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General recombination in ionization chambers with spatially non-uniform ionization intensity—two special cases
Int. J. Radiat. Phys. Chem. 1975, Vol. 7, pp. 243--249. Pergamon Press. Printed in Great Britain
GENERAL RECOMBINATION IN IONIZATION CHAMBERS WITH SPATIALLY NON-UNIFORM IONIZATION INTENSITYuTWO SPECIAL CASES J. W. BoA~ Institute of Cancer Research, Sutton, Surrey SM2 5PX, England
(Received 13 May 1974) Abstract--The proportion of general recombination which occurs in an ionization chamber with non-uniform distribution of ionization has been examined for two cases of practical interest: (1) when the ionization intensity declines exponentially through the gas layer, and (2) when the ioniT~ion is concentrated in cylindrical columns which retain their form as they drift to the electrodes. In both these cases, it is shown that the simple formula for general recombination based on the average ionization intensiW over the whole volume of the chamber is a good first
approximation in the near-saturation region and the extent of correction necessaryfor extremely non-uniform distribution is investigated. INTRODUCTION
Mucx~ A ~ O N has been paid over the years to the problems of current collection in ionization chambers. The various types of ionic recombination--preferential, initial and general--have been recognized and their effects on ionization measurements carefully investigated for the conditions which normally obtain in well-designed ionization chambers cx~. When general recombination is considered, it has nearly always been assumed that the ionization is uniform throughout the gas volume of the chamber, whatever the geometrical form of the latter may be. Recently, however, ionization chambers have become interesting as image-forming devices for diagnostic radiology~, 8~. For this purpose a layer of gas of high atomic number at a pressure of several atmospheres is the absorber of the primary quanta and the range of the photoelectrons is small compared with the thickness of this gas layer. The primary X-ray beam may be considerably attenuated in traversing the gas so that the assumption of uniform ionization is no longer valid. Another case in which the assumption of uniform ionization is incorrect is general recombination occurring in a gas ionized by heavy ions or by recoil nuclei from fast neutron irradiation. Here the columns of positive and negative ions which escape initial recombination retain some vestiges of their cylindrical form as they meet and interact with other columns of oppositely charged ions in crossing the gas space. These two special cases are examined below, from the point of view of their effect on the conditions necessary to achieve saturation. No attempt is made to set up a theoretical model for the saturation curve down to low collection efficiency.
GENERAL RECOMBINATION WHEN THE IONIZATION INTENSITY FALLS EXPONENTIALLY THROUGH THE GAS
Let us assume plane-parallel geometry with the ionization intensity falling exponentially as the X-ray beam traverses the gas. We are interested in the approach to saturation and shall first derive the distribution of ion concentrations across the 243
244
J.W. BOAO
chamber when general recombination c a n be ignored. From the interaction between these positive and negative ion distributions we then calculate the small proportion of recombination which would occur. The ideal concentrations can be corrected to allow for this and a second approximation obtained. This method has been shown to give excellent agreement with experiment and with more exact theories when the ionization is uniform through the chamber provided that the collection efficiency is higher than about 70 per cent m. In the present case let the ionization intensity be given by
(1)
I = I0 e-~'x,
where x is the distance from the electrode through which the beam enters the gas space, l i s the ionization intensity in coulombs cm-* s -t at distance x,/0 that for x = 0 a n d / , is an attenuation coetf~ent. Figure I shows the geometry and nomenclature. Then the amount of positive charge (Ccm -t s -x) crossing the plane at x is
O
V
.Ioe-/zx ~
.
+ ve Ion
/ d F I G . 1. N o m e n c l a t u r e
(2)
used in the theoretical discussion.
x L I 0 e - a * d x = ~ (1 - e-#Z), d0 |
and since the velocity of the ions is kx V/d, V being the collecting voltage, kl the ionic mobility (cm 2 V -1 s -x) and d the distance between plates (Fig. 1), we can write down the volume concentration of positive charge at x in the steady state. This is (3)
Iod xql = #-klV (1 -e--t' ).
