- Email: [email protected]
Some aberration and other properties of an Enge split-pole spectrograph
272
Nuclear Instruments and Methods m Physics Research A306 (1991) 272-277 North-Holland
Some aberration and other properties of an Enge split-pole spectrograph P.H. Barker, S.C. Baker, S.A . Brindhaban and M.J. Brown Physics Department, Auckland University, Auckland, New Zealand
Received 11 February 1991
By accelerating a beam of heavy ions through a precisely measured and adjustable potential difference, image shapes in an Enge split-pole spectrograph have been measured, and the second order aberration coefficients determined . In addition, the sensitivity of the orbit magnetic rigidity to variations in the spectrograph nominal temperature and to the elapsed time since the last current setting has been studied.
1 . Introduction For some time our laboratory has been involved in a programme which is aimed at determining decay parameters of superallowed nuclear beta decays . In particular, we have developed a system, heavy ion source system (HISS), for measuring the decay energies very accurately, and this has involved measuring the average kinetic energy of the accelerator proton beam with a precision of around ten parts per million (ppm). The basic method of HISS has been described in detail elsewhere [1], but the essential elements are that the beam from our accelerator, AURA2, passes on a tightly collimated horizontal path round an Enge splitpole spectrograph, and, after emerging, it is used to perform the nuclear physics experiment of interest . The spectrograph field is monitored with an NMR probe in the second pole gap, and the orbit is defined by narrow, vertical object and image slits. During an experiment, the magnetic rigidity of the orbit is calibrated by accelerating a beam of heavy ions, normally either 39 K + or 133 Cs', through a voltage difference V so that the ions follow the same path as the accelerator beam had done, without altering the magnetic field. The voltage V is then determined by dividing it down and comparing with a 1 V standard . After various small effects at the . have been taken into account, the level of < 100 ppm proton beam energy may be determined fairly straightforwardly to ± 10 ppm, and the nuclear reaction energy of interest somewhat more imprecisely, depending on the yield of the reaction . A typical reaction might involve a proton beam of energy 6.5 MeV, in which case 133CS+, the heavy ion beam would be and the voltage V to be measured would be approximately 49 kV .
In an ideal situation, which is sometimes realisable, the path through the spectrograph is very stringently collimated, down to a full width at half maximum (FWHM) in energy of around 50 ppm, which, since the accelerator beam is never better than 200 ppm wide, effectively constitutes throwing most of the proton beam away. This is acceptable provided that the yield of the nuclear reaction is large enough, see for example ref. [1]. Often however the lack of nuclear reaction yield is the overwhelming factor in determining the magnitude of the final error in a reaction energy measurement, and so there is a strong motivation to relax the collimation and increase the intensity of the beam, either by allowing a larger angular spread to enter the spectrograph or by simply widening the object and image slits. In both cases one needs to have a good understanding of the beam energy profile which is being accepted, as this will bear on the detailed analysis of the nuclear reaction . Consequently, we have studied the aberration behaviour of the spectrograph for small and medium beam entrance angles. A second feature of the spectrograph behaviour which will affect the magnetic rigidity of the particle orbits is expansion of the physical structure due to thermal effects. Although rapid temperature changes are probably unlikely due to the mass of the instrument, nearly 11 tonnes, it cannot be assumed that effects are completely negligible, since the coefficients of linear expansion of aluminium and iron, two of the main constituents, are 23 and 12 ppm/K respectively . By installing an appropriate temperature monitor and using the HISS, we have investigated the temperature-dependent behaviour of the system . At an earlier stage of our system, before we devel-
0168-9002/91/$03 .50 © 1991 - Elsevier Science Publishers B.V . (North-Holland)
P.H Barker et al. / Properties of an Enge split-pole spectrograph
oped HISS, proton energy calibrations were made by altering the magnetic field until alpha particles from a radioactive source were brought to the same point on the focal plane as had been struck by the proton beam . In this method, one of the largest potential sources of error was the occurrence of differential hysterisis, where this may be regarded as the lack of proportionality between the average magnetic field strength over the particle orbits and that at the position of the NMR probe. This was investigated and a limit set whose contribution to the error ascribed to an energy measurement was relatively minor in comparison with other factors [2]. With HISS, the magnetic field remains essentially unaltered, and so such effects will be much smaller and, if present, would be only the variation with time of the differential hysterisis, so that a calibration performed with HISS would not be exactly correct for a nuclear reaction performed a few hours later. Despite the expectation that these effects for HISS would be small, they were investigated . The properties of the Enge spectrograph in the three areas mentioned above have, of course, been investigated because we need to understand them better . In addition, however, there are many of these instruments in use in nuclear physics laboratories throughout the world, and we hope the results presented here will be of value to their users. 