- Email: [email protected]
On pellet target luminosity modulation
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 556 (2006) 641–643 www.elsevier.com/locate/nima
On pellet target luminosity modulation V. Ziemann The Svedberg Laboratory, Uppsala University, Thunbergsvagen 5a, Box 533, 75121 Uppsala, Sweden Received 11 November 2005; received in revised form 15 November 2005; accepted 17 November 2005 Available online 9 December 2005
Abstract We calculate the variations in the target thickness for a pellet target interacting with a circulating beam and the influence on luminosity or trigger rate. r 2005 Elsevier B.V. All rights reserved. PACS: 29.25Pj; 29.27.a Keywords: Beam dynamics; Beam–target interaction; Pellet target
In order to achieve luminosities in the 1032 =cm2 s range in the PANDA detector [1] proposed for the anti-proton storage ring HESR at the FAIR facility [2] targets with a thickness in excess of 1015 =cm2 are required. Presently pellet targets [3] appear to be the only viable solution to fulfill these requirements but, at the same time, due to the granular character of the pellets, this will cause fluctuations in the overlap between beam and pellets resulting in a temporally varying luminosity and trigger rate for the detector. In such cases we can expect times with high trigger rate and many simultaneous events that may be difficult to discern and times of tranquility. An evenly distributed trigger rate is much preferred. In this report we will parameterize the variability of the luminosity by the ratio of average to peak value of the overlap of target and beam distributions. For the beam we can assume that the combined effects of electron cooling, target and intra-beam scattering will result in a Gaussian distribution at the target location with rms widths sx and sy in the horizontal and vertical direction, respectively. The pellets are spheres of frozen hydrogen or deuterium with diameter of about 30 mm, but for the purpose of this report we assume that they are point-like and that the pellets are injected from above into the scattering chamber. This results in an even distribution of pellets over a circle of Tel.: +46 18 471 3867; fax: +46 18 471 3833.
E-mail address: [email protected]. 0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.11.126
radius R in the plane of the ion beam. The spacing between subsequent pellets is assumed to be a distance d which is given by the product of pellet velocity and repetition rate. Typically, both d and R are on the order of mm. The effect of the mismatch of the horizontal beam size sx and the pellet stream can be described by averaging a Gaussian expðx2 =2s2x Þ over the circle in the horizontal– longitudinal (x2z) plane. We do not normalize the Gaussian, because we want to compare to the peak value, which is unity for the un-normalized Gaussian. Then the ratio between average and peak value rh is given by the integral 2 pffiffiffiffiffiffiffiffiffiffi 3 Z R Z R2 z2 1 2 2 4 pffiffiffiffiffiffiffiffiffiffi ex =2sx dx5dz. rh ¼ (1) pR2 R R2 z2 Noting that the integral is symmetric in x and in z we can substitute the lower integral boundaries by zero and multiply by 4 instead. Moreover, the Gaussian can be integrated leading to an error function [4]. After substituting z ¼ Rs we arrive at pffiffiffi ! Z 1 2sx 2 R pffiffiffiffiffiffiffiffiffiffiffiffi2ffi rh ¼ pffiffiffi erf pffiffiffi 1 s ds. (2) R p 2sx 0 The remaining integral can be easily integrated numerically and in Fig. 1 we show rh as a function of the ratio of beam size to circle radius sx =R and find a linear increase for small sx but once the beam size sx becomes comparable to
ARTICLE IN PRESS V. Ziemann / Nuclear Instruments and Methods in Physics Research A 556 (2006) 641–643
642
deduced from Eq. (5) by observing that for small beam sizes only the term with n ¼ 0 contributes, that the error function is anti-symmetric and approaches unity for large argument values. We thus find pffiffiffiffiffiffi 2psy for sy o0:3R (6) rv d
0.9 0.8
average/peak
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5
sigmax/R Fig. 1. Ratio of average and peak target thickness as a function of the horizontal ion beam size.
the radius the peak to average ratio rh saturates at unity. It is instructive to approximate Eq. (2) for small sx by pffiffiffi 2s x for sx oR=2. rh (3) R This approximation is shown as the straight line through the origin in Fig. 1 and can be used for estimating the modulation due to the finite horizontal width of the ion beam. Next we discuss the ratio of average to peak density due to finite vertical beam size sy and spacing d between consecutive pellets. This system is equivalent to a sequence of Gaussians expðy2 =2s2y Þ along the vertical y-direction separated by d. If the distance d is large compared to the width the peak value is equal to unity, but if the separation is small the contributions of neighboring Gaussians add up and will increase the peak value. The accurate value for the peak pv is given by pv ¼
1 X
eðndÞ
2
=2s2y
which is shown as the straight line through the origin in Fig. 2. We can summarize our findings that the modulation of the target density and thereby the luminosity and trigger rate can be approximated for beam sizes less than approximately half the target radius R or the pellet distance is given by the product of the respective modulations for the horizontal and vertical dimension pffiffiffi 2 psx sy rh rv for sx oR=2 and sy o0:3R (7) Rd
1
0.8 average/peak
1
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sigmay/d Fig. 2. Ratio of average and peak target thickness as a function of the vertical ion beam size.