Similarly, the volume concentration of negative ions at the position x is (4)
/od ~x ~qz ----_ L ~ ( e --e- ~). /~2r
Recombination in ionization chambers with non-uniform ionization intensity
245
The rate of charge recombination per unit volume at the plane x is therefore dq1 dqg. . .dt . . . dt.
(5)
o~ e ql q2,
where o~is the recombination coefficient and e the charge on an electron. That is, dq
(6)
a
-'~ =
e\t~V]
)
kxk~"
""
"
Hence the total recombination which occurs throughout the chamber per unit area per second is (7)
adq ~dx R= fodt
• ,~ - M fo (e-~Z-e-~X--e-/*d + e-#(a+x)) dx, -
where
(8)
M = ekxkaIj~V]
or
(9)
1 R = M ~ {(1 - e-2a a) - 2/M e-~d},
while the total ionization per second in the same space is
(lO)
Q=
10e-~xdx = I-°(1-e-~%
Hence the collection efficiency, f, is
R oL Io d~ / 1 - e - ~ a - 2 / u / e - ~ d t f = 1 - - ~ = 1 ek-~k2I~~ V ~ [ ~ J (11)
a0d'\r 3 t ~ 2~ae-,~] - 1 6e~k2~2z~-~)[~-~ll+e-'______._ (1-e-~d) jJ"
NOW the quantity/o in (11) is the ionization intensity where the rmliation entexs the gas layer. It is appropriate to replace this as the reference quantity by the mean ionization intensity:throughout the chamber, which corresponds to the rate at which charge could be collected if there were no recombination. From (10) this is /t) 1 - e =~
,
and on introduCing this quantity I into (11) and simplifying the algebra, we find
(13)
A=I
6ekl k ~
(,ff) d' Vz .
.
where (14)
3 (1 - e-~, "~- 2 ~ e,,'~) ' (1-e:,O'
•
.
.
246
J.w. BoAG
The expression for f in (13) is exactly similar to the first approximation for the collection efficiency in a chamber uniformly irradiated at ionization intensity ~7I(1). The reduction factor ~7 allows for the non-uniformity when there is exponential absorption in the gas. This factor ~7(Fig. 2) remains close to unity over the range of values likely to occur in image-forming ionization chambers. We can therefore assume that the improved formula, derived for the uniformly irradiated chamber, which makes some allowance for field distortion due to space charge will be valid in the present case also. This more accurate formula is(1) f = {1 +
(15)
1.00
a
/~lcl*\1-1
0.5
-~
"
~d-.~
1.0 1.00
\ 0.99
"N
0.99
\
0.98
~
0.98
\ 0.97
~ 0
0.5
~d-~
0.97 1.0
FIG. 2. Reduction factor, ~, to allow for cxpon©ntial absorption of the primary radiation in the gas. GENERAL RECOMBINATION WHEN THE INITIAL DISTRIBUTION IS IN COLUMNS
Another situation of some interest is the recombination between the ions from different columns, once they have escaped initial (columnar) recombination. Diffusion is a slow process compared with drift in a strong electric field and the ions of one sign from a particular column may retain a quasicolumnar configuration and cross other similar formations of oppositely charged ions as they traverse the chamber gap. Does this interaction between different spatially correlated groups of ions lead to more or less recombination than would occur if the same number of ions were being formed at random throughout the gas space ? We can approach this question by considering first the fate of a single ion formed in any one track as it drifts through a random assortment of columnar distributions-the "ghosts" of other tracks. Does its chance of recombining with an oppositely charged ion depend on whether there are few columns of high ion density or many columns of low ion density--shading off, in the limit, to a homogeneous non-track distribution containing the same number of ions per unit volume of the gas ? To take an extreme case we shah consider cylindrical columns of ions, having uniform concentration within each column and retaining this uniform concentration as they
