Notes on Doppler-shift lifetime measurements

Nuclear Physics 88 (1966) 501--512; (~) North.Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

NOTES ON DOPPLER-SHIFT LIFETIME MEASUREMENTS A. E. B L A U G R U N D t

Argonne National Laboratory, Argonne, Illinois tt Received 26 May 1966 Abstract: By use of the universal scattering cross section derived by Lindhard et al., the velocity

of an ion in a scattering medium is calculated as a function of time. The effect of both electronic and atomic collisions is included. In addition, the average scattering angle (in the lab system) resulting from multiple atomic collisions is derived. It is shown that in most cases the attenuation of the Doppler shift as a result of the changed direction of the scattered nucleus is more important than the effect of slowing down due to atomic collisions. In fact, at low velocities and for heavy scatterers the change in the direction of motion is the dominating factor. The results are given in a form amenable to numerical analysis. In the limiting cases o f (i) very short lifetimes and (ii) predominantly electronic scattering, the average Doppler shifts are given in an analytical form.

The Doppler-shift technique has been used by many investigators 1- 3) for determining nuclear level lifetimes in the region 10-14-10-Us. In the most commonly used version of this technique, a level is populated in a nuclear reaction and decays by emission of gamma radiation. This gamma ray is emitted from a nucleus recoiling in matter, and its energy is shifted by an amount depending on the instantaneous velocity of the nucleus at the time of emission. The nucleus is slowed down by the matter through which is moves, and under suitable circumstances the average Doppler shift will depend on the lifetime of the level and on the slowing-down time of the nucleus in the stopping material. Thus, the lifetime of the level can be calculated from a measurement of the attenuated Doppler shift if the slowing-down time is known. With the poor energy resolution of scintillation counters, reasonably accurate measurements of Doppler shifts required a high initial recoil velocity (about 1% or more of the velocity of light). At these high velocities the moving ion loses most of its energy in collisions with electrons of the stopping material and collisions with nuclei can usually be. neglected 4). With the advent of the lithium-drifted germanium counters, gamma-ray detectors with much better energy resolution have become available. Doppler shifts associated with much lower initial recoil velocities have thus become measurable. In this low-velocity region, the effect of atomic collisions can no longer be neglected. Atomic collisions affect the Doppler shift in two ways; (i) the moving ion loses part of the kinetic energy in the collision and (ii) the direction of motion of the moving t On leave of absence from the Weizmann Institute of Science, Rehovoth, Israel. tt Work performed under the auspices of the U.S. Atomic Energy Commission. 501

502

A. E. BLAUGRUND

ion is changed through a scattering angle ~b and consequently the observed Doppler shift is changed by a factor cos q~. In the past, attempts had been made to include the effect of energy loss by atomic collisions but the effect of scattering angle had been neglected 2). The present paper is an attempt to show how electronic stopping and these two effects of atomic collisions influence the observed Doppler shift. It will be shown that in most cases, the scattering angle affects the Doppler shift more than does the energy loss due to atomic collisions. This derivation is based on the theory of atomic collisions developed by Lindhard, Scharff and Schiott s). These authors introduce dimensionless variables e for the energy and p for the path length traversed by the moving ion. These variables are defined as =

aM2 E, Z, Z2 e2(M1 + M2)

P = 41ra2N

MI M2

x,

(1)

(M1 + M2) 2

where the subscripts 1 and 2 refer to the moving atom and the atoms of the stopping medium, respectively. In these expressions a = 0.8853 x (h2/rneZ)(Z ~ + Z~2)-*, m and e are the mass and charge of the electron, Z the atomic number, M the atomic mass, N the number of scattering atoms per unit volume, E the kinetic energy of the moving atom and x the distance traveled along its path. Using a screened potential, Lindhard et al. arrive at a universal differential atomic-scattering cross section da(e) =

2 dt

ira ~f~f(t ),

(2)

where t ½ = e sin ½0, and O is the deflection angle in the centre-of-mass system. The functionf(t *) is given in fig. 1 of ref. s). From (2), it follows that the specific energy loss due to atomic collisions is

= -eff(x)dx.

(3)

1 ~ (where c is the speed of light), the For recoil velocities v < (e2/h)Z~1 = i 3~cZ1 specific energy loss due to collisions with electrons is given by (de/dp) e = ke½,

(4)

where, as shown in ref. 5),

k = Z~ O'0793Z~Z~2(A' +A2)~ (Z1 +Z2) AIA2 According to Northcliff 6), experimental values of the constant k are approximately 20 % higher than predicted by (4). Also, eq. (4) seems to be valid only for v < 1-~7c. It seems advisable to use experimentally obtained values of k where such values are available.

