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The estimation of asymmetries from incomplete azimuthal angular distributions
NUCLEAR
INSTRUMENTS
AND METHODS
164 ( 1 9 7 9 ) 1 7 5 - 1 8 2 ;
(~) N O R T H - H O L L A N D
PUBLISHING
CO.
THE ESTIMATION OF ASYMMETRIES F R O M INCOMPLETE AZIMUTHAL ANGULAR DISTRIBUTIONS J.A. EDGINGTON
Department of Physics, Queen Mary College, Mile End Road, London E1 4NS, England Received 2 November 1978 The problem of estimating the parameters of an angular distribution in scattering experiments is discussed. The context is an experimental situation that artificially restricts the available angular range in a manner that differs from one scattering event to the next. The formal solution, which is non-linear in the parameters, is presented, and its implementation in a particular case is described.
1. Statement of the problem The central problem in scattering experiments is that of estimating, from a measured angular distribution, the parameters that govern that distribution. The case considered here is that of a detector whose angular acceptance differs from, one scattering event to the next, so that each event is a sample from a unique probability distribution. Though the problem is no doubt commonly met with, its practical solution does not seem to have been discussed. In order to set the problem in perspective, it is necessary to describe briefly the detector and its use. The case considered is that of a polarimeter used to measure the polarisation of a beam of protons. This polarimeter has been used in experiments on nucleon-nucleon scattering carded out at TRIUMF, Vancouver'). The construction of the instrument, and the calibration of its analysing power, has been described previously2). As the problem discussed here is of a very general nature and does not depend on operational details, only a brief outline of the experiments will be given. The polarimeter comprises a thin plate of carbon sandwiched between arrays of scintillation and multi-wire proportional counters, as shown in fig. 1. Protons enter from the front, having been scattered from a liquid hydrogen target by a beam of neutrons. The polarisation vector of the neutron beam can be directed by means of magnets either parallel or perpendicular to its direction of travel. Suppose e is the polarisation vector of the recoil protons. Then the various polarisation transfer parameters in neutron-proton scattering can be found by measuring the components of tr orthogonal to the proton's momentum; in the general case these components ax, try are both non-zero though specific experimental situations may cause one or other to be very small. Consider a proton which scatters in the carbon at angle (0, ~), the azimuthal angle ¢ being measured from the horizontal direction. We might expect the probability distribution for ~ to be of the form P(~b) ,,~ 1 + A(O) (try cos~b + trx sint~),
(1)
where A (0), the analysing power of the scattering reaction, has been determined previously. However, this expression is a poor approximation to the azimuthal distribution actually observed, since the likelihood of detecting scattered particles may vary with azimuth, and this efficiency variation must be convoluted with P(O). The procedure for estimating a is standard when the detectors used are small, spanning only a small range of ~. [For example, suppose that two small counters were placed at angles of approximately ¢ = 0 and ~ = 7r, and that their counting efficiencies differed. The effects of this difference, and of small angular misalignments, could both be overcome by reversing the direction of polarisation of the neutron beam. If the number of counts were No+-, N~+- where + , - indicate the two polarisation directions, then the best estimator, try, of try, correct to second order in the angular acceptance A¢ would be 3) ar
1 N +- NA (0) N + + N -
where N ~ --=~/N~ N~. However, our polarimeter takes data at many azimuthal angles concurrently, and the elimination of
176
J.A. EDGINGTON ,"
i"
|
pROT., ,0NSNC.OEN T 1\ ON POLARIMETER
FRONT CARBON REAR MWPCs P L A T E MWPCs
Fig. 1. A sketch of the polarimeter, showing also the polarization vectors of an incident proton.
B
I CARBON PLATE
REAR DETECTOR
Fig. 2. The two classes of events, distinguished by the intersection of their scattering trajectories with the rear detector. ~1= 0 is to the left.
instrumental bias, without loss of statistical precision, is less simple. Consider first a possible slow variation in efficiency as a function of ¢, due to small variations in the performance of the multiwire chambers. Since their sense wires run vertically and horizontally, one expects such variations to show certain simple symmetries, and we therefore choose to represent this efficiency variation by a trigonometric series truncated at a low order; we multiply the distribution of eq. (1) by the function 1 + e l cos~b + r h sinq~ +e2 cos2~b +r/2 sin2~b + e4 cos4~b. The coefficients s~, rh, e2, rh, e4, represent respectively a variation in efficiency from left to right; up to down; horizontal to vertical; one diagonal to the other; and side to comer. We anticipate (and find) that these coefficients will be small, of order 0.01 or less, though this is not essential to the argument. The second effect is due to the finite area of the back of the detector, which results in regions of ¢, for a given 0, for which the efficiency drops to zero. In fig. 2 we illustrate this by constructing cones of semi-angle 0, with axis along the incident trajectory and vertex at the carbon plate, for two events A and B having