Atomic ionoluminescence used in differential mesurement of excited state lifetimes

NUCLEAR I N S T R U M E N T S

AND METHODS

157 ( 1 9 7 8 )

99-108 : ©

NORTH-HOLLAND

PUBLISHING CO.

ATOMIC IONOLUMINESCENCE USED IN DIFFERENTIAL MEASUREMENT OF EXCITED STATE LIFETIMES

J.

AZENCOT and

R.

GOUTTE

Institut National des Sciences Appliqudes de Lyon, Laboratoire d'Optique Cotpusculaire et d'UItrasons, 69621 - Villeurbanne, France Received 9 March 1978 and in revised form 10 July 1978 We describe a new method of measuring the excited state lifetimes using a new excitation process: atomic ionolumi nescence. The measuring technique itself corresponds to the phase shift method oriented towards a differential measurement. In this article which is more concerned with the theoretic study of the method, we give the essential relations which define the measuring technique. We then supply some indications concerning the systematic errors which derive from this technique and we propose in particular two methods of calculating the cascade corrections. To illustrate this method we also present the results of lithium lifetime measurements. We finish this first study by comparing the advantages and disadvantages of this technique with other known methods. We show its great simplicity of use, which allows the study of solid conducting or insulating samples.

1. Introduction When the surface of a solid target is bombarded by an ionic beam of a few keV energy, the analysis of the particles emitted from this collision give: a) atoms and ions originating from the incident beam t ), 13) atoms and ions originating from the target2), c) atoms and ions originating from the adsorbed gases on the target3), d) secondary electrons4), e) photonsS-7). In the spectral analysis of the photonic emission we can observe two sorts of distributions: a) a continuous spectrum corresponding to the crystalline luminescence caused by the de-excitations of the atoms still in the interior of the targel:8), 13) a line spectrum whose wavelengths correspond to energy transitions of sputtered excited atoms. These atoms could come from either the incident atomic beam (excited by their impact on the target then reflected)9), or from the targed°). This last photonic emission caused by the neutral target atoms sputtered in the excited state characterises the general phenomenon known as atomic ionoluminescence ~l). In this article we will begin by developing the principles of a new method of measuring the excited state lifetimes using the atomic ionoluminescence as an excitation process. The measurement itself uses the well known phase shift methodl2), and we present this as a differential measurement

technique since it adapts best in this form to the atomic lifetime measurement. We will then give some indications of the different kinds of systematic errors attached to this method and we specially insist on the correction of cascade phenomena. We present some results concerning the lithium lifetime measurement derived from this technique. We will finish this first analysis by comparing the advantages and disadvantages of this method with other methods generally used for atomic lifetime measurement. Later we will present a second article specially reserved for the experimental results of lifetime measurements given by this technique.

2. Measurement principles 2.1. EXPERIMENTALARRANGEMENT A solid target T (see fig. 1) containing atoms

B

t~lt) I .

L /o.g. P.M.

. iv(t)

Fig. 1. Schematic of the experimental set-up.

I00

J. A Z E N C O T

E~£

,/, / ,'cascades / / r e p o p u l a t ions

Ek

AND R. G O U T T E

upper levels

It will be necessary therefore, if we wish to observe a given luminous emission Zkj whose luminance wili be written /~kj (or more simply l~ or even Iv, introducing the frequency v), to use both a light collector represented by the lens L, and an optical filter 0.F. (interference filter or monochromator) adapted to the radiation 2kj. At the end of the apparatus we put a photoelectric current converter P.M. (photomuttiplier) this allows us to measure the luminance l,.(t) by means of the current i,,(t).

/ /

// // //

//

level t/ under stu

Ej lower levels Fig. 2. Transitions between atomic energy levels.

whith lifetimes rk relative to a given energy level Ek (see fig. 2) that we wish to measure, is bombarded by an accelerated and modulated ionic beam B. The area unit time density of incident particles near the target is denoted as j(t). Because of collisions between the incident ions and target atoms, the latter are sputtered and leave the target in an excited state. This state is characterised by different unequally probable levels E]
2.2. SYSTEM TRANSFER FUNCTION

If in the first study we ignore the cascade repopulation of the k level [whose population is nk(t)], it can be shown (see appendix 1) that the evolution law of nk (t) and the luminance/v (t) satisfy the equations (the k term being understood): dn(t) n(t) - + = CN jCt), (2) dt z

lvCt) = _ff J 1Ak hvnCt).

Relations in which Akj represents the atomic transition probability for unit time from level Ek to level Ej ; r (or rk) the lifetime of level k given by the r e l a t i o n 1 rk = ~ AkJ , (4) j
v(or vkj) the observed radiation f r e q u e n c y C (or Ck) the average efficient section of incident ion interaction with target atoms, to populate level k; N (or Nk) the global population of all the levels below E k. This value is given by the relation: =

study). The emitted photons in fact make a luminous pocket of a certain thickness (about 100 ~tm) surrounding the targetl4).

J(P)

Ek level deexcitation

observation of Xkj radiation NIP)

"~"Akj

Fig. 3. Partial transfer function of the studied system.

(s)

If taking the Laplace transforms of relations (2) and (3) [these we symbolise by using capitals: for examplej(t) fJ(p)], we easily arrive at the transfer function in fig. 3. This transfer function introduces three terms which are respectively" a) CN: parameters connected with the shock excitation caused by the incident ionic beam.

(t) represents the emitted speed of the atom under

excitation parameters

nj. j
(1)

vt

(3)

l

~- L,,(P)

ATOMIC

IONOLUMINESCENCE

b) r / 1 + r p : spontaneous radiative de excitation law of the studied Ek level. c) 1/ H A k/h v : parameters related to the observation of a given 2~j radiation wavelength. In grouping the coefficients which have no particular role in lifetime m e a s u r e m e n t , and if we put: k = -~ C N A k j h V ,

(7)

This shows that this first analysis leads to a simplified first order system and that the phase shift tit) between/v (t) and./(t) will be for a sinusoidal modulation of ./(t) at the angular frequency 09o: 4) = - arctg (Doz, which implies that: 1 z -tgq5 6Oo

ment of r~ by means of the knowledge of z r as in the relation below: 1

zi -

(D O (Dr =

tg(qS~ + qSf),

(lO)

arctg 6ooTr"

--

2.3. OPTIMISATION OF THE DIFFERENTIAL MEASUREMENT

(6)

fig:. 3 implies: H(p) - Lv(p) _ kz J (p) 1 + zp "

101

(8)

This relation is the basis of the phase shift method and allows us to establish r by means of the phase shift m e a s u r e m e n t 0 between /~(t) and ./(t). The main inconvenience of this m e t h o d which relies on a phase m e a s u r e m e n t , comes from the necessity to determine precisely all the phase shift errors attached to the experiment. It seems to us more advantageous to develop this m e t h o d by directing it towards a differential type measurement. The principle consists of measuring the phase shift 0j on a reference radiation 2 r corresponding to a transition of a known lifetime rr. The phase shift 02 o f the radiation 2~ corresponding to an u n k n o w n lifetime transition r,, is then measured. During these two m e a s u r e m e n t s o~ly the wavelength of the filtered radiation by the m o n o c h r o m a t o r passing from 2~ to 2~ has been modified. The result is that the phase shift error 0p does not vary and consequently the following relations are satisfied:

2.3.1. C h o i c e o f the working f r e q u e n c y This choice is influenced by the sensitivity of the phase shift m e a s u r e m e n t 0~ which can be characterised by the ratio: d~bo Si =

dzi/.fi

,

which with the help of relation (9) imples that: (11)

(D°zl

Si

=

1 + ((DoZi) ~

The variation of S, represented in fig. 4 shows that the best choice of oJ0 is given by 6ooZl = 1.

(12)

As we do not in fact know the exact value of t,, we choose ~)0 so that:

Si((Do) ~

1 ~SiMax,

which implies that: 0.27 <~ 6oo zi ~< 1.

(13)

[We exclude co0r~>l which in the present case has no particular adverse effect, but however enables the reduction of certain error factors15)]. The modulation frequency must therefore be adapted to the lifetimes to be measured, this calls for an apparatus which works at variable frequencies. W e can see however as a result of relation (13) that an apparatus working between 1 and 10 MHz allows the m e a s u r e m e n t of lifetimes between 4.4 and 160 ns; this corresponds to the most usual atomic lifetime v a l u e s ] 6 ) ] .