Recombination in ionization chambers with non-uniform ionization intensity
247
drift across the chamber. Any softening of this rigid model such as the assumption of Oaussian radial distributions or the recognition that the columns expand by diffusion as they drift across the chamber will tend to bring the model that much closer to the uniform distribution with which we are comparing it. It is, of course, necessary to assume that the radiation dose rate is high enough to ensure that at any instant there are many such columns present in the gas space. For convenience, we assume that the tracks are short compared with the spacing between plates so we need not consider end-effects. However, if we derive the collection efficiency for short columns it is an immediate corollary that it will hold also for longer columns, as may be proved by imagining that short segments having equal inclination 0 are assembled to form long tracks, a procedure which should not change collection efficiency at all. Suppose that n tracks are formed in the gas per unit volume per unit time, each containing m ions of either sign and lying randomly oriented at angles 0 to the electr/c field• The area of any particular column of ions of length I and diameter a, when projected on to the plane of either collecting plate is a/sin 0 (ignoring endeffects). If a single ion of the opposite sign, driven by the electric field normal to the plates, passes through this column of ions its average transit distance across the column is ~ra cosec 0 since ~ra is the mean length of random chords of a circle of diameter a and cosec 0 allows for the inclination of the track to the lines of force. Now i f j (ions cm -a) is the uniform ion density inside the column then (16)
J = m/~ra21 = ~r-~l"
The probability per unit time that a single ion drifting through a density, A of ions of oppoSite sign shall recombine is (17)
. d--~-=q dp or p=cLjT,
where T is the time of transit through the column, assuming p ~ 1. That is • ~ra c o s e c 0 (18)
p
=
~J4(kl+kOX,
where kl, k= are the mobilities of positive and negative ions, respectively (cm=s -x V-X), and X is the electric field strength (Vcm-X). During continuous irradiation the columns of positive ions produced at random throughout the gas are swept at velocity kl X towards the negative plate and, to maintain a steady state, when recombination is negligible, nd of these columns must reach the negative plate cm -2 s -x. The density of positive columns therefore rises linearly from zero at the positive plate to nd/k 1Xcm -z at the negative plate. A single negative ion will move through this distribution with a relative velocity (k 1+ k O X and, when it is at distance x from the positive plate, it will therefore pass by, or pass through, a number (nx/kl X) (kl + kO X of positive columns per cm s per sec. The total number of such columns which it encounters or passes by per cm s on its passage from x = xo to the positive plate, which lasts for a time (xo/k2 X), is therefore h where (19)
h = f=,pt~ [ nx ~(kl+ks)Xd t
Jo
~kz X]
"
248
J.W. BoAo
Inserting x ffi X o - k 2 X t into the above integral we find _ kl+k~
(20)
h-
klk2
nxo ~
2X"
The number of these columns which the negative ion actually traverses is given by their projected area per cm ~ on a plane parallel to the electrodes, i.e. by ha/sin 0, and so the probability of the negative ion recombining in moving from x 0 to the positive plate is P(xo) where (21)
P(xo) = phalsin 0.
Inserting the value o f p from equation (18) and of h from equation (20) we find (22)
P(xo) = oannxo2/2kl k 2 X ~.
