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Instrumental limit and number of effective granules in superheated superconducting detectors
Nuclear Instruments and Methods 214 (1983) 415-418 North-Holland Publishing Company
415
I N S T R U M E N T A L L I M I T AND N U M B E R O F EFFECTIVE G R A N U L E S IN S U P E R H E A T E D SUPERCONDUCTING DETECTORS A. H R I S O H O Laboratoire de I'Acc~l~rateur Lin~aire, Universit~ Paris Sud, Bat. 200, 91405 Orsay, France
G. W A Y S A N D Groupe de Physique des Solides de I'Ecole Normale Supbrieure, Tour 23, Universitb Paris VIL 75251 Paris Cedex 05, France
Received 14 January 1983
A relation has been established between the number of effective granules in a superheated superconducting detector and the noise factor of the electronic channel which is expressed in terms of the magnetic flux involved. Consequences for detector design are discussed.
1. Introduction
2. Number of effective granules
When a small granular sphere of a type I superconductor is placed in a magnetic field, the Meissner effect (expulsion of the magnetic field out of the granular sphere) remains in effect from above the thermodynamical critical magnetic field He, up to Hsh the superheated critical field [1]. In this magnetic field range H c < H ~< Hsh, the superheated sphere can irreversibly go to the normal state when it is traversed by a charged particle, provided that the incident particle loses enough energy in the sphere to overcome the energy threshold of the sphere. Then the magnetic flux penetrates into the sphere and this flux change can be detected by a surrounding pick-up coil connected to a convenient charge amplifier [2]. Because the energy threshold is strongly size dependent, in a collection of granules larger than 5 ttm in diameter only a fraction of the granule above H can be flipped to the normal state by a given irradiation [3]. Section 2 is devoted to establishing that the number, Neff, of these effective granules is a strongly decreasing function of their radius. Therefore, Neff is limited by the value rinst , the radius of the smallest detectable granule, for a given channel of a detector. In section 3, it is assumed that rinst corresponds to a granule which produces a flux change equal to the total noise of a channel. This allows the number of effective granules to be expressed as a function of the physical parameters representative of the superconducting material, the incident particles, and the electronic channel noise performance. The consequences of these results on the design of superheated superconducting detectors are discussed in section 4.
Let L 0 be the inductance of the pick-up coil surrounding the granules and H the magnetic field applied to them. The coil is connected by one or two transformers to a preamplifier followed by a bipolar shaping filter. By sweeping the magnetic field H from 0 to Hsh one can count the number of pulses at the output of the electronic channel and get a superheating curve as in fig. I. The number of counts obtained when the field reaches Hsh is the number of metastable granules in the coil. Due to screening of the magnetic field by percolating granules, the beginning of the curve is not linear [4]. The applied field is always taken in the linear portion of the superheating curve. The slope of this linear part can be expressed by:
0167-5087/83/0000-0000/$03.00 © 1983 North-Holland
ON OH
kN H~h -- He'
(1)
where k is a numerical constant. gr~a,de N=282'14 "100 ~ ~
"" I •" ," •"
..1"
•' I 0
.r
............
~.....~
...............
""
.
[ [ ]
] i
Hc
"H~ H
I
.-
H orb.un
Fig. I. A typica] superheated superconducting curve for a
collection of tin granules. N is the registered number of granules when the magnetic field is increased from 0 G.
A. Hrisoho, G. Waysand / Superheated superconducting detectors
416
The irradiation can be characterized by the mean energy deposition in a granule of radius r:
C-;rDE .
AE=J0
Drdr'
(2)
approximation, is independent of r, in which case Nef f varies as 1 / r 6. More frequently expression (2) varies as 1 / r 5 and for high energy particles, we can write: 4 DE A E = 3r D~-
(11)
where DE~Dr is the energy loss per unit length in the metal, 4r is the average thickness of metal crossed by a parallel flux of particles irradiating a granule of radius
and
F.
Neff = k f V Hsh - H c Dr 3 ~ 2 r 5 C p "
a E is converted into heat raising the temperature of a fraction of the granule by a T. During the elevation of temperature the superheated critical magnetic field Hsh(T ) is reduced to:
Hsh(T+AT)=Hsh--
~
AT
(4)
the transition occurs. For the transition to occur, it is sufficient that condition (3) be satisfied only at one point within the equatorial circle of the sphere perpendicular to the direction of the magnetic field. For tin and indium, the volume to be heated up in order to satisfy (3) is half that of the sphere stated above [5]. Thus, AE
-~r3C.