equal values of 0. The heavy lines show the scatter trajectory actually observed, while the ellipses show the intercept of the possible trajectories with the back of the detector (the rear scintillator). For events like A, the complete range is in principle accessible, whereas for those like B it is cut off by the detector edges. In other words, only for class A events does the intercepting ellipse lie wholly within the boundary, and hence the acceptance, of the back of the detector. Thus each class A event may be considered as sampling the same azimuthal probability distribution, whereas each event of class B is a sample from a separate, unique distribution, whose normalisation differs from event to event as the accessible fraction of the complete 2rt azimuth varies. Because protons incident on our polarimeter are not confined to the centre of the carbon but can strike it anywhere, we would disqualify many events from analysis if we restricted ourselves to class A events alone. We therefore had to develop a procedure for dealing with class B events also. The formal solution for the ith event is to multiply P(~) by a function g~ which is zero for inaccessible regions of azimuth, and 1 elsewhere. In the next section we develop this and display the results for the estimators of the various parameters and their statistical errors; in order to clarify the somewhat opaque algebra, we give both the general formulae applicable to class B events, and their simplification for class A events. The final result is a set of non-linear equations, and in the third section we describe an iterative procedure we have used for solving these, and present some illustrative results. 2. The formal solution As mentioned, the ith event is taken to be a sample from an azimuthal probability distribution of the form gi 1-1 + A(O) (try cos~b + trx sin ~b)] x (1 + e1 cost# + r/1 sin~b + e2 cos2~ + r/2 cos2~b + e4 cos4~),(2) P'@) = V, where the quantity Cj is present to ensure the correct normalisation. Note that the intercept of a cone with a rectangular detector allows up to eight values of azimuthal angle (say ~i, J = 1, ..o, 8) which divide the
'
ESTIMATION
OF ASYMMETRIES
177
azimuthal range into a maximum of four regions with g~= 1, and an equal number (or none for class A events) with gj = 0. We expand P,.(4') as a trigonometric series Pi(4') = ~Oi (ao + al cos4, + a2 cos24, + a3 cos34, + a, cos44, + as cos54' + bi sin4, + b2 sin24, + + b3 sin 34, + b5 sin 54,),
(3)
whose coefficients are: ao = 1 + ½A(0) (0-y~l + 0-xr/1), aa = 51 + ½A(O) (20-y + 0-y~2+ 0-xr/2), a 2 = e2 +
½A(0) (0-y~1
bl = ~h + ½A(O) (20-~ - 0-x~2 + 0-yr/2), b2 = 1~2 "~ ½A(O) (0-y~1 -t- 0-x~1),
-- 0-x~1),
(4) a 3 = ½A(O) (o'yg 2 + 0-y~4 - 0-xr/2),
b 3 = ½A(O) (0",1'/2 + 0-xE2 - 0-x~4),
a s = ½A(O) 0-,e,~,
bs = ½A(O) 0-xe,,.
By requiring o" P,(4')d4' = 1, we find the normalisation constant C~: C, = ao {4'}i + al {sin4'}, +-~-a2 {sin24'}, + ~a3 {sin 34,}, +--~ a4 {sin44,}, + -5-a5{sin 54,}' _ bl {cos4,}, b2 Icos24,},cos34,},os54,}, 2 Here we have introduced the notation { }~ to stand for the definite integral {f(4')}~ -=
~o'~
(5)
9,d [f(4,)] •
For example, {4,}, =
o
o, d4, -- 4 , 1 - 4 , ~ + 4 , ~ - 4 , , + 4 , ~ - 4 , ~ + 4 , ~ - 4 , ~ .
For class A events, eq. (5) reduces to G = 2~[1 + 3.4(0) (0-,~ + 0 - ~ ) ] , since only {0}~ is non-zero. One can now write down a set of expectation values for the ith probability distribution, for example:
{
A(O) (A(O)cos4,), = 2a o {sin4,}, + al 4' + sin24'/2 j+a2
+
{ ~
+ sin54'/
-b3[
/ c°s24' 2
+ c°s44'/
4 J,-b5
The general form of these expectation values is:
{
sin4, + -+" aT3- sJ i, n 3 4 ' / + sin64'/
{~_.~
+
C°S64'/
6 J,"
+ sin44'/4j , + {cos 4,
cos34,~
178
J. A. EDGINGTON
Ci
c
c
c
c
c
c
c
c
c
A(O) (A(O) cos n~b)l = aoXo. + al xln -.[-a2X2n + a3X3n + a4 x~. + asxs. + bl Yln + b2Y2n + b3Yan + bsy5., (6a) 2 C~ ~ ~ ~ s A(O) (A(O) sin nq~)~ = aoXo. + a l x l , , + a2x2, , + a3x3. + a4x~4. + asx~, ' + bly~, ' + b2Y~2,, + b3Y~3,, + bsys.,(6b)
where _ /sin(m_~-n)q~ Xm.
[
m--n
sin(m+n)q~ 1 +
-m+ n
Ji'
_ {cos(m-n)~b -t c°s(m +n)~b/ , m-n
Ymn ="
{ xS" -
[
j~
c o s ( m - n ) ~ b + cos(m+n)~b / m- n m ~}--n "Ji'
-
_ /sin(m-n)~b Ymn
~
(7)
sin(m_+n)~b /
-m ---n
m+n
ji'
and by convention we take sin P~b P cos P~b I P
----0,
for
P=0.
Eqs. (6a) and (6b) are the general forms; for class A events they simplify drastically to 2 A(O) (A(O) cosn~b) = a.,
2
A(O)
(A(0) sinn~b) = b . ,
which are included in the general forms as the leading terms. It must now be stressed that, since the inefficiency coefficients el, rh, e2, rh, e4 are small, we have to a first approximation a0 ~ 1, al ~A(0)ay, bl ~A(0)ax and all other coefficients are small. Likewise, for those class B events which lose only a small portion of the total azimuthal range, x ~ , ~ y ~ , , ~ 2n and the other definite integrals are small. We therefore re-write eqs. (6a) and (6b), projecting out the large parameters and the large definite integrals; we also divide through by A(O). The result is
2Ci
A2(0 ) (A(O) cos~b)
c
= exP~ +tryQ~ + trxR ~ + $1,
2 Ci
A2(O) (A(O) c0s24,) = 82P~ + o~Q~ + oxR~ + S2, 2~i
¢
e
-~2-'~) (A(O) cos4~b) = 84P~ + tryQ4 + a~R~, + $4, 2 Ci
A2(0 ) (A(O) sin~b) 2 C~
~
= qxPx + % Q ~ + axR~ + Sx, ~
~
(A(O) sin2q~) = r]2P 2 + ayQ2 + axR~2 + $2,
where P.~ = x~./,4 (0) and p s = y S / A (0) exactly; Q~ and R s contain x.C. and y s respectively as the leading
(8)
ESTIMATION OF ASYMMETRIES
179
terms; and all the terms in R e, S~, Q~, S~ are products of small parameters and/or small definite integrals. For class A events, eqs. (8) reduce further to 2 (A(0)cosqb > 242(0)
el
(~.)
