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Space Charge in Channel Multipliers
Space Charge in Channel Multipliers W. BAUMGARTNER and B . GILLIARD Imtitut fur Technwche Phyaik, E T H Zurich, Switzerland
INTRODUCTION In an earlier paper1 formulae relating to a theory of space charge saturation in channel multipliers were deduced. Their evaluation consists mainly in carrying out intricate multiple integrals. It is the aim of the present paper to report some results of the numerical determinations and to compare them with experimental findings. The phenomenon of space charge saturation which occurs in channel multipliers can be described as follows: a cloud of secondary electrons is built up from one electron by the process of electron multiplication. The cloud travels along the tube growing steadily. Within it a space charge potential is created which reduces the time of flight and the gain in energy of the secondary electrons. The space charge density has its maximum value when (on the average and for a given axial field E ) one secondary electron is emitted from the wall at each impact of an electron from the cloud. The theoretical considerations are based mainly on three assumptions : (i) The wall potential is a linear function of the axial coordinate z. (ii) The axial extent of the cloud is large compared to the tube diameter R,or, in other words neither the space charge density nor the space charge potential depends on x . Consequently the theory does not hold in an intermediate range between space charge free and space charge limited operation. (iii) The initial velocities of the secondary electrons exhibit a Maxwellian distribution. THETIMEOF FLIGHT Figure 1depicts the circular tube section of a channel multiplier. The radial and the tangential coordinates are denoted by r and 8 respec113
114
W. BAUMGARTNER AND B. OILLIARD
tively. The path of a secondary electron projected transversely will be a straight line if there is no space charge. The latter causes a deflection and a,point of return at r = rm.
FIQ.1. Section of channel multiplier tube with trajectory of e secondary electron (not to scale).
The components to, Rd, and 2, of the initial velocity are measured in and are denoted by the dimensionless units of its RMS value ~'(5) variables p, a and E with the limits 0
co; -
00 < a <
00; -03
03.
Then the fraction d3j(R) of the secondary electron current density, j ( R ) leaving the wall with definite values p, a, E is given by
+ +
d3j(R; p, a, E) = j ( R )8/7r exp[ -2( pa u2 E 2 ) l p dpdudE. (1) Expressions for the space charge density ~ ( rand ) the space charge potential +(r)have been deduced.l In both cases a parameter /3 appears. Its significance becomes clear by calculating the space charge per unit of tube length. One finds where /3
> 1 and el',
=
U, = ntcO2/2). ~
115
SPACE CHARQE IN CHANNEL MULTIPLIERS
As ,6 approaches unity, the space charge increases monotonically. The charge distribution is approximately homogeneous for 8 > 3 but becomes strongly concentrated near the tube wall for ,6 < 3. The dominant magnitude in the theoretical considerations is represented by the time of flight ~ ( i ”8,;, 8). With 2
= (v/R)2,2m = r,/R and + ( P , a, 8) =
d/(coa)/R7 (i0 , 8 0 ; 8)
we have, according to Baumgartner and Schniid,l I
dx
Xm
1
dx
(4)
xln
N(z; p ,
,6) is represented as a function of x in Fig. 2.
0;
N
X\k
P‘
/
-
-U‘
\ X
FIQ.2. N agein&,z in the interval ( 0 , l )for the following values of 8: (8’ = 00,
8’’ > 8”’)
116
W. BAUMGARTNER AND B . GILLIARD
Despite the zero a t z = xm the integral is finite for u # 0. For there exists a critical value pc, defined by Pc2 = In
8/(8 -
u =
0
(5)
11,
yielding i = co. In fact a secondary with p < pc will reach r = rm and then return. With p > pc it is retarded but it crosses the axis and lands on the opposite wall. Finally a secondary with p = pc (and u = O ! ) will stop at the axis and remain there. The following approximations may easily be deduced i.
= 2 / d [ p 2 - hi P/(8 - I)],
+
x 2(13
- l)p,
p a - In P2@
B/(B -
1)
- 1) Q 1.
