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An exact solution of the doppler-shift attenuation problem
Nuclear Phyalca A2T1 (1977) 317-336 ; © NortIb~Flolland Publitltinp Co., Antrterdanr Not to be reproduced by photoprlnt o~r microfilm without wrhtm permisdon from the publisher
AN EXACT SOLUTION OF THE DOPPLER-SKIFT ATTENUATION PROBLEM M . M . R. WILLIAMS
Nuclem Engtneerirtg Departniertt, Quern Mmy College, University of London Received 8 June 1976 (Revised 12 October 1976) Abshact : A method for calculating the average Doppler shift of y-rays emitted by excited particles slowing down in a host medium is developed based oa the direct, or forward, form of the 13oltzmann equation . The purpose is to connect the measured Doppler shift to the lifetime of the excited nuclear state. The Boltzmann equation is expanded in energy~epeadeat spherical harmonic moments which are then calculated exactly by the extension of an analytical method developed for the limoenergy distribution of slowed down neutrons . The method uses Mellin transforms and relies upon separability of the scattering law in the c.m . system . We are able to obtain concise, closed form solutions for the fractional shift F(r) for a variety of simple models, and series expansions for more complicated situations . Finally, the angular moments of the differential y-ray spectrum are calculated explicitly for hard-sphere scattering . In all cases, electronic stopping can be included, and the importance of this effect on the fractional shift is shown to depend on the energy dependence of the transport cross section for recoil-host atom collisions . It should be stressed that the emphasis throughout is on simple models and their exact representation ; it is unlikely that the results of the work reported here will be more than a ganeral guide to the cxperimeater . Comparison of the structure of the forward equation with that of the backward equation shows that, while some advantage is offered by the former in analytical studies with simple potentials, a numerical approach has to be adopted for both equations when non-separable scattering laws are involved . The greater familiarity of the forward equation for solving transport problems does, however, favour its usage .
1. Introdactiao The Doppler-shift method for measuring nuclear lifetimes has been in use for many years t ) as an accurate means of obtaining lifetimes in the range 10- t t s to 10 - t< s. The excited particle is allowed to slow down in a host medium and during its slowing down history it will emit a y-ray whose energy relative to the lab system will depend upon the velocity of the recoil nucleus at the time of emission . Quite clearly, the method will be most effective for lifetimes of the order of the slowing down time and, moreover, to obtain an accurate measure of the Doppler shift due to motion, a detailed statistical analysis of the slowing down mechanism must be employed. Because slowing down takes place by electronic and atomic collisions, it seems that a formulation in terms of a velocity distribution function is most appropriate and this may be obtained directly as a solution of the Holtzmann transport oquation 2). 317
31 8
M. M . R. WILLIAMS
Early work on the theory of this method used the stopping power concept which had the disadvantage of neglecting deflections arising from collisions . More recently, Blaugrund a) has improved the theory by including multiple scattering effects. However, an approach fully consistent with the Boltzmann equation is due to Winterbon4).Winterbon has formulated a balance equation for the functions, v, ~), which is the probability density for a Doppler shift s from a recoil having initial velocity v and direction 8 = cos -1~ with respect to the detector axis . The equation forf(s, v, ~.) was derived on the basis ofthe backward form ofthe Smoluchowski master equation s) and includes electronic stopping through the conventional friction factor and also nuclear stopping using Lindhard's method . In principle, the method of Winterbon can give the mean fractional shift F(T) of a nucleus with lifetime i, and the associated Doppler spectrum, with high accuracy and it must be regarded as a very satisfactory theory . Nevertheless, it has been shown by the author e .') that a number of simplifications, both analytical and physical, arise when the same type of problem is reformulated using the direct, or forward, form ofthe transport equation where, instead of the final recoil variables being fixed, as in the backward equation, it is the initial variables that are fixed. It is the purpose ofthis paper to explore the theory ofthe Doppler-shift experiment using the forward equation, and to make use ofa number ofresults derived in the area of neutron transport by nuclear reactor physicists. One of the main advantages of this method is that it avoids in many instances complicated series expansions and can give, for some special models, closed form expressions for F(ti). This will enable the convergence of the series arising for more realistic cases to be assessed and also an estimate to be made of the sensitivity of this function to various features of the scattering~law . In our formulation of this problem we include electronic stopping in the normal way but, in order to obtain analytical solutions, we must use the power law approximation for nuclear stopping. However, a rather more general form is used for this than is normal and it therefore provides greater generality in assessingthe effect ofthe scattering law on the interpretation of the basic experiment. The analysis employs Mellin transforms and a novel method ofsolving theresulting differential-difference equations. Finally, we wish to stress that our work should be regarded as complementary to Winterbon's results rather than in any way supplanting them . To put it otherwise, Winterbon's calculations deal with accurate scattering laws usingnumerical methods, whereas in the present work, simple scattering laws are used but exact solutions are obtained by means ofsophisticated analytical procedures . 2. Tre basic egnatioe of trseeport The experiment which our theory attempts to explain consists of studying the history ofexcited nuclei injected into a host slowing down medium. Thus, in principle, it should involve a spatial, as well as a time-dependent, description. However, it has
DOPPLER-SHIFT ATTENUATION
31 9
been argued that if the host medium is several times thicker than the range of the excited particles (10_ a -~ 10 - s cm) but small compared with the mean free path of a y-ray, then it is valid to regard the medium as infinite, with the source uniformly distributed throughout the body 1 ). This assumption is implicit in all of the more advanoed theories on this subject and we will continue to use it. Nevertheless, it should be emphasised that its range of validity has not been accurately asse.4sed, particularly as regards the degradation of the Doppler shifted photon after its emission. On the assumption of an infinite medium, we define the function N(E, La~Eo, La°, tkiEdLa as the number of recoil particles (i.e. y-emitters) at time t with energies between E and E+dE travelling in the solid angle dLa centred about the unit vector La, due to an initial pulse of particles at time zero whose initial energies were Eo and directions Lao. The equation describing this function when slowing down takes place in a multi-species host medium can be written as follows, where we define the new function ¢(E, ä1~Eo, La o, t) by ~ = vN, v being the speed of the slowing down particle corresponding to energy E [refs . 2" e,')] v ôt
~(E' ~~ E t) °' ~°' - ~ J 1~J J
e~a E' ' ,~~~°
x CQJ(E~, 6~(E/E~)~(E~, La~IEo+ ~o, t)-
+
ôE
-g~ElE~))
É~
Q~(E, e~(ElE~~E, nlEo, Lao,
(S~(~~E, nlEo, ~o, t))+a(E-Eo~(~-~o~(t~
t)J (1)
where
with AJ = MJ /M, nJ the number density of host species oftypej, QJ(E, B~ the scattering cross section in the c.m. system for a particle of energy E being deflected through an angle B~ and aJ = (AJ-1)2/(AJ+ 1)2. Here Mis the mass of the recoil atom and MJ the mass of the host atom. Further, Se(E) is the electronic stopping cross section. Now during its slowing down period, the recoil has a probability of emission at time t (assuming a single level) of a y-ray given by P(t) = exp(-t/T)/s, where T is the nuclear lifetime. Therefore the number of y-rays emitted with a Doppler shift (E, La) superimposed upon the normal rest value is G(E, La~Eo, Lao) =
~dtP(t)N(E, n~Eo, Lao, t). J0
(3)
If we write ~ = vG then this may be obtained from eq . (1) by means of eq . (3) to
320
M . M . R. WILLIAMS
give
e~a~ dE' 1 2n ~ ~(E, ~IEo, ~o) _ ~ 1-a1c ~ É, ~~ô -9~(ElE')) J i 8 x CQ~E , , B~(ElE'))~(E', ~IEo, ~o) Q(E, e~(ElE~)~(E, ~IEo, ~o) É, J + âE (Se(~(E, ~Eo, ~o))+ z S(E-Eo)S(~-~o) " The energy Er of a y-ray emitted from a moving recoil of velocity v is given in the non-relativistic limit by Er = E,,b(1 + v~/c~ where vx is the component ofvelocity along the detector axis. Ifvx is a random variable, as it will be due to the statistical nature of the slowing down process, we must write the mean observed energy of the y-ray as
~r =Ero
so r 1 ~ dE d[d vxG(E, ~~Eo, ~o) 0 J 1+ Bo (' c ~ dE j dl1 G(E, L1~Eo,1ï o) ,/o
If the particle were not slowed down at all by the medium the energy ofthe observed y-ray would be ~r = Ero ~ l + v,A/c}, where v,~ corresponds to (Eo, !~o) . Thus we may define the mean fractional shift F(T) as F(i) --
v
0
dE dl~ v~G(E, fè~Eo, Slo) J
Bo
,Jo i.e. in the slowing down medium the mean energy of the detected y-ray will be vxo . ~r - Erc ~1 + c F(2)~
(9)
Thus knowing Em, v,~ and c and measuring ~,, we may calculate F(r). It is the purpose of the present work to show how the lifetime r is related to the measured fractional shift F(s). To simplify the analysis, we take the distribution to be symmetrical about the axis through the detector (e .g. the x-axis) and hence abbreviate ~/r(E, U~Eo, no) by ~r(E,
DOPPLER-SHIFT ATTENUATION
32 1
~~Eo, ~) where u and ~o are the cosines of the angles made by the particles with the x-axis. We see therefore that eq. (8) becomes 1
~
(10) du~~(E, u~Eo+ uo)~ vo~o o i where we have noted that, by integrating eq . (4) over all E and âï, the denominator in eq. (8) is unity. To obtain ~r(E, u~Eo, ho) we write F(T) _
~ ~~
~(E, alEo, ~o) _ ~ }z(2l+ 1 )P,(Et)PdIb~~EI Eo) 1=0
(l l)
and insert it into eq. (4). It is then clear that the angular moments ~r,(E~Eo) are given by solutions of the following equation : sia, , 1 ~e dE C~J
I+
~
E
(Se(~)~~~)+ j ~E - Eo)~
(12)
where we have suppressed the dependence of ~, on Eo . From eq. (10) we may now write
1 B" _ - ~ dE~r l(E7. (13) vo 0 Thus only the equation for ~i(E) need be solved for this particular quantity . However, since the complete spectrum of emitted y-rays is also of interest and also because it involves little extra labour, we obtain solutions for all values of l. F(T)
3. Solution of the transport equation
An accurate form for the differential cross section a(E, 8) has been given by Lindhard e) and is based upon a rational fit to the Fermi-Thomas model of the atom. Lindhard finds that 2n sin
e~(E, e)ae =
nag a~f(t~),
(14a)
where a is an atomic screening length, t = lz sine ~B and ~ = E/E U E L being a conventional reference energy, andf(t~) is a universal function of t. It is clear that with the scattering law (14aß it is not possible to solve eqs. (12) analytically. However, a number of very efficient numerical txhniques are available for solving the integral equations for the angular moments ~y,(E) and there is no doubt that such a procedureis equally as good as that employed by Winterbon in his solution of the backward equation.
