Els principis estatutaris sobre finançament

. * _ ¿ » 2 + 1*2 , a»2 it(s*2 + I»2_

^ I 21*

y « o b = = A i i i = - < ·"> n ,

^ I * 21*2 ^

Nov/ v:rlte the denominator 21 (l + l ) ; >Uobs “ ^x. w ill be given vihen mj takes on the value I .

11 max. = J?*2 + 1*2 _ g*2 + k (a *2 + i*2 _ »2 ) 1 .2 5

y ^o b s * ,

2 (1 + 1 ) nuclear magnetons We shall now use 1 .25 to .-.©t the first of the four ex­

pressions on p. 14, 1 .1 7

I = t + 1 / 2 ; 1 = 1 - 1 /2

max = (I- i ) ( l + - i ) + K l + l ) - ^A + k L ^ /^+ I ( I + D - ( I - i ) ( l + i )1

/ o b s 2 ( I + l )

_ I2-¿ + i2+ I _ | + h;(| j2 ^ J _ ι 2 ^ .¿ ) 2 (1 + 1 )

_ i 2 ( 2 ) + I ( l + k ) - 1 + k 2 (1+1 )

= 1 — + 3«2 9 I + 2 .2 g = 1 + 2 .2 9 Nuclear

I + 1 Magneton

The other equations are obtained I n the same manner, remem­

bering, of course, that the fir s t three terms In the numerator of 1 .2 5 . above, are m ultlnlled by zero for the case of the neutron.

APPENDIX 1 . 2 : ELECTRIC QUADRUPOLE MOMENT

Look at P ig . 1 . 8 . The polynomials o f 1 . 2 0 , p. 1 5 , are unity for 0=0, so 1 .2 0 gives, along the z-axls,

^ 1.26

I f we can calculate 0 along the z- a x ls , by any method at a l l , and expand It as a convergent series PIQ·. 1 . 8 : Whole nucleusj In Integral pov/ers of z, we can get

symmetric about I . the c o e ffic ie n ts , a^^, by comparison.

We are Interested only In a2· By Coulomb's law

^(^') dtl

22 Appendix 1.2: Electric Quadrupole Moment Ch. I

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 / p3.

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, M3, 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 M3^·

2 S' a

“ if. T * r

OQ

*Mayer and Mayer ,"Stat 1st leal Mechanics," p 3 7 6.

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.

Ch. I P R O B L E M S 23

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

Q cU· + 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

28 Bohr Formula Ch. I l momentxam acquired by the electron, electron is then

Oh. I I Bohr Formula

^max chosen so that >*7^’

range Integrated. Thus we may set V

b \ [ T ^ V

(cm)

29 i s v a lid over the

I I . 6 where V is an appropriate average frequency fo r electrons in

the absorbing m ateria l.

Problem: Discuss the statement that for \/zf ^ o energy transfer to electrons i s n e g lig ib le (p r in c ip le of adiabatic in v a r ia n c e ).

Consider the component of the motion of the electron J- to the path of the incident p a r t ic l e . Let the coordinate of the electron be y . y = b + d a l m / t . ÿ = 7/ d cos z/ t . b » d .

A (e n e r g y ) = J ( j component of f o r c e )X (v e l o c it y ) dt

Since cos Ô ^ 1 ACenerrjr)

cos yt

■dt =

ÿe®r^®cl r” cos^,· cju

-a.

V=

-ut

_ i r e '

V* ~

_ 7T

h V ^

a s

The limit bminS ( l ) C l a s s ic a l l y , the maxlmam velocity that can be Imparted to the electron (in head-on c o llis io n ) Is less than 2 V .* The energy given cannot exceed ■¿m(2V)2. Therefore b cannot have values that Imply a greater energy transfer per c ollision than 2mV2 As a function of b , the energy transferred per c o llis io n Is 33-2S-i . Values of b smaller than the solution

i S ! s i ,

V-of mvV*b* “ must be excluded In the Integration.

