On the accuracy of the photon-difference method used in high-energy photonuclear experiments

NUCLEAR

INSTRUMENTS

AND

METHODS

95 0 9 7 0

245-252;

© NORTH-HOLLAND

O N THE A C C U R A C Y OF THE P H O T O N - D I F F E R E N C E

METHOD

PUBLISHINGCO.

USED

IN HIGH-ENERGY PHOTONUCLEAR EXPERIMENTS K. T E S C H

Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany Received 15 M a r c h 197l

The accuracy in calculating the cross sections per photon, ak, from a set of measured cross sections per equivalent 7 quantum, 6,, is discussed. Four commonly used approximations for the bremsstrahlung spectrum and corrections for finite target thicknesses and collimator angles are reviewed. For the calculation of ~k a simple unfolding method is proposed which is known for solving the Fredholm integral equation of the first kind, and in

which the necessary smoothing of the resulting curve is derived from the mean experimental error of ~,. For some examples of cross sections which may be encountered in photonuclear research above 100 MeV, the mean error of elk caused by inaccurate knowledge of the bremsstrahlung spectra, by the unfolding method and by the experimental errors of c~,~,is calculated.

1. Introduction

2). Although both a k curves are essentially different, they give very similar aq curves, which can only be separated experimentally with high measuring accuracies. Yet the calculation of ak is necessary, since the primary measured cross section per equivalent ? q u a n t u m is without any physical interest and depends on the special set-up of the experiment. In most publications the Penfold-Leiss method is used for calculating ak. Penfold and Leiss ~0) calculated sets of constants for every m a x i m u m energy up to 1 GeV assuming a Schiff spectrum, and by means of these constants the ak'S can be calculated as linear combinations of the aq'S. Once the constants are available, the m e t h o d is simple. But if experiments are extended to energies higher than 1 GeV, or if it is worthwhile to use a better representation o f the spectrum than Schiff's formula, the constants have to be recalculated anyway. Therefore it seems to be more convenient for solving eq. (1) to use more m o d e r n mathematical methods which are more flexible and transparent and, in addition, allow a satisfactory solution o f the problem of smoothing the resulting ak curves. To keep the error o f a k small a good approximation o f the actually used spectrum must be known ; secondly, a mathematical method must be available which is unlikely to introduce additional errors in ak; thirdly, the measuring inaccuracies in aq must be kept very small. Therefore, in the following section firstly the different approximations for the shape of the bremsstrahlung spectrum and the possible corrections due to finite collimator angles and target thicknesses will be reviewed. In section 3 a known method for solving the F r e d h o l m integral equation o f the first kind will be applied to eq. (l). Section 4 shows practical examples o f the resulting errors in ak.

It is well known that the measurement of nuclear reactions induced by high-energy t' quanta is difficult because monoenergetic p h o t o n s in the high-energy range are not available. Recently methods for the prod uction o f m o n o c h r o m a t i c 7 beams have been published (e.g., scattering of laser light by high-energy electrons), but such sources do not have large intensities, so that future research of photonuclear reactions will mainly be carried out with bremsstrahlung beams. The continuous,; spectrum is especially inconvenient in the simple activation experiments where the reaction products are not kinematically determined but identified after irradiation by nuclear physics methods. Here the only possibility for the determination of cross sections is to measure firstly the cross section per equivalent 7 q u a n t u m , ~rq, for a n u m b e r o f m a x i m u m energies of the bremsstrahlung spectrum, and then to calculate the cross section per photon, crk, as a function of the p h o t o n energy (photon-difference method). In recent years more such research in the energy range above 100 MeV has been done. Therefore this method and its principal errors will be examined in deta:[1 for this energy range. The connection between the magnitudes aq and ak is given by .... / =

f

ko'"x N ( k ..... k ) o k ( k ) d k ,

(1)

where N(k .......k) is the bremsstrahlung spectrum as a function of the ~ energy k and the m a x i m u m ?, energy k ...... normalized to 1 equivalent ? quantum. The difficulty in calculating ak(k ) f r o m a measured set of aq is shown in fig. 1. Two curves for ak were assumed and the corresponding values of aq(k .... ) for both curves calculated using a Schiff spectrum (see section

245

246

K. TESCH

~kq[mb]

~0

~------~~--~'-~A~'~''~

70

60

50

J ~/,~/r/

l i

i 40

"

'~/ .//o' / /J ~,.,

O'q,2

//

C;q~

\

"'1/',/ [Jx/",

,

'

\

.

Uk,2

30 /' ,t / / ?

20

~k,l

/' ~°// /

"/

•

/

.

, ,,\

*

I0

/

I

I

]

I

I

I

I

I

I

I00

200

300

400

500

600

700

800

900

!

I000

kkma,[MeV]

Fig. 1. Two curves ~,,1 and cq~,2calculated from curves 6k,~ and crk,2,respectively.

2. The shapeof the bremsstrahlungspectrum Firstly we consider the ideal uncollimated bremsstrahlung spectrum produced by means of an infinitely thin target and integrated over all production angles. Four different approximations for the shape of this spectrum can be found in experimental papers of recent years. The simplest representation is of course the 1/k spectrum which is independent of the maximum energy. The next best solution of the Bethe Heitler equation by means of the Born approximation is obtained if the atomic Thomas Fermi potential is assumed but only the special case of complete screening is considered. This results in an especially simple mathematical expression, which is given in the wellknown textbook of Rossi'), and will be called the Rossi spectrum in the following. The Rossi spectrum is a good approximation for very high energies and for Z,~ 137/2n ( Z = a t o m i c number of the target). The calculation by Schiff 2) gives a better approximation. Here a simplified exponential potential is assumed; thus Schiff obtained a simple analytic expression for the integrated spectrum instead of the Thomas-Fermi potential. This approximation should also give good results for low Z's only because of the use of the Born approximation. A more accurate expression for the shape of the integrated spectrum without application of the Born approximation was given by Bethe and Maximon 3'4) (BM spectrum). This approximation is

valid for all target materials; for energies above 50 MeV the error should be less than 1%4). Fig. 2 shows the four spectra, all normalized to one equivalent ? quantum, for an energy of 0.5 GeV and Z =73. Table 1 gives a quantitative comparison with the most accurate BM spectrum for the energies 0.1, 1 and 6 GeV; here the ideal BM spectrum was set equal to unity for all 7 energies. This shows that the simple Schiff spectrum represents an astonishingly good approximation, especially at higher energies. The ideal spectrum is modified by the finite thickness of the bremsstrahlung target and by the collimator (or, instead of the collimator, by an experimental target smaller than the natural diameter of the bremsstrahlung beam). These modifications were examined in detail by Lutz and SchulzS). They are due to the fact that both the flight path of the electrons in the target (mainly by multiple Coulomb scattering) and their energies (mainly by bremsstrahlung production) are changed. As table 1 shows, these corrections can be neglected for thin targets (to ~ 0.05 r.1.). For thicker targets (t o ~ 0.12 r.l.) and collimator angles 0o practically used they should be allowed for where a higher accuracy is wanted. The corrections increase with larger collimator angles, The spectrum is further modified in the experimental target. The calculation is simple, because only pair production has to be considered (hadronic effects

ON

THE

ACCURACY

OF

THE

PHOTON-DIFFERENCE

247

METHOD

N(k)-k 1.4

1.3 •\ \

1.2

\~,\

1.1

":')~.~._~

1.0

Vk Rossi ~ . /

0.9

0.8 0.7

t 0.1

I 0.2

0,3

I

I

I

l

I

I

0,4

0.5

0.6

0,7

0.8

0.9

k/Eo

1

1.0

Fig. 2. Four approximations for the bremsstrahlung spectrum. E0 = 500 MeV, Z : 73. account for less than 5%). In the case of a target of 0.2 r.1. thickness, the intensity of the bremsstrahlung beam will be reduced by approximately 5 to 8%, almost independent of the y energy. The shape of the spectrum remains unchanged. In the examples of section 4 we assume that the produced pairs still hit the quantameter and therefore are included in the quantameter reading normalizing the spectrum. If' actually a real collimator is used, the shape of the spectrum may be changed by the electromagnetic cascades initiated by interactions with the inner walls of the collimator. Unfortunately, this effect cannot be calculated. To keep it small the distance between collJimator and experimental target should be made as long as possible.

3. An unfolding method The photon-difference method requires the solution of eq. (1), which is known as a Fredholm integral equation of the first kind. The solving of this equation leads into difficulties if the aq(k .... ) are known with only a modest accuracy. Then one obtains strongly oscillating solutions which have to be excluded for the physical problem on hand. One can easily show 6) that this tendency towards oscillating solutions is due to the mathematical structure of the equation. This property is the principal disadvantage of the photon-difference method. Twomey 7) has shown how oscillating solutions can be .avoided by means of additional conditions. His method is simple and transparent and has the additional advantage making it possible to derive the

necessary smoothing of the solutions from the mean errors of the Oq'S. The method will be presented here in a way which is especially suitable for the problem in question and convenient for the user. Similar methods were applied by Cook s) and Routtig). I f we put ak(k) = f(k) a n d O - q ( k m a x ) = g ( k m a x ) , eq. (1) can be written

~

kmaxN(k . . . . . k)f(k)dk = g(km,×) +

8(kmax)~

(2)

0

where e is the error in the cross section per equivalent 7 quantum. We assume that aq has been measured at n end-point energies, and ak shall be calculated at m energies. Then we substitute the integral equation (2) by the linear system:

i=1

aji,f i = g j + £ j ;

j = l...n,

(3)

or in matrix form: Af = g + e.

(3a)

Generally ajl is the product of the spectrum at point (Li) and a constant which depends on the quadrature formula used (e.g. Simpson's rule). The ej's are of course unknown, but the mean error from the experiment is known, and thus we have the additional condition n

Z eJ2 : e2,

(4)

j=l

e is a known constant. Therefore t h e f [ s must be found

248

K. o

7

m

×

e-

IESCH in such a way that the ej's fulfil condition (4). Furthermore the resulting curve f(k) should be as smooth as possible; nonphysical oscillating solutions are avoided by minimizing the second differences at the i points. Therefore it is additionally required that the expression

QQQQ~%~

× tt~

<

>

7

m-

QQQQQQ~ ×

i

II gggm=~EE

~g ==e

~ o

<5

.

.

.

.

1

.

.

2

Z gJ2

(5)

j-I

should be small:

dC/df~ = 0,

(6)

OF

w

(m equations).

A*~: + 7 H f = 0

Here a*i

=

I

-2

× ¢.q

1

H =

x

0

0

0

0 •

0

1

0

0

0

0

1

0

0

0

l

0

0

5 -4

1 -4

7

(7)

aii and H is a constant:

1 -2 m ~

n

C = Z (J}-,-2J}+J}+,)2+Y-'

6 -4

0

I -4

6 -4

0

0

0

1 -4

0

0

0

0

7 × II

m

e-

×

?

©

~ ~

m m

f=(A*A +2H)-~'(A*g), g=Af -g.

×

?

ggggggg~

×

m

~

ii

N m

e-

a

H

.r~

II

g

1 - 2

I

The free parameter 7 has to be determined in such a way that eqs. (3a), (4) and (7) are fulfilled. These are n + 1 + m equations for the unknowns f , . . "fro, r'l"" .r,,, and 7.7 is a measure for the applied smoothing. A larger y causes a stronger smoothing of the resulting curve. The solution becomes much simpler when we take y as known. Then the final equations are:

~

-¢s

_

5 -2

(s)

In practice one will solve the equation system (8) for several values of 7, calculate the expression Ee 2 for each 7, and then choose that 7 for which this expression fulfils eq. (4) in the best way. Sometimes 7 has still to be taken a little larger to prevent non-physical negative solutions. The amount of smoothing is no longer arbitrary like in the Penfold-Leiss method but is connected in a transparent way with the mean experimental error. It is advantageous not to calculate the matrix elements asl in the way generally necessary but to make use of the fact that an approximation, the 1/k spectrum, is known which is independent of the maximum energy. For our special unfolding problem, and for a (small)

ON T H E A C C U R A C Y

OF T H E P H O T O N - D I F F E R E N C E

METHOD

249

O"k a s s u m e d

ak[ b] O"k r e c a l c u l a t e d :

60

•

50

o Y : 1 10"5 , Z Ej 2 = 9.9 ~/:1 10-4 , Z~j2= 0.086

~ / : 1.10 .6 , : C % 2 = 3-103

y= 3

•

40 •

30

°

2O

l

,

~EEj2= 35

"y: 1-102 , ~ E j 2 : 2 . 8 . 1 0 2

&

\o o o o\~

A

-.