(5)
where Co is the specific heat of the superconductor in the normal state. The maximum reduction of Hsh is DHsh AH= --~AT=
Dnsh A E DT -~rr3C p .
(6)
U n d e r irradiation with an energy loss A E, only the granules which are in the range IH, H + A H I of the superheating curve will undergo a change of state. They are the effective granules and their number is given by:
kN Hs h _ H c A H ,
=
(12)
Nef f varies as 1/r 5. The smaller rinst is, the electronic
threshold, the larger Nef f will be. The next section is devoted to the evaluation of rinst.
3. Smallest detectable granule and estimation of Nef f
Hsh(T + A T ) <~H
Nef f
1
(3)
and if
aT=
DHsh/OT OE
(7)
3.1. Noise The lumped parameter equivalent circuit for the pick up coil and the transmission line connecting it to the preamplifier is represented by fig. 2 following ref. 2. The input impedance of a charge sensitive amplifier, for the frequencies of interest, can be considered as real. For noise calculations Ca, the input capacitance and C t the capacitance of the transmission line can be neglected. As usual we separate the noise between a noise voltage generator e , ( t ) with a resistance r, representing the series noise of the input F E T and a noise current generator in(t ) in parallel with the inductance and with a resistance Rp corresponding to the noise in the passive elements. This gives the representation of fig. 3 where
L = n2Lg.
(13)
Lg is equal to the inductance of the pick up coil because it is larger than that of the connecting line and the signal is given by: e s = Aq~/A t,
(14)
where
DHsh/DT NAE Nef f = k Hs h _ Hc 2~r3Cp.
(8)
Aq) = ~ r ( r 3 / l ) H
(15)
Let f be the filling factor of the granule in the volume V of the pick up coil.
and l is the width of the pick up coil when it is a U-shaped one, or is the loop diameter when it is a circular loop. We shall define
N=
en(t)=qornN(t),
,
fV
3~r
3'
. ~ DHsh/DT, ~ 1 Nef f = gJv Hs h _ He ~.l/Z _89¢r2r6Cp .
(16)
(9)
(10)
This relation shows that Neff is a strongly decreasing function of the radius of the granules. If the detected particles are "t rays, their energy loss is deposited in one spot of the granule and, in first
N
Fig. 2. Equivalent circuit of a channel in a superheated superconducting detector, n is the total transformation ratio.
A. Hrisoho, G. Waysand / Superheated superconducting detectors Q
This noise evaluation is valid if A t, the penetration time, is smaller than t M. On the other hand, in order to detect all the flux change, we need:
= "b/CCW----~
e. (t) -
417
~R d
L / R d >i t M.
(25)
Since we want the maximum counting rate, we choose in (25) the equality condition: (26)
Fig. 3. Signal and noise generator of the electronic channel where L = nZLg; Lg inductance of the pick up coil surrounding the superconducting granules.
t M = L//R d
where q0 is an electric charge, 8 ( 0 the delta function and we associate it with a repetition rate n(rn). Similarly, we define
This quantity is a minimum when the two terms between brackets are equal; this occurs when
in(t ) = qoS(t)
and
n (Rp).
(17)
EN4,2 - 8 k r
R d= (GRp/3)
½r.
+ ~-
1/2
,
(27)
.
(28)
The open circuit voltage shown in fig. 4 is given by
On the other hand, we know that in this type of preamplifier [2]
eoc(t)=L di"(t) dt
R d
Lq°8'(t)
withn(Rp).
(18)
With this result, the sources of signal and noise can be represented by voltage generators in series (fig. 4). The noise spectral density for the noise voltage generator is given by
e~ = 2( qorn)Zn (rn) = 4kTr n
(19)
because as usual
n ( r, ) = (2/q2o) kT/r.
(20)
e2oc= 2( Lqo)2n(rp) = 4kTL2/Rp.