'P~ = 4 ~ r + q ~ , ,Si

~

= ~,+~, 0.5

--' q~. = ~ : - ¢ ~

= ~--¢r"

Then ~bo = - (arctg 6OoZ~- arctg 090 r~).

0.25

(9) .27

This last relation shows us that the differential phase shift m e a s u r e m e n t 0~ allows the establish-

3.73

Fig. 4. Phase shift sensitivity as a function of frequency.

102

J.

AZENCOT

AND

R.

3.1.

2.3.2. Choice of the re,@rence l(fetime

It can be shown from ref. 17 that the greatest undesirable p h e n o m e n o n in lifetime measurements using atomic ionoluminescence as an excitation process, is the cascade repopulation (see fig. 2). The initial transfer function is then modified as shown in fig. 5 (see appendix 2). If we put:

Tr

from relation (10) we can deduce: 1 'ca [tg(~ba + qSr) -- tg qSr].

(14)

(DO

This last relation can also be written: 1

"Ca --

(1 +(D02"Ci'Cr) t g 0 a .

E R R O R S C O N N E C T E D TO THE ATOMIC IONOLUMINESCENCE: CASCADE P H E N O M E N A

If we introduce the quantity r 6 given by: T 6 -- T i

GOUTTE

(15)

(D O

We then see that the lifetime difference r a depends not only on the differential phase shift measurement 06, but also on the reference lifetime rr. This can have an undesirable influence in certain cases and this comes from the fact that the functions under study are of non-linear type (tg x function). We can however overcome these inconveniences by carefully choosing the reference rr. In fact if

gt

--

Cz Nl Atkzl, C k Nk

we can see that the new transfer function H c(p) in the presence of cascade p h e n o m e n a related to H(p) is given by:

He(P) - 1 + Y' g l . H (p) '-i" 1 + zt p 4'~ = - arctg (Do%,

+(Do'c,) =

then 'ca ~ - -

(19)

If then :

(02 'ci Tr <~ ] ,

1

(18)

;

(20) + (port)

tgqS6'

(16)

l

(DO

we have from eq. (19)

r6 depends now only on 06, and we find an identical relation in (8) but corresponding this time to a differential measurement. The choice of rr will be made in consideration of relation (12) so that: (D O 17r ~ 1 . (17)

'c -

t g ( 0 c - qS~)

(21)

°90

(0c represents the measured phase shift in the presence of cascade phenomena). The relation (21) shows that measurement of only 0c is insufficient and that it is necessary to estimate r,: in order to determine r. Since this is an error estimation, we can use the astrophysical constants which are found in the tables, for example those given in ref. 16. The estimation of the quantity e~ still remains difficult since, considering eq. (18), it makes use of parameters which charac-

3. Error e s t i m a t i o n We can distinguish two types of error, those due to the physical observed p h e n o m e n a which have a tendency to modify the transfer function previously defined, and those due to the measuring apparatus itself.

1

CASCADE EFFECTS

Fig. 5. Transfer function in the presence of cascades.

1

- I

Akjh

Lvtp,

ATOMIC

IONOLUMINESCENCE

terise the excitation process (CN). For this we propose two estimation methods of e,, depending on whether it is possible or not to observe a 2, radiation wavelength coming from the upper cascading level. The first m e t h o d allows us to correct the lifetime m e a s u r e m e n t and the second m e t h o d enables the determination of the m a x i m u m uncertainty of lifetime m e a s u r e m e n t caused by this cascade p h e n o m e n o n . a) The situation when a ray /lu can be observed The luminance ratio L , / L k j in continuous mode will be (see fig. 3):

Lu

Au 2kj CtNt z,

i = A

c kN

This allows us to determine elmax by using the usual astrophysical constants. It can be noted in conclusion of this cascade study that t h e most usual case is when certain lines such as 2, can be observed but other lines can not ; we can then estimate an intermediate value of r, (and therefore of q~) by simultaneously using both calculation methods. In order that a lifetime m e a s u r e m e n t is as near as possible to its real value, we must verify that in all cases the following condition is satisfied: % ,~ z. (28) Following the relations (20) and (21) this implies that the correction or the error will always be very low compared to the measured lifetime. Finally let us draw attention to other cascade correction methods using numerical identification 19) or deconvolution 2°) techniques.