LIFETIME MEASUREMENTS

503

The specific energy loss due to atomic collisions (de/dp), is shown in fig. 1 as a function of ~. The same figure also shows the electronic part of the specific energy loss (de/dp), for several values of the parameter k. At this point it is convenient to introduce two additional dimensionless quantities. Let ~ ---- (hc/e2)(v/c) be the dimensionless variable corresponding to the velocity v ,

f o , o ~ - ~ ~

~

:

I ~

o.,~~,.~ ~

o.oli

........

1

o.I

~.

....

_~L~

l.o

........

to.o

I loo.o

£ Fig. I. The universal specific energy loss (de/dp)n from atomic collisions as a function o f energy e. In the energy range e > I this function is closely fitted by (de/dp)n = ( ~ ) - ~ { 0 . 3 + I n [(0.6+e~)/e]}. In the interval 1.2 < e < 20, the expression (de/dp)n = 0.4e-~r is a reasonably good approximation. In addition, the specific energy loss (de/dp) = kei due to electronic collisions is shown for four values of k.

of the moving ion, and let the parameter ~¢ be defined by the equation s From the above definitions is follows that

=

½~ttm3v2.

1.63 x 103A1 A 2

~¢t'=

Z1 Z2(Z~ + Z~)½(A, + A2) " In addition, a dimensionless time variable 0 is introduced in such a way that dp/dO = ~ . It can easily be shown that 0 = t/T, where t is the time and h

T-

(A1 + A 2 ) 2

e2 4xa2NA1 A 2

A useful relation for calculating values of s is =

10.2-

dr' At

E mc 2"

Figs. 2 and 3 show the quantities k, ~t' and T as a function of Z 1 and Z2. Here, the

504

A. E. BLAUGRUND

t °l !

i

'

I

'

1

'

I

60~

I

40

2O

0 0

20

40 Z,

60

80

Fig. 2. The parameters k and ~,' on a plot o f Z1 versus Z2. The approximate relation A = 2 Z - k ~ o Z 2 has been used in calculating these parameters.

'

I

TD 0nd

'

!

'

I

'

I

aD are in s.g/cm 3 %

,,

.

0/ 0

i

I

20

I

1

=tO Z=

i

I

60

I

I

80

Fig. 3. The products TD and =D on a plot of Z1 versus Z=, where D is the density of the stopping materia| in g/cm a and = is the electronic slowing-down time in sec. For definition of = and T, see the text.

LIr~TIME MEASUREMENTS

505

approximate expression A = 2Z+T~-aZ 2 for stable nuclei has been used. These numbers should be used as a guideline only. The energy and the velocity of an ion passing through an absorber can be calculated as a function of time from the expression

fl°e½(de/dp),

0 =

(5)

where de/dp = (de/dp)n+(de/dp), and e o is the energy of the ion at 0 = 0. Using the values of (de/dp) shown in fig. 1 and numerically performing the integration in expression (5), one gets e and therefore ~ as a function of 0. In order to account for the attenuation of the Doppler shift by the scattering through angle q5 in the lab system, cos 4, has to be calculated as a function of time. The derivation presented here is based on the multiple-scattering theory developed by Goudsmit and Saunderson 7) and extended by Lewis a). Both Goudsmit and Saunderson and Lewis assume the recoil to be negligible and make no distinction between centre-of-mass and lab angles. For the application considered here, this assumption is not valid and the theory has to be extended to t h e general case in which the scattered ion and the scatterer have comparable masses. According to eq. (7) in Lewis's publication, the angular distribution function F(~, x) for an originally parallel beam of ions after traversing a distance x in a scattering medium is given by F(~b, x) = ~1 l=o(2l + 1)PI(cos q~) exp

(--f oxl,) dx

,

(6)

where f, xt¢, = N J [1 -P,(cos (k)]da(~b), the Pt(cos ~) are the unnormalized Legendre polynomials and a(~b) the differential cross section for scattering given as a function of the lab angle q~[da(~b) - a(~b)df2#]. It follows immediately from eq. (6) that cos~b = exp

(-f oxl÷dx )

,

(7)

and in the general case P,(cos q ~ ) = e x p ( - f ~ x , ¢ , d x ) .

