Nuclear Instruments and Methods m Physics Research 227 (1984) 327-334 327 North-Holland. Amsterdam

A T I M E R E S O L V I N G S P E C T R O M E T R Y M E T H O D F O R P A R T I C L E S E M I T T E D IN I N T E N S E

B U R S T S

M~adhltna V L A D

lnatttute of Phvstcs and Technology of Radtatton Devtces, P 0 Box MG-7, M(tgurele, Bucharest, Rumama

Received 19 May 1982 and m revised form 19 October 1983

A theoretical description of and some numerical tests for a new experimental method for determining the time-resolved energy spectrum of particles emitted in long duration, intense bursts are presented Th~s method ,s based on the free propagation of particles. being an extension of the t~me-of-fllght method It is applicable to some experimental condmons (especmlly m intense neutron flux measurements) for wh,ch none of the existing methods g,ves sat,sfactory results

1. Introduction

The t ,me-of-fhght method is a very powerful spec- trometry method which can be used for particles emitted one by one as well as m intense bursts [1] It ~s used especmlly for neutrons, be:ng based on the umque rela- t,on existing between the particle energy and the experi- mentally measurable txme wh,ch ,s reqmred by the particle to travel in free motion a gxven distance. In the case of particles ematted m bursts, the entire energy

spectrum can be obtained in one measurement by re- cording the t~me dependent rate of part,cles that are m o d e n t on a detector placed at a distance x from the part,cle source The best accuracy of such a measure- ment ,s hmlted by the value of the emission duration AT, being given by the ratio A E / E = 2AT / ' r , where "r = x / ~ ~s the average particle time of fl,ght from the source to the detector and b is the average velooty. An examination of th~s energy resolution shows that the t~me-of-fl,ght method can be used for sources em,tt ing short particle pulses (or for short pulses obtained by a chopping method) w,th a broad energy distribution and w~th a not too large average energy. Th,s theoretical energy resolution can be improved by increasing the fl~ght path x. but for und~rectloned particle sources, thxs means also the decreasing of the number of particles that str~ke the detector and, consequently, the increasing of the statistical errors of the measurement, so that the maximum value of the fhght path x is hm,ted by the

intensity of the emission For part,cle sources characterlsed by long emission

durauons or high average velocmes and narrow spectra th~s method cannot be used and m some cases (espe- emily m intense neutron flux measurements) none of the existing spectrometry methods gives sansfactory results

0 1 6 8 - 9 0 0 2 / 8 4 / $ 0 3 00 © Elsevier Science Pubhshers B V (North-Hol land Physics Pubhshlng Dwls,on)

For this class of experimental conditions, a new spec- trometry method, also based on the free propagation of emitted particles, is presented m this paper. It is a time-resolving method which determines the velocity spectrum of particles from the modifications of the time dependence of the particle flux with the flight distance It means that the experiment consists in recording the particle flux using a cham of detectors (not one, as in the standard tlme-of-fhght method) placed at various distances from the particle source in order to obtam a good approximation of the flux dependence on the

flight distance. Both from the experimental and from the data

processing point of view, this method ,s more difficult

than the time-of-flight method but it has the advantage of providing more detailed reformation about the phe- nomenon the time dependence of the veloctty spectrum of the emitted particles The duration of the particle emlsston can have, in prmcxple, any value because it does not affect the energy resolutxon of the method However, certain limitations determined by the char- actenstlcs of the detection systems (e g. detector rise- time) will appear m practical sltuat,ons When apphed m experiments where the standard time-of-flight method can stdl be used, this new method reqmres a much shorther flight path for the same energy resolution

The idea of the method and the equation for the t~me-dependent velocity distribution function of the particles are presented in sectxon 2. The numerical methods for solving th~s equaHon and some numerical tests of the method are the subject of section 3 Some general aspects concerning the design of an experiment, i e. the general methods of estimating the parameters of the detecting and recording systems, t h o r number and pos~t~ons are contained m section 4. A short description

328 M Vlad / Timing resoh,mg spectrometry method

of the experiment for which this method was prepared (for a plasma focus neutron source) and suggestions for some other applications are presented in section 5

2, Statement of the problem

Let us consider a point source emitting In a given direction a high number of identical particles in a burst

of duration AT. The particles fly in space with constant velocities (they do not interact and are not scattered by the medium)

The rate of particles incident on a detector placed at

a distance x from the source can be determined from the following considerations, a particle having the veloc- ity v will be at a time T at the distance x only if it was emitted at the time t (t < T) given by the relationship

t = T - x / ~ , ( 1 )

The total number of particles detected at distance x m the umt time interval, l e the particle flux, S(x, T). through the detector surface IS obtained by adding the contributions of any emission time moment, namely by Integrating the particle velocity spectrum, f ( u, t), In the (u, t) plane on the hyperbolae arc of eq. (1)

0 "v~

,~<,/~ ,,,) ~- , (2)

where vl, v 2 are the hm~ts of the particle velocity range When the emission is not dlrectxoned, the rxght hand

side of relationship (2) must have a coefficient equal to the solid angle determined by the surface of the detec- tor If the rate of incident partxcles is dlwded by this coefficient an expression similar to eq (2) but in which

S(x, T) is the time dependence at the distance x of the flux of parhcles emitted m the unit sohd angle, IS obtained. With this modification of the meaning of S(x, T) all the following considerations will apply to undirectloned particle sources too

For the condit ions of the s tandard time-of-flight method [i e., Instantaneous emission f (v , t )= f ) (v )8 ( t - t 0)] relationship (2) becomes.

S , , , ( x ,T )= f l ~ ( T - t o ) 2" (3)

so that the velocity spectrum is found to be

f , ( ~ ) = s,,,( x , x / ~ + t o ) X / V 2 (4)

Therefore, the time dependent particle flux, St.( x, T), recorded at any distance from the point source, is m fact the particle velocity spectrum (fig. la). In practice, this method is used for short pulses of parhcles if the flight path x is taken long enough so that the burst

</.",

to T ~

0 AT T

Fig 1 Propagation of bursts of nonmteractmg part)ties [a short (a) and a finite (b) duration burst] Two elementary short bursts are indicated m fig lb to show the superposmon of spectra of parhcles emitted at various tmles m the particle flux detected at any distance x

durat ion AT becomes much shorter than the flight time of any particle and it can be neglected.

For long bursts of particles, the particle flux mea- sured at any distance x represents a superposltlOn (de- pending on x) of the velocity spectra of particles emitted at various time moments (fig lb) The method de- scribed in this paper makes possible the separation, from this superposltion, of time-resolved velocity spec- tra. The time resolution is determined by the available detection systems in the sense that f ( v , t) represents the velocity spectrum averaged over a time interval of the order of the detector time constant

This separation can be performed using the follow- lng two observations concerning particle bursts emitted instantaneously at a time t o [f(v, t ) = f l ( v ) 8 ( t - to) ]'

(1) The time integral of the rate of particles recorded at a distance x from the source ~s independent of this distance, being equal to the total number of particles emitted in that direction

The relationship (3) and the variable change v = x / ( T - to) were used to obtain eq (5) It means that, integrat- ing the function S,.(x, T) over any straight hne x = a one obtains the same value

(2) This result can be generahsed for any family of parallel straight hnes m the.plane of variables x and T the hne integral of the function S,,,(x, T) has the same

M Vlad / Ttmmg resolving spectrometry method 329

~x,rl x ¢

to t T

Fig 2 Line integrals of the particle flux on a famdy of parallel straight lines of equatmn x = u ( T - t) for a short burst emitted at a t~me t o

value on any straight line of the famdy (fig 2). In order to demonstrate this statement we consider the parallel strmght hnes of equation x = u ( T - t ) , where t is a

parameter and u a constant, u ~ [v l, v2]. The surface dehmated by the lntersectmn of the

surface S,o(x, T) with a plane normal to the (x , T ) plane on the hne x = u ( T - t) (see fig 2), has the area

I,,, = ..:,n the hne dlS ' ° (x ' T ) ~ = u ( T t )

= f fdxdrS,,,(x, r ) a ( r - t - x / . )

= ( t + x / . ) f0Xd xS, o x ,

X IS the maximum distance for which the value of the function St,,(x, T ) on the strmght hne x = u ( T - t) is not zero (fig. 2). Usmg relationship (3) and performing

the variable change v = x / ( t - t o + x / u ) one obtains

:"~d uv 1,o = v f l ( v ) , (6) U - - U

which shows that the value of the area It,, is independent of the parameter t of the line faintly. It depends only on the velocity d~strtbutmn function of the particle burst, f l ( 9 ) and on the slope u of parallel hnes.

For a fimte duration of particle emission, the hne integral of the particle flux on the path of equation x = u ( T - t ) (see fig 3) is obtained by adding the con tnbu tmn of any emission t~me t o in the interval [0, t]

l ( t . u ) = f o X d X S ( x , t + x / u ) = f0'dto)t,, (7)

The upper limit of the first integral is

x = u . ~ (8) U - - U 2 '

and it is finite m the definition range of the variable u As follows from observatmn (2), the contr ibunon of

the pamcles emitted at a moment t 0, t o < t, to the value of the flux hne integral on the path (1) m fig. 3 is equal to the contribution of the same particles to the value of

Six,T)

0 to t' AT r" Fig 3 Line integrals of the pamcle flux on two parallel straight lines (1) and (2) for a long duration pamcle burst The propa- gation of the parncles emitted at a ttrne t o and the contribution of these particles to the two line integrals are md~cated

the line integral on the path (2) in fig 3. Takmg the difference between the two hne integrals of the functton S ( x , T), the contribution of the emission time t o ~s cancelled The same happens to any emission time m the interval [0, t] in fig. 3 and when t ' is very close to t, the difference of the two integrals contains only the contr lbuuon of the p i rac ies emitted at the moment t

This means that by differentiating relationship (7) with respect to t and by using relationship (6) for It, , one succeeds in separating the two variables of the d~stnbu- tlon function, f ( v , t), and obtains a one-dimensional equation for the velocity dlstr lbutmn function of the particles emitted at the t~me t:

f,, d fo x ( x ) "2 uv ~ d x S x , t + v f ( v , t ) u - v Ot

~ G ( u , t ) (9)

Th~s IS a Fredholm-type integral equatmn of the first kind in the variable v (t is just a parameter) which can

be solved numerically. It follows that the t~me-resolved velocity spectrum of

particles emitted m long, intense bursts can be de- termined by the time dependent flux of particles known as a function of the flight path [S(x , T)], i e by experi- mentally recording the particle flux in a sequence of spatial posmons that begins near the particle source and lasts to a value Xma ,. In principle, the total flight path, Xma x, IS arbitrary and Jt fixes the range of the varmble u of the integral eq. (9). This equatmn can be solved for any range of u (i.e. for any value of x . . . . ) if ~ts right hand side G(u, t) is exactly known For an approximate knowledge of ~t the soluUon can be obtained only for values of the total fl~ght path, xm,,x, exceeding a mini- mum value which depends on the error m G(u, t) (determined by the errors in the experimental values of the parucle flux and by the number and posmons of the detectors) and on the characteristic parameters of the unknown function f ( v , t) (average velocity and spec-

330 M Vlad / Ttmmg resolvmg spectrometry method

t rum width). Intmtlvely, this means that the total fhght pa th must be long enough so that the t ime dependence modlficaUons of the parucle flux t ravehng from the source to the last detector considerably exceed the ex- per imenta l errors. A method for evaluat ing the mm,- mum value of th~s dis tance is described m section 4

3. N u m e r i c a l so lu t ion and tes t s

The numerical code developed for ob ta in ing the velocity spectrum using this method conta ins two parts the computa t ion of the right hand side of eq. (9), G(u, t), from the exper imental values of the part ,cle flux and the solution of eq. (9)

The algori thm for comput ing the funct ion G(u, t) takes advantage of the fact that the particle flux S(x, T) is experimental ly obta ined as a cont inuous funct ion of t ime at some discrete values of the d,stance, %, ~ = 1, n, where n ts the n u m b e r of detectors The hne integral of the particle flux on the pa th x = u ( T - t) was calculated using a family of contour Itnes on the surface S(x, T). Determin ing the intersections of the integrat ion pa th with the contour lines it was possible to find the values of the particle flux S(x, T) at a larger n u m b e r of points on the path of ]ntegraUon and to have a bet ter accuracy of the results. The t tme denva te m G(u, t) was obta ined f rom values of the integral calculated at a set of points t,

using an ,nterpolat~on with the least-squares method This part of the code has been tested using some known velocity d~stnbutlon functions, f(v, t), to compute the part icle flux, S(x,, T), i = 1, n [relationship (2)], and then the right hand side G(u, t) of eq (9). The results were compared w~th the direct computa t ion of G(u , t) from the test d l s t r lbuuon function using the integral on the left hand side of eq (9) This compar ison shows that, ff the ve looty spectrum has a simple shape, the code determines the funct ion G(u, t) with an error of about 1% even when the particle flux xs recorded at only four distances

The soluuon of eq. (9) cannot be obta ined by means of s tandard numerical methods (discreuzat~on and solv- ing of a hnea r algebraic system) since Fredholm-type integral equat ,ons of the first kind, and especially those having a smooth kernel like eq. (9), are " i l l -posed" problems m the sense of Hadamard . A problem of thts class has a umque solutton, but ~t is not stable to the small vanaUons of the known functions m the equat ion [G( u, t ) for eq. (9)]. It means that, when these funct ions are not exactly known (because of the experimental errors or just because of the small t runcat ion errors of the computer) , the solution one ob tams by s tandard methods is not even an approximat ion of the so luhon of the exact equation.

The fact that Fredholm-type integral equat ions of the first kind are "al l-posed" problems results from the

f(v,t) 00 1 t . a )

" I 6, 100- • • /

/ - • • / 3

1 0 1 • • I"5[°Vu'] 2

-100 - • •• l 1

-200 - • |

J 0

b

1.5 ~[a.u J 1

Ftg 4 Results of the classical numerical method of solving eq (9) (a) and of a stabthsat~on method (b) for the test functton represented by the continuous hne

M Vlad / Ttming resolving spectrometry method 331

following observation, a large amphtude varmtlon 6 f of

the function f is t ransformed by the integral operator m eq (9) into an arbitrarily small variation 8G of G, if ~t is

sufficiently fast-oscdlatmg. Conversely, a small error 8G m G can correspond to arbltrardy large varmtlons 8 f m the computed solution f

Special numerical methods for solving such problems [2-13] and numerical codes [3,4] based on them were

developed during the last years. We have chosen for eq (9) mlmmlsat lon methods [2,4,9,10,13] using a smooth- mg regulanzer of the form f,',~ d v [ q t f 2 + q2(~f/av)2], where ql and q2 are continuous and positive functions of velocity We have used only the condition of noise attenuation, hut better results could be obtained ~f the

constraints resulting for some aprlort knowledge of the solutton are ~mposed. This means that such a method g~ves the solution of the integral equatton for larger errors m G(u, t)

The numerical method of solving the integral eq. (9)

was tested usmg gtven velocity spectra to compute the right hand side term, G(u, t). In fig 4 the efficiency of such a stabthsat~on method is presented an at tempt of solvmg eq. (9) by means classtcal numerical methods gives disastrous results (fig. 4a), whde the stabdisatlon method gets a very good approximation of the test function (fig. 4b). The results obtained for some test functions are presented in fig 5 For any simulation the total fl~ght path xm,~x and the number n of detecting

f(v.t) f(v,t)

X.o~= 5 a.U.

n-'5 • d5 =035%

3 t t ~' d5 =0.95% 2

1 f x xx

~'0 I 5 flv,t) a v f(~

2i

1

o-: 2O

X,,,ox= 4.OJJ. =

1%

I Z 23 24 V i a u,1

, , ~ . X ,,~,, = 5 IZU.

n = 5 %

'~t) '-'" b 15

I I O'

1.9

xoa,=lO a.u. n=4 5 =0.03%

. e a I .

2,23 2 6 VE,~cL]

d

Fig 5 Various tests of the stable method for solving the particle burst propagation eq (9) The continuous hne represents the test velocity distribution function at a fixed value of emission t~me The maximum d~stance Xma x of particle flux detection and the number n of detecting systems are mdtcated 8G is the maximum relatwe error on the right hand side of eq (9) for which st is stdl possible to obtain an accurate numerical solution of the problem for the specified values of x,la~ and n

332 M Vlad / Ttmmg resolving spectrometry method

systems are indicated. Because ill-posed problems are strongly affected by experimental errors, numerical studies dealing wath the solution of eq. (9) conta in ing a statastlcal f luctuat ion 8G in the right hand side term G were performed The max imum relative value of this error, ~G, for which the solution can stall be obta ined (for specified values of Xm~ ~ and n) was de termined In the next section it is proved that the velocity spectrum can be obta ined for larger values of the error 8G of G(u, t) if the flight pa th X n ~ IS increased. In all tests in fig. 5 the burs t du rauon Is one t ime umt We have to point out tha t this value reqmres in the s tandard time- of-flight method distances of 45 length units in exam- ples a and b and 75 length units in c and d for a velocity resolutaon of only 0.05 velocity units The values of xm~ x required by the new method are much smaller than

these

4. T h e e x p e r i m e n t

As was demons t ra ted in section 2 an exper iment of de te rmining the velocity spectrum of particles emit ted in intense pulses consasts in recording the time depen- dent flux of particles with a n u m b e r of identical, tem- porally correlated detect ing systems placed on the prop- agat lon direction in a sequence of posi t ions that begins near the particle source The n u m b e r and characterist ics of the detect ing systems that have to be used and the posi t ions at which they have to be placed are related parameters which depend on the par t icular condi t ions of each kind of experament. This section conta ins some general methods for their evaluat ion

The most impor tan t parameter for choosing the de- tectors and the assocmted electronics is the t ime resolu- tion. Its value must be smaller than the t ime intervals of significant variat ion In the time dependence of the part icle flux, but large enough to have small statistical errors of the measurement The ampl i tude cal ibrat ion of the detect ing and recording systems is not necessary, because the no rmahsa t ion condi t ion (5) can be used for a prel iminary processing of experimental da ta

We have considered so far that one experimental ly obta ins the t ime dependen t rate of particles incident on the detector surface, l e that the detect ing systems have the same sensitivity for all values of the partacle energy. This is not an essential condit ion. When the detector sensitivity is a known funct ion of the Incadent particle energy, ko(E) [where o(E) is the same for all detect ing systems one uses in this exper iment and k is a constant , which can be unknown and different for each detector], only the kernel of the particle burs t propagat ion eq. (9) is modified by a factor o, so that this becomes

f"2dvf(v,t)[ uv o ( v ) l = G ( u , t ) . (10) cl U - - U J

The numbers and the posit ions of the detectors depend on the errors in the particle flux records and on the characterist ic parameters of the particle velocity dis t r ibut ion funct ion (average velocity and spectrum width) The distance between two successive detectors is de termined by the condi t ion that the difference of the temporal shapes of the particle flux in the two records exceeds the experimental errors but is small enough for providing a good approximat ion of the flight distance dependence of the particle flux

In order to est imate the distance from the particle source to the posi t ion of the farthest detector, we change the parameter u in eq (9) into y = l / u and obta in the following equivalent equat ion

