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Kinematic broadening in two-body collisions at relativistic energies
NUCLEAR
INSTRUMENTS
KINEMATIC
AND
BROADENING
METHODS
63 (1968)
IN TWO-BODY
23-28;
‘g NORTH-HOLLAND
COLLISIONS
AT RELATIVISTIC
PUBLISHING
co.
ENERGIES
D. L. SMITH U. S. .-1rrr!~* Missile
Corw~mrrd, Rcvistone Arserlal, Received
15 February
Alabama
35809, U.S. A.
1968
Kinematic broadening is studied for two-body collisions in u hich the kinetic energies of the interacting particles are large enough to require relativistic analysis. Three overlapping categories of problems are treated. For each category, a formula is derived which can be used in the calculation of kinematic broadening. The first category includes collisions in which the kinetic energies of the particles may be of the same order of magnitude as their rest-mass energies. The formula which is derived for this category is very general and may be used regardless of the energies of the interacting particles. The second category consists of ultrarelativistic collisions. Here it is assumed that the kinetic energies of
the incident and emitted particles are considerably larger than their rest-mass energies. Finally, the third category includes collisions which are only slightly relativistic in the sense that the kinetic energies of the interacting particles are considerably smaller than their rest-mass energies. Each of the formulas derived is an expansion in the small angular deviations from the mean direction of emission of the observed particles. These formulas may be applied to problems in atomic, nuclear and high-energy physics. Their use is demonstrated by means of three numerical examples.
1. Introduction
in an experiment in which 5.5-MeV protons were scatdetected tered elastically from 63Cu and subsequently at 45” in the laboratory by a magnetic spectrometer with a solid angle of 3.2 x lop2 steradian’). Kinematic broadening can be determined for a specific experiment by directly calculating the energies of the particles emitted in various directions encompassed by the detector. In practice, this method is generally somewhat inaccurate since one is calculating arelatively small number by subtracting two much larger numbers. An alternative method for calculating kinematic broadening will now be described. Fig. I shows the coordinate system which will be used in treating this problem. It should be noted that this is not a conventional spherical system. One may think of 8 as an azimuthal angle in the plane TPB and 4 as the angle of elevation
Most two-body collis;on experiment5 are conducted under the following conditions: A nominally monoenergetic beam of particles with mass t?zL and laboratory kinetic energy T, bombards target particles with mass mz which are at rest in the laboratory (neglecting thermal motion). Two types of particles, with masses m3 and ma, respectively, are emitted as a result of the collision process. These particles may or may not be the same as the beam and target particles depending on whether or not a transmutation has occurred. The kinematics for a two-body collision is uniquely specified by the requirements that momentum and total energy be conserved. If sufficient kinetic energy is provided by the incident particles to permit a particular collision process (with a specific Q-value) to take place, then all of the particles of the same type, which are detected in a given direction in the laboratory, will have the same kinetic energy. The only exception to this rule is that for some types of collision processes, near threshold, the kinetic energy of the detected particles may assume one or the other of two allowed values’). The unobserved (recoil) particles carry away the remaining available kinetic energy. However, if the same particles are now detected in a slightly different direction from the previous one, their measured kinetic energy will differ somewhat from its previous value. In some experiments it is necessary to use detectors which subtend appreciable solid angles in order to compensate for low yield. The spread in the energies of groups of detected particles (kinematic broadening), which is thereby introduced by kinematics, can be a major contributor to reduced resolution. Kinematic broadening has been clearly observed
Fig. 1. Coordinate system for the calculation of kinematic broadening: Point T locates the target position. Line TB identifies the incident beam direction. Line TD identifies the direction in which the scattered particles are detected. The plane TPD is perpendicular to the plane TPB. The angles @,O, and 4 are related by the equation cos0 = COS~COS~.
23
24
D. L. SMITH
of the line TD relative to the plane TPB. This particular coordinate system was chosen because it happens to be a convenient one to use in several types of problems involving finite detectors. The detector-fixed coordinate systems, which are conventionally used to describe the operation of some of these detectors, can be easily matched with the coordinate system we have chosen at the detector-to-space interface. Assume that the particles with mass nf3 are detected by a detector whose axis lies along the direction(H.4). Kinematic broadening may be calculated with greater precision if one employs formulas which express the shift AT, in the kinetic energy T, of the emitted particles in terms of a shift dfI and d4 in the emission direction. A more important advantage in having formulas for AT, which are expansions in A0 and Aq5 is that it may be possible to design detection apparatus with achromatic compensation for the kinematic broadening associated with one or more terms in the expansion2). A nonrelativistic expansion of AT,/T, to third order in A0 and A4 was derived in an earlier communication [hereafter referred to as I “)I. The objective of this present work is to investigate kinematic broadening for collisions involving relativistic energies. Expansion formulas will be derived for AT,,IT, which will permit kinematic broadening to be determined for most twobody collision processes. Three categories of collisions will be studied : I. Gene/.al collisions: The kinetic energies of the incident and emitted particles may be of the same order of magnitude as their rest-mass energies. 2. Ultrurelrrtiristic collisiotts: The kinetic energies of the incident and emitted particles are taken to be considerably larger than their rest-mass energies. 3. Sliglttlr relrrtiristic collisions: The kinetic energies of the incident and emitted particles are assumed to be considerably smaller than their rest-mass energies, but not small enough so that the nonrelativistic formula derived in I can be used.
function T, of the angles 0 and 4 varies so smoothly that rapid convergence of the expansion can be anticipated (if we avoid regions where T, vanishes) for realistic A0 and A$. For this reason, the expansion in eq. (I) is not extended beyond third order. Define the following quantities which are associated with the I?‘~ particle (II = I,..., 4): nt,: mass of the particle, u,: velocity of the particle (of = ~1,~. u,), 1’”E (1 -$!c2)-f, where c is the speed of light in a vacuum, pn = ~~nz,+t,,: momentum of the particle (y,Z = P,,* p,), E, = y,tn,c2 = (c’c): + tnics)f: total energy of the particle, T,, = E,, -tn,,c’ : kinetic energy of the particle. If Q is the energy released in the collision, then Q = T,+T,-T,. Conservation tion
of momentum P4 = Pl -P3r
(T, = 0). is expressed
(2) by the equa(3)
(Pr =O).