/

A

/o

•

10

".. \'--"

I~0

' 200

~ ~ ~-----~-~--:~a-

;

;

,

,

,

,

z~ L3

3 0

4 0

S00

600

700

800

800

~000 k [ MeV ]

Fig. 3 . 2 0 values o r aq ( n o t shown) were calculated f r o m the assumed ak curve. F r o m these values, 30 ak's were recalculated w i t h 5 different values of the s m o o t h i n g p a r a m e t e r 7- F o r each 7 the c o r r e s p o n d i n g m e a n error o f aq, ~ej 2, is also given.

interval kl... formed:

ki+,, the right side of eq. (1) can be trans-

f kk~+,N(kma, j,k)f(k)dk=f(kx,i)Jk['k~ ' +,N(kma, i,k)dk,

(9)

i

where kx,i lies in the interval kl...ki+r If we assume for a moment a l/k spectrum, and that f(k) is linear in the interval kz...ki+ ~, we obtain

ki+l-kl kx,i - ln(ki+ l/kl)"

(lO)

Then the matrix A we can calculate as

a j, =

k i+!

N(km,,j,k)dk,

(ll)

i

provided that f(k) is taken at point f(kx,,) in the interval ki... ki +l; N(kmax j, k) is the spectrum actually used. In this way smaller unfolding errors are obtained. This is important in all cases where the experimental errors are small, since for the smoothing of oscillations due to inaccuracies of the ai~ the expression Z~2 cannot be used as a criterion for choosing the smoothing parameter 7. The method permits to choose i >j. This also helps to reduce the errors of the unfolding procedure. Fig. 3 shows an example for the unfolding method. The assumed cross section shows a resonance at 200

MeV and some structure around 400 MeV. Under the assumption of a Schiff spectrum, aq was calculated for 20 bremsstrahlung energies, and from this curve 30 values for ak were recalculated. As no experimental errors for ak were assumed, the smoothing value for the parameter 7 must be chosen in such way that the calculated value for E ~ is as small as possible. 4. Error s o u r c e s

Instead of a general error discussion based on eq. (1) it is simpler and more instructive to study the influence of inaccuracies in a k by means of practical examples. Similar calculations can then be made in actual cases. The error caused by the unfolding method described is small and usually not higher than a few percent, provided that in calculating the matrix A the integration errors are smaller than 1 x 10 -3. Larger errors of ak can be expected if narrow peaks show up in the cross section, because in the calculation the second derivatives are minimized. In our field sharp resonances are interesting if the accelerator allows measurements not only in the high-energy range but also in the giantresonance region. Fig. 4 shows an example, in which the cross section from fig. 3 was supplemented by a sharp resonance at 15 MeV. 30 measurements of aq between 10 and 1000 MeV were assumed. Obviously the unfolding method described in section 3 gives quite satisfactory results (indicated by crosses in fig. 4) also in this giant-resonance region. If, however, the aq mea-

250

K. T E S C H

o [mbl ~40

120

?\

60 i -50

100

80

6C

-30

4G

~-20 )

i

o /~ o ~

2(3 0 0

/

I

*-----4 *!/ 10 20 3

/-,0

I

~ 100

I

200

I

300

I

400

l

500

l

600

I

700

I

800

I

900

1000 k [NeV]

Fig. 4. The unfolding method in the giant-resonance region, x : 30 rr,~ measurements between 10 and 1000 M e V are assumed; O : 30 or,, measurements between 60 and 1000 M e V are assumed. surements can only be m a d e at higher energies, one obtains wrong results for the resonance region: the circles in fig. 4 show the calculated crk values for 30 O'q m e a s u r e m e n t s between 60 and 1000 MeV. However, in b o t h cases above 100 M e V the mean error in c% is smaller than 2%. A n example for the errors due to inaccurate app r o x i m a t i o n s of the b r e m s s t r a h l u n g is shown in fig. 5. First the same cross section a~ as in fig. 3 was assumed

(continuous curve in fig. 5), and then 20 values o f Oq between 100 and 1000 MeV were calculated. This calculation was based on the best known a p p r o x i m a t i o n for the bremsstrahlung spectrum: BM spectrum with corrections for the finite target thickness in the accelerator (in our example 0.12 r.1.), c o l l i m a t o r angle (5 × 10 -3 rad), and thickness o f experimental target (0.1 r,l.). F r o m these Oq values, 30 values for ak were then recalculated using the following spectra: the same

mb] 6O

.\

50 40

30

20

\:

~o

" ~

/

~ ,~+

•

%

•

•

a

•

•

•

•

•

•

•

•

X 0 + \X,~ ° ~ -

10 X~X~x~ X

1 O0

200

I

[

300

400

.