> u2,
J
(6)
?( p , a; /I) has been computed by numerical integration on Computer CDC 6500. Representative results are given in Fig. 3. The ourves show a sharp maximum at p = pc for low values of a. They become smoother for higher values of a.
10' -
10-21
I
I
I
, I
1 4 1
P
FIQ.3. Time of flight against initial radial velocity p for various values of the parameters (I arid B.
117
SPACE C H U U E IN CHANNEL MULTIPLIERS
THESATURATION CURRENTIs A general formula for I S suitable in the context of a numerical determination follows by combining equations (4.8) and ( 5 . 6 ) of Baumgartner and Schmid: One gets 1 s = g[y/(nUo)/rnlVn[~~~(B)]/S; where VR = E R , .*.
Is
==
4.13.10-51/IroV~[~'(B)]IBA.
1
(7)
Here V , and VR are measured in volts, and m is the mass of the electron. The parameter p2(8) in Eq. (7)is defined by
of course closely related to the relative dispersion of the time of is given by the simple relation1 flight. Whereas
p2(B) is
~m
__
f(S) = 2/W8
-.
1)iP;
(10)
must be computed. p 2 ( # l )is depicted as a function of (#l - 1) in Fig. 4.
I
10-1
I 100
CP-1) Bra. 4. pZ(j3) 8s ti function of ( p -- 1).
I
lo'
I 102
118
W. BAUMQARTNER AND B. QILLIARD
It is interesting to notice the minimum of p2(8) a t /I x 4-5; in fact the relative dispersion will be particularly small when the domain of moderate variation of ? ( p , U ; /3) near po2 coincides with the domain of the highest frequency in the velocity distribution p exp( --Bp2). Hence pc2=ln/I/(/3
.'.fi = 4-52.
- 1) = 1/4,
(11)
To find the value of Is corresponding to a given field Id we introduce the coefficient of secondary emission 6( U ) of the tube wall where U I U (p, u, I ;P), denotes the energy of impact and U R = e VR
u(P,0, 8) =
UR2/4Uo * ' ( P ,
0;
8)
+ URl?(P,
/I)
0;
+ uo.
(12)
The explicit dependence of 6 ( U ) on the angle of impact is neglected for the sake of simplicity. Furthermore the condition of self-consistency of the space charge regime is --_
S ( U ) = 1. (13) Equation (13) determines the value B P", characterising the space charge mode for a given wall material and a given V R . First we apply Eq. (13) to a simple linear dependence of 8 on U ;=
Making use of Eq. (12) we get immediately
VR/ad[
vo(vi - Vo)] = 1/1/[+'(8*)1.
The right hand side decreases monotonically with increasing
Table I.
'FABLE
~
m-a
~~
1.85 101 7.11 loo 2.69 100 1.62 loo 9.12 10-1
1-01 1.03 1.10 1.20
1.50 1-60 2.00
8.34 6.53
10-1 10-1
Inserting the minimum value of VR
3
(15) see
1
0)-+B
B
8,
~~
3.00 5.00
10.00
20-00 30.00 50.00 a,
?'(@
5.10 4.28 3.78 3.48 3.39 3.28 3.06
10-1 10-1 1010-1 10-1 10-1 10-1
into Eq. (15) gives
2/([3vo(vi- vo)]/S) =
vR*.
(16)
SPACE CHARGE IN CHANNEL MULTIPLIERS
Fin. 5.
119
l3against the voltage ratio V~/t’a*.The numbers along t.he curve indicat.ethe respective values j3*, of the space charge parameter j3,
FIG.6 . Computed values of U / U Magainst (j3 - 1 ) with ~ V Rl ~( l . ’ ~ )as - l aparametdr. The intersection between the ordinate A and the curve for a definitevalueof 2 Vit( VoVy)-h yields the corresponding value of fi*.
W. BAUMQARTNER AND B . OILLIARD
180
We therefore recognise the existence of a threshold value VR* of VR below which a self consistent space charge operation is impossible. A definite value of p* belongs to each VR beyond VR* which, after its introduction into Eq. (7) yields the corresponding magnitude of I s (Fig. 5):
1 s = z/C(4/mlV,(V, - V,)* = 8.27 10-5V0(V1 - V,)t TA.
