The contribution to the frequency of visual sensations induced in the dark-adapted eye by the virtual photon field was calculated, this field is associated with the heavy nuclei that exist in space beyond the geomagnetic field. In order to determine the probability that the virtual photon field induces a light flash, only the portion of the virtual photon spectrum that corresponds to the known frequency dependence of the sensitivity of human rods to visible light was utilized. The results can be expressed as a curve of the mean frequency of light flashes induced by the absorption of at least R virtual photons versus the threshold number R. The contribution to the light flash frequency from the virtual photon field of heavy cosmic ray nuclei is smaller than that from Cerenkov photons. The flux and energy spectra of galactic cosmic ray nuclei helium to iron were used

Madey, R.

Mcnulty, P. J.

English

NASA Technical Reports Server

DIRECT STIMULATION OF THE RETINA BY THE METHOD OF VIRTUAL-QUANTA
FOR HEAVY COSMIC-RAY NUCLEI*
by
P.J. McNulty
Physics Department
Clarkson College of Technology
Potsdam, N.Y. 13676
and
Air Force Cambridge Research Laboratories +
Bedford, Mass. 01730
and
Richard Madey
Physics Department
Clarkson College of Technology
Potsdam, N.Y. 13676
The method of vlrtual-quanta is based on the fact that the electro-
magnetic field of a fast charged particle approaches the field of a plane
wave as the particle speed approaches the speed of light. The electro-
magnetic field of a particle that passes through the retina appears to a
rod in the retina as an electromagnetic pulse in time. The quantlzatlon
of the components of the Fourier frequency spectrum of this pulse of
electromagnetic radiation yields the virtual quanta. This method assumes
that the vlrtual-quanta interact with the retina in the same way as ordinary
photons.
We have calculated the contribution to the frequency of visual sen-
sations induced in the dark-adapted eye by the virtual-photon field asso-
ciated with the heavy-nuclel that exist in the region of space beyond the
geomagnetic field. In order to determine the probability that the virtual-
photon field induces a light-flash, we utilize only the portion of the
vlrtual-photon spectrum that corresponds to the known frequency dependence
of the sensitivity of human rods to visible light. The results of our
calculation can be expressed as a curve of the mean frequency of light-
flashes induced by the absorption of at least R vlrtual-photons versus the
threshold number R. The contribution to the llght-flash frequency from the
vlrtual-photon field of heavy cosmlc-ray nuclei is smaller than that from
Cerenkov photons. We have used the flux and energy spectra of galactic
cosmic-ray nuclei helium to iron measured by Comstock, Fan and Simpson.
INTRODUCTION
i
In the preceding paper, we calculated the
frequency of inducing visual sensations by Ceren-
kov radiation from heavy cosmlc-ray nuclei tra-
versing the retina. The results of that calcu-
lation for high detection efficiencies corres-
ponding to states of relatively complete dark-
adaptlon, are in general agreement with the fre-
quencies reported by astronauts on Apollo missions
11-14. However, it has been shown that these sen-
sations can be induced by means other than free or
Cerenkov photons. Light-flashes have been induced
by a number of stimuli including mechanical pres-
sure, electric currents, X-rays, and energetic
neutrons._l_ In the case of the energetlc
4neutrons, , the flashes probably result from
recoil protons or heavier nuclei produced in the
nelghborln_.medla; more particularly, the results
of Fremlin _ seem to rule out any significant
contribution from Cerenkov radiation.
In this paper we have modified the well known
Method of Virtual-Quanta (MVQ) to provide a theo-
retical formalism for calculating the frequency of
llght-flashes to be expected from direct excitation
ofthe photo-receptor elements by charged par-
ticles. The results of our calculation agree with
the data of Tobias et a113 and Fremlln 14 in the
sense that one can produce visual sensations by
non-relatlvlstlc particles; however, the frequency
of visual sensations calculated by this method for
astronauts exposed to the spectrum of galactic
cosmlc-ray nuclei at solar minimum is at least an
order of magnitude lower than the highest rates
observed. The MVQ as applied in this paper calcu-
lates only the contributions from direct exclta-
tion of the rhodopsin molecules through the ex-
change of vlrtual-quanta in a narrow energy range
corresponding to optical frequencies; it should,
therefore, be viewed as a minimum estimate of the
*Supported in part by funds from the Research Corporation and Air Force Cambridge Research Laboratories.