(21)
Following the method and the notation of [7], we have the total equivalent noise EN,b2 expressed in volt. seconds: EN~b2
i 2 ~enaF2-4~i e o2c a F l ,
(22)
CO
= gmC f •
This means that for a given field-effect transistor (conductance gin) and a capacity Cf which is always at the minimum compatible with the feedback loop (typically under 1 pF) the minimum noise level is determined by the choice of c o. It might seem surprising that (28) is independent of L. However, we must recall that L is determined by (26): t M = L / R d. For a given detector, the value of the inductance of the pick up coil Lg is given. Since L is fixed, this imphes from (13) and (26) that n the ratio of transformation must be optimized: nopt =
[/I
~ \1/2
(RdtM/Ls) '/2 = ~[3r, Kp)
,.
[ 3 r ~1/2 11/2
aF1 = 4.
(31)
(23)
3.2. Smallest detectable granule and number of effective granules
If t M is the time when the signal is maximum ENC#2 =
4kT[ rnaF2tM + Rp L-~2 aF---L M]
Assuming that the minimum detectable granule gives a flux change equal to E----'~d/)opt,we get:
=8kT(½rntu + ~-~tM ].
(24)
7rr?nstn/l = [ ~ k T ( 3 r n / R p )1/2 Lg ] 1 / 2
ri,,t=(l/grH ) L ~
'(t) [n(Rp)]
(30)
and
where aF1 and aF2 represent the weighting function constants of the filter stage (see fig. 7 of ref. 7), here aF2 = ~
~1/2
tM/Lg )
rqo 6(t)
[oCRp)]
I/3
16
(ykT(3r,/Rp)
1/2
Lg)
(32) 1/6
(33)
and finally, introducing this result in (12), one gets:
r.
Ro
V 2 H2~Hsh/OT
e,(t)(")
AE Fig. 4. Representation of the electronic channel in terms of
voltage sources.
× [~-krCp(3rn/Rp) 16
1/2 Lg] I/2 '
(34)
418
A. Hrisoho, G. Waysand / Superheated superconducting detectors
with
V = lED.
(35)
D = length of the pick up coil
9 ~112OHsh/OT AE Nef f = ~X]l) --~sh-~---~c [ ~ k T C p ( 3 r n / R p ) L , ] ' / 2 " (36)
4. Discussion Leaving aside the numerical constants, Nef f is the product of four quantities. (1) The first one G ~kfD/(Lg) 1/2 is a product of geometrical factors. The values Lg of the pick-up coil is proportional to the length D of the coil when it is a U-shaped one.
is related to the superconducting properties of the metal and is dependent on the temperature of the helium bath. The detailed discussion of this term is outside the scope of this paper. (4) Finally the number Nef f is also proportional to A E which is evident. In conclusion, we have detailed the noise calculation of the charge preamplifier in terms of flux noise: this defines the minimum detectable granule. The size rinst of this granule allows us to evaluate the number of effective granules as a product of four quantities related to the geometry of the detector, the superconducting properties, the F E T characteristics and the incident irradiation.
=
G = AfrO.
(37)
F o r a given inductance Nef f varies like D j/2. (2) The second factor to which Nef f is proportional is
h = IkT(3rn/Rp)
1/2
1
l- •
(38)
Characteristic of the preamplifier. This unit must have a large R p / r . that is, actually, a field effect transistor with a large gin" However, in decreasing r,, t i is shortened, therefore we need not only a high gm but also a good high-frequency response for the FET. (3) The third factor
M =
H2OHsh/OT
(Hsh _/.,/c) (Cp)
(39)
References [1] C. Valette, Th6se Orsay (1971) unpublished. [2] R.L. Chase, Ch. Gruhn, A. Hrisoho, C. Valette and G. Waysand, Proc. 2nd Ispra Nuclear Electronics Symp., Publication n ° EUR 537 Oe (C.E.C., Luxembourg, 1975) p. 29. [3] D. Hueber, C. Valette and G. Waysand, Nucl. Instr. and Meth. 167 (1979) 201. [4] D. Stauffer, C. Valette and G. Waysand, Solid State Comm. 41 (1982) 305. [5] D. Hueber, Th~se de 3~me cycle, Orsay (1980) unpublished. [6] M.O. Deigton, Nucl. Instr. and Meth. 58 (1968) 201. [7] A. Hrisoho, Nucl. Instr. and Meth. 185 (1981) 213.