2. Measurements The Enge split-pole spectrograph is described in detail by Spencer and Enge in ref. [3]. Our instrument is identical to that described in ref. [3] except that the dimensions are reduced by two-thirds . The beam collimation consists of two adjustable vertical slits, the first normally being 2 mm high and 0.075 mm wide, at the object point of the spectrograph ; the second 20 mm high and 0.025 mm wide, at a position in the focal plane which is at a radius of around 90% of the maximum. The entrance angles are limited by a rectangular aperture, 170 mm downstream of the object point, whose horizontal and vertical extents, called H and V, can be up to 17 mm and 10 mm respectively, but which we have never contemplated setting to values greater than 1 .6 mm and 1 .6 mm in an experiment, and which were set only to a maximum of 6 .4 mm and 6 .4 mm in the present investigations . The spectrograph itself was designed as a broad range instrument, with high resolution but small dispersion, which could be rotated to look at reaction particles from -90' to 130' relative to the beam axis . In our application the angle is fixed at 0 ° and only a very small part of the 1 m long focal plane (where the image slits are) is used . Other changes which have been made are that the NMR probe is 10 cm further into the
273
second pole gap than in the original design, to place it very close to the particle orbits, and that the original current supply and NMR system have been replaced by more reliable units with greater stability. 2 .1 . Aberration behaviour
In ref. [3], the focusing properties of the spectrograph are specified as 2 2 y2/p = A 1 0 + A2tp2 + A 303 +A 4 00 , z21P
= A64) + A74,
0' = BIA5,
where y2 and z 2 are the horizontal and vertical coordinates in the image plane, perpendicular to the central ray, of a ray which left the object point at horizontal and vertical angles of 6 and 0 respectively . Distances are made dimensionless by dividing by p, the radius of curvature of the particles in the magnetic field. The horizontal exit angle is 0' and A, to A, are the most important aberration coefficients, with A S being the linear horizontal magnification. The A-coefficients were calculated theoretically in ref. [3], but probably have never been obtained experimentally [4]. In the present case, the beam from HISS was of 133 Cs' ions, accelerated to approximately 15 keV energy. An energy modulation of the beam was unavoidably introduced by the accelerating voltage power supply, and was approximately triangular of frequency 100 Hz, with a peak to peak width of 20 ppm. The beam entered the spectrograph through the object slit, and its angles were constrained by H and V. The aberration coefficients are most easily obtained with large angles . Nevertheless, H and V in these tests were restricted to 6.4 mm, as it was established that only up to this size did the ion beam completely and uniformly fill the entrance aperture . This corresponded to maximum horizontal and vertical entrance angles of ±0 .019 rad. The image shape, for a particular setting of H and V, was obtained by scanning the outgoing beam across the image slit, and plotting a graph of transmitted beam current as a function of accelerating voltage. A typical such curve had a maximum intensity of 100 pA, was a few hundred ppm total width at the base, and took ten minutes to record. During this time the magnetic field, as monitored by the NMR, was held constant to better than 5 ppm. The basic shape for paraxial Cs' ions entering through the 0.075 mm wide object slit, with kinetic energy modulated by a 20 ppm triangular waveform, and scanned across a 0.025 mm exit slit, is the convolution of these three, essentially square, contributions . In image space the 0.075 mm object width becomes (0 .075 x 0 .34 = 0.026 mm) and then using the dispersion of 1.75 from ref. [3], and a radius of 550 mm, the final
274
P. H. Barker et al. / Properties of an Enge split-pole spectrograph
calculated image shape is roughly Gaussian with a FWHM of 50 ppm in energy . This then is the shape which should be obtained near the first order focus for rays leaving the object at small angles . It is also the shape which is convoluted in with the theoretical image derived from the above equations to produce the image shapes which are compared with experiment . The first task was to locate the position of the first order focus. This was accomplished by taking beam profile scans as outlined above, for the two cases H = V = 0.8 mm and H = V = 6.4 mm, and as the 0.025 mm wide image slit was moved in small steps in a line approximately along the outgoing beam axis . These data are represented in fig. 1, where the full width at half maximum (FWHM) for the small angle case and the full width at quarter maximum (FWQM) for the larger angle case are plotted as a function of the position of the slit . Using these results, it was decided to put the slit at a setting corresponding to an image position of 7 mm. We note in passing that the FWHM for the near-paraxial case is 55 ppm, close to the calculated figure . In ref. [3], the At through A7 coefficients for a radius of 90% of maximum are calculated to be 0, -1 .1, -16.5, 2.0, 0.34, - 0.1 and - 8.5, but towards the end of the paper a brief mention of an experimental test expresses regret that A1 was found to be closer to -1 .0 . For the range of angles in the present study, the second order coefficients A 1 and A 2 and the horizontal magnification AS play the most significant roles, and the last was estimated fairly well by using H to limit the input angle and then measuring the angular spread of the outgoing beam . Good agreement with an angular magnification of 3 .0, i.e . a linear magnification of about 0.34, as quoted in ref. [3] was found. The importance of the second order coefficients, A t and A2, could be seen