J
(17)
A more realistic approximation to the behaviour of S(U) may be stated formally as follows
I"
n
Vn
/v:
FIQ.7. Saturation current against, V R / V R * . The solid and tho broken line hold for &(V) and &(U)respectively. The dotted line applies to & ( U ) with the additions1 assumption pa@) = pa(,). I
SPACE CHARGE IN CHANNEL MULTTPLIERS
121
The condition of Eq. (13) then entails
Eq. (18) is not reducible to a simple relationship. But putting A again yields a threshold voltage VR* (Fig. 6).
<1
I.26mm
FIG.8. Experimental single pulse output of two channel multipliers operating at space chesge sat,urst>ion.
In addition, a glance at Fig. 6 shows a definite value /I*for each value of V X .> V s * . T o facilitate the comparison of Isl and I s 2 corresponding to a,( U ) and a,( U ) respectively they have both been normalised to one for V,/ VR* = 4 (Fig. 7 ) . The solid line related to a,( U) is apparently less bowed than the broken line representing the cam of 6,(U).
122
W. BAUMGARTNER AND B. QJLLIARD
A better fit to the shape of 6( U ) reveals only minor changes of the slope and bowing of the corresponding Is - V R curve. Expressions for S(U) showing a secondary emission maximum may allow two operating modes with different space charge densities, only one of which is stable. Figure 8 illustrates by two examples the general behaviour of the experimental results. The existence of VR* is apparent, as is the almost straight form of the curves. Moreover the actual values of I S fit remarkably well with the statements of the theory.
CONCLUSIONS The exact numerical computation of the output current of channel multipliers operating at space charge saturation corroborates previously published results1 and supplements them on various points. The theory predicts the experimental facts to a satisfactory degree provided allowances are made for the respective properties of the tube wall. ACKNOWLEIDQMENTS The contents of this paper contribute to developmental work sponsored by the Schweizerischer Nationalfond. The experimental results are taken from meamwements made by M. Lorenzi.
REFERENCE 1. Baurngartner, W. and Schmid, J., J . Phya. D 6, 1769 (1972).
INTRODUCTION In an earlier paper1 formulae relating to a theory of space charge saturation in channel multipliers were deduced. Their evaluation consists mainly in carrying out intricate multiple integrals. It is the aim of the present paper to report some results of the numerical determinations and to compare them with experimental findings. The phenomenon of space charge saturation which occurs in channel multipliers can be described as follows: a cloud of secondary electrons is built up from one electron by the process of electron multiplication. The cloud travels along the tube growing steadily. Within it a space charge potential is created which reduces the time of flight and the gain in energy of the secondary electrons. The space charge density has its maximum value when (on the average and for a given axial field E ) one secondary electron is emitted from the wall at each impact of an electron from the cloud. The theoretical considerations are based mainly on three assumptions : (i) The wall potential is a linear function of the axial coordinate z. (ii) The axial extent of the cloud is large compared to the tube diameter R,or, in other words neither the space charge density nor the space charge potential depends on x . Consequently the theory does not hold in an intermediate range between space charge free and space charge limited operation. (iii) The initial velocities of the secondary electrons exhibit a Maxwellian distribution. THETIMEOF FLIGHT Figure 1depicts the circular tube section of a channel multiplier. The radial and the tangential coordinates are denoted by r and 8 respec113
114
W. BAUMGARTNER AND B. OILLIARD
tively. The path of a secondary electron projected transversely will be a straight line if there is no space charge. The latter causes a deflection and a,point of return at r = rm.
FIQ.1. Section of channel multiplier tube with trajectory of e secondary electron (not to scale).
The components to, Rd, and 2, of the initial velocity are measured in and are denoted by the dimensionless units of its RMS value ~'(5) variables p, a and E with the limits 0
co; -
00 < a <
00; -03
03.
