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Kinematics for the decay of π0 and K0 beams
NUCLEAR
INSTRUMENTS
AND
METHODS
98 (I972) 5 7 3 - 5 7 5 ;
©
NORTH-HOLLAND
PUBLISHING
CO.
K I N E M A T I C S F O R T H E DECAY OF n ° AND K ° BEAMS* V.L. HIGHLAND
Department of Physics, Temple University, Philadelphia, Pa, U.S.A. Received 24 September 1971 Several simple relations appropriate to the detection o f the 2~, decay o f a parallel :r ° or K ° b e a m are derived.
Using the equation for sin 0 equivalent to eq. (1):
1. Introduction The kinematics of the decay of the neutral pion, n o ~ y + ~, are particularly simple and give rise to m a n y elegant relationships. Some of these, such as the opening angle distribution, are well known, and have been exploited by many workers 1-s). The relations are so simple that one can even attempt to compute the detection efficiency of a counter system by analytically integrating over the detector aperturesS). Most of this work has been done for n°'s produced at a localized target. This note presents several simple relationships appropriate to the detection of a parallel beam of n °'s. Such a situation arises in the t-decay in flight of the charged pion, 7z+ ~ n ° + e ÷ + v. The neutral long-lived K meson is also normally studied by its decays in a parallel beam, so that this work would also apply to such currently interesting decays as K L a n + ~ and K L ~ e + + e -. Unfortunately the /t is not sufficiently light for the results to be applicable to KL--.p + +/~-. 2. Results First recall some well-known results. The relation between the photon lab angle, 0, and its cm angle O' is: cos0' =
cos0-fl 1 -- fl cos 0"
(1)
sin 0,, b =
sin 0' ~(1 + /~ COS0')
,
(3)
one can show rigorously that sinOa sinOb = sin2em
1 --
~2 (1
_~2
COS 2 0'
. (4)
I f the second term may be dropped, we have the simple hyperbolic relation: sin 0a sin 0b = sin EO(m,
(5)
and in the small angle approximation 0a 0b = am.2 This approximation is justified if 0' is not too far from 90 °, say 0 ' > 45 ° for ~ = 4. In practice one predominantly detects the symmetrical decays near 0 ' = 90 °, and rarely would one detect a large enough fraction of the cm solid angle to violate this condition. Next consider a pair of symmetrical counters on either side of, and perpendicular to, the line offtight of a n o decaying at z (fig. la). The inside and outside edges of the counters subtend angles 01 and 02 at z. We assume that the counters together cover the full azimuthal angle of 27c; if not, the following results are multiplied by an obvious factor. To detect both photons the system must have
Defining the lab opening angle between the two photons as 2~, then ~ has a minimum value ~m, corresponding to decay with 0' = 90 °, given by sinct m = l / y ,
tan
a m
=
1/fl~.
ol
(2)
fl and ~ have their usual significance as the parameters of the Lorentz transformation from lab to cm. With this preparation we derive a new relationship between the two photon lab angles, 0, and 0b (fig. 1a). * 'Work s u p p o r t e d in part by the U.S. A t o m i c Energy C o m mission.
~r, r2 ]
(a)
Fig. 1. (a) Typical decay of ~0 at z into two p h o t o n s detected by identical symmetrical counters. (b) Special case o f symmetrical decay into angles am illustrating m e a n i n g o f the angle ~.
573
574
v.L.
HIGHLAND
0l < am < 02. In terms of the actual counter dimensions rl, r2 and the decay point, this is exactly flTrl < z < flTr2. We see that if any decays are detected they include at least the symmetrical decay cos 0' = 0 corresponding to ct = ¢~m- Additional asymmetrical decays, cos 0 ' > 0, are detected until the forward photon misses the counter, i.e. until 0a < 01. [We assume for the present that 02 is always large enough to catch the wide angle photon. If this is not so, we will later increase the effective value of 01 using eq. (5)]. The fraction of decays detected, F, is given by the fraction of the cm solid angle detected, cos 0~. Hence F-
cOS0a-fi 1 - fi cos 0,
(6)
We now define the angle 6 = ~m--01 (fig. lb). Note that 6 is small, ranging from 0 to at most am, which is approximately 1/7. Expanding eq. (6) we have through second order terms:
F=76(
1-fi715/2) | -- ]~-~at--~2~2/2
(7)
F(z)
= ( 1 - T r , / z ) [1 + f l ( l - T r l / z ) / 2 ] , 7 v / ( r l r 2 ) > z > 7r,
F(z)
= (l
-z/Tr2)
(lo)
[l+fl(1-z/Tr2)/2], 7r2 > z > 7 v / ( r l
re).