415
I N S T R U M E N T A L L I M I T AND N U M B E R O F EFFECTIVE G R A N U L E S IN S U P E R H E A T E D SUPERCONDUCTING DETECTORS A. H R I S O H O Laboratoire de I'Acc~l~rateur Lin~aire, Universit~ Paris Sud, Bat. 200, 91405 Orsay, France
G. W A Y S A N D Groupe de Physique des Solides de I'Ecole Normale Supbrieure, Tour 23, Universitb Paris VIL 75251 Paris Cedex 05, France
Received 14 January 1983
A relation has been established between the number of effective granules in a superheated superconducting detector and the noise factor of the electronic channel which is expressed in terms of the magnetic flux involved. Consequences for detector design are discussed.
1. Introduction
2. Number of effective granules
When a small granular sphere of a type I superconductor is placed in a magnetic field, the Meissner effect (expulsion of the magnetic field out of the granular sphere) remains in effect from above the thermodynamical critical magnetic field He, up to Hsh the superheated critical field [1]. In this magnetic field range H c < H ~< Hsh, the superheated sphere can irreversibly go to the normal state when it is traversed by a charged particle, provided that the incident particle loses enough energy in the sphere to overcome the energy threshold of the sphere. Then the magnetic flux penetrates into the sphere and this flux change can be detected by a surrounding pick-up coil connected to a convenient charge amplifier [2]. Because the energy threshold is strongly size dependent, in a collection of granules larger than 5 ttm in diameter only a fraction of the granule above H can be flipped to the normal state by a given irradiation [3]. Section 2 is devoted to establishing that the number, Neff, of these effective granules is a strongly decreasing function of their radius. Therefore, Neff is limited by the value rinst , the radius of the smallest detectable granule, for a given channel of a detector. In section 3, it is assumed that rinst corresponds to a granule which produces a flux change equal to the total noise of a channel. This allows the number of effective granules to be expressed as a function of the physical parameters representative of the superconducting material, the incident particles, and the electronic channel noise performance. The consequences of these results on the design of superheated superconducting detectors are discussed in section 4.
Let L 0 be the inductance of the pick-up coil surrounding the granules and H the magnetic field applied to them. The coil is connected by one or two transformers to a preamplifier followed by a bipolar shaping filter. By sweeping the magnetic field H from 0 to Hsh one can count the number of pulses at the output of the electronic channel and get a superheating curve as in fig. I. The number of counts obtained when the field reaches Hsh is the number of metastable granules in the coil. Due to screening of the magnetic field by percolating granules, the beginning of the curve is not linear [4]. The applied field is always taken in the linear portion of the superheating curve. The slope of this linear part can be expressed by:
0167-5087/83/0000-0000/$03.00 © 1983 North-Holland
ON OH
kN H~h -- He'
(1)
where k is a numerical constant. gr~a,de N=282'14 "100 ~ ~
"" I •" ," •"
..1"
•' I 0
.r
............
~.....~
...............
""
.
[ [ ]
] i
Hc
"H~ H
I
.-
H orb.un
Fig. I. A typica] superheated superconducting curve for a
collection of tin granules. N is the registered number of granules when the magnetic field is increased from 0 G.
A. Hrisoho, G. Waysand / Superheated superconducting detectors
416
The irradiation can be characterized by the mean energy deposition in a granule of radius r:
C-;rDE .
AE=J0
Drdr'
(2)
approximation, is independent of r, in which case Nef f varies as 1 / r 6. More frequently expression (2) varies as 1 / r 5 and for high energy particles, we can write: 4 DE A E = 3r D~-
(11)
where DE~Dr is the energy loss per unit length in the metal, 4r is the average thickness of metal crossed by a parallel flux of particles irradiating a granule of radius
and
F.
Neff = k f V Hsh - H c Dr 3 ~ 2 r 5 C p "
a E is converted into heat raising the temperature of a fraction of the granule by a T. During the elevation of temperature the superheated critical magnetic field Hsh(T ) is reduced to:
Hsh(T+AT)=Hsh--
~
AT
(4)
the transition occurs. For the transition to occur, it is sufficient that condition (3) be satisfied only at one point within the equatorial circle of the sphere perpendicular to the direction of the magnetic field. For tin and indium, the volume to be heated up in order to satisfy (3) is half that of the sphere stated above [5]. Thus, AE
-~r3C.
