note on wave amplification by interaction with a stream of electrons

2
LETTERS TO THE EDITOR 1721 different ranges need to be assumed for the central and the tensor part of the interaction in triplet states. Assuming such different ranges, however, Castillejo and Richardson 1 have been unable to account for the observed total cross section for neutrons of energy about 90 Mev. To investigate whether better agreement could be obtained with a different well shape, similar calculations have been made employing a two-range Yukawa-well with ranges 0.80X10~ 13 cm and 1.60 -13 cm for the central and tensor forces respectively for the triplet interaction. The appropriate constants were determined by the iteration method for these ranges to fit the data on the binding energy and the electrical quadrupole moment of the deuteron, and the proportion of Z>-wave thus obtained is 3.8 percent—in good agreement with that needed to explain the observed magnetic moment of the deuteron. For the interaction in singlet states a range of 1.18X 10~ 13 cm was used to fit the data on p-p scattering. The low energy scattering total cross sections have been ob- tained for incident neutrons of energy 0-3 Mev in agreement with the experimental results. But the high energy (83 Mev) total cross sections are large as usual, viz., 14.6X10 -26 cm 2 and 12.1 XlO" 26 cm 2 for the symmetrical and Serber (interaction in even states only) types of interaction respectively. The following table gives the angular distribution in the two cases, in lO" 26 cm 2 : Sym. <r(6) Serber <r(0) 0 1.49 1.30 15 1.43 1.30 30 1.30 1.27 60 0.97 0.89 90 0.81 0.73 120 1.06 0.89 150 2.08 1.27 165 2.65 1.30 180 2.97 1.30 The validity of Born's approximation for high energy scattering in the triplet states has also been examined and it is found that in the present case it gives much smaller value for the cross section than that obtained by the exact method. The detailed account will be published elsewhere. I wish to express my indebtedness to Professor H. S. W. Massey, F.R.S. for his keen interest in the calculation. I also wish to thank Dr. T. M. Hu and Mr. K. N. Hsu for many valuable discussions. Finally I express my thanks to the Government of Bihar (India) for extension of the scholarship enabling me to complete this work. i L. Castillejo and H. T. Richardson, Phys. Rev. 76, 1732 (1949). Structure of the Line 0^4686 of He II K. MURAKAWA, S. SUWA, AND T . KAMEI Institute of Science and Technology, Komaba, Meguro-ku, Tokyo, Japan October 10, 1949 I N order to test the theory of level shift in hydrogen-like atoms, 1,2 the structure of the line X4686 of He II was studied. The light source was a cool aluminum hollow cathode discharge tube, which was filled with helium of about 1 mm Hg pressure. The fine structure was examined by the use of a silvered Fabry-Perot etalon and a glass Lummer plate of thickness 4.7 mm. The structure of the line X4686 calculated according to the recent theory, 2 together with the observed structure, is shown in Fig. 1. Figure 2 is a reproduction of a photograph taken with a 2-mm etalon. Mack and Austern 3 had previously resolved the components j and k } and obtained the value 0.16±0.02 cm -1 as the mutual distance. The value obtained by us is 0.18=b0.006 cm -1 , and is nearer the one required by the theory (0.182 cm -1 ). In addition to the resolution of the components^' and k, the components / and m were also resolved in the present work, and the mutual distance was observed to be 0.14±0.01 cm -1 , which is to be compared with the theoretical value 0.130 5 cm -1 . The observed structure is therefore in harmony with the theory, 1 ' 2 but the accuracy of measurement is still insufficient for testing the level shift of the terms other than S terms. One weak component not required by the theory was observed at —0.87 cm -1 . This was also observed by Paschen 4 in his former aj_c fa J j_* . . .. 519-.470-122 0.211\ /.456 5 1.084 1.266 1.816 1.947 CM"' CALC- 1 _ 365 >2j4 >455 0.49 AV 0BS : 0 0.456 1.087 1.27 1.81 1.950 CM FIG. 1. Transition scheme and calculated and observed structure of He II X4686. I FIG. 2. Inference pattern of lie II X4686 taken with a 2-mm etalon. work, but rejected as doubtful in his later work. The line breadth of the component seems to be due to a light element, but the possibility of attributing it to the aluminum spectrum is not so definitely excluded. In conclusion we wish to thank Professor M. Kiuchi who took deep interest in our work and kindly loaned us the helium gas. i H. A. Bethe, Phys. Rev. 72, 33 (1947). F. J. Dyson, Phys. Rev. 73, 617 (1948). *N. M. Kroll and W. E. Lamb, Phys. Rev. 75, 388 (1949). J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949). a J. E. Mack and N. Austern, Phys. Rev. 72, 972 (1947). See also G. R. Fowles, Phys. Rev. 73, 639 (1948); 74, 219 (1948). <F. Paschen, Ann. d. Physik 50, 901 (1916); 82, 689 (1927). Note on Wave Amplification by Interaction with a Stream of Electrons L. R. WALKER Bell Telephone Laboratories, Murray Hill, New Jersey October 10, 1949 I N a recent paper, J. A. Roberts, 1 extending the work of V. A. Bailey 2 on the propagation of plane electromagnetic waves in an ionized gas, purports to show that plane waves with gain exist in infinite space filled with a uniform density of heavy positive ions and an equal density of electrons which move with uniform velocity in a given direction. The paper appears to be open to criticism, some of which extends to Bailey's original work. Roberts* Eq. (1), a special case of one derived by Bailey, is said to be non-relativistic, but contains terms of order V 2 /C*, w T here V is the velocity of the electron stream. When the equation is derived again using the relativistic equations of motion for the electrons, further terms of order V 2 /0 appear. The equation thus found may be readily factored and the resultant plane wave solu- tions show no gain. A similar conclusion may be reached by making a Lorentz transformation to a system in which the elec- trons are at rest. In this case the permissible plane wave solutions are found very simply and they may then be transformed back into the rest system. It would appear that an inconsistent neglect

