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Ionization loss of fast charged particles in detectors of different thickness
Volume 85A, number 3
21 September
PHYSICS LETTERS
1981
IONIZATION LOSS OF FAST CHARGED PARTICLES IN DETECTORS OF DIFFERENT THICKNESS A.S. AMBARTSUMIAN, G.M. GARIBIAN and C. YANG YerevanPhysics Institute, Armenia, USSR Received 27 April 1981
The dependence of the total energy loss (in the range of atomic frequencies) on both the thickness of the plate and the Lorentz factor of the charged particle is discussed. Special attention is given to the conditions under which tile Fermi density effect in the plate is absent. The results obtained are compared with a recent experiment.
The total energy loss of a charged particle, uniformly passing through a plate of arbitrary thickness
0.~
in vacuum, consists of loss due to the particle’s own field in matter, i.e. ionization loss [1], and additional
//
A
loss;s, from the fields arising due to the boundaries of The dependence of the total energy loss (in the range of atomic frequencies) on the thickness of the plate ,a,and the Lorentz factor of the charged particle, ‘y, is discussed in the present paper. Let us start with a plate of infinite thickness (a ~ oo). As is well known [1], the ionization loss in infinite media (~erenkov loss is also accounted for) as a function of y increases logarithmically after a minimum and then approaches an asymptotic value (the Fermi plateau). That is, when the velocity of the particle v is not close to the light velocity in vacuum c (fig. 1, region A), the loss is defined by 2e2a / W W0— “2 ~
UK
D g
Fig. 1.The range of values of (a, lg’y) for different ‘y-dependences of the ionization loss: A: a minimum and
logarithmic increase [formula(1)]; B: Fermi plateau [formula (2)]; C:
the region of sharper ~~dependence; D: logarithmic increase
[formula(6)]. taken into account, one must replace, in the formulae given above, ,c~by (2mE1 )1/2 where E1 is the max-
(1)
imum energy in the collision, and m is the electron mass. If the thickness of the plate is finite, an additional
where wp is plasma frequency of the matter, ~ is the minimum distance at which macroscopic electrodynamics is applicable, j3 = u/c, ~3is some mean atomic frequency [1]. When the velocity v is close enough to c (more ex-
loss arises, which can be interpreted as consisting of loss on transition radiation with a wide frequency spectrum and of a correction to the ionization loss, mainly concentrated in the atomic frequency range. The transition radiation generated at atomic frequen-
actly, if ~ ~‘ 1, fig. 1, region B), the loss is defined by
cies can then be absorbed by the matter of the plate,
0’)’
,
U
C’)
2e2a
w ~
=
p2 \
~
/
~‘2 c
i<
ln
0c —
in this way seemingly increasing the ionization loss.
(2)
~‘
(Fermi plateau). If the short-range collisions are also 188
The total loss was studied by Garibian et al. [2—4] in plates of finite thicknesses. It was shown that when the plate thickness
0 031—9163/81/0000—0000/$ 02.50 © North-Holland Publishing Company
Volume 85A, number 3
PHYSICS LETTERS
a’~acr
(3)
21 September 1981
3
where acr (c/W~)ln7(for a more precise formula see below), the additional loss completely eliminates the density effect, i.e. the loss will increase with 7. This effect was found in refs. [2—4]by expanding
S
‘~
________
________
________
-~
2. /
—
(
the additional loss in powers of the plate thickness, a. It was shown that when ~ ~‘ 1, the linear term in a
/
a,
has a negative sign and is determined by 7 — ~32)
W(
______
2
1)=—
c
&‘~
________
~‘)p (in—
~,r(Q~)
~c~(Q’)
(4)
“ 2 where ~i is defined in ref. [3]. Together with formula (2) the total loss logarith-
Fig. 3. The
mically increases with 7 (fig. 1, region D):
7-dependence of the total loss W0, normalized to the Fermi plateau: a1 = 1O~cm,ycr(al) = 55;a2 = 2 X cm,3.‘Ycr(a2) = 5.5 X 10~a3= 3 X iO~cm, ‘y~(a3)
~
c~2e2a
~2\
\ ~i “2 (ln_____~~-). (5) It should be noted that this formula differs from c
(1) only by the definition of the mean frequency. The term proportional to a2 is positive, i.e. it de-
creases the linear one: W(
2a2/4u~&2, 2)
=
4O~
iO~
~.