(22)

rE'

and using eq. (18) Akj "~li Lll gl = AlkTk At i 2k j Lk ~.

(23)

The relation (23) allows the estimation of e, by using the usual astrophysical constants and the luminance ratio, this being of course measurable. b) The situation when a ray 2u cannot be observed This case is quite frequent and it corresponds to infra-red light emitted from cascading levels. This light spectrum is generally outside the capabilities of the measuring equipment used ( m o n o c h r o m a t o r and photomultiplier working in ultra-violet and visible light). Let us calculate the population ratio n,/nk in continuous mode from / to k level. From fig. 3 we have n, _ Ct Nlz, nk Ck Nk "Ck '

(24)

arid using eq. (18) we have: nt

el = A l k Z k - - . nk

(25)

3.2.

ERRORS CONNECTED WITH THE MEASURING APPARATUS

The experimental lay-out shown in fig. 1 shows three kinds of error: The error caused by insufficient light collection (L error). - T h e error caused by insufficient selectivity of the optical filter (O.F. error). - The error caused by the transfer function of the photomultiplier (P.M. error). Finally, as we have to measure the phaseshift between j ( t ) and iv (t), we must also take into account the errors due to the phasemeter that is being used. In what follows we will only consider the two intermediate errors because it can be shown 17) that it is not difficult to adapt a lens position to collect nearly all the emitted light. This makes the first type error of negligible. As for errors of phase shift m e a s u r e m e n t , we can make these also negligible -

Selectivity

We can therefore estimate the m a x i m u m value of ~, by supposing that the ionoluminescence excitation intensity is very strong (which is not true) and equivalent to an infinite t h e r m o d y n a m i c equilibrium temperature. The M a x w e l l - B o l t z m a n n statistics allows us to write ~8) nt/nk = g/gk

(26)

(g, and gk: statistic weights of levels / and k). T h e relation (25) in this hypothesis becomes gl . . . . = Alk'Ck gt/gk. (27)

103

Fig. 6. Graph of optical filter response.

104

J. A Z E N C O T

AND R. G O U T T E

Fig. 7. Transfer function in the case of an insufficient selectivity of the optical filter.

by using the relevant phasemeter (a lock-in phase meter for example). This subject was studied in a previous publication 2t). a) The case o f an insufficiently selective optical .filter This happens when a supplementary radiation 2s is sufficiently near to the studied radiation 2, so that the o~ attenuation caused by the filter (see fig. 6) has an appreciable value. The photomultiplier will then add the two luminances which correspond to these two radiations. This gives us the transfer function represented in fig. 7. If fl _ C S Ns Asr 2kS (29) C N AkS 2~r' and =

zs/z ,

we obtain

H~(p) _ 1 + c¢fl~ l + zp H (p) l+zsp"

(30)

I~l

(31)

tg (qSs - q~')

(32)

COo (0~ represents the measured phaseshift in the presence of ;t~). The relative error made when this correction is not considered can be calculated. We find that (A:'] (1 +coo2z 2) z'~ (33) --

(34)

with

we can deduce z -

-

(1 +~OoZ )z, (1 - COO 2 "UC,E)'C , ,

PM

c,/~(7-1) = z~ l + ~ f l y + c o o Z2 ~2 ( 1 + ~ / ~ ) ' 1

2 2

Az

If we decide that ~b~ = - arctg cooz'~, ~.,

A numerical calculation in an exaggerated case will give for ~z-0.5, /~=1, 7 = 2 and co0 r = l , ( d r / r ) s = 29%. This large error shows that we must make sure in the greatest possible degree that the optical filter selectivity always remains sufficient. b) P.M. transfer .function In including the RC integrating circuit connected in the P.M. output we c a n s h o w 17) that the P.M. transfer function corresponds to that represented in fig. 8. In this figure we see that the only terms influencing the phase measurement are caused by the transit time in the P.M. and by the RC integrating circuit, whereas the transit time fluctuation has no adverse effect on this type of measurement. Nevertheless since our measurement is of a differential type, only the term d 0 p M / d ) ~ will create an error (0PM: phaseshift caused by the P.M.). It can be shown by a similar calculation that the relative error (Az/r)pM is given by:

2 ! (1-COoZZ~) z

"

P.M. gain

P.M. d e l a y

Fig. 8. P.M. transfer function.

Transit time fluctuation

O9oZ~' = tg [ coo ~alto (2i-2r) 1 •

(35)

If we carry out a numerical calculation in an exaggerated case : dto COoZ = 1;~Z = 0.2 ps/./k; )]'i--2r = 0.5 /~m, we find the following relative errors: for 1 MHz,

Fo =

1 9/0,

3 MHz, (Az/'OpM = 4 ~ , 10 MHz, 13 9/0.

RC circuit

ATOMIC

6p ," 5p i S •

~Q..h.....

IONOLUMINESCENCE

~ ~-'s~, ~

5(:1 /

--

6

105

f sf

"f,?~. 4f

;;"

A.

4s

3s

2f 2s

Fig. 9. Lil transitions used in our measurements. T h e s e e x t r e m e cases show that the errors still remain acceptable and therefore that in the majority of cases the errors introduced by the P.M. could be ignored.

4. Lithium lifetime measurements As an example of the application of this m e t h o d we give the results of lifetime m e a s u r e m e n t s on lithium. Our experimental conditions were as follows: - I o n b e a m : K +, 1 5 k e V , density 2 . 5 ~ A / m m 2 modulated at 3 MHz. - Pressure in the m e a s u r i n g c h a m b e r : 5x x 10 7 torr. - Target material: monocrystal of LiF, then pure Li. -Grating m o n o c h r o m a t o r (Baush and Lomb): 3 3 , ~ / m m dispersion, working between 2000 and 7000 ,~. - Photomultiplier EMI 6256 B: usable between 2000 and 5000,~. Fig. 9 represents the LiI energy levels and also tile transitions (in continuous lines) which interest us. T h e dotted lines show the cascade repopulations which correspond totally to infra-red light

emissions outside the spectral range of our optical detection system. Table 1 gives results of lifetime m e a s u r e m e n t s concerning these energy levels, carried out by other experimenters w h o used different techniques; Buchet et al. 22) ( " b e a m - g a s " ) , Bickel et al. z3) ( " b e a m - f o i l " ) , Karstensen et al? 4) ("delayed coincidence m e t h o d " ) . C o l u m n s theory 1 and 2 represent the theoretical results calculated using the following m e t h ods: theory 1: " C o u l o m b a p p r o x i m a t i o n " 25), theory 2: " S e l f consistent field method"26). In table 2 we r e s u m e the results of a "statistical a n a l y s i s " on the previous m e a s u r e m e n t s . It can be said that the relative r.m.s, error between the various experimenters is about 10%. Table 3, finally, restates our own lifetime meas u r e m e n t s using atomic ionoluminescence as an excitation process obtained from Li and LiF solid samples (the results were confirmed to within 5% in the two cases). T h e chosen lifetime reference is that concerning the level 3d z D, since it is the lower of the four lifetimes studied. In this table we have reported the m a x i m u m error estimations caused by the cascade repopulation

T~BLE 1

Lithium lifetimes (in ns) obtained by different methods. Energy level

4s2S 3d2D 4d2D 5d2D

J. P. Buchet et al.

W.S. Bickel et al.