32 2
M. M. R. WILLIAMS
It is not our purpose here, however, to consider the numerical aspects ofthe problem but rather to examine the analytical properties of the equations . In order to do this it is necessary to make certain assumptions about a(E, B). We therefore assume that Q~(E, 9) =
Q ~J~(B), 4
(14b)
where Q,(E) is the transport cross section and f(B) the angular distribution . We shall discuss the nature of these functions below, but basically they are slightly more general than Lindhard's power law approximation e ) . To simplify notation, we define the total macroscopic cross section (15)
_ ~ n~~~~
(16)
p~~ = n,~~~(~l E~(~~
Further, (17) E E~(~ = Ec(Eo) ~É
(18)
and 9~,(E) = E~(E~,(E) . Then with p~ independent of energy, p = q+~k, K ~. = K/E,(Eo)E~` and To 1 = vo E,(Eo), eq . (12) can be written K
To (É~
= ~(1-k),
~~~ = s~, aÉ {E~SP,(~Î +~ 1~Ja,
~`J~x)
~~~
Cx) P~(9~x))-x~`~~J
+ 8(E - Eo),
(19)
where rc = ~-2m in Lindhard's notation. In arriving at eq. (19) we have made some simple substitutionsto change the limits on the integral and also have setf(x) _ ,R9(x)). It remains to solve the integro-differential equation (19). This we do by defining the Mellin transform, viz.
which after application to eq . (19) reduces it to toEô~,(p-x) _ -iD,(p)~~P)-Z~IP-1)~~P+~ - 1)+Eg-1,
where
(21)
323
DOPPLER-SHIFT ATTENUATION
In terms ofthe Mellin transform, it is readily shown that F(i) =
E~
R
(23)
~~(~ - x)~
Eq. (21) is a difference equation in a form which we have already solved in a previous publication') concerned with the slowing down of fast particles. In that paper two cases were considered, (a) P = 1-x which if y takes the usual value of i leaves k arbitrary, and (b) P = 1 when y = 1- Zk. Taking the solution in case (a) we find that
where
ecl- K)~~=o 1 Eo rßt(P~-v~liu -~ dP ~-~ 2ni L E T o Eo f
G(t)
_i
~,0 x (1
(
t _f 1 _
, Y- °~
e_Kxor~w~
(2~
and ß,(p) is a function defined by In case (b~ we have an expression identical with (26~ except that ~ o = 0, G(t) = t and ~,(p) is defined by H,(p+x~,(p+x) _ ~i(p~
(29)
where H,(p) = Di(p)+~(p-1~ In principle, therefore, subject to a knowledge of ßß(p) we have a complete solution to the problem. 4. Cakolatioo of the fnctMeal Dopier shift Let us now calculate F(r) for a variety ofsituations. We note from eq. (23) that
324
M. M. R. WILLIAMS
which can be written as follows for case (a) F(T)
= T r1(~-x)J 0~dtexp ~-
CT
+ °° ~ 2 0 tJ (~1
(
_G(t) _1
dpY-P
(31)
For case (b), eq. (31) remains valid ifwe set ~o = 0, G(t) = t and use eq. (29) for ~1(p). To proceed further it is necessary to consider separately the cages x ? 0 and x < 0. In.the former case, we may expand the exponential in G(t) in eq. (31), integrate over y and evaluate the conto.r integral over the simple poles. The result is that in case (b) i F(i) = 1+ ~ ~- -~ ~ H1(~ - x+mx~ e=1
TO
~n=1
(32)
where we have used the readily proved relationship _
~11Y)
r1ll~+nx) -
~ H1(P+mx).
iw=1
(33)
For case (a) the problem is more difïicult and we find ~dtexp ~+ °° ~ t G(tr ~ D1(~-x+mx). (34) ~ n!>~0 J 0 1 + .~l0t ~T 0 + n=1 CT 20 J m=1 But if we note that F(s)
we can set y = exp(-x~, o t/TO) and find F(s)
~ ~ (n~ 1+ .1 °1Tn s° +A_1 ~ (~ xr~a1
D1(~-rc+mx)
We defer consideration ofthe case x < 0 until later. Let us now consider some speoific cases which yield closed form expressions for F(i) . Firstly, we note that D1(P) _ ~~ 1~f,(x)Cl-X°- ' {~(A,+1)x~_~,q,_l~C-~} ], ~ 1-a~J ar
(36)
which for power law scattering where
r(x)
_
(2-Sxl-ar
ul -Xr
(3~
DOPPLER-SHIFT ATTENUATION
325
can be evaluated in terms of incomplete beta functions, viz. ~,,
1
D1(P) _ ~ zP,~2 - s~) ~ 1- s1 - ~(A~+
lxl -a~r~ -1B1 _a~(1-s1 ; P+i)+~{A!-1) x( 1-
a~x'_
1B1-ai(1 - s~ ;P -i)t~
Note that s = 1 +m in Lindhard's notation. In the special case of hard-sphere scattering s = 0, and D1(p) becomes A~+1 (1-a~ +~) A~-1 (1-aj-~) D1(P) _ ~P1{1-
1-a~
2p+1
+ 1-a~
Similarly, in the case when A = 1 and we have a single species 2-s ~
2p-1
(38)
(39)
r(2-s)r(p+Z)l
which for isotropic scattering in the c.m. system can be written (s = 0) as
With the appropriate values of s~ and a1 we can now evaluate F(T) by direct use of eqs. (32) and (35). However, to obtain a simple closed form for F(T~ we consider the case given by eq. (41) and set x = ~, i.e. a cross section independent of energy. Then D1 (i-x+mx) = D 1 (1+gym) and we easily show that 6 D1( 1 +zm) = (n+2xn+3)~ ~_1
(42)
Thus we can evaluate F(T) very easily by direct term by term summation. If~u = 0, i.e. no electronic stopping, the series can be written Fs ( ) = 1+6
which sums to
_T ~ s=1 ~ To ~ L~~r
1
(n+2xn+3)'
6 - 6 (1 F(t) _ ~ + +T) ln (1 +T~
(43)
(44)
where 4 A further exact solution can be obtainod when x = 0 which is valid for arbitrary D1(p). We note that setting x = 0 in either (32) or (34) leads to =
T/TO .
(45)
326
M. M. R. WILLIAMS
which can be summed to give
It is interesting to note that this is the same functional form as would be obtained from simple stopping power theory although naturally the coefficient multiplying T differs. The case for x < 0 is rather more difficult to deal with sincethe expansionin powers of y" in eq . (31) is no longer valid. Let us consider case (b) first. Then we may perform the integration over t and obtain 1 ~ K r p F(T) = ßl(i-x) 2ni,~L ~~ 1~)Jo dl+TCOK
Now setting x = - ~, where ~ > 0, we can write(47) as 1 F(T) = ßl(~+~)2ni
dp
1 ~~~+z { -r
fL~Jo
i+ca~
Expanding the denominator of the second integrand as a series in cup/i, integrating over m and inverting the transform, leads to ~ (-i F(T) _ -o ~+1
ßl(~+ ~) 1(~+~+n+ 1~)
(49)
which from the difference equation for ß1(p) allows us to write
A-o
~=o
(50)
This is an asymptotic series in inverse powers of T and is the analytic continuation of eq. (32) . For the special case ~ = 0 it can be summed to give agreement with eq. (46). In addition, if we invoke the hard-sphere assumption and equal masses, it can also be summed to give, for ~ = i and ~o = 0 F(T)=
lOTZ+15T+6 . 6(1 +T)3
(51)
There are some disadvantages in using the general asymptotic series for models where an exact summation cannot be performed because it precludes analytic continuation of the series to small values of T. We therefore propose the following alternative procedure. The integral in (48) is written "1 dwW~ + +zr-v (52) o (1+4)-(1-m~'
327
DOPPLER-SHIFT ATTENUATTON
and expanded in powers of(1-cot)/(1 +T). Then using eq. (34a) weget (53) 1 +~ ~ ( ~ Hl(i+m+1~)-1, v ~,= =o (1 +ip ,,= o -)yCn/ o which converges for all i. Case (a~ as described by eq. (31~ is more difficult and at the present time no way has been found of reducing it to a series in terms of Dl(p). The problem stems from the inability to perform the integral over t or to find a convergent expansion which will enable this to be done . There appears no alternative, therefore, but to find Î' 1 (p) directly and evaluate the integrals term by term. We show in the appendix how to construct the functions ß1(p) from the difference equation. Using ß1Y) [as given by (A.5)] in eq . (31), we may invert the Laplace transform using tables la) to find that F(T) _ ~
F(s) =
dt exp ~- ~~ + ~~ ~ t- G(t)J 1F1
CK, x
;
G(t)I .
