Appendix 1.2: Electric Quadrupole Moment Ch

Dewtiii^ rtt< urfil- vtt&r, , I’y S ^

1^ - a'I = (/I®· ♦ A'* - a * · * ' ) = /L I - (■ _>1. ) 3 >t{l - (»)}'/x.

I I- -X

$0

r< 1

For comparison with the ag term in 1.35 , we are interested

only in the two terms in l / p

3

.

and when r l i e s along the z-axis

which, by comparison with 1 .2 6 , Ju s t ifie s 1 . 2 1 , p 15.

APPENDIX 1 . 3 : MASS CORRECTION FOR NEUTRON EXCESS, M

3

, p

6

. Equation V I I I . 46, p./^<^, shows that nuclei can be described as cold Feiml-Dlrac gases, with a Perml-energy (for the neutrons) Now, the total enercy of the neutrons,measured from the bottom of

the w ell. Is ^ -

E. · ! » ( « .

So _and

Term p .

6

, corrects e x p lic itly for electrostatic enercy,

so we Ignore It here. Then E]j+

2

;(>“ln) w il l occur when li=Z=^/

2

. The mass correction will be proportional to g.

~ ~ + ? ’ - ^ (

2

) ]

Lel· A 3

rV--

I-

Tayloi^expand the first two binomials as far as terms In

as assumed in M

3

·

2 S' a

“ if. T * r

OQ

The solutions, references, e t c ., are not due to Dr. Fermi. 1. Design a mass spectrocraph to measure the mass difference

between Hydrocen and Deuterium. Measure the separation of

the close lines (H2)+ and D+·

References: Mattaucli and Fluegge, ’’Nuclear Physics Tables” 1942; Harn-well and Livingood, »’ Experimental Atooiic Physics”

1933; M.Cj. chapter on Modern Mass Spectroscopy in

’’A^ivances in Electronics” 1948.

2. Use the semi-empirical mass foimila to calculate the energy of

a-particle emitted from 9211^^5. Compare th is with the obser­

ved value.

Calculate the binding energy of a proton and a neutron in U 2 3 5 .

Answers: a-particle— theoretical, 4 . 1 4 Mev; experimental, 4 .5 2 . BE(N) = 6 . 8 ; BE(P) = 4 .8 5 * See Metropolis, "Table of Atomic

masses^ Oak Ridge.

3 . Desic^n a molecular beam apparatus to determine the atomic magnetic moment of Na in the ground state, "-Sx.

Reference: RG-J Fraser, "Molecular Beams" 1937.

Consider: Temperature of furnace, slit dimensions, magnet

dimensions, pressure, beam separation after splitting, width of beam.

4. Problem on r e lativ ity . A cosmic ray meson, mass = 2l6m,

passes throurili two G-eiger counters, 10 m apart. ¥hat error,

dt, in the time, z^t, between the two pulses, is allowable, i f we wish the uncertainty in energy to be less than 10^?. Consider the cases where the "energy" ( i . e . kinetic energy) of the meson is 5 0 , 100, 1000, Mev.

Solution: Be sure to calculate

^ < 1 0 %

where T is the kinetic , not the total^ energy. For lov7-T par-

tiolos, dE/E has little importance.

Ansv:ers: For T rr 50 Mev, dt = l.::!9 x 1 0 “ ^ sec.

1 0 0 " 0 . 7 1 '*

1000 " 0 .0 2 9 6 ”

5 . Design a 10 Mev Betatron. (Three assif^nments; one v/eek‘ s homev;ork)

Points to c o n s i d e r S h a p e of pole pieces (taper, position of stable o r b i t ), ampere turns required and pov/er supply, fre­ quency·, laniinntion, vacuum, d .c . b ia s , injection, extraction. Referencos: W. Bosley, "Betatrons," a review. Jour. S c i . In s t . ,

2 ^ , 277 (1946) ■ , , X

D.W. Kerst, "2 0 Mev Betatron',' R e v . S c i . I n a t . 13 , 387 (1942) alao Phva. R e v. 6 0 . 47 and

63

(1941)

W .F. Weatendorf. se of d . c . in Induction Accelerators," Jo u r. A n p . Phya. 1 6 . 657 (1 9 4 5 ). See also page 581.

