I L NUOV0 CIMENT0 VoL. 65 B, N. 1 11 Set tembre 198i

General Theory of the Ionization of an Atom

by an Electrostatic Field.

I~. FONDA

International Centre ]or Theoretical Physics - Trieste, Italy Scuola Intevnazionale Superiore di Studi Avanzati - Trieste, Italia

(rieevuto i l 25 Maggio 1981)

Summary . - - The ionization of an a tom by an external electrostat ic field is reconsidered by taking into account the interactions of the system with the measuring apparatus. The exper imental ionization ra te is drast ical ly different from the expression obtained when no measure- ments are present. A dependence on the mean frequency of measure- ments is found. This fact can be used to determine this quant i ty once the ionization ra te is determined experimental ly.

1. - I n t r o d u c t i o n .

R e c e n t l y t h e r e has b e e n a r e v i v a l of i n t e r e s t in t h e p r o b l e m of t h e i o n i z a t i o n

of a n a t o m e m b e d d e d in a n e x t e r n a l e l e c t r o s t a t i c f ield, a p r o b l e m f i rs t t a c k l e d

b y OPP~,NHEI~ER (1) in 1928 (see, for e x a m p l e , (2-,)). T h e p h e n o m e n o n is a lso

k n o w n as S t a r k effect a n d i t is s i m i l a r t o t h e d e c a y of a n u n s t a b l e q u a n t u m

(1) J , R. OPPENHEIMER: Phys. Rev., 31, 66 (1928). (2) L. D. LANDAU and :E. M. LIFSmTZ: Quantum Meehanies (London, 1959), p. 257. (3) 1~. Jr. ALEXANDER: Phys. Rev., 178, 34 (1969). (~) J . 0 . HmSCHF~LD~.R and L. A. CURTISS: J. Chem. Phys., 55, 1395 (1971). (5) M. HV.H~NBV.RG~R, H. V. MCI~TOSH and E. BRhNDAS : Phys. l~ev. A, 10, 1494 (1974). (s) N . A . GUSmNA and V. K. NIKULIN: Chem. Phys., 10, 23 (1975). (~) R . J . DAMBUR~ and V. V. KoLosov: J. Phys. B, 9, 3149 (1976). (8) T. YAMAaE, A. TACHIBANA and H. J. SILV~RSTOI~E: Phys. ~ev. A, 16, 877 (1977). (9) S. GET.T~N: J, Phys. B, 11, 3323 (1978).

45

46 L. 1~()xI).~

system, since the discrete energy eigenstates of the a tom become resonances

in the presence of the external electrostatic field. This is due to the fact that ,

when the external potential Vo:t-: eE ~ z~ (e# > 0) is switched on, the elec-

trons find it possible to tunnel through the potential barrier in the direction

z - + - - co, so tha t the a tom disintegrates. The lifetime is, of course, charac-

terized t)y the length of the potential barrier the electrons have to travel through,

in the sense that the shorter the tmmel is, the quicker the a tom will decay.

The tunnel is, of course, shorter the greater is the electric field # (*). The typical t reatments obtain the ionization rate usually for the hydrogen

a tom in the ground state. E.g., following" the W K B approximat ion (see, for

example, (o)), one finds for the ionization rate of this a tom

(1.1) 1 1 ~ 2 1 [ 3 e ] r,,-,r ~ \ a ] ecfa exp - - ~ ,

where a = lie'me 2 - - 0.529"10 -s o111 is the Bohr radius.

This result is, of course, obtained by assuming that the ionization rate is constant in time, which means that the probabil i ty of decay is exponentiM

ia time with lifetime T,,~u. We know theft this is not true. Deviations from the

exponential are present in the decay law (see, for example, (~0) and references

therein). GELT3L~=~-(9), who solved exact ly the SchrSdinger equation for a

simple atomic model embedded in an electrostatic field (**), found for his par-

t icular model tha t the decay probabil i ty shows a parabolic start at t = 0 fol- lowed by an essentially flat behaviour at intermediate times which then finally

merges into an exponential. The flat, almost constant, ionization probabil i ty

at intermediate times has been interpreted by GELT3IAN aS a typical quan tum

effect related to the electronic configuration in the tunnel region. The deviations front the exponential law studied by GELT3IAN give, of course, rise globally to a departure of the ionization rate f rom (1.1). GELT)IA:~*, however, and also all pre-v-ious authors who treated this problem have not taken into consideration

the fact tha t the decaying system is actual ly subject to repeated measurements

during its lifetime, whose effect is tha t of radically changing the survival prob-

