BRAGG-GRAY DOSIMETRY: THEORY OF BURCH
Gudrun Alm Carlsson
Dept of Radiation Physics, IMV
Faculty of Health Sciences
Linköping University

I. Introduction
The theoretical approach to Bragg-Gray dosimetry is: a Bragg-Gray cavity is a cavity
(detector) so small that, when inserted into a medium, it does not disturb the fluence of
charged particles existing in the medium.
This means that the ideal Bragg-Gray cavity (detector) is one of infinitesimal
dimensions, a "point" detector. In practice, such detectors do not exist but many real
detectors may, in a first approximation, be treated as Bragg-Gray detectors to a high
degree of accuracy. Corrections needed (so called perturbation corrections) to account
for the deviation of the signal from a practical detector from that of an ideal one has
been treated by, e.g., ICRU 1984, Alm Carlsson, 1985, Svensson and Brahme 1986,
Alm Carlsson 1987.
Derivation of "perturbation corrections" needs careful consideration and under-standing
of the ideal case, i.e., that from which deviations are to be corrected for. The ideal case
of a Bragg-Gray detector has been treated by Bragg 1912, Gray 1936, Laurence 1937,
Spencer and Attix 1955 and Burch 1955.
The formulation of Bragg-Gray theory by Spencer and Attix has found wide practical
application and has been treated in detail elsewhere (Alm Carlsson,1978). The theory of
Burch treats the same problem as did Spencer and Attix, viz., the significance of
generation and slowing down of delta-particles in both medium and detector. Burch
treated the problem in considerable detail but didn't find a solution for practical
calculations. From a physical point of view, however, there is much to learn from
Burch's approach. Also, his treatment of so called track ends, evaluated in some detail
2
by Burch 1957, has been adapted in later versions of the Spencer-Attix formulation of
Bragg-Gray theory (Nahum 1978, ICRU 1984).
II. Short review of earlier theories
Bragg
Bragg 1912 discussed the possibility to use the ionization in a small air volume as a
measure of the electron fluence (or in the terminology of Bragg "the density of b-rays")
in the surrounding medium. Bragg was interested in estimating the ranges of electrons
in different media. He argued that the electron fluence in a photon irradiated medium
was equal to the product of the number of electrons emitted per unit volume and their
ranges and the ionization in a small air volume to be proportional to this product.
Note that the last statement above requires that the contribution to the ionization from
electrons liberated by photons in the air is negligible. This volume only "senses" the
electrons generated in the surrounding medium which is possible provided it is
sufficiently small: Let the dimensions of the air cavity be Dl (a cavity diameter). The
ionization from electrons liberated by photons in the cavity is proportional to Dl3
(number of electrons released) times Dl (length of travel in the cavity before escaping
from it). The ionization from electrons entering the cavity from the surrounding medium
is proportional to Dl2 ( the cross-section area of the cavity) times Dl (length of travel in
the cavity). Thus the quotient between the ionization caused by electrons released by
photons within it and those entering from outside is proportional to(Dl)4/(Dl)3 = Dl
which approaches zero as Dl approaches zero.
Gray
Gray 1929 was the first to formulate a quantitative theory for the relation between the
ionization per unit volume of a small gas cavity and that in the surrounding medium.
Gray based his derivation of this relation on comparing the gas cavity with an
equivalent volume in the undisturbed medium, Fig 1.
3
Fig 1: Gray based his theory on comparing the gas cavity (detector) with an
equivalent volume in the undisturbed medium: electrons entering the two
volumes at equivalent positions travel in straight lines and lose the same
energy in traversing them.
The electrons were assumed to travel in straight lines through the volumes, Fig 1. The
ratio of the equivalent straight lines ldet and lmed in the detector and the equivalent
medium volume respectively could be identified with the inverse ratio of the linear
stopping powers

where dT/dx is the linear stopping power.