0 TJ 0 0 For ~ o = 0, the integral may be carried out explicitly, and we obtain _ TU (1_0 2 4 F(i) T+T ZFl lK' l' K ' T+T 0/ ' 0
(54)
(55)
where the hypergeometric functions are to be interpreted as
1 2 ~ (57) lim zFl -, 1 ; - +e ; x , K CK respectively. For K = i, eq. (55) reduces to eq . (44~ for K = 0, eq . (54) reduces to eq . (46) and for K = -i, eq. (55) reduces to eq. (51). When ~ o ~ 0, we note that for K = -~ 1 2 _G _ 1 _G ~ 1+2~
and for K = -1
_1 GZ 0 +12T2, 0
1F1(1C'1C't 0
1 2 G
Cx,K
1G ;~~=1+2r
(58)
(59)
The integrals for F(T) must then be evaluated numerically. We also note that when K = -1 and ~ o = 0, eq. (55) gives _ 2+3i F(t) 2(1 +=)s
328
M. M. R. WILLIAMS S. Numerical evaluation and discussion of F(s)
For x ? 0, we have obtained exact expressions for F(T) for the generalized inverse power law model. These results are given by eqs. (32) and (35) and for the special case of x = 0 by eq. (46). In addition, we can give a simple closed form expression for F(t) in the case ofhard spheres when the target and recoil particle masses are equal, i.e. eq. (44) . When x < 0, the problem is more difficult and we can only obtain a satisfactory expression for F(i} incase (b), i.e. when theelectronic stopping cross section bears a direct relation to the nuclear stopping. However, a general expression for F(t) can be obtained when the scattering is isotropic in the centre of mass system, i.e. eq. (54). Thus the dependence of F(T) on a wide range of parameters connected with the scattering law can be studied. T,~a~1 Fractional Doppler shift F(r) as a function of the reduced lifetime 4 = s/ro, for isotropic scattering in the c.m . system, equal target and recoil masses and varying suss-sectional energy dependence, K = }-2m K
i 0 0.01 0.05 0.1 0.2 0.5 1 2 5 10 20 50 100
0 1 0.995 0.976 0.953 0.911 0.807 0.682 0.528 0.324 0.202 0.117 0.0528 0.0278
1 0.995 0.976 0.952 0.909 0.800 0.667 0.500 0.286 0.167 0.0909 0.0385 0.01%
-i 1 0.995 0.975 0.952 0.907 0.790 0.646 0.469 0.255 0.145 0.0775 0.0324 0.0164
-1 1 0.995 0.975 0.950 0.903 0.778 0.625 0.444 0.236 0.132 0.0703 0.0292 0.0148
Our first example of the sensitivity of F(T) to the scattering law behaviour is shown in table 1 . Here, we assume isotropic scattering in the c.m . system, i.e. s = 0 in eq . (40), equal masses of target and recoil, and (a) x = ~, m = 0, i.e. constant cross section, (b) K = 0, m = ;, cross section inversely proportional to velocity, (c) rc = -}, m = 4, cross suction inversely proportional to E}. In this case we also neglect electronic stopping . It is clear from the table that F(s) becomes more sensitive to the value of K as 4 increases. For example, at i = 0.1, the dißerences in the values of F are less than 0.3 ~ between rc = ~ and -1. At T = 1, the fractional dißerence increases to 8.3 whilst at i = 10 it is 35 ~. Clearly, therefore, in the experimentally important range of values of 4, an accurate value of K is required . In order to examine the sensitivity of F to the angular distribution of scattering in
DOPPLER-SHIFT' ATTENUATION
329
thec.m. system, we consider eq . (46) with D i (i) given by eq. (38). It is readily shown that Di(i) _ ~ ~~(A~+ 1~
(61)
i.e. independent of s. Thus the fractional shift for x = 0 can be written (62) which depends on s via ~o and T o. Before proceeding with our discussion it is as well to note that T, the reduced lifetime, is scaled in units of To, which is the mean free time between collisions for source energy particles. By definition, i~' = voE,(Eo), where E,(Eo) is the transport cross section. In terms of the Lindhard formalism this is readily shown to be l') nnaz~. E~(Eo) = 1-
Ec
m CEo)
z~
(63)
where n is the number oftarget atoms per unit volume, a the Thomas-Fermi shielding length, EL a characteristic energy and ~i , a parameter of order unity. This formula is valid for 0 5 m ~ 1. On the other hand, Winterbon ~) has soled his lifetime in terms of T m where To -_ ~{1-mxl-aÏ"Tm. We note, therefore, that by using z~ as the scaling parameter, eq. (62) becomes 1)z~- i + .lo(A + 1)z~ i (A + 1 1 = 1+ (65) We feel that so is a more logical choice, although in principle it would be better to scale in terms of
E
Numerically, Ri o varies between zero and about unity. To illustrate the combined effect ofchanges in ~o and mass number A, we show in table 2 values of F(T) for A = 10, 1 and 0.1 and values ~lo = 0 and 0.5. F.q. (62) is employed and the results therefore apply to m = ~. It is clear that the effect of changes in ~o has its greatest influence for small values of A, i.e. when the recoil is massive compared with the target atom. A further measure of the sensitivity of F(T) to system parameters is shown in figs
330
M. M. R. WILLIAMS T~>li.~ 2
The fractional Doppler shift for m = ~ with variable recoil mass and electronic stopping factor ~ u .4
10
0.01 0.05 0.1 0.2 O.S 1 2 5 10 20 50 100
1
0.1
0
0.5
0
0.5
0
0.5
0.993 0.967 0.936 0.879 0.744 0.592 0.420 0.225 0.127 0.0676 0.0282 0.0143
0.999 0.955 0.914 0.842 0.680 0.516 0.347 0.176 0.0962 0.050 0.0208 0.0105
0.995 0.976 0.952 0.909 0.800 0.667 0.500 0.286 0.167 0.0909 0.0285 0.0196
0.993 0.974 0.930 0.870 0.727 0.571 0.400 0.211 0.118 0.0625 0.0260 0.0132
0.999 0.997 0.993 0.986 0.967 0.936 0.879 0.744 O.S92 0.420 0.225 0.127
0.997 0.984 0.969 0.940 0.862 0.758 0.611 0.385 0.239 0.136 0.0590 0.0304
t-0
s ~7 ~6 ~5 .4 3 ß "1 0 10-'
1
10
~E
10 '
Fig. 1 . The influence of electronic stopping on the fractional Doppler shift, F, as a function of normalised nuclear lifetime 4. The model assumes isotropic scattering in the c.m. system with aoss sxtion inversely proportional to the energy (K = -}) . The recoil to host atom mesa ratio is unity. The curves correspond to values of Zo = 0, 0.1, O .S, 1 .0 .
O1 10 '
1
10
;
10
Fig. 2. The fractional Doppler shift as in fig. 1 but with cross sxtion inversely proportional to the cube of velocity (K = -1) .
DOPPLER-SHIFT ATTENUATION
33 1
1 and 2. Here we show how, with isotropic scattering in the c.m. system, F(T) varies with d o for different functional forms of the total cross section. In all cases A = 1. We observe a strong influence as ~o goes from zero to unity, with the effect being greater when the total cross section is a more rapidly varying function of energy, i.e. fig. 2. The general influence of electronic stopping is a reducion in the Doppler shift for a given lifetime . This result is expected physically owing to the shorter slowing down time experienced by the recoil. A summary ofthe various analytical results obtained for F(z) is given in table 3. TABLE
3
The various analytical results obtained for F(s) Equation
Parameters crie (b), K , 0, all Zo, s, a case (a), x ~ 0, all zo , s, a
(32)
(35)
K= },zo=O,s=O,a=0 K = 0, all z o, s, a K < 0, all Zo, s, a K =- },zo=O,s=0,a=0 rc= -l,zo =O,s=O,a=0 crie (a), K < 0, s = 0, a = 0, all Zo K=},0,-}, -l,s=O,a=O,zo=O K=O,s=4,zo=0,0 .5, .l= 10,1,0 .1
(53) (51) table 1 table 2
6. The y-ray energy spectrum We have defined in eq . (3) the energy and direction shift superimposed upon the emitted y-ray as a result of the recoil slowing down . In terms of our present co-ordinate system, this is given by eq . (11) in the following form: GtE, ~IEo, po) _ ~~ ~ ît2l+ 1)P,(~t)Pr~o)~a(EIEo)~ i-o
where the ~P,(E/Eo) are found from eq. (24). Let us first note that eq . (26) can, by the transformation C = (EolE)y, be cast into the form ~`~E't)
_ Ott-K )xoUiu
s 0E0
1 dp~~~_vxdko 2rri,fL (68)
where s(E) = to(Eo/E~` is the collision time for particles with energy E. Now for x ? 0, it is readily shown by writing _ {"`ii _
p~
~
Tt~} -
e
_ ac
~
7C~4
1 - h~p ~ 1 - h~'
332
M . M, R. WILLIAMS
where p = 1-~-`, and using the generating function for Laguerre polynomials, that eq. (68) may be written as
where x = G(t)/z(E) and eu-R~~/to "
V Cn/ TOEO v=0
Thus integrating over t and using eq. (24), we find
x ~ D - (p+mx) =o
0
dx 1-
e- L (x).
