text: Figures

text: and

text: we

text: present

text: a

text: comparison

text: between

text: the

text: between

text: rise

text: and

text: fall

text: time

text: scales.

text: However,

text: due

text: to

text: the

text: large

text: distributions

text: for

text: the

text: number

text: of

text: spikes

text: per

text: burst

text: for

text: each

text: set

text: scatter

text: in

text: the

text: distribution,

text: it

text: is

text: not

text: possible

text: to

text: say

text: whether

text: this

text: of

text: priors,

text: and

text: the

text: differential

text: distributions

text: for

text: the

text: most

text: impor-

text: correlation

text: deviates

text: from

text: a

text: linear

text: relationship

text: (as

text: expected

text: for

text: tant

text: quantities

text: for

text: both

text: priors.

text: Overall,

text: there

text: is

text: a

text: signiﬁcant

text: a

text: ﬁreball

text: formation)

text: in

text: the

text: limit

text: of

text: large

text: peak

text: ﬂux.

text: increase

text: in

text: the

text: number

text: of

text: components

text: per

text: burst

text: when

text: using

text: a

text: log-normal

text: prior,

text: related

text: to

text: an

text: excess

text: in

text: short-duration

text: spikes

text: for

text: a

text: log-normal

text: prior

text: in

text: the

text: differential

text: distribution

text: for

text: dura-

text: distribution

text: of

text: the

text: number

text: of

text: components

text: per

text: burst

text: tion,

text: while

text: the

text: distributions

text: for

text: amplitude

text: and

text: ﬂuence

text: remain

text: exponential

text: prior

text: qualitatively

text: the

text: same

text: (Figure

text: and

text: have

text: an

text: overall

text: excess

text: lognormal

text: prior

text: in

text: peaks

text: at

text: the

text: mode

text: of

text: both

text: distributions

text: compared

text: to

text: what

text: is

text: observed

text: in

text: the

text: exponential

text: prior.

text: Much

text: like

text: the

text: differen-

text: tial

text: distributions

text: shown

text: in

text: Figure

text: both

text: the

text: waiting

text: time

text: distribution

text: presented

text: in

text: Figure

text: as

text: well

text: as

text: the

text: correlations

text: between

text: various

text: parameters

text: presented

text: in

text: Figures

text: and

text: re-

text: main

text: statistically

text: compatible

text: within

text: bounds

text: of

text: the

text: intra-model

text: variance

text: (comparisons

text: of

text: priors

text: for

text: these

text: choices

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text: us

text: to

text: remain

text: conﬁdent

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text: our

text: results

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text: the

text: face

text: of

text: different

text: prior

text: choices.

text: Why

text: the

text: log-normal

text: prior

text: leads

text: to

text: a

text: larger

text: number

text: of

text: model

text: components

text: is

text: not

text: immediately

text: obvious

text: and

text: merits

text: further

text: exploration.

text: However,

text: given

text: that

text: the

text: minor

text: changes

text: in

text: distributions

text: do

text: little

text: to

text: change

text: our

text: con-

text: clusions

text: from

text: the

text: data,

text: we

text: defer

text: this

text: in-depth

text: exploration

text: of

text: priors

text: to

text: future

text: work.

text: Number

text: of

text: components

text: We

text: also

text: tested

text: whether

text: using

text: binned

text: data

text: rather

text: than

text: the

text: full

text: unbinned

text: Poisson

text: likelihood

text: would

text: have

text: any

text: effect

text: on

text: the

text: FIG.

text: The

text: number

text: of

text: components

text: (spikes)

text: per

text: burst

text: for

text: bursts

text: for

text: derived

text: results.

text: For

text: randomly

text: chosen

text: bursts,

text: we

text: used

text: the

text: both

text: exponential

text: priors

text: (blue)

text: for

text: amplitude

text: and

text: rise

text: time

text: scale,

text: the

text: standard

text: same

text: prior

text: assumptions

text: and

text: the

text: same

text: generative

text: model,

text: but

text: an

text: prior

text: assumed

text: in

text: deriving

text: the

text: results

text: of

text: this

text: work,

text: and

text: log-normal

text: priors

text: (green)

text: as

text: an

text: alternative

text: hypothesis

text: for

text: the

text: prior

text: distributions

text: on

text: amplitude

text: unbinned

text: Poisson

text: likelihood

text: of

text: the

text: form

text: and

text: rise

text: time

text: scale.

text: Changing

text: the

text: prior

text: to

text: a

text: log-normal

text: distribution

text: leads

text: to

text: an

text: increase

text: in

text: the

text: number

text: of

text: components

text: per

text: burst.

text: L(N,

text: α,

text: {θ

text: })=p(t

text: N,

text: α,

text: {θ

text: H)

text: n

text: n

text: n

text: Testing

text: the

text: Prior

text: Assumptions

text: =exp(−

text: λ(t)dt)

text: λ(tn)

text: Because

text: we

text: have

text: little

text: prior

text: information

text: about

text: the

text: nature

text: n

text: of

text: magnetar

text: bursts

text: and

text: the

text: details

text: of

text: the

text: underlying

text: physical

text: processes

text: producing

text: them,

text: we

text: choose

text: mostly

text: uninformative

text: where

text: t

text: is

text: the

text: nth

text: photon

text: arrival

text: time

text: and

text: λ(t

text: is

text: the

text: n

text: n

text: priors

text: for

text: all

text: model

text: parameters.

text: There

text: remains

text: some

text: leeway

text: model

text: ﬂux

text: as

text: deﬁned

text: in

text: Equation

text: The

text: posterior

text: distributions

text: in

text: the

text: choice

text: of

text: these

text: prior

text: distributions,

text: and

text: thus

text: it

text: merits

text: derived

text: with

text: either

text: the

text: binned

text: or

text: the

text: unbinned

text: likelihood

text: are

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