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Page 1: Note on Wave Amplification by Interaction with a Stream of Electrons

L E T T E R S T O T H E E D I T O R 1721

different ranges need to be assumed for the central and the tensor part of the interaction in triplet states. Assuming such different ranges, however, Castillejo and Richardson1 have been unable to account for the observed total cross section for neutrons of energy about 90 Mev.

To investigate whether better agreement could be obtained with a different well shape, similar calculations have been made employing a two-range Yukawa-well with ranges 0.80X10~13 cm and 1.60-13 cm for the central and tensor forces respectively for the triplet interaction. The appropriate constants were determined by the iteration method for these ranges to fit the data on the binding energy and the electrical quadrupole moment of the deuteron, and the proportion of Z>-wave thus obtained is 3.8 percent—in good agreement with that needed to explain the observed magnetic moment of the deuteron. For the interaction in singlet states a range of 1.18X 10~13 cm was used to fit the data on p-p scattering.

The low energy scattering total cross sections have been ob­tained for incident neutrons of energy 0-3 Mev in agreement with the experimental results. But the high energy (83 Mev) total cross sections are large as usual, viz., 14.6X10-26 cm2 and 12.1 XlO"26 cm2 for the symmetrical and Serber (interaction in even states only) types of interaction respectively. The following table gives the angular distribution in the two cases, in lO"26 cm2:

e° Sym. <r(6) Serber <r(0)

0 1.49

1.30

15 1.43

1.30

30

1.30

1.27

60

0.97 0.89

90 0.81

0.73

120 1.06 0.89

150

2.08 1.27

165 2.65

1.30

180 2.97

1.30

The validity of Born's approximation for high energy scattering in the triplet states has also been examined and it is found that in the present case it gives much smaller value for the cross section than that obtained by the exact method. The detailed account will be published elsewhere.

I wish to express my indebtedness to Professor H. S. W. Massey, F.R.S. for his keen interest in the calculation. I also wish to thank Dr. T. M. Hu and Mr. K. N. Hsu for many valuable discussions. Finally I express my thanks to the Government of Bihar (India) for extension of the scholarship enabling me to complete this work.

i L. Castillejo and H. T. Richardson, Phys. Rev. 76, 1732 (1949).

Structure of the Line 0̂ 4686 of He II K. MURAKAWA, S. SUWA, AND T . KAMEI

Institute of Science and Technology, Komaba, Meguro-ku, Tokyo, Japan October 10, 1949

IN order to test the theory of level shift in hydrogen-like atoms,1,2

the structure of the line X4686 of He II was studied. The light source was a cool aluminum hollow cathode discharge tube, which was filled with helium of about 1 mm Hg pressure. The fine structure was examined by the use of a silvered Fabry-Perot etalon and a glass Lummer plate of thickness 4.7 mm.

The structure of the line X4686 calculated according to the recent theory,2 together with the observed structure, is shown in Fig. 1. Figure 2 is a reproduction of a photograph taken with a 2-mm etalon.