0
(71)
This expression is rather tedious to evaluate and so we consider the limit ~.° ~ 0 when the integral over x can be performed and is equal to z_ z(~ "+ i (E) ([+z(E))
If case (b) is acceptable we may still use (72) but replace D, in eq. (71) by H,. In the special case of x = 0, we may write 1
E°1 P
1
(73) JL \ In order to evaluate the inverse Mellin transforms, it is necessary to examine the behaviour ofthe integand as ~p~ -~ oo. Clearly, since H,(oo) -~ oo as ~p~ -" oo, all ofthe integrals in eq. (71) for case (b) will converge. On the other hand, if we neglect eleotropic stopping, ~i,o = 0, and D,(p) -~ D,(oo) as ~p~ -" oo. Thus using the arguments of sect. 5 we must recast the oocpression for ~,(E~E° ) as follows : 0 0
Dd~h+zo ~`(E~E°) + z(~Eo "~ CD~(~)z+z(E))
x
" Y n _ 1 E° P ' 1 LdP ~(-) (v ) 2ni ,1 (E ) ~ ,~ D~P+mx)
1 D~(oo)" },
(74)
DOPPLER-SHIFT ATTENUATION
33 3
where D,(oo)=
(2-si) ~Pi2(1 -si) , oo,
si < 1 si ? 1.
It is readily seen that for s = 0, i.e. zero lifetime ~,(E/E o) = 8(E- Eo)/To and, from eq. (67), that the emergent spectrum is equal to S(E- Eo)S(ü - uo), i.e. unal%cted by the medium. This is a self-evident result but emerges from our analysis in a consistent fashion. When ~o # 0 there is no `uncollided', i.e. S(E-Eon-~o), component unless i = 0 because electronic stopping slows the particle down continuously at the instant it enters the medium . Nuclear stopping on the other hand is a discrete process and in general there are always some particles which have not collided, although as time proceeds, the fraction of these is reduced. However, even for some cases of nuclear stopping, where D~(oo) = oo, there is no uncollided contribution because ofthe long range nature ofthe power law forces. This has the effect ofmaking the total cross section divergent. For most practical cases, therefore, where s ? 1, the uncollided component will be absent and the discrete picture of nuclear scattering events must be abandoned. For the case x = - ~, ~ > 0, we again encounter difficulties in treating eq . (26). However, setting G = t and do = 0, we readily find that
CEo,° ~~ 1 1 ~ _ 1 ~E- Eo) x 1 ~ dP 2ni L E fI ~ D,(p+m~) D;(oo) + To 1+iD,(ao)'
(75)
where in the case (b) formalism, D, is replaced by H, and there is no uncollided contribution to SY,. Eqs . (74) and (77) are particularly well illustrated by the cases of ~ = x = 0, when for ~ o = 0, they both reduce to E 1 1 E ° 1 _ ~ - Eo + ~P EI E o )1 dP i TD~~) ioEo 2ai,fL ~ E ~ ~1 +TD~(p) 1 +iD,(oo)} ~
( 76 )
Further progress in evaluating ~Y,(EIEo) requires knowledge of D,(p). As we see from eq. (22), these functions can be evaluated once the scattering lawf(x) is given . To illustrate the problem we consider eq. (76) for hard-sphere scattering with A = 1, when the D,(p) are particularly easy to use in the inversion integral . They can then be defined as
f0
D~P) = I-2 1dYY2°-1 p~Y~
(77a)
334
M . M . R. WILLIAMS
or more explicitly 1D~P) =
from which Do(P) _ (P - 1)lP,
r(P)r(P+i) r(~+2-Z~r(l'+1+Z~'
(77b)
Di(P) _ (2P- 1)l(2P+ 1),
DZ(P) _ (PZ+~)/PCP+I),
D3(p) _ (4p2+4p+7)/(2p+3x2p+1),
Da(P) _ (P 3 +2p2 +4 - â)/PCP+1xP+2),
etc .
In all cases D,(oo) = 1 . The functions ~P'o and ~ 1 are easily evaluated, the results being (78) cr-1~/zu +r~
ToEo~i(EIEo) _ (1 +i)2 ( Eo)
(79)
From the total number density spectrum ~`o it can be observed that as the lifetime of recoils increases, the differential energy shift of the emitted y-ray moves towards lower energies. This fact is obvious from the physical situation. The current spectrum Y'1 is rather more interesting since it is closer to the measured quantity . This indicates that the detailed nature of the differential energy shift spectrum depends crucially on the value of the lifetime . For example when T < 1, the spectrum is rich in high energy y-rays, whilst for T > 1, it is the low energy part of the spectrum which predominates . The sensitivity of the measured spectrum to the value of z has been noted by experimental workers 12). Naturally, the very simple model considered here would not be expected to give good agreement with experiment, however it does give the general trend. More realistic calculations would require detailed evaluation of eqs. (74) and (75). Finally, we mention that 3Y 2 and 9T S can be evaluated easily, although for certain values of T the poles become complex and give rise to an oscillatory behaviour with (Eo/E). Since 9' 2 and SP3 do not have any special physical significance this behaviour is not unexpected. To compute the complete angle-energy spectrum, G(E, h~Eo, ~o)+ requires a rather large number of the moments ~P',(E~Eo), especially near E = Eo. However if G(E, p~Eo, uo) is partially integrated over p as it might be when accounting for the geometry of a detector, then the number of moments required for good convergence is considerably reduced. However, we are still faced with the basic problem, similar to that met with the backward equation, of summing the angular components . Thus the forward equation offers no advantage in this instance.
DOPPLER-SHIFT ATTENUATION
33 5
7. Summary and general discussion The main purpose ofour work has been to demonstrate the alternative, Boltzmann equation approach to the calculation of the slowing down of particles in matter . It is self-evident from the foregoing work that it is a viable technique with certain advantages over the more familiar backward formulation : not the least of these being the availability ofa large number ofmathematical techniques (and also computer codes) developed by nuclear reactor physicists. These would be particularly useful when more realistic scattering potentials are used . We do not conclude, however, that the forward equation has any over-riding advantage over the backward equation, except for the fact that it has been studied more frequently and its properties, both analytical and numerical, are better understood. Some very convenient closed form expressions have been obtained for the fractional Doppler shift based on simple, but non trivial, models ofthe slowing down mechanism In addition, some expressions which are directly comparable with experiment are obtained which can be evaluated at the expense of computational effort. The angular components of the differential energy shift of y-rays is also obtained and the crucial role played by the nuclear lifetime on the shape ofthis function is illustrated by means of a hard-sphere model of scattering. Further work in this area, should concentrate on direct numerical solutions of eq. (4) using the most sophisticated models of a(E, B) available. Finally, it is recommended that the errors arising from spatial variation in a slab geometry be examined. Appendix SOLUTION OF THE DIFFERENCE EQUATION
It is shown in Levy and Lessman' 3) that the solution of the difference equation g(x)y(x + 1) = f(x)y(x~
(A.1)
where g(x) and J(x) are polynomials of degree r and s, respectively, can be written in the form l'(x)
where
a a a: r _ ~-~ ~ I'(x - a.)l ~ l'(x - l'al+ b ~-i i-i
(A .2)
a
r=i For scattering that is isotropic in the c.m. system, we have shown that Di(p)
336
M. M. R. WILLIAMS
_ (2p-1)/(2p+1). Thus the dü%rence equation (28) can be cast into the above canonical form and the solution is easily shown to be ~t(p) ° l'
p
Cx
+ 1+
1 p 1 ~~ + l 2K)'r 2rc) ~
(A.5)
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
S. Devons et al., Proc. Phys. Soc. A68 (1955) .18 M. M. R. Williams, J. of Phys . A9 (1976) 771 A. E. Blaugrund, Nucl. Phys. 88 (1966) 501 K. B. Winterbon, Nucl . Phys . A246 (1975) 293 S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1 M. M. R. Williams, J. of Phys . D9 (1976) 1279 M . M. R. Williams, Rad. Effects 30 (1976) 47 J. Lindhard et al., Mat. Fys. Medd. Dan. Vid. Selek. 33 (1963) no. 14 I. Waller, Ark. Fys. 37 (1968) 569 M. J. Lighthill, Fourier analysis and generalised fondions (1959), Camb. Univ. Press P. Sigmund, Rev. Rouen. Phys. 17 (1972), 823, 969, 1079 A. E. Litherland et al., Nucl . Phys . 44 (1963) 220 H. Levy and F. Leasman, Finite difference equations (Pitman, London, 1959) G. E. Roberte and H. Kaufuran, Tables of Laplace transforms (W. B. Saunders, Philadelphia and London, 1966)
AN EXACT SOLUTION OF THE DOPPLER-SKIFT ATTENUATION PROBLEM M . M . R. WILLIAMS
Nuclem Engtneerirtg Departniertt, Quern Mmy College, University of London Received 8 June 1976 (Revised 12 October 1976) Abshact : A method for calculating the average Doppler shift of y-rays emitted by excited particles slowing down in a host medium is developed based oa the direct, or forward, form of the 13oltzmann equation . The purpose is to connect the measured Doppler shift to the lifetime of the excited nuclear state. The Boltzmann equation is expanded in energy~epeadeat spherical harmonic moments which are then calculated exactly by the extension of an analytical method developed for the limoenergy distribution of slowed down neutrons . The method uses Mellin transforms and relies upon separability of the scattering law in the c.m . system . We are able to obtain concise, closed form solutions for the fractional shift F(r) for a variety of simple models, and series expansions for more complicated situations . Finally, the angular moments of the differential y-ray spectrum are calculated explicitly for hard-sphere scattering . In all cases, electronic stopping can be included, and the importance of this effect on the fractional shift is shown to depend on the energy dependence of the transport cross section for recoil-host atom collisions . It should be stressed that the emphasis throughout is on simple models and their exact representation ; it is unlikely that the results of the work reported here will be more than a ganeral guide to the cxperimeater . Comparison of the structure of the forward equation with that of the backward equation shows that, while some advantage is offered by the former in analytical studies with simple potentials, a numerical approach has to be adopted for both equations when non-separable scattering laws are involved . The greater familiarity of the forward equation for solving transport problems does, however, favour its usage .
