Line shape calculation for nuclear scattering from gaseous targets

NUCLEAR INSTRUMENTS

A N D M E T H O D S 88 ( I 9 7 0 ) 2 8 9 - 2 9 3 ; © N O R T H - H O L L A N D

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L I N E S H A P E C A L C U L A T I O N FOR NUCLEAR SCATTERING FROM GASEOUS TARGETS* K. W. KEMPER, D. S. HAYNES and N. R. FLETCHER

Department of Physics, Florida State University, Tallahassee, Florida 32306, U.S.A. Received 20 July 1970 A line shape calculation combining geometric and straggling effects for the energy spectrum of nuclear scattering from a gas target is described. The calculated full widths at half maximum are in agreement with values extracted from scattering data to within about 1%. The nearly Gaussian calculated line shape is

shown to accurately describe a measured spectral line shape over an energy range of + 2a of a fitted Gaussian. The effect of large angle double scattering originating outside the geometric target volume, which is the principle contribution near three standard deviations from the centroid, has not been included.

1. Introduction Much attention has been given in the past to gas target chamber design, solid angle determination in gas target experiments, and accuracy of the extracted cross sections, but a detailed calculation of the energy distribution within an observed spectral line from a nuclear scattering experiment seems to have not been of sufficient importance to undertake, even though for some time digital computers have been available which actually make this task quite straightforward. For any experimental situation in which accurate Q-value determination or separation of an unresolved multiplet is of interest it is essential that one has some knowledge of the shapes of the individual energy groups that form the energy spectrum. Also in the initial design of a gas target chamber it is highly desirable to optimize the energy resolution, E/AE (fwhm), for a given counting rate or to maximize the counting rate within the bounds of a certain required energy resolution. To these ends we have designed a F O R T R A N computer code for the numerical calculation of spectral line shapes. Herein the calculation is described, the result of parameter variations is shown, and the application to Z°Ne + 3He reactions is demonstrated. Some possible revisions for improving the calculation are also discussed. The calculation divides naturally into two parts. The geometric line shape calculation includes the energy spreading effects of energy loss in the target, finite target volume, and finite detector along with dE/dO of the scattered particle. The Gaussian broadening calculation includes energy and angle straggling and the effect of electronic and detector resolution. The straggling and resolution effects are then convoluted into the geometric line shape to produce the

final spectral distribution. Since the final line shape is very nearly Gaussian, a Gaussian peak-fitting subroutine which employs a direct search method I) is used to extract a calculated fwhm and centroid. This procedure gives realistic comparison with data which is also extracted from spectra using the same Gaussian fitting routine. 2. Line shape calculation

* Research sponsored in part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force under AFOSR Grant No. AF-AFOSR-69-1674, and the National Science Foundation Grant No. NSF-GJ-367.

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2.1. GEOMETRIC LINE SHAPE

A schematic representation of the important geometric parameters is illustrated in fig. 1. The active target volume is determined by the beam width, w, the detector slit positions d 1 and d2, measured from a coordinate origin, and the detector slit widths, Sl

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Fig. I. Schematic illustration of the geometric parameters important to this calculation.

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and s 2. The coordinate origin is located at the intersection of the incident beam center line and the center line of the detector collimation. Following the practice used of labelling particles in a nuclear reaction, M2(Ma,M3)M4, one designates the incident beam energy at a reaction site (x,y) within the target volume by

E,(x,y) = E,(0,0) + {dEl/d(px)} [E,(o,o)'P x,

(1)

where E~(0,0) is the beam energy of the coordinate origin and p is the gas density. In practice of course E~(0,0) is dependent on gas pressure and therefore is calculated in this program. The energy, E3, of a reaction particle emitted from point (x,y) at the angle 0 is determined by a kinematic calculation based on El(x,y), the Q-value and the masses of particles involved. Energy losses from the reaction point to the detector will differ depending on the coordinates (x,y). This differential energy loss correction is made to first order z) by writing

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E 3 ( 0 ) --[-{dE3/d(px)}

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(2)

where x' = xcos0 + ysin0. The reaction angle is not, however, confined to 0, but rather it can be anything consistent with the slit geometry. The resulting maximum and minimum particle energies are given by

cluding the storage of unity in all channels between Emj. and Em,x. In this manner a complete line shape spectrum is built up with any desired accuracy. The limiting values of x are given by, {(ay -- y)dx - (dr - y)ax } I(a~,Xm,,= {(by-y)cx-(cy-Y)bx}/(by-cy

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(4)

A typical triangular geometric line shape is shown in fig. 2, curve (a). Most such curves are triangular to trapezoidal with a degree of edge rounding depending primarily on target thickness. The centroid of the geometric line shape is always within 1 keV of E3(0,0) for the geometries and pressures considered. 2.2. ENERGY BROADENING OF THE GEOMETRIC LINE SHAVE