+NAS-NR_ Resident Research Associate
767
https://ntrs.nasa.gov/search.jsp?R=19720010054 2020-03-17T03:46:45+00:00Z
contribution due to direct excitation.
METHOD OF VIETUAL-QUANTA
The mechanism for the electromagnetic inter-
action between two charged particles is the ex-
change of a virtual-quantum. If the momentum
transferred during the collision is small, then
the exchange can be looked upon as the absorption
by the target particle at rest of a virtual-quan-
tum from the electromagnetic field accompanying
the moving particle. The number and configuration
of the virtual-quanta change dramatically as the
speed of the incident particle approaches the
speed of light. Fig.(1) is a schematic diagram
showing how the magnitude of the electric field
at a given distance from its charged particle
source changes as the particle's speed approaches
the speed of light. At rest ( B = O),the elec-
tric field at a given distance has the same mag-
nitude in all directions; whereas for 8 = 0.5,the
field has increased in directions transverse to
the particle's motion,and it has decreased
slightly in the direction parallel to the motion.
This elongation of the electric field in the
transverse direction increases as 8÷ i. In cal-
culating the effect on the target particle of
absorbing one of these virtual-quanta, it is
possible to ignore the incident particle itself
and consider only the cloud of virtual-quanta
that accompanies it. Weizsacker 15 and Williams 16
have developed the details of this method for
charged particles moving through a vacuum. A
quantum of angular frequency _ has an energy k
given by the Planck-Einstein formula
k = n_ (1)
where _ is Planck's constant divided by 2 w .
The energy spectrum of the photons in the equiva-
lent pulse of virtual-quanta can be obtained
directly from the frequency spectrum of the elec-
tromagnetic fields in the pulse. The differential
cross-section for a given inelastic interaction
(e.g. pion production, nuclear excitation, or, as
in our case, excitation of a molecule) for events
in which a photon of energy between _ and
_ + dT_ is emitted by the incident par-
ticle and absorbed in the target system is given
by the MVQ as the product of the number of vir-
tual-quanta in the incident pulse with energies
between _ and _ + d_ and the experimental
cross-section doo(_)/d_ for the analo-
gous photon-induce_ interaction.
dOp(1_)/d_ = N(1%_)doQ(1_)/d_ (2)
Visual sensations induced by light incident
on the retina result ultimately from the absorp-
tion of individual light quanta by rhodopsin
molecules that are contained in the outer seg-
ments of the rod cells. Although a rod responds
to a single absorbed photon, a single absorption
does not seem to be sufficient to activate the
optic nerve to produce a visual sensation. The
threshold for a visual sensation even under opti-
m_l physiological conditions requires from two to
0.95
0.50
Figure I. Magnitude of the electric field a given
distance from a point source that moves
with speed _c. The distance from a
point on a curve to the source repre-
sents the magnitude of the field at
that orientation.
perhaps as high as fourteen absorptions _ photons
in coincidence by rods in a cluster served by a
single optic nerve flber.17,18 Previous exposure
to light increases the threshold required for the
production of light-flashes.19, 20
The rhodopsin molecules are contained in disk-
shaped sacs that are stacked at intervals of a few
hundred Angstroms in the outer segments of the rod
cells. The rhodopsin molecule is sensitive to
light at wave lengths that are longer than dis-
tances between successive sacs and much longer than
the inter-mo!ee_lar distances of the medium; there-
fore, it is necessary to modify the MVQ to take
into account the polarizability of the molecules
in the rod. The virtual-quanta field polarizes
the molecules of the medium which in turn change
the field of virtual-quanta experienced by the
rhodopsin molecule. For optical wavelengths, we
can modify the MR to include polarization by
using the characteristic macroscopic fields in a
dielectric instead of the free space fields.