-- A ( O ) -I- try
8.._.L_2 tryst cos2qg> - A(O) 4 2
A~
trot/2
1 +
+ ----~, trxrll 2 '
2 ~4 A2(0) - A(O)' 2 A2(0 ) (A(O) sinq~> 2
th -
(8a) -~
A(O) +
2
+
t/2 tr~i = A(O--'~+ ~ +
(1 :---~ try\2]'
tr~ 2
"
If, furthermore, the inefficiency coefficients are taken to be zero, we arrive at the well-known expressions (see, for example, ref. 4): 2
A2(O) 2 ~-~
= tr,,
= trx,
whence the (unbiassed) estimators of ay, ax are: ,
2
1
N
tr, = ~2-'~ ~ , ~ t a(0) cos~bi, ,
2
1
trx = " ~ - ~
~ A(O) i=l
sin~b~,
with variances given by: var ( : )
= ~
1[ A-~
var (a*) = .~
_ (:)2
1,
- (a*) 2 .
Eqs. (8) may now be re-written so as to yield similar, but more complex, expressions for the general case. For example, from the first of these equations we deduce that
:,'
:
with a variance var (tr*) = ~
_ (try,)2 ,
(9)
and also -
__2,
,el + tr.a
+
180
i.A. EDG1NGTON
with variance 1[var (~*) =
2
.27
J
Notice that the correction terms to the estimators, which are in principle exactly calculable, do not contribute to the statistical uncertainties. These eqs. (9) and the similar ones for O~x, r/~*, e~*, r/~ and e~* are the formal solution to the problem. Because the normalisation constants C,, and the coefficients pg,s, Q~,S, R~,' and S~,' themselves contain the parameters ~rx, try, el, r/l, e2, r/2, e4 which we are seeking, these equations are non-linear. It is this non-linearity which is the essential difference between eqs. (8), representing the general solution, and eqs. (8a) which refer to the special case in which the whole of the azimuth is, in principle, accessible. If the non-linearity is neglected, and the equations solved directly by taking the C,, etc. to be constants, then the resulting estimators are biassed. To obtain a set of unbiassed estimators, we have solved eqs. (9) iteratively.
3. Implementation of the solution We reconstruct the kinematics of each detected event off-line so that the scattering angles (0, 0), and the intercept azimuth angles e j , j = 1,..., 8, are known• We then calculate for each event the definite integrals of eq. (7). In the first approximation we assume that all the inefficiency coefficients and polarisation components are zero, with the consequence that all the coefficients given by eq. (4) are zero, except for a0 which is 1 • The normalisation constant C~ and the quantities P,C , S , Q,C , S , R ,C,S and S,C,S of eq. (8) are then calculated. Finally we form the sums necessary to evaluate the estimators of eq. (9). For class A events this is an extremely good approximation since for such events C~ = 2 zra0 = 2n[1 + ½A(O)(a~e~ + ax rh)] = 2rr(1 + di). Inserting typical values [A(0) -~ 0.5, ax ~ % ~. 0.5, el N rh ~ 0.02] we find an upper limit for the correction term 6 of order 0.005. A similar argument holds for the quantitiesP~ ,s etc., and we find that the solutions for class A events are inaccurate by a few percent at most. They therefore provide an excellent starting point for a first iteration in which we recalculate all the coefficients of eq. (4) and then repeat the calculations and summations event by event as before. Convergence is rapid and a second iteration is not usually needed. In practice the instrumental biases are so small compared with their statistical errors that the values for the inefficiency coefficients have to be found by averaging over many data runs taken during a period of uniform chamber operation. In table 1 we list average values of the inefficiency coefficients obtained in this way for a 5-day period of steady operation. The measurement here was of the polarization transfer parameter Dt, and so trx was expected to be zero. The results obtained after the first and second iterations were found to agree almost exactly, and show that not only trx, but all the instrumental asymmetry coefficients, are indeed consistent with zero. The analysis described here is capable of resolving instrumental biases inherent in the detector operation, but it can neither resolve nor eliminate any biases introduced at the data reduction stages. One such bias appears for those protons which scatter close to the edge of the carbon plate. There is always some uncertainty in the calculation of trajectories (due to multiple scattering, and to digitisation effects in the multiwire chambers) but, except near the edges of the acceptance region, there is an approximate equality between the numbers of trajectories assigned incorrectly to one side and to the other of the 'correct' position. At the edges an incorrect assignment to a position outside the detector is not allowed, and this can have a dramatic effect on the apparent asymmetry. Consider, for example, the range of azimuth available to two groups of particles, those scattering from just to the left of centre of the carbon plate and those scattering from the extreme left hand edge. Suppose the actual azimuthal distribution to be flat. The group of particles striking near the centre of the carbon plate includes, besides those with correctly assigned trajectories (C), others with incident trajectories assigned incorrectly either to the left (L) or right (R) (fig. 3). All these particles sum to yield the correct azimuthal distribution. A particle from amongst the group striking near the le•hand edge, however, cannot possibly be assigned a trajectory even further to the left, so such events are absent. Since they occupy a smaller azimuthal range than the correctly assigned events, or those further to the right, the effect is a 'hole' around ~ = n, corresponding to an artificial
E S T I M A T I O N OF A S Y M M E T R I E S i
181
TABLE 1
i
C
Values of trx and of the inefficiency coefficients during a 5-day period of chamber operation. The null hypothesis that the true values of all these coefficients are zero leads to a X2 of 5.09 on 6 d.f. I
I
0
"IT
ax = - 0.0045 _+0.0122
2"Tr
Fig. 3. A hypothetical flat azimuthal distribution, showing the portions accessible to scattered particles whose incident trajectory has been assigned correctly (C) and also, to the left (L), and to the right (R) of its actual position.