The kinetic energy T, may be eliminated from eq. (2) by using eq. (3) and appropriate relationships between the total energy, kinetic energy and momentum. The result is Q = T,-T,
+[Tf
+2m,c’T,
+ T~+~~H,C~T,+~~I~C~-
-2cosOCOS~(T:+7/H,cZT,)+(T~+2t?13C2T3)+]+- I?1,$C2.
(4)
Eq. (4) may be used to calculate T,. Furthermore, this equation is the starting point for deriving the expansion of At. The function Q does not depend on the direction in which the particles are detected in the laboratory; therefore Q = T3+AT3-T,+-jTf+h1c2T~+(T3+AT3)*+ +2tJ?,C’(T3+AT3)+t~~~c4-
2. General collisions
-Zcos(~+A0)cos(~+A~)(Tf+?tn,c2T,)*x
We wish to derive a third-order of the form
expansion
formula
x [(T3 + AT,)’
At-AT3/T3=(At~AO)AU+(At~A~)A~+ +(At~AU2)AO’+(At~
AOA~)AOAc~+
+(At/A~‘)A~‘+(At~A03)A03+ +(Atl
A02A~)AH’A~+
+(Ar 1AOA~2)AOA~~2
+(Ar 1A43)A43,(
1)
where the coefficients (_._I .) are functions of the collision parameters and do not depend on A0 or A4. The
+2tn3c2(T3
+ AT,)]+,+
- tu4c2.
(5)
If we equate the right-hand sides of eqs. (4) and (j), we obtain an implicit relation between AT, (or At), AU and A+. The next step in the calculation is to expand all the functions ofAt,A6,andA4 in this relation to third order in the manner described in 1. No generality is lost by taking 4 = zero degree since one is free to choose the orientation of the scattering plane:in a two-body collision. After performing a considerable amount of algebra, one obtains a relation of the form
KlNEMATlC
BROADENING
IN
At = aAO+hA02+cA~2+tfA~Ar+eAt2+fA83+
notation
E, = tt,,,cz/Tn,
(6)
13824$a”ty’k:SjsinOcost)+
2 = (2$a3 - 2$a” + 2$a”ki - 2li/a”k:) +(3$atl-
/12 = (,?&?I:)&:,
- 6$a’ki
- 3IC/at/k: - 60$a3qk:
5 = p2k:+k:+p2-2pk,k,cos0. a = ii-+
- 60$a3tlk:
I,//= [I +~+ak~+(-24at\k~-24at~k~)cosO,-‘.
+ (- 288rC/a3tl’k: + 288$a”rl’k:
the coefficients
(9)
+ 96$a3tlk;
+
-2304$a3r12k:2)sinOcos0,
(10)
- 6$aqk;
- 96$a”qk;
-
-4311/a3t/k;)cosO+(576$a3t~2k;+ + 1 I 52$03ty2k:0 + 576i,ha3tl”k:‘)cos2
0.
(11)
(8$atlk~)sinO+(2304$a3~2k:‘)sinfIcosO+ (12)
g = (24~at~k~)sinI~+(2304~~03t~~k~‘)sinOcos0,
(13)
+
0 + (230411/a”t12k:0 + 13824$a5t12k:20 + ( - I I 52$a3t12k:0 -
- 1 152t/xa3t~2k~2)cos20 +(33l 776$asq3ki6
(14)
+ 48$a3tlk:
+48$a3~k~)cosO+(
+
- 1152$a3q2k:’
-
- I 151~a3t~2k~2)cos20,
(15)
x = (6IC/aqk: - 12$avlk: +48$a3qk: + 1921C/a3qk; - 288$a’tlk: - 576$a’qkt
+ 6$atlk:
+4811/a3tlk! -
- 28811/a’qk:‘)sinQ+
+ 27648t,ha-it/3k\8)cos3
0.
(17)
+~c1~+2abe+a’.~+33~~~tle+u~-_+2~z~e’)A~~+
+
(18)
When calculating the coefficients appearing in expansion formulas for At, one should use the kinetic energy T, of the particles etnitted in the direction corresponding to the axis of the detector. collisions
The formalism of the preceding section still applies when the kinetic energies of the incident and emitted particles are large relative to their rest-mass energies. Under these conditions, several of the parameters defined in the previous section approach limit values. These limits are k,,k, -+ I, (19) iI+ p/24.
+
+331776~a’~3k~8)sin’Ocos0, lZ$atlk:
-
0+
We can obtain a solution for At in the form of eq. (1) if we apply an iteration procedure to eq. (6). This technique is described in 1. The result is the formula
3. Ultrarelativistic
+(-2311841j~~t~~k~*)sin~O,
+48tia3qk:
+ 82944$a”t/“k:’
+(g+al+cd+2nce)AOAqS’.
e = (--1+!1a+$a~+2$a~k:+$a~k:)+(-6$atlk:+
I = (- 12$aqk:-
288$a3q’ki2
- 3456$a”ty’k:‘)cos’
Ar = aAfl+(b+~d+~Ze)AOZ+cA~Z+(f+ah+bd+
+96tj~3tlk~)sinO+(-2304rl/o”tjZk:0-
- I 3824$a’r12k:‘)sin2
- 10368$a5tl’ii:‘-
+288$a”rl”k:’
- 345611/a’t12kt +
(8)
c = ( - 2411/ar/k;)cos 0,
+2304$~s~q~k:‘-
+ I2V/a”tlk~ +
+(27648$a’t~3k:‘+82944$a”t\3k:J+ (7)
b = (- 24Gorlk;) cosO+(2304~03t~2k~2)sin20,
+48t,ha3tlk;)cos
- 6$(r”k:-
144$a”tjk~“)cosO+
- 10368$a”g2k:
~1= (-48tiorfkS)sinO.
(I= (- 24$arlk’: - 24tiatlk;
ap-
(16)
+ 144$a5tlk: + 3t,hatlkS -
+ 432tia’tlk:
+4321C/astjkt+
Expressed in terms of this notation, pearing in eq. (6) are
I~I/JcJP/~:- I ?$ot&
-
3t,hatlk: + I ?rba”tlki -
t1 = pk, /(24kZ),
h = (-
+
- 165888$a5t~3k~s)sinOcos’0.
(II = 1.3),
+ 12ll/otpk’: -48$a3qk’:
+
+( - I 658881//a”r13k:” - 33 I776$a”~“P:’
P = TIP,,
f=
25
+ I 3824$a5r12k:0 -576rl/a3t12k:2 +27648tia’t12k:2+
applies to eq. (6): (II = 1.3).
k,5 = 1 +?&,,,
COLLISIONS
+( - 576t+ha3tj2kt -3456t+ha3tl’k:’
+gAOA~2+ltA02At+lA~2At+sAOAt2+-At3. The following
TWO-BODY
(20)
~~(~‘+1+~~‘-‘pcos0),
(21)
+(1+2a-ZapcosO-‘.
(22)
Eq. (18) is still used to calculate At in the ultrarelativistic limit; however, we may now use the following limit values for the parameters which appear in this equation: cI + (- Z$ap)sinO, (23) b+(-Il/ap)cosO+(4$a3p2)sin’0,
(24)
D. L. SMITH
26
(29
c+(-l+hp)cose, n-+(-2$ap+8$(r3p)sin8+(-8$03p2)sinBcos6r,
(26)
e+(-$o+4$a3)+(-8~a3p)cos0+(4t//a3p2)cos2~,
(27)
~f+(+~crp)sinO+(4$a3p2)sin~cos8+ +( - 16$05p3)sin3
0,
g + ($0~) sin 8 + (41C/a3p2)sin Bcos 0,
(28) (29)
A+(-$ap+411/03p)cos0+(8$a3p2-481//05p2)sinZO+
+(-41C/a3p2)cos2bl+(48$o”p3)sin20cos0,
(30) (31)
i - (- Ic/op+ 4$03p) c0Se+(-4tj&+0S20, ?c-t(121C/a3p-481j/osp)sin8+(
-8$(r”p’+
+96$~5p2)sin~cos~+(-48~~5p3)sinBcos’0,
(32)
i+(4t+!/a3-16$a’)+(-4$a3p+48$05p)co~O+ +( -48$c5p2)cos2 Eq. (4) should
0 + ( 16$a5p3)cos3
(33)
T3.
be used to calculate
4. Slightly relativistic
0.
collisions
In this limit,
T,,
(II =
-$ I,
p;/1n,2c2 4 1.
1,3,4),
(II = 1,3,4).
{P:/(~+~)L
(35)
1 + Sii,),
(n = 1,3,4),
(36)
(H = 1,3,4).
(37)
Eqs. (36) and (37) may be used to eliminate Ta from eq. (2). The result, to first order in 6, and S,, is Q = {l+(m3/JJ7,)}T3-
{l-(ml/JJJ~)}T~
-(2/JJ2,)(JJtIT,iJz3T3)~COse+ +(~~,/JJ~~)[(JJ~,~J~~-JJ~~)T~-~z~JJ~~T~+(4JJ+/JJ3(JJz,T,JJz3T3)+cosU-2JJ1~nJ3T,cosZfl]+ +(~S3/nz:)[(JJ13m:-nz:)T3-JJz,rJJ:T,+
-(JJJ1/JJ3T,/T3)+cos0]-‘+ +ii,
1[(-,/Jf;/?z,/J?l:,+ \
+(/JJ://JJ~-$)(JiJ1JJJ3T~/T3)~cose-
-
x
x [MS + 1H3- (JJyJJ3T,/T3)%osc)]-2+ +[(21?1://~l:-_t)(1?1~111~T~/T~)3sinO-(2JJ~fJJz,/JJJ$si1leCose]
where the first term is the familiar nonrelativistic kinetic energy and the second term is the lowest-order relativistic correction. Another approximation which is useful in the slightly relativistic limit is P,, = (2,n,T,)+(
(38)
The d, and d, terms are the lowest-order relativistic corrections to the familiar nonrelativistic Q-equation (e.g. see I). The assumption will now be made that for slightly relativistic collisions one only needs to calculate the lowest-order relativistic contribution for the first-order (AU) term of the expansion in eq. (1). We will use the nonrelativistic expressions derived in I for the secondorder terms and discard the third-order terms. The resulting formula will be sufficiently accurate for many applications. The derivation of (At/A@, including the lowest-order relativistic contribution, is initiated by calculating the differential dQ of Q given in eq. (38) with 0 and T3 as the variables. When we set dQ = 0, we obtain a useful relation between dT, and de. To first order, the relation between AT, and A0 is the same as the relation between the differentials. If all but the first-order terms in 5, and 6, are eliminated, we are left with the following formula for (At/Al?):
-(,,,~m,/~~~~)~os~U][2(r~1,r~~~T,/T~)~sine] (34)
approaches
T, = {~,z/(2m,))-
-2rJl,rJl~z;~~~‘e]*.
(At1 AO) = -~(JJz,JJJ,T,/T,)~S~~~I[JJI,+JJI,-
A somewhat different approach will now be used to derive a formula for At applicable to the study of collisions which are only slightly relativistic. The slightly relativistic limit may be defined by the following inequali ties : 6, = T,/(JJJ,c2)
+(4111:--i1:)(IJIITI~i13T3)tC0Se-
x
X [IJJa + /JJ3 -(/7J,/JZ3TL~~f3)~COS~]-1
I +b,[(JJJ,
- JJJ:/JJJ:)-
X [Jill
,fi13T,/T3)+sine]
+
LJ~J:Tl /(2J71:T3)}
(JJI
- (JJJ~IJz~T~/(JJz~T~))cos~U] x [2(/J?