Mack and Austern3 had previously resolved the components j and k} and obtained the value 0.16±0.02 cm -1 as the mutual distance. The value obtained by us is 0.18=b0.006 cm-1, and is nearer the one required by the theory (0.182 cm-1). In addition to the resolution of the components^' and k, the components / and m were also resolved in the present work, and the mutual distance was observed to be 0.14±0.01 cm-1, which is to be compared with the theoretical value 0.1305 cm-1. The observed structure is therefore in harmony with the theory,1'2 but the accuracy of measurement is still insufficient for testing the level shift of the terms other than S terms.

One weak component not required by the theory was observed at —0.87 cm-1. This was also observed by Paschen4 in his former

aj_c fa J j _ * . . ..519-.470-122 0.211\ /.4565 1.084 1.266 1.816 1.947 CM"'

CALC- 1 _ 3 6 5 > 2 j 4 >455

0.49 AV 0 B S : 0 0.456 1.087 1.27 1.81 1.950 CM

FIG. 1. Transition scheme and calculated and observed structure of He II X4686.

I

FIG. 2. Inference pattern of lie II X4686 taken with a 2-mm etalon.

work, but rejected as doubtful in his later work. The line breadth of the component seems to be due to a light element, but the possibility of attributing it to the aluminum spectrum is not so definitely excluded.

In conclusion we wish to thank Professor M. Kiuchi who took deep interest in our work and kindly loaned us the helium gas.

i H. A. Bethe, Phys. Rev. 72, 33 (1947). F. J. Dyson, Phys. Rev. 73, 617 (1948).

*N. M. Kroll and W. E. Lamb, Phys. Rev. 75, 388 (1949). J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949).

a J. E. Mack and N. Austern, Phys. Rev. 72, 972 (1947). See also G. R. Fowles, Phys. Rev. 73, 639 (1948); 74, 219 (1948).

<F. Paschen, Ann. d. Physik 50, 901 (1916); 82, 689 (1927).

Note on Wave Amplification by Interaction with a Stream of Electrons

L. R. WALKER Bell Telephone Laboratories, Murray Hill, New Jersey

October 10, 1949

IN a recent paper, J. A. Roberts,1 extending the work of V. A. Bailey2 on the propagation of plane electromagnetic waves in

an ionized gas, purports to show that plane waves with gain exist in infinite space filled with a uniform density of heavy positive ions and an equal density of electrons which move with uniform velocity in a given direction. The paper appears to be open to criticism, some of which extends to Bailey's original work.

Roberts* Eq. (1), a special case of one derived by Bailey, is said to be non-relativistic, but contains terms of order V2/C*, wThere V is the velocity of the electron stream. When the equation is derived again using the relativistic equations of motion for the electrons, further terms of order V2/0 appear. The equation thus found may be readily factored and the resultant plane wave solu­tions show no gain. A similar conclusion may be reached by making a Lorentz transformation to a system in which the elec­trons are at rest. In this case the permissible plane wave solutions are found very simply and they may then be transformed back into the rest system. It would appear that an inconsistent neglect

Page 2: Note on Wave Amplification by Interaction with a Stream of Electrons

1722 L E T T E R S T O T H E E D I T O R

of relativistic terms affects the nature of the plane wave solutions quite radically. A note will be submitted deriving the relativistic equation and its solutions.

The remarks of Roberts concerning growing waves in a cylin­drical pipe filled with electrons moving parallel to the axis and upon traveling wave tubes seem to be based upon a misconception. It is stated that Ramo3 in a discussion of this problem failed to discover a set of waves with negative attenuation. Ramo pointed out that two types of wave exist in such a tube; there are two "space charge" waves moving roughly with the electron stream velocity and two fast "field" waves. The latter solutions according to Ramo are cut off at a certain frequency. This is not strictly correct in the sense that the solutions below this frequency are not pure exponentials. However, such solutions are simply cut-off modes in the pipe perturbed by the electron stream, one increasing and decreasing in the direction of electron flow. The direction of power flow is opposed to the electron flow for these "field" waves. If a situation were to prevail in which the wave increasing in the direction of electron motion dominated the other three waves, clearly power would have to be supplied at the high level end. J. R. Pierce has shown in some unpublished calculations on traveling wave tubes, where a backward increasing wave may also exist, that it is possible by modulating the input stream and arranging the boundary conditions at the two ends of the tube correctly to obtain the four waves excited in such a way that the signal level at the upstream end is greater than that at the down­stream end and power flows against the electron flow. However, the gain in such cases is very small compared to the exponential increase of the single backward cut-off wave and bears no simple relation to it. It is clear that the enormous gains cited by Roberts for the single wave would be quite unobtainable in practice. The use of retarding structures to produce approximate synchronism between the circuit wave and electron stream, with a consequent wave showing gain and power flow in the direction of electron flow is not simply a superfluous complication.