I

I

I

r

[

500

600

700

800

900

~. 1000 k [MeV]

Fig. 5. F r o m the assumed c~, curve (continuous line) 20 values of ~,~ (not shown) were calculated using a corrected BM spectrum.

From these values, 30 ~,'s were recalculated by means of a corrected BM spectrum (×), an ideal BM spectrum (O), a Rossi spectrum (+), and a Ilk spectrum (O).

ON T H E A C C U R A C Y

OF T H E P H O T O N - D I F F E R E N C E

251

METHOD

Ok [mb] 60

50

40

30

20

10

I

100

200

Ok

[MW]

Fig. 6. F r o m the a s s u m e d crk curve ( c o n t i n u o u s line) 20 values of aq (not s h o w n ) were calculated, to which statistical errors with a m e a n o f 1 ~ a n d 1 0 ~ were applied. A crk curve recalculated f r o m these values lies statistically distributed in the hatched area a n d in the grey area, respectively.

"best" spectrum like above, BM spectrum without corrections or Schiff spectrum (both give practically the same results), Rossi spectrum and 1/k spectrum. The mean values of the errors of the resulting ak'S in our four cases are: 2.7%, 15%, 21% and 80%. These values are practically independent of the distribution of the energy points at which the aq and ak values were calculated, and also rather independent of the number of tJhe calculated crk's. It should be mentioned that the given example is especially unfavourable, and that the discrepancies in using different spectra become smaller if the resonance is situated at the high-energy end of the energy range (e.g. at 850 MeV). The errors o f a k caused by statistical inaccuracies are more serious. It seems that this error source is sometimes underestimated. We studied the error propagation Z~O'q/O'q-"~AO'k/O"k for the described method in the following examples: First the same cross section ak as in fig. 3 was assumed and aq calculated using a Schiff spectrum. By means of random numbers and on the assumption of a Gaussian distribution, a statistical error was applied to each aq and then the unfolding procedure was carried through. Fig. 6 shows an example. The assumed mean statistical errors of the aq'S were 1% and 10%. For both cases, 20 values for O'q between 60 and 1000 MeV, and 30 values for crk between 40 and 1000 MeV were chosen, all equally distributed in the respective intervals; the calculations were performed for 5 to 10 sets of random numbers. The resulting crk curves were found to be statistically distributed in the

hatched area (mean error 1%) and in the grey area (mean error 10%). ak being known in our examples, the smallest error averaged over all calculated ak's and all sets of random numbers can be calculated, if for every unfolding the smoothing parameter y is not chosen by using the criterion eq. (4) but by looking for the best agreement with the crk curve known a priori. These mean errors of ~k are 20% and 44% in our example. Here the error due to the unfolding method is small and amounts to 3%. In practice Ck is of course unknown and the criterion eq. (4) must be applied for choosing the most probable curve. Then the mean errors are still larger: 26°/; and 48%. These numbers, by the way, are not very dependent on the number of the % measurements;if, e.g., an experiment with a mean statistical error of 10% is repeated the reduction of the ~rk error due to the doubled number of Cq measurements is hardly visible. The examples show that for the application of the photon-difference method the statistical errors of the measured cross sections per equivalent ? quantum should not be much larger than 1%. If this is achieved, a good approximation for the actually available bremsstrahlung spectrum should be used. But even under these favourable conditions the errors of ~rk will be high, and they can be studied in the simple way outlined above. The measurement of a fine structure or the shape of resonances of the cross section Ck will turn out to be difficult anyway.

252

K. TESCH

R eferences 1) B. Rossi, High-energyparticles(Prentice Hall Inc., Englewood Cliffs, N.Y., 1956). e) L. I. Schiff, Phys. Rev. 83 (11951) 252. :)) H. A. Bethe and L. C. Maximon, Phys. Rev. 93 (1954) 768; H. Olsen, Phys. Rev. 99 (1955) 1335. 1) D. Lublow, DESY A2.96 (1963) unpublished.

5) G. Lutz and HI. D. Schulz, DESY 67/29 (1967). (~) D. L. Phillips, J. Assoc. Comput. Mach. 9 (1962) 84. 7) S. Twomey, J. Assoc. Comput. Mach. 10 (1963) 97. x) B. C. Cook, Nucl. Instr. and Metb. 24 (1963) 256. ~) J. T. Routti, UCRL-18514 (Lawrence Rad. Lab., Berkeley, 1969). io) A. S. Penfold and J. E. Leiss, Phys. Rev. 114 (1959) 1332.