(*) If the external field is made extremely strong, tile energy levels of the atom pop out of the atomic well and the atom disintegrates instantly. Before that occurs, elec- trons will be stripped off the cathode. We are, of course, interested ill comparatively weaker fields, so that this does not occur. (10) L. FONDA, G. C. GHIRARDI and A. Ri)iIXI: I~ep. Prog. Phys., 41, 587 (1978). (**) The model is one-dimensionM and the atomic potential is replaced by a &function at the origin. I t has been previously considered by CRAIGIn (11). Tile first treatment actually belongs to TITCIIMARSH aS one can find at p. 271 of his book Eigen]unction Expansions Associated to Second-Order Differential Equations (London, 1958). (11) N. S. CRAmI~: B. Sc. Dissertation, University College, London (1967).

GENER A L THEORY OF THE IONIZATION OF AN ATOM ETC. ~ 7

abil i ty of the atomic system. In this paper we shall consider the effect of such measurements on the ionization of an a tomic sys tem embedded in an electro- s tat ic field. I n sect. 2 the general t r ea tmen t is laid down. Section 3 is devoted to the ob ta inment of the results and to the analysis of the changes with respect to the usual approaches. We shall see tha t the exper imenta l ionization ra te is drastically different f rom expression (1.1). A dependence on the mean fre- quency of measurements will be found. The formula which one obtains can ac tual ly be reversed in the sense tha t , by knowing the applied external field g~, a measurement of the ionization ra te is able to provide a de terminat ion of the f requency of measurements , a quan t i ty which in this case is difficult to guess a priori.

2. - Interactions of the atomic system with the measuring apparatus.

As originally pointed on by EKSTEIN and SIEGERT (12) and b y :FONDA, GmRARDI, R I ~ I and WEBER (13) (for a review see (lo)), a decaying sys tem which lives long enough to meri t an invest igat ion of its t ime development is never left isolated, bu t unavoidably interacts repea tedly with its environ- ment . B y envi ronment we mean here the measuring apparatus , used b y the exper imenter to detect the decay products. Exper imenta l ly the unstable sys tem is subjected to r andom measurement processes of the yes-no t y p e which ascertain whether the sys tem has decayed or not. This is more easily unders tood in a decay exper iment performed in a bubble chamber. The de- caying (charged) part icle t ravel l ing along the chamber leaves a t rack of bub- bles. Each bubble is interpreted, according to its position, as represent ing the part icle not decayed or one of its decay products. I f the bubble shows the par t ic le undecayed, this means tha t a quan tu m measurement has been per- formed and the wave funct ion of the sys tem has collapsed onto a subspace or thogonal to the subspace of the decay products . The wave funct ion is then localized within a sphere of radius R of the order of magni tude of molecular distances.

When an isolated resonance characterizes the unstable system, the wave funct ion evolves f rom V~u,t,b~ at t ~ 0 up to a measurement t ime t in a causal way and then at t i t jumps back essentially to the same s ta te V~,t~b~o, and a sort of s ta t ionary si tuat ion is established. I n fact~ the s ta te a t t ime t before the measurement will be

(2.1)

(12) tL EKSTEX~ and A. J. F. SIV.GV.R~: Ann. Phys. (~V. Y.), 68, 509 (1971) (13) L. FONDA, G. C. GHIRARDI, _A. RIMINI and T. WEB~R: _NU0V0 Cimento A, 15, 689 (1973); 18, 805 (1973).

48 L. FONDX

where A(t) is the nondecay probabil i ty ampli tude

A(t) ~ (V~t~b~e,exp [-- ~ Ht] ,v~t~b~} (2.2)

and ~(t) is the state representing the decay products and satisfies

<~oo~b,ol,~(t)> = O.

I f one neglects the par t of ~(t) which is present inside the sphere of localization,

after the measurement performed at t ime t in which the system is found un-

decayed, one has the following collapse of the state vector :

A(t) (2.3) ]~(t)) . . . . . . . . . . t) ]A(t)[ lVu.~t~o) �9

This state follows then the same his tory as before ~nd so on, until, of course,

the ~ctual decay occurs which provokes the quan tum elimination of the state

V..~t~b~., while the detectors reveal the decay products. For our Stark effect a similar si tuation occurs. I n fact, in practice the

sys tem will be pu t between two oppositely charged plates and the decay of

the system will be registered by a counter which will reveal the arrival of the

electron at the ~node. By arrival of the electron at the anode we mean tha t a quan tum measurement has abrupt ly reduced the electron wave funct ion in

the region outside the plates. But, before tha t occurs, the electron wave func-

t ion has interacted repeatedly with the counter. I n fact, as soon as the ex-

ternal electrostatic potential is switched on, the electron wave function, leaking

through the Coulomb barrier, starts overlapping the anode. Then either the counter reveals the electron reducing its wave function outside the plates

(the ~tom is found decayed), or the counter does not find any th ing and, as a consequence, the electron wave funct ion is reduced in the region inside the plates (the a tom is found undecayed). Since the a tom possesses an infinity

of bound states, in general the wave funct ion will be a superposition of these

and of a free state inside the plates. ~Ve face here a case in which the unstable state is characterized b y a superposition of resonances and not by a unique

quan tum state. Both because of this reason and because of the fact tha t in any case for the state ~'ector it is impossible to write down a causal evolution equation, we have necessarily to resort to the formalism of the densi ty op-

erator. Let us summarize the general t rea tment (~4-~v).

(14) L. FONDA, G. C. GHIICARDI and A. RIMINI: •uovo Cimento J~, 18, 1 (1973). (15) S. T. ALI, L. FONDA and G. C. GIIIRAI~DI: ~OVO Cimento A, 25, 134 (1975). (16) L, FONDA, G. C. C~HI~])I, C. OM~RO, A. RI~IINI and T. W~B:E~: Proceedings o] the Meeting, Mathematical Problems in the Quantum Theory o/ Irreversible Processes, Laboratorio Cibernetica C.N.R., Arco Fclice (Napoli) (1978). (17) L. FONDA, N. MANKoh-BoRgTNIK and M. ROSlNA: Nuovo Cimento A, 50, 229 (1979).

G E N E R A L T H E O R Y O F T H E I O N I Z A T I O N O F A N A T O M E T C . 4 ~

If one is interested only in the history of the decaying atom and then dis- regards the destiny of its decay products, the effect of a measurement at time t on the e-operator can be visualized as the transformation

(2.4) eft) . . . . . . . . . . ,> e~(t) = P~ eft) P . ,

/DR being the operator which projects the wave function within the plates. On the other hand, the evolution without measurements through the time

interval (0, t) is given by exp [--iHt] e(0)exp [iHt], where H is the total Hamiltonian of the system:

(2.5) H = H~to, n + ee ~ z,. t

There follows that in general, when both evolutions with and without meas- urements are taken into consideration, one can write the following equation for e(t):

r i 1 r 1 i (2.6) de ( t )= / td te ' ( t ) + (1--~dt)exPl--rHdtle(t)expl~'dtl'L'~ J t J

Here 2 is the mean frequency of the random measurements; 2dr represents then the probability of the occurrence of a measurement in the interval dt of time. Equation (2.6) leads to the differential equation

(2.7) dQ(t) i dt = - -~ [H, e(t)] + 2(e*(t)--e(t))

which in turn is equivalent to the integral equation

(2.s) 0 ( t ) = e x p [ - - 2 t ] e x p ~ H e(0)exp ~Ht + t

o

exp )] Since one assumes that the state ~un.*~b,e is realized at t = 0, one has the boundary condition

(2.9) o ( o ) = l v , = . , ~ , o > < ~ ~ �9

The main question now is the determination of the operator PR. If one disregards the presence of the decay products within the plates, which in our opinion is an excellent approximation, one can safely assume that PR is given

- Z l Nuovo G l m e n t o B.

50 L. FONDA

by the sum of the projection operators on tile bound states of H to~:

(2.10) /}R = ~ l~,~) (~f.], n

By bracketing (2.8) O i l < ~ n ] a n d ] ~ m ) , o n e gets

(2.11)

where

( 2 . 1 2 )

o,,,~(t) = exp [-- ).t] ~L.~(t) + t

0

exp [-- ).(t--6)]P ........ ,~,(t-- 6) 0 . . . . ( ( ~ ) ,

,) , , , ,(t) = @ . [ , , ( t ) l ~ , , . ) ,

P] 52,,(t) ~ (v~[exp --~Ht o(0) exp ~Ht ]V,~),

i P,,,,,,,,,,,~,(t) ~ @,.iexp f--~ Htl]Y~.,,) (V,.,exp [--~ Ht]]~,.,)~.

We are interested i~t the s ta t ionary situation which establishes itself after

a ~ime of a few 1/).. For this purpose we shift the initial t ime f rom t := 0 to t ----- - - T. By defining with or(t) and A'r(t) the operators obtained in such a ease, we get in place of (2.11)

(2.]3) r t o.~( ) = exp [-- 2(t + T)] Nr.m(t) + t

, , f o .w( ) . + n ~ 2(ldexp[--2(t--O)]P ....... .,,(t--(~) r O --1"

We see now tha t for T greater than, say, 5/). the first te rm on the r.h.s. (.an be dropped, while the integral can safely be extended to - - c~. Indica t ing by

the 0-operator which one gets in this s ta t ionary situation, after a change of variable one has

(2.14) co

0

The solution of (2.14) can immediately be guessed to be exponential. Let us, in fact, t ry

( 2 . J 5 ) 5 , , 4 t ) = B (~) exp [-- t/v~]. fr~n

Substituting (2.15) i~1 (2.14), we see th,qt (2.14) is s~tisfied provided the coef- ficients B s,~tisfy the following set of eqnMions:

co

n'~n' nn ' - - = 0 . n ' m ' ~

0

The condition for the existence of ~ solution of (2.16) is the v~nishing of the determinant of the coefficients

r m ~

[~/~,] ~ 0 ,

0

which is ~n implicit equation for the unknown vz. In general, the solution v~ will not be unique, so that ~(t) will actually be a superposition of exponentials:

(2.18) ~,,,,(~) = ~ B'~ exp [-- t/vz] .

The probability/~(t) that ~t the time t the atom is found undecayed will be given by the trace of the @-operator

(2.19) F(t) ~ ~ 5..(t) = ~/~.~ exp [-- t/vt]. n n~

F(t) tells us how the atomic system survives, inclusion made of all interactions with its environment, up to the time t. From (2.17), we see that the quantities ~, are actually implicit functions of i :

so that a dependence of the lifetime, and consequently of the survival prob- ability, on the frequency of measurements is actually realized. This will be discussed in more detail in the next section.

3. - The i o n i z a t i o n rate in the presence o f m e a s u r e m e n t s .

In order to evaluate the experimental ionization rate, we have first of all to understand the properties of the probabilities P~, ,~,( t) . There is a slight difference between the standard decay theory discussed in the literature (see (lo)) and the ionization of an atom embedded in an electrostatic field. In this last case, in f~ct, the energy spectrum has no lower bound.

As already pointed out in sect. 1, due to the fact that the potential energy eSz~ of an electron in an external electrostatic field takes arbitrarily large

5 2 L . I ? O N D A

negative values as z ~ - + - - c o , electrons with a rb i t ra ry negative energy can move at large distances f rom the a tom in the direction of the anode. The to ta l Hamil tonian of the system H = H tom + eo ~ ~ z~ exhibits then a purely eon-

i

t inuous spect rum not bounded f rom below, i.e. stretching f rom - - do to + o0, while the original discrete levels of H,to~ have disappeared, being subs t i tu ted by resonant states of finite ( # 0) width.

] f we denote by (H, e) the complete set of commuting observables which includes H and indicate by CE,~ the common eigenstates of these operators ,

we have then

/ HICE.~" = E[r176 - ~ < E < + ~ , (3.1)

Since the set of states {r is complete and ortholmrmal, we can use it

to expand the nondecay probabi l i ty ampli tudes A..,(t):

(3.2) i ] ,

A,. .( t) ~ @,. 'ex 1) - - ~ Ht ',/',,, '

where the V,'s are eigenstates of H~tom (see (2.10)) and o),,,,(E) is given by

(3.3) c,,,,,, (E) = . i @ <v'~ Ir <r l ~',,,> �9

:Let us smmm~rize briefly the general propert ies of A...,(t). At the origin of t imes one gets A.,,,(0) = 6 ...... The modulus IA.m(t)] is always less t han 1:

" e x p [ i IV,~ (3.4) IA.,,,(t)[ < [iv-,] - - ~ H t = 1 .

Applicat ion of the Riemann-Lebesgue lemma shows tha t

(3.5) lira A,,~(/) = 0 . t ~ + c o

To prove (3.5) one needs only the absolute integrabil i ty of co,,,(E):

q-co q-co

fdEIo m(E)I fdEfd.l< '= < --co --co

< [ffdF + [ffd o>l ] +

I f (o,,,~(E) is a purely Brei t-Wigner energy denominator

(3.6) o) . . (E) = ~ ( E - - E R . ) 2 q- ,

= 1 .

GENERAL T H E O R Y OF THE IONIZATION OF AN AT01~ ETC. 5 3

then A. . ( t ) is a pure exponential. In this case, in fact, after contour integration the contribution at the pole E = E R . - i7~/2 would give for (3.2)

(3.7) A ~ ( t ) : exp ~ ER, exp [-- 7,t/2]g].

The result (3.7) would be possible just because of the fact that the energy spectrum does not have a lower bound in our case. The original argument by KH~PI~ (18), according to which the decay does not follow an exponential law if o)(E) vanishes identically on a finite energy interval, clearly does not apply in this case (see also the review article by FO~DA et al. (lo)). In general~ however, since the contour integration will be prevented by the presence of singularities in the complex E-plane of co~n(E), the exponential law will not be possible for all times, as in fact was found by G ] ~ L ~ (g) in his model. We expect that the exponential be established for sure over an intermediate time region.

However, since the integrals appearing in (2.17) are drastically cut off by the presence of the exponential exp [-- ,~], the physically relevant region for the probabilities P . . . . ,~,(~) is that for small times. By expanding these quan- tities around ~ ~ 0, to the second order one gets

(3.8) ~,~,~,(~) = ~.,~, + ~ (O..,H~,~ -- O~.H..,) - -

I t is then possible to perform the integrals appearing in (2.17). One gets in this way an algebraic equation for vz whose order is 3N ~, if N is the number of atomic bound states one takes into consideration in the evaluation of the de- terminant (2.17).

Let us solve (2.17) for the hydrogen atom. If one takes 8 such tha t no transition away from the ground state is possible by means of the process of measurement, one is able to describe the process as an isolated energy resonance characterized by a width 7 and by a unique quantum state, i.e. the ground state of the atom; in this case N---- 1 and (2.17) becomes a cubic equation of the type (for a discussion of this case, see also ref. (13))

(3.9) x ~ - x ~ l § 0,

(xs) L. A. KHALI~IN: Soy. Phys. JETP, 6, 1053 (1958). (19) L. FONDA and T. P~a~si: ICTP/81/15, Trieste.

54 L. rO~D.~

where x = 2 - 1It, ~" is g iven b y (*)

1 (3.10) (~2 = ]7 @]EH --/7]=lW>

and V is the g r o u n d s ta te of the a tom. /~ is the average value of the t o t a l

H a m i l t o n i a n (2.5) for the s ta te V. E q u a t i o n (3.10) is easi ly eva lua ted for ou r

ease : e 2

(3 .11) r = _ _ O*r - - ~2)

which for the g r o u n d s ta te of the h y d r o g e n a t o m reads

~2 (3.12) r 2 ~2

Of the three solut ions of (3.9), we shall tal ;e as the phys icMly re levan t one

t h a t which is ana ly t i c a l l y eonnec ted wi th the on ly rea l solut ion of the ease of

pos i t ive d i sc r iminan t 21. This solut ion, wr i t t en in t e rms of l / r , is g iven b y

(3.13)

where

1 2

~. = ~ ( ~ ; 2 _ ~ ) ~ r(;., g),

A=(~).: ~--~).~ ~A(),#).

We can dis t inguish three cases:

~) 1/). > l / s , b) 1/), comparab l e wi th 1 /~ ,

c) 1/~ << 1/~.

a) I n this case the m e a n t ime 1/). be tween two m e a s u r e m e n t s falls on the exponen t i a l i n t e rmed ia t e t ime region, therefore outs ide the region of va l id i ty

of expans ion (3.8). I n this ease the e x p e r i m e n t a l l y de tec tab le l i fet ime v is

v e r y close to the inverse of t he resonance w i d t h (for a deta i led discussion of this poin t , see (20)):

1 1

(*) From (3.8), one has in this case P ( d ) = 1--~2d2. (20) A. D•GASPERIS, L. FONDA and G. C. GHIRAI~DI: _~'UOVO Cimento A, 21, 471 (1974).

C~EN~RAL TI:I~ORY OF T H E IONIZATIOI~ r OF AN ATOM ETC. 5 5

and one reobta ins the resul ts of the approaches which do no t t ake into account the process of measurement~ in par t i cu la r eq. (1.1). This will occur for r a the r

s t rong fields.

b) Le t us now consider the case in which 1/2 is comparab le wi th 1/a. I n th is case the l i fe t ime is no longer re la ted to the wid th ~ of the resonance. F r o m eq. (3.13) we see t h a t 1Iv becomes a funct ion of the mean f requency of meas-

u remen t s 2. The dependence on the electric field # is of the t y p e

(3.15) [1 + ad ~ ~ $r -~ vg~2)�89 t .

The depar tu re f rom (1.1), ob ta ined when no measu remen t s are present , is drastic.

c) Consider now the case 1/2 << 1/a. I n this case eq. (3.13) can be wr i t t en

in a more compac t fo rm:

[ 0] (3.16) -1 = 2_ ~ 1 - - cos , 3 ~ < a '

where the angle 0 is defined b y

(3.17) 0 ---- a r c tg - - r

E x p a n d i n g cos (0/3) for large 2/a finally yields

(3.18) 1 2~ 2 1 1 7-~ ~ , ~<<~,

which for the hydrogen a t o m reads

(3.19) 1 2 e 2 a 2 # 2 1 7 ~ ~2---F, ~ << e ~ '

which shows a quadra t ic dependence of the exper imenta l ion iza t ion r a t e on

the electric field d'. The depar tu re f rom (1.1) is of course re levant for weak fields. Increas ing the electric field, one goes f rom (3.19) to (3.13) and then, for sufficiently s t rong fields, one should recover the old resul t (1.1) (*).

(*) GmRA~DX et al. (21) have recently discussed the limit ~-~ ~ in the context of the so-called Zeno paradox (~2.22). They have pointed out that, as a consequence of the time-energy uncertainty relation, in the limit ~ --~ c~ the lifetime becomes again ~7-1. (21) G. C. GHIRARDr, C. OMv.RO, T. W~BER and A. RIMINI: Nuovo Cimento A, 52, 421 (1979). (22) B. MIs~A and E. C. G. SVDA~SKA~: J. Math..Phys. (s Y.), 18, 756 (1977). (2a) C. CHIV, E. C. G. SVD~SrrA~r and B. MISRA: Phys. t~ev. D, 16, 520 (1977).

5 6 L. FO.~'DA

4 . - C o n c l u s i o n s .

We h,~ve seen tha t the interactions of the physical sys tem with the meas-

urement ~ppar~tus change drzstically the ionization rate of ~n a tom embedded

in ,~n external electrostatic field 6 ~. I n p,~rtieular, besides obtailfing ~ depen-

dence on 6 ~ very different f rom (1.1), the ionization rate becomes ~ funct ion of

the nlea.n frequency of measurements 2. However, the evaluzt ion of 2 is r~ther

difficult in this c,~se. Approximate evaluations of this quant i ty have been

given previously ('-'0) for dec~ys in bubble chamber and for r~dioactive ma-

teri,~ls. However, in both these types of experiments one cai1 single out ,~

typical periodicity which c~n then be related to 2. This periodicity is missing here for the case we are considering. One e~n, instead, reverse our a rgument

in the sense tha t experimental ly one should check whether (1.1) or (3.13)-(3.19)

hold good. I n this way, once the external electrostatic field 6 ~ is fixed ~nd the

ionization r~te 1/z is measured, one ca.n get the value of the mean frequency

of measurements 2. To our knowledge, this would be the first measurement of this physica.1 quant i ty .

�9 R I A S S U N T O

Si riconsider,~ la ionizzazione di un a~omo immerso in un campo elettrostatico est~erno prendcndo in considerazione l'inter~zione presente tra it sistema fisico e l'apparato di misura. L~ vita media del sistema dipende dalla frequenza media delle mism'e. Questo f,~tto pub essere usato per determin,~re spcrimentalmente questa quantith.

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