In many cases one might prefer to drop the small second term in the bracket for simplicity. That is the same as using eq. (9) rather than eq. (8). The function F(z) gives the detection efficiency as a function of position. The shape o f the curve is immediately apparent, and is plotted in fig. 3. F(z) goes to zero at 7rl and 7r2 rather than the rigorous flTrl, flTr2. Evidently, the approximations involved shift the whole curve by a factor fl, a few percent discrepancy of the order o f the expected accuracy. F has a peak at z = 7x/(q r2) given by fpe,k = [ 1 - - v / ( r , / r 2 ) ] { l + f l [ 1 - - ½ x / ( r , / r 2 ) ] } ,
(11)
"
This expression is accurate to order ~y 2 or about 1% for 7 = 4. If/~76 ~ 1, we approximate further through second order in fl76: F = 76(1+fl76/2).
dimensions, rl and r2, and the decay point z:
F(~)
(8)
Clearly the approximation requires that 0~ be a substantial fraction of am. F o r experimental reasons this will often be the case. Finally, we can make the further gross approximation F=
7~ = 1 - 7 0 1 .
(9)
Although the approximation is crude, eq. (9) differs from eq. (7) by less than 20% over the whole range of a, and for large 6 it is a better approximation than eq. (8). For m a n y calculations 20% accuracy is certainly sufficient, so this outstandingly simple result should be quite useful. The results in eqs. (7), (8) and (9) are compared in fig. 2. N o t e that the results imply that the forward p h o t o n is nearly uniformly distributed in angle between 0 and :tm. We now must consider the values of 02 for which the results so far are correct. In order for the wide angle photon to be caught, 0 b < 02. Using eq. (5) in the small angle approximation, this requires 0, >am2/02 . So eqs. (7)-(9) apply for ~ 2 / 0 2 < 01 < 0(m. If 02 is not suffi2 ciently large, ~m < 02 < ~m/01, then the effective value of 0, is increased. We summarize these results and re-express them more usefully in terms o f the counter
i .2
i .6
.4
i .8
I,
Fig. 2. Comparison of three expressions for detection efficiency F given by: (a) nearly exact eq. (7), (b) approximate eq. (8), (c) approximate eq. (9). F(Z)
Z
Fig. 3. Detection efficiency F as a function of z for re = 4r] at energy E1 and energy Ee = 2 Et.
KINEMATICS
FOR
THE
DECAY
independent of energy• As the energy increases, the curve broadens but stays the same height. Integrating eq. (10) over the decay region, one finds the detection rate per pion decaying into the mode of interest: R = b/'(2cz)((l-fi)(r2-r,)+r,
I4--~fi3 x
J
x [ r 2 / r I -- x/ ( r l / r 2 ) ] -- (1 +fi) log(r2/rl)
l) , (12)
where r is the beam particle lifetime and b the relevant branching ratio. If the approximation of eq. (9) is made, one finds a much simpler result for R: h
R = ~ [(rz-rl) 2cz
- rl l o g ( r 2 / r O ]
•
(13)
Despite appearances, eqs. (12) and (13) differ by very lktle, typically only 10%. One can carry the analysis further in several directions. We shall restrict ourselves here to two more points. The total opening angle angle 2~ = 0,+0b. Using eq. (5), 2c~ = ( 0 2 + ~2m)/0a, and substituting the angle 6 we find
2~ = 2~m 1 + ½ 1 _ y 6 /
•
(14)
As before, 76 is usually small so that the opening angle has very nearly the constant value 2 ~ m. Finally, suppose the two p h o t o n s hit the detector at ra and rb. Using the above result, one can deduce the point o f decay z to a very g o o d approximation: z = fly(raq-rb)/2.