from the fact that as the aperture was opened out, the image shapes became asymmetric . The following method was used to determine the best values of A l and A 2 . Beam profiles were taken for 12 combinations of the sizes of H and V, chosen between the minimum and maximum values of 0.8 and 6.4 mm. Now the limit on the width of the image for very small angles is given by the convolution of the effects of the energy modulation, and the geometrical widths of the image and object slits, as outlined above. This is essentially the data shown in fig. 2a . Then, for larger angles, the image formed by computing the y2 positions of 10000 rays uniformly illuminating the entrance aperture was convoluted with this basic shape, its area normalised to be the same as the corresponding data curve, and a goodness of fit criterion, S2, evaluated. The S2 was simply the sum of the squares of the differences between the magnitudes of the data points and the calculated points. This procedure was repeated for different assumed values of At and A2, and the "best" values of the coefficients found. Figs . 2b, 2c and 2d show, as examples, the calculated shapes for the cases (H = 0.8 mm, V = 6 .4 mm), (H = 6.4 mm, V= 0.8 mm) and (H = 6.4 mm, V = 6.4 mm) using the best overall values of A, and A2 . The quality of the fits is generally quite good, although there is some underestimation of the intensity of the tail for the maximum horizontal angle. The minima in S2 as functions of A, and A2 were quite sharp and led to final values, derived from all 12 curves, of A, = -0 .4+0 .05 and AZ = -0 .1 ± 0.05. Although these are quite different from those estimated in ref. [3], this is perhaps not too disturbing, as, even there, the size of A, is calculated as 0.0, estimated as -1 .0 and then in a later figure re-estimated to be -0 .38. The effects of altering the sizes of A3, A4, A6 and A 7 away from the values of ref. [3] were completely negligible because of the relatively small angles involved . We feel confident that, as a result of these investigations we can understand the effects on our nuclear physics experiments of opening out the entrance angles to the spectrograph, at least up to a reasonable fraction of the largest size discussed above, and of widening the object and image slits. 2 .2 . Temperature dependence
Image Displacement (mm)
Fig. l. Location of the first order focus. See text . The lines are to guide the eye.
When a HISS calibration is performed, a calibration number X corresponding to the central orbit is calculated. For a 15 keV 133 Cs' ion, which is nonrelativistic, X is essentially the mass of the ion multiplied by its kinetic energy and divided by the square of the NMR frequency, in units of (u eV MHz -2 ). For light ions the equivalent calculation must be performed relativistically. The dependences of X on spectrograph temperature
P.H. Barker et al / Properties of an Enge spltt-pole spectrograph 200
200 H=0 .8 mm V=0 .8 mm
a
Through Current (pA)
275
Through Current (pA)
100
100
60-40
-20
0
20
40
"
L
60
-0100
Energy Offset ( ppm ) 240
H=6 .4 mm V=0 .8 mm
Through Current (pA)
0
-
100
Energy Offset ( ppm )
Through Current (pA)
160
80
-I00
0
100
200
300
400
Energy Offset ( ppm )
Energy Offset ( ppm )
Fig. 2. (a) The image shape for entrance angles 0 <_ ±0.0024, p<_ ±0 .0024 rad. The solid line is the calculated shape. See text. (b) The image shape for entrance angles 0 _< ±0 .0024, (5 ±0.019 rad. The solid line is the calculated shape. See text . (c) The image shape for entrance angles B < ± 0.019, ¢ _< ±0 .0024 rad. The solid line is the calculated shape. See text . (d) The image shape for entrance angles 0 < ±0.019,
and on the time elapsed since the last magnetic field change are to some extent intertwined, but experience
accumulated over more than 10 years had convinced us that at least if the spectrograph had been at nominally the same field for a week, then we could assume that any further monotonic time dependency of X would be
negligible, and so we could then look for temperature effects. Obviously it is not strictly possible to speak of
"the temperature"
of a large object made of many
materials and weighing 11 tonnes, but it was hoped to obtain a reliable temperature monitor by embedding an electronic temperature sensor in the aluminium vacuum
box, and a convenient place was chosen just above the centre of the outer boundary of the second pole. The magnet cooling water is a doubly-deionised sealed system whose temperature excursions are limited by passing it through a secondary heat exchanger. In this secondary system, thermal switches allow either cold or warm water from the local building supply to enter, and they are normally set II'C apart.
Fig. 3 shows the results of measuring the X calibraa few weeks, as a function of the
tion value over
monitor temperature. The X values have been assigned an error of 5 ppm, which is probably rather generous,
particularly since it is really only the relative changes to X which are being studied, and the horizontal error bars
represent ±0 .1 K. A correlation of X with temperature is apparent, and this may be well represented over the likely range of interest as a linear dependency with a slope of 33 ppm/K.
As a consequence of the above, all calibration and nuclear physics experiments now have a temperature associated with them, and their energies are normalised to a nominal 24'C . 2.3 . Differential hysterisis time dependence To explore the time dependence of the differential hysterisis of the spectrograph when used in the way discussed above, all that is necessary is to determine the
276
P.H. Barker et al. / Properties of an Enge spht-pole spectrograph (X-7777) ppm
60 a 5 MeV Day 1
30
E o. a c0
0
.m .m
x
-30
-60 -
1 0'
10 ,
to ,
Temperature (°C)
101
1
01
to ,
Time (minutes)
Fig 3. The X calibration value plotted against nominal spectrograph temperature . (X-7773) ppm
80
b
2 MeV Day 7
40
X calibration number as a function of time. This was done three times and the results are shown in fig. 4. The first point of each data set was taken about ten minutes after the current through the coils had been set, and then the X value followed for about one week . All the points have been corrected for temperature variation, as outlined in section 2.2 . Before Day 1 the field was zero, and then for fig. 4a the field was set to transmit 5 MeV protons and the X values measured with a 133Cs' beam of about 38 keV. Fig. 4b corresponds to 2 MeV protons and the data were taken with 51 keV 39 K+ ions, and then fig. 4c reverts to the same conditions as fig. 4a . Two things are immediately apparent . The first is that the variation of X with time is very slight and therefore the average of two calibrations taken half a day apart and bracketing a proton measurement will give the correct value of X for the protons certainly to better than 10 ppm, without having to wait for several days after a field change. The second is that there seem to be some anomalous points, two in fig. 4a and one in 4c, and this is of course of some concem. After much investigation the cause of the anomalies was found to be in the secondary cooling water system . With the field set for a particular energy proton beam, the secondary cooling water cuts in at regular intervals: every two or three hours for 5 MeV protons and once or twice a day for 2 MeV. By overriding the automatic temperature sensor, we could force the cooling to cut in and in this way it was discovered that if the X value was measured within about 10 minutes of this cut-in, then it could be anomalous; that is, the temperaturecorrelation correction was invalid. But outside this period no anomalies were found .
0
-4U
0,
10 1
10 2
10 2
10'
10'
10'
10 1
Time (min)
80 (X-7777) ppm
C 5 MeV
Day 16
40
0
-40 -
10'
10 ,
10 ,
10 ,
Time (minutes)
Fig. 4. (a) Changes in the X calibration value plotted against time, with the magnetic field set for 5 MeV protons. (b) Changes in the X calibration value plotted against time, with the magnetic field set for 2 MeV protons. (c) As (a), but 16 days later.
P. H Barker et al. / Properties of an Enge split-pole spectrograph
3. Conclusion In the light of the above discussion, we feel that the behaviour of our Enge split-pole spectrograph with respect to temperature and time, and the aberration properties up to entrance angles of ± 0.02 rad are now adequately understood . Consequently, when we combine these factors with the other tests dealt with in ref. [1], we have confidence in quoting the average kinetic energy of the proton beam which we use in our nuclear physics experiments to ± 10 ppm. In addition, we hope that our experience will be of benefit to other spectrograph users.
277
Acknowledgements We would like to thank R.E. White for helpful discussion and the staff of the AURA2 accelerator laboratory for their continued and enthusiastic support. References [1] R.E. White, P.H . Barker and D.M .J . Lovelock, Metrologia 21 (1985) 193. [2] P.H. Barker, H. Naylor and R.E. White, Nucl . Instr. and Meth. 150 (1978) 537. [3] J.E . Spencer and H.A . Enge, Nucl . Instr. and Meth. 49 (1967) 181. [4] H.A . Enge, Private communication (1990) .