This determines bmin ^3

I I . 7 (2) This c l a s s ic a l treatment Is v a l id only I f the Coulomb fie ld of the In cid ent p artic le v a r ie s n e g lig ib l y over the dimen­

sion ^ of the quantum mechanical wave packet representing the

* This la easiest to see I n the rest system o f the incident pai^

t i d e . Then the e lectron appears to c o l l id e with something like a r ig id w a l l .

30 Bohr Formula Ch. I I electron. A Is approximately the de Broglie wavelength of the electron as seen from the incident p article. In a coordinate system in which the incident particle is at rest (this nearly coincides with the center of mass system, for a heavy incident p artic le ), the electron has velocity of about V , assiiming its orbital velocity is much less than V . Th^ momentum of the electron in this coordinate system is and therefor©

> _ , J.

~/m Y · Only values of have meaning, and therefore another criterion for is

I I . 8 The larger of(bjjjin)c< (binin)oM 't'® used in the intecration*. For values of V where bm ax> l^mln. (^min) > (^m in L and therefore I I . 8 whould be used. UslTng I I . o in I I .B s * ”

d £ 4-ir3-^e^^ a

"oOx 'wiV*

Lng

(erg cm“ ^ ) I I . 9 where is a suitable average of the oscillation frequencies of the electrons.

More precise calculation** leads to the following formula for heavy particles, i . e . , not electrons:

11.10

where I Is the average ionization potential of the electrons of t h ^ aTj^sorber. in ergs. The In term as 9 for 1 mev protons in

3 . Electrons. There are two main reasons why 1 1 .1 0 cannot apply to electrons, (l) The derivation assumes that the incident particle Is practically undeflected. But the Incident particle acquires a transverse component of momentum per collision approx­

imately equal to that given to an electron In the absorber, and i f the incident particle is an electron, the transverse velocity corresponding to this momentum w il l not be ne g ligib le . (2) For collisions between identical particles exchange phenomena must be taken Into account**·*. Bethe * * gives the following formula for energy loss by electrons:

'oL-X 'tn.

2Tre

■ V

^ -1 +173 “ J

_________________________________________ (electrons)

11.11

where T la the average ionization potential of the atoms of the absorber and ~f~ = relatlvistlc kinetic energy of the electron.

* I n cutting the Integral of I I . 4 o f f at "bjjjjLn'^O, we have neglected a term , This is Ju stifie d in "Lecture Series in Nuclear P h y s ic s ", LA 2 4 , Lecture X I , printed edition p. 2 7 .

* * Bethe, Handbuch der Ph v sik. p . 519

* **M o t t , Proc. Rov. See. 1 2 5 . 2 2 2 , 1 2 6 , 259 (1929)

Enercy, ln

incident p articles of identical charge moving in l ik e absorbers

32 Bohr Fonnula, Range Ch. I I

Em p irical range-energy formulas: A rough formula giving the range of alpha p articles I n a i r at 15°C . and atmospheric (for Al, but ie close for all other substajices

7. Polg.rlzation e ffe c t s . I n the

Ch. II Ionization of a G-as 35

Me* 10 Me»

FIG. I I . 4 \OOMc* E N E R G Y The curve BCD ßives the 1/V dependence. At relativistic

energies V chanj^es l it t l e and CD is asymptotic to V = c. At relativistic energies, the log term in (V v l- ß ^ ) changes, and

increases as V — c , giving the rise in the curve from C to E.

At very lov7 energies (region AB) equation 1 1 .1 0 breaks down because the particle has velocity comparable to that of the orbital electrons in the absorber, and the efficiency of energy exchange is much lower. The particle i t s e l f captures electrons and spends part of its time with reduced charge.

9. Ionization of a p:as. I f ionizatio n is produced in a gas the ions may be collected by charged electrodes, and the amount

of charge collected w ill be proportional to the number of ions produced. The change of potential of one of the electrodes will depend on the charge collected (and the external circuit) and therefore on the number of ions produced. This voltage pulse may be amplified linearly and measured q u antitatively , as with an oscillograph. A gas chamber for this purpose is called an ion­

ization chamber^*·.