The expression for the number of virtual-
quanta that have energies between?_ and _+d_
has been shown 21 to be of the form
(3)
2
z P -91
where _ is the flne-structure constant,Z is the
atomic number of the incident particle,and v = 8c
is its speed_ the function e(_) represents
the dielectric permeability of the medium; the
constant a is the minimum value of the impact
parameter which we take equal to one Bohr radius;
ande'(_) represents the real part of £(_) . The
number of visible virtual-quanta can be obtained
by integrating Eq.(3) over the complete optical
spectrum. Since the rhodopsin molecules are not
equally sensitive to all photons in the visible
768
range,wehavetakeninto accountthe rhodopsin
sensitivity byreplacingtheactualabsorption
spectrumbya narrowrectangulardistribution of
widthA?lw= 0.52eV centered about the energy of
peak sensitivity, ?_o = 2.48 eV. All photons in
this reduced optical region are assumed to have
the maximum probability (20%) 22 of being absorbed
by a rhodopsin molecule upon traversing the re-
tina; photons outside of this region are assumed
to have zero probability of absorption.
The number of vlrtual-quanta with energies in
this restricted visible region that accompany a
charged particle as it passes through the retina
is given by
M
The dielectric permeability g(m) is now repre-
sented by a complex dielectric constant e O which
can be represented in terms of the real quantities
I t!
e o and g o as
, TI
¢o = to + ito (5)
The real and imaginary parts of the dielec-
tric constant are related to the index of refrac-
tion n and the absorption coefficient K by
2 2 ' 'I
n - K + 2 in < = E O + ie o (6)
For transparent materials, therefore, t O can be
replaced by the square of the index of refraction.
Absorption of visible light passing through the
retina occurs primarily in the outer segments of
the rod and cone cells. Individual outer segments
transmit about 70% of the light incident upon
them, resulting in at least 80% transmission for
the retina as a whole. Taking advantage of the
relative transparency of the components of the
retina,, the known v_lues of n 2 were substituted
for t o . Sidman 23 has measured the index of
refraction to be 1.41 and 1.39 for the outer seg-
ments of rods and cones respectively; the extra
cellular fluid which separates the outer segments
in the living retina presumably has the same index
of refraction as physiological saline, about
I. 3324
The term E O depends upon the amount of
absorption in the medium. ,If the retina were
completely transparent, c o would be zero; since
the retina is only slightly opaque, it can be
expected to be small. The number of visible vlr-
tual-quanta obtained from Eq. (4) is not sensitive
to the exact value of g o used. A crude estimate
of g o can be obtained from the relatlonshlp 21
Tr
_o = (_I do_(_ Id00 (7)
where N is the number of rhodopsln molecules per
unit volume in the medium. Eq, (7) can be re-
written in terms of the ratio of the transmitted
light intensity I to the incident intensity Io as
II
E O = -(C/_ 0 X) In (I/Io) (8)
The quantity x in Eq.(8) is the equivalent thick-
ness of the rhodopsln molecules in the retina.
Dartnell et a125 have shown that if the outer seg-
ment of the rod is 2.5 _ in diameter, there must
be N I = 2xlO 8 rhodopsin molecules in an outer seg-
ment for 30% probability of absorption for photons
passing through the outer segment. If we assume
a cylindrical outer segment of radius r and with
a thickness of rhodopsin x we have
N1/ _r 2 x = N O p/A (9)
where N O is Avogadro's number, A is the molecular
weight of rhodopsln (_ 40,O00),and 0 is its den-
sity ( _ 1.27 gm/cm_). The value of x obtained
from Eq.(9) is 2,13 _ . Substituting this value
into Eq.(8) for 70% transmission through an outer
segment yields £ O = 0.0133. This value was used
in Eq.(4) for our calculations.
The number of virtual-quanta M _ccompanylng
a nucleus of atomic number Z is plotted in Fig.(2)
as M/Z 2 versus the velocity of the nucleus. There
la a maximum value of the velocity above which
less than M vlrtual-quanta accompany the nucleus
through the layer of outer segments. The values
of this maximum velocity are pl6tted in Fig.(3)
Pot M = 2, I0, 20, 50 and i00. If the threshold
number of photons R required for a visual sensa-
tion is greater than or equal to M then, ignoring
fluctuations in M, particles moving faster than
x will not produce visual sensations. When
M m_, the probability that at least R out of the
M incident vlrtual-quanta will be absorbed is
given by
T=N-R M: P_M-T)(I-p)T
QM (_ R) = I (I0)
T=0 T! (_-T)!