For a general reference see M .S . Livingston, chapter on par­ ticle accelerators in "Advances in Electronics," Academic Press, 1948

6 . Design a 200 Mev synchro-cycloti^Dn ( i .e . fm cyclotron). Also one v;eek's homework.

Points to consider: Dimensions, frequency, frequency of modu­

lation, radial decrease of > phase stability, voltage on

dees, electrostatic focussinc, injection, extraction, vacuum. References: Chapter by Pickavance in "Progress in Nuclear

Physics, 1" by 0. Frisch, 1950. The Berlceley machine is

described by Chew and Moyer, Am» Jour. Phvs. 1 8 . 125 (1 9 5 0 ). The Chicago machiEe in the "170- in. synchro-cyclol?rcn,

Progress Report" Institute for Nuclear Studies, Univ. of Chicago, 1950.

24

P R O B L E M S

Oh. I

7 . Design a one Mev Cockroft-Walton accelerator.

References: Proc. Roy. Se e . L o n d . A136 610 , 619 ('3 2 ) To reduce the expense to l /l O by using radiofrequency, see Rev. S c i . In s t. 20, 216 (1 9 4 9 ).

8 . Describe the precautions and apparatus necessary to carry out simple chemical operations upon a one curie sample.

One should not approach within about ten meters of the un­ shielded sample. Thus about 5"Pb would be a reasonable thickness of shield.

9. The activity of a sample is the total number of processes counted per unit time.

A = Z A i = Z / ? i n i

where A is the decay constant, and nj_ the number, of the 1-th sort of disinteciratins nucleus.

Plot, acainst time, the activity of a two-element radio­ active chain.

Solution. The equations are

n, = ”>

1

, e

--n, -

W ritt hhU in the farm

h - j i d t = t ? [ ^ n ^ -fPcU- K

= e

iPdUr t

QcU· + c]

Ch. I

P R O B L E M S

25

f -Xb ^ ]

Je

dL·

A , 4.

It can be ahovm by straichforward substitution that the curves of parent and dauchter a c t iv it y cross at the exact time that the daufhter ac tiv ity is a maximum. This is il l u s t ­ rated in the curves below.

’ Parertl· AdUv't'Cy, A ,

D a u ^h tir AcJiVily, A ^^

•Z. Fcrsi·^ Pau^htir^ / ^ » /¿> » 1 -z,

o.s

I.S i.s

PARENT and DAUGHTER A C TIVITIES Two statements can be made about the curves above: 1 . The f in a l decay rate depends upon the lonr.er X·, X ^

A

2 . The simultaneous Ap max, and cross-over point occurs at a time rouchly of the order of magnitude of the smaller X.

This is shown as follov^s: I f the cross-over time is

called 0 , then straichtforward alff.ebra gives

e

Since the lo g . is a slowly v a r y in c function, O varies roughly as the loncer "K ; that i s , as the shorter -C .

26

1 0.

P R O B L E M S

t2 3 5

Ch.

I

At time t = 0 , U'·''"' Is stripped of a l l its decay products.

Plot the build^xp and decay of Actlnlvun X .

-7 Solution. The chain, v.'ith decay constants,/^ , In s e c "^ . Is as follovis (/^ = 0 .6 9 3 /T , vfhere T i s the h a lf l i f e ) .

i / £ 3 5

3.l7i^’'' 7.6%I0~^ l.sxio'^ 4.2 X to 7.4% to

Tlie f i r s t daurii-tor, Th, vjlll Into secular equlllbrluni

with the U vrlthln a matter of d ay s . This Is illustrated by the

*‘fast-dau^>liter" (type I ) curve of i:^robleiii 9. A fter a fevr df^.ys we

may consider the Th activity equal to the U activity , and .'^.o on

to consider hov; the Pa crows in .

The Pa viill build up in a period ^ 10^3 sec (105 y e a r s ).

'We may thus neglect the comparatively short Th f,rovth period, and write

4'i^p clt ra

I f vre now restrict ourselves to a time considerable less

than ti/, we may c o n s i d e r a constant, equal to . The

solution to the d ifferential equation is then

^Pa ^Pa

0 -

Pa has a T much longer than any of its dau 'hters. Thus, as

the Pa Grows in G^i3.ually, so do a l l its daur'hters, in secular

e q uilibriu m . This is another v;ay of sayinc that the AcX activity

w ill always equal the Pa ac tiv ity .

We have nov; completed the problem, except for a description of the disappearance of the vrhole chain (v;hich is by then in per­ fect eq uilibriu m ) as the U decays.

During, this time the AcX activity»· is r.iven by

M

J.

O — ^

The coniplete curve Is riven belovi.

Notice that the asyn^totlc lncreo.se of the Pa ac tiv ity after a time on the order of Its h a lf l i f e is important in an anala- ,^ous problem: irra d ia tio n . One accomplishes l i t t l e by irradl- atin,·^ c?. sample for a time lo'icer than, say, twice the h a lf life under c o n sid e ra tio n .

CHAPTER I I ■ INTERACTION OF RADIATION VnCTH MATTER A . ENERGY LOSS BY CHAR&ED PARTICLES

1

. A charged p a r tic le moving through m atter lo ses energy

by electromagnetic in te rac tio n s which r ais e electrons of the matter to excited energy states. I f an e xc ite d le v e l Is I n the continuum of states the electron I s io n ize d ; I f not, the electron is In an excited bound s ta te . I n e it h e r case the Increment of energy Is taken from the k in e tic energy of the in c id e n t p a r t ic l e . I n the following section "i o n i z a t io n " w i l l r e f e r to both degrees o f exc itatio n .

Range = to tal distance traveled by the p a r t ic le u n t i l i t s

kinetic energy is 0 . Before a formula fo r the range o f a par­

tic le can be d e r iv e d , the rate of energy lo ss per u n it path must be calculated. The f ir s t such c a lc u latio n i s due to Bohr**·, and is essentially classical., i . e . , non-quantum m ec h a n ic a l,

2 . Bohr Formula. Consider one electron o f mass m at a

distance b from the path o f an incident p a r t ic le having charge ^e, mass M and ve lo c ity V .

m (e le c tr o n ) *— b= impact parameter mass M charge velocity V F IG . I I

. 1

Assume the electron is free and I n i t i a l l y at r e s t, and moves so sllgjitly d urin g the c o l l is io n that the e le c t r ic f i e l d acting on the electron due to the p a r t ic le can be c alculated at the In it i a l lo c atio n of the electron. The l a s t assumption I s not v a lid fo r an Incident p artic le of v e l o c it y comparable to that acquired by the electron.

We shall calculate fir s t the momentum acquired by an e le c ­ tron during a c o l l i s i o n , and from th is fin d the energy ac q u ire d. As the p article p asse s, the e le ctr o s ta tic force F changes

d irection. By symmetry the Impulse cLt p a r a lle l to the

path Is zero, since for each p o sitio n o f the p a r t ic le to the le ft of A , y ie ld in g a forward contrib ution to the Im pulse, there Is a p osition at equal distance to the rig h t of A g iv in g an equal but opposite c ontrib ution .

The Impulse-L to the path la . We f i r s t estimate

the order o f magnitude of :

L . “ , « (e l e c t r o s t a t i c fo rce)X (tlm e of c o l l i s i o n )* « ·^ :

More exact computation: Consider a c irc u la r' c y lin d e r

centered on the path and passing through the p o s itio n o f the electron. F i g . I I . 2 . Let & be the e le c t r o s ta tic f i e l d i n ­

tensity due to the particle" The e le c t r ic f lu x i s

.,<¿

5

· = 4Tf (independent o f V)

* Bohr. R i l l . M a g . 2 4 . 10 C 1913). 3 0 . 581 (1 9 1 5 )

27

/ ê -

CYU.

28

Bohr Formula Ch. I l by Gauss’

then the flux =s theorem,

to the path, I f Sj. = component of £ -L

J d x z r b e x ^ 4 r T fe . T h e r e f o r e ^

The varlatloiT of £ l with time at the electron is the same

function as would be found by keeping fixed and observing at

a point moving with velocity V along the cylinder surface.

Therefore 2^ e

Yb

reiore Y

= Ÿ J^ejx)cL·

_{I I . 1}

... ^

_■

Ì

A

r

^

I

/

u

FIG. :I I . 2 00 *2,,

The impulse IjL“ -/»

_Y

=

momentxam acquired by the electron, electron is then

J£-

2An, o u Y * b “

where p is-the I I . 2

The energy acquired by one

I I . 3

In document Enrico Fermi - Nuclear Physics - Course Notes (Page 31-37)