Hence the mean value of P(x o) for ions starting at any point between the plates is (23)
1 fo P(x°)dx° ~ ~ = ~1 = 6k~mnd 1 k 2 X2 2"
The foregoing expression (23) is identical with the formula for the proportion of recombination which occurs in an ionization chamber in which the ionization intensity is uniform and equal to q = mne coulombs cm-% -I (1). Evidently, if our assumptions are valid, the persistence of the columnar ionization pattern during ion collection will not affect the probability of recombination for an isolated ion crossing the gap. Thus far we have been considering the fate of a single ion, and therefore of any random distribution of single ions, when they drift through a random distribution of columns of ions of the opposite sign. In fact, what we wish to discover is what will happen when two distributions of columns of ions of opposite sign drift through one another. From the point of view of a single negative ion the presence of neighbouring negative ions in the same column will affect its chance of recombination only by depleting the density of positive ions in any positive column which it traverses. The extent of this depletion depends upon many circumstances--the ion density in both columns, the field strength, the angle of crossing, etc. We can estimate it, however, from the amount of initial recombination which occurred in the original tracks, for every time a negative column and a positive column meet, the situation in the overlapping sections is analogous to reconstituting part of an original mixed track, albeit with ion density reduced by diffusion in the practical case, although not in our rigid model. The proportion of ions in the overlapping sections which evade recombination during the interpenetration and separation of the two columns will be roughly the square of the proportion escaping initial recombination in a track of the same ion density. This estimate is based on the fact that when two columns meet and cross one another there is both an approach phase and a separation phase, whereas only the separation phase exists in an original track. From the point of view of our single typical negative (or positive) ion, however, the average depletion it encounters in Crossing the positive (or negative) column is likely to be about equal to that for the approach phase only. Its chance of recombining will be correspondingly reduced and the loss of ions by this type of "gvneral" recombination between columns will be slightly less than that predicted by equation (23). If the ions escaping initial
Recombination in ionization chambers with non-nnlform ionization intensity
249
recombination can be calculated from the various parameters of the radiation and ionization chamber to be a fraction z of all the ions formed then the proportion of these ions which subsequently recombine in the "general" mode must lie between the value P given by equation (23) and the modified value (~)' obtained by inserting (zq) for q into equation (23). In view of the expansion of columns by diffusion the true value of recombination will usually lie closer to the upper bound P than to the lower
bound (P)'. REFERENCES 1. F. H. Almx and W. C. ROESCH, in Radiation Dosimetry, Academic Press, New York, 1966, Vol. II, Chapter 9. 2. K. H. REIns and (3. LANOE,Phys. Med. Bio1.1973, 18, 695. 3. H. E. JOHNS, A. FENSTEX,D. PLEWES, J. W. BOAG and P. N. JE~xY, Br. J. Radiol. 1974, in press.
GENERAL RECOMBINATION IN IONIZATION CHAMBERS WITH SPATIALLY NON-UNIFORM IONIZATION INTENSITYuTWO SPECIAL CASES J. W. BoA~ Institute of Cancer Research, Sutton, Surrey SM2 5PX, England
(Received 13 May 1974) Abstract--The proportion of general recombination which occurs in an ionization chamber with non-uniform distribution of ionization has been examined for two cases of practical interest: (1) when the ionization intensity declines exponentially through the gas layer, and (2) when the ioniT~ion is concentrated in cylindrical columns which retain their form as they drift to the electrodes. In both these cases, it is shown that the simple formula for general recombination based on the average ionization intensiW over the whole volume of the chamber is a good first
approximation in the near-saturation region and the extent of correction necessaryfor extremely non-uniform distribution is investigated. INTRODUCTION
Mucx~ A ~ O N has been paid over the years to the problems of current collection in ionization chambers. The various types of ionic recombination--preferential, initial and general--have been recognized and their effects on ionization measurements carefully investigated for the conditions which normally obtain in well-designed ionization chambers cx~. When general recombination is considered, it has nearly always been assumed that the ionization is uniform throughout the gas volume of the chamber, whatever the geometrical form of the latter may be. Recently, however, ionization chambers have become interesting as image-forming devices for diagnostic radiology~, 8~. For this purpose a layer of gas of high atomic number at a pressure of several atmospheres is the absorber of the primary quanta and the range of the photoelectrons is small compared with the thickness of this gas layer. The primary X-ray beam may be considerably attenuated in traversing the gas so that the assumption of uniform ionization is no longer valid. Another case in which the assumption of uniform ionization is incorrect is general recombination occurring in a gas ionized by heavy ions or by recoil nuclei from fast neutron irradiation. Here the columns of positive and negative ions which escape initial recombination retain some vestiges of their cylindrical form as they meet and interact with other columns of oppositely charged ions in crossing the gas space. These two special cases are examined below, from the point of view of their effect on the conditions necessary to achieve saturation. No attempt is made to set up a theoretical model for the saturation curve down to low collection efficiency.
