Temperature dependence of slow positron reemission from metals

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by  Reza Rahemi

Temperature dependence of slow positron reemission from metals
∗
Reza Rahemi
(Dated: November14, 2014)
Intensity of positron reemission from a metal is related to its work function. This property
is dependent on temperature and can be modified with temperature control. In this article a
simple model is proposed to explain and predict the variation of the positron work function with
temperature. Thedependenceofslowpositronyieldofmetalsontemperatureispredictedbasedon
thismodelandthecriticaltemperaturesatwhichtheslowpositronyieldismaximumiscalculated.
6
The proposed model is consistent with experimental observations.
1
0 Keywords: WorkFunction; Positron;Temperature;Metals
2

n
INTRODUCTION author and a T-dependent electron work function model
a
J has been proposed [28].

7 Positron is often used in many applications and in- The objective of this study is to establish a tempera-
2 vestigationmethods,e.g,PositronEmissionTomography ture dependent work function relationship for positron,
(PET), Positronium Annihilation Lifetime Spectroscopy based on the previous study regarding that of the elec-
]
i (PALS) and positron reemission microscopy (PRM). tronworkfunctionandthe existingknowledgeregarding
c
the temperature dependence of the positronium atom.
s Positron Emission Tomography (PET) is a medi-
-
cal imaging technique which uses positrons from the
l
r
decay of a radio-isotope which is inserted into the
t VARIATION IN POSITRON WORK FUNCTION
m body by the means of injection or inhalation of radio-
WITH TEMPERATURE
. pharmaceuticals. Decaying positrons have energies
t
a greater than their rest mass. When positrons interact
m According to Lang and Kohen [29], the electron work
with the nearby electrons, they annihilate by emitting
function consists of two separate contributions. One is
- photonswhicharedetectable. Thisresultsinproduction
d thebulkelectronchemicalpotential, µ− whichisthedif-
of high resolution images of the tissues [1, 2].
n
ferenceintheenergyofgroundstatebeforeandafterthe
o Another important application is the use of the
removal of an electron from the solid. The other is the
c positronbeamasaprobetoanalyzedefectsinmaterials,
[ surface dipole moment, ∆, which is the potential caused
since around a vacancy-type defect, the electron density
by electrons spilling out beyond the metal surface.
1 islocallyreducedwhichresultsinpositronlifetimebeing
v longerthanthatindefect-free lattice[3]. This methodis ϕ− =−µ−+∆ (1)
9 oftenusedtoinvestigatethestructuresofdefectsinsemi-
0
conductors,polymersandnano-structuredmaterialsand
7 and
also to study temperature induced phase transitions in
1
0 High Tc superconductors [4–17].
2. Furthermore, positron reemission microscopes (PRM) ϕ+ =−µ+−∆ (2)
0 areusedforsurfacestudiesinbiologicaltissuesandsemi-
6 conductors [18–26]. Understanding the temperature de- where ϕ+ and ϕ− are work functions of positron and
1 pendenceofthespontaneousreemissionofpositronsfrom electron, and µ+ and µ− are their chemical potentials,
:
v themetalsurfacescanguideselectingordesigningbetter respectively.
i materials for relevant applications. Positronium is a short-lived hydrogen-like atom com-
X
Positron reemission intensity is related to the work posed of an electron and a positron (the antiparticle of
r
a function of positrons which is dependent on tempera- theelectron). Itdecaysbyannihilationtoproducetwoor
ture. However,the effect of temperature on the position more (less often) photons. This exotic atom has a ther-
work function is not well understood. It is possible to malization time of approximately 10−12s in condensed
investigate this effect by looking at the dependence of matter [30]. Since the positronium work function is re-
the work function of its anti-particle, electron, on tem- lated to the electron work function, varies with temper-
perature. The electron work function is a barrier for ature, and is a general property of metals rather than a
the electrons to be moved from inside a solid to a point characteristicofaparticularmetal[31],itcanbeusedas
in vacuum right outside the solid surface [27]. As the a probe to investigate the variationin the positron work
temperature increases, this barrier decreases, since elec- function of metals with temperature.