(15)
OF
7~° A N D
K ° BEAMS
575
3. C o n c l u s i o n
We have found approximate analytic expressions for the detection efficiency for a n ° beam as a function of z. By integrating that expression along the beam one finds a closed form for the counting rate of a given counter configuration. A hyperbolic relation between the decay p h o t o n angles and the near constancy of the opening angle have also been established. These results are offered, in part, as a reminder o f the powers of mathematics in a world rife with Monte Carlo programs. Of course for a final result one will do a Monte Carlo calculation which includes all the complications of the actual equipment. These equations provide the always necessary check on the Monte Carlo results. Furthermore, they provide quick, easy numbers useful in the design o f an experiment. The greatest value o f analytic formulas is that they provide a direct insight into the dependence o f the rate on energy, dimensions, etc., and so provide estimates of accuracy that are hard to obtain from Monte Carlo work. Finally, results such as eq. (15) provide a simple algorithm useful for on-line monitoring o f an experiment by a small computer. By quickly reconstructing the decay point z, one can check whether the observed distribution has the expected shape F(z). References
1) W. K. H. Panofsky, J. Steinberger and J. Steller, Phys. Rev. 86 (1952) 180. 2) W. Chinowgky and J. Steinberger, Phys. Rev. 93 (1953) 586. a) j.D. Prentice, E. H. Bellamy and W. S. C. Williams, Proc. Phys. Soc. (London) 74 (1959) 124. 4) R. M. Sternheimer, Phys. Rev. 99 (1955) 277. 5) V. L. Highland and J.W. DeWire, Phys. Rev. 132 (1963), 1293; for a detailed discussion of ~0 kinematics, see V.L. Highland, Ph. D. Thesis (Cornell University, 1963) unpublished.
INSTRUMENTS
AND
METHODS
98 (I972) 5 7 3 - 5 7 5 ;
©
NORTH-HOLLAND
PUBLISHING
CO.
K I N E M A T I C S F O R T H E DECAY OF n ° AND K ° BEAMS* V.L. HIGHLAND
Department of Physics, Temple University, Philadelphia, Pa, U.S.A. Received 24 September 1971 Several simple relations appropriate to the detection o f the 2~, decay o f a parallel :r ° or K ° b e a m are derived.
Using the equation for sin 0 equivalent to eq. (1):
1. Introduction The kinematics of the decay of the neutral pion, n o ~ y + ~, are particularly simple and give rise to m a n y elegant relationships. Some of these, such as the opening angle distribution, are well known, and have been exploited by many workers 1-s). The relations are so simple that one can even attempt to compute the detection efficiency of a counter system by analytically integrating over the detector aperturesS). Most of this work has been done for n°'s produced at a localized target. This note presents several simple relationships appropriate to the detection of a parallel beam of n °'s. Such a situation arises in the t-decay in flight of the charged pion, 7z+ ~ n ° + e ÷ + v. The neutral long-lived K meson is also normally studied by its decays in a parallel beam, so that this work would also apply to such currently interesting decays as K L a n + ~ and K L ~ e + + e -. Unfortunately the /t is not sufficiently light for the results to be applicable to KL--.p + +/~-. 2. Results First recall some well-known results. The relation between the photon lab angle, 0, and its cm angle O' is: cos0' =
cos0-fl 1 -- fl cos 0"
(1)
sin 0,, b =
sin 0' ~(1 + /~ COS0')
,
(3)
one can show rigorously that sinOa sinOb = sin2em
1 --
~2 (1
_~2
COS 2 0'
. (4)
I f the second term may be dropped, we have the simple hyperbolic relation: sin 0a sin 0b = sin EO(m,
(5)
and in the small angle approximation 0a 0b = am.2 This approximation is justified if 0' is not too far from 90 °, say 0 ' > 45 ° for ~ = 4. In practice one predominantly detects the symmetrical decays near 0 ' = 90 °, and rarely would one detect a large enough fraction of the cm solid angle to violate this condition. Next consider a pair of symmetrical counters on either side of, and perpendicular to, the line offtight of a n o decaying at z (fig. la). The inside and outside edges of the counters subtend angles 01 and 02 at z. We assume that the counters together cover the full azimuthal angle of 27c; if not, the following results are multiplied by an obvious factor. To detect both photons the system must have
Defining the lab opening angle between the two photons as 2~, then ~ has a minimum value ~m, corresponding to decay with 0' = 90 °, given by sinct m = l / y ,
tan
a m
=
1/fl~.