(5)
where Co is the specific heat of the superconductor in the normal state. The maximum reduction of Hsh is DHsh AH= --~AT=
Dnsh A E DT -~rr3C p .
(6)
U n d e r irradiation with an energy loss A E, only the granules which are in the range IH, H + A H I of the superheating curve will undergo a change of state. They are the effective granules and their number is given by:
kN Hs h _ H c A H ,
=
(12)
Nef f varies as 1/r 5. The smaller rinst is, the electronic
threshold, the larger Nef f will be. The next section is devoted to the evaluation of rinst.
3. Smallest detectable granule and estimation of Nef f
Hsh(T + A T ) <~H
Nef f
1
(3)
and if
aT=
DHsh/OT OE
(7)
3.1. Noise The lumped parameter equivalent circuit for the pick up coil and the transmission line connecting it to the preamplifier is represented by fig. 2 following ref. 2. The input impedance of a charge sensitive amplifier, for the frequencies of interest, can be considered as real. For noise calculations Ca, the input capacitance and C t the capacitance of the transmission line can be neglected. As usual we separate the noise between a noise voltage generator e , ( t ) with a resistance r, representing the series noise of the input F E T and a noise current generator in(t ) in parallel with the inductance and with a resistance Rp corresponding to the noise in the passive elements. This gives the representation of fig. 3 where
L = n2Lg.
(13)
Lg is equal to the inductance of the pick up coil because it is larger than that of the connecting line and the signal is given by: e s = Aq~/A t,
(14)
where
DHsh/DT NAE Nef f = k Hs h _ Hc 2~r3Cp.
(8)
Aq) = ~ r ( r 3 / l ) H
(15)
Let f be the filling factor of the granule in the volume V of the pick up coil.
and l is the width of the pick up coil when it is a U-shaped one, or is the loop diameter when it is a circular loop. We shall define
N=
en(t)=qornN(t),
,
fV
3~r
3'
. ~ DHsh/DT, ~ 1 Nef f = gJv Hs h _ He ~.l/Z _89¢r2r6Cp .
(16)
(9)
(10)
This relation shows that Neff is a strongly decreasing function of the radius of the granules. If the detected particles are "t rays, their energy loss is deposited in one spot of the granule and, in first
N
Fig. 2. Equivalent circuit of a channel in a superheated superconducting detector, n is the total transformation ratio.
A. Hrisoho, G. Waysand / Superheated superconducting detectors Q
This noise evaluation is valid if A t, the penetration time, is smaller than t M. On the other hand, in order to detect all the flux change, we need:
= "b/CCW----~
e. (t) -
417
~R d
L / R d >i t M.
(25)
Since we want the maximum counting rate, we choose in (25) the equality condition: (26)
Fig. 3. Signal and noise generator of the electronic channel where L = nZLg; Lg inductance of the pick up coil surrounding the superconducting granules.
t M = L//R d
where q0 is an electric charge, 8 ( 0 the delta function and we associate it with a repetition rate n(rn). Similarly, we define
This quantity is a minimum when the two terms between brackets are equal; this occurs when
in(t ) = qoS(t)
and
n (Rp).
(17)
EN4,2 - 8 k r
R d= (GRp/3)
½r.
+ ~-
1/2
,
(27)
.
(28)
The open circuit voltage shown in fig. 4 is given by
On the other hand, we know that in this type of preamplifier [2]
eoc(t)=L di"(t) dt
R d
Lq°8'(t)
withn(Rp).
(18)
With this result, the sources of signal and noise can be represented by voltage generators in series (fig. 4). The noise spectral density for the noise voltage generator is given by
e~ = 2( qorn)Zn (rn) = 4kTr n
(19)
because as usual
n ( r, ) = (2/q2o) kT/r.
(20)
e2oc= 2( Lqo)2n(rp) = 4kTL2/Rp.