Electrons were assumed to lose their energies continuously such that the energy lost =
imparted energy. Moreover, the stopping power ratio was assumed to be independent on
energy.
det
med
4
The gas volume ”senses” more incident electrons than the equivalent volume: the ratio
of the number of electrons entering the two volumes equals the ratio of the equivalent
straight lines squared: (ldet/lmed)2 (the area of projection is proportional to the square
of the linear dimension of a volume). Since each electron entering one of the volumes
(at equivalent positions) imparts the same energy to this volume, the ratio of the
energies imparted per unit volume is given by:
[ ]
[ ] det
3
det
2
det det
/
/
l
l
l
l
l
l
V
V med med
med med
= ÷ ÷
ø
ö
ç çè
æ
÷ ÷ø
ö
ç çè
æ
=
e
e
(2)
or in terms of absorbed dose
( )
( )med S med
S
D
D
r
r
/
/ det = det
(3)
where S/r is the mass stopping power
1
r
dT
dx
æ
è
ç ö
ø
÷ .
Gray also discussed the contribution to the ionization in the gas cavity from electrons
generated by photons in it. He demonstrated (cf the discussion above) that this could be
reduced to a negligible fraction provided the dimensions of the cavity are sufficiently
small. In addition, he argued that Eq(2) (and consequently Eq(3)) is valid independent
of cavity size provided it acts as a Bragg-Gray detector. Inversely, if a linear
relationship between the ionization in a gas cavity and the volume of this cavity is
found, this is a demonstration of the validity of the theory as well as an indication that
the detector behaves as a Bragg-Gray detector: one that does not disturb the fluence of
electrons (charged particles) in the medium.
Note that as soon as electrons liberated by photons in the detector volume contribute a
significant part of the total energy imparted to it, the fluence of electrons in the detector
can no longer be identical to that in the undisturbed medium (provided it is not medium
equivalent with respect to atomic composition in which case Fano’s theorem may
invalidate the statement). Therefore, a prerequisite for applying Bragg-Gray theory to a
detector in a photon irradiated medium is that the contribution to the absorbed dose
from electrons liberated by photons in it is negligible.
5
Experimental findings
Gray was aware that the theory, Eq (2), could never be exactly valid. He did some
experiments to test the constancy of the ionization obtained per unit volume of an air
cavity. He found that the requirements for such a cavity to behave like an ideal Bragg-
Gray detector depend on photon energy. With unfiltered g-radiation (from a Ra-source),
a 3 cm3 air volume fulfills Eq (2) with an accuracy of about 1% while with 100 kV Xrays
the corresponding volume must not exceed 0.1 cm3 (air within graphite).
Gray 1937 also measured the ionization in an 0.1 cm3 air volume within walls of
differing atomic numbers. When irradiated in the same photon beam (g-rays from a Rasource)
and with electronic equilibrium in the surrounding wall, the ionization per unit
volume increased with increasing atomic number of the wall. Gray demonstrated that
this is caused by a decreasing stopping power per electron with increasing atomic
number of the stopping medium. Thus, even in cases when Compton scattering is the
predominant interaction process, i.e. when the number of secondary electrons released
per unit mass is proportional to the number of electrons per unit mass, the equilibrium
fluence in a medium of high atomic number is larger than that in a medium of a lower
atomic number.
In varying the gas pressure in the ionization chambers, the ionization increased linearly
with the pressure in the graphite but not in the lead chamber (Gray 1936). Similar
experiments were later carried through by Attix and De la Vergne (subsequently
published by Attix et al 1958) using plane parallel chambers and varying the air volume
by varying the plate separation from 1 mm to 12 mm. The results were in accordance
with those of Gray: the ionization per unit volume of air was a constant with chamber
walls of low atomic numbers but increased with decreasing air volume with walls of
high atomic numbers. The theoretical explanation of this had to await the theories of
Spencer and Attix 1955 (Alm Carlsson 1978) and Burch 1955 taking the effects of delta
particle production into account.