The calculations in the previous section were based on an average value for the energy losses through matter, however, due to the statistical nature of the energy loss process the particle energy may be described by a probability distribution P(E). Symon a) and Rossi 4) have shown that P(E) is nearly a Gaussian distribution when the total energy losses are very large compared to the maximum energy loss (el) suffered in one dec"

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E3(x,Y)min = E3(x,y,O ) -t- ( d E 3 / d 0 ) " [ Omax(X,y ) -- 0]. The minimum angle of acceptance of the detector for a reaction at (x,y) is Omin(X,y) and is the greater of Ob(x,y) and O,j(x,y), while the maximum angle of acceptance, Omax(X,y ) is the lesser of O.(x,y) and Oc(x,y). The angles 0,, 0b, 0c and 0a are determined from the coordinates of the reaction point (x,y) and the coordinates of the slit edge points a, b, c and d, shown in fig. 1. A section of computer memory is set aside for the purpose of multichannel spectrum generation. In the present case 1000 channels at a calibration of 1 keV per channel were used with a contribution at energy E3(0,0 ) stored in channel 500. For the reaction point (x,y) then, one count 2) is stored in each channel between the channels designating E3(x,y)max and E3(x,Y)min. The integration over the entire active target volume is effected by stepping the x-coordinate from Xml. to X~.ax and stepping the y-coordinate through to limits set by the input parameter, w, and for each new reaction point repeating all the above calculation in-

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dz=0.625" and d~=4.0". Curve (a) is the geometric line shape; (b) is the convolution of (a) and the Gaussian broadening term; (c) is a Gaussian fitted to (b); (d) is curve (c) plus an exponential fitted by magnitude variation only to the experimental points which show statistical errors.

LINE SHAPE CALCULATION FOR NUCLEAR SCATTERING

~ = ((60)2).(dE3/dO) 2, from angle straggling of incident beam in the beam entrance foil; a 2 = ((60)2).(dE3/dO) 2, from angle straggling of incident beam in the target gas; a6z, effective energy spread from electronic and detector noise. Terms 2 and 3 are pressure dependent; terms 4 and 5 are angle dependent; term 6 is assumed to be constant; and all energy dependence is contained in e 1 and dE3/dO. The detectors were contained within the gas volume, hence there is no contribution from an exit foil. The broadening from electronic and detector noise was measured directly by use of an Z41Am, a-particle source giving a fwhm contribution of ~ 18 keV [-fwhm = 2(21n2)~a]. The Gaussian broadening term of the form of eq. (5) with a 2 replaced by a 2 is then numerically convoluted into the geometric line shape spectrum. The resulting spectrum is not Gaussian as illustrated in fig. 2, curve (b). The curve will be very nearly Gaussian when the energy broadening terms are large compared to the geometric width. This has been demonstrated in accurate line shape measurements by Mancusi et al.7).

tronic collision, yet small compared to the initial particle energy. These conditions are met in the present experiment therefore for passage through a single stopping medium we have

P(E) = a - 1exp{ -I(E--Eo) 2/a2}, where a 2 = ~ [ 1 +}(I/e,)ln(e,/I)],

a 2 = 4nzZe4NZx.

(5)

The quantities x, N, I and Z refer to the stopping material: distance traveled, number of atoms per unit volume, ionization potential and charge number. The quantity z is the charge number of the initial particle. In addition to the energy straggling contributions, there is an energy spreading contribution from small angle double scattering in the incident beam. This is given by a2(0)=((6O)2)'(dE3/dO) z, where the mean square angular spread in the incident beam has been given by Williams 5) and Segr66). Since all energy broadening contributions are of a Gaussian form we can write 6

a2 = Z a?,

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3. Parameter optimization, comparison with data, extensions

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where the terms in the summation originate from the following: %1, energy straggling of incident beam in the entrance foil; a z, energy straggling of incident beam in the target gas; ch2, energy straggling of exit particle in the target gas;

The complete line shape calculation is very useful for determining optimum gas pressure and geometrical parameters for gas cell experiments. For example, holding dl fixed and requiring a constant count rate one can find the minimum line width by making

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Fig. 4. Calculated f w h m vs the particle energy at the detector for a variety o f Q-values a n d reaction angles, 0; Sl=S2--w=0.040", d l = 0.625". Solid curves connect points o f c o n s t a n t 0; d a s h e d curves connect points of c o n s t a n t Q-value.