(7a)

Eqs. (7) and (7a) are also given in ref. s). The relation between the lab and centre-of-mass angles q~ and O is tg 4, -

sin O r + cos O

(8)

506

A. E. BLAUGRUND

where r = A x / A 2. In principle, this relation makes it possible to calculate da(¢) from da(O) given in eq. (2). However, in order to calculate the quantity ~:1¢ = N f [ l - c o s

(¢)]da(¢),

(9)

d

it is more convenient to express the integrand of (9) as a function of O. By making use of eq. (8), cos ~b can be expanded in a series of Legendre polynomials P,(cos O). The result is n+l

cos ¢ = P,(cos O)+Z, (-1)"n= ~

[Pn+l(COSO)-Pn-l(cos O)]rn'

for

r < 1,

(lOa) co_ cos¢ = 1+~(-1) ,=1

"+1

n -

-

2n+l

[en+l(COS O ) - P . _ , ( c o s O)]

[1~ "+1

~,r]

for

'

r>l. (10b)

Eqs. (10) give cos ~b as a function of the centre-of-mass angle O. By inserting these expressions into (9) and using dcr(O) instead of do(C), ~l~ and therefore cos ¢ can be calculated to any desired order in r (or 1/r). This can most conveniently be done by introducing the quantities Pt(cos O) defined by Pl(cos O) -- exp where

(fo) -

xlodx

(11)

,

tote = N f, ) [1-Pt(cos O)]do'(O). It should be pointed out that P~(cos O) becomes identical with P~(cos if) when the recoil can be neglected, i.e., when A2 >> A1 or r << 1. In particular Pt(cos O) = cos O = cos ¢ in this case. Through eqs. (7), (9) and (10), cos ~b can be expressed in terms of the quantities Pt(cos O) in a very simple manner / . . . .

oo

~

F" ~

z

~x

-](--1)n[(n+l)/(2n+l)]

/ r . + ~ t c o s -~) /

[rltc°s "~)II / ~ ~ / / "=' L P._,(cos O) ] COS (/) =

~|

oo

!.°1

i" ~ / /

- ~-~ " l ( - 1 ) n + * [ n / ( 2 n + l ) ] ( 1 / r )

! I-[ / r . + 1(cos t~) !

rn

for

r < 1

for

r>

"+1

1.

(12)

It should be noted that eqs. (12) are exact. For the scattering cross section given by eq. (2), the values of P~(cos O) can be

LIFETIMEMEASUREMENTS

507

calculated as a function of e from the relation (13) where

I = ~o (de/dp). de, e(de/dp) and from

P,(cos 6)) ,~ (cos 6))'.

(14)

Eq. (14) is only approximately valid. The approximation involves integrals of the form S~ x"f(x)dx in which f ( x ) has been assumed constant and equal to an average value (l/a) ~

f(x)dx = (de/dp),. However, for r << 1 the quantities Pl> t(cos O)

enter in the form of small correction factors only. For r = 1, ~b = ½0 and cos q~ can be calculated without the above approximations. For r >> 1, the scattering angles th are very small and their effect on the Doppler shift can be neglected. For the particular case of the scattering law given by eq. (2), it follows from (12) and (14) that

f cos O n~__l-[I1cos 6)] (-t,"[(2n+2)/(2n+l)]r", cos q~ =

r < 1

=

®_ _ ]--I [cos 6)](-1)"+'t2./(2,,+l)Xt/.)-+~,

(12a) r > 1,

",n=l and through eq. (13)

cos ~p = exp [ - 1 A 2- - G(r)II 2At where

I = (to (de/dp). de e(de/dp) '

(15)

l + ~ } r _ 1_~_r2+8 ~,

(-r)"

. =a (2n + 1)(2n- 1)(2n c(r)

, -

r < 1

3)

=

8 1

+i3--8 r

¢; (-1/0"-' ~3 , = (2n + 1)(2n- 1)(2n- 3)

r>l.

The sums in the expression for G(r) can usually be neglected even for r ,,~ 1. Fig. 4 shows G(r) as a function of r = A 1/A2. The only approximations involved in eq. (15) are those stemming from eq. (14). If this approximation is not used, G(r) becomes also a function of e. However, the dependence of G(r) on e is very weak for 8 > 0.1. This can be seen in fig. 4 which shows G(r) calculated numerically for several values o f 8.

508

A.E.

BLAUGRUND

For calculating the average Doppler shift of a gamma ray, the knowledge of ~ and of cos ~b as a function of time is necessary. The dependence of ~ on time follows from eq. (5). Eq. (15) can now be used to obtain cos ~b as a function ofe and thus, through eq. (5), also as a function of time. Very often the recoiling nucleus is slowed down in a medium containing several types of atoms. Let ~¢, Z 2, A2, etc. refer to the heaviest of these atoms and let ~ , Z2~, A21, etc., refer to the ith type of lighter atoms. In such a case, eq. (5) is replaced 1.4 1,2

i

I

I

~

I i~l]

I

I

i

~

I I =

/

1.0

.