Nuclear Instruments and Methods m Physics Research A306 (1991) 272-277 North-Holland
Some aberration and other properties of an Enge split-pole spectrograph P.H. Barker, S.C. Baker, S.A . Brindhaban and M.J. Brown Physics Department, Auckland University, Auckland, New Zealand
Received 11 February 1991
By accelerating a beam of heavy ions through a precisely measured and adjustable potential difference, image shapes in an Enge split-pole spectrograph have been measured, and the second order aberration coefficients determined . In addition, the sensitivity of the orbit magnetic rigidity to variations in the spectrograph nominal temperature and to the elapsed time since the last current setting has been studied.
1 . Introduction For some time our laboratory has been involved in a programme which is aimed at determining decay parameters of superallowed nuclear beta decays . In particular, we have developed a system, heavy ion source system (HISS), for measuring the decay energies very accurately, and this has involved measuring the average kinetic energy of the accelerator proton beam with a precision of around ten parts per million (ppm). The basic method of HISS has been described in detail elsewhere [1], but the essential elements are that the beam from our accelerator, AURA2, passes on a tightly collimated horizontal path round an Enge splitpole spectrograph, and, after emerging, it is used to perform the nuclear physics experiment of interest . The spectrograph field is monitored with an NMR probe in the second pole gap, and the orbit is defined by narrow, vertical object and image slits. During an experiment, the magnetic rigidity of the orbit is calibrated by accelerating a beam of heavy ions, normally either 39 K + or 133 Cs', through a voltage difference V so that the ions follow the same path as the accelerator beam had done, without altering the magnetic field. The voltage V is then determined by dividing it down and comparing with a 1 V standard . After various small effects at the . have been taken into account, the level of < 100 ppm proton beam energy may be determined fairly straightforwardly to ± 10 ppm, and the nuclear reaction energy of interest somewhat more imprecisely, depending on the yield of the reaction . A typical reaction might involve a proton beam of energy 6.5 MeV, in which case 133CS+, the heavy ion beam would be and the voltage V to be measured would be approximately 49 kV .
In an ideal situation, which is sometimes realisable, the path through the spectrograph is very stringently collimated, down to a full width at half maximum (FWHM) in energy of around 50 ppm, which, since the accelerator beam is never better than 200 ppm wide, effectively constitutes throwing most of the proton beam away. This is acceptable provided that the yield of the nuclear reaction is large enough, see for example ref. [1]. Often however the lack of nuclear reaction yield is the overwhelming factor in determining the magnitude of the final error in a reaction energy measurement, and so there is a strong motivation to relax the collimation and increase the intensity of the beam, either by allowing a larger angular spread to enter the spectrograph or by simply widening the object and image slits. In both cases one needs to have a good understanding of the beam energy profile which is being accepted, as this will bear on the detailed analysis of the nuclear reaction . Consequently, we have studied the aberration behaviour of the spectrograph for small and medium beam entrance angles. A second feature of the spectrograph behaviour which will affect the magnetic rigidity of the particle orbits is expansion of the physical structure due to thermal effects. Although rapid temperature changes are probably unlikely due to the mass of the instrument, nearly 11 tonnes, it cannot be assumed that effects are completely negligible, since the coefficients of linear expansion of aluminium and iron, two of the main constituents, are 23 and 12 ppm/K respectively . By installing an appropriate temperature monitor and using the HISS, we have investigated the temperature-dependent behaviour of the system . At an earlier stage of our system, before we devel-
0168-9002/91/$03 .50 © 1991 - Elsevier Science Publishers B.V . (North-Holland)
P.H Barker et al. / Properties of an Enge split-pole spectrograph
oped HISS, proton energy calibrations were made by altering the magnetic field until alpha particles from a radioactive source were brought to the same point on the focal plane as had been struck by the proton beam . In this method, one of the largest potential sources of error was the occurrence of differential hysterisis, where this may be regarded as the lack of proportionality between the average magnetic field strength over the particle orbits and that at the position of the NMR probe. This was investigated and a limit set whose contribution to the error ascribed to an energy measurement was relatively minor in comparison with other factors [2]. With HISS, the magnetic field remains essentially unaltered, and so such effects will be much smaller and, if present, would be only the variation with time of the differential hysterisis, so that a calibration performed with HISS would not be exactly correct for a nuclear reaction performed a few hours later. Despite the expectation that these effects for HISS would be small, they were investigated . The properties of the Enge spectrograph in the three areas mentioned above have, of course, been investigated because we need to understand them better . In addition, however, there are many of these instruments in use in nuclear physics laboratories throughout the world, and we hope the results presented here will be of value to their users. 2. Measurements The Enge split-pole spectrograph is described in detail by Spencer and Enge in ref. [3]. Our instrument is identical to that described in ref. [3] except that the dimensions are reduced by two-thirds . The beam collimation consists of two adjustable vertical slits, the first normally being 2 mm high and 0.075 mm wide, at the object point of the spectrograph ; the second 20 mm high and 0.025 mm wide, at a position in the focal plane which is at a radius of around 90% of the maximum. The entrance angles are limited by a rectangular aperture, 170 mm downstream of the object point, whose horizontal and vertical extents, called H and V, can be up to 17 mm and 10 mm respectively, but which we have never contemplated setting to values greater than 1 .6 mm and 1 .6 mm in an experiment, and which were set only to a maximum of 6 .4 mm and 6 .4 mm in the present investigations . The spectrograph itself was designed as a broad range instrument, with high resolution but small dispersion, which could be rotated to look at reaction particles from -90' to 130' relative to the beam axis . In our application the angle is fixed at 0 ° and only a very small part of the 1 m long focal plane (where the image slits are) is used . Other changes which have been made are that the NMR probe is 10 cm further into the
273
second pole gap than in the original design, to place it very close to the particle orbits, and that the original current supply and NMR system have been replaced by more reliable units with greater stability. 2 .1 . Aberration behaviour
In ref. [3], the focusing properties of the spectrograph are specified as 2 2 y2/p = A 1 0 + A2tp2 + A 303 +A 4 00 , z21P
= A64) + A74,
0' = BIA5,
where y2 and z 2 are the horizontal and vertical coordinates in the image plane, perpendicular to the central ray, of a ray which left the object point at horizontal and vertical angles of 6 and 0 respectively . Distances are made dimensionless by dividing by p, the radius of curvature of the particles in the magnetic field. The horizontal exit angle is 0' and A, to A, are the most important aberration coefficients, with A S being the linear horizontal magnification. The A-coefficients were calculated theoretically in ref. [3], but probably have never been obtained experimentally [4]. In the present case, the beam from HISS was of 133 Cs' ions, accelerated to approximately 15 keV energy. An energy modulation of the beam was unavoidably introduced by the accelerating voltage power supply, and was approximately triangular of frequency 100 Hz, with a peak to peak width of 20 ppm. The beam entered the spectrograph through the object slit, and its angles were constrained by H and V. The aberration coefficients are most easily obtained with large angles . Nevertheless, H and V in these tests were restricted to 6.4 mm, as it was established that only up to this size did the ion beam completely and uniformly fill the entrance aperture . This corresponded to maximum horizontal and vertical entrance angles of ±0 .019 rad. The image shape, for a particular setting of H and V, was obtained by scanning the outgoing beam across the image slit, and plotting a graph of transmitted beam current as a function of accelerating voltage. A typical such curve had a maximum intensity of 100 pA, was a few hundred ppm total width at the base, and took ten minutes to record. During this time the magnetic field, as monitored by the NMR, was held constant to better than 5 ppm. The basic shape for paraxial Cs' ions entering through the 0.075 mm wide object slit, with kinetic energy modulated by a 20 ppm triangular waveform, and scanned across a 0.025 mm exit slit, is the convolution of these three, essentially square, contributions . In image space the 0.075 mm object width becomes (0 .075 x 0 .34 = 0.026 mm) and then using the dispersion of 1.75 from ref. [3], and a radius of 550 mm, the final
274
P. H. Barker et al. / Properties of an Enge split-pole spectrograph
calculated image shape is roughly Gaussian with a FWHM of 50 ppm in energy . This then is the shape which should be obtained near the first order focus for rays leaving the object at small angles . It is also the shape which is convoluted in with the theoretical image derived from the above equations to produce the image shapes which are compared with experiment . The first task was to locate the position of the first order focus. This was accomplished by taking beam profile scans as outlined above, for the two cases H = V = 0.8 mm and H = V = 6.4 mm, and as the 0.025 mm wide image slit was moved in small steps in a line approximately along the outgoing beam axis . These data are represented in fig. 1, where the full width at half maximum (FWHM) for the small angle case and the full width at quarter maximum (FWQM) for the larger angle case are plotted as a function of the position of the slit . Using these results, it was decided to put the slit at a setting corresponding to an image position of 7 mm. We note in passing that the FWHM for the near-paraxial case is 55 ppm, close to the calculated figure . In ref. [3], the At through A7 coefficients for a radius of 90% of maximum are calculated to be 0, -1 .1, -16.5, 2.0, 0.34, - 0.1 and - 8.5, but towards the end of the paper a brief mention of an experimental test expresses regret that A1 was found to be closer to -1 .0 . For the range of angles in the present study, the second order coefficients A 1 and A 2 and the horizontal magnification AS play the most significant roles, and the last was estimated fairly well by using H to limit the input angle and then measuring the angular spread of the outgoing beam . Good agreement with an angular magnification of 3 .0, i.e . a linear magnification of about 0.34, as quoted in ref. [3] was found. The importance of the second order coefficients, A t and A2, could be seen