I
+
x x
+ JJJ3 -(JJIIJi13T,/T3)+COS~]-2+
+II(~JJJ:/JJJS-~)(JJJ,J~J~T,/T~)~S~~~- ~~JJ~~~J~~~~/(JJJ~~~))S~I~~COS~] X [JJJ5 + JJZ3 -(IJI~JJJ~T~/T~)+cos~]-~
X
)
.
(39)
The dependence of Q on 4 is not included in eq. (38) because it was shown in the two preceding sections that the &dependence is at most second order near & = 0”. It will be seen that we will have to concern ourselves only with the calculation of the first-order (33) term in the present section.
KINEMATIC
From I we obtain tivistic) : (dtld0’)
the second-order
BROADENING
terms
= -~cos0+2~2sin20-/?3sin28cos8, (AtlA0~4) (AC~A@) =
= 0, -~cos~.
(nonrela-
(40)
5. Applications
(43)
[eq. (38) with 6, = 6, = T3.
of the formulas for At
Two important related applications for the formulas which were derived in the preceding section will now be described. The first application is the calculation of overall kinematic broadening for a specific detector geometry. The direction (0,4 = 0) is assumed to lie along the axis of the detector. Suppose At assumes its minimum value Atmin (usually less than or equal to zero) when A0 = AO, and A$J = A@, and its maximum value At,,, (greater than zero) when At? = AO, and A+ = Adz. Then the overall fractional kinematic broadening is given by At,,,,,,,
= At,,,,-At,i”.
COLLISIONS
27
initial conditions vector when the kinetic energies of the particles are kinematically broadened. It is appropriate to use this form for At since A% and A4 also appear in the initial conditions vector to specify the particle’s direction of motion. 6. Numerical examples
(44
p 3 (~“,,?z3T~/T~)tsinO[~llf+/)z3-
The nonrelativistic Q-equation = 0] may be used to calculate
TWO-BODY
(41)
where
-(In,m,T,/T3)~cosn]-‘.
IN
(44)
It is generally not too difficult to determine AO,, A$,, AO,, and A42 by inspection. Of course, there are many other factors which contribute to the peak shapes in spectra recorded by a detector. The formula in eq. (44) merely gives an upper limit for the contribution one can expect from kinematic broadening. A second application of the formulas for At is in matrix calculations for ion optical elements. The motion of a particle entering an optical element may be characterized mathematically by an initial conditions vector whose components are functions of the parameters of motion: position, direction of motion, and kinetic energy. One obtains a new vector, which describes the motion of the particle as it leaves the optical element, by multiplying the initial conditions vector by a matrix which represents the optical element”). The kinetic energy of a particle is usually specified in terms of its fractional difference AT/T from some average value T for the group. The parameter ATJT appears in the initial conditions vector. An optical element is said to be achromatic whenever its effect upon a particle is independent of AT/T. One may substitute At, in the form of an expansion like eq. (I), for AT/T in the
Three numerical examples will be treated in this section. Coefficients of terms in the appropriate expansions for At will be calculated for collisions which might be studied in laboratory experiments. E.~amnple 1: A slightly relativistic collision. The experiment involves the detection of 50-MeV protons which are scattered elastically from 40Ca at a mean laboratory angle of 8 = 90”. For this collision, T3 =47.56 MeV. Eqs. (39-42) and the fact that (AflA4) = 0 lead to the result: At z (-0.0500-0.0013)Ad+(0.0013)A~12.
(45)
The second term of the AO coefficient in eq. (45) is the relativistic contribution. Even though the relativistic contribution to the first-order term is small (2.6:&), it is still larger than the second-order term, since Ad is always less than unity. E,uanzp/e 2: A moderately relativistic collision. In this experiment 500-MeV protons are elastically scattered from hydrogen (protons) at a mean laboratory angle of 0 = 45’. The average kinetic energy of the scattered protons is 220.47 MeV. Eqs. (7-l 1) and (18) yield : At z (-2.237)40+(0.527)AO
+( - 1.582)4$‘.
(46)
When A0 z A@ 2 0.1 rad, the magnitude of the secondorder contribution is about 6.4:; of the magnitude of the first-order term. The third-order terms have not been calculated. Example 3 : An ultrarelativistic collision. Accelerators capable of producing beams of particles with energies an order of magnitude larger than those attainable today may be built in the not-too-distant future. One can conceive of an experiment in which a hydrogen target is bombarded with lOO-GeV gamma rays in order to produce neutral pions via the reaction y + p+ rro + p (Q = - 135 MeV). Assume that the pions are to be detected at a mean laboratory angle of 0 = 45’. Their average kinetic energy will be 2.96 GeV. Eqs. (19-27) and (18) yield: At z (-2.344jAO+(4.449)A02+(-
l.L72)A42.
(47)
When AO IT A$% 0.1 rad, the magnitude of the secondorder contribution is about 15.800 of the magnitude of
28
D. L. SMITH
the first-order term. It is doubtful whether the precision of an experiment such as this one would warrant consideration of the third-order terms. The author would like to thank Dr. T. G. Miller for proofreading this paper during the early stages of preparing the manuscript. The manuscript was expertly typed by Mrs. Sherry M. Ray.
References ‘) J. B. Marion science,
and
New York,
?) D. L. Smith 169.
J. L. Fowler,
Fast
nerrfro~
ph_rsics (lnter-
1960) p. 49.
and H. A. Enge, Nucl.
Instr.
and Meth. 51 (1967)
“) D. L. Smith. Nucl. Instr. and Meth. 58 (1968) 315. “) K. L. Brown, R. Belbeoch and P. Bounin, Rev. Sci. Instr. 35 (1964) 481.