FIG . I I . 5

In the arranr.ement in FiG- I I . 5 , electrons are collected at the top plate. A negative pulse, of duration determined by R and the capacity of the ionization chamber and associated circuit, is produced at the ^rid of the linear a m p lifie r.

It turns out that there is a close proportionality between number of ions produced and total energy lost by the incident particle. For most ^ases one ion pair (electron plus ionized atom) is produced for each 32-34 e .v . lost by the particle, (see table on following p ag e ). Although empirically the result is a

References on ionizatio n chambers are: K o r f f , Elec'tron and Nuclear Counters (Yan Nostrejid), Rossi and Staub, Ionization

chambers and Counters (McGraw-Hill).

34 Scattering Ch. I I simple proportionality betv/een Enerp;;^;^ for one ion p a ir ^ number of ions and energy spent, G-as Energy spent for the explanation is very compli- ____________ one ion pair>e_.v.

cated. Theoretical prediction

culating what fraction of energy

carried away by primary ionized electrons i s used in producing angle scattering event. Electrons are acattered much more fre­

quently, and th e ir tracks are as ahown:

Ch. II Scattering 35

n u c l e u s o f

1 1 .1 7 This formula ia v a lid at non- relativistic v e l o c i t i e s , V · ^ · ^ c.

A relativistiosLlly correct version of 1 1 .1 9 for small angles

Q

is given in the paragraph containing 1 1 . 2 0 .

Exact quantum mechanical c alculatio n g iv es the same formula provided the nuclear fie ld is exactly a Coulomb f i e l d . Both classically and quantum m echanically the formula is v a l i d only i f the distance of nearest approach of the p artic le to the nucleus is larger than the nuclear radius.

The cross-section for scattering of the incident p a r t ic le at an angle 0 in the range d© is defined to be the total area _ L to the i n it ia l path o f the p a rtic le such that i f the p a r t ic le

passes through this area i t is deflected by anaangle 0 in d0 . Since for given particle and nucleus, b is a function of 0 o n ly , the area corresponding to a given 0 l i e s at a certain radius b ( 0 ) , and has magnitude dC^« 2.Tr

bc®>

^db. S u bstitu tin g for b i n terms of the corresponding angle 0 : dC^ = 27fb(0)b' {Q)dQ. D iv id e by the element of solid angle 2Trsin0d0 to find the. cross section per unit steradian, and substitute for b ( 0 ) it s value from solving

1 1 .1 7 for b . Then the cross-section per u n it solid angle at 0 is

cLoj 4-\M V® / 1 1 .1 8

Note that most p articles are scattered at small eCngles.

A r e la t iv is t ic a lly correct equation for 0 as a function of b when 0 is small can be derived e a s i l y , u s in g the same argu­

ments used to derive I I . 3 . Since now we deal with nu clear charge Z a n d incident particle of charge we must multipy I I . 2 by the nuclear charge in order to get the transverse Impulse impart­

ed to the incident particle in the c o l l i s io n . T h is gives

, =

2 2 i e "

V to

If y p is the r e la tiv is tic momentum o f the inc id ent p a r t ic l e , the angle of d eflection is very nearly , i f

A / j p = 1 1 .1 9

11.20

I f we put ^ = M V (n o n - r e la tiv is tic ) t h is becomes id e n t ic a l to 1 1 .1 7 when 0 is small and tan 0 / 2 =«? 0 / 2 . In these formulae

»See, for example. Lindsay. Physical M e c h an ic s, p . 7 6 .

3 6 Multiple Scattering Gh. I I b la limited to dlatancea from the nucleus within which the nu­

clear charre can be felt, I . e . , has not been screened by nearby electrons.

2 . Multiple Scatterinc· Particles, particularly electrons, are deflected many times in passing through a foil of metal.

Since st&tistlcally the individual events do not differ, 11.23 ornasos gradually with distance, but fortunately the log term

11.26

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