1 J . A. Roberts , Phys . Rev. 76, 340 (1949). 2 V . A. Bailey, N a t u r e 161, 599 (1940); J . Roy. Soc. N.S.VV. 82, 107

(1948); Austr . J . Sci. Research 1, 351 (1948). 3 S . Ramo, Phys . Rev. 56, 276 (1939).

5001

A Naturally Occurring Odd-Odd Isotope of Vanadium

W A L L A C E T. L E L A N D *

Department of Physics, University of Minnesota, Minneapolis, Minnesota October 24, 1949

\ S part of a program to investigate the isotope abundances of *\> the heavier elements, an isotope of vanadium of mass 50 having an abundance of 0.23±0.01 percent was discovered. The instrument used in the investigation was a 60 degree mass spec­trometer of a design similar to one already described bv NierJ

ions were obtained by electron bombardment of vanadium vapor produced by evaporation of the metal placed on a heated tungsten filament. That the isotope at mass 50 is actually vanadium and not due to an impurity is indicated by the following observa­tions. (1) The ratio of the mass 50 peak to V61 remained constant while varying the temperature of the tungsten filament and con­sequently the rate of vanadium evaporation. Figure 1 shows the results obtained with two separate samples of vanadium. The intensity ratio of V51 to mass 50 is plotted against intensity of V51. The scattering of the points is due to measurement errors intro­duced by time fluctuations in the vanadium vapor supply. I t is seen that over the range of 20 in intensity covered, the ratio remained constant at about 425. (2) As indicated in Fig. 2, the variation in intensity of the mass 50 peak with energy of electrons producing the ions is the same as the variation characteristic of vanadium. (3) The mass spectrum in the region surrounding

J54O0P

300

200

100

gQJf Ufe ° x -e-e

X-SAMPLE NO. I O-SAMPLE NO. 2

_J I I I ' I I 1 I

4 8 12 16 20 24 INTENSITY OF V51

F I G . 1. The ratio of t h e V « peak to the mass 50 peak is shown to be independent of the V61 intensi ty which in effect demons t ra tes the one to one correspondence of the mass 50 peak to the vapor pressure of v a n a d i u m .

masses 50 and 51 was void of peaks, thereby indicating the absence of impurities such as hydrocarbons or chromium and titanium which also have isotopes of mass 50. (4) Doubly charged ions were observed which gave the same ratio of intensities, namelv 51++/50++ e q u a ] t 0 425.

The existence of the odd-odd V50 with an abundance of 0.23 percent is unexpected. Aston2 examined vanadium with his mass spectrograph in 1924 and reported it to be 100 percent V51. His apparatus at that time, however, was not very sensitive, and the failure to detect V50 not surprising. In checking the literature, con­flicting reports are found in regards to radioactive vanadium isotopes, and in the latest table of isotopes by Seaborg and Perlman3 V50 is not listed. A recent article by Cork, Keller, and Stoddard4 reports a 635-day activity produced in vanadium irra­diated in a pile. This activity may be due to V49 previously reported as having a half-life of 600 days5 and produced in their experiment by a (n,2n) reaction on V50. It would be interesting to examine chromium produced by a (d,n) reaction on vanadium since V50

would go into the 26.6-day Cr51 while V51 would end up as stable Cr52.

Although the relatively high abundance of V50 speaks against it, one might expect that as in the case of K40 and other odd-odd nuclei heavier than N14,** V50 would be unstable. A check with a thin-walled beta-counter, however, failed to*reveal any activity. A subsequent experiment wherein vanadium was placed on photographic plates for a period of 8 days also failed to reveal any ionizing events in excess of the cosmic-ray background.

6h

<

or <

> 2

o-v51

X - V 5 0 * 4 2 5

J — i i I l i » i

20 40 60 80 ELECTRON ENERGY (VOLTS)

F I G . 2. The ionization efficiency curve for the mass 50 peak is shown to have the same shape as t h a t of vanadium by the congruence of the two curves, the one for V51 and the other for the mass 50 peak times 425.