INSTRUMENTS
KINEMATIC
AND
BROADENING
METHODS
63 (1968)
IN TWO-BODY
23-28;
‘g NORTH-HOLLAND
COLLISIONS
AT RELATIVISTIC
PUBLISHING
co.
ENERGIES
D. L. SMITH U. S. .-1rrr!~* Missile
Corw~mrrd, Rcvistone Arserlal, Received
15 February
Alabama
35809, U.S. A.
1968
Kinematic broadening is studied for two-body collisions in u hich the kinetic energies of the interacting particles are large enough to require relativistic analysis. Three overlapping categories of problems are treated. For each category, a formula is derived which can be used in the calculation of kinematic broadening. The first category includes collisions in which the kinetic energies of the particles may be of the same order of magnitude as their rest-mass energies. The formula which is derived for this category is very general and may be used regardless of the energies of the interacting particles. The second category consists of ultrarelativistic collisions. Here it is assumed that the kinetic energies of
the incident and emitted particles are considerably larger than their rest-mass energies. Finally, the third category includes collisions which are only slightly relativistic in the sense that the kinetic energies of the interacting particles are considerably smaller than their rest-mass energies. Each of the formulas derived is an expansion in the small angular deviations from the mean direction of emission of the observed particles. These formulas may be applied to problems in atomic, nuclear and high-energy physics. Their use is demonstrated by means of three numerical examples.
1. Introduction
in an experiment in which 5.5-MeV protons were scatdetected tered elastically from 63Cu and subsequently at 45” in the laboratory by a magnetic spectrometer with a solid angle of 3.2 x lop2 steradian’). Kinematic broadening can be determined for a specific experiment by directly calculating the energies of the particles emitted in various directions encompassed by the detector. In practice, this method is generally somewhat inaccurate since one is calculating arelatively small number by subtracting two much larger numbers. An alternative method for calculating kinematic broadening will now be described. Fig. I shows the coordinate system which will be used in treating this problem. It should be noted that this is not a conventional spherical system. One may think of 8 as an azimuthal angle in the plane TPB and 4 as the angle of elevation
Most two-body collis;on experiment5 are conducted under the following conditions: A nominally monoenergetic beam of particles with mass t?zL and laboratory kinetic energy T, bombards target particles with mass mz which are at rest in the laboratory (neglecting thermal motion). Two types of particles, with masses m3 and ma, respectively, are emitted as a result of the collision process. These particles may or may not be the same as the beam and target particles depending on whether or not a transmutation has occurred. The kinematics for a two-body collision is uniquely specified by the requirements that momentum and total energy be conserved. If sufficient kinetic energy is provided by the incident particles to permit a particular collision process (with a specific Q-value) to take place, then all of the particles of the same type, which are detected in a given direction in the laboratory, will have the same kinetic energy. The only exception to this rule is that for some types of collision processes, near threshold, the kinetic energy of the detected particles may assume one or the other of two allowed values’). The unobserved (recoil) particles carry away the remaining available kinetic energy. However, if the same particles are now detected in a slightly different direction from the previous one, their measured kinetic energy will differ somewhat from its previous value. In some experiments it is necessary to use detectors which subtend appreciable solid angles in order to compensate for low yield. The spread in the energies of groups of detected particles (kinematic broadening), which is thereby introduced by kinematics, can be a major contributor to reduced resolution. Kinematic broadening has been clearly observed
Fig. 1. Coordinate system for the calculation of kinematic broadening: Point T locates the target position. Line TB identifies the incident beam direction. Line TD identifies the direction in which the scattered particles are detected. The plane TPD is perpendicular to the plane TPB. The angles @,O, and 4 are related by the equation cos0 = COS~COS~.
23
24
D. L. SMITH
of the line TD relative to the plane TPB. This particular coordinate system was chosen because it happens to be a convenient one to use in several types of problems involving finite detectors. The detector-fixed coordinate systems, which are conventionally used to describe the operation of some of these detectors, can be easily matched with the coordinate system we have chosen at the detector-to-space interface. Assume that the particles with mass nf3 are detected by a detector whose axis lies along the direction(H.4). Kinematic broadening may be calculated with greater precision if one employs formulas which express the shift AT, in the kinetic energy T, of the emitted particles in terms of a shift dfI and d4 in the emission direction. A more important advantage in having formulas for AT, which are expansions in A0 and Aq5 is that it may be possible to design detection apparatus with achromatic compensation for the kinematic broadening associated with one or more terms in the expansion2). A nonrelativistic expansion of AT,/T, to third order in A0 and A4 was derived in an earlier communication [hereafter referred to as I “)I. The objective of this present work is to investigate kinematic broadening for collisions involving relativistic energies. Expansion formulas will be derived for AT,,IT, which will permit kinematic broadening to be determined for most twobody collision processes. Three categories of collisions will be studied : I. Gene/.al collisions: The kinetic energies of the incident and emitted particles may be of the same order of magnitude as their rest-mass energies. 2. Ultrurelrrtiristic collisiotts: The kinetic energies of the incident and emitted particles are taken to be considerably larger than their rest-mass energies. 3. Sliglttlr relrrtiristic collisions: The kinetic energies of the incident and emitted particles are assumed to be considerably smaller than their rest-mass energies, but not small enough so that the nonrelativistic formula derived in I can be used.
function T, of the angles 0 and 4 varies so smoothly that rapid convergence of the expansion can be anticipated (if we avoid regions where T, vanishes) for realistic A0 and A$. For this reason, the expansion in eq. (I) is not extended beyond third order. Define the following quantities which are associated with the I?‘~ particle (II = I,..., 4): nt,: mass of the particle, u,: velocity of the particle (of = ~1,~. u,), 1’”E (1 -$!c2)-f, where c is the speed of light in a vacuum, pn = ~~nz,+t,,: momentum of the particle (y,Z = P,,* p,), E, = y,tn,c2 = (c’c): + tnics)f: total energy of the particle, T,, = E,, -tn,,c’ : kinetic energy of the particle. If Q is the energy released in the collision, then Q = T,+T,-T,. Conservation tion
of momentum P4 = Pl -P3r
(T, = 0). is expressed
(2) by the equa(3)
(Pr =O).