1. Introdactiao The Doppler-shift method for measuring nuclear lifetimes has been in use for many years t ) as an accurate means of obtaining lifetimes in the range 10- t t s to 10 - t< s. The excited particle is allowed to slow down in a host medium and during its slowing down history it will emit a y-ray whose energy relative to the lab system will depend upon the velocity of the recoil nucleus at the time of emission . Quite clearly, the method will be most effective for lifetimes of the order of the slowing down time and, moreover, to obtain an accurate measure of the Doppler shift due to motion, a detailed statistical analysis of the slowing down mechanism must be employed. Because slowing down takes place by electronic and atomic collisions, it seems that a formulation in terms of a velocity distribution function is most appropriate and this may be obtained directly as a solution of the Holtzmann transport oquation 2). 317
31 8
M. M . R. WILLIAMS
Early work on the theory of this method used the stopping power concept which had the disadvantage of neglecting deflections arising from collisions . More recently, Blaugrund a) has improved the theory by including multiple scattering effects. However, an approach fully consistent with the Boltzmann equation is due to Winterbon4).Winterbon has formulated a balance equation for the functions, v, ~), which is the probability density for a Doppler shift s from a recoil having initial velocity v and direction 8 = cos -1~ with respect to the detector axis . The equation forf(s, v, ~.) was derived on the basis ofthe backward form ofthe Smoluchowski master equation s) and includes electronic stopping through the conventional friction factor and also nuclear stopping using Lindhard's method . In principle, the method of Winterbon can give the mean fractional shift F(T) of a nucleus with lifetime i, and the associated Doppler spectrum, with high accuracy and it must be regarded as a very satisfactory theory . Nevertheless, it has been shown by the author e .') that a number of simplifications, both analytical and physical, arise when the same type of problem is reformulated using the direct, or forward, form ofthe transport equation where, instead of the final recoil variables being fixed, as in the backward equation, it is the initial variables that are fixed. It is the purpose ofthis paper to explore the theory ofthe Doppler-shift experiment using the forward equation, and to make use ofa number ofresults derived in the area of neutron transport by nuclear reactor physicists. One of the main advantages of this method is that it avoids in many instances complicated series expansions and can give, for some special models, closed form expressions for F(ti). This will enable the convergence of the series arising for more realistic cases to be assessed and also an estimate to be made of the sensitivity of this function to various features of the scattering~law . In our formulation of this problem we include electronic stopping in the normal way but, in order to obtain analytical solutions, we must use the power law approximation for nuclear stopping. However, a rather more general form is used for this than is normal and it therefore provides greater generality in assessingthe effect ofthe scattering law on the interpretation of the basic experiment. The analysis employs Mellin transforms and a novel method ofsolving theresulting differential-difference equations. Finally, we wish to stress that our work should be regarded as complementary to Winterbon's results rather than in any way supplanting them . To put it otherwise, Winterbon's calculations deal with accurate scattering laws usingnumerical methods, whereas in the present work, simple scattering laws are used but exact solutions are obtained by means ofsophisticated analytical procedures . 2. Tre basic egnatioe of trseeport The experiment which our theory attempts to explain consists of studying the history ofexcited nuclei injected into a host slowing down medium. Thus, in principle, it should involve a spatial, as well as a time-dependent, description. However, it has
DOPPLER-SHIFT ATTENUATION
31 9
been argued that if the host medium is several times thicker than the range of the excited particles (10_ a -~ 10 - s cm) but small compared with the mean free path of a y-ray, then it is valid to regard the medium as infinite, with the source uniformly distributed throughout the body 1 ). This assumption is implicit in all of the more advanoed theories on this subject and we will continue to use it. Nevertheless, it should be emphasised that its range of validity has not been accurately asse.4sed, particularly as regards the degradation of the Doppler shifted photon after its emission. On the assumption of an infinite medium, we define the function N(E, La~Eo, La°, tkiEdLa as the number of recoil particles (i.e. y-emitters) at time t with energies between E and E+dE travelling in the solid angle dLa centred about the unit vector La, due to an initial pulse of particles at time zero whose initial energies were Eo and directions Lao. The equation describing this function when slowing down takes place in a multi-species host medium can be written as follows, where we define the new function ¢(E, ä1~Eo, La o, t) by ~ = vN, v being the speed of the slowing down particle corresponding to energy E [refs . 2" e,')] v ôt
~(E' ~~ E t) °' ~°' - ~ J 1~J J
e~a E' ' ,~~~°
x CQJ(E~, 6~(E/E~)~(E~, La~IEo+ ~o, t)-
+
ôE
-g~ElE~))
É~
Q~(E, e~(ElE~~E, nlEo, Lao,
(S~(~~E, nlEo, ~o, t))+a(E-Eo~(~-~o~(t~
t)J (1)
where
with AJ = MJ /M, nJ the number density of host species oftypej, QJ(E, B~ the scattering cross section in the c.m. system for a particle of energy E being deflected through an angle B~ and aJ = (AJ-1)2/(AJ+ 1)2. Here Mis the mass of the recoil atom and MJ the mass of the host atom. Further, Se(E) is the electronic stopping cross section. Now during its slowing down period, the recoil has a probability of emission at time t (assuming a single level) of a y-ray given by P(t) = exp(-t/T)/s, where T is the nuclear lifetime. Therefore the number of y-rays emitted with a Doppler shift (E, La) superimposed upon the normal rest value is G(E, La~Eo, Lao) =
~dtP(t)N(E, n~Eo, Lao, t). J0
(3)
If we write ~ = vG then this may be obtained from eq . (1) by means of eq . (3) to
320
M . M . R. WILLIAMS
give
e~a~ dE' 1 2n ~ ~(E, ~IEo, ~o) _ ~ 1-a1c ~ É, ~~ô -9~(ElE')) J i 8 x CQ~E , , B~(ElE'))~(E', ~IEo, ~o) Q(E, e~(ElE~)~(E, ~IEo, ~o) É, J + âE (Se(~(E, ~Eo, ~o))+ z S(E-Eo)S(~-~o) " The energy Er of a y-ray emitted from a moving recoil of velocity v is given in the non-relativistic limit by Er = E,,b(1 + v~/c~ where vx is the component ofvelocity along the detector axis. Ifvx is a random variable, as it will be due to the statistical nature of the slowing down process, we must write the mean observed energy of the y-ray as
~r =Ero
so r 1 ~ dE d[d vxG(E, ~~Eo, ~o) 0 J 1+ Bo (' c ~ dE j dl1 G(E, L1~Eo,1ï o) ,/o
If the particle were not slowed down at all by the medium the energy ofthe observed y-ray would be ~r = Ero ~ l + v,A/c}, where v,~ corresponds to (Eo, !~o) . Thus we may define the mean fractional shift F(T) as F(i) --
v
0
dE dl~ v~G(E, fè~Eo, Slo) J
Bo
,Jo i.e. in the slowing down medium the mean energy of the detected y-ray will be vxo . ~r - Erc ~1 + c F(2)~
(9)
Thus knowing Em, v,~ and c and measuring ~,, we may calculate F(r). It is the purpose of the present work to show how the lifetime r is related to the measured fractional shift F(s). To simplify the analysis, we take the distribution to be symmetrical about the axis through the detector (e .g. the x-axis) and hence abbreviate ~/r(E, U~Eo, no) by ~r(E,
DOPPLER-SHIFT ATTENUATION
32 1
~~Eo, ~) where u and ~o are the cosines of the angles made by the particles with the x-axis. We see therefore that eq. (8) becomes 1
~
(10) du~~(E, u~Eo+ uo)~ vo~o o i where we have noted that, by integrating eq . (4) over all E and âï, the denominator in eq. (8) is unity. To obtain ~r(E, u~Eo, ho) we write F(T) _
~ ~~
~(E, alEo, ~o) _ ~ }z(2l+ 1 )P,(Et)PdIb~~EI Eo) 1=0
(l l)
and insert it into eq. (4). It is then clear that the angular moments ~r,(E~Eo) are given by solutions of the following equation : sia, , 1 ~e dE C~J
I+
~
E
(Se(~)~~~)+ j ~E - Eo)~
(12)
where we have suppressed the dependence of ~, on Eo . From eq. (10) we may now write
1 B" _ - ~ dE~r l(E7. (13) vo 0 Thus only the equation for ~i(E) need be solved for this particular quantity . However, since the complete spectrum of emitted y-rays is also of interest and also because it involves little extra labour, we obtain solutions for all values of l. F(T)
3. Solution of the transport equation
An accurate form for the differential cross section a(E, 8) has been given by Lindhard e) and is based upon a rational fit to the Fermi-Thomas model of the atom. Lindhard finds that 2n sin
e~(E, e)ae =
nag a~f(t~),
(14a)
where a is an atomic screening length, t = lz sine ~B and ~ = E/E U E L being a conventional reference energy, andf(t~) is a universal function of t. It is clear that with the scattering law (14aß it is not possible to solve eqs. (12) analytically. However, a number of very efficient numerical txhniques are available for solving the integral equations for the angular moments ~y,(E) and there is no doubt that such a procedureis equally as good as that employed by Winterbon in his solution of the backward equation.