from the fact that as the aperture was opened out, the image shapes became asymmetric . The following method was used to determine the best values of A l and A 2 . Beam profiles were taken for 12 combinations of the sizes of H and V, chosen between the minimum and maximum values of 0.8 and 6.4 mm. Now the limit on the width of the image for very small angles is given by the convolution of the effects of the energy modulation, and the geometrical widths of the image and object slits, as outlined above. This is essentially the data shown in fig. 2a . Then, for larger angles, the image formed by computing the y2 positions of 10000 rays uniformly illuminating the entrance aperture was convoluted with this basic shape, its area normalised to be the same as the corresponding data curve, and a goodness of fit criterion, S2, evaluated. The S2 was simply the sum of the squares of the differences between the magnitudes of the data points and the calculated points. This procedure was repeated for different assumed values of At and A2, and the "best" values of the coefficients found. Figs . 2b, 2c and 2d show, as examples, the calculated shapes for the cases (H = 0.8 mm, V = 6 .4 mm), (H = 6.4 mm, V= 0.8 mm) and (H = 6.4 mm, V = 6.4 mm) using the best overall values of A, and A2 . The quality of the fits is generally quite good, although there is some underestimation of the intensity of the tail for the maximum horizontal angle. The minima in S2 as functions of A, and A2 were quite sharp and led to final values, derived from all 12 curves, of A, = -0 .4+0 .05 and AZ = -0 .1 ± 0.05. Although these are quite different from those estimated in ref. [3], this is perhaps not too disturbing, as, even there, the size of A, is calculated as 0.0, estimated as -1 .0 and then in a later figure re-estimated to be -0 .38. The effects of altering the sizes of A3, A4, A6 and A 7 away from the values of ref. [3] were completely negligible because of the relatively small angles involved . We feel confident that, as a result of these investigations we can understand the effects on our nuclear physics experiments of opening out the entrance angles to the spectrograph, at least up to a reasonable fraction of the largest size discussed above, and of widening the object and image slits. 2 .2 . Temperature dependence
Image Displacement (mm)
Fig. l. Location of the first order focus. See text . The lines are to guide the eye.
When a HISS calibration is performed, a calibration number X corresponding to the central orbit is calculated. For a 15 keV 133 Cs' ion, which is nonrelativistic, X is essentially the mass of the ion multiplied by its kinetic energy and divided by the square of the NMR frequency, in units of (u eV MHz -2 ). For light ions the equivalent calculation must be performed relativistically. The dependences of X on spectrograph temperature
P.H. Barker et al / Properties of an Enge spltt-pole spectrograph 200
200 H=0 .8 mm V=0 .8 mm
a
Through Current (pA)
275
Through Current (pA)
100
100
60-40
-20
0
20
40
"
L
60
-0100
Energy Offset ( ppm ) 240
H=6 .4 mm V=0 .8 mm
Through Current (pA)
0
-
100
Energy Offset ( ppm )
Through Current (pA)
160
80
-I00
0
100
200
300
400
Energy Offset ( ppm )
Energy Offset ( ppm )
Fig. 2. (a) The image shape for entrance angles 0 <_ ±0.0024, p<_ ±0 .0024 rad. The solid line is the calculated shape. See text. (b) The image shape for entrance angles 0 _< ±0 .0024, (5 ±0.019 rad. The solid line is the calculated shape. See text . (c) The image shape for entrance angles B < ± 0.019, ¢ _< ±0 .0024 rad. The solid line is the calculated shape. See text . (d) The image shape for entrance angles 0 < ±0.019,
and on the time elapsed since the last magnetic field change are to some extent intertwined, but experience
accumulated over more than 10 years had convinced us that at least if the spectrograph had been at nominally the same field for a week, then we could assume that any further monotonic time dependency of X would be
negligible, and so we could then look for temperature effects. Obviously it is not strictly possible to speak of
"the temperature"
of a large object made of many
materials and weighing 11 tonnes, but it was hoped to obtain a reliable temperature monitor by embedding an electronic temperature sensor in the aluminium vacuum
box, and a convenient place was chosen just above the centre of the outer boundary of the second pole. The magnet cooling water is a doubly-deionised sealed system whose temperature excursions are limited by passing it through a secondary heat exchanger. In this secondary system, thermal switches allow either cold or warm water from the local building supply to enter, and they are normally set II'C apart.