The kinetic energy T, may be eliminated from eq. (2) by using eq. (3) and appropriate relationships between the total energy, kinetic energy and momentum. The result is Q = T,-T,
+[Tf
+2m,c’T,
+ T~+~~H,C~T,+~~I~C~-
-2cosOCOS~(T:+7/H,cZT,)+(T~+2t?13C2T3)+]+- I?1,$C2.
(4)
Eq. (4) may be used to calculate T,. Furthermore, this equation is the starting point for deriving the expansion of At. The function Q does not depend on the direction in which the particles are detected in the laboratory; therefore Q = T3+AT3-T,+-jTf+h1c2T~+(T3+AT3)*+ +2tJ?,C’(T3+AT3)+t~~~c4-
2. General collisions
-Zcos(~+A0)cos(~+A~)(Tf+?tn,c2T,)*x
We wish to derive a third-order of the form
expansion
formula
x [(T3 + AT,)’
At-AT3/T3=(At~AO)AU+(At~A~)A~+ +(At~AU2)AO’+(At~
AOA~)AOAc~+
+(At/A~‘)A~‘+(At~A03)A03+ +(Atl
A02A~)AH’A~+
+(Ar 1AOA~2)AOA~~2
+(Ar 1A43)A43,(
1)
where the coefficients (_._I .) are functions of the collision parameters and do not depend on A0 or A4. The
+2tn3c2(T3
+ AT,)]+,+
- tu4c2.
(5)
If we equate the right-hand sides of eqs. (4) and (j), we obtain an implicit relation between AT, (or At), AU and A+. The next step in the calculation is to expand all the functions ofAt,A6,andA4 in this relation to third order in the manner described in 1. No generality is lost by taking 4 = zero degree since one is free to choose the orientation of the scattering plane:in a two-body collision. After performing a considerable amount of algebra, one obtains a relation of the form
KlNEMATlC
BROADENING
IN
At = aAO+hA02+cA~2+tfA~Ar+eAt2+fA83+
notation
E, = tt,,,cz/Tn,
(6)
13824$a”ty’k:SjsinOcost)+
2 = (2$a3 - 2$a” + 2$a”ki - 2li/a”k:) +(3$atl-
/12 = (,?&?I:)&:,
- 6$a’ki
- 3IC/at/k: - 60$a3qk:
5 = p2k:+k:+p2-2pk,k,cos0. a = ii-+
- 60$a3tlk:
I,//= [I +~+ak~+(-24at\k~-24at~k~)cosO,-‘.
+ (- 288rC/a3tl’k: + 288$a”rl’k:
the coefficients
(9)
+ 96$a3tlk;
+
-2304$a3r12k:2)sinOcos0,
(10)
- 6$aqk;
- 96$a”qk;
-
-4311/a3t/k;)cosO+(576$a3t~2k;+ + 1 I 52$03ty2k:0 + 576i,ha3tl”k:‘)cos2
0.
(11)
(8$atlk~)sinO+(2304$a3~2k:‘)sinfIcosO+ (12)
g = (24~at~k~)sinI~+(2304~~03t~~k~‘)sinOcos0,
(13)
+
0 + (230411/a”t12k:0 + 13824$a5t12k:20 + ( - I I 52$a3t12k:0 -
- 1 152t/xa3t~2k~2)cos20 +(33l 776$asq3ki6
(14)
+ 48$a3tlk:
+48$a3~k~)cosO+(
+
- 1152$a3q2k:’
-
- I 151~a3t~2k~2)cos20,
(15)
x = (6IC/aqk: - 12$avlk: +48$a3qk: + 1921C/a3qk; - 288$a’tlk: - 576$a’qkt
+ 6$atlk:
+4811/a3tlk! -
- 28811/a’qk:‘)sinQ+
+ 27648t,ha-it/3k\8)cos3
0.
(17)
+~c1~+2abe+a’.~+33~~~tle+u~-_+2~z~e’)A~~+
+
(18)
When calculating the coefficients appearing in expansion formulas for At, one should use the kinetic energy T, of the particles etnitted in the direction corresponding to the axis of the detector. collisions
The formalism of the preceding section still applies when the kinetic energies of the incident and emitted particles are large relative to their rest-mass energies. Under these conditions, several of the parameters defined in the previous section approach limit values. These limits are k,,k, -+ I, (19) iI+ p/24.
+
+331776~a’~3k~8)sin’Ocos0, lZ$atlk:
-
0+
We can obtain a solution for At in the form of eq. (1) if we apply an iteration procedure to eq. (6). This technique is described in 1. The result is the formula
3. Ultrarelativistic
+(-2311841j~~t~~k~*)sin~O,
+48tia3qk:
+ 82944$a”t/“k:’
+(g+al+cd+2nce)AOAqS’.
e = (--1+!1a+$a~+2$a~k:+$a~k:)+(-6$atlk:+
I = (- 12$aqk:-
288$a3q’ki2
- 3456$a”ty’k:‘)cos’
Ar = aAfl+(b+~d+~Ze)AOZ+cA~Z+(f+ah+bd+
+96tj~3tlk~)sinO+(-2304rl/o”tjZk:0-
- I 3824$a’r12k:‘)sin2
- 10368$a5tl’ii:‘-
+288$a”rl”k:’
- 345611/a’t12kt +
(8)
c = ( - 2411/ar/k;)cos 0,
+2304$~s~q~k:‘-
+ I2V/a”tlk~ +
+(27648$a’t~3k:‘+82944$a”t\3k:J+ (7)
b = (- 24Gorlk;) cosO+(2304~03t~2k~2)sin20,
+48t,ha3tlk;)cos
- 6$(r”k:-
144$a”tjk~“)cosO+
- 10368$a”g2k:
~1= (-48tiorfkS)sinO.