40’
5.5X i0
K 0C7
40
‘
(6)
irW~e
where &‘~is some twice averaged atomic frequency (see ref. [3]). The ratio of (6) to (4) is
acr
_4c~( —
ITWp
ln
W~7 ~\ —
—
~
(8)
The dependence is plotted in fig. 2. of acr Ofl ‘y for different materials It is seen from the above stated that under condition (3) the second-order term can be neglected, compared with the linear one. But the worse this condition is satisfied the greater is the role of the secondorder term. When a ‘‘acr,this term compensates the linear one.
IW(
2)/W(1) I a/acr ,
(7)
=
where
When a acr and 7 ~‘
~‘
1, the ionization loss does
not depend on 7. As to the energy of the transition
radiation absorbed in the layer at atomic frequencies, it can be evaluated as [5]: 2Ic)&’)ln7,
icc’ (Cm)
______
______
______
_____
_____
_____
,)—.‘
~
_____
_____
~
______
o
(9)
Wtr’~’(e
~E~EEEEE 10
lO~
!0~
{O~
I I
Wt
11
I
c~wlny
acr a a—
where a ~w/W~ is some numerical coefficient of the order of unity. When a aaci, the role of the transition radiation absorbed at atomic frequencies is small, and the loss localized in the matter practically ~-
~
Fig. 2. The y-dependence ofa~for different materials 1: lithium, 2: toluene, 3: silicon, 4: graphite.
where i~wis soi~eeffective interval of atomic frequencies determined mainly by strong absorption in the ultraviolet frequency region. The value of ~w will be of the order of several tens of hundreds of eV. From (2)and (9)we have
does not depend on ‘y (fig. 1, region B). Let us now consider the 7-dependence of the loss,
189
Volume 85A, number 3
PHYSICS LETTERS
when the plate thickness a is given. Since acr depends 7cr’ or on 7, the same plate can be “thick” if 7 ~ “thin” if 7 ~ 1cr’ where 7cr follows from (8) by sub-
stituting a for acr. When 7cr ~‘ 7 (fig. 1, regions A and B), the loss is
determined by formula (l)or (2), and when 1~ 7cr
21 September 1981
in ref. [7] loss was considered in a 100 mcm (10—2 cm) silicon foil in which the effect of the logarithmic increase with ‘y (with the electron energies available from accelerators) should not be observed because of the large thickness of the foil (nevertheless, see refs. [7,8]).
(fig. 1 , region D), by formula (5). And when 7
(fig. 1 region C), there is evidently a smooth transition between these two formulae. The 7-dependence
References
of the total loss, normalized to the plateau value, is plotted in fig. 3 (for silicon). It is seen that in the intermediate area y —‘ 7cr the 7-dependence ~5 sharper
[11 L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media (Addison-Wesley, 1957). [2] Zh. M.M. Eksp.Muradian, Teor. Fir.Izv. 37 (1959) 527. [3] G.M. G.M. Garibian, Garibian and Akad. Nauk Arm. SSR Fir. 1 (1966) 310; Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12 (1969) 1362.
,
than the logarithmic one, defined by formula (5). The effect predicted in ref. [2] was considered experimentally in ref. [6], where the ionization loss of electrons with energies from 20 to 86 MeV in foils of
polystyrene was observed. It was found that in foils with thicknesses of l0~ cm the loss did not increase
withy, and in foils with thicknesses of 10_6105 cm the increase was logarithmic. On the other hand, the measurements of ref. [6] were not absolute, but relative. Absolute measurements were carried out recently [7]. Unfortunately,
190
[4] G.M. Garibian and M.P. Lorikian, Dokl. Akad. Nauk
Arm. SSR 40 (1965) 21. [5] G.M. Garibian and G.A. Tchallkian, Zh. Eksp. Teor. Fir. 35 (1958) 1281; Izv. Nauki 12 (1959) 49. Akad. Nauk Arm. SSR Fiz.-Mat. [6] A.I. Allkhanian et al., Zh. Eksp. Teor. Fir. 44 (1963)
1122;46 (1964) 1212. [7] W. Oegle et al., Phys. Rev. Lett. 40 (1978) 1243. [8] W. Oegle, LASL Report LA-UR-76-1398.