Then the fraction d3j(R) of the secondary electron current density, j ( R ) leaving the wall with definite values p, a, E is given by
+ +
d3j(R; p, a, E) = j ( R )8/7r exp[ -2( pa u2 E 2 ) l p dpdudE. (1) Expressions for the space charge density ~ ( rand ) the space charge potential +(r)have been deduced.l In both cases a parameter /3 appears. Its significance becomes clear by calculating the space charge per unit of tube length. One finds where /3
> 1 and el',
=
U, = ntcO2/2). ~
115
SPACE CHARQE IN CHANNEL MULTIPLIERS
As ,6 approaches unity, the space charge increases monotonically. The charge distribution is approximately homogeneous for 8 > 3 but becomes strongly concentrated near the tube wall for ,6 < 3. The dominant magnitude in the theoretical considerations is represented by the time of flight ~ ( i ”8,;, 8). With 2
= (v/R)2,2m = r,/R and + ( P , a, 8) =
d/(coa)/R7 (i0 , 8 0 ; 8)
we have, according to Baumgartner and Schniid,l I
dx
Xm
1
dx
(4)
xln
N(z; p ,
,6) is represented as a function of x in Fig. 2.
0;
N
X\k
P‘
/
-
-U‘
\ X
FIQ.2. N agein&,z in the interval ( 0 , l )for the following values of 8: (8’ = 00,
8’’ > 8”’)
116
W. BAUMGARTNER AND B . GILLIARD
Despite the zero a t z = xm the integral is finite for u # 0. For there exists a critical value pc, defined by Pc2 = In
8/(8 -
u =
0
(5)
11,
yielding i = co. In fact a secondary with p < pc will reach r = rm and then return. With p > pc it is retarded but it crosses the axis and lands on the opposite wall. Finally a secondary with p = pc (and u = O ! ) will stop at the axis and remain there. The following approximations may easily be deduced i.
= 2 / d [ p 2 - hi P/(8 - I)],
+
x 2(13
- l)p,
p a - In P2@
B/(B -
1)
- 1) Q 1.
> u2,
J
(6)
?( p , a; /I) has been computed by numerical integration on Computer CDC 6500. Representative results are given in Fig. 3. The ourves show a sharp maximum at p = pc for low values of a. They become smoother for higher values of a.
10' -
10-21
I
I
I
, I
1 4 1
P
FIQ.3. Time of flight against initial radial velocity p for various values of the parameters (I arid B.
117
SPACE C H U U E IN CHANNEL MULTIPLIERS
THESATURATION CURRENTIs A general formula for I S suitable in the context of a numerical determination follows by combining equations (4.8) and ( 5 . 6 ) of Baumgartner and Schmid: One gets 1 s = g[y/(nUo)/rnlVn[~~~(B)]/S; where VR = E R , .*.
Is
==
4.13.10-51/IroV~[~'(B)]IBA.
1
(7)
Here V , and VR are measured in volts, and m is the mass of the electron. The parameter p2(8) in Eq. (7)is defined by
of course closely related to the relative dispersion of the time of is given by the simple relation1 flight. Whereas
p2(B) is
~m
__
f(S) = 2/W8
-.
1)iP;
(10)
must be computed. p 2 ( # l )is depicted as a function of (#l - 1) in Fig. 4.
I
10-1
I 100
CP-1) Bra. 4. pZ(j3) 8s ti function of ( p -- 1).
I
lo'
I 102
118
W. BAUMQARTNER AND B. QILLIARD
It is interesting to notice the minimum of p2(8) a t /I x 4-5; in fact the relative dispersion will be particularly small when the domain of moderate variation of ? ( p , U ; /3) near po2 coincides with the domain of the highest frequency in the velocity distribution p exp( --Bp2). Hence pc2=ln/I/(/3
.'.fi = 4-52.
- 1) = 1/4,
(11)
To find the value of Is corresponding to a given field Id we introduce the coefficient of secondary emission 6( U ) of the tube wall where U I U (p, u, I ;P), denotes the energy of impact and U R = e VR
u(P,0, 8) =
UR2/4Uo * ' ( P ,
0;
8)
+ URl?(P,
/I)
0;
+ uo.