(I= (- 24$arlk’: - 24tiatlk;
ap-
(16)
+ 144$a5tlk: + 3t,hatlkS -
+ 432tia’tlk:
+4321C/astjkt+
Expressed in terms of this notation, pearing in eq. (6) are
I~I/JcJP/~:- I ?$ot&
-
3t,hatlk: + I ?rba”tlki -
t1 = pk, /(24kZ),
h = (-
+
- 165888$a5t~3k~s)sinOcos’0.
(II = 1.3),
+ 12ll/otpk’: -48$a3qk’:
+
+( - I 658881//a”r13k:” - 33 I776$a”~“P:’
P = TIP,,
f=
25
+ I 3824$a5r12k:0 -576rl/a3t12k:2 +27648tia’t12k:2+
applies to eq. (6): (II = 1.3).
k,5 = 1 +?&,,,
COLLISIONS
+( - 576t+ha3tj2kt -3456t+ha3tl’k:’
+gAOA~2+ltA02At+lA~2At+sAOAt2+-At3. The following
TWO-BODY
(20)
~~(~‘+1+~~‘-‘pcos0),
(21)
+(1+2a-ZapcosO-‘.
(22)
Eq. (18) is still used to calculate At in the ultrarelativistic limit; however, we may now use the following limit values for the parameters which appear in this equation: cI + (- Z$ap)sinO, (23) b+(-Il/ap)cosO+(4$a3p2)sin’0,
(24)
D. L. SMITH
26
(29
c+(-l+hp)cose, n-+(-2$ap+8$(r3p)sin8+(-8$03p2)sinBcos6r,
(26)
e+(-$o+4$a3)+(-8~a3p)cos0+(4t//a3p2)cos2~,
(27)
~f+(+~crp)sinO+(4$a3p2)sin~cos8+ +( - 16$05p3)sin3
0,
g + ($0~) sin 8 + (41C/a3p2)sin Bcos 0,
(28) (29)
A+(-$ap+411/03p)cos0+(8$a3p2-481//05p2)sinZO+
+(-41C/a3p2)cos2bl+(48$o”p3)sin20cos0,
(30) (31)
i - (- Ic/op+ 4$03p) c0Se+(-4tj&+0S20, ?c-t(121C/a3p-481j/osp)sin8+(
-8$(r”p’+
+96$~5p2)sin~cos~+(-48~~5p3)sinBcos’0,
(32)
i+(4t+!/a3-16$a’)+(-4$a3p+48$05p)co~O+ +( -48$c5p2)cos2 Eq. (4) should
0 + ( 16$a5p3)cos3
(33)
T3.
be used to calculate
4. Slightly relativistic
0.
collisions
In this limit,
T,,
(II =
-$ I,
p;/1n,2c2 4 1.
1,3,4),
(II = 1,3,4).
{P:/(~+~)L
(35)
1 + Sii,),
(n = 1,3,4),
(36)
(H = 1,3,4).
(37)
Eqs. (36) and (37) may be used to eliminate Ta from eq. (2). The result, to first order in 6, and S,, is Q = {l+(m3/JJ7,)}T3-
{l-(ml/JJJ~)}T~
-(2/JJ2,)(JJtIT,iJz3T3)~COse+ +(~~,/JJ~~)[(JJ~,~J~~-JJ~~)T~-~z~JJ~~T~+(4JJ+/JJ3(JJz,T,JJz3T3)+cosU-2JJ1~nJ3T,cosZfl]+ +(~S3/nz:)[(JJ13m:-nz:)T3-JJz,rJJ:T,+
-(JJJ1/JJ3T,/T3)+cos0]-‘+ +ii,
1[(-,/Jf;/?z,/J?l:,+ \
+(/JJ://JJ~-$)(JiJ1JJJ3T~/T3)~cose-
-
x
x [MS + 1H3- (JJyJJ3T,/T3)%osc)]-2+ +[(21?1://~l:-_t)(1?1~111~T~/T~)3sinO-(2JJ~fJJz,/JJJ$si1leCose]
where the first term is the familiar nonrelativistic kinetic energy and the second term is the lowest-order relativistic correction. Another approximation which is useful in the slightly relativistic limit is P,, = (2,n,T,)+(
(38)
The d, and d, terms are the lowest-order relativistic corrections to the familiar nonrelativistic Q-equation (e.g. see I). The assumption will now be made that for slightly relativistic collisions one only needs to calculate the lowest-order relativistic contribution for the first-order (AU) term of the expansion in eq. (1). We will use the nonrelativistic expressions derived in I for the secondorder terms and discard the third-order terms. The resulting formula will be sufficiently accurate for many applications. The derivation of (At/A@, including the lowest-order relativistic contribution, is initiated by calculating the differential dQ of Q given in eq. (38) with 0 and T3 as the variables. When we set dQ = 0, we obtain a useful relation between dT, and de. To first order, the relation between AT, and A0 is the same as the relation between the differentials. If all but the first-order terms in 5, and 6, are eliminated, we are left with the following formula for (At/Al?):
-(,,,~m,/~~~~)~os~U][2(r~1,r~~~T,/T~)~sine] (34)
approaches
T, = {~,z/(2m,))-
-2rJl,rJl~z;~~~‘e]*.
(At1 AO) = -~(JJz,JJJ,T,/T,)~S~~~I[JJI,+JJI,-
A somewhat different approach will now be used to derive a formula for At applicable to the study of collisions which are only slightly relativistic. The slightly relativistic limit may be defined by the following inequali ties : 6, = T,/(JJJ,c2)
+(4111:--i1:)(IJIITI~i13T3)tC0Se-
x
X [IJJa + /JJ3 -(/7J,/JZ3TL~~f3)~COS~]-1
I +b,[(JJJ,
- JJJ:/JJJ:)-
X [Jill
,fi13T,/T3)+sine]
+
LJ~J:Tl /(2J71:T3)}
(JJI
- (JJJ~IJz~T~/(JJz~T~))cos~U] x [2(/J?