32 2
M. M. R. WILLIAMS
It is not our purpose here, however, to consider the numerical aspects ofthe problem but rather to examine the analytical properties of the equations . In order to do this it is necessary to make certain assumptions about a(E, B). We therefore assume that Q~(E, 9) =
Q ~J~(B), 4
(14b)
where Q,(E) is the transport cross section and f(B) the angular distribution . We shall discuss the nature of these functions below, but basically they are slightly more general than Lindhard's power law approximation e ) . To simplify notation, we define the total macroscopic cross section (15)
_ ~ n~~~~
(16)
p~~ = n,~~~(~l E~(~~
Further, (17) E E~(~ = Ec(Eo) ~É
(18)
and 9~,(E) = E~(E~,(E) . Then with p~ independent of energy, p = q+~k, K ~. = K/E,(Eo)E~` and To 1 = vo E,(Eo), eq . (12) can be written K
To (É~
= ~(1-k),
~~~ = s~, aÉ {E~SP,(~Î +~ 1~Ja,
~`J~x)
~~~
Cx) P~(9~x))-x~`~~J
+ 8(E - Eo),
(19)
where rc = ~-2m in Lindhard's notation. In arriving at eq. (19) we have made some simple substitutionsto change the limits on the integral and also have setf(x) _ ,R9(x)). It remains to solve the integro-differential equation (19). This we do by defining the Mellin transform, viz.
which after application to eq . (19) reduces it to toEô~,(p-x) _ -iD,(p)~~P)-Z~IP-1)~~P+~ - 1)+Eg-1,
where
(21)
323
DOPPLER-SHIFT ATTENUATION
In terms ofthe Mellin transform, it is readily shown that F(i) =
E~
R
(23)
~~(~ - x)~
Eq. (21) is a difference equation in a form which we have already solved in a previous publication') concerned with the slowing down of fast particles. In that paper two cases were considered, (a) P = 1-x which if y takes the usual value of i leaves k arbitrary, and (b) P = 1 when y = 1- Zk. Taking the solution in case (a) we find that
where
ecl- K)~~=o 1 Eo rßt(P~-v~liu -~ dP ~-~ 2ni L E T o Eo f
G(t)
_i
~,0 x (1
(
t _f 1 _
, Y- °~
e_Kxor~w~
(2~
and ß,(p) is a function defined by In case (b~ we have an expression identical with (26~ except that ~ o = 0, G(t) = t and ~,(p) is defined by H,(p+x~,(p+x) _ ~i(p~
(29)
where H,(p) = Di(p)+~(p-1~ In principle, therefore, subject to a knowledge of ßß(p) we have a complete solution to the problem. 4. Cakolatioo of the fnctMeal Dopier shift Let us now calculate F(r) for a variety ofsituations. We note from eq. (23) that
324
M. M. R. WILLIAMS
which can be written as follows for case (a) F(T)
= T r1(~-x)J 0~dtexp ~-
CT
+ °° ~ 2 0 tJ (~1
(
_G(t) _1
dpY-P
(31)
For case (b), eq. (31) remains valid ifwe set ~o = 0, G(t) = t and use eq. (29) for ~1(p). To proceed further it is necessary to consider separately the cages x ? 0 and x < 0. In.the former case, we may expand the exponential in G(t) in eq. (31), integrate over y and evaluate the conto.r integral over the simple poles. The result is that in case (b) i F(i) = 1+ ~ ~- -~ ~ H1(~ - x+mx~ e=1
TO
~n=1
(32)
where we have used the readily proved relationship _
~11Y)
r1ll~+nx) -
~ H1(P+mx).
iw=1
(33)
For case (a) the problem is more difïicult and we find ~dtexp ~+ °° ~ t G(tr ~ D1(~-x+mx). (34) ~ n!>~0 J 0 1 + .~l0t ~T 0 + n=1 CT 20 J m=1 But if we note that F(s)
we can set y = exp(-x~, o t/TO) and find F(s)
~ ~ (n~ 1+ .1 °1Tn s° +A_1 ~ (~ xr~a1
D1(~-rc+mx)
We defer consideration ofthe case x < 0 until later. Let us now consider some speoific cases which yield closed form expressions for F(i) . Firstly, we note that D1(P) _ ~~ 1~f,(x)Cl-X°- ' {~(A,+1)x~_~,q,_l~C-~} ], ~ 1-a~J ar
(36)
which for power law scattering where
r(x)
_
(2-Sxl-ar
ul -Xr
(3~
DOPPLER-SHIFT ATTENUATION
325
can be evaluated in terms of incomplete beta functions, viz. ~,,
1
D1(P) _ ~ zP,~2 - s~) ~ 1- s1 - ~(A~+
lxl -a~r~ -1B1 _a~(1-s1 ; P+i)+~{A!-1) x( 1-
a~x'_
1B1-ai(1 - s~ ;P -i)t~
Note that s = 1 +m in Lindhard's notation. In the special case of hard-sphere scattering s = 0, and D1(p) becomes A~+1 (1-a~ +~) A~-1 (1-aj-~) D1(P) _ ~P1{1-
1-a~
2p+1
+ 1-a~
Similarly, in the case when A = 1 and we have a single species 2-s ~
2p-1
(38)
(39)
r(2-s)r(p+Z)l
which for isotropic scattering in the c.m. system can be written (s = 0) as
With the appropriate values of s~ and a1 we can now evaluate F(T) by direct use of eqs. (32) and (35). However, to obtain a simple closed form for F(T~ we consider the case given by eq. (41) and set x = ~, i.e. a cross section independent of energy. Then D1 (i-x+mx) = D 1 (1+gym) and we easily show that 6 D1( 1 +zm) = (n+2xn+3)~ ~_1
(42)
Thus we can evaluate F(T) very easily by direct term by term summation. If~u = 0, i.e. no electronic stopping, the series can be written Fs ( ) = 1+6
which sums to
_T ~ s=1 ~ To ~ L~~r
1
(n+2xn+3)'
6 - 6 (1 F(t) _ ~ + +T) ln (1 +T~
(43)
(44)
where 4 A further exact solution can be obtainod when x = 0 which is valid for arbitrary D1(p). We note that setting x = 0 in either (32) or (34) leads to =
T/TO .
(45)
326
M. M. R. WILLIAMS
which can be summed to give
It is interesting to note that this is the same functional form as would be obtained from simple stopping power theory although naturally the coefficient multiplying T differs. The case for x < 0 is rather more difficult to deal with sincethe expansionin powers of y" in eq . (31) is no longer valid. Let us consider case (b) first. Then we may perform the integration over t and obtain 1 ~ K r p F(T) = ßl(i-x) 2ni,~L ~~ 1~)Jo dl+TCOK
Now setting x = - ~, where ~ > 0, we can write(47) as 1 F(T) = ßl(~+~)2ni
dp
1 ~~~+z { -r
fL~Jo
i+ca~
Expanding the denominator of the second integrand as a series in cup/i, integrating over m and inverting the transform, leads to ~ (-i F(T) _ -o ~+1
ßl(~+ ~) 1(~+~+n+ 1~)
(49)
which from the difference equation for ß1(p) allows us to write
A-o
~=o
(50)
This is an asymptotic series in inverse powers of T and is the analytic continuation of eq. (32) . For the special case ~ = 0 it can be summed to give agreement with eq. (46). In addition, if we invoke the hard-sphere assumption and equal masses, it can also be summed to give, for ~ = i and ~o = 0 F(T)=
lOTZ+15T+6 . 6(1 +T)3
(51)
There are some disadvantages in using the general asymptotic series for models where an exact summation cannot be performed because it precludes analytic continuation of the series to small values of T. We therefore propose the following alternative procedure. The integral in (48) is written "1 dwW~ + +zr-v (52) o (1+4)-(1-m~'
327
DOPPLER-SHIFT ATTENUATTON
and expanded in powers of(1-cot)/(1 +T). Then using eq. (34a) weget (53) 1 +~ ~ ( ~ Hl(i+m+1~)-1, v ~,= =o (1 +ip ,,= o -)yCn/ o which converges for all i. Case (a~ as described by eq. (31~ is more difficult and at the present time no way has been found of reducing it to a series in terms of Dl(p). The problem stems from the inability to perform the integral over t or to find a convergent expansion which will enable this to be done . There appears no alternative, therefore, but to find Î' 1 (p) directly and evaluate the integrals term by term. We show in the appendix how to construct the functions ß1(p) from the difference equation. Using ß1Y) [as given by (A.5)] in eq . (31), we may invert the Laplace transform using tables la) to find that F(T) _ ~
F(s) =
dt exp ~- ~~ + ~~ ~ t- G(t)J 1F1
CK, x
;
G(t)I .