F. Karstensen et al.

Theory 1

Theory 2

55.8_+2.8 14.0±0.7 39.2_+2.0 72.0±3.6

48_+2 15± 1 33_+ 1 56 ± 2

16.7-+1.0 42.3_+ 1.5 -

56.9 14.5 35.2 65.0

57.0 14.0 33.5 64.8

J. A Z E N C O T

106

AND

R. G O U T T E

TABLE 2 Statistical analysis of table 1. Energy level

N u m b e r of different values

Mean value (in ns)

Root m e a n square error (ns)

Relative r.m.s, error (%)

4 5 5 4

54.4 14.8 36.6 64.5

4.1 1.5 4.0 5.1

8 10 11 8

4 s2S 3d2D 4 d 2D 5d2D

TABLE 3 Results of our lifetime m e a s u r e m e n t s . Energy level

Observed wavelength (A)

4 s2S 3 d2D 4 d2 D 5 d2D

4971.7 6103.6 4602.9 4132.6

Cascade corrections (ZJTc/~')max (Z~Tc/'r)measure d (%) (%) 4 13 8 4

phenomena, these are calculated by means of Wiese's atomic constant values27). Concerning the level of 4s2S, we were able to determine the corresponding cascade correction with greater precision, since the transitions from the cascading level could be observed: 4pZP ~ 2s/S (2741.19/~), 5p2P --+ 2s2S (2562.31/~), 6p2P ~ 2s2S (2475.06/~). We can see that the relative correction obtained in this case (2%) is half as large as that corresponding to an infinite excitation intensity. Concerning the three other levels we were not able to do this kind of calculation, since radiations coming from cascading levels 4f, 5f and 6f are all situated in the infra-red spectrum. A partial calculation concerning the other cascading levels 4p, 5p and 6p shows however that the m a x i m u m relative uncertainties due to the cascade repopulation given in the preceding column are certainly very exaggerated. These are mostly about 1%, and will therefore be negligible. Nevertheless, we have given in the last column of table 3 the m a x i m u m uncertainties caused by the cascade repopulation in all cases, and taking into account that these are more increased by the reference uncertainty. This

2 -

Our results Nominal value Cascade errors (ns) (ns) 66.4 14.8(ref.) 40.1 59.9

±2.3 ±1.9 ±2.8 +3.0

is because, as we have already said, our measurements are of differential type. If we then compare our results with the average values in table 2, we can see that the relative r.m.s, error is about 10%, which shows that the precision obtained in this case is not better than that given by the other known methods. 5. Conclusion To conclude this study we can try to classify the advantages and disadvantages attached to this new lifetime measurement method. The disadvantages are of two sorts. Those connected with the atomic ionoluminescence come from the non-selective excitation process. We have seen that this implies the introduction of cascade corrections and it is this in our opinion which is the most critical part of our method. This problem is not particular however to our technique since even very sophisticated methods such as beam-foil spectroscopy 28) also need this type of correction. The second inconvenience is connected to the phase shift method itself and comes from the fact that this technique is more orientated towards a differential measurement rather than an absolute one. This then needs the previous knowledge of a reference atomic lifetime. In fact this is a minor

ATOMIC

inconvenience and for a given element we rarely do not know at least one atomic lifetime which corresponds to a particular excitation state. Even if this happened, this method would still allow us to establish an atomic lifetime list of the same element, scaled to a particular lifetime taken as a reference value. Concerning the advantages we can also distinguish those coming from the atomic ionoluminescence and those connected with the phase shift method. A m o n g the former we note the possibility of directly analysing the atoms of a solid sample, whether this is a conducting or an insulating material. This is a new possibility when compared to all the other methods which either require work on a gas 29) or produce an ion source from the element under study2a). A second technological advantage is derived from a simplification of the ion source which only needs a low energy incident beam (a few keV). We can finally note, in another field, the important advantage of not having, as in working on gases, to correct the lifetime m e a s u r e m e n t because of' the uncertainly of the excitation m o m e n t caused by a finite thickness of the gas layer under study29). In fact, in our case, because the atomic ionoluminescence is a surface phenomenonS), all the target atoms are excited at the sametime, on condition of course that the ionic beam is sufficiently focused on the target17). It is for this reason also that the imprisonment radiation p h e n o m e n a 29) are absolutely negligible in this technique and that it is not necessary to make the relevant corrections. It should be noted however that depending on the particular excitation process, certain non-radiative transitions such as the Auger or tunnel effects 3°) can in certain cases superimpose on the radiative spontaneous deexcitation of the sputtered Ek , nk 2 : Shock

repopu[ation/

/

/

i

/

e

/

/

Appendix 1 1) Equation law of population evolution n(t) Let us consider the two energy levels Ek and E~ shown in fig. 10, whose populations are respectively nk and n/. T h e nk population is a function of two sorts of variations: one is a decrease caused by the spontaneous radiative deexcitation of atoms taking path 1, the other is an increase brought about by the excitation repopulation as in path 2. W e can therefore write:

dnk

dlnk

dt-

dt

+

d2nk dt "