Laurence
In his theory, Gray assumed the stopping power ratio to be independent of electron
energy. Gray was himself aware of this being an approximation. Laurence 1937
6
improved the theory by taking into account the energy dependence of the stopping
power ratio (continuous slowing down was still assumed). This in turn requires
derivation of the energy distribution of the electrons in the medium. In cases with
photon irradiated media and with electronic equilibrium existing at the site of the
detector, calculation of the energy distribution of the electron fluence is manageable.
This was a common presumption in the early theories (Bragg 1912, Gray 1929, 1936),
Laurence 1937, Spencer and Attix 1955, Burch 1955) before use of high energy photon
and electron beams started. For the latter cases, calculations of electron fluence energy
distributions at various points in a medium are now performed using Monte Carlo
methods (Berger 1963, Berger and Seltzer 1969, Nahum 1978).
Assuming (with Laurence) continuous slowing down and electronic equilibrium, the
differential fluence FT can be identified with the differential track length y(T) of the
emitted electrons (see, e.g. Alm Carlsson 1985, Eq(63) on p 49), Fig 2
T T-dT
dV
( ) dx
dT
dT
dL =
Fig 2: An electron with initial kinetic energy To (released from volume element dV)
slows down continuously losing energy dT while passing the track length
dL = dT/(dT/dx); dT/dx is the linear stopping power for an electron of kinetic
energy T.
When dS/dV electrons of kinetic energy To are emitted per unit volume and
continuously slowed down, one has
( ) ( ) ( )(dT dx)
dT
T dT dS dV y T dT dS dV /
F = / = / (4)
dL
7
where y(T) dT = dL in Fig 2.
Assuming continuous slowing down, the absorbed dose is the product of charged
particle fluence and mass collision stopping power. Thus, for Ddet and Dmed one has in
Bragg-Gray conditions (the same charged particle fluence in detector and medium)
D (S ) dT col
T
T det
0
det /
max
= òF r (5a)
D (S ) dT col med
T
med T / r
max
0 ò
= F (5b)
The quotient Ddet/Dmed is the quotient between the integrals in Eqs (5a) and (5b). This
is the way the so called Bragg-Gray-Laurence theory is depicted (ICRU 1984) as
extended also to cases with high energy photon (lacking electronic equilibrium) and
electron beams.
Going back to the more specific conditions presupposed by Laurence: electronic
equilibrium, negligible bremsstrahlung energy losses and monoenergetic electrons of
kinetic energy To emitted in the medium, Eqs (5a) and (5b) can be written (with FT
from Eq(4))
( ) ( )det
0
det /
/
S r
dT dx
dT
dV
dS
D
To
med = ò (6a)
( ) ( ) o
med
T
med
med
med T
dV
dS
S
dT dx
dT
dV
dS
D
o
r
r = = ò
0
/
/ (6b)
The quotient Ddet/Dmed finally is
Ddet / Dmed =
1
To
(S/ r)det
(S/ r)0 med
To
ò dT = (S/ r)med
det (7)
8
where
det
med
S
ú úû
ù
ê êë
é
÷ ÷ø
ö
ç çè
æ
r is a weighted mean (weighting factor is the function 1/To) of the mass
stopping power ratio ( S/ r)med
det for detector and medium. Since photons liberate
electrons with varying initial kinetic energies, the conversion factor Ddet/Dmed is a
suitably weighted mean of that in Eq (7). Values of weighted means of mass stopping
power ratios with air as detector material are given by, e.g., Burlin 1968, for various
media and monoenergetic electrons, Eq (7), as well as for the energy distributions of
electrons liberated by monoenergetic photons.
The conversion factor Ddet/Dmed derived from Bragg-Gray theory is commonly called
the "stopping power ratio". The significance of this is demonstrated in Eqs (3) and (7)
under two specific conditions. Taking the quotient of the integrals in Eqs (5a) and (5b),
the generalized Bragg-Gray-Laurence relation is obtained as a quotient of weighted
means of stopping powers: Scol ( / r)det / Scol ( / r)med . The weighting factor for both
averages is the relative energy distribution of the charged particle fluence in the medium
at the site of the detector. It may be of some interest to note that the generalized Bragg-
Gray-Laurence relation may also be derived as a weighted mean of the stopping power
ratio as in Eq (7):
DT,det dT = FT Scol ( / r)det dT (8a)
DT, med dT = FT Scol ( / r)med dT (8b)
Here, DT dT is the absorbed dose from electrons with kinetic energies in the interval dT
around T.