(21)
Following the method and the notation of [7], we have the total equivalent noise EN,b2 expressed in volt. seconds: EN~b2
i 2 ~enaF2-4~i e o2c a F l ,
(22)
CO
= gmC f •
This means that for a given field-effect transistor (conductance gin) and a capacity Cf which is always at the minimum compatible with the feedback loop (typically under 1 pF) the minimum noise level is determined by the choice of c o. It might seem surprising that (28) is independent of L. However, we must recall that L is determined by (26): t M = L / R d. For a given detector, the value of the inductance of the pick up coil Lg is given. Since L is fixed, this imphes from (13) and (26) that n the ratio of transformation must be optimized: nopt =
[/I
~ \1/2
(RdtM/Ls) '/2 = ~[3r, Kp)
,.
[ 3 r ~1/2 11/2
aF1 = 4.
(31)
(23)
3.2. Smallest detectable granule and number of effective granules
If t M is the time when the signal is maximum ENC#2 =
4kT[ rnaF2tM + Rp L-~2 aF---L M]
Assuming that the minimum detectable granule gives a flux change equal to E----'~d/)opt,we get:
=8kT(½rntu + ~-~tM ].
(24)
7rr?nstn/l = [ ~ k T ( 3 r n / R p )1/2 Lg ] 1 / 2
ri,,t=(l/grH ) L ~
'(t) [n(Rp)]
(30)
and
where aF1 and aF2 represent the weighting function constants of the filter stage (see fig. 7 of ref. 7), here aF2 = ~
~1/2
tM/Lg )
rqo 6(t)
[oCRp)]
I/3
16
(ykT(3r,/Rp)
1/2
Lg)
(32) 1/6
(33)
and finally, introducing this result in (12), one gets:
r.
Ro
V 2 H2~Hsh/OT
e,(t)(")
AE Fig. 4. Representation of the electronic channel in terms of
voltage sources.
× [~-krCp(3rn/Rp) 16
1/2 Lg] I/2 '
(34)
418
A. Hrisoho, G. Waysand / Superheated superconducting detectors
with
V = lED.
(35)
D = length of the pick up coil
9 ~112OHsh/OT AE Nef f = ~X]l) --~sh-~---~c [ ~ k T C p ( 3 r n / R p ) L , ] ' / 2 " (36)
4. Discussion Leaving aside the numerical constants, Nef f is the product of four quantities. (1) The first one G ~kfD/(Lg) 1/2 is a product of geometrical factors. The values Lg of the pick-up coil is proportional to the length D of the coil when it is a U-shaped one.
is related to the superconducting properties of the metal and is dependent on the temperature of the helium bath. The detailed discussion of this term is outside the scope of this paper. (4) Finally the number Nef f is also proportional to A E which is evident. In conclusion, we have detailed the noise calculation of the charge preamplifier in terms of flux noise: this defines the minimum detectable granule. The size rinst of this granule allows us to evaluate the number of effective granules as a product of four quantities related to the geometry of the detector, the superconducting properties, the F E T characteristics and the incident irradiation.
=
G = AfrO.
(37)
F o r a given inductance Nef f varies like D j/2. (2) The second factor to which Nef f is proportional is
h = IkT(3rn/Rp)
1/2
1
l- •
(38)
Characteristic of the preamplifier. This unit must have a large R p / r . that is, actually, a field effect transistor with a large gin" However, in decreasing r,, t i is shortened, therefore we need not only a high gm but also a good high-frequency response for the FET. (3) The third factor
M =
H2OHsh/OT
(Hsh _/.,/c) (Cp)
(39)
References [1] C. Valette, Th6se Orsay (1971) unpublished. [2] R.L. Chase, Ch. Gruhn, A. Hrisoho, C. Valette and G. Waysand, Proc. 2nd Ispra Nuclear Electronics Symp., Publication n ° EUR 537 Oe (C.E.C., Luxembourg, 1975) p. 29. [3] D. Hueber, C. Valette and G. Waysand, Nucl. Instr. and Meth. 167 (1979) 201. [4] D. Stauffer, C. Valette and G. Waysand, Solid State Comm. 41 (1982) 305. [5] D. Hueber, Th~se de 3~me cycle, Orsay (1980) unpublished. [6] M.O. Deigton, Nucl. Instr. and Meth. 58 (1968) 201. [7] A. Hrisoho, Nucl. Instr. and Meth. 185 (1981) 213.