I- UNITED STATES A To M I C E N E R G Y COMMISSION
AECU-1389
THE INDEX OF REFRACTION FOR NEUTRONS
By
David Kleinman
George Snow
JUN l ū |So,
April 17, 1951 UNIV. Of WAC'ſ
Brookhaven National Laboratory
– Technical Inform a ti on Service, Oak Ridge, Te n n e s see
|NIVERSITY OF MI HIGAN
#3 III.III.ii. |
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PHYSICS
AEC, Oak Ridge, Tenn., 11-23-51--385-W5075
THE INDEX OF REFRACTION FOR NEUTRONS*
By David Kleinman and George Snow
In a recent paper on variational principles in scattering theory,
Lippmann and Schwinger* have treated the scattering of slow neutrons by
bound protons as an application of their time independent formulation, which
they obtain from time dependent scattering theory by means of several de-
vices including the adiabatic reduction of the interaction to zero at t = toe.
The merits of this formulation are (a) the exact transition rate is obtained
in the form wba = 2 iſ 6 (Eb - Ea) Tba | * where T is an operator which
in Born approximation is the interaction energy, (b) a variational principle
is given for the T - matrix, (c) a sum rule is given which states that the
total rate of transition out of the initial state q^a is - 2 im Taa.
By a simple generalization from the neutron scattering by one bound pro-
ton We have considered the scattering by a thin slab of material containing
many nuclei. The pa component of the final state S @a, where S is the
Heisenberg S - matrix, is ( l - i. (m)/2T)Taa) ºpa. When the sum rule is
invoked the imaginary part of Taa gives the customary amplitude extinction
factor (l-# NOx ) & exp (-# NOx), where x is the thickness of the slab and
N is the nuclear density. Similarly, the real part of Taa represents a
phase shift, which one might interpret as due to the index of refraction of
the slab. When the Fermi approximation to Taa is used the phase shift is
in agreement with the usual formula for the index of refraction n = 1 - NA *ac/
27 , where ac is the isotopic average of the bound coherent scattering am-
plitudes of the nuclei.
One cannot expect an actual derivation of the index of refraction to
follow from time dependent scattering theory, because, from the time depend-
ent view point, S pa is the final state after the interaction has been adia-
batically reduced to zero, and it is not clear why the phase shift in S qìa
should necessarily be the same as that which the interaction Would produce
in a stationary Wave function. To overcome this objection. We have found a
formal solution to the stationary eigenvalue problem identical With the pa")
s of Lippmann and Schwinger, which we show represents an incoming plane wave
: and an outgoing scattered Wave. This stationary formulation is similar to
... that of Mollerº, who introduced the S - matrix into stationary scattering
tº theory. The connection between the S - matrix and the wave function is that
*º
º:
f,
... the outgoing part of (S-l ) pa is identical with the scattered Wave. On the
... basis of this theory we arrive at a general derivation of the index of re-
: fraction for gases, liquids, and solids.
*
wº *Research carried out under contract with AEC
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AECU-1389 - l
2 AECU-1389
The derivation of the index clearly shows that there is no Doppler
effect due to the motion of the individual nuclei, because the A in the
formula is the neutron Wavelength relative to the boundary of the slab.
The variational principle gives, beside the Fermi approximation, a tempera-
ture dependent correction to the simple formula, thereby taking into account
the effect of virtual inelastic interactions of the neutron With the mate-
rial. We have not as yet evaluated the correction in a specific case, but
Lippmann; finds a correction of 0.2 - 0.3% to the scattering cross section
of parahydrogen in its lowest rotational state for neutrons of Zero energy.
In liquids or solids at fairly high temperature and for neutrons of non-
Zero energy it would seem possible for the correction to be of the order of
1%. A change in the critical angle of reflection of a mirror of this amount
Would be significant in the experiment of Hughes, Burgy, and Ringo", and
might cause a detectable change in the intensity of a reflected beam of pile
neutrons.
REFERENCES
l. Lippmann and Schwinger, Phys. Rev. 79, 169 (195)
2. C. Moller, D. Kgl. Danske Widensk. Selskab, Mat-fys. Medd. XXIII,
Nr. 1 (1915)
3. B. A. Lippmann, Phys. Rev. 79, 181 (1950)
3. Hughes, Burgy, and Ringo, Phys. Rev. 77, 291 (1950)
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