1
(4)
n¼1
0.9 0.8 0.7 sigmay/d
and the average value av can be calculated by 1 Z d=2 2 1 X 2 eðyndÞ =2sy dy av ¼ d n¼1 d=2 " ! pffiffiffi pffiffiffi ! X 1 2sy p 1 d pffiffiffi ¼ erf nþ 2 2 d 2s y n¼1 !# 1 d pffiffiffi erf n . 2 2sy
0.6
ð5Þ
Numerically evaluating the remaining sum over n for n ¼1000; . . . ; 1000 allows us to show the ratio of average to peak rv ¼ av =pv as a function of the vertical beam size sy normalized to the pellet spacing in Fig. 2. Again we observe a linear behavior for small beam sizes and saturation once d and sy become comparable. The linear part is easily
0.6 0.5 0.9 0.4 0.3 0.5 0.2 0.1
0.1 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
sigmax/R Fig. 3. Contour plot displaying lines of constant rh rv in steps of 0.1.
ARTICLE IN PRESS V. Ziemann / Nuclear Instruments and Methods in Physics Research A 556 (2006) 641–643
which can be used for quick estimates. If the limits are exceeded, Fig. 3, which shows a contour plot of the modulation as a function of sx =R and sy =d in steps of 0.1, can be used to estimate the modulation. Using parameters d ¼ R ¼ 1 mm and beam sizes of sx ¼ pffiffiffiffiffi sy ¼ eb with b ¼ 1 m and e ¼ 108 m rad we find that we have sx=y ¼ 0:1 mm such that the average to peak value is approximately 1/30, resulting in a very uneven luminosity and trigger rate and potentially causing difficulties for the detector. The discussion whether to deal with this in the detector, the target, or the accelerator is beyond the scope of this report.
643
Discussions with Dag Reistad, TSL, Oliver BoineFrankenheim, GSI, and A. Lehrach, FZJ are gratefully acknowledged. References [1] D. Bettoni, J. Phys. Conf. Ser. 9 (2005) 309. [2] W. Henning (Ed.), An international accelerator facility for beams of ions and antiprotons, Conceptual Design Report, GSI, November 2001. [3] C. Ekstro¨m, et al., Nucl. Instr. and Meth. A 371 (1996) 572. [4] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1970.
Nuclear Instruments and Methods in Physics Research A 556 (2006) 641–643 www.elsevier.com/locate/nima
On pellet target luminosity modulation V. Ziemann The Svedberg Laboratory, Uppsala University, Thunbergsvagen 5a, Box 533, 75121 Uppsala, Sweden Received 11 November 2005; received in revised form 15 November 2005; accepted 17 November 2005 Available online 9 December 2005
Abstract We calculate the variations in the target thickness for a pellet target interacting with a circulating beam and the influence on luminosity or trigger rate. r 2005 Elsevier B.V. All rights reserved. PACS: 29.25Pj; 29.27.a Keywords: Beam dynamics; Beam–target interaction; Pellet target
In order to achieve luminosities in the 1032 =cm2 s range in the PANDA detector [1] proposed for the anti-proton storage ring HESR at the FAIR facility [2] targets with a thickness in excess of 1015 =cm2 are required. Presently pellet targets [3] appear to be the only viable solution to fulfill these requirements but, at the same time, due to the granular character of the pellets, this will cause fluctuations in the overlap between beam and pellets resulting in a temporally varying luminosity and trigger rate for the detector. In such cases we can expect times with high trigger rate and many simultaneous events that may be difficult to discern and times of tranquility. An evenly distributed trigger rate is much preferred. In this report we will parameterize the variability of the luminosity by the ratio of average to peak value of the overlap of target and beam distributions. For the beam we can assume that the combined effects of electron cooling, target and intra-beam scattering will result in a Gaussian distribution at the target location with rms widths sx and sy in the horizontal and vertical direction, respectively. The pellets are spheres of frozen hydrogen or deuterium with diameter of about 30 mm, but for the purpose of this report we assume that they are point-like and that the pellets are injected from above into the scattering chamber. This results in an even distribution of pellets over a circle of Tel.: +46 18 471 3867; fax: +46 18 471 3833.
E-mail address: [email protected]. 0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.11.126
radius R in the plane of the ion beam. The spacing between subsequent pellets is assumed to be a distance d which is given by the product of pellet velocity and repetition rate. Typically, both d and R are on the order of mm. The effect of the mismatch of the horizontal beam size sx and the pellet stream can be described by averaging a Gaussian expðx2 =2s2x Þ over the circle in the horizontal– longitudinal (x2z) plane. We do not normalize the Gaussian, because we want to compare to the peak value, which is unity for the un-normalized Gaussian. Then the ratio between average and peak value rh is given by the integral 2 pffiffiffiffiffiffiffiffiffiffi 3 Z R Z R2 z2 1 2 2 4 pffiffiffiffiffiffiffiffiffiffi ex =2sx dx5dz. rh ¼ (1) pR2 R R2 z2 Noting that the integral is symmetric in x and in z we can substitute the lower integral boundaries by zero and multiply by 4 instead. Moreover, the Gaussian can be integrated leading to an error function [4]. After substituting z ¼ Rs we arrive at pffiffiffi ! Z 1 2sx 2 R pffiffiffiffiffiffiffiffiffiffiffiffi2ffi rh ¼ pffiffiffi erf pffiffiffi 1 s ds. (2) R p 2sx 0 The remaining integral can be easily integrated numerically and in Fig. 1 we show rh as a function of the ratio of beam size to circle radius sx =R and find a linear increase for small sx but once the beam size sx becomes comparable to
ARTICLE IN PRESS V. Ziemann / Nuclear Instruments and Methods in Physics Research A 556 (2006) 641–643
642
deduced from Eq. (5) by observing that for small beam sizes only the term with n ¼ 0 contributes, that the error function is anti-symmetric and approaches unity for large argument values. We thus find pffiffiffiffiffiffi 2psy for sy o0:3R (6) rv d
0.9 0.8
average/peak
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5
sigmax/R Fig. 1. Ratio of average and peak target thickness as a function of the horizontal ion beam size.
the radius the peak to average ratio rh saturates at unity. It is instructive to approximate Eq. (2) for small sx by pffiffiffi 2s x for sx oR=2. rh (3) R This approximation is shown as the straight line through the origin in Fig. 1 and can be used for estimating the modulation due to the finite horizontal width of the ion beam. Next we discuss the ratio of average to peak density due to finite vertical beam size sy and spacing d between consecutive pellets. This system is equivalent to a sequence of Gaussians expðy2 =2s2y Þ along the vertical y-direction separated by d. If the distance d is large compared to the width the peak value is equal to unity, but if the separation is small the contributions of neighboring Gaussians add up and will increase the peak value. The accurate value for the peak pv is given by pv ¼
1 X
eðndÞ
2
=2s2y
which is shown as the straight line through the origin in Fig. 2. We can summarize our findings that the modulation of the target density and thereby the luminosity and trigger rate can be approximated for beam sizes less than approximately half the target radius R or the pellet distance is given by the product of the respective modulations for the horizontal and vertical dimension pffiffiffi 2 psx sy rh rv for sx oR=2 and sy o0:3R (7) Rd
1
0.8 average/peak
1
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sigmay/d Fig. 2. Ratio of average and peak target thickness as a function of the vertical ion beam size.
1
(4)
n¼1
0.9 0.8 0.7 sigmay/d
and the average value av can be calculated by 1 Z d=2 2 1 X 2 eðyndÞ =2sy dy av ¼ d n¼1 d=2 " ! pffiffiffi pffiffiffi ! X 1 2sy p 1 d pffiffiffi ¼ erf nþ 2 2 d 2s y n¼1 !# 1 d pffiffiffi erf n . 2 2sy
0.6
ð5Þ
Numerically evaluating the remaining sum over n for n ¼1000; . . . ; 1000 allows us to show the ratio of average to peak rv ¼ av =pv as a function of the vertical beam size sy normalized to the pellet spacing in Fig. 2. Again we observe a linear behavior for small beam sizes and saturation once d and sy become comparable. The linear part is easily
0.6 0.5 0.9 0.4 0.3 0.5 0.2 0.1
0.1 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
sigmax/R Fig. 3. Contour plot displaying lines of constant rh rv in steps of 0.1.
ARTICLE IN PRESS V. Ziemann / Nuclear Instruments and Methods in Physics Research A 556 (2006) 641–643
which can be used for quick estimates. If the limits are exceeded, Fig. 3, which shows a contour plot of the modulation as a function of sx =R and sy =d in steps of 0.1, can be used to estimate the modulation. Using parameters d ¼ R ¼ 1 mm and beam sizes of sx ¼ pffiffiffiffiffi sy ¼ eb with b ¼ 1 m and e ¼ 108 m rad we find that we have sx=y ¼ 0:1 mm such that the average to peak value is approximately 1/30, resulting in a very uneven luminosity and trigger rate and potentially causing difficulties for the detector. The discussion whether to deal with this in the detector, the target, or the accelerator is beyond the scope of this report.
643
Discussions with Dag Reistad, TSL, Oliver BoineFrankenheim, GSI, and A. Lehrach, FZJ are gratefully acknowledged. References [1] D. Bettoni, J. Phys. Conf. Ser. 9 (2005) 309. [2] W. Henning (Ed.), An international accelerator facility for beams of ions and antiprotons, Conceptual Design Report, GSI, November 2001. [3] C. Ekstro¨m, et al., Nucl. Instr. and Meth. A 371 (1996) 572. [4] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1970.