21 September
PHYSICS LETTERS
1981
IONIZATION LOSS OF FAST CHARGED PARTICLES IN DETECTORS OF DIFFERENT THICKNESS A.S. AMBARTSUMIAN, G.M. GARIBIAN and C. YANG YerevanPhysics Institute, Armenia, USSR Received 27 April 1981
The dependence of the total energy loss (in the range of atomic frequencies) on both the thickness of the plate and the Lorentz factor of the charged particle is discussed. Special attention is given to the conditions under which tile Fermi density effect in the plate is absent. The results obtained are compared with a recent experiment.
The total energy loss of a charged particle, uniformly passing through a plate of arbitrary thickness
0.~
in vacuum, consists of loss due to the particle’s own field in matter, i.e. ionization loss [1], and additional
//
A
loss;s, from the fields arising due to the boundaries of The dependence of the total energy loss (in the range of atomic frequencies) on the thickness of the plate ,a,and the Lorentz factor of the charged particle, ‘y, is discussed in the present paper. Let us start with a plate of infinite thickness (a ~ oo). As is well known [1], the ionization loss in infinite media (~erenkov loss is also accounted for) as a function of y increases logarithmically after a minimum and then approaches an asymptotic value (the Fermi plateau). That is, when the velocity of the particle v is not close to the light velocity in vacuum c (fig. 1, region A), the loss is defined by 2e2a / W W0— “2 ~
UK
D g
Fig. 1.The range of values of (a, lg’y) for different ‘y-dependences of the ionization loss: A: a minimum and
logarithmic increase [formula(1)]; B: Fermi plateau [formula (2)]; C:
the region of sharper ~~dependence; D: logarithmic increase
[formula(6)]. taken into account, one must replace, in the formulae given above, ,c~by (2mE1 )1/2 where E1 is the max-
(1)
imum energy in the collision, and m is the electron mass. If the thickness of the plate is finite, an additional
where wp is plasma frequency of the matter, ~ is the minimum distance at which macroscopic electrodynamics is applicable, j3 = u/c, ~3is some mean atomic frequency [1]. When the velocity v is close enough to c (more ex-
loss arises, which can be interpreted as consisting of loss on transition radiation with a wide frequency spectrum and of a correction to the ionization loss, mainly concentrated in the atomic frequency range. The transition radiation generated at atomic frequen-
actly, if ~ ~‘ 1, fig. 1, region B), the loss is defined by
cies can then be absorbed by the matter of the plate,
0’)’
,
U
C’)
2e2a
w ~
=
p2 \
~
/
~‘2 c
i<
ln
0c —
in this way seemingly increasing the ionization loss.
(2)
~‘
(Fermi plateau). If the short-range collisions are also 188
The total loss was studied by Garibian et al. [2—4] in plates of finite thicknesses. It was shown that when the plate thickness
0 031—9163/81/0000—0000/$ 02.50 © North-Holland Publishing Company
Volume 85A, number 3
PHYSICS LETTERS
a’~acr
(3)
21 September 1981
3
where acr (c/W~)ln7(for a more precise formula see below), the additional loss completely eliminates the density effect, i.e. the loss will increase with 7. This effect was found in refs. [2—4]by expanding
S
‘~
________
________
________
-~
2. /
—
(
the additional loss in powers of the plate thickness, a. It was shown that when ~ ~‘ 1, the linear term in a
/
a,
has a negative sign and is determined by 7 — ~32)
W(
______
2
1)=—
c
&‘~
________
~‘)p (in—
~,r(Q~)
~c~(Q’)
(4)
“ 2 where ~i is defined in ref. [3]. Together with formula (2) the total loss logarith-
Fig. 3. The
mically increases with 7 (fig. 1, region D):
7-dependence of the total loss W0, normalized to the Fermi plateau: a1 = 1O~cm,ycr(al) = 55;a2 = 2 X cm,3.‘Ycr(a2) = 5.5 X 10~a3= 3 X iO~cm, ‘y~(a3)
~
c~2e2a
~2\
\ ~i “2 (ln_____~~-). (5) It should be noted that this formula differs from c
(1) only by the definition of the mean frequency. The term proportional to a2 is positive, i.e. it de-
creases the linear one: W(
2a2/4u~&2, 2)
=
4O~
iO~
~.