(36)

And following the transition probability definition of Akj and the section efficiency Cj~ we can write 3~): da nk/dt = -- nk Ak~' (37)

dz nk/dt = nj Cjk j(t). If we consider this time all the levels j < k together, which can exchange transitions with k, the relation (36) must be added for all j < k values and, taking eq. (37) into account, becomes: dnk d--7 = "k Z Ak; + j(t) y~ ,,j Cjk. (38) j
If nj Cjk

/

Ck = LS~____, nj

]:Spontaneous

deexcitation

Ei ° i Fig. 10. Transitions between two energy levels. I

atoms. These modify the " n a t u r a l " lifetime measurement obtained from a " p u r e " de-excitation process. W e will study this important measuring problem in more detail in our next article. Concerning the advantages connected to the phase shift method: these are a result of the simplicity of operating this technique and also the speed of m e a s u r e m e n t , since only two points (in differential measurement) are sufficient to establish a given atomic lifetime. Certain other methods need in fact an entire curve statement - for example the impulse response curve2S,32). W e can therefore finish this study by remarking that, if this m e t h o d has no pretention of enabling a greater precision to be brought to present known methods, it can be profitable to use it thanks to its simplicity in the lifetime m e a s u r e m e n t of solid conducting or insulating samples.

j
/ /

107

1ONOLUMINESCENCE

.,

j
Z

j
(39)

108

J.

i

AZENCOT

AND

El , n I

/

GOUTTE

t o g e t h e r eq. (42) b e c o m e s :

dnk q- nk = dt Tk

3:Cascade

iI

R.

II

C k N

k

j ( t ) + ~ nlAlk ,

(44)

l>k

iI

w h e r e a s we s u p p o s e that the u p p e r / levels do not r e p o p u l a t e by cascade p h e n o m e n a , t h e y satisfy the " u s u a l " differential e q u a t i o n :

Ek , nk

iI

tt iI

iI

iI ,

i it

/

iI

/2:Shock tt e x c i t a t i o n : i**

J

/' 2,, "t /i t sS

t

1: S p o n t a n e o u s

dnl k n! = Cl)~z j ( t ) .

deexcitation

dt

T h e s e two relations allow us to arrive at the t r a n s fer f u n c t i o n o f fig. 5.

kj , nj

Fig. 11. Level k repopulation by cascade effects.

References

relation (38) b e c o m e s , by t a k i n g eq. (4) into account'

dnk + nk = CkNk j ( t ) . dt Zk

(2)

2) L u m i n a n c e calculation In the t r a n s i t i o n Ek ~L~ there are nkSA~i a t o m s which de-excite in u n i t t i m e from the target w h o s e area equals S. T h e total e m i t t e d flux 0u in a solid angle of 2/7 sr is therefore:

d?~ = nkSAkjhV,

(40)

If we s u p p o s e that the target radiates following L a m b e r t ' s law (which is not a b s o l u t e l y exact, this h o w e v e r is of n o i m p o r t a n c e b e c a u s e this coeffic i e n t does not affect the phase shift m e a s u r e m e n t ) the l u m i n a n c e /v is g i v e n by: c~

1~ -

(41)

HS"

T h i s , t a k i n g into a c c o u n t eq. (40), i m p l i e s :

1

l~ = FI AkjhVnk"

(3)

Appendix 2 Trans/er ./imction in the presence 0/' cascade repopulations It is easy to see from fig. 11 that the nk e v o l u tion is a f u n c t i o n of t h r e e t e r m s :

rink

dl nk

dt = d - ~ - + ~

d2 nk d3 nk - - + dt "

(42)

T h e first two t e r m s are g i v e n by eq. (37); t h e third is d3 nk

dt

= nl Alk"

(45)

zl

(43)

I n c o n s i d e r i n g all the u p p e r / a n d l o w e r . / levels

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