(12)
The explicit dependence of 6 ( U ) on the angle of impact is neglected for the sake of simplicity. Furthermore the condition of self-consistency of the space charge regime is --_
S ( U ) = 1. (13) Equation (13) determines the value B P", characterising the space charge mode for a given wall material and a given V R . First we apply Eq. (13) to a simple linear dependence of 8 on U ;=
Making use of Eq. (12) we get immediately
VR/ad[
vo(vi - Vo)] = 1/1/[+'(8*)1.
The right hand side decreases monotonically with increasing
Table I.
'FABLE
~
m-a
~~
1.85 101 7.11 loo 2.69 100 1.62 loo 9.12 10-1
1-01 1.03 1.10 1.20
1.50 1-60 2.00
8.34 6.53
10-1 10-1
Inserting the minimum value of VR
3
(15) see
1
0)-+B
B
8,
~~
3.00 5.00
10.00
20-00 30.00 50.00 a,
?'(@
5.10 4.28 3.78 3.48 3.39 3.28 3.06
10-1 10-1 1010-1 10-1 10-1 10-1
into Eq. (15) gives
2/([3vo(vi- vo)]/S) =
vR*.
(16)
SPACE CHARGE IN CHANNEL MULTIPLIERS
Fin. 5.
119
l3against the voltage ratio V~/t’a*.The numbers along t.he curve indicat.ethe respective values j3*, of the space charge parameter j3,
FIG.6 . Computed values of U / U Magainst (j3 - 1 ) with ~ V Rl ~( l . ’ ~ )as - l aparametdr. The intersection between the ordinate A and the curve for a definitevalueof 2 Vit( VoVy)-h yields the corresponding value of fi*.
W. BAUMQARTNER AND B . OILLIARD
180
We therefore recognise the existence of a threshold value VR* of VR below which a self consistent space charge operation is impossible. A definite value of p* belongs to each VR beyond VR* which, after its introduction into Eq. (7) yields the corresponding magnitude of I s (Fig. 5):
1 s = z/C(4/mlV,(V, - V,)* = 8.27 10-5V0(V1 - V,)t TA.
J
(17)
A more realistic approximation to the behaviour of S(U) may be stated formally as follows
I"
n
Vn
/v:
FIQ.7. Saturation current against, V R / V R * . The solid and tho broken line hold for &(V) and &(U)respectively. The dotted line applies to & ( U ) with the additions1 assumption pa@) = pa(,). I
SPACE CHARGE IN CHANNEL MULTTPLIERS
121
The condition of Eq. (13) then entails
Eq. (18) is not reducible to a simple relationship. But putting A again yields a threshold voltage VR* (Fig. 6).
<1
I.26mm
FIG.8. Experimental single pulse output of two channel multipliers operating at space chesge sat,urst>ion.
In addition, a glance at Fig. 6 shows a definite value /I*for each value of V X .> V s * . T o facilitate the comparison of Isl and I s 2 corresponding to a,( U ) and a,( U ) respectively they have both been normalised to one for V,/ VR* = 4 (Fig. 7 ) . The solid line related to a,( U) is apparently less bowed than the broken line representing the cam of 6,(U).
122
W. BAUMGARTNER AND B. QJLLIARD
A better fit to the shape of 6( U ) reveals only minor changes of the slope and bowing of the corresponding Is - V R curve. Expressions for S(U) showing a secondary emission maximum may allow two operating modes with different space charge densities, only one of which is stable. Figure 8 illustrates by two examples the general behaviour of the experimental results. The existence of VR* is apparent, as is the almost straight form of the curves. Moreover the actual values of I S fit remarkably well with the statements of the theory.
CONCLUSIONS The exact numerical computation of the output current of channel multipliers operating at space charge saturation corroborates previously published results1 and supplements them on various points. The theory predicts the experimental facts to a satisfactory degree provided allowances are made for the respective properties of the tube wall. ACKNOWLEIDQMENTS The contents of this paper contribute to developmental work sponsored by the Schweizerischer Nationalfond. The experimental results are taken from meamwements made by M. Lorenzi.
REFERENCE 1. Baurngartner, W. and Schmid, J., J . Phya. D 6, 1769 (1972).