trons are thermally excited and easier to escape from The positronium work function, composed of electron
the metal surface. The effect of temperature on electron work function (ϕ−) and positron work function (ϕ+), is
workfunctionwaspreviouslyinvestigatedbythe present expressed as
2
PositronemissionfromCopperversustemperature
ϕPs =ϕ++ϕ−−6.8eV (3) %D 21
@
L1
1
where 6.8 eV is the binding energy of the positronium Hu120
atom [31–33]. Differentiating equation 3 with respect to C
or
temperature gives eldf19
yi
n
o
dϕPs dϕ+ dϕ− ositr18
dT = dT + dT (4) -wP TCritical= 271K
o17
Sl
A temperature-dependent work function has been de-
rived via generalization of Lennard-Jones potential and 0 50 100 150 200 250 300 350
is expressed as [28]: Temperature@KD
FIG. 1. Variations in slow positron yield with temperature.
(k T)2 The data points are from ref. [36] and the solid line is pre-
ϕ− =ϕ−−γ B (5)
0 − dicted based on eq. 9.
ϕ
0
−
where ϕ is the electron work function of a material at
0
Experimental results from Schultz and Lynn [36] and
T =0 K and γ is a calculable material coefficient. More
predicted values using eq. (9) are plotted and presented
details about calculation of γ is given in ref. [28] and
some values are presented in Table I. in Fig. 1. Values of c and y0 are obtained by fitting
Differentiating eq. (5), yields the experimental data and the average value for dϕPs is
dT
used from available experimental measurements. dϕPs
dT
dϕ− k2 is a constant and a general property of metals, equal to
=−2γ BT (6) −(6±3)k [31]. Theexperimentalandtheoreticalcurves
dT ϕ− B
0 are in good agreement.
Combining equations 4 and 6 we have:
MORE INFORMATION FROM THE MODEL
dϕ+ dϕPs 2γ
=( + k2T) (7)
dT dT ϕ−0 B It is evident that the slow positron yield increases as
the temperature rises. However, after reaching a critical
Integrating equation 7 yields
temperature, the yield starts to decrease with continu-
ously increasing temperature. Such phenomenon can be
T dϕPs 2γ dϕPs γ explainedbasedontheenergylossofpositronatsurface.
ϕ+ = ( + k2T′)dT′ = T + (k T)2
Z dT ϕ− B dT ϕ− B The proposed model also agrees with this observation.
0 0 0
As eq. (2) suggests, the positron work function is con-
(8)
tributedbyitsbulkchemicalpotentialandsurfacedipole
Experimental results of reemission of implanted slow
moment. Differentiatingeq. (2)withrespecttothe tem-
positron(withenergiesintheorderofeVtoKeV)suggest
thatasthetemperaturerisesfrom50K toabout200K, perature gives
the positron yield should also rise [34–36] The positron
yield (YPs) is related to the positron work function [36]. dϕ+ dµ+ d∆
This can be realized as a decrease in the work function =− − (10)
dT dT dT
of positron and subsequently an increase in the positron
reemission intensity given that the positrons with work Value of dµ+ is negative, since the thermal positrons
functionsequaltothatofametalarere-emittedfromthe dT
lowertheirenergiesbyadjustingtotheionicdensityfluc-
surface[37]. Assumingthatthisrelationshipislinear,the
tuations [38]. The positron chemical potential has three
positron yield may be written as
majorcontributors: zero-pointenergy due to the motion
of the positron in the ionic lattice, the positron-electron
dϕPs γ correlations and the positron-phonon coupling, which is
Yps =C( dT T + ϕ−(kBT)2)+y0 (9) negligible compared to the other two [31].
0 Theterm d∆ ispositive,sinceasmoreelectronsareex-
dT
where c is a constant of proportionality (negative) and cited thermally, the attraction of positrons into the vac-
y0 is the initial value for yield at the absolute zero. uum by the electrostatic dipole potential increases [33].
3
Metal γ ϕ−295K[eV] ϕ−0[eV] TC(Cal.)[K] TC(Exp.)[K] model and the relationship between positron yield and
positron work function.
Al 583 4.28 4.37 261±130 273±150 The author is grateful for financial support from the
Ag 478 4.26 4.33 315±157 200±100 Natural Sciences and Engineering Research Council of
Cu 307 4.5 4.54 515±257 300±50 Canada (NSERC) and to Prof. Dongyang Li for taking
time to review the introduction.
TABLE I. Calculated values for the critical temperature,
TC(Cal.),atwhichthepositronyieldbecomesmaximum,us-
ing equation (11). The experimental values, TC(Exp.), for
alumnium, silver and copper are from experimental results
∗
reported by Ref. [34], [35] and [36] respectively. γ values are [email protected]
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