For the retina the probability P of absorption by
a rhodopsln molecule for photons in the "visible"
wavelength region defined earlier is 0.2.
IO z --
IO t -
I0°
I0=_
i0-=
I
IO-= I0-' I0°
Figure 2. Plot. of M/Z 2 as a function of speed
at the retina.
769
I0
.01 I •
I I I ! I
5 iO 15 20 25
Z
Figure 3. Maximum velocity at the retina for at
least M vlrtual-quanta.
EXPECTED FREQUENCIES OF LIGHT FLASHES
The flux and energy spectra of the cosmlc-ray
nuclei from boron to nickel measured by Comstock,
Fan and Simpson 26 were used in calculations. Each
energy spectrum was separated into kinetic energies
-_ _,,_I_ ah_v_ ._d below a narticular value TI:
the spectrum fur that atomic number Z then being
represented as
nZ (T) = C 1 (O<T< TI) (Ii)
nz (T) -C (T +me2) x (T>T l) (Z2)
where me 2 is the proton rest mass. The values of
TI, C I and C used for the nuclei included in our
calculations are listed in Table I. The value of
used for all nuclei was 2.5. The integral
flux spectra corrasponding to Eqs.(ll-12) can be
written (l -y)
N (> T) = C(T + mc 2) (T>TI) (13)
and ( ¥ -i)
(I-
N (> T) = C 1 (TI-T)+C (Tl_mC2)_._) (O<T<TI)(14)
To compute the mean frequency of light-flashes
tO be expected from direct excitations in the
optical region defined earlier, we have prepared
a program that calculates the contribution to the
the light-flash frequency _rom each nucl_ Of
atomic number Z. We sketch the steps in the pro-
gram:
TABLE 1
Parameters for Representin_ _he Differential FI_ Spectra of Galactic
Co_tc-Rays at a T_e of Mtn_ Solar Hodulatloa
cI c T1
NucIeus p/m2 _s_sr_(BeV.nue-1) p/m2 .s.sr. KBeV.nue-1) - + I
C12, O16 5.4 12 ,437
BII 1.7 4.3 .513
Ne,_,Si 1.0 1.4 .207
P-Mn 1.0 1._ .2O7
Fe-N_ 1.0 1.4 .207
a. Find from Eq.(4) the maximum kinetic
!
energy per nucleon T M at the retina that each
cosmic ray nucleus can have and still be accom-
panied by M visible virtual-quanta.
b. Calculate the range in tissue R; of a
proton with this kinetic energy. For thls purpose
we used the power-law I
' = k T_a (15)Rp
The values k = 4.25xi0 "3 gm/cm 2 H20 - MeV a and
e = 1.627"were used.
c. The maximum kinetic energy per nucleon T M
in the incident spectrum that a particle can have
and be accompanied by M virtual-quanta after it
has traversed a shielding thickness _ is given by
' Z 2 I/u
TM = ((Rp + Z/A)/k) (167
d. The unidirectional flux density N Z ,M_TM) of
nuclei of atomic number Z with kinetic energies
per nucleon greater than T M can be obtained from
Eqs.(13-14); the differential unidirectional flux
of nuclei of atomic number Z that are accompanied
by M visible vlrtual-quanta can then be written:
ANz, s = NZ,M._I/2( > TM+$/2 ) _Nz,M.I/2 (>TM_I/2) (17)
The number of nuclei of charge Z to pass through
each eye per minute is obtained by multiplying
ANz ,M by
Y " 120 G (cos_l - cOS_ 2) (18)
where /_I, _2 and G are given in Ref. i.
e. The number of llght-flashes per minute
in both eyes caused by direct excitation by cos-
mic-ray nuclei of atomic number Z is given by
CZ (>R) - [ 2 Y ANz,M QM (>R) (19)
M-R
770
In Fig.(4) the relative contributions to the
llght-flash frequency from the most abundant of
the heavy cosmic-ray nuclei are given for a
threshold of i0 absorbed "visible" vlrtual-quanta.
Similar results were obtained for R = 2 and 20.
f. The total frequency of light flashes
induced by at least R direct excitations of
rhodopsin molecules is given by
C(> R) = I I 2 Y ANz, