Nuclear spectroscopy with energy resolution of a part per million

NUCLEAR INSTRUMENTS AND METHODS 59 (I968) I97-213; © NORTH-HOLLAND P U B L I S H I N G CO.

N U C L E A R S P E C T R O S C O P Y W I T H E N E R G Y R E S O L U T I O N OF A P A R T P E R M I L L I O N * C. J. KAPADIA, V. E. MICHALK and J. A. McINTYRE Texas A&M University, College Station, Texas, U.S.A.

Received 28 September 1967 An instrument is described for producing a variable energy gamma ray beam with energy resolution of a part per million utilizing neutron capture gamma rays. The variation in energy is achieved through the recoil velocity of the radiating nucleus which has been excited by a gamma ray through the resonance fluorescence process. The instrument has been tested by exciting

an energy level at 7.277 MeV in s0spb, also through the resonance fluorescence process, and it is shown that the line width/" of the level can be measured. It is estimated that with a one MW reactor (neutron flux of 1013/cmS'sec)one energy level per month can be studied using such an instrument.

1. Introduction

2. Theoretical background

In the preceding 1) paper an instrument was described for producing a variable energy g a m m a ray beam to be used for resonance fluorescence studies of nuclei. The instrument, however, is limited to the excitation of energy levels below 2 MeV. This limitation can be avoided, as has been shown by Knowles et al. 2) by using an intense source of neutron capture g a m m a rays to provide a primary beam energy of 9 MeV. Energy levels as high as 7.8 MeV have been studied with this instrument. The energy resolution of this device, however, is determined by the resolution of the lithiumdrifted germanium g a m m a ray detector which is in the neighborhood of 10 keV. This value is larger than the energy level spacing found in the neighborhood of the neutron binding energy of most nuclei a); therefore, this type of instrument cannot, in general, be used to study individual nuclear energy levels in this energy region. The purpose of this paper is to describe an instrument 4) which is capable of studying energy levels in the 5-9 MeV region with an energy resolution limited only by the Doppler-broadened line width of the level itself, which is of the order of a few eV. However, the range of the instrument is of the order of about 1 keV for a particular neutron capture g a m m a ray source. Thus, the instrument can be considered as an energy vernier for the broad range but poorer resolution instrument of Knowles 2) that was referred to in the last paragraph. Some introductory comments on the resonance fluorescence process will be made in section 2 before describing the instrument in section 3. Its operation and the measurement of a line width will be explained in section 4. An analysis of experimental data obtained with the instrument will be given in section 5, and a discussion of the applicability and limitations of the device will appear in section 6.

The operation of the instrument depends on the resonance fluorescence process. In this process a g a m m a ray of energy E and wavelength 2 excites a nuclear energy level of energy Er. A g a m m a ray of t h e same energy is then emitted by the excited nucleus. The differential cross section for emission of the g a m ~ ray into the solid angle dO at an angle 0 with respect to the direction of the initial g a m m a ray may be writtenS): dtr/df2 = {w(o)/4,r}~o(ro/r) 2 {[2(E-Er)/ r] 2 + 1}-a, where

(1) fro = (22[2rt)(2Jt + 1)[(2J o + 1).

(2)

The total cross section for the absorption of g a m m a rays by the resonance fluorescence process, aa, will be then ~ = ao(ro/r) { [ 2 ( E - Er)/F] 2 + 1}- 1.

(3)

The notation is as follows: W(O) = Po + a2P2 (cos 0) + a4P4 (cos 0) where the P/(cos 0) are Legendre polynomials normalized so that P0 = 1; F is the total line width of the excited state, F 0 is the partial width for a g a m m a ray transition to the ground state, J1 is the spin of the excited state, and 3"0 the spin of the ground state. The F parameter is termed the natural width of the excited nuclear state. Eqs. (1)and (3)show that the cross section has, for an energy dependence, the well-known "Lorentz" shape. However, these equations were derived with the assumption that the excited nuclei were all at rest. In fact, the nuclei are in motion and the effect of this motion is to broaden the energy dependent peak in the cross section. The cross sections can be written then as

da/dg2 =

(4a)

and

* Supported by the Robert A. Welch Foundation. =

197

(4b)

198

c . J . KAPADIA et al.

where ao is defined in eq. (2) and O(x,t)=½(nt) ½

[exp{-l(x-y)2/t}](l+y2)-'dy. -

(5)

The new parameters x, y, and t are defined as x = 2(E-E3/F,

(6a)

y = 2(E' - Er) / F,

(6b)

t = (A/r)

(6c)

and A = (2kTeff/Mc2)½Er .

(6d)

Here, k is Boltzmann's constant, Toff is an effective temperature of the solid which is related s) to the laboratory and the Debye temperature, M c 2 is the rest energy of the nucleus, and E, is the resonance energy of excitation. The parameter, A, is called the Doppler width of the emitted gamma ray spectrum. The essential points for the following work is that the resonance fluorescence cross section has a narrow peak of width F when F >>A or width A when A >> F. In the first case, the cross section has the "Lorentz" dependence on energy, for the last, a "Gauss" or "Doppler" dependence as can be seen by inspection of eq. (5). For the nucleus to be studied here the peak cross section has a magnitude of about 25 barn. The line width parameters F and A have values of 0.8eV and 3.76eV respectively. It has been necessary, therefore, in the analysis of the data to use the exact expression in eq. (5). Necessary calculations have been performed with an IBM 7094 computer. For purposes of planning experiments and for thinking about the meaning of data it is convenient to have a simpler expression for the cross section than that in eqs. (4) and (5). Such an expression can be obtained by assumingthat A >>/7 even though, in the experiments under consideration, A is only 5F. For A >>/7, then, the resonance scattering differential cross section of eq. (4a) becomes da/d~2 = {W(O)/(8n½)}ao(Fo/A) • •( r o / r ) e x p { - ( e - e , ) 2 / A 2 } ,

for J1, the spin of the excited nuclear state. This spin can be found also by resonance fluorescence if necessary, since W(O) depends on J1- In the following, J1 will be assumed to be known.

3. Description of the instrument A schematic drawing of the instrument is shown in fig. 1. A source material for the production of neutron capture gamma rays is placed next to a reactor core so that a large flux of thermal neutrons passes through the source material. Neutron-capture gamma rays originating in this material pass through the shield wall of the reactor and strike a target. To be specific, we select the 7.277 MeV capture gamma ray 6) from S7Fe to strike a 2°apb target. Then, as has been shownT), the 7.277 MeV gamma ray is scattered by resonance fluorescence from a level in 2°apb with an average cross section of about 4 barn. The existence of this large cross section provides the basis for the instrument shown in fig. 1. Since the lifetime of the excited 2°aPb level is about 10-15 see the 2°aPb nucleus still has its velocity of recoil when it re-emits the gamma ray. The resonance scattering process can be considered kinematically, therefore, to be the same kind of process as that for Compton scattering except that in the resonance fluorescence case the recoil particle is a 2°spb nucleus instead of an electron. The energy, E', of the scattered gamma ray can thus be expressed E' = E [ I + ( E / # ) ( I - c o s 0 ) ]

-x,

(9)

where E is the energy of the incoming gamma ray, 0 is the angle between the directions of the incoming and outgoing gamma rays, and # -- M e 2 is the rest energy of

Reactor Shield Wall (n,)') S0ur7

nO/ E-SOOeV

Absorber

E-2OeV

Reactor (7) E=Z277 MeV Gamma Rays

while the total absorption cross section of eq. (4b) becomes a= = ½n½ao(Fo/A) exp ( - ( E - Er)21A 2 },

with ao defined The purpose is to determine eqs. (7) and (8)

(8)

by eq. (2). of resonance fluorescence experiments F o and F since the other quantities in are ordinarily known; except possibly

o TypicalSpectrum

Fig. 1. Schematic diagram illustrating the principle of operation of the instrument.

NUCLEAR

SPECTROSCOPY

the 2°spb nucleus• For E = 7.277 MeV, the value of E ' varies between E(O = 0 °) and E - 545 eV (0 = 180°). An arrangement has therefore been achieved which produces g a m m a rays of variable energy. The g a m m a rays are emitted by the target and the energy is determined by the angle 0. The energy resolution of the variable energy beam is obtained by differentiating eq. (9), dE(eV) = - 273 sin 0 dO (rad),

(10a)

dE(eV) = - 4 . 7 6 s i n 0 d0(deg).

(10b)

or

An energy resolution of a few eV/deg is achieved. At r o o m temperature the Doppler-broadened line width of the g a m m a rays emitted by the Z°Spb nucleus is 3.76 eV which is comparable, then, to the natural energy resolution of the apparatus. A method for utilizing this variable energy beam is indicated also in fig. 1. A detector is placed at an angle 0 with respect to the beam. Between the detector and the target an absorber is located containing the nuclear species to be studied. A hypothetical detector counting rate as a function of 0 is indicated at the lower right of fig. 1. A smooth curve is obtained except at angles corresponding to energies which coincide with nuclear energy levels. At these angles the resonance absorption cross section, eq. (8), rises rapidly so that a dip in the counting rate results. Since, as shown in eq. (8), the magnitude of the absorption cross section is proportional to F o, such a measurement can be used to determine F o. [-It should be recalled that eq. (8) holds only for A >> F so that, insofar as this inequality is not valid, the cross section will depend also somewhat on F.] In addition to determining the value of F 0 for the various nuclear levels excited, the spacing of the dips in the counting rate curve in fig. 1 will give also the spacing between the energy levels that are excited by an electromagnetic transition between the ground state and the excited state. In this paper we report on an instrument constructed on the basis of the principles just described. The purpose of the instrument was, first of all, to test the feasibility of the instrument under actual experimental conditions. It was decided, therefore, not to jeopardize the success of the test by searching for a line for study in the 545 eV energy range of the instrument. Rather, a study could be made of the energy level in 2°spb that is responsible for the resonance scattering o f the 7.277 MeV neutron capture g a m m a ray from 57Fe. Since it was known s) that the energy of the 57Fe g a m m a ray was above that of the 2°Spb energy level, an absorption dip in a lead absorber corresponding to that for the excited level in 2°spb should appear at an angle near 0 °. The fact that

WITH

ENERGY

199

RESOLUTION

the scattered energy shifts very slowly near 0 °, eq. (10), helps to move the dip away from the 0 ° position. The amount of the shift of the dip away from 0 ° is easily calculated using the Doppler form of the resonance scattering cross section given in eq. (7). The situation is shown in fig. 2 where the spectrum of the incoming beam of g a m m a rays N(E) is seen to be displaced an amount of energy 6 above the resonance energy of the excited nuclear state in 2°spb. Since there is no nuclear recoil in the 0 = 0 ° direction, we calculate here the spectrum, Nx(E, 0 = 0 °) of the g a m m a rays at 0 °. Writing N(E) as

N(E) = {N O/(n½Ab)}exp ( -- [(E - E,)-- 612/A~ }, (I I) where 5 = E b - E r , E b being the mean energy of the g a m m a ray beam, and A b being the Doppler width of the beam, NI(E,O°) is found to be, from eqs. (7) and (11), NI(E,O °) =

KN(E)da(E,O = O°)/dt2,

N,(E,O °) = K (W(O°)/(8~Ab)}aoNo(Fo/A) (ro/r). • e x p ( - ~2/A o2). e x p { - [ ( E -

E,) - C]2/A ~},

(12) where Kis a constant depending on the target geometry. The quantities Ao a n d / t 1 are defined as Ao = (A z +A2)~

(laa)

A, = AbfA/Ao),

(13b)

while C, the energy displacement of the scattered g a m m a ray spectrum, is

C = 6(A/Ao).

dt~/dn~~

Er

Eb

I I

t I

(13c)

~ \

ENERGY

Fig. 2. Schematic diagram illustrating the energy relationship between the neutron capture gamma ray beam with energy centred at Eb aqd the resonance scattering differential cross section with energy centered at E,. The Doppler widths for the two spectra, A and Ab, are also shown.

200

c.J.

KAPADIA

Eq. (12) exhibits the following features of the spectrum of the gamma rays scattered at 0 °, N~(E,0°): 1. The intensity is proportional to exp (-62/A2o). 2. The mean energy of the gamma ray beam is displaced from Er + di to E~ + C. 3. The Doppler width of the spectrum is decreased from Ab to A 1. It is the spectrum NI(E,O °) of eq. (12) that is shifted in energy at the different scattering angles according to the relation given in eq. (9). For the experiment under consideration, 6 has been measured 9) to be about 6 eV, while Ab ~ 2A with A = 3.76 eV. The value of C, then, is 2.7 eV. From eq. (9), the energy ~hift of the scattered gamma ray (E-E'), may be written E - E' = (E2/#) (½02),

(laa)

( E - E') (eV) = 0.042 02(deg).

(14b)

or,

Consider now, a lead absorber to be used in fig. 1. The minimum in the absorption dip will occur at an angle 0 where the energy has been shifted by C = 2.7 eV. Using eq. (14a), 0 = 8°. The energy width of the dip Ao is related to the energy width of the scattered beam A and the energy width of the resonance line A by the expression A 2 = A~+A 2. (14c) This result can be obtained from the following considerations. For the angle 0, the maximum value of N~ in eq. (12) no longer occurs at the energy Em = E~+ C but shifts to the energy E'(O)= E~+ C'(O) where E'(O) is calculated from Em using eq. (14a) and Em-E~(O) = C-C'(O). We, therefore, obtain for NI(E',O), the scattered energy spectrum at the angle 0,

NI(E', O) = r l exp { - [ ( E ' - E~)- C'(O)] 2/A 2,

(lad)

where K 1 is a constant. The number of gamma rays penetrating the absorber at the angle 0 in fig. 1 is

Na(O) =

NI(E', O) exp { - noa,(E')x} dE', 0

where no is the number of resonant nuclei per unit volume, a, is the resonance absorption cross section of eq. (8) and x is the absorber thickness. For a thin absorber, using eqs. (8) and (14d),

f

Na(O) = K2 - K3 ~dE' exp { - [(F_f- E r ) - C'(0)]21A 21). .exp{-(E'-E,)2/A2}, or

N,(0) = K2 - K4exp { - (C'(O)/Aa) 2).

(14e)

et

aL

Here, the Kl are constants while da is given by eq. (14c). Eq. (14e) shows that the width of the absorption dip is Ad. To convert the width of the dip from the energy scale to the angular scale a crude estimate can be obtained by usingeq. (10b) at 0 = 8 °, the minimum of the absorption dip. From the above values of A 1 and A, Ad = 5 eV anc!, using eq. (10b), Ad = 7.5 °. Thus, the absorption dip should extend from its minimum at 8° to angles beyond ! 5°. A measurelnent has already been made and reported 4) on this absorption dip. It was found that data could be obtained for angles greater than 13°. For smaller angles the energy of the Compton-scattered gamma rays had increased to the point that they interfered Seriously in the NaI(T1) detector with the elastic resonance-scattered gamma rays. In the work to be reported here this limitation was avoided by detecting the presence of the resonance-scattered gamma rays by a second resor~ance scattering rather than by the use of an absorber. Since the Compton-scattered gamma rays have been shifted off the resonance line even at the smallest scattering angles, they will be eliminated by the second resonance scattering process. Vi The apparaIus used to perform the double scattering experiment is ~hown to scale in fig. 3. The reactor core at the right is that of the 100 kW swimming pool Texas A & M University Research Reactor. The natural-iron capture-gamma-ray source was placed in an aluminum tube of 11.5 cm i.d., the tube being supported from the bottom of th e pool and placed adjacent to the core as shown. The neutron flux was about 1012 neutrons/cm 2. sec. The source material consisted of two iron plates 12.5 cm long by 9 cm high by 1.25 cm thick set at 45 ° with respect to the axis of the tube. A 12.5 cm-thick bismuth plug was placed at the right end of the tube to stop any neutron-capture gamma rays originating in the aluminum cap at the end of the tube. A narrow water gap (1 cm) separated the tube in the pool from the tube through the reactor wall. Boron-loaded paraffin absorbers in the wall tube of 20 cm thickness were used to reduce the neotron flux passing through the wall without greatly BStenuating the desired gamma ray beam. Paraffin as well as lead collimators were used to reduce both the neqCron and the gamma ray flux to negligible amounts outside the collimated beam. The lead collimators were so placed that the experimental targets could not "s~e" the wall of the aluminum tube near the reactor core. The number of gamma rays from neutron capture in aluminum was thereby reduced to a small value. The beam of gamma rays to be used for the experiment emergvs from the tube in the reactor wall and

NUCLEAR SPECTROSCOPY WITH ENERGY RESOLUTION

/..:.~

..,.~

. ~.~/

(,:~ A .~.~ o ' , b-° ~ . . / I ~ " ° ,' "j . 0 / I~" t

~' " ,~ "

BORAL

PLATE--....

SECOND TARGET7

FIRST

~;~:;:~ ~ ! ~

~ L-L~'AD

LEAD

~

PARAFFIN ~

WATER--~&

~" ' / ~ I- ~ / . - I REACTOR

:' :° °°'o/ CORE--N'[--BISMUTH "-4

I ~ ~.~.-,~' o o : ~°:,, ,1 . . .

" ~ ,,

i ~/~

~-~-=

¢ ~' ~

~ ~ v

~

. .0 - o /

~;

~'j,.\

~ ~ ~ L ' ~

COLLIMATORS ~-DETECTOR SHIELDING

201

0 '~

. ~

IX,

I ~

~

LIRON

~ ~

ISHIELD7

I

CAPTURE

GAMMA RAY SOURCE

A.~~, ~' ~" " - - C O N C R E T E

\ ~ I METER

'

Fig. 31 Scale drawing of experimental arrangement.

strikes the first target of the experiment. The beam dimensions are determined by the 5-cm wide by 10-cm high collimator apertures. A lead wall hdjacent to the first target acts as a collimator and permits the angle 0 to be as small as 5 ° without the secofid target being struck by gamma rays in the beam. The first target is mounted on the vertical axis of the instrument. The second target is mounted on a heavy steel beam which can be rotated about this axis. The shielded gamma ray detector shown adjacent to the second target is supported by a heavy table which can be rofated about the vertical axis of the second target. As shotVn in fig. 3, the detector is placed so that the angle of scattering detected at the second target is 90 ° . The dimensions of the first lead tariget were 5 cm wide by 10 cm high by 1.1 cm thick, the thickness being chosen to optimize scattering while minimizing absorption of the scattered gamma rays. The number of 7.277 MeV photons striking this target ~kas 8 x 106/sec. A bismuth dummy target of identical c r ~ s section with the same number of nuclei was also Used for background measurements. The second target was placed at a distance of 137 cm from the first target; its dimensions were 10 cm x 10 cm x 1.97 cm thi~-k. This target was placed at a 45 ° angle with respect tO the beam and oriented so that the scattered g a m m a rays emerged through the same face as those enterin[ the target. A 7.5 cm long by 7.5 cm dia. NaI(TI) scintillator was used to detect the scattered gamma rays. It Was placed with its front face 7.5 cm from the vertical axis of the second

target. The detector was shielded by 15 cm of lead and 10 cm of boron-loaded paraffin except for an aperture between the detector and the target. A 1.85 cm thick boron-loaded paraffin plug was placed in the aperture to shield the detector from slow neutrons. Gamma rays resonantly scattered by the first target will also resonantly scatter from the second target for small values of 0 and will be detected by the shielded NaI(TI) detector placed at a 90 ° angle with; respect to the gamma rays striking the second target. These gamma rays will have their full energy of 7.277 MeV. On the other hand, gamma rays Compton-scattered by the first target will enter the second target off-resonance and so will scatter from the second target only by the Compton effect; their energy, after scattering through 90 °, will be only 480 keV. Thus, the gamma rays that were scattered by the resonance process by the first target have been effectively separated from gamma rays that were scattered by the Compton process by the first target. The small-angle limit of 13° experienced in the absorption measurement has, therefore, been removed and it was possible to obtain data down to an angle of 0 = 5 °. A price was paid, of course, for this separation of the resonance scattering, namely, a large reduction in counting rate. By judicious shielding (fig. 3) the background counting rate was reduced to about twice the cosmic ray level which was 2 counts/min (integrating the counts under the two highest energy peaks in the 7.277 MeV pulse height spectrum of the NaI(TI)

202

c . J . KAPADIA et al.

detector). The background, however, was still about four times that of the number of double scattering events. The term "background" is used here to mean the counting rate obtained when bismuth rather than lead targets were placed in the first and second target positions. It was found that the low background obtained depended on the fact that a "tangential" beam port '°) was used, i.e. a port placed tangential to the reactor core as shown in fig. 3. Initially, a "radial" beam port was used for the experiment, i.e. a port with its axis passing through the reactor core. The lowest background achieved with the radial beam port was 40 counts/min as compared to the 4 counts/min achieved with the tangential port. From the results of various shielding tests it appeared that fast neutrons from the reactor core passing through the beam port accounted for the background obtained with the radial port. This

conclusion is based on the negative evidence that slow neutrons and gamma rays were eliminated as the source of background by the shielding measurements. Photographs of the apparatus are shown in figs. 4 and 5. The base of the spectrometer has four extended legs. The first target is placed on the spectrometer axis; the second target (hidden behind the paraffin-shielded detector at the left) rotates about the axis and is supported by the two wheels at the bottom left. The reactor wall is at the right. The cave of concrete blocks formed at the left is used as the gamma ray "beam catcher". In fig. 5 the view of the apparatus is from atop the beam catcher and looking toward the reactor wall. Two experiments are visible, being separated by a dividing wall of shielding. The experiment at the right is the one being described. The aperture in the reactor wall, the first target, and the second target are clearly visible. Not shown in any of the figures is the Geiger tube monitor

Fig. 4. Photograph of the apparatus from the side.

NUCLEAR

SPECTROSCOPY

WITH ENERGY RESOLUTION

203

Fig. 5. Photograph of the apparatus from atop the beam catcher and looking along the beam line toward the reactor. which is placed near the reactor wall and counts gamma rays scattered by the beam collimator.

4. Experimental results 4.1. INSTRUMENTCALIBRATION

4.1.1. Scattering angle Because of the requirement that the instrument measures small changes in angle near 0 °, a careful measurement of the position of the 0 ° angle was made. This calibration was made by detecting, in the NaI(TI) detector, gamma rays scattered from the first target by the Compton effect. [The experimental arrangement for this measurement was easily obtained by removing the second target and swinging the NaI(T1) detector around to face the first target, fig. 3.-I Because of the complicated iron capture-gamma-ray spectrum, sharp peaks were not obtained in the NaI(T1) pulse height distribution. However, the well-defined spectra shown in fig. 6

were obtained at the indicated scattering angles on either side of 0 = 0 °. The data show that at + 13.5 ° and - 13.5 ° the energy spectra of the scattered gamma rays are indistinguishable, whereas the spectra at + 12.5 ° are considerably different. From the large energy shift corresponding to a 1o change in scattering angle and the good agreement for the __+_13.5 ° measurements, the accuracy of the 0 = 0 ° determination is O.1°.

4.1.2. Energy scale The calibration of the energy scale for the NaI(TI) detector was made by removing the first target (fig. 3) and by setting the spectrometer at 0 = 0 ° with a lead target in the second target position so that it was struck by the gamma ray beam from the reactor. The detector was at 90 ° with respect to the spectrometer as shown in fig. 3. Resonance scattering by the lead target was then detected in the NaI(TI) detector. The spectrum obtained

204

c . J . KAPADIA et al.

5000

, =..." "~'~ ,"l'~ff1''l~.," ," ,,,.,,., ix • x•x•

z

=x,

matltlgtl=ll

•

IX

2000

o

''' H= roll

I000

•

-- 1 3 - 5 ° + 135 ° "=" 1 2 . 5 °

x

• =m =l x= x=

I 1 =l

z

.500

x

•

8

x,

m== o'• x

200

i

418

4 5

I 5. I

I 5.4

ENERGY

IN

I 5.7

I 6.0

lo ,o

6.5

MeV

Fig. 6. Compton scattering spectra used to determine the

0 = 0° direction. 1800

I

OI

I

I

n

1500

el

1200

go

•

Z

=E o

oO

"

~

7 . 2 7 7 MeV

""

*

900 •

p • e

0.

-

.

600 l i B•

z :3

o 500

0 5.5

5.6

.9

6.2

6.5 6.8

7.1

7.4 2 7

8.0

ENERGY IN MeV

Fig. 7. Pulse height spectrum of scattered 7.277 MeV gamma rays.

is shown in fig. 7. The characteristic three-peak spectrum for a gamma ray at the 7.277 MeV resonance energy was obtained, the peaks corresponding to 7.28, 6.57, and 6.26 MeV. 4.2. DOUBLE SCATTERING MEASUREMENT

The procedure used for this basic measurement was to set the angle 0 at the desired value and place a lead target in both the first and second target positions (fig. 3). A pulse height spectrum was then obtained in the detector. The second lead target was then replaced

by bismuth and a second spectrum obtained. The difference between the spectra is assumed to result from the resonance scattering in the second target. Such a difference spectrum is shown in fig. 8 where 0 = 6 °. The curve drawn through the spectrum was obtained from the experimental pulse height spectrum of fig. 7 and represents the spectrum for 7.277 MeV gamma rays. It fits the experimental points in fig. 8 very well. An important check that the resonance scattering was indeed the result of double scattering was made by replacing the first lead target with a bismuth target. For, there is the possibility that the resonance scattering by the second target could be caused by gamma rays at the edge of the incident beam hitting the second target. The difference of lead and bismuth scattering by the second target for a bismuth first target is shown in fig. 9. Here, in contrast to fig. 8, there is no evidence ot resonance scattering. Hence, the first lead target, present for fig. 8 and absent for fig. 9, is essential for the production of the resonance scattering. Having proved that the resonance scattering in fig. 8 is the result of double scattering, similar spectra were obtained for other values of the scattering angle, 0. The number of counts in the two highest energy peaks for each angle was obtained. This number is plotted in fig. 10 as a function of the scattering angle 0 and also as a function of the energy shift from that at 0 °, these quantities being related through the Compton scattering relation given in eq. (9). A small residual background of 0.11 counts/min was subtracted from all points shown so that the large angle data would be normalized to zero. Because the points in fig. 1 are obtained from small differences between lead and bismuth targets (at the peak near 8 ° the bismuth subtraction is four times the signal that is plotted) it is not surprising that a 2% mismatch occurred between the lead and bismuth targets thereby causing the 0.11 residual background which was subtracted. The curves shown in fig. 10 will be discussed in section 5. 4.3. DETERMINATIONOF OTHER PARAMETERS The detailed relationship between the yield, Y, for the double scattering experiment and the desired quantities, Fo, F and 6 is worked out in appendix A. According to eq. (A5) of appendix A, an integral containing the parameters F o, F and & can be determined from the measurement of Y providing that the quantities ~, no, No, db, J1, ztQt and ztfl are known. The quantities easily calculated are no, the number of 2°apb atoms per cm a in the lead target; A t, the area of the first target perpendicular to the gamma ray beam; Ab, the Doppler

N U C L E A R S P E C T R O S C O P Y W I T H ENERGY R E S O L U T I O N

205

200

150 MeV I00 z

50

0 0 n,W 13.

0

o3

Iz::~ of._)

- 50

-I00 I

I

I--

5.50 5.60

I

I

(

I

I

I

I

I

I

5.90 6.20 6,50 6,80 7.10 7.40 7.70 8.00

I

8.30

ENERGY IN MeV Fig. 8. Pulse height spectrum obtained after double scatteringfrom lead first target and the differenceof lead and bismuth second targets. The curve through the points was determinedby the spectral shapeof the 7.277 MeV gamma rays found in fig. 7.

it_

7.277 MeV

1

50

z o o I.Y hi Q.

1,1i[!] [,IT TtIT

I]~ ]-TTT~!TT ]~I;TT~! r ~ ! , L , ~ ,

@4

-50 Z

0 -I00

-150 50

5.60

5.90

' 6.20

' 6.50

' 6.80

7.'10

r 7.40

i 7.70

J 8.00

ENERGY (MeV) Fig. 9. Pulse height spectrum obtained after double scattering from bismuth first target and the difference of lead and bismuth second targets.

c . J . KAPADIA et al.

206

width of the energy spectrum of the gamma ray beam, which can be calculated for the 57Fe source using eq. (6d); and A t2t, the solid angle subtended by the second target, A ~ t = A 2 / d 2,

TI V-l

N~

L_J

DET

c~° ]

"

'

L_J

I

I

YI

@ =0 °

(a) TI

where A 2 is the projected area of the second target perpendicular to the scattered beam while d is the distance between the two target vertical axes. Both quantities are known. The other quantities, e, N o, and A f2, must be determined by subsidiary measurements. Here, e is the efficiency of the gamma ray detector for 7.277 MeV gamma rays while No is the flux of 7.277 MeV gamma rays at the first target, i.e. the number/ cm 2. sec. To accomplish the measurement of No, the two targets in fig. 3 were removed and the detector placed in the gamma ray beam as indicated schematically in fig. 1la. The reactor power was reduced to a low level to obtain a reasonable counting rate in the detector. Since the 7.277 MeV line is too weak to appear in the 57Fe capture gamma ray spectrum, the highest

No

,'--I I

I

L-I

/ / F ~ D ET

6) =0 °

hiJ Y2 I

(b)

r [, I

1

L_J

T,

, Y2

%___ I J (¢)

1.4

I

I

I

I

i

I

i

i

i

i

i

i

I'= 0.60 eV 8 = 7 95 eV

1.2

TI

NO =-

1.0

~ROOM

TEMPERATURE

08 i

Q6

~78

DOPPLER ~ / ~ (78°K APPROX)

0.4

*K

\

I

I

I

4

I

[

I

8

\\\f

I

I

2

I

DET

T2/v! ~

Y3

Y'I = e(9 )N'o(9)Ad,

I

I

,2

I

,;

I

(9 (DEGREES) 0

(d)

r? ~

energy 9.295 MeV line was used to calibrate the beam. Thus, the quantity measured was

\

-02o

" ~ / r

Fig. 11. Schematic drawings illustrating the various kinds of experiments performed. The angle 0 is the scattering angle while is the angle between the beam entering the target and the normal to the target. Dotted symbols show that the indicated object has been removed.

\

O2

~ ~ = 2 5 o

[

I

4 6 8 A E (eV)

I

12

I

16

Fig. 10. Angular distribution of the double scattered gamma rays plotted as a function of the angle 0 of the first scattering. The ordinates of the points are determined by the area under spectra such as the one shown in fig. 8.

(15)

where e(9) and N~(9) are the efficiency and the beam intensity respectively of the 9.295 MeV gamma rays striking the detector which has an area of cross section A~. By multiplying by R e = e/e(9), the ratio 11) of the detector efficiency at 7.277 MeV to that at 9.295 MeV, and by RN = N6/N~(9), the ratio 12) of the beam intensity at 7.277 MeV to that at 9.295 MeV, the quantity

Y, = eN'oAd = ReRNY',

(16)

was obtained. Here, N~ is the beam intensity at the detector. This is the quantity utilized in appendix A [eq. (A11)]. It must be emphasized, however, that the

NUCLEAR

SPECTROSCOPY

measured quantity is Y; and that the accuracy of the quantity Yx depends on the accuracy of the ratios Re and RN. Since the quantity desired is eNo and not eNd, the ratio No/N ~ had to be determined experimentally. This was accomplished, as shown in figs. (1 lb) and (1 lc) by measuring the resonance scattering at 0 = 90 ° from a lead target placed in the second target position (fig. 1 lb) and then in the first target position (fig. 1 lc). A special target was fabricated so that it covered, when placed at 45 ° with respect to the beam, the same area perpendicular to the beam as the usual first target. The same portion of the beam was sampled, then, as the first target. (Measurements were always made with a dummy bismuth target as well as the lead target to obtain the difference between the lead and bismuth yields which represents the resonance scattering.) To accomplish the measurement at the second target position, the detector was moved back so that its face was 50 cm away from the center-line of the target instead of 7.5 cm. This was done so that the solid angle, A f2', subtended by the detector could be calculated accurately, taking into account the average penetration 13) of the gamma rays into the detector. A yield measurement, Y~, was thus obtained. A similar measurement of the scattering from the same target placed in the first target position gave a yield, Y2 (fig. I lc). Since the detector was now 147 cm from the target a different solid angle A t2" was calculated. Since

WITH

ENERGY

RESOLUTION

AQ = (Y2/Y'2)A~'.

5. Analysis of the data 5.1. D O U B L E SCATTERING EXPERIMENT The experimental data in fig. 10 were compared to the theoretical expression for the yield, Y, given in eq. (A5) of the appendix. As explained in the last section, the coefficients of the integral I(Fo,F,6) in eq. (A5) have been determined. The expression for I itself can be obtained by comparing eq. (A5) to eq. (A4) and its antecedents eqs. (A1), (A2) and (A3). In order to fit the expression for Y(Fo,F,6) to the experimental data, F0, F and 6 were varied to give a least-squares fit to the experimental data in fig. 10. The curve giving the leastsquares fit to the data is plotted in fig. 10 as the " r o o m temperature" curve. The goodness-of-fit of the curve was found by calculating Z2 where

{(Y,-E,)/o',} ~.

(18)

i

the ratio No/N ~ could be obtained from the measurement of Y~ and Y~. The quantity eN o can then be expressed as

eNo = (YaY~/Y'2)(AtT/AQ")/Ad

(17a)

4.3.2. Subsidiary scattering measurement It is shown in appendix A, eq. (A12), that the yield from the double scattering measurement, Y, can be combined with a measurement of the beam intensity Y1, and two single scattering measurements Y2 and Y3 to place a condition on integrals/, 12, and I 3 which are known functions of the quantities, F 0' F and 6. The quantity Y3 was therefore measured (fig. 1 ld) for the scattering angle, 0 = 2 5 ° . The other quantities in eq. (A12), A d, A2, Af2t, Af23 and A b a r e all known.

Z~ = ~

Y~/Y'2 = (No/N'o)(Af2"/Af2'),

207

(17)

and can be inserted into eq. (A5). 4.3.1. Measurement of Af2, the detector solid angle Since the face of the 7.5 cm dia. by 7.5 cm thick NaI(T1) scintillator is only 7.5 cm away from the vertical axis of the second target, it is difficult to calculate the effective solid angle of the detector. A measurement was made, therefore, of the magnitude of this quantity by the method indicated in fig. 1 lb. As just described above, the quantity Y~ was measured while A I2', the solid angle o f the detector at the distance o f 50 cm, was calculated. The counting rate or yield I12, with the detector in its usual position was also measured (fig. 1lb). Then, the effective solid angle of the detector in this position, A t2, can be written,

Here Yi is the value of Y for the angle 0i in fig. 10, E i is the experimental value at 01, and al is the statistical uncertainty in E i. The calculations of the Yi were made for the case of F o = F since these two parameters have been shown to be equal .4) to an accuracy of 20%. The application of this assumption greatly reduces the number of calculations; since these measurements are for illustrative purposes only (Fo, F and 6 have been measured several times before) the calculations are limited to the F o = F plane in F o, F, 6 space. The evaluation of the corresponding magnitudes of Z2 were performed with an IBM 7094 computer. A value of ~2 = 13 was found for the " r o o m temperature" curve in fig. 1 corresponding to a confidence limit of 0.60. The standard deviations of the values o f F and 6 were then obtained by defining the standard deviation of a parameter as being that value which increases the magnitude of Z2 from its minimum value, ~(min' 2 to Z2mi.+ 1 (appendix B). The plot in fig. 12 shows values for F and 6 as an ellipse which represents the area inside the standard deviations of F and 6.

c . J . KAPADIA et al.

208 10.5

i

,o.o .-

i

fig. 12. The upper and lower limits of this line are found from the experimental standard deviation of the ratio I2Ia/I by determining the values of F and 6 corresponding to the values of I2Ia/I = 2.86 _+ 0.51. These limiting lines are also plotted in fig. 12. 5.3. EFFECT OF THE UNCERTAINTY IN ~ N0

95

CONSI .TENC¥

A

> v

90

U"v

n,, uJ Z hl 8.5 Z 0 I-- 8.0 tr I

~o,,Nf..R .-"~-)J L ~ ~ C ~ - - DOUBLE-----.~/" ""'

w

7.57.0 ~

/

;CATTERIN(// / / "

~

/bUI////// ~V~/'//~/// -

In addition to the statistical errors which account for the size of the double scattering ellipse and the width of the internal consistency band, there are systematic errors to be considered. The largest error of this type is the 20% uncertainty in the value of eN o. This uncertainty appears directly in the coefficient of Y in the double scattering experiment, eq. (A5). While it cancels out in the ratio YYt/(}'2 Y3), it does not cancel in the ratio actually measured, YY~/(Y2 Y3). Thus, this error will affect the positions of both the double scattering ellipse and the internal consistency band in fig. 12. The magnitude of this effect is indicated in fig. 12 by plotting the ellipse and the lower limit of the band for the case of eNo being reduced by 20%. 5.4. FINAL RESULTS

/

Q50

z/

/

/"

0.55 0.60 0.65 LEVEL WIDTH, E' (eV)

0.70

Fig. 12. The two ellipses plot the locus forz 2 = ~ m l n 2 + 1 for the fit to the double scattering data in fig. 10. The solid ellipse is that obtained for the expected value of eNo,the dashed curve for eNo decreased by 20%. The "center" solid line represents the values of t5 and F that are compatible with the "internal consistency" relationship between the single and double scattering experiments. The "upper" and "lower limit" curves are deduced from the statistical uncertainties in these experiments. The dashed "lower limit" curve is obtained from the solid "lower limit" curve by reducing eNo by 20%. 5.2. INTERNAL CONSISTENCY DETERMINATION In this determination the relation among the yields of the various types of experiments as given in appendix A, eq. (A12) is used to obtain an independent relation a m o n g the parameters F and 6. The values used for the Y's in eq. (A12) are the experimental values for the various measurements described in section 4. For the experimental value for Y, the value at 0 = 6 ° in fig. 10 was used. No X2 test could be used here since only one value for YY1/(Y2Y3) was measured. The ratio I213/I was found to be 2.86 4- 0.51, the standard deviation resulting from the standard deviations in the measuremerits of Y, Y1, }'2 and Ya. Values of F and 6 were then found which would satisfy the ratio I2Ia/I = 2.86. These values determine a line in F--6 space which is plotted in

If the minimum value for eN o is used, then the center of the double scattering ellipse in fig. 12 lies about two standard deviations away from the center of the internal consistence band which gives a reasonably satisfactory confidence limit of 0.05 for the consistency of the two measurements. The values of F and 5 corresponding to the center of the dotted ellipse are F = 0.64 + 0.01 eV,

J = 7.30 + 0.60 eV,

as compared to the values of F = 0.56 _ 0.01,

5 = 7.50 4- 0.60,

for the original ellipse. The systematic error dominated in the determination of F while the statistical error is the most important one for 6. We, therefore, quote for our final values, F = 0.56 4- 0.08,

6 = 7.50 4- 0.60.

It should be recalled that Fo has been assumed to be equal to F in obtaining this result. The effect on these values of letting F 0 = 0.8 F, as is allowed by other experiments, has not been investigated since the present data will not lead to more accurate parameter determinations than those already published. The purpose of the present calculation is only to obtain an indication of the sensitivity of the small angle scattering experiment to the parameters, Fo, F, and 6. A comparison of these results with those of other investigations is given in table 1.

209

N U C L E A R S P E C T R O S C O P Y W I T H ENERGY R E S O L U T I O N

6. Applicability and fimitations

The original purpose of the instrument described here was to produce a gamma ray beam of variable energy for the study of nuclear energy levels in the 5 to 9 MeV energy range. It was for the purpose of testing the instrument that experiments near 0 ° were performed. These experiments have shown that the parameters Fo, F, and 5 can be determined for the case of the 7.277 MeV neutron-capture gamma ray from STFe scattering by resonance fluorescence from 2°spb near 0 °. Thus, two types of experiments are possible: 1. scattering of neutron-capture gamma rays near 0 ° and 2. scattering of a variable energy beam at large angles as illustrated in fig. 1. These two experiments will be discussed separately in the following. 6.1. EXCITATION OF NUCLEAR ENERGY LEVELS BY NEUTRON-CAPTURE GAMMA RAYS

This technique is the one used to obtain the data presented in this paper. For the energy level excited it has given values ofFo, F and 5 with accuracy in the 10% region. The accuracies of the different parameters depend on the data shown in fig. 10. The dependence of these parameters on the properties of the peak can be understood by reference to eqs. (12) and (13) and the associated comments in the text. The quantity, NI(E,

0°), is the scattered beam intensity at 0 °. As 0 is increased from 0 °, E shifts to E ' according to eq. (14). Now, the energy Em~(0°), at the energy maximum of the scattered beam is E r + C, eq. (12). The maximum in the double scatteringyield will occur when E'ax(O ) = Er, since E r is the energy at the maximum of da/df2. Thus, using eq. (13c),

Em,,-E'm,x(O)

=

C

=

6(AIAo).

(18a)

Since, from eq. (14a), Emax-E'ax = (EZI#)(½02J,

5 = (E2Ao/2#A)O~ax .

(19)

Thus, the value of fi depends on the value of 0rex. Experimentally, then, a determination of 0m~x, or equivalently, the horizontal position of the experimental peak in fig. 10 will yield a value of 5. On the other hand, the determination of F o and F depends on the amplitude of the peak in fig. 10. This follows from the fact that the double scattering yield is a product of the scattered beam intensity, eq. (12), and the resonance cross section, eq. (7). Considering the double scattering yield at the angle of the peak in fig. 10, the exponentials in energy are unity and the yield Y (peak) has the form

r (peak) = const. 2exp((20) If F o = cF is assumed during the search procedure over TABLE 1 F and 5 where c is a constant, then the amplitude of the experimental peak is proportional to F 2. Values of Fo and t~, for the 7.277 MeV level in 20aPb as obtained t o date. Finally, information about F can be obtained from the shape of the experimental peak providing F is not Case no. Fo (eV) ~(eV) Reference too much smaller than A 1. This follows from the fact no. that the ~b-function of eq. (5) represents the shape of the energy spectrum of the differential cross section for 1 0.80 _+ 0.08 4.8 + 0.4 a) scattering. Since ~k is a fold of the Doppler (or Gauss) 2 0.1 < F < 4 < 26 eV b) shape of width A t and the Lorentz shape of width F it 3 0.8 + 0.03 8.0 + 1 e) has, for F < A 1, a Doppler form for E ~ Er. However, 4 6.5 + 1 a) 5 0.86 __. 0.06 5.0 + 0.5 e) the Lorentz "tails" will eventually dominate for energies 6 0.7 _+ 0.2 f) sufficiently displaced from Er because of the exponen7 0.68 _+ 0.09 8.00 _ 0.14 8) tial cut-off of the Doppler energy function. This effect 8 0.56 _ 0.08 7.50 + 0.60 h) can be enhanced by cooling the scatterers so that d is (F0 = F) decreased. Curves have been drawn in fig. 10 for the lead scatterers cooled to liquid nitrogen temperature • ) H. H. Fleischmann and F. W. Stanek, Z. Phys. 175 (1963) 172. (78°K). The difference in shape between a Doppler b) C. S. Young and D. J. Donahue, Phys. Rev. 132 (1963) 1724. curve (corresponding to F~.A1) and a curve core) B. Arad, G. Ben-David, I. Pelah and Y. Schlesinger, Phys. Rev. responding to the actual curve (F = 0.60 eV and A 1 = Rev. 133 (1964) B684. a) B. Arad, G. Ben-David and Y. Schlesinger, Phys. Rev. 136 3.4 eV) is easily seen; however, the maximum difference (1964) B370. is still comparable to the error bars on the experimental e) M. Giannini, P. Oliva, D. Prosperi and S. Sciuti, Nuel. Physics points. Thus, even at the low temperature the shape of 65 (1965) 344. the curve by itself is not sufficient to determine F very f) J . A . Mdntyre and J. D. Randall, Phys. Letters 17 (1965) 137. well. Such a shape determination would be most cong) S. Ramehandran, private communication. h) This work. venient since it would eliminate the need for measuring

210

c . J . KAPADIA et al.

beam intensity values, solid sngles, and detector effi-

ciencies. Other factors to be considered when planning these experiments are the following. 1. There are ~ 50 levels ~s) that have been excited by neutron capture g a m m a rays. Thus, there appear to be many possibilities for study. However, there are only a few cases where the cross section for excitation is greater than a tenth of the cross section of the case studied here. When consideration is given to the low counting rates obtained (fig. 8), and it is recalled that the square of the cross section determines the counting rate in a double scattering experiment, even an increase of a factor of ten above the present 100 kW reactor power level would not make many such experiments appear to be feasible. 2. A considerable amount of computing time is required to evaluate the many integrals involved in the analysis of the data. Each X2 fit to the data in fig. 10 for a set of the parameters F o, F and 6 requires 6 min of computing time on the IBM 7094 computer. 3. The type of information obtained here can also be obtained by single-scattering experiments (table 1) which have much higher counting rates or by using a high speed rotor x6) to shift the energy spectrum. For example, an absolute value scattering experiment, an absorption experiment, and a low temperature experiment will permit the determination of F o, F and ~. In summary, then, the double scattering experiment has the advantage that the Doppler-broadened line shape is mapped out experimentally so that values for Fo, F, and 6 are over-determined. However, for the usual case, the Doppler width, A, is considerably larger than the natural (Lorentz) width, F, so that the line shape is rather insensitive to F and the determination of F depends on an absolute measurement of the scattering cross section. In addition, counting rates are low and considerable computing time is required. It would appear that the parameters F o, F and 6 can more profitably be determined, at present, by using single scattering experiments or the rotor technique. 6.2. EXCITATIONOF NUCLEAR ENERGY LEVELSBY VARIABLE ENERGY GAMMARAYS17) In this type of experiment (fig. 1) absorption, rather than a second scattering, can be used to detect the resonance since the g a m m a rays scattered by the Compton effect in the first target have sufficiently low energy to be rejected for 0 > 15 °. Also, the first scatterer can be chosen to have a large cross section while many different absorbing materials are studied. Thus, the counting rate problem of the small angle studies of section 6.1.

above is avoided in two ways: 1. by eliminating the loss in solid angle introduced by the second scattering; 2. by being able to utilize a large cross section for the first scattering when studying many nuclei. For the purpose of orientation, an evaluation will be made of the properties of a system using 7.277 MeV 57Fe neutron capture g a m m a rays scattered by 2°apb. For a thick Pb scatterer with its normal set at half the scattering angle, and with the scattered beam emerging from the same face as the incident beam, the scattered intensity dI/dQ is related to the beam intensity I o by

dI/df2.,~ ¼(Io/4n) (aJa), where a s is the resonance ~cattering cross section, and a is the total absorption cross section. The factor ¼ appears because: 1.2°apb is only about 50% abundant; 2. scattering occurs from the ingoing beam path while absorption occurs for both the ingoing and outcoming b e a m paths. Since tr s ~ 4 barn while a ~ 20 barn,

dI/dt2 ~ 5 x 10- al o. Using the same beam and scatterer described in section 2, Io ~ 107/sec so that dI/dt2 ,~ 5 x 104/see • sterad. A 1° angular resolution with energy resolution, AE(eV)~ 5sin0 is obtained using a 1 cm wide scatterer and a 1 cm wide slit in front of the NaI(T1) detector placed at a distance of 60 cm from the scatterer. A factor of 5 is lost by the beam striking the narrow scatterer while the solid angle d • = 7 . 5 / 6 0 2 ~ 0.002. Thus, dI--20/sec. The use of a Soller slit 17) at the detector with the full size scatterer will recover a factor of about 3.5 so that d I = 70/sec. For a small absorption dip, the optimum absorber thickness, based on counting statistics, is 2 mean free paths (appendix C) so that the counting rate is reduced another factor of 7.5. Finally, the detection efficiency of the NaI(TI) detector is about 20% in the three-peak spectrum so that the counting rate is reduced to 2/see. Since neutron capture data 3) indicate that g a m m a ray line widths F for energy levels near neutron threshold are in the 0.1 eV region, the counting period at each angle should be sufficient to yield statistics which will reveal absorption dips corresponding to line widths of this magnitude. F r o m eq. (C5) in appendix C, the accuracy of the measurement is Percent error = 100(e / ~/2) (#/Pr) / C~o•

(21)

Now, P/Pr = a/aa, where a is the total absorption cross section and a, is the resonance absorption cross section. Let a ~ 10 barn while a, ~ 1 barn for F = 0.1 eV. Then, from eq. (21), for a 30% error in the measurement,

NUCLEAR SPECTROSCOPY WITH ENERGY RESOLUTION Co "~ 4000 counts. At the rate calculated of 2/sec, the time required at each angle is about 30 min. Covering the angular range from 30 ° to 150° in 1° intervals the time required to cover a range of 470 eV is 60 h, or 4 days at 15 h per day. From previous studies~7), the spacing of levels that are excited by the resonance fluorescence process is about 104 eV. Thus, a scan period of about 200 days or 10 months would be required to locate and measure one line of width F = 0.1 eV, to an accuracy of 30%. A line width of F = 0.3 eV would be measured with an accuracy of 10%. Finally, for a 1 MW reactor power level (which will be available within the year), the counting times will be reduced a factor o f t e n so that 12 levels per year should be discovered usingthis technique. Thus, the variable energy beam should be a useful tool for finding and measuring the line widths of energy levels in the 5-9 MeV excitation region. As shown before, the absorption measurement determines F 0 unless the line width F becomes comparable in magnitude to A, the Doppler width. Since the exciting radiation can be made to coincide in energy with the absorption cross-section, the parameter 5 is known to be zero and need not be considered in the analysis of the data.

Appendix A 1. DOUBLE SCATTERING MEASUREMENT The expression for the counting rate, Y, for the double scattering experiment is calculated in three steps. First, the energy spectrum of the gamma rays scattered by the first target, N~(E) is calculated. The energy shift of the gamma ray from E to E'(O) is then calculated using the Compton scattering recoil relation of eq. (9). The spectrum NI(E'), at the scattering angle, 0, is then scattered through 90 ° by the second target. An integration over all energies of the scattered gamma rays then gives the counting rate, or yield, Y. The expression for NI(E) is

N,(E,O) = noA~N(E) [d~(e,O)/dO] (AO,

)fxodx.

211

N(E)

has been assumed to have a Doppler energy dependence because, for the STFe source, Ab = 7.68 eV while the natural line width for gamma ray emission is usually about 0.1 eV. The expression used for da(E,a)/ dO in eq. (A1) is that given by eq. (4a) in the text, while

W(O)=

l +

(A3)

P2(cos 0).

The function used for W(O) is determined by the fact that the resonance fluorescence in 2°apb is known to proceed by a dipole transition. The term A O t is the solid angle subtended by the second target, the integral in eq. (A1) sums the contributions to the scattering and absorption of the target at different depths in the scatterer. The parameter, Pna, is the non-resonant absorption coefficient. The resonance absorption cross section, a,, has a different value for the incoming gamma ray of energy E than for the outgoing gamma ray of energy E' which is related to E through eq. (9) of the text. The target geometry defining the angle 0 is shown in fig. 13. The energy spectrum of eq. (A1) is next converted to the actual spectrum at the angle 0 by including the effect of the nuclear recoil. This conversion is accomplished by shifting E to E' according to eq. (9) of the text so that NI(E,O)is converted to NI(E',O) which is the energy spectrum striking the second target. The scattering from the second target is similar to that from the first target except for three changes: the incoming beam is NI(E',O) instead of N(E), there is no resonance absorption of the outgoing gamma rays by the target because they are scattered through a large angle and thus off the resonance energy, and the yield, Y, of the scattered gamma rays is the integral over all energies of the number of gamma rays scattered. Thus, / ~~ T A R G E 1

I

~/----

TARGET

o x 0 -- x~

"exp {--Pna ( x + - ,o

+

COS0

] --

]

Xo~j}.-x

(A1)

The quantities in eq. (A1) are: no, the number of resonant nuclei per cm a in the scatterer, A~, the area of the first target perpendicular to the gamma ray beam, and N(E), the number of gamma rays in the beam striking the target per cm 2 per second, where

N(E) = (No/n½Ab)exp{- [(E-E,)-5]21A~}.

(A2)

Fig. 13. Drawing defining the scattering parameters.

2

212

c.J. K A P A D I A

Y=

(A4)

,,,of

fiidz/cos=)

•exp { - noah.[(z/cos ct) + ( z / c o s , 8 ) ' / - noa.(E')z/cos ct}. The quantities not yet defined in eq. (A4) are: e, the efficiency of the gamma ray detector, A f2, the solid angle of the detector, and z, ct and ,8 which are defined in fig. 13, the angles ~ and ,8 being measured from the normal to the second target. Combining eqs. (A1), (A2), and (A4), Y may also be expressed as

Y = en2(NofiriAb)Al(dQt)(AE2)I(Fo,F,&),

(A5)

where I is an integral containing known quantities except for F0, F, and 6. 2. SINGLE SCATTERING MEASUREMENT 1. From the first target: The yield obtained for a single scattering measurement, Y3 (fig. l ld), is found by integrating NI(E,O) in eq. (A1) over all energies, and multiplying by, first, (A~t'~3/AQt) where d ~ 3 is the detector solid angle and, second, e, where e is the detector efficiency. Thus,

Y3(0) =

et al. Y1 = 8N'oAd,

where A d is the area of the detector. 4. RELATION AMONG Y, Y1, Y2 AND Y3

By combining eqs. (A5), (A7), (A10) and (A11), the following relationship among the Y's is found

YY, /(Y2Y3) = (Ad/Au)(A~JAt23)TriAb{I /(I213)). (A12) Appendix B We show here a simple means of determining the uncertainty in parameters a s that have been adjusted to give a least-squares fit between a mathematical function and a set of experimental datatS). Let E t + a i be the experimental values with their uncertainties, and let F~(aj) be the values of the function corresponding to the experimental points• The likelihood function, L, is then defined as the probability of any particular fit occurring between the experimental points and the function, Ft. It will now be assumed that each experimental distribution follows a Gaussian function centered at E~ with a standard deviation at. Then the unnormalized likelihood of F~ occurring is

(A6)

L t = exp {

o

Combining eq. (A6) with eq. (A1), Y3 may be expressed as

Y3 = ,no(No/rC*ZJb)Ax(.~a3)13(ro.r.~).

(A7)

where I 3 is a function of the unknown quantities F o, F, and & • 2. From the second target: The yield, ]I2 (fig. 1lb) is Y2--

(A8)

enoA2f: dEN'(E)[da(E,O=90°)/dE2]AE~f~dz/c°scO" • exp { -

noan.[(zlcos ct) + ( z / c o s , 8 ) / - noa.(E)zlcos o~}.

The only quantity not defined before is N'(E) which is the gamma ray beam spectrum at the second target:

N'(E) = (N'o/No)N(E),

(A9) with N(E) defined in eq. (A2) and N~ being the gamma ray beam flux (number/cm 2" see) at the second target when the second target with a r e a A 2 is placed in the beam. Combining eqs. (A2), (A8) and (A9) ]"2 may be expressed as

Y2 = 8no(N~/~Ab)A2(Aa)12(Fo,F,6) •

(A10)

3. BEAMINTENSITYMEASUREMENT The beam intensity, Y1, is measured with the geometry shown schematically in fig. 1la. Thus,

(All)

-

½(Et - Fi)2/a~ }

and the unnormalized likelihood L for all the F t occurring is the product, L = ]--[exp [ - ½{(El -- Fl) / trl} 2] t

= e x p ( - ½X2),

(B1)

where Z2 = E f r E t - t J / a 3 2 :

(BE)

i

The criterion for the least-squares fit is that L is maximum, i.e. that X2 is a minimum (X2 = X~in)"This condition is achieved by adjusting the parameters aj. The problem at hand is to find a measure for the accuracy of this determination of the aj. The standard deviation 6aj in the a s will be defined to be that change in a i from its best value aoj which removes the L one standard deviation from its maximum value. Assuming that the likelihood function L has a Gaussian form as a function of each a s, i.e.

L = Cexp[-½{(a~-aoj)/6as}2],

(B3)

this expression can be compared to eq. (B1), where L = exp( - ½Xz) = exp { - ½(X2min+ n)} = exp (-- ½Z21n)exp(-- ½A).

(B4)

NUCLEAR

SPECTROSCOPY

WITH

ENERGY

RESOLUTION

213

F r o m eq. (B3) it is clear t h a t L/Lmax = e x p ( - ½ ) when a j - a o j = ~Saj. Eq. (B4) shows, then, t h a t for a change o f one s t a n d a r d d e v i a t i o n in a j, A = 1 a n d

ground. T h e effect o f these counts will be t o reduce the o p t i m u m a b s o r b e r thickness.

Z2 = )~min 2 -[- 1.

W e are greatly i n d e b t e d to Mr. E. E. Vezey w h o d i d m o s t of the design a n d c o n s t r u c t i o n o f the s p e c t r o m e t e r a n d b e a m port. W e are grateful for m a n y discussions with Professor R. A. Kenefick a n d a c k n o w l e d g e the large c o n t r i b u t i o n o f Dr. J. D. R a n d a l l in directing t h e o p e r a t i o n o f the reactor.

(B5)

This expression is m o s t c o n v e n i e n t for d e t e r m i n i n g

~Saj since )~ml, 2 is usually f o u n d f r o m a search p r o g r a m which evaluates )Cz as a f u n c t i o n o f t h e aj. Thus, d a t a are usually a l r e a d y available for finding the a j values c o r r e s p o n d i n g to )~2 = Xmin 2 "-[-1.

References

Appendix C T h e o p t i m u m thickness for the a b s o r b e r in fig. 1 is calculated. It is a s s u m e d t h a t the d i p in the c o u n t i n g rate due to t h e r e s o n a n c e a b s o r p t i o n is small c o m p a r e d to t h e c o u n t i n g r a t e itself. A t t h e a b s o r p t i o n m i n i m u m , C r = C Oexp { - (p + pr)x},

(C1)

where C, is t h e c o u n t i n g rate with a b s o r b e r , C 0, t h e c o u n t i n g rate w i t h o u t a b s o r b e r , / t is t h e n o n - r e s o n a n c e a b s o r p t i o n coefficient, Pr is the r e s o n a n c e a b s o r p t i o n coefficient, a n d x is the a b s o r b e r thickness. A t a n angle a w a y f r o m the r e s o n a n c e angle, C = Coexp {-#x}.

(C2)

T h e r a t i o R is t h e n R = C/Cr = exp {#,x) = 1 + #rx,

(C3)

since t h e a s s u m p t i o n is being m a d e t h a t the r e s o n a n c e a b s o r p t i o n is small c o m p a r e d to t h e n o n - r e s o n a n c e absorption. The o p t i m i z a t i o n o f the a b s o r b e r thickness d e p e n d s o n the effect o f the a b s o r b e r o n the c o u n t i n g statistics o f the detector. Since the r e s o n a n c e a b s o r p t i o n is small, C ~ Cr, a n d A R / R , t h e p e r c e n t a g e e r r o r in R, i s x / 2 times AC/C, t h e percentage e r r o r in C. Thus,

A R = (R~/2)(AC/C) = ( R x / 2 ) / x / C . Using

eqs. (C2)

and

(C3)

and

remembering

that

~ , x ~ 1, A R = exp(½/~x)(2/Co) ~r.

(C4)

T h e o p t i m u m a b s o r b e r thickness is o b t a i n e d when 1 is k n o w n with the best precision, i.e. when

i~,x = R -

( R - 1)~(dR) = p ~ x [ e x p ( - ½#x)'] (½C0) t" is a m a x i m u m . This occurs w h e n x = 2/#, i.e. when the a b s o r b e r is t w o m e a n - f r e e p a t h s thick. Thus, t h e accuracy of the absorption experiment under optimum c o n d i t i o n s is

AR/(#~x) = (e/~/2)(#/lar)/C~o.

(C5)

It s h o u l d be n o t e d t h a t the a b o v e analysis has i g n o r e d d e t e c t o r c o u n t s resulting f r o m general b a c k -

1) G. K. Tandon and J. A. Mclntyre, Nucl. Instr. and Meth. 59 (1968) 181. 2) A. M. Khan and J. W. Knowles, Bull. Am. Phys. Soc. 12 (1967) 538 ; J. W. Knowles and N. M. Ahmed, Atomic Energy of Canada Report, AECL 2535 (March 1966). a) BNL-325, Neutron cross-sections (Physics-TID-4500). available from Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards, U.S. Department of Commerce, Springfield, Virginia, 22151. 4) j. A. McIntyre and J. D. Randall, Phys. Letters 17 (1965) 137. 5) F. R. Metzger, in Progress in Nuclear Physics 7 (ed. O. R. Frisch; Pergamon Press, New York, 1959). a) The energy determination of this level has been made recently by H. E. Jackson (private Communication). 7) The important discovery that neutron capture gamma rays will excite energy levels by resonance fluorescence was made independently by H. H. Fleischmann and by C. S. Young and D. J. Donahue. The first report of Fleischmann's work was made by H. Maier-Leibniz at the Conference on Programming and Utilization of Research Reactors, Vienna, Austria, Oct., 1961 (Conf. Proc. 3, p. 145). The journal paper reporting the work is H. H. Fleischmann, Ann. Phys. 12 (1963) 133. A report of the Young and Donahue work is given in Bull. Am. Phys. Soc. 8 (1963) 61 while their paper is C. S. Young and D. J. Donahue, Phys. Rev. 132 (1963) 1724. The resonance studied by Young and Donahue was the 7.277 MeV level in 2°spb. This excitation has since been studied by many investigators (table 1). 8) This fact was kindly communicated to us by G. Ben-David. a) References of table 1. 10) L. Jarczyk, H. Knoepfel, J. Lang, R. Mailer and W. W61fli, Nucl. Instr. and Meth. 13 (1961) 287 have described the advantages of the "tangential" beam port compared to a "radial" beam port. 11) L. Jarczyk, H. Knoepfel, J. Lang. R. MOiler and W. W61fli, Nucl. Instr. and Meth. 17 (1962) 310. 12) L. V. Groshev, V. N. Lutsenko, A. M. Demidov and V. I. Pelekov, Atlas of gamma ray spectra from radiative capture of thermal neutrons (Pergamon Press, New York, 1959). 13) E. G. Fuller and E. Hayward, Phys. Rev. 101 (1956) 692. 14) M. Giannini, P. Oliva, D. Prosperi and S. Sciuti, Nucl. Physics 65 (1965) 344. 15) G. Ben-David, B. Arad, J. Balderman and Y. Schlesinger, Phys. Rev. 146 (1966) 852. le) B. Arad, G. Ben-David and Y. Sehlesinger, Phys. Rev. 136 (1964) B370. 17) R. Moreh and G. Ben-Yaacov, Nuclear Research CenterNegev, NRCN-180 (1967). This paper reports on work using a system such as that described here. is) This method was shown to us by Professor S. D. Baker.