.E=IO

0.8-O.6--

i

0.4

I

I

l

I

I I II]

0.I

l

I

I

I

I I l

1.0 r = A;/A

I0,0

z

Fig. 4. The approximate function G(r). The curve obtained from the series expression in eq. (15} is represented by the solid line. Numerically calculated values of G(r) for e = 0. I-1.0 do not deviate significantly from this curve. Values of G(r) for e = 2 and for e = 10 are given by the dashed curves. The very slight dependence o f G(r) on e is evident.

by the modified expression 0 =

~[(d~/dp)+ E C,(de/dp),]'

(5a}

where 0 and e refer to the heaviest scattering atom C~-

Ni Z2i ai A I + A 2

N Z 2 a AI+A2i

and (de/dp)i is the total specific energy loss calculated at the energy (dt'i/~()e. In a similar way cos q~ can be calculated from (15) by using I = f f ° (de/dp). + E C,(A2i/Az)(Gi/G)(de/dP).i de. ,~i-(de/dp)+ X C,(de/dp),-I

(15a)

It can be seen from fig. 1 that in the region 1.2 < e < 20 (de/dp)~ is fairly well approximated by (de/dp). ,,~ 0.4e -~, 1.2 < e < 20. (16) Using (de/dp) = 0 . 4 e - i + k e i, one can readily perform the integrations in (5) and (15) and obtain e l+0.4/ke _ exp ( - x / 2 ~ k O ) ,

eo l +0.4/keo

(17)

LII~T1ME MEASURE~NTS

509

[_1 + 0.41(k~) ] -6/2, cos ~b = LI +0.4/(k%)1

(18)

I f several types of atoms are present in the stopping material, these equations take the form

8 1 +dJ(d,e) = exp ( - ~ / 2 ~ d , O ) , ~o l +dJ(deao) cos q~ = [ 1 + d,/(d,s)] -(~/2,)(a'n/a.o

(17a) (18a)

Li-+do/(doso-----))_]

where

d= = k+ ~ C,k,(c/g,/...CL)½, dn = 0.4['14 ~ Ci(.~l..l[,)~], d'n = 0.411 + ~ C,(A2,1A2)(G,/G)(.X[/..¢[,)½]. Expressions (17) and (18) lend themselves easily tO numerical calculations. Once ,~ and cos tk are known as a function of the time 0, the calculation of the Doppler shift of a gamma ray emitted by a nucleus moving through an absorber is straightforward. Consider a point counter detecting gamma rays emitted along the initial direction of motion of the excited nucleus. Let E~ denote the unshifted energy of the gamma radiation and AEro the full Doppler shift observed for gamma rays emitted by nuclei moving with the initial velocity ~ o in the direction of the counter. The mean energy E-~ measured by the counter is t

-E r =

Ev [e1+ -hc- fo

° exp

-

~ co~d0

1,

(19)

where z is the mean life of the nuclear level, and where ~ and the mean cosine of the scattering angle are functions of 0. The quantities most commonly measured are in the case of short lifetimes

AE'°-AE' - Tfo exp (AE~o

(1-

~o

dO,

(19a)

and in the case of long lifetimes ~-~o

= ~

exp

- ~

~oo

cos~bd0,

(19b)

where AEr = E~-E~ is the attenuated Doppler shift. In the case of very short lifetimes [z << ~t = (2Jt')*T/d,] and in the case in which the electronic stopping dominates [(de/dp)= >> (de/dp)n for 0 < z/T], analytical expressions can be obtained for the Doppler shift given by eqs. (19a) and (19b). Let us first consider the case of predominantly electronic collisions for which dn/(doso) < d~,/(d,s) << 1. If it can be assumed that (A2/A1)[d,J(d,%)]exp(2"r/~) << 1, t In this expression the quantity ~cos~ should be used rather than rt~cos~. However, the difference between these two quantities is small. For Aa = At and (de/dp)n >> (de[dp)e, the value of I should be reduced by 5-10% in order to take account of the correlation between ~ and ~b. In other cases this correction is even smaller.

510

A.E.