appropriate variations of d2 and gas pressure. The result of such a study is shown in fig. 3 for the reaction 2°Ne(3He,aHe)19Ne at Q = - 2 . 0 MeV. The dashed curve describes the pressure necessary to maintain a constant count rate for different values of d z, while d l = 0.625". Changes in active target volume have been included. The solid curves record the calculated fwhm as a function of d2, under the restriction of constant count rate. At large d2 and high pressure the straggling terms dominate the calculated width, whereas at small dE and low pressure the geometric width is most important. It should be noted that as the count rate requirement is relaxed the minima in fig. 3 will lower

and m o v e to larger values of d E, while the opposite happens when a greater count rate is required. From fig. 3 the best values seem to be d2 ~ 3.5" and p ~ 75 tort. Having selected values for dE a n d p it is advantageous to know expected observable line widths in an experimental spectrum as a function of E a, 0 and Q. The result of such calculations is presented in fig. 4, again for 2°Ne(aHe,4He)19Ne, showing fwhm vs detected energy. The dashed lines connect points of constant Q-value and the solid lines connect points of constant 0. The primary contribution to a changing fwhm is the value of dEa/dO since the actual energy dependence in

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%(0,0) I MeV) Fig. 5. M e a s u r e d a n d calculated f w h m for the elastic a n d inelastic scattering of SHe by 2°Ne. Solid curves connect calculated points at c o n s t a n t 0; dashed lines connect calculated points o f c o n s t a n t Q-value (MeV). T h e dashed lines are not s m o o t h curves because o f gas pressure variations <~ 3 % between r u n s recorded at different times; p ~ 100 torr, Sl = s~ = w = 0.040", dl = 0.625", d z = 4.0".

LINE SHAPE C A L C U L A T I O N FOR N U C L E A R S C A T T E R I N G

straggling is slight. At the lowest energy, back angle calculation, the target thickness effect is apparent. A comparison of calculated and observed line shape has been shown in fig. 2. The Gaussian curve (c) fitted to the calculated line shape gives a good representation of the observed scattering as far as 2a -- 75 keV from the maximum. Deviations the order of 1% of maximum near 3~ are probably due to second scattering in the gasV). The reliability of the calculated fwhm at different angles is illustrated in fig. 5 for elastic and inelastic scattering of 3He b y / ° N e . The experimental widths for elastic scattering (3He0) are very reliable because of good statistical accuracy in the data and the lack of significant interfering contributions from (3He,gHe) reactions. The latter made direct extraction of 3He~ at 57.5 ° impossible. The high degree of correspondence between calculated and observed widths for isolated particle groups in the energy spectra has demonstrated the reliability of using the calculated fwhm for extraction of data from spectral regions not completely resolved. The calculation described up to this point is quite adequate for the slit geometry and gas pressures used. The object was to extract yields from complex spectra and to establish Q-values from spectral line positions. The most important revision of the line shape calculation for the present conditions is the calculation of the contribution from a scattering outside the active target volume followed by a second scattering which directs the particle of interest toward the detector. A crude calculation which would yield a shape for the double scattering effect could be done quite easily, but a calculation to include the absolute magnitude as well as an accurate shape would increase the running time of the geometric line shape part of the program by at least an order of magnitude in order to describe what is in the present case only a 1% effect. When slit and beam geometry are considerably less restrictive there are a number of changes which must

293

be introduced to calculate an accurate line shape. In addition to an obvious asymmetry in the line shape itself, a shift of the calculated centroid from E3(0,0 ) will result. Some corrections could be introduced without an appreciable increase in program running time by altering the number, previously unity, which is stored in each channel between E3(x,y ) .... and E3(x,y)mi,. A store of unity ignores the variation of the scattering cross section over the finite detector. The store could also be made a function of the beam intensity distribution, I(y), with I(y) either measured or calculated for an ideal caseS). Lastly the detection efficiency is usually oc 1/r 2, but only an additional factor of 1/r need be introduced in the store since a factor of 1/r is inherent in the present method of establishing the number of stores from eqs. (3) through 0 max(X,y)--Omin(X,y). Approximate methods for handling the above corrections will depend on the geometry employed. When it is required to numerically integrate over the detector face as well as the target volume the calculation becomes very cumbersome. The linear eqs. (2) and (3) are no longer applicable and the overall calculation cannot be separated into geometric and straggling parts. In this sense the calculations of fwhm for d2 < 1.5", shown in fig. 3 are probably not accurate. References 1) j. p. Chandler, STEPIT, Quantum chemistry program exchange (Chemistry Department, Indiana University, Bloomington, Indiana). 2) An additional refinement in the calculation may be made at this point. Such refinements are discussed in sec. 3. 3) K. R. Symon, Thesis (Harvard University, 1948) unpublished. 4) B. Rossi, High-energy particles (Prentice-Hall Inc., New York, 1952) p. 31. 5) E. J. Williams, Rev. Mod. Phys. 17 (1945) 217. 6) E. Segr6, Nuclei and particles (W. A. Benjamin, Inc., New York, 1964) p. 39. 7) M. D. Mancusi, J. K. Bair, C. M. Jones, S. T. Thornton and H. B. Willard, Nucl. Instr. and Meth. 68 (1969) 70. s) A. P. Banford, The transport of charged particle beams (E. and F. N. Spon Ltd., London, 1966) p. 22.