XIII. Two Notes Relating to Stopping Power

XIII. TWO NOTES RELATING TO STOPPING POWER UNPUBLISHED MANUSCRIPTS FROM FOLDER P E N E T R A T I O N

October 1940 DANISH TEXT AND TRANSLATION

See Introduction to Part 11, sect. 6.

P A R T 11: G E N E R A L T H E O R Y O F P E N E T R A T I O N

The material filed under the title “Penetration” consists of a vast amount of notes, outlines, figures and drafts for Bohr’s paper of 1948 on penetration (document XXII). There are altogether approximately 1500 pages. There are typewritten manuscripts, carbon copies, graph pages and handwritten pages, mainly in the handwriting of Bohr, Aage Bohr and Rozental. The languages are English and Danish. The material has been divided into 11 folders as follows (our numbering): Folder 1 “Ahandlinger om Stopping Power of Fission Fragments” (Papers on Stopping Power of Fission Fragments). Manuscripts and proofs of papers from the Institute relating to stopping power of fission fragments from 1940 and 1941. Folder 2 “Optegnelser ang. Stopping Power. 2” (Notes Concerning Stopping Power 2). 20 short notes from September and October 1940 relating to stopping phenomena. Folder 3 “Mskr. ti1 den store Ahandling. Kladder” (Manuscripts for the Large Paper. Drafts). Mainly drafts from November 1940 and February and March 1941 for the introductory chapters. Folder 4 “Optegnelser 3” (Notes 3). 16 short notes from November 1940, February and March 1941, and May-June 1942, mainly relating to chapters 4 and 5. Folder 5 “Penetration. Section 4. Capture and Loss”. Material from May to August 1942 and March and August-September 1943, relating to chapter 4. Folder 6 “Spild ti1 V” (Waste from V). Material from April to May 1943, relating to chapter 5. Folder 7 “Rozental”. Almost complete manuscript as of September 1943. Folder 8 “Penetration. Ny Udgave. Introduction” (Penetration. New Version. Introduction). Material relating to the Introduction, mainly from January and March 1946. Folder 9 “Nyt Spild ti1 2” (New Waste from 2). Material relating to chapter 2, mainly from the spring and summer of 1943 and from March-April 1946. Almost 300 pages. Folder 10 “Nyeste Spild ti1 Chapter-3. Indviet 3/9-43” (Latest Waste from Chapter 3. Inaugurated 3/9-43). Material from the years 1943,1946 and 1947, relating to chapters 3 and 4. More than 500 pages.

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Folder 11 Envelope containing the 3rd proofs of the paper, dated November 1948, and handwritten pages with the formulae. For additional material from this file, see documents XIV, XV, XVI, XVIII, XIX and XX. Cf. also the material described in the editorial note on p. [336]. This material has not yet been microfilmed.

The notes reproduced here are contained in folder 2. We have reproduced the table of contents of this folder as well as the notes numbered 2.11 and 2.15. The former consists of 2 pages dated 21 October 1940 and the latter of 3 pages dated 23 October 1940. The material reproduced is in ink in Rozental’s handwriting. Except for a few lines in English in the table of contents the language is Danish. The symbols used are essentially the same as in Bohr’s paper of 1915 (Part I, document VIII).

P A R T 11: G E N E R A L T H E O R Y O F P E X E T R A T I O N

[Table of Contents]

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.16" 2.17 2.18 2.19 C2.20

Simplified representation of stopping formula Stopning af a-Partikler af tunge Atomer Contribution of electronic interaction to stopping power Contribution of nuclear collisions to stopping power 1 Tathedsfordelingen svarende ti1 et Potential P = r Beregningen af et Integral ti1 2.8 Relativt Potential af to Kugleskaller med Ladninger Z 1 og Z , , Radier R , og R,, Afstanden mellem Centrene a < R1,R , Udledning af Potential imellem to Kerner Z , e og Z,e, hver omgivet af en Z . e e-'Iai Elektronfordeling svarende ti1 et afskzrmet Potential r Beregningen af Indflydelse af Elektronafskarmning ved Stoppeeffekten ved Kernesammenstsd [These pages deal with the stopping formula for fission fragments. They also contain calculations and a graph of dV/dx as a function of V.] Sandsynlighed for, at en Kerne med Ladning Z , e og Hastighed V fanger en Elektron fra en Kerne Z , e Elektrostatisk Tydning af hurtig bevregede Partiklers Stopning Energioverfsrelse og Ionisation ved Sammenst~dmellem to H-Atomer Energioverfsrelse og Ionisation ved Sammenstsd mellem to Kerner, hver med en Elektron Ioniseringen ved hurtige ladede Partikler Stopping-Power in Collisions between two Neutral Heavy Atoms of Atomic Number Z , and Z , Closer Considerations of Energy Exchange by forces proportional to inverse cube Tvarsnit for Sammenstsd med forskellige Kraftfunktioner Probable Angle Deviation in Nuclear Collisions Comparison between behaviour of fission fragments in light and heavy gases Effect of stripping of fragments on nuclear collisions]

P A R T 11: G E N E R A L T H E O R Y O F P E N E T R A T I O N

TRANSLATION [Table of Contents] 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

2.9 2.10 2.1 1 2.12 2.13 2.14 2.15 2.16 2.1 6a 2.17 2.18 2.19 c2.20

Simplified representation of stopping formula Stopping of a particles by heavy atoms Contribution of'electronic interaction to stopping power Contribution of nuclear collisions to stopping power 1 Density distribution corresponding to a potential P = -e-ria r Calculation of an integral for 2.8 Relative potential of two spherical surfaces of charges 2, and Z,, radii R , and R,, the distance between their centres a < R , , R, Derivation of potential between two nuclei Z , e and Z,e, each surrounded by an electronic distribution corresponding to a screened potential r Calculation of the influence of electronic screening on stopping power in nuclear collisions [These pages deal with the stopping formula for fission fragments. They also contain calculations and a graph of d V / d x as a function of V.] Probability of capture of an electron from a nucleus of charge Z,e by a nucleus of charge Z , e and velocity V Electrostatic interpretation of the stopping of swiftly moving particles Energy transfer and ionization in collisions between two H-atoms Energy transfer and ionization in collisions between two nuclei with one electron each Ionization by swiftly moving charged particles Stopping power in collisions between two neutral heavy atoms of atomic number 2, and 2, Closer considerations of energy exchange by forces proportional to inverse cube Cross sections for collisions with different force functions Probable angle deviation in nuclear collisions Comparison between behaviour of fission fragments in light and heavy gases Effect of stripping _ .- of fragments on nuclear collisions]

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MS,p. I 21 Oct 40

Sandsynlighed for, at en Kerne med Ladning Z , e og Hastighed V fanger en Elektron fra en Kerne Z2e.

Tvzersnittet for Sammenstsd, hvorved Elektronen faar en Hastighedsmdring af samme Stsrrelsesorden som V er lig

For at capture kan finde Sted, maa Sammenstsdet finde Sted indenfor en Afstand il fra 2,. Sandsynligheden derfor er

Sandsynligheden for, at et saadant Sammenstsd leder ti1 en Omsatning af Impuls mellem Elektronen og Z 2 er

hvor

derfor er E2

2

= z:(+)

Endelig er Sandsynligheden for, at den relative Hastighed efter det dobbelte Sammenstsd* mellem Z , og Elektronen er 5 VoZ, lig Eg

N

(+)

3

z:

I Stsrrelsesorden er derfor Tvarsnittet for Indfangningen : a

* [See document XXII, p. 111.1

= a"'EtE2E3 =

(

z:z: ;)I2

P A R T 11: G E N E R A L T H E O R Y O F P E N E T R A T I O N

hvilket stemmer overens med Kramers direkte Regning med Benyttelse af Borns Approximation. Betingelsen for en saadan Regning er, at saavel K1=--

4nZ1e2 - 2--.z1 Vo hV V

lc2

MS, p, 2

VO V

= 2--.z2

er mindre end 1. Ved klassisk Regning finder Thomas (Proc. Roy. SOC.114,561, 1927)

(

aI prop. med a ~ Z ~ / 2 Z ~'/2

7 ) 1 1

=

aF(xl)- 3/2(x2)1/2

TRANSLATION Probability of Capture of an Electron from a Nucleus of Charge Z 2 e by a Nucleus of Charge Z,e and Velocity T/:

The cross section for a collision in which the electron obtains a change in velocity of the same order of magnitude as V is equal to Z2e4

a,=-1_-

(m1/2)2

- aoz:($-)

4

.

In order for capture to take place, the collision must take place within a distance A from Z 2 . The probability of this happening is

The probability of such a collision leading to an exchange of momentum between the electron and 2, is

(s) 2

&2 =

MS,P. I 21 Oct 40

P A R T 11: G E N E R A L T H E O R Y O F P E N E T R A T I O N

where

hence

Finally, the probability of the relative velocity being collision* between 2, and the electron is

VoZ, after the double

The order of magnitude of the cross section for capture is therefore

which agrees with Kramers’ direct calculation based on the Born approximation. The condition for such a calculation is that both q

MS,p, 2

4 x 2 e2 V, L =2--.21 V hV

=

and lc2

VO V

= 2--.z2

are less than 1. By a classical calculation Thomas finds (Proc. Roy. SOC. London A114 (1927) 561-576)

* [See document XXII, p. 111.3

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Ioniseringen ved hurtige ladede Partikler. Efter den klassiske Behandling (J.J. Thomson) er Problemet overordentlig simpelt. Ioniseringen fremkommer ved Sammenstsd med Elektroner, ved hvilke Stsdparameteren p er mindre end p r , bestemt ved

I=

2 ~ : ~ 4 mV’(p3 a’)

+

hvor

Vi har derfor

Tvarsnittet for Ionisation er simpelthen

Denne Beregning hviler paa et veldefineret Grundlag, naar

men selv i dette Tilfielde maa vi tage Hensyn til, at der ved Sammensterd, hvor p > p r [kan forekomme Ionisering], idet vi efter Kvanteteoriens Elementer her ikke kan beregne Udfaldet af det enkelte Stsd, men alligevel finde de statistiske Virkninger ved at tanke 0s den tilferrte Energi akkumuleret indtil den er tilstrakkelig for Ionisationen eller Excitationen. Idet vi for Simpelheds Skyld ser bort fra den sidste og betragter alle tilladelige Energioverfsrelser at viere lig med I , finder vi

hvor p v er lig med den adiabatiske Stsdgranse V 2nv

pv=-=-

hV 2nI

MS,p. I 23 Oct 40

P A R T 11: G E N E R A L T H E O R Y O F P E N E T R A T I O N

altsaa

MS,p. 2

Dersom K < 1, er de klassiske Beregninger uholdbare, men vi ved fra en simpel Betragtning, at vi ved Sammenstsd mellem Elektron og en afskarmet Kerne, hvor Feltet forsvinder for p > p s , at alle Virkninger er statistisk set ntajagtig de samme som i den klassiske Teori for en Afskarmning ved p = p s ~For . oI faar vi derfor nu

Denne Beregning begrundes ved, at

Ved de paagaldende Sammenstsd er derfor Afstanden mellem Elektron og Kerne stsrre end Usikkerheden i Elektronens Sted i Atomet. I Normaltilstand er Atomradien a, netop lig

a, =

~

h 2nm V,

For hrajere Kvantetilstande er Atomradien vel sttarre, men Stedsbestemmelsen alligevel tilladelig statistisk set. TRANSLATION MS,p. I 23 Oct 40

Ionization by Swiftly Moving Charged Particles. According to the classical treatment (J.J. Thomson) the problem is extremely simple. The ionization takes place in collisions with electrons for which the impact parameter p is less than p r , which is determined by

P A R T 11: G E N E R A L T H E O R Y O F P E N E T R A T I O N

I=

2z:e4 mV2(’p; + a’) ’

where

We therefore have

The cross section for ionization is simply

This calculation rests on a well-defined basis when

but even in this case we must take into consideration that [ionization may take place] in collisions for which p > p I : According to the principles of the quantum theory we cannot in this case calculate the outcome of the individual collisions; yet we can find the statistical effects by imagining that the energy transferred is accumulated until it is sufficient for ionization or excitation. Neglecting the latter for simplicity and considering all permissible energy transfers to be equal to I , we find

where py is equal to the adiabatic cut-off parameter

v

pv=-=2nv

1.e.:

hV 2711

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MS,p. 2

If K < 1, classical calculations are untenable, but we know from a simple consideration that in collisions between an electron and a screened nucleus, where the field vanishes for p > p s , all effects are statistically exactly the same as in the classical theory for a screening distance p = pSlc. For cIwe therefore now get

This calculation is motivated by the fact that

By the collisions in question the distance between the electron and the nucleus is therefore larger than the uncertainty in the position of the electron within the atom. In the normal state the atomic radius a,, is precisely equal to a,=-.

h 2nm V,

True enough, for higher quantum states the atomic radius is larger, but nevertheless the determination of the position is statistically permissible.