- d v f ( v , t ) l _ v v ~ dxS (x , t+xy )

=-F(y,t), y E [ y l , y z ] (11)

The max imum value Yz of the variable y corresponds to the max imum detect ion distance, Xrnax. because the upper limit of the integral in F(y, t) is an increasing funct ion of y (see fig 3)

t;2t X (12)

1 - v2y

The average particle velocity can be determined as a funct ion of t ime without solving eq (9), from the value in y = 0 of the funct ion F(y, t) as can be seen from the form (11) of eq. (9) and from relat ionship (2)

f,:'2dvf(v," t)v F(y, t ) , v= 0

S(x,)l _o (13)

If all particles emit ted at a time t have the velocity ~( t ) , [i.e., a monoenerglc spectrum f ( v , t ) = S(O, t)8(v - ~)], the funct ion F(y, t) In eq (11) will be

~(t) (14) Fmo,o(y, t)= S(O, t) 1 - 5(t)y

A measure of the velocity dependence of the right hand side of eq (11) due to the spectrum width a round the average velocity as the difference'

d(y, t)= r (y , t ) - Fmo,o(y, t ) (15)

When the right hand side F(y, t) of eq (11) is known with an error 6F, the solution of the problem can be found only if this error is, for any time t, much smaller than the var ia t ion Ad of the function d(y, t), in its definit ion range, [Yl, Y2]: 8F << Ad (16)

The variat ion Ad is, for any velocity dis t r ibut ion funct ion f ( v , t), an increasing funct ion of Y2, for Y2 > 0 (see appendix) This means that the variat ion A d is increasing with the flight pa th Xm~ x, SO that, for a given

M Vlad / Tmung resoh,mg spe~tromet O, method 333

error 8 F m F(y , t), the c o n d m o n {16) of solv,ng the problem can be fulfilled by ,ncreasmg the dis tance x . . . . For a fixed value of the flight path x ..... , the v a n a u o n ,.ld ,s an increasing function of the spectrum w~dth, At, = tj 2 - - t , 1 , and a decreasing functmn of the particle average velocity (see appen&x) This means that for a fixed experimental error, the flight path ,c,,,.,, required by this method decreases for smaller values of the average particle velocity and for larger values of the spect rum width In such expernnental c o n d m o n s the t ime dependence of the particle flux is modified after t ravehng small flight paths

One can observe these facts m the s lmulanons pre- sented m fig. 5, where the max imum allowed error 8G m the right hand side of eq (9) (which was determined for each test c o n d m o n by numerical experiments) has larger values for smaller average veloclues (b compared to c) or for larger spectrum width (b compared to a), or for longer distances of p ropaga tmn (d compared to c)

Thus, the flight path for this method can be found for any experimental condmons from the relat ionship (16) after evaluat ing the variat ion Ad as a functmn of the distance x using an est lmaUon of the velocity spec- t rum of the emit ted particles and after evaluat ing the error m the right hand side of eq (9) or (10) from the experimental errors in the records of the parncle flux

Having an est lmaUon of the fhght path and of the dis tance between two successive detectors, the n u m b e r of detect ,ng systems ~s obta ,ned as the quotient of these dtstances

5. Appl icat ions

The method presented in th,s paper was developed for neut ron emission studms on deuter ium plasmas pro- duced in plasma focus devices [14] These studies are very ~mportant, because they give &rect i n fo rmauon (not influenced by the electromc plasma componen t or by magnet ic and electric fields) about plasma fusion processes Due to the p lasma focus neut ron emissmn parameters (high neut ron emission rate, 101~-1018 n/s , em~ssmn duraUon of about 1 0 - 7 S w~th v a n a t m n m ume intervals of 10 -9 s, neut ron average energms of 2 5 MeV and spectrum width of some hundred keV), only a httle amoun t of thts reformat ion IS available by means of the exper imental methods used so far The energy spectrum of the neutrons emit ted m a given &rect .on has been measured only by means of two experimental tech- niques whmh are not very accurate, the ume-of-fl ight and nuclear emulsion methods, nmther method Is time resolv,ng It should be pointed out that the two methods give different results for the same &scharge, mdmatmg that there is a ume dependence of the neut ron energy spect rum dur ing the emission

The eshmaUons described m secuon 4 show that the

energy spectrum of the neutrons emit ted m a p lasma focus experiment can be determined w~th time resolu- t ion by means of this method using at least four detect- mg systems placed at equal &stance intervals on a flight pa th Xm, x which must exceed 5 m, but is not necessarily longer than 20 m The value of the flight pa th that has to be chosen in this interval depends on actual experi- mental condmons ( the recording errors, the number of available recording systems) It should be ment ioned that the s tandard time-of-flight method apphed to this exper iment reqmres flight paths of about 100 m for an energy resolution of only 100 keV [15,16]

This method can also be used m other experiments m which intense bursts (having finite time width) of non- in terac t ing particles that fly freely m space are emitted. It is very useful for neutral particles because the existing experimental methods are cumbersome and inaccurate. We ment ion only some examples other fu- sion experiments (laser fusion, electron beam fusion), fission and thermal neut ron studms, measurements of the energy spectra of neutral a tom sources.

Appendix

We study the dependence of Ad defined m relauon- ship (15) on the max imum distance of the pamc le flux detection, on the particle average velocity and on the spect rum width Eq (11) is used to compute the dif-

ference.

d ( y , t ) = F ( y , , ) - F . ~ o . o ( y , t )

f, , , v - ~ 1 - d v f ( v , t ) 1----~' " (17) 1 vy

In this r e l a t ionsh ip i5 is rep laced by ~ = f,'."

dv'f( v', t )v' /S(O, t)

1 d ( y , , ) =

S(0 , t ) ( 1 - ~)')

"' ~ - ~'-' ( 1 8 ) × f f dvdv'f(v,t)f(v',t)~Tvv t l

The double integral can be writ ten as a sum of an integral over the range v > v' and an mtegral over the range v ' > v In this last integral v and v' are rater- changed so that the range of this integral ~s t ransformed into the range of the first one

d( ),, t) = >' 2 S(0 . t ) ( 1 - O y )

ff" - dvdv ' f ( v t ) f ( v ' , t ) ×

× (~ ' - ' " )~ (19) (1 - v v ) ( l - v 'y )

33d M Vlad / Ttmmg resolomg spectrometry method

Dlfferent latmg it with respect to y, one obtains.

3 d ( y , t ) > 0 for y > 0 . (20) 0y

It follows that the variation of the functton d(y, t) in

the interval [Yl, Y2] is equal to.

A d = d(y2, t ) - d ( y , , t ) , (21)

and 3Ad/~y 2 = Od(y2, t)/3y2 > 0 for Y2 > 0, so that it is an increasing function of Y2 or, eqmvalently, of the maxtmum recording distance Xma x, for Y2 > 0. The con- dltlon Y2 > 0 corresponds, for any emission ttme t, to dxstances that are travelled by all the particles in t~me intervals longer than t.

The relationship (17) shows that the difference d(y , t) for a fixed, postlve value of y and a fixed average velocity ~s an increasing functton of the spec-

trum width Av = v 2 - - V 1 In order to study the dependence of the variation a d

on the average velocity ~, the upper limit X in the integral on the right hand side of eq (11) is expressed as a funcuon of

t(b + kAy) (22) x= 1 - ( ~ + kav )y '

where k is a constant, 0 < k < 1. The derivative Oy/3Ox=ct can be computed from this relationship and it is equal to - 1 / v ~ The denvatwe of the function d(y , t) with respect to ~ is'

Od(y, t) Od(y, t) Oy O~ Oy O~

Od(y, t) 1 - < 0 f o r y > 0. (23)

Oy vz 2

Therefore, the variation Ad IS a decreasing function of the average velocity of the particles, for fixed values of the maximum detecting dts'iance and of the spectrum

width

References

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