GENERAL RECOMBINATION WHEN THE IONIZATION INTENSITY FALLS EXPONENTIALLY THROUGH THE GAS
Let us assume plane-parallel geometry with the ionization intensity falling exponentially as the X-ray beam traverses the gas. We are interested in the approach to saturation and shall first derive the distribution of ion concentrations across the 243
244
J.W. BOAO
chamber when general recombination c a n be ignored. From the interaction between these positive and negative ion distributions we then calculate the small proportion of recombination which would occur. The ideal concentrations can be corrected to allow for this and a second approximation obtained. This method has been shown to give excellent agreement with experiment and with more exact theories when the ionization is uniform through the chamber provided that the collection efficiency is higher than about 70 per cent m. In the present case let the ionization intensity be given by
(1)
I = I0 e-~'x,
where x is the distance from the electrode through which the beam enters the gas space, l i s the ionization intensity in coulombs cm-* s -t at distance x,/0 that for x = 0 a n d / , is an attenuation coetf~ent. Figure I shows the geometry and nomenclature. Then the amount of positive charge (Ccm -t s -x) crossing the plane at x is
O
V
.Ioe-/zx ~
.
+ ve Ion
/ d F I G . 1. N o m e n c l a t u r e
(2)
used in the theoretical discussion.
x L I 0 e - a * d x = ~ (1 - e-#Z), d0 |
and since the velocity of the ions is kx V/d, V being the collecting voltage, kl the ionic mobility (cm 2 V -1 s -x) and d the distance between plates (Fig. 1), we can write down the volume concentration of positive charge at x in the steady state. This is (3)
Iod xql = #-klV (1 -e--t' ).
Similarly, the volume concentration of negative ions at the position x is (4)
/od ~x ~qz ----_ L ~ ( e --e- ~). /~2r
Recombination in ionization chambers with non-uniform ionization intensity
245
The rate of charge recombination per unit volume at the plane x is therefore dq1 dqg. . .dt . . . dt.
(5)
o~ e ql q2,
where o~is the recombination coefficient and e the charge on an electron. That is, dq
(6)
a
-'~ =
e\t~V]
)
kxk~"
""
"
Hence the total recombination which occurs throughout the chamber per unit area per second is (7)
adq ~dx R= fodt
• ,~ - M fo (e-~Z-e-~X--e-/*d + e-#(a+x)) dx, -
where
(8)
M = ekxkaIj~V]
or
(9)
1 R = M ~ {(1 - e-2a a) - 2/M e-~d},
while the total ionization per second in the same space is
(lO)
Q=
10e-~xdx = I-°(1-e-~%
Hence the collection efficiency, f, is
R oL Io d~ / 1 - e - ~ a - 2 / u / e - ~ d t f = 1 - - ~ = 1 ek-~k2I~~ V ~ [ ~ J (11)
a0d'\r 3 t ~ 2~ae-,~] - 1 6e~k2~2z~-~)[~-~ll+e-'______._ (1-e-~d) jJ"
NOW the quantity/o in (11) is the ionization intensity where the rmliation entexs the gas layer. It is appropriate to replace this as the reference quantity by the mean ionization intensity:throughout the chamber, which corresponds to the rate at which charge could be collected if there were no recombination. From (10) this is /t) 1 - e =~
,
and on introduCing this quantity I into (11) and simplifying the algebra, we find
(13)
A=I
6ekl k ~
(,ff) d' Vz .
.
where (14)
3 (1 - e-~, "~- 2 ~ e,,'~) ' (1-e:,O'
•
.
.
246
J.w. BoAG
The expression for f in (13) is exactly similar to the first approximation for the collection efficiency in a chamber uniformly irradiated at ionization intensity ~7I(1). The reduction factor ~7 allows for the non-uniformity when there is exponential absorption in the gas. This factor ~7(Fig. 2) remains close to unity over the range of values likely to occur in image-forming ionization chambers. We can therefore assume that the improved formula, derived for the uniformly irradiated chamber, which makes some allowance for field distortion due to space charge will be valid in the present case also. This more accurate formula is(1) f = {1 +
(15)
1.00
a
/~lcl*\1-1
0.5
-~
"
~d-.~
1.0 1.00
\ 0.99
"N
0.99
\
0.98
~
0.98
\ 0.97
~ 0
0.5
~d-~
0.97 1.0
FIG. 2. Reduction factor, ~, to allow for cxpon©ntial absorption of the primary radiation in the gas. GENERAL RECOMBINATION WHEN THE INITIAL DISTRIBUTION IS IN COLUMNS
Another situation of some interest is the recombination between the ions from different columns, once they have escaped initial (columnar) recombination. Diffusion is a slow process compared with drift in a strong electric field and the ions of one sign from a particular column may retain a quasicolumnar configuration and cross other similar formations of oppositely charged ions as they traverse the chamber gap. Does this interaction between different spatially correlated groups of ions lead to more or less recombination than would occur if the same number of ions were being formed at random throughout the gas space ? We can approach this question by considering first the fate of a single ion formed in any one track as it drifts through a random assortment of columnar distributions-the "ghosts" of other tracks. Does its chance of recombining with an oppositely charged ion depend on whether there are few columns of high ion density or many columns of low ion density--shading off, in the limit, to a homogeneous non-track distribution containing the same number of ions per unit volume of the gas ? To take an extreme case we shah consider cylindrical columns of ions, having uniform concentration within each column and retaining this uniform concentration as they
Recombination in ionization chambers with non-uniform ionization intensity
247
drift across the chamber. Any softening of this rigid model such as the assumption of Oaussian radial distributions or the recognition that the columns expand by diffusion as they drift across the chamber will tend to bring the model that much closer to the uniform distribution with which we are comparing it. It is, of course, necessary to assume that the radiation dose rate is high enough to ensure that at any instant there are many such columns present in the gas space. For convenience, we assume that the tracks are short compared with the spacing between plates so we need not consider end-effects. However, if we derive the collection efficiency for short columns it is an immediate corollary that it will hold also for longer columns, as may be proved by imagining that short segments having equal inclination 0 are assembled to form long tracks, a procedure which should not change collection efficiency at all. Suppose that n tracks are formed in the gas per unit volume per unit time, each containing m ions of either sign and lying randomly oriented at angles 0 to the electr/c field• The area of any particular column of ions of length I and diameter a, when projected on to the plane of either collecting plate is a/sin 0 (ignoring endeffects). If a single ion of the opposite sign, driven by the electric field normal to the plates, passes through this column of ions its average transit distance across the column is ~ra cosec 0 since ~ra is the mean length of random chords of a circle of diameter a and cosec 0 allows for the inclination of the track to the lines of force. Now i f j (ions cm -a) is the uniform ion density inside the column then (16)
J = m/~ra21 = ~r-~l"
The probability per unit time that a single ion drifting through a density, A of ions of oppoSite sign shall recombine is (17)
. d--~-=q dp or p=cLjT,
where T is the time of transit through the column, assuming p ~ 1. That is • ~ra c o s e c 0 (18)
p
=
~J4(kl+kOX,
where kl, k= are the mobilities of positive and negative ions, respectively (cm=s -x V-X), and X is the electric field strength (Vcm-X). During continuous irradiation the columns of positive ions produced at random throughout the gas are swept at velocity kl X towards the negative plate and, to maintain a steady state, when recombination is negligible, nd of these columns must reach the negative plate cm -2 s -x. The density of positive columns therefore rises linearly from zero at the positive plate to nd/k 1Xcm -z at the negative plate. A single negative ion will move through this distribution with a relative velocity (k 1+ k O X and, when it is at distance x from the positive plate, it will therefore pass by, or pass through, a number (nx/kl X) (kl + kO X of positive columns per cm s per sec. The total number of such columns which it encounters or passes by per cm s on its passage from x = xo to the positive plate, which lasts for a time (xo/k2 X), is therefore h where (19)
h = f=,pt~ [ nx ~(kl+ks)Xd t
Jo
~kz X]
"
248
J.W. BoAo
Inserting x ffi X o - k 2 X t into the above integral we find _ kl+k~
(20)
h-
klk2
nxo ~
2X"
The number of these columns which the negative ion actually traverses is given by their projected area per cm ~ on a plane parallel to the electrodes, i.e. by ha/sin 0, and so the probability of the negative ion recombining in moving from x 0 to the positive plate is P(xo) where (21)
P(xo) = phalsin 0.
Inserting the value o f p from equation (18) and of h from equation (20) we find (22)
P(xo) = oannxo2/2kl k 2 X ~.
Hence the mean value of P(x o) for ions starting at any point between the plates is (23)
1 fo P(x°)dx° ~ ~ = ~1 = 6k~mnd 1 k 2 X2 2"
The foregoing expression (23) is identical with the formula for the proportion of recombination which occurs in an ionization chamber in which the ionization intensity is uniform and equal to q = mne coulombs cm-% -I (1). Evidently, if our assumptions are valid, the persistence of the columnar ionization pattern during ion collection will not affect the probability of recombination for an isolated ion crossing the gap. Thus far we have been considering the fate of a single ion, and therefore of any random distribution of single ions, when they drift through a random distribution of columns of ions of the opposite sign. In fact, what we wish to discover is what will happen when two distributions of columns of ions of opposite sign drift through one another. From the point of view of a single negative ion the presence of neighbouring negative ions in the same column will affect its chance of recombination only by depleting the density of positive ions in any positive column which it traverses. The extent of this depletion depends upon many circumstances--the ion density in both columns, the field strength, the angle of crossing, etc. We can estimate it, however, from the amount of initial recombination which occurred in the original tracks, for every time a negative column and a positive column meet, the situation in the overlapping sections is analogous to reconstituting part of an original mixed track, albeit with ion density reduced by diffusion in the practical case, although not in our rigid model. The proportion of ions in the overlapping sections which evade recombination during the interpenetration and separation of the two columns will be roughly the square of the proportion escaping initial recombination in a track of the same ion density. This estimate is based on the fact that when two columns meet and cross one another there is both an approach phase and a separation phase, whereas only the separation phase exists in an original track. From the point of view of our single typical negative (or positive) ion, however, the average depletion it encounters in Crossing the positive (or negative) column is likely to be about equal to that for the approach phase only. Its chance of recombining will be correspondingly reduced and the loss of ions by this type of "gvneral" recombination between columns will be slightly less than that predicted by equation (23). If the ions escaping initial
Recombination in ionization chambers with non-nnlform ionization intensity
249
recombination can be calculated from the various parameters of the radiation and ionization chamber to be a fraction z of all the ions formed then the proportion of these ions which subsequently recombine in the "general" mode must lie between the value P given by equation (23) and the modified value (~)' obtained by inserting (zq) for q into equation (23). In view of the expansion of columns by diffusion the true value of recombination will usually lie closer to the upper bound P than to the lower
bound (P)'. REFERENCES 1. F. H. Almx and W. C. ROESCH, in Radiation Dosimetry, Academic Press, New York, 1966, Vol. II, Chapter 9. 2. K. H. REIns and (3. LANOE,Phys. Med. Bio1.1973, 18, 695. 3. H. E. JOHNS, A. FENSTEX,D. PLEWES, J. W. BOAG and P. N. JE~xY, Br. J. Radiol. 1974, in press.