Fig. 3 shows the results of measuring the X calibraa few weeks, as a function of the
tion value over
monitor temperature. The X values have been assigned an error of 5 ppm, which is probably rather generous,
particularly since it is really only the relative changes to X which are being studied, and the horizontal error bars
represent ±0 .1 K. A correlation of X with temperature is apparent, and this may be well represented over the likely range of interest as a linear dependency with a slope of 33 ppm/K.
As a consequence of the above, all calibration and nuclear physics experiments now have a temperature associated with them, and their energies are normalised to a nominal 24'C . 2.3 . Differential hysterisis time dependence To explore the time dependence of the differential hysterisis of the spectrograph when used in the way discussed above, all that is necessary is to determine the
276
P.H. Barker et al. / Properties of an Enge spht-pole spectrograph (X-7777) ppm
60 a 5 MeV Day 1
30
E o. a c0
0
.m .m
x
-30
-60 -
1 0'
10 ,
to ,
Temperature (°C)
101
1
01
to ,
Time (minutes)
Fig 3. The X calibration value plotted against nominal spectrograph temperature . (X-7773) ppm
80
b
2 MeV Day 7
40
X calibration number as a function of time. This was done three times and the results are shown in fig. 4. The first point of each data set was taken about ten minutes after the current through the coils had been set, and then the X value followed for about one week . All the points have been corrected for temperature variation, as outlined in section 2.2 . Before Day 1 the field was zero, and then for fig. 4a the field was set to transmit 5 MeV protons and the X values measured with a 133Cs' beam of about 38 keV. Fig. 4b corresponds to 2 MeV protons and the data were taken with 51 keV 39 K+ ions, and then fig. 4c reverts to the same conditions as fig. 4a . Two things are immediately apparent . The first is that the variation of X with time is very slight and therefore the average of two calibrations taken half a day apart and bracketing a proton measurement will give the correct value of X for the protons certainly to better than 10 ppm, without having to wait for several days after a field change. The second is that there seem to be some anomalous points, two in fig. 4a and one in 4c, and this is of course of some concem. After much investigation the cause of the anomalies was found to be in the secondary cooling water system . With the field set for a particular energy proton beam, the secondary cooling water cuts in at regular intervals: every two or three hours for 5 MeV protons and once or twice a day for 2 MeV. By overriding the automatic temperature sensor, we could force the cooling to cut in and in this way it was discovered that if the X value was measured within about 10 minutes of this cut-in, then it could be anomalous; that is, the temperaturecorrelation correction was invalid. But outside this period no anomalies were found .
0
-4U
0,
10 1
10 2
10 2
10'
10'
10'
10 1
Time (min)
80 (X-7777) ppm
C 5 MeV
Day 16
40
0
-40 -
10'
10 ,
10 ,
10 ,
Time (minutes)
Fig. 4. (a) Changes in the X calibration value plotted against time, with the magnetic field set for 5 MeV protons. (b) Changes in the X calibration value plotted against time, with the magnetic field set for 2 MeV protons. (c) As (a), but 16 days later.
P. H Barker et al. / Properties of an Enge split-pole spectrograph
3. Conclusion In the light of the above discussion, we feel that the behaviour of our Enge split-pole spectrograph with respect to temperature and time, and the aberration properties up to entrance angles of ± 0.02 rad are now adequately understood . Consequently, when we combine these factors with the other tests dealt with in ref. [1], we have confidence in quoting the average kinetic energy of the proton beam which we use in our nuclear physics experiments to ± 10 ppm. In addition, we hope that our experience will be of benefit to other spectrograph users.
277
Acknowledgements We would like to thank R.E. White for helpful discussion and the staff of the AURA2 accelerator laboratory for their continued and enthusiastic support. References [1] R.E. White, P.H . Barker and D.M .J . Lovelock, Metrologia 21 (1985) 193. [2] P.H. Barker, H. Naylor and R.E. White, Nucl . Instr. and Meth. 150 (1978) 537. [3] J.E . Spencer and H.A . Enge, Nucl . Instr. and Meth. 49 (1967) 181. [4] H.A . Enge, Private communication (1990) .