I
+
x x
+ JJJ3 -(JJIIJi13T,/T3)+COS~]-2+
+II(~JJJ:/JJJS-~)(JJJ,J~J~T,/T~)~S~~~- ~~JJ~~~J~~~~/(JJJ~~~))S~I~~COS~] X [JJJ5 + JJZ3 -(IJI~JJJ~T~/T~)+cos~]-~
X
)
.
(39)
The dependence of Q on 4 is not included in eq. (38) because it was shown in the two preceding sections that the &dependence is at most second order near & = 0”. It will be seen that we will have to concern ourselves only with the calculation of the first-order (33) term in the present section.
KINEMATIC
From I we obtain tivistic) : (dtld0’)
the second-order
BROADENING
terms
= -~cos0+2~2sin20-/?3sin28cos8, (AtlA0~4) (AC~A@) =
= 0, -~cos~.
(nonrela-
(40)
5. Applications
(43)
[eq. (38) with 6, = 6, = T3.
of the formulas for At
Two important related applications for the formulas which were derived in the preceding section will now be described. The first application is the calculation of overall kinematic broadening for a specific detector geometry. The direction (0,4 = 0) is assumed to lie along the axis of the detector. Suppose At assumes its minimum value Atmin (usually less than or equal to zero) when A0 = AO, and A$J = A@, and its maximum value At,,, (greater than zero) when At? = AO, and A+ = Adz. Then the overall fractional kinematic broadening is given by At,,,,,,,
= At,,,,-At,i”.
COLLISIONS
27
initial conditions vector when the kinetic energies of the particles are kinematically broadened. It is appropriate to use this form for At since A% and A4 also appear in the initial conditions vector to specify the particle’s direction of motion. 6. Numerical examples
(44
p 3 (~“,,?z3T~/T~)tsinO[~llf+/)z3-
The nonrelativistic Q-equation = 0] may be used to calculate
TWO-BODY
(41)
where
-(In,m,T,/T3)~cosn]-‘.
IN
(44)
It is generally not too difficult to determine AO,, A$,, AO,, and A42 by inspection. Of course, there are many other factors which contribute to the peak shapes in spectra recorded by a detector. The formula in eq. (44) merely gives an upper limit for the contribution one can expect from kinematic broadening. A second application of the formulas for At is in matrix calculations for ion optical elements. The motion of a particle entering an optical element may be characterized mathematically by an initial conditions vector whose components are functions of the parameters of motion: position, direction of motion, and kinetic energy. One obtains a new vector, which describes the motion of the particle as it leaves the optical element, by multiplying the initial conditions vector by a matrix which represents the optical element”). The kinetic energy of a particle is usually specified in terms of its fractional difference AT/T from some average value T for the group. The parameter ATJT appears in the initial conditions vector. An optical element is said to be achromatic whenever its effect upon a particle is independent of AT/T. One may substitute At, in the form of an expansion like eq. (I), for AT/T in the
Three numerical examples will be treated in this section. Coefficients of terms in the appropriate expansions for At will be calculated for collisions which might be studied in laboratory experiments. E.~amnple 1: A slightly relativistic collision. The experiment involves the detection of 50-MeV protons which are scattered elastically from 40Ca at a mean laboratory angle of 8 = 90”. For this collision, T3 =47.56 MeV. Eqs. (39-42) and the fact that (AflA4) = 0 lead to the result: At z (-0.0500-0.0013)Ad+(0.0013)A~12.
(45)
The second term of the AO coefficient in eq. (45) is the relativistic contribution. Even though the relativistic contribution to the first-order term is small (2.6:&), it is still larger than the second-order term, since Ad is always less than unity. E,uanzp/e 2: A moderately relativistic collision. In this experiment 500-MeV protons are elastically scattered from hydrogen (protons) at a mean laboratory angle of 0 = 45’. The average kinetic energy of the scattered protons is 220.47 MeV. Eqs. (7-l 1) and (18) yield : At z (-2.237)40+(0.527)AO
+( - 1.582)4$‘.
(46)
When A0 z A@ 2 0.1 rad, the magnitude of the secondorder contribution is about 6.4:; of the magnitude of the first-order term. The third-order terms have not been calculated. Example 3 : An ultrarelativistic collision. Accelerators capable of producing beams of particles with energies an order of magnitude larger than those attainable today may be built in the not-too-distant future. One can conceive of an experiment in which a hydrogen target is bombarded with lOO-GeV gamma rays in order to produce neutral pions via the reaction y + p+ rro + p (Q = - 135 MeV). Assume that the pions are to be detected at a mean laboratory angle of 0 = 45’. Their average kinetic energy will be 2.96 GeV. Eqs. (19-27) and (18) yield: At z (-2.344jAO+(4.449)A02+(-
l.L72)A42.
(47)
When AO IT A$% 0.1 rad, the magnitude of the secondorder contribution is about 15.800 of the magnitude of
28
D. L. SMITH
the first-order term. It is doubtful whether the precision of an experiment such as this one would warrant consideration of the third-order terms. The author would like to thank Dr. T. G. Miller for proofreading this paper during the early stages of preparing the manuscript. The manuscript was expertly typed by Mrs. Sherry M. Ray.
References ‘) J. B. Marion science,
and
New York,
?) D. L. Smith 169.
J. L. Fowler,
Fast
nerrfro~
ph_rsics (lnter-
1960) p. 49.
and H. A. Enge, Nucl.
Instr.
and Meth. 51 (1967)
“) D. L. Smith. Nucl. Instr. and Meth. 58 (1968) 315. “) K. L. Brown, R. Belbeoch and P. Bounin, Rev. Sci. Instr. 35 (1964) 481.