40’
5.5X i0
K 0C7
40
‘
(6)
irW~e
where &‘~is some twice averaged atomic frequency (see ref. [3]). The ratio of (6) to (4) is
acr
_4c~( —
ITWp
ln
W~7 ~\ —
—
~
(8)
The dependence is plotted in fig. 2. of acr Ofl ‘y for different materials It is seen from the above stated that under condition (3) the second-order term can be neglected, compared with the linear one. But the worse this condition is satisfied the greater is the role of the secondorder term. When a ‘‘acr,this term compensates the linear one.
IW(
2)/W(1) I a/acr ,
(7)
=
where
When a acr and 7 ~‘
~‘
1, the ionization loss does
not depend on 7. As to the energy of the transition
radiation absorbed in the layer at atomic frequencies, it can be evaluated as [5]: 2Ic)&’)ln7,
icc’ (Cm)
______
______
______
_____
_____
_____
,)—.‘
~
_____
_____
~
______
o
(9)
Wtr’~’(e
~E~EEEEE 10
lO~
!0~
{O~
I I
Wt
11
I
c~wlny
acr a a—
where a ~w/W~ is some numerical coefficient of the order of unity. When a aaci, the role of the transition radiation absorbed at atomic frequencies is small, and the loss localized in the matter practically ~-
~
Fig. 2. The y-dependence ofa~for different materials 1: lithium, 2: toluene, 3: silicon, 4: graphite.
where i~wis soi~eeffective interval of atomic frequencies determined mainly by strong absorption in the ultraviolet frequency region. The value of ~w will be of the order of several tens of hundreds of eV. From (2)and (9)we have
does not depend on ‘y (fig. 1, region B). Let us now consider the 7-dependence of the loss,
189
Volume 85A, number 3
PHYSICS LETTERS
when the plate thickness a is given. Since acr depends 7cr’ or on 7, the same plate can be “thick” if 7 ~ “thin” if 7 ~ 1cr’ where 7cr follows from (8) by sub-
stituting a for acr. When 7cr ~‘ 7 (fig. 1, regions A and B), the loss is
determined by formula (l)or (2), and when 1~ 7cr
21 September 1981
in ref. [7] loss was considered in a 100 mcm (10—2 cm) silicon foil in which the effect of the logarithmic increase with ‘y (with the electron energies available from accelerators) should not be observed because of the large thickness of the foil (nevertheless, see refs. [7,8]).
(fig. 1 , region D), by formula (5). And when 7
(fig. 1 region C), there is evidently a smooth transition between these two formulae. The 7-dependence
References
of the total loss, normalized to the plateau value, is plotted in fig. 3 (for silicon). It is seen that in the intermediate area y —‘ 7cr the 7-dependence ~5 sharper
[11 L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media (Addison-Wesley, 1957). [2] Zh. M.M. Eksp.Muradian, Teor. Fir.Izv. 37 (1959) 527. [3] G.M. G.M. Garibian, Garibian and Akad. Nauk Arm. SSR Fir. 1 (1966) 310; Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12 (1969) 1362.
,
than the logarithmic one, defined by formula (5). The effect predicted in ref. [2] was considered experimentally in ref. [6], where the ionization loss of electrons with energies from 20 to 86 MeV in foils of
polystyrene was observed. It was found that in foils with thicknesses of l0~ cm the loss did not increase
withy, and in foils with thicknesses of 10_6105 cm the increase was logarithmic. On the other hand, the measurements of ref. [6] were not absolute, but relative. Absolute measurements were carried out recently [7]. Unfortunately,
190
[4] G.M. Garibian and M.P. Lorikian, Dokl. Akad. Nauk
Arm. SSR 40 (1965) 21. [5] G.M. Garibian and G.A. Tchallkian, Zh. Eksp. Teor. Fir. 35 (1958) 1281; Izv. Nauki 12 (1959) 49. Akad. Nauk Arm. SSR Fiz.-Mat. [6] A.I. Allkhanian et al., Zh. Eksp. Teor. Fir. 44 (1963)
1122;46 (1964) 1212. [7] W. Oegle et al., Phys. Rev. Lett. 40 (1978) 1243. [8] W. Oegle, LASL Report LA-UR-76-1398.