ol
(2)
fl and ~ have their usual significance as the parameters of the Lorentz transformation from lab to cm. With this preparation we derive a new relationship between the two photon lab angles, 0, and 0b (fig. 1a). * 'Work s u p p o r t e d in part by the U.S. A t o m i c Energy C o m mission.
~r, r2 ]
(a)
Fig. 1. (a) Typical decay of ~0 at z into two p h o t o n s detected by identical symmetrical counters. (b) Special case o f symmetrical decay into angles am illustrating m e a n i n g o f the angle ~.
573
574
v.L.
HIGHLAND
0l < am < 02. In terms of the actual counter dimensions rl, r2 and the decay point, this is exactly flTrl < z < flTr2. We see that if any decays are detected they include at least the symmetrical decay cos 0' = 0 corresponding to ct = ¢~m- Additional asymmetrical decays, cos 0 ' > 0, are detected until the forward photon misses the counter, i.e. until 0a < 01. [We assume for the present that 02 is always large enough to catch the wide angle photon. If this is not so, we will later increase the effective value of 01 using eq. (5)]. The fraction of decays detected, F, is given by the fraction of the cm solid angle detected, cos 0~. Hence F-
cOS0a-fi 1 - fi cos 0,
(6)
We now define the angle 6 = ~m--01 (fig. lb). Note that 6 is small, ranging from 0 to at most am, which is approximately 1/7. Expanding eq. (6) we have through second order terms:
F=76(
1-fi715/2) | -- ]~-~at--~2~2/2
(7)
F(z)
= ( 1 - T r , / z ) [1 + f l ( l - T r l / z ) / 2 ] , 7 v / ( r l r 2 ) > z > 7r,
F(z)
= (l
-z/Tr2)
(lo)
[l+fl(1-z/Tr2)/2], 7r2 > z > 7 v / ( r l
re).
In many cases one might prefer to drop the small second term in the bracket for simplicity. That is the same as using eq. (9) rather than eq. (8). The function F(z) gives the detection efficiency as a function of position. The shape o f the curve is immediately apparent, and is plotted in fig. 3. F(z) goes to zero at 7rl and 7r2 rather than the rigorous flTrl, flTr2. Evidently, the approximations involved shift the whole curve by a factor fl, a few percent discrepancy of the order o f the expected accuracy. F has a peak at z = 7x/(q r2) given by fpe,k = [ 1 - - v / ( r , / r 2 ) ] { l + f l [ 1 - - ½ x / ( r , / r 2 ) ] } ,
(11)
"
This expression is accurate to order ~y 2 or about 1% for 7 = 4. If/~76 ~ 1, we approximate further through second order in fl76: F = 76(1+fl76/2).
dimensions, rl and r2, and the decay point z:
F(~)
(8)
Clearly the approximation requires that 0~ be a substantial fraction of am. F o r experimental reasons this will often be the case. Finally, we can make the further gross approximation F=
7~ = 1 - 7 0 1 .