eI r/1 ~2 r/2 e4
= 0.0080_+0.0077 = - 0.0064___ 0.0077 = 0.0072_+0.0077 = -0.0117_+0.0077 = 0.0001 _+0.0077
positive value of the computed coefficient of cos ¢. Conversely the effect at the right hand edge will simulate a negative value of the coefficient. A similar effect will modify the sin ~ coefficient at top and bottom edges. This bias seems inevitable at the edges of detectors with incomplete angular acceptance and finite angular resolution. We expect, and find, that it is small or absent for interactions occurring near the centre of the carbon. We analysed separately the data from different neutron-proton scattering angles corresponding to different vertical strips of the carbon plate, and the polarimeter was moved round so that at three different settings events at a given n - p scattering angle fell to the left, in the centre and to the right of the carbon. In table 2 we show a typical set of asymmetry coefficients got in this way, both for class A events alone and for all events. The effect is clearly present in the latter group but not in the former. It must be stressed that this bias in no way affects the results of the final analysis. For example, in our case, data were taken in alternate runs with the neutron beam polarisation reversed, and the quantity of physical interest was obtained from the difference of asymmetry coefficients in the two cases. The artificial bias exactly cancels. The data of table 2 were taken in a measurement of the polarisation transfer parameter D r , in which the polarisation component tr~ is given by try=
-- Pnp + Dt (O'n) 1--P.p(an) '
P,p b e i n g t h e p o l a r i s a t i o n p a r a m e t e r in n - p s c a t t e r i n g a n d ( a , ) t h e p o l a r i s a t i o n o f t h e n e u t r o n b e a m . T h u s t h e d i f f e r e n c e b e t w e e n try m e a s u r e d in t h e t w o s p i n o r i e n t a t i o n s is Arty=
2 (a.)
(O t - P ~ p )
1 - p2p (tr,)2
,
w h e n c e D t c a n b e c a l c u l a t e d . V a l u e s o f D t a r e l i s t e d in t a b l e 2 a n d g o o d a g r e e m e n t is s e e n b e t w e e n t h o s e f o u n d f r o m class A e v e n t s a l o n e , a n d t h o s e g o t f r o m all e v e n t s . I n t h i s p a r t i c u l a r set o f d a t a , t h e g a i n in TABLE 2 Values of try under various experimental conditions, showing the bias occurring near the edges of the polarimeter. The derived values of Dt are also listed, and are seen not to depend on position. Position of scatter
Left hand edge Centre Right hand edge
Weighted averages
Event class
Values of try: Spin normal Spin raversed
Derived values of D t
A A+B A A+B A A+B
0.465_+0.109 0.831 _ + 0 . 0 8 8 0.597_+0.108 0.563_+0.094 0.430 _+0.084 0.050_+0.073
-0.280_+0.118 - 0.267_+ 0.095 - 0.200_+ 0.089 - 0.200_+ 0.078 - 0.197 _+0.096 - 0.174_+0.084
A A+B
0.011_+0.103 0.393_+0.083 0.165_+0.087 0.132_+ 0.075 0.084 _+0.088 - 0.266_+ 0.076
-0.218_+0.057 -0.209_+0.049
182
J. A. EDGINGTON
statistical precision from using all events, rather than class A events only, is quite significant. Overall we found that class B events comprised some 3596 of all events, leading to an improvement in the statistical errors of about 18 96 when these events were included.
4. Summary The problem discussed in this paper arose during the analysis of data taken with a large polarimeter in which it was not possible to constrain events to a uniform central region of complete angular acceptance. Rejection of all events associated with incomplete azimuthal acceptance would have meant a significant loss in statistical precision. It was therefore necessary to devise a method for incorporating into the analysis both the usual small instrumental biases, and also the gross effect of incomplete acceptance. Eq. (9) represents the formal solution to the problem, using the method of point (moment) estimation. These non-linear equations were solved iteratively, and we had the advantage that those events which did have a complete angular acceptance region provided an approximate solution which lead to rapid convergence. Certain systematic biases, which seem unavoidable, remain for scatters occurring close to the edges of the carbon analyser plate, but these are entirely eliminated when the physical quantities of interest are calculated from the diff6rence between data taken with reversed beam polarisations. The programme ASYM, written to perform this calculation, is available from the author. I thank all my colleagues in the BASQUE group at TRIUMF for helpful discussions. Throughout this paper the first person plural refers to them. References 1) j. A. Edgington, Proc. Vancouver Conf. on Nucleon--nucleon interactions (American Institute of Physics, N.Y., 1978) p. 19. 2) G. Waters, I.M. Blair, G.A. Ludgate, N.M. Stewart, C. Amsler, R.C. Brown, D.V. Bugg, J.A. Edgington, C.J. Oram, K. Shakarchi, A. S. Clough, D. A. Axen, S. Jaccard and J. Vavra, Nucl. Instr. and Meth. 153 (1978) 401. 3) G. G. Ohlsen, Proc. Fourth Int. Symp. on Polarization phenomena in nuclear reactions, ed W. Grtiebler and V. Konig (Birkhatiser Verlag, Basel, 1976) p. 287. 4) F.T. Solmitz, Ann. Rev. Nucl. Sci. 14 (1964) 375.