BLAUGRUND

it follows from (17a) and (18a) that --cos~

~exp

--

"2~o

0 ---

~

1+ - - G

de eo

A1

sinh

0 ,

dnl

and from (19b) that

AEro

l +'r/~z

de% \

A1

dn] 1-('r/~z) 2'

(20)

where ~ = (2..¢/)½T/de is a measure of the electronic slowing-down time. Expression (20) is valid only for

"r/~x< ¼1n (A~ de~°l ~ 0.3-0.5. dn ! Fig. 3 shows ~ as a function of Z 1 and Z 2. The two terms in eq. (20) represent the contributions of the electronic and the nuclear stopping. The nuclear part contained in the second term consists itself of two parts. One of them is due to energy loss in atomic collisions and the other to atomic scattering. Since, usually, G(d~,/dn) g 1, it is clear that the contribution from scattering cannot be neglected. On the contrary, for A2/AI >> 1 the scattering angle contributes more to attenuating the Doppler shift than does the process of slowing down by nuclear collisions. Let us now consider the other approximation in which "r/~ << 1. The quantity dn/(de%) need not necessarily be small. In this approximation one can expand ~" and cos ~b in a Taylor series as a function of 0 and retain the linear terms only. One then gets --

=

1-

~'Y~o

- -

o,

"-~o

(20

o

c°sq~=l-A-AzG(d~)Ax nO(~--~o) 0,

(22)

where (de/dp)o and (de/dp)no are the total specific energy loss and its nuclear part taken at the time 0 = 0. Eqs. (17a) and (18a) now become

__ ~o~ = 1 - (1 + d n ]

de 0,

(21a)

cos ~b = 1 - Az- G d'. 1 0. As So x/2~¢/

(22a)

Then by combining eq. (19b) with (21a) and (22a), one gets

AE~o

o:

dee o

AI

dn]

I

I00'01

I

I

I

I

I

8=0

I

I

8=6.0(~/2) =/z

I0.0

I

0.1 ........ I 0 2

1

I

4

6

I

I

8 I0 (2/~)1/2

I

I

I

12

i4

16

18

G

Fig. 5. The energy e as a function of time 0 with e0 = 40 for three values of the parameter k. The procedure for finding 0 for other values of eo is illustrated in the figure. F o r e0 = 14.8 and k = 0.15, the value of 0 for e = 4.4 is 0(e = 4.4) = 6.0(½~t¢)$. t

I

r,~t~ill

I

1.0

0.1

I

I IIIII

I

4'..,

!

i

i

i i ill

--

-

.---%,.,0.01

0.001 0.1

hO

I0.0

I00.0

E Fig. 6. Plot o f I as a function of e for different values o f k with e0 = 40. The procedure for finding I for other values o f e0 is similar to that outlined in fig. 5. The value o f c ~ can be obtained t h r o u g h eq. (15).

512

A. E. BLAUGRUND

AEy°-AEr ~1 + dn A2 d'n ~ "¢ AE~o - \ dee0 + --A, G deeo/~-"

(23a)

Since d'/(d, eo) need not be small in this approximation, it is clear that for A2 >> A1 the effect of scattering angle may be the most important term in attenuating the Doppler shift. Eqs. (20) and (23) apply only in the limiting cases of predominantly electronic collisions or very short lifetimes. In the general case, Doppler shifts should be calculated by starting from eqs. (17a) and (18a) if 1.2 < e < e o < 20 or from eqs. (5) and (15) if the energy is not confined to these limits. Figs. 5 and 6 show e as a function of 0 and I as a function of e for three values of the parameter k. The accuracy of the Doppler shifts calculated with the aid of the above expressions depends on the accuracy of the expression for the specific energy loss given by Lindhard et aL 5). These authors estimate this expression to be accurate within 20 ~o for e > 0.1. The author is indebted to Drs A. E. Litherland and S. S. Hanna for helpful comments. References 1) 2) 3) 4) 5)

S. Devon, G. Manning and D. St. P. Banbury, Proc. Phys. Soc. A68 (1955) 18 E. K. Warburton, D. E. Alburger and D. H. Wilkinson, Phys. Rev. 129 (1963) 2180 A. E. Litherland, M, J. L. Yates, B. M. Hinds and D. Eccleshall, Nuclear Physics 44 (1963) 220 N. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 18, No. 8 (1948) J. Lindhard, M. Scharff and H. E. Schiott, Mat. Fys. Medd. Dan. Vid. Selsk. 33, No. 14 (1963); J. Lindhard, Nucl. Science Series, Report no. 39, Publication 1133, p. 1 (1964) 6) L. C. Northcliff, Ann. Rev. Nucl. Sci. 13 (1963) 67 7) S. Goudsmit and J. L. Saunderson, Phys. Rev. 57 (1940) 24, 58 (1940) 36 8) H. W. Lewis, Phys. Rev. 78 (1950) 526