(9)
Although the approximation is crude, eq. (9) differs from eq. (7) by less than 20% over the whole range of a, and for large 6 it is a better approximation than eq. (8). For m a n y calculations 20% accuracy is certainly sufficient, so this outstandingly simple result should be quite useful. The results in eqs. (7), (8) and (9) are compared in fig. 2. N o t e that the results imply that the forward p h o t o n is nearly uniformly distributed in angle between 0 and :tm. We now must consider the values of 02 for which the results so far are correct. In order for the wide angle photon to be caught, 0 b < 02. Using eq. (5) in the small angle approximation, this requires 0, >am2/02 . So eqs. (7)-(9) apply for ~ 2 / 0 2 < 01 < 0(m. If 02 is not suffi2 ciently large, ~m < 02 < ~m/01, then the effective value of 0, is increased. We summarize these results and re-express them more usefully in terms o f the counter
i .2
i .6
.4
i .8
I,
Fig. 2. Comparison of three expressions for detection efficiency F given by: (a) nearly exact eq. (7), (b) approximate eq. (8), (c) approximate eq. (9). F(Z)
Z
Fig. 3. Detection efficiency F as a function of z for re = 4r] at energy E1 and energy Ee = 2 Et.
KINEMATICS
FOR
THE
DECAY
independent of energy• As the energy increases, the curve broadens but stays the same height. Integrating eq. (10) over the decay region, one finds the detection rate per pion decaying into the mode of interest: R = b/'(2cz)((l-fi)(r2-r,)+r,
I4--~fi3 x
J
x [ r 2 / r I -- x/ ( r l / r 2 ) ] -- (1 +fi) log(r2/rl)
l) , (12)
where r is the beam particle lifetime and b the relevant branching ratio. If the approximation of eq. (9) is made, one finds a much simpler result for R: h
R = ~ [(rz-rl) 2cz
- rl l o g ( r 2 / r O ]
•
(13)
Despite appearances, eqs. (12) and (13) differ by very lktle, typically only 10%. One can carry the analysis further in several directions. We shall restrict ourselves here to two more points. The total opening angle angle 2~ = 0,+0b. Using eq. (5), 2c~ = ( 0 2 + ~2m)/0a, and substituting the angle 6 we find
2~ = 2~m 1 + ½ 1 _ y 6 /
•
(14)
As before, 76 is usually small so that the opening angle has very nearly the constant value 2 ~ m. Finally, suppose the two p h o t o n s hit the detector at ra and rb. Using the above result, one can deduce the point o f decay z to a very g o o d approximation: z = fly(raq-rb)/2.
(15)
OF
7~° A N D
K ° BEAMS
575
3. C o n c l u s i o n
We have found approximate analytic expressions for the detection efficiency for a n ° beam as a function of z. By integrating that expression along the beam one finds a closed form for the counting rate of a given counter configuration. A hyperbolic relation between the decay p h o t o n angles and the near constancy of the opening angle have also been established. These results are offered, in part, as a reminder o f the powers of mathematics in a world rife with Monte Carlo programs. Of course for a final result one will do a Monte Carlo calculation which includes all the complications of the actual equipment. These equations provide the always necessary check on the Monte Carlo results. Furthermore, they provide quick, easy numbers useful in the design o f an experiment. The greatest value o f analytic formulas is that they provide a direct insight into the dependence o f the rate on energy, dimensions, etc., and so provide estimates of accuracy that are hard to obtain from Monte Carlo work. Finally, results such as eq. (15) provide a simple algorithm useful for on-line monitoring o f an experiment by a small computer. By quickly reconstructing the decay point z, one can check whether the observed distribution has the expected shape F(z). References
1) W. K. H. Panofsky, J. Steinberger and J. Steller, Phys. Rev. 86 (1952) 180. 2) W. Chinowgky and J. Steinberger, Phys. Rev. 93 (1953) 586. a) j.D. Prentice, E. H. Bellamy and W. S. C. Williams, Proc. Phys. Soc. (London) 74 (1959) 124. 4) R. M. Sternheimer, Phys. Rev. 99 (1955) 277. 5) V. L. Highland and J.W. DeWire, Phys. Rev. 132 (1963), 1293; for a detailed discussion of ~0 kinematics, see V.L. Highland, Ph. D. Thesis (Cornell University, 1963) unpublished.