^{1}, W. W. Moses

^{1}, J. Singh

^{2,a)}, A. N. Vasil’ev

^{3}and R. T. Williams

^{4}

### Abstract

Analytical expressions for the local light yield as a function of the local deposited energy and total scintillation yield integrated over the track of an electron of initial energy are derived from radiative and/or nonradiative rates of first through third order in density of electronic excitations. The model is formulated in terms of rate constants, some of which can be determined independently from time-resolved spectroscopy and others estimated from measured light yield efficiency as a constraint assumed to apply in each kinetic order. The rates and parameters are used in the theory to calculate scintillation yield versus primary electron energy for comparison to published experimental results on four scintillators. Influence of the track radius on the yield is also discussed. Results are found to be qualitatively consistent with the observed scintillation light yield. The theory can be applied to any scintillator if the rates of the radiative and nonradiative processes are known.

This work was supported by the National Nuclear Security Administration, Office of Defense Nuclear Nonproliferation, Office of Nuclear Nonproliferation Research and Engineering (NA-22) of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098 and Grant No. NNSA LB06-316-PD05/NN2001000. We gratefully acknowledge the technical assistance from Ms Francesca Morlino in preparing the manuscript. We would like to thank Steve Payne of Lawrence Livermore National Laboratory for sharing his own comprehensive framework prior to publication in terms of describing nonproportionality of scintillators.

I. INTRODUCTION

II. RATE EQUATIONS

A. Local light yield

B. Total scintillation yield

C. Form of

D. Strictly excitonic excited state

E. Strictly electron and hole excited state

III. EXTRACTION OF RATES FROM EXPERIMENTS AND RESULTS

A. NaI:Tl scintillator

B. (core-valence) scintillator

C. and scintillators

IV. DISCUSSIONS

A. Discussions of results of analytical model

B. Estimation of track radius: Consideration of carrier diffusion in NaI:Tl

C. Additional comments on the axial and radial distribution of excitations in the track region

V. CONCLUSION

### Key Topics

- Excitons
- 42.0
- Electron scattering
- 28.0
- Collisional energy loss
- 23.0
- Excitation energies
- 19.0
- Diffusion
- 18.0

## Figures

The relative light output (total light yield) vs incident energy measured by Mengesha *et al.* (Ref. 11) (a) for seven materials, , LSO, YAP, BGO, GSO, , which show no hump in the yield and (b) for three materials, NaI:Tl, CsI:Tl, and CsI:Na which show a hump in the yield.

The relative light output (total light yield) vs incident energy measured by Mengesha *et al.* (Ref. 11) (a) for seven materials, , LSO, YAP, BGO, GSO, , which show no hump in the yield and (b) for three materials, NaI:Tl, CsI:Tl, and CsI:Na which show a hump in the yield.

(a) The local light yield plotted as a function of (keV/cm) from Eq. (8) for NaI:Tl (1), (2), GSO:Ce (3), and (4). Rates used for different materials are given in Table I.

(a) The local light yield plotted as a function of (keV/cm) from Eq. (8) for NaI:Tl (1), (2), GSO:Ce (3), and (4). Rates used for different materials are given in Table I.

The total light yield plotted as a function of the total energy (keV) from Eq. (11) for NaI:Tl (1), (2), GSO:Ce (3), and (4). Rates used for different materials are given in Table I.

The total light yield plotted as a function of the total energy (keV) from Eq. (11) for NaI:Tl (1), (2), GSO:Ce (3), and (4). Rates used for different materials are given in Table I.

(a) The local light yield plotted as a function of calculated using Eq. (8) and (b) total light yield as a function of the total energy calculated using Eq. (11) for NaI:Tl scintillating crystals with , average track radius 6 nm and using rates from Table I. The dotted line curve corresponds to and solid line corresponds to .

(a) The local light yield plotted as a function of calculated using Eq. (8) and (b) total light yield as a function of the total energy calculated using Eq. (11) for NaI:Tl scintillating crystals with , average track radius 6 nm and using rates from Table I. The dotted line curve corresponds to and solid line corresponds to .

Energy loss function multiplied by the energy loss vs energy loss for different values of . Upper panel is for and lower panel is for .

Energy loss function multiplied by the energy loss vs energy loss for different values of . Upper panel is for and lower panel is for .

Top panel: electron stopping power (left -axis) for NaI crystal ; right axis shows the corresponding density of excitations at the track axis for mean themalization length . Bottom panel: mean free path for electron-electron scattering for NaI (black curve) and mean energy loss per a scattering (gray curve).

Top panel: electron stopping power (left -axis) for NaI crystal ; right axis shows the corresponding density of excitations at the track axis for mean themalization length . Bottom panel: mean free path for electron-electron scattering for NaI (black curve) and mean energy loss per a scattering (gray curve).

## Tables

Rate constants and other parameters used to calculate the scintillation light yield vs particle energy curves for four materials. Values in bold type were determined from independent measurements and considerations as discussed. Values in normal type were constrained as constants common to all four materials since they could only be roughly estimated. Values in italics are varied as fitting parameters among the four materials. References and estimations used in deducing or declaring these parameters are given in the text discussion.

Rate constants and other parameters used to calculate the scintillation light yield vs particle energy curves for four materials. Values in bold type were determined from independent measurements and considerations as discussed. Values in normal type were constrained as constants common to all four materials since they could only be roughly estimated. Values in italics are varied as fitting parameters among the four materials. References and estimations used in deducing or declaring these parameters are given in the text discussion.

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