0 TJ 0 0 For ~ o = 0, the integral may be carried out explicitly, and we obtain _ TU (1_0 2 4 F(i) T+T ZFl lK' l' K ' T+T 0/ ' 0
(54)
(55)
where the hypergeometric functions are to be interpreted as
1 2 ~ (57) lim zFl -, 1 ; - +e ; x , K CK respectively. For K = i, eq. (55) reduces to eq . (44~ for K = 0, eq . (54) reduces to eq . (46) and for K = -i, eq. (55) reduces to eq. (51). When ~ o ~ 0, we note that for K = -~ 1 2 _G _ 1 _G ~ 1+2~
and for K = -1
_1 GZ 0 +12T2, 0
1F1(1C'1C't 0
1 2 G
Cx,K
1G ;~~=1+2r
(58)
(59)
The integrals for F(T) must then be evaluated numerically. We also note that when K = -1 and ~ o = 0, eq. (55) gives _ 2+3i F(t) 2(1 +=)s
328
M. M. R. WILLIAMS S. Numerical evaluation and discussion of F(s)
For x ? 0, we have obtained exact expressions for F(T) for the generalized inverse power law model. These results are given by eqs. (32) and (35) and for the special case of x = 0 by eq. (46). In addition, we can give a simple closed form expression for F(t) in the case ofhard spheres when the target and recoil particle masses are equal, i.e. eq. (44) . When x < 0, the problem is more difficult and we can only obtain a satisfactory expression for F(i} incase (b), i.e. when theelectronic stopping cross section bears a direct relation to the nuclear stopping. However, a general expression for F(t) can be obtained when the scattering is isotropic in the centre of mass system, i.e. eq. (54). Thus the dependence of F(T) on a wide range of parameters connected with the scattering law can be studied. T,~a~1 Fractional Doppler shift F(r) as a function of the reduced lifetime 4 = s/ro, for isotropic scattering in the c.m . system, equal target and recoil masses and varying suss-sectional energy dependence, K = }-2m K
i 0 0.01 0.05 0.1 0.2 0.5 1 2 5 10 20 50 100
0 1 0.995 0.976 0.953 0.911 0.807 0.682 0.528 0.324 0.202 0.117 0.0528 0.0278
1 0.995 0.976 0.952 0.909 0.800 0.667 0.500 0.286 0.167 0.0909 0.0385 0.01%
-i 1 0.995 0.975 0.952 0.907 0.790 0.646 0.469 0.255 0.145 0.0775 0.0324 0.0164
-1 1 0.995 0.975 0.950 0.903 0.778 0.625 0.444 0.236 0.132 0.0703 0.0292 0.0148
Our first example of the sensitivity of F(T) to the scattering law behaviour is shown in table 1 . Here, we assume isotropic scattering in the c.m . system, i.e. s = 0 in eq . (40), equal masses of target and recoil, and (a) x = ~, m = 0, i.e. constant cross section, (b) K = 0, m = ;, cross section inversely proportional to velocity, (c) rc = -}, m = 4, cross suction inversely proportional to E}. In this case we also neglect electronic stopping . It is clear from the table that F(s) becomes more sensitive to the value of K as 4 increases. For example, at i = 0.1, the dißerences in the values of F are less than 0.3 ~ between rc = ~ and -1. At T = 1, the fractional dißerence increases to 8.3 whilst at i = 10 it is 35 ~. Clearly, therefore, in the experimentally important range of values of 4, an accurate value of K is required . In order to examine the sensitivity of F to the angular distribution of scattering in
DOPPLER-SHIFT' ATTENUATION
329
thec.m. system, we consider eq . (46) with D i (i) given by eq. (38). It is readily shown that Di(i) _ ~ ~~(A~+ 1~
(61)
i.e. independent of s. Thus the fractional shift for x = 0 can be written (62) which depends on s via ~o and T o. Before proceeding with our discussion it is as well to note that T, the reduced lifetime, is scaled in units of To, which is the mean free time between collisions for source energy particles. By definition, i~' = voE,(Eo), where E,(Eo) is the transport cross section. In terms of the Lindhard formalism this is readily shown to be l') nnaz~. E~(Eo) = 1-
Ec
m CEo)
z~
(63)
where n is the number oftarget atoms per unit volume, a the Thomas-Fermi shielding length, EL a characteristic energy and ~i , a parameter of order unity. This formula is valid for 0 5 m ~ 1. On the other hand, Winterbon ~) has soled his lifetime in terms of T m where To -_ ~{1-mxl-aÏ"Tm. We note, therefore, that by using z~ as the scaling parameter, eq. (62) becomes 1)z~- i + .lo(A + 1)z~ i (A + 1 1 = 1+ (65) We feel that so is a more logical choice, although in principle it would be better to scale in terms of
E
Numerically, Ri o varies between zero and about unity. To illustrate the combined effect ofchanges in ~o and mass number A, we show in table 2 values of F(T) for A = 10, 1 and 0.1 and values ~lo = 0 and 0.5. F.q. (62) is employed and the results therefore apply to m = ~. It is clear that the effect of changes in ~o has its greatest influence for small values of A, i.e. when the recoil is massive compared with the target atom. A further measure of the sensitivity of F(T) to system parameters is shown in figs
330
M. M. R. WILLIAMS T~>li.~ 2
The fractional Doppler shift for m = ~ with variable recoil mass and electronic stopping factor ~ u .4
10
0.01 0.05 0.1 0.2 O.S 1 2 5 10 20 50 100
1
0.1
0
0.5
0
0.5
0
0.5
0.993 0.967 0.936 0.879 0.744 0.592 0.420 0.225 0.127 0.0676 0.0282 0.0143
0.999 0.955 0.914 0.842 0.680 0.516 0.347 0.176 0.0962 0.050 0.0208 0.0105
0.995 0.976 0.952 0.909 0.800 0.667 0.500 0.286 0.167 0.0909 0.0285 0.0196
0.993 0.974 0.930 0.870 0.727 0.571 0.400 0.211 0.118 0.0625 0.0260 0.0132
0.999 0.997 0.993 0.986 0.967 0.936 0.879 0.744 O.S92 0.420 0.225 0.127
0.997 0.984 0.969 0.940 0.862 0.758 0.611 0.385 0.239 0.136 0.0590 0.0304
t-0
s ~7 ~6 ~5 .4 3 ß "1 0 10-'
1
10
~E
10 '
Fig. 1 . The influence of electronic stopping on the fractional Doppler shift, F, as a function of normalised nuclear lifetime 4. The model assumes isotropic scattering in the c.m. system with aoss sxtion inversely proportional to the energy (K = -}) . The recoil to host atom mesa ratio is unity. The curves correspond to values of Zo = 0, 0.1, O .S, 1 .0 .
O1 10 '
1
10
;
10
Fig. 2. The fractional Doppler shift as in fig. 1 but with cross sxtion inversely proportional to the cube of velocity (K = -1) .
DOPPLER-SHIFT ATTENUATION
33 1
1 and 2. Here we show how, with isotropic scattering in the c.m. system, F(T) varies with d o for different functional forms of the total cross section. In all cases A = 1. We observe a strong influence as ~o goes from zero to unity, with the effect being greater when the total cross section is a more rapidly varying function of energy, i.e. fig. 2. The general influence of electronic stopping is a reducion in the Doppler shift for a given lifetime . This result is expected physically owing to the shorter slowing down time experienced by the recoil. A summary ofthe various analytical results obtained for F(z) is given in table 3. TABLE
3
The various analytical results obtained for F(s) Equation
Parameters crie (b), K , 0, all Zo, s, a case (a), x ~ 0, all zo , s, a
(32)
(35)
K= },zo=O,s=O,a=0 K = 0, all z o, s, a K < 0, all Zo, s, a K =- },zo=O,s=0,a=0 rc= -l,zo =O,s=O,a=0 crie (a), K < 0, s = 0, a = 0, all Zo K=},0,-}, -l,s=O,a=O,zo=O K=O,s=4,zo=0,0 .5, .l= 10,1,0 .1
(53) (51) table 1 table 2
6. The y-ray energy spectrum We have defined in eq . (3) the energy and direction shift superimposed upon the emitted y-ray as a result of the recoil slowing down . In terms of our present co-ordinate system, this is given by eq . (11) in the following form: GtE, ~IEo, po) _ ~~ ~ ît2l+ 1)P,(~t)Pr~o)~a(EIEo)~ i-o
where the ~P,(E/Eo) are found from eq. (24). Let us first note that eq . (26) can, by the transformation C = (EolE)y, be cast into the form ~`~E't)
_ Ott-K )xoUiu
s 0E0
1 dp~~~_vxdko 2rri,fL (68)
where s(E) = to(Eo/E~` is the collision time for particles with energy E. Now for x ? 0, it is readily shown by writing _ {"`ii _
p~
~
Tt~} -
e
_ ac
~
7C~4
1 - h~p ~ 1 - h~'
332
M . M, R. WILLIAMS
where p = 1-~-`, and using the generating function for Laguerre polynomials, that eq. (68) may be written as
where x = G(t)/z(E) and eu-R~~/to "
V Cn/ TOEO v=0
Thus integrating over t and using eq. (24), we find
x ~ D - (p+mx) =o
0
dx 1-
e- L (x).