From Eq (8b), FT can be solved as
col med
T med
S
D
( / )
,
r . Substituted for FT in
Eq (8a), one has for
Ddet = Dò T,det dT
D (S ) D dT T med
T
col med ,
0
det
det
max
= ò / r (9)
and finally
9
( ) , ( )det
0
det
det / / /
max
col med
med
T med
T
med col med dT S
D
D
D D = ò S r = r (10)
Eq (10) has the same form as Eq (7). Weighting factor in averaging the mass collision
stopping power ratio is (in both cases) the relative absorbed dose to the medium from
electrons with kinetic energies in the interval dT around T at the site of the detector.
III. Theory of Burch
Electrons do not lose their energies continuously in slowing down but can occasionally
produce d-particles of high kinetic energies. The Bragg-Gray-Laurence relation, Eq (7),
can be interpreted as being valid in cases when d-particle equilibrium (Alm Carlsson
1985) exists in both medium and detector. This is a bad approximation when detector
and medium differ considerably in atomic composition. Burch 1955 like Spencer and
Attix 1955 suggested that the production of d-particles in both medium and detector
must be considered in the theory of Bragg-Gray detectors.
Definition of "infinitesimal" cavity
Burch starts with a careful description of an "infinitesimal" cavity yielding the initial
assumptions of the theory:
1) the cavity is so small that the number of electrons passing into it with a range less
than the cavity dimensions is a negligible fraction of the total number traversing it
2) the energy imparted ("ionization" in Burch's paper) to the cavity by electrons
liberated by photons in it is a negligible proportion of the total
3) the energy imparted per unit mass (absorbed dose) in the medium in the immediate
neighborhood of the cavity is assumed to be sensibly constant
Note that assumption 3) means that the cavity dimensions are small compared to any
absorbed dose gradients in the medium. This is equivalent to saying that the dimensions
of the cavity are small with respect to the ranges of the charged particles in the medium:
As seen from assumption 3), electronic equilibrium in a photon irradiated medium is
not a prerequisite for Bragg-Gray theory. Such assumption as used by Laurence 1937 in
10
deriving Eq (7) and in the numerical calculations by Spencer and Attix 1955 only serve
(as pointed out above) the purpose to make calculations of differential fluences
manageable.
Analysis
Burch discusses the case with a gas detector in a solid medium: His arguments are here
generalized to an arbitrary cavity (detector). The following quantities are used in the
analysis:
nT,cdT = number of electrons crossing the cavity with kinetic
energies in the interval dT around T
lc(T) = average path length traversed in the cavity by
electrons entering with kinetic energy T
(dT/dx)'c = average energy imparted to the cavity per unit
pathlength traversed by an electron with kinetic
energy T
Note that (dT/dx)'c is not the same as the electron (charged particle) stopping power: it
does not include that part of the energy lost which is subsequently carried out of the
cavity via photons (bremsstrahlung, characteristic roentgen rays) or energetic secondary
electrons (d-particles). Its value depends on the cavity size and shape.
The mean absorbed dose in the cavity DT ,cdT
from electrons with kinetic energies in the interval dT around T is:
D T,cdT =
1
DMc
e T ,cdT =
1
DMc
n T,c l c(T)(dT/dx)'c dT (11)
where DMc is the mass of the cavity and e c is the mean energy imparted to the cavity.
In the following, Burch takes the same approach as Gray comparing the cavity with an
equivalent medium volume (Fig 2) such that
l c(T)(dT/dx)'c = l m(T)(dT/dx)'m (12)
11
i.e., the mean energy imparted to the imaginary medium volume when traversed by an
electron of kinetic energy T equals that to the cavity when traversed by the
corresponding electron (l m and (dT/dx)'m has the same significance as l c and (dT/dx)'c
but refer to the medium volume).
Assuming the electrons to travel in straight lines through the cavity and medium
volumes, the mass DMm of the latter can be derived
c
c
m
c
m
c
c
m
m m m m M
l T
l T
V
l T
l T
V M D ú
û
ù
êë
é
= ú
û
ù
êë
é
D = =
3 3
( )
( )
( )
( )
r
r
r r
(13)
Moreover, n T,c and n T,m are related through
T c
c
m
T m n
l T
l T
n ,
2
, ( )
( )
úû
ù
êë
é
= (14)
The mean absorbed dose to the medium D T,mdT from electrons with kinetic energies in
the interval dT around T is given by Eq (11) substituting index m for index c. Utilizing
Eqs (12) - (14), one has
( )( )
( )
( ) ( )( )
dT
dx
dT
dx
dT
D
l T dT dx dT
dT dx
dT dx
M
n
l T dT dx dT
l T
l T
M
n
D dT
c
m
T c
c c
m
c
c
m
c
T c
m m
m
c
m
c
c
T c
T m
'
'
,
,
,
,
1
1
/ '
/ '
/ '
/ '
( )
( )
÷ ÷ø
ö
ç çè
æ
÷ ÷ø
ö
ç çè
æ
=
D
=
=
D
=
r
r
r
r
r
r
(15)
Integrating over T and taking the ratio D c / D m one has
12
( ) dT R
D
D
dT R T
D
D
dx
dT
dx
dT
D D
m
T m
T
m
T m
T
m
c
c m = =
÷ ÷ø
ö
ç çè
æ
÷ ÷ø
ö
ç çè
æ
= ò ò ,
0
,
0
'
'
max max
1
1
/
r
r
(16)
Burch argues that the mass stopping power ratio in the earlier theories (Gray 1929, 1936
and Laurence 1937) should be replaced by a "mass energy dissipation ratio" R(T) in a
theory taking the incontinuous energy losses into account.
Note, that it was assumed that any absorbed dose gradient in the medium in the
immediate neighborhood of the cavity is negligible: D m in Eq (16) can be replaced by
Dm for a point at the center of the cavity in the undisturbed medium.
Difficulties in determining R(T)
Burch discusses in some detail a method to calculate the electron fluence energy
distribution (related to nT,c and nT,m) in electronic equilibrium taking into account the
discontinuous energy losses (generation of d-particles).
The main difficulty in evaluating the theory quantitatively is determination of the
quantities (dT/dx)'c and (dT/dx)'m, i.e., determination of the mass energy impartation
ratio R(T) ("mass energy dissipation ratio" in the terminology by Burch).
Burch discussed the possibility to approximate (dT/dx)' with a restricted stopping
power: (dT/dx)'c » (dT/dx) hc. But how should the energy restriction h be chosen? One
difficulty is that h will probably not be the same for cavity and medium. This is
elucidated in Fig 3.
13
Fig 3: A fast electron enters the cavity and passes along the straight line CD. An
electron of the same energy enters the equivalent medium volume along the
equivalent straight line AB. In both volumes a d-particle is produced at
equivalent positions. Elastic scattering is larger in the cavity and causes the d-
particle to be completely absorbed in it. In the medium volume, the d-particle
escapes carrying some energy out of the volume.
In Fig 3, CD and AB are chosen such that the "local" energy impartation along the
tracks is identical. In both volumes, a d-particle is generated. While it is completely
absorbed in the cavity, it is not in the medium volume (due to, e.g., larger elastic
scattering in the cavity material). Consequently, the total energy impartation will be
larger in the cavity than in the medium volume. However, the prerequisite for choosing
the equivalent medium volume was to make the total energy impartation the same as in
the cavity. The dimensions of the medium volume must be increased. They must,
however, not be increased so much that the d-particle gets completely absorbed since
increasing AB means that the "local" energy impartation along the high energy particle
track in the volume is also increased. The cut off energy hm for the restricted medium
stopping power should in this case be hm < m =" ò"> hm since
h'm » the kinetic energy of an electron entering from outside the cavity which can just
pass the cavity while hm equals the kinetic energy of a d-particle released within the
hypothetical medium volume which terminates at the volume boundary.
The track end problem was recently revived in the calculations of stopping power ratios
by Nahum 1978. Spencer and Attix 1955 do take the track end problem into account but
the concept of a track end is not quite the same as that discussed by Burch. Nahum
combined calculations of mass stopping power ratios according to the Spencer-Attix
formulation with the track end contribution according to Burch 1957. His expression for
D det is (cf ICRU 1984).
= F ( ) + F (D)( (D) ) D D
D ò
det , det , det / /
max
D L r dT S r T m
T
T m (18)
15
where r
LD is a restricted mass collision stopping power, ( ) , F D T m is the differential
electron fluence in the medium evaluated at T = D and S(D)/r is the mass collision
stopping power for T = D.
Problem (Exercise)
Discuss the relation between Eqs (17) and (18). How can the track end term in Eq (18)
be derived from that in Eq (17) ?
Guidance: In Nahum 1978 the track end term is given as NDD where ND is the number
of electrons per unit mass in the medium which drop below the energy limit D. Discuss
the conditions to be fulfilled for equating
N N dT
m
D = ò T m D
'
0
,
h
Derive the relation between NDD and the track end term in Eq (18).
16
REFERENCES
1. Alm Carlsson G (1978). Spencer Attix kavitetsteori. Stencil. In Swedish.
2. Alm Carlsson G (1985). Theoretical basis for dosimetry. In: The dosimetry of
ionizing radiation. Vol I., pp 1-75. Ed by K R Kase, B E Bjärngard and F H Attix.
Academic Press. New York.
3. Alm Carlsson G (1987). Cavity Theory: a theoretical formalism for perturbation
correction factors. Stencil.
4. Attix, F H, De La Vergne L and Ritz V H (1958). Cavity ionization as a function of
wall material. J Res Nat Bur Stand 60, 235-243.
5. Berger M J (1963). Monte Carlo calculation of the penetration and diffusion of fast
charged particles. In: Methods of computational physics. Vol 1. Ed by B Alder, S
Fernbach and M Rotenberg. Academic Press, New York, pp 135-215.
6. Berger J M and Seltzer S M (1969). Calculation of energy and charge deposition and
of the electron flux in a water medium bombarded with 20 MeV electrons. Ann NY
Acad Sci 161, 8-23.
7. Bragg W H (1912). Studies in radioactivity. Mac Millan and Co, London, pp 91-99,
161-169.
8. Burch P R J (1955). Cavity ion chamber theory. Radiat Res 3, 361-378.
9. Burch P R J (1957). Comment on recent cavity ionization theories. Radiat Res 6, 79-
84.
10. Burlin T E (1968). Cavity-chamber theory. In: Radiation Dosimetry. Vol 1. Ed by F
H Attix, W C Roesch and E Tochilin. Academic Press, New York, pp 331-392.
11. Gray L H (1929). The absorption of penetrating radiation. Proc Roy Soc A 122, 647-
668.
12. Gray L H (1936). An ionization method for the absolute measurement of g-ray
energy. Proc Roy Soc A156, 578-596.
13. International Commission on Radiation Units and Measurements (1984). Radiation
dosimetry: electron beams with energies between 1 and 50 MeV. ICRU Rep 35.
ICRU, Washington, D.C.
14. Laurence G C (1937). The measurement of extra hard X-rays and gamma-rays in
roentgens. Can J Res A 15, 67-
15. Nahum A E (1978). Water/air mass stopping power ratios for megavoltage photon
and electron beams. Phys Med Biol 23, 24-38.
17
16. Spencer L V and Attix F H (1955). A theory of cavity ionization.
Radiat Res 3, 239-254.
17. Svensson H and Brahme A (1986). Recent advances in electron and photon
dosimetry. In: Radiation dosimetry. Physical and Biological Aspects , pp 87-170. Ed
by C G Orton. Plenum Press, New York.