INSTRUMENTS
AND METHODS
164 ( 1 9 7 9 ) 1 7 5 - 1 8 2 ;
(~) N O R T H - H O L L A N D
PUBLISHING
CO.
THE ESTIMATION OF ASYMMETRIES F R O M INCOMPLETE AZIMUTHAL ANGULAR DISTRIBUTIONS J.A. EDGINGTON
Department of Physics, Queen Mary College, Mile End Road, London E1 4NS, England Received 2 November 1978 The problem of estimating the parameters of an angular distribution in scattering experiments is discussed. The context is an experimental situation that artificially restricts the available angular range in a manner that differs from one scattering event to the next. The formal solution, which is non-linear in the parameters, is presented, and its implementation in a particular case is described.
1. Statement of the problem The central problem in scattering experiments is that of estimating, from a measured angular distribution, the parameters that govern that distribution. The case considered here is that of a detector whose angular acceptance differs from, one scattering event to the next, so that each event is a sample from a unique probability distribution. Though the problem is no doubt commonly met with, its practical solution does not seem to have been discussed. In order to set the problem in perspective, it is necessary to describe briefly the detector and its use. The case considered is that of a polarimeter used to measure the polarisation of a beam of protons. This polarimeter has been used in experiments on nucleon-nucleon scattering carded out at TRIUMF, Vancouver'). The construction of the instrument, and the calibration of its analysing power, has been described previously2). As the problem discussed here is of a very general nature and does not depend on operational details, only a brief outline of the experiments will be given. The polarimeter comprises a thin plate of carbon sandwiched between arrays of scintillation and multi-wire proportional counters, as shown in fig. 1. Protons enter from the front, having been scattered from a liquid hydrogen target by a beam of neutrons. The polarisation vector of the neutron beam can be directed by means of magnets either parallel or perpendicular to its direction of travel. Suppose e is the polarisation vector of the recoil protons. Then the various polarisation transfer parameters in neutron-proton scattering can be found by measuring the components of tr orthogonal to the proton's momentum; in the general case these components ax, try are both non-zero though specific experimental situations may cause one or other to be very small. Consider a proton which scatters in the carbon at angle (0, ~), the azimuthal angle ¢ being measured from the horizontal direction. We might expect the probability distribution for ~ to be of the form P(~b) ,,~ 1 + A(O) (try cos~b + trx sint~),
(1)
where A (0), the analysing power of the scattering reaction, has been determined previously. However, this expression is a poor approximation to the azimuthal distribution actually observed, since the likelihood of detecting scattered particles may vary with azimuth, and this efficiency variation must be convoluted with P(O). The procedure for estimating a is standard when the detectors used are small, spanning only a small range of ~. [For example, suppose that two small counters were placed at angles of approximately ¢ = 0 and ~ = 7r, and that their counting efficiencies differed. The effects of this difference, and of small angular misalignments, could both be overcome by reversing the direction of polarisation of the neutron beam. If the number of counts were No+-, N~+- where + , - indicate the two polarisation directions, then the best estimator, try, of try, correct to second order in the angular acceptance A¢ would be 3) ar
1 N +- NA (0) N + + N -
where N ~ --=~/N~ N~. However, our polarimeter takes data at many azimuthal angles concurrently, and the elimination of
176
J.A. EDGINGTON ,"
i"
|
pROT., ,0NSNC.OEN T 1\ ON POLARIMETER
FRONT CARBON REAR MWPCs P L A T E MWPCs
Fig. 1. A sketch of the polarimeter, showing also the polarization vectors of an incident proton.
B
I CARBON PLATE
REAR DETECTOR
Fig. 2. The two classes of events, distinguished by the intersection of their scattering trajectories with the rear detector. ~1= 0 is to the left.
instrumental bias, without loss of statistical precision, is less simple. Consider first a possible slow variation in efficiency as a function of ¢, due to small variations in the performance of the multiwire chambers. Since their sense wires run vertically and horizontally, one expects such variations to show certain simple symmetries, and we therefore choose to represent this efficiency variation by a trigonometric series truncated at a low order; we multiply the distribution of eq. (1) by the function 1 + e l cos~b + r h sinq~ +e2 cos2~b +r/2 sin2~b + e4 cos4~b. The coefficients s~, rh, e2, rh, e4, represent respectively a variation in efficiency from left to right; up to down; horizontal to vertical; one diagonal to the other; and side to comer. We anticipate (and find) that these coefficients will be small, of order 0.01 or less, though this is not essential to the argument. The second effect is due to the finite area of the back of the detector, which results in regions of ¢, for a given 0, for which the efficiency drops to zero. In fig. 2 we illustrate this by constructing cones of semi-angle 0, with axis along the incident trajectory and vertex at the carbon plate, for two events A and B having equal values of 0. The heavy lines show the scatter trajectory actually observed, while the ellipses show the intercept of the possible trajectories with the back of the detector (the rear scintillator). For events like A, the complete range is in principle accessible, whereas for those like B it is cut off by the detector edges. In other words, only for class A events does the intercepting ellipse lie wholly within the boundary, and hence the acceptance, of the back of the detector. Thus each class A event may be considered as sampling the same azimuthal probability distribution, whereas each event of class B is a sample from a separate, unique distribution, whose normalisation differs from event to event as the accessible fraction of the complete 2rt azimuth varies. Because protons incident on our polarimeter are not confined to the centre of the carbon but can strike it anywhere, we would disqualify many events from analysis if we restricted ourselves to class A events alone. We therefore had to develop a procedure for dealing with class B events also. The formal solution for the ith event is to multiply P(~) by a function g~ which is zero for inaccessible regions of azimuth, and 1 elsewhere. In the next section we develop this and display the results for the estimators of the various parameters and their statistical errors; in order to clarify the somewhat opaque algebra, we give both the general formulae applicable to class B events, and their simplification for class A events. The final result is a set of non-linear equations, and in the third section we describe an iterative procedure we have used for solving these, and present some illustrative results. 2. The formal solution As mentioned, the ith event is taken to be a sample from an azimuthal probability distribution of the form gi 1-1 + A(O) (try cos~b + trx sin ~b)] x (1 + e1 cost# + r/1 sin~b + e2 cos2~ + r/2 cos2~b + e4 cos4~),(2) P'@) = V, where the quantity Cj is present to ensure the correct normalisation. Note that the intercept of a cone with a rectangular detector allows up to eight values of azimuthal angle (say ~i, J = 1, ..o, 8) which divide the
'
ESTIMATION
OF ASYMMETRIES
177
azimuthal range into a maximum of four regions with g~= 1, and an equal number (or none for class A events) with gj = 0. We expand P,.(4') as a trigonometric series Pi(4') = ~Oi (ao + al cos4, + a2 cos24, + a3 cos34, + a, cos44, + as cos54' + bi sin4, + b2 sin24, + + b3 sin 34, + b5 sin 54,),
(3)
whose coefficients are: ao = 1 + ½A(0) (0-y~l + 0-xr/1), aa = 51 + ½A(O) (20-y + 0-y~2+ 0-xr/2), a 2 = e2 +
½A(0) (0-y~1
bl = ~h + ½A(O) (20-~ - 0-x~2 + 0-yr/2), b2 = 1~2 "~ ½A(O) (0-y~1 -t- 0-x~1),
-- 0-x~1),
(4) a 3 = ½A(O) (o'yg 2 + 0-y~4 - 0-xr/2),
b 3 = ½A(O) (0",1'/2 + 0-xE2 - 0-x~4),
a s = ½A(O) 0-,e,~,
bs = ½A(O) 0-xe,,.
By requiring o" P,(4')d4' = 1, we find the normalisation constant C~: C, = ao {4'}i + al {sin4'}, +-~-a2 {sin24'}, + ~a3 {sin 34,}, +--~ a4 {sin44,}, + -5-a5{sin 54,}' _ bl {cos4,}, b2 Icos24,},cos34,},os54,}, 2 Here we have introduced the notation { }~ to stand for the definite integral {f(4')}~ -=
~o'~
(5)
9,d [f(4,)] •
For example, {4,}, =
o
o, d4, -- 4 , 1 - 4 , ~ + 4 , ~ - 4 , , + 4 , ~ - 4 , ~ + 4 , ~ - 4 , ~ .
For class A events, eq. (5) reduces to G = 2~[1 + 3.4(0) (0-,~ + 0 - ~ ) ] , since only {0}~ is non-zero. One can now write down a set of expectation values for the ith probability distribution, for example:
{
A(O) (A(O)cos4,), = 2a o {sin4,}, + al 4' + sin24'/2 j+a2
+
{ ~
+ sin54'/
-b3[
/ c°s24' 2
+ c°s44'/
4 J,-b5
The general form of these expectation values is:
{
sin4, + -+" aT3- sJ i, n 3 4 ' / + sin64'/
{~_.~
+
C°S64'/
6 J,"
+ sin44'/4j , + {cos 4,
cos34,~
178
J. A. EDGINGTON
Ci
c
c
c
c
c
c
c
c
c
A(O) (A(O) cos n~b)l = aoXo. + al xln -.[-a2X2n + a3X3n + a4 x~. + asxs. + bl Yln + b2Y2n + b3Yan + bsy5., (6a) 2 C~ ~ ~ ~ s A(O) (A(O) sin nq~)~ = aoXo. + a l x l , , + a2x2, , + a3x3. + a4x~4. + asx~, ' + bly~, ' + b2Y~2,, + b3Y~3,, + bsys.,(6b)
where _ /sin(m_~-n)q~ Xm.
[
m--n
sin(m+n)q~ 1 +
-m+ n
Ji'
_ {cos(m-n)~b -t c°s(m +n)~b/ , m-n
Ymn ="
{ xS" -
[
j~
c o s ( m - n ) ~ b + cos(m+n)~b / m- n m ~}--n "Ji'
-
_ /sin(m-n)~b Ymn
~
(7)
sin(m_+n)~b /
-m ---n
m+n
ji'
and by convention we take sin P~b P cos P~b I P
----0,
for
P=0.
Eqs. (6a) and (6b) are the general forms; for class A events they simplify drastically to 2 A(O) (A(O) cosn~b) = a.,
2
A(O)
(A(0) sinn~b) = b . ,
which are included in the general forms as the leading terms. It must now be stressed that, since the inefficiency coefficients el, rh, e2, rh, e4 are small, we have to a first approximation a0 ~ 1, al ~A(0)ay, bl ~A(0)ax and all other coefficients are small. Likewise, for those class B events which lose only a small portion of the total azimuthal range, x ~ , ~ y ~ , , ~ 2n and the other definite integrals are small. We therefore re-write eqs. (6a) and (6b), projecting out the large parameters and the large definite integrals; we also divide through by A(O). The result is
2Ci
A2(0 ) (A(O) cos~b)
c
= exP~ +tryQ~ + trxR ~ + $1,
2 Ci
A2(O) (A(O) c0s24,) = 82P~ + o~Q~ + oxR~ + S2, 2~i
¢
e
-~2-'~) (A(O) cos4~b) = 84P~ + tryQ4 + a~R~, + $4, 2 Ci
A2(0 ) (A(O) sin~b) 2 C~
~
= qxPx + % Q ~ + axR~ + Sx, ~
~
(A(O) sin2q~) = r]2P 2 + ayQ2 + axR~2 + $2,
where P.~ = x~./,4 (0) and p s = y S / A (0) exactly; Q~ and R s contain x.C. and y s respectively as the leading
(8)
ESTIMATION OF ASYMMETRIES
179
terms; and all the terms in R e, S~, Q~, S~ are products of small parameters and/or small definite integrals. For class A events, eqs. (8) reduce further to 2 (A(0)cosqb > 242(0)
el
(~.)
-- A ( O ) -I- try
8.._.L_2 tryst cos2qg> - A(O) 4 2
A~
trot/2
1 +
+ ----~, trxrll 2 '
2 ~4 A2(0)
th -
(8a) -~
A(O) +
2
+
t/2 tr~i
(1 :---~ try\2]'
tr~ 2
"
If, furthermore, the inefficiency coefficients are taken to be zero, we arrive at the well-known expressions (see, for example, ref. 4): 2
A2(O)
= tr,,
whence the (unbiassed) estimators of ay, ax are: ,
2
1
N
tr, = ~2-'~ ~ , ~ t a(0) cos~bi, ,
2
1
trx = " ~ - ~
~ A(O) i=l
sin~b~,
with variances given by: var ( : )
= ~
1[ A-~
var (a*) = .~
_ (:)2
1,
- (a*) 2 .
Eqs. (8) may now be re-written so as to yield similar, but more complex, expressions for the general case. For example, from the first of these equations we deduce that
:,'
:
with a variance var (tr*) = ~
_ (try,)2 ,
(9)
and also -
__2,
,el + tr.a
+
180
i.A. EDG1NGTON
with variance 1[var (~*) =
2
.27
J
Notice that the correction terms to the estimators, which are in principle exactly calculable, do not contribute to the statistical uncertainties. These eqs. (9) and the similar ones for O~x, r/~*, e~*, r/~ and e~* are the formal solution to the problem. Because the normalisation constants C,, and the coefficients pg,s, Q~,S, R~,' and S~,' themselves contain the parameters ~rx, try, el, r/l, e2, r/2, e4 which we are seeking, these equations are non-linear. It is this non-linearity which is the essential difference between eqs. (8), representing the general solution, and eqs. (8a) which refer to the special case in which the whole of the azimuth is, in principle, accessible. If the non-linearity is neglected, and the equations solved directly by taking the C,, etc. to be constants, then the resulting estimators are biassed. To obtain a set of unbiassed estimators, we have solved eqs. (9) iteratively.
3. Implementation of the solution We reconstruct the kinematics of each detected event off-line so that the scattering angles (0, 0), and the intercept azimuth angles e j , j = 1,..., 8, are known• We then calculate for each event the definite integrals of eq. (7). In the first approximation we assume that all the inefficiency coefficients and polarisation components are zero, with the consequence that all the coefficients given by eq. (4) are zero, except for a0 which is 1 • The normalisation constant C~ and the quantities P,C , S , Q,C , S , R ,C,S and S,C,S of eq. (8) are then calculated. Finally we form the sums necessary to evaluate the estimators of eq. (9). For class A events this is an extremely good approximation since for such events C~ = 2 zra0 = 2n[1 + ½A(O)(a~e~ + ax rh)] = 2rr(1 + di). Inserting typical values [A(0) -~ 0.5, ax ~ % ~. 0.5, el N rh ~ 0.02] we find an upper limit for the correction term 6 of order 0.005. A similar argument holds for the quantitiesP~ ,s etc., and we find that the solutions for class A events are inaccurate by a few percent at most. They therefore provide an excellent starting point for a first iteration in which we recalculate all the coefficients of eq. (4) and then repeat the calculations and summations event by event as before. Convergence is rapid and a second iteration is not usually needed. In practice the instrumental biases are so small compared with their statistical errors that the values for the inefficiency coefficients have to be found by averaging over many data runs taken during a period of uniform chamber operation. In table 1 we list average values of the inefficiency coefficients obtained in this way for a 5-day period of steady operation. The measurement here was of the polarization transfer parameter Dt, and so trx was expected to be zero. The results obtained after the first and second iterations were found to agree almost exactly, and show that not only trx, but all the instrumental asymmetry coefficients, are indeed consistent with zero. The analysis described here is capable of resolving instrumental biases inherent in the detector operation, but it can neither resolve nor eliminate any biases introduced at the data reduction stages. One such bias appears for those protons which scatter close to the edge of the carbon plate. There is always some uncertainty in the calculation of trajectories (due to multiple scattering, and to digitisation effects in the multiwire chambers) but, except near the edges of the acceptance region, there is an approximate equality between the numbers of trajectories assigned incorrectly to one side and to the other of the 'correct' position. At the edges an incorrect assignment to a position outside the detector is not allowed, and this can have a dramatic effect on the apparent asymmetry. Consider, for example, the range of azimuth available to two groups of particles, those scattering from just to the left of centre of the carbon plate and those scattering from the extreme left hand edge. Suppose the actual azimuthal distribution to be flat. The group of particles striking near the centre of the carbon plate includes, besides those with correctly assigned trajectories (C), others with incident trajectories assigned incorrectly either to the left (L) or right (R) (fig. 3). All these particles sum to yield the correct azimuthal distribution. A particle from amongst the group striking near the le•hand edge, however, cannot possibly be assigned a trajectory even further to the left, so such events are absent. Since they occupy a smaller azimuthal range than the correctly assigned events, or those further to the right, the effect is a 'hole' around ~ = n, corresponding to an artificial
E S T I M A T I O N OF A S Y M M E T R I E S i
181
TABLE 1
i
C
Values of trx and of the inefficiency coefficients during a 5-day period of chamber operation. The null hypothesis that the true values of all these coefficients are zero leads to a X2 of 5.09 on 6 d.f. I
I
0
"IT
ax = - 0.0045 _+0.0122
2"Tr
Fig. 3. A hypothetical flat azimuthal distribution, showing the portions accessible to scattered particles whose incident trajectory has been assigned correctly (C) and also, to the left (L), and to the right (R) of its actual position.
eI r/1 ~2 r/2 e4
= 0.0080_+0.0077 = - 0.0064___ 0.0077 = 0.0072_+0.0077 = -0.0117_+0.0077 = 0.0001 _+0.0077
positive value of the computed coefficient of cos ¢. Conversely the effect at the right hand edge will simulate a negative value of the coefficient. A similar effect will modify the sin ~ coefficient at top and bottom edges. This bias seems inevitable at the edges of detectors with incomplete angular acceptance and finite angular resolution. We expect, and find, that it is small or absent for interactions occurring near the centre of the carbon. We analysed separately the data from different neutron-proton scattering angles corresponding to different vertical strips of the carbon plate, and the polarimeter was moved round so that at three different settings events at a given n - p scattering angle fell to the left, in the centre and to the right of the carbon. In table 2 we show a typical set of asymmetry coefficients got in this way, both for class A events alone and for all events. The effect is clearly present in the latter group but not in the former. It must be stressed that this bias in no way affects the results of the final analysis. For example, in our case, data were taken in alternate runs with the neutron beam polarisation reversed, and the quantity of physical interest was obtained from the difference of asymmetry coefficients in the two cases. The artificial bias exactly cancels. The data of table 2 were taken in a measurement of the polarisation transfer parameter D r , in which the polarisation component tr~ is given by try=
-- Pnp + Dt (O'n) 1--P.p(an) '
P,p b e i n g t h e p o l a r i s a t i o n p a r a m e t e r in n - p s c a t t e r i n g a n d ( a , ) t h e p o l a r i s a t i o n o f t h e n e u t r o n b e a m . T h u s t h e d i f f e r e n c e b e t w e e n try m e a s u r e d in t h e t w o s p i n o r i e n t a t i o n s is Arty=
2 (a.)
(O t - P ~ p )
1 - p2p (tr,)2
,
w h e n c e D t c a n b e c a l c u l a t e d . V a l u e s o f D t a r e l i s t e d in t a b l e 2 a n d g o o d a g r e e m e n t is s e e n b e t w e e n t h o s e f o u n d f r o m class A e v e n t s a l o n e , a n d t h o s e g o t f r o m all e v e n t s . I n t h i s p a r t i c u l a r set o f d a t a , t h e g a i n in TABLE 2 Values of try under various experimental conditions, showing the bias occurring near the edges of the polarimeter. The derived values of Dt are also listed, and are seen not to depend on position. Position of scatter
Left hand edge Centre Right hand edge
Weighted averages
Event class
Values of try: Spin normal Spin raversed
Derived values of D t
A A+B A A+B A A+B
0.465_+0.109 0.831 _ + 0 . 0 8 8 0.597_+0.108 0.563_+0.094 0.430 _+0.084 0.050_+0.073
-0.280_+0.118 - 0.267_+ 0.095 - 0.200_+ 0.089 - 0.200_+ 0.078 - 0.197 _+0.096 - 0.174_+0.084
A A+B
0.011_+0.103 0.393_+0.083 0.165_+0.087 0.132_+ 0.075 0.084 _+0.088 - 0.266_+ 0.076
-0.218_+0.057 -0.209_+0.049
182
J. A. EDGINGTON
statistical precision from using all events, rather than class A events only, is quite significant. Overall we found that class B events comprised some 3596 of all events, leading to an improvement in the statistical errors of about 18 96 when these events were included.
4. Summary The problem discussed in this paper arose during the analysis of data taken with a large polarimeter in which it was not possible to constrain events to a uniform central region of complete angular acceptance. Rejection of all events associated with incomplete azimuthal acceptance would have meant a significant loss in statistical precision. It was therefore necessary to devise a method for incorporating into the analysis both the usual small instrumental biases, and also the gross effect of incomplete acceptance. Eq. (9) represents the formal solution to the problem, using the method of point (moment) estimation. These non-linear equations were solved iteratively, and we had the advantage that those events which did have a complete angular acceptance region provided an approximate solution which lead to rapid convergence. Certain systematic biases, which seem unavoidable, remain for scatters occurring close to the edges of the carbon analyser plate, but these are entirely eliminated when the physical quantities of interest are calculated from the diff6rence between data taken with reversed beam polarisations. The programme ASYM, written to perform this calculation, is available from the author. I thank all my colleagues in the BASQUE group at TRIUMF for helpful discussions. Throughout this paper the first person plural refers to them. References 1) j. A. Edgington, Proc. Vancouver Conf. on Nucleon--nucleon interactions (American Institute of Physics, N.Y., 1978) p. 19. 2) G. Waters, I.M. Blair, G.A. Ludgate, N.M. Stewart, C. Amsler, R.C. Brown, D.V. Bugg, J.A. Edgington, C.J. Oram, K. Shakarchi, A. S. Clough, D. A. Axen, S. Jaccard and J. Vavra, Nucl. Instr. and Meth. 153 (1978) 401. 3) G. G. Ohlsen, Proc. Fourth Int. Symp. on Polarization phenomena in nuclear reactions, ed W. Grtiebler and V. Konig (Birkhatiser Verlag, Basel, 1976) p. 287. 4) F.T. Solmitz, Ann. Rev. Nucl. Sci. 14 (1964) 375.