0
(71)
This expression is rather tedious to evaluate and so we consider the limit ~.° ~ 0 when the integral over x can be performed and is equal to z_ z(~ "+ i (E) ([+z(E))
If case (b) is acceptable we may still use (72) but replace D, in eq. (71) by H,. In the special case of x = 0, we may write 1
E°1 P
1
(73) JL \ In order to evaluate the inverse Mellin transforms, it is necessary to examine the behaviour ofthe integand as ~p~ -~ oo. Clearly, since H,(oo) -~ oo as ~p~ -" oo, all ofthe integrals in eq. (71) for case (b) will converge. On the other hand, if we neglect eleotropic stopping, ~i,o = 0, and D,(p) -~ D,(oo) as ~p~ -" oo. Thus using the arguments of sect. 5 we must recast the oocpression for ~,(E~E° ) as follows : 0 0
Dd~h+zo ~`(E~E°) + z(~Eo "~ CD~(~)z+z(E))
x
" Y n _ 1 E° P ' 1 LdP ~(-) (v ) 2ni ,1 (E ) ~ ,~ D~P+mx)
1 D~(oo)" },
(74)
DOPPLER-SHIFT ATTENUATION
33 3
where D,(oo)=
(2-si) ~Pi2(1 -si) , oo,
si < 1 si ? 1.
It is readily seen that for s = 0, i.e. zero lifetime ~,(E/E o) = 8(E- Eo)/To and, from eq. (67), that the emergent spectrum is equal to S(E- Eo)S(ü - uo), i.e. unal%cted by the medium. This is a self-evident result but emerges from our analysis in a consistent fashion. When ~o # 0 there is no `uncollided', i.e. S(E-Eon-~o), component unless i = 0 because electronic stopping slows the particle down continuously at the instant it enters the medium . Nuclear stopping on the other hand is a discrete process and in general there are always some particles which have not collided, although as time proceeds, the fraction of these is reduced. However, even for some cases of nuclear stopping, where D~(oo) = oo, there is no uncollided contribution because ofthe long range nature ofthe power law forces. This has the effect ofmaking the total cross section divergent. For most practical cases, therefore, where s ? 1, the uncollided component will be absent and the discrete picture of nuclear scattering events must be abandoned. For the case x = - ~, ~ > 0, we again encounter difficulties in treating eq . (26). However, setting G = t and do = 0, we readily find that
CEo,° ~~ 1 1 ~ _ 1 ~E- Eo) x 1 ~ dP 2ni L E fI ~ D,(p+m~) D;(oo) + To 1+iD,(ao)'
(75)
where in the case (b) formalism, D, is replaced by H, and there is no uncollided contribution to SY,. Eqs . (74) and (77) are particularly well illustrated by the cases of ~ = x = 0, when for ~ o = 0, they both reduce to E 1 1 E ° 1 _ ~ - Eo + ~P EI E o )1 dP i TD~~) ioEo 2ai,fL ~ E ~ ~1 +TD~(p) 1 +iD,(oo)} ~
( 76 )
Further progress in evaluating ~Y,(EIEo) requires knowledge of D,(p). As we see from eq. (22), these functions can be evaluated once the scattering lawf(x) is given . To illustrate the problem we consider eq. (76) for hard-sphere scattering with A = 1, when the D,(p) are particularly easy to use in the inversion integral . They can then be defined as
f0
D~P) = I-2 1dYY2°-1 p~Y~
(77a)
334
M . M . R. WILLIAMS
or more explicitly 1D~P) =
from which Do(P) _ (P - 1)lP,
r(P)r(P+i) r(~+2-Z~r(l'+1+Z~'
(77b)
Di(P) _ (2P- 1)l(2P+ 1),
DZ(P) _ (PZ+~)/PCP+I),
D3(p) _ (4p2+4p+7)/(2p+3x2p+1),
Da(P) _ (P 3 +2p2 +4 - â)/PCP+1xP+2),
etc .
In all cases D,(oo) = 1 . The functions ~P'o and ~ 1 are easily evaluated, the results being (78) cr-1~/zu +r~
ToEo~i(EIEo) _ (1 +i)2 ( Eo)
(79)
From the total number density spectrum ~`o it can be observed that as the lifetime of recoils increases, the differential energy shift of the emitted y-ray moves towards lower energies. This fact is obvious from the physical situation. The current spectrum Y'1 is rather more interesting since it is closer to the measured quantity . This indicates that the detailed nature of the differential energy shift spectrum depends crucially on the value of the lifetime . For example when T < 1, the spectrum is rich in high energy y-rays, whilst for T > 1, it is the low energy part of the spectrum which predominates . The sensitivity of the measured spectrum to the value of z has been noted by experimental workers 12). Naturally, the very simple model considered here would not be expected to give good agreement with experiment, however it does give the general trend. More realistic calculations would require detailed evaluation of eqs. (74) and (75). Finally, we mention that 3Y 2 and 9T S can be evaluated easily, although for certain values of T the poles become complex and give rise to an oscillatory behaviour with (Eo/E). Since 9' 2 and SP3 do not have any special physical significance this behaviour is not unexpected. To compute the complete angle-energy spectrum, G(E, h~Eo, ~o)+ requires a rather large number of the moments ~P',(E~Eo), especially near E = Eo. However if G(E, p~Eo, uo) is partially integrated over p as it might be when accounting for the geometry of a detector, then the number of moments required for good convergence is considerably reduced. However, we are still faced with the basic problem, similar to that met with the backward equation, of summing the angular components . Thus the forward equation offers no advantage in this instance.
DOPPLER-SHIFT ATTENUATION
33 5
7. Summary and general discussion The main purpose ofour work has been to demonstrate the alternative, Boltzmann equation approach to the calculation of the slowing down of particles in matter . It is self-evident from the foregoing work that it is a viable technique with certain advantages over the more familiar backward formulation : not the least of these being the availability ofa large number ofmathematical techniques (and also computer codes) developed by nuclear reactor physicists. These would be particularly useful when more realistic scattering potentials are used . We do not conclude, however, that the forward equation has any over-riding advantage over the backward equation, except for the fact that it has been studied more frequently and its properties, both analytical and numerical, are better understood. Some very convenient closed form expressions have been obtained for the fractional Doppler shift based on simple, but non trivial, models ofthe slowing down mechanism In addition, some expressions which are directly comparable with experiment are obtained which can be evaluated at the expense of computational effort. The angular components of the differential energy shift of y-rays is also obtained and the crucial role played by the nuclear lifetime on the shape ofthis function is illustrated by means of a hard-sphere model of scattering. Further work in this area, should concentrate on direct numerical solutions of eq. (4) using the most sophisticated models of a(E, B) available. Finally, it is recommended that the errors arising from spatial variation in a slab geometry be examined. Appendix SOLUTION OF THE DIFFERENCE EQUATION
It is shown in Levy and Lessman' 3) that the solution of the difference equation g(x)y(x + 1) = f(x)y(x~
(A.1)
where g(x) and J(x) are polynomials of degree r and s, respectively, can be written in the form l'(x)
where
a a a: r _ ~-~ ~ I'(x - a.)l ~ l'(x - l'al+ b ~-i i-i
(A .2)
a
r=i For scattering that is isotropic in the c.m. system, we have shown that Di(p)
336
M. M. R. WILLIAMS
_ (2p-1)/(2p+1). Thus the dü%rence equation (28) can be cast into the above canonical form and the solution is easily shown to be ~t(p) ° l'
p
Cx
+ 1+
1 p 1 ~~ + l 2K)'r 2rc) ~
(A.5)
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
S. Devons et al., Proc. Phys. Soc. A68 (1955) .18 M. M. R. Williams, J. of Phys . A9 (1976) 771 A. E. Blaugrund, Nucl. Phys. 88 (1966) 501 K. B. Winterbon, Nucl . Phys . A246 (1975) 293 S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1 M. M. R. Williams, J. of Phys . D9 (1976) 1279 M . M. R. Williams, Rad. Effects 30 (1976) 47 J. Lindhard et al., Mat. Fys. Medd. Dan. Vid. Selek. 33 (1963) no. 14 I. Waller, Ark. Fys. 37 (1968) 569 M. J. Lighthill, Fourier analysis and generalised fondions (1959), Camb. Univ. Press P. Sigmund, Rev. Rouen. Phys. 17 (1972), 823, 969, 1079 A. E. Litherland et al., Nucl . Phys . 44 (1963) 220 H. Levy and F. Leasman, Finite difference equations (Pitman, London, 1959) G. E. Roberte and H. Kaufuran, Tables of Laplace transforms (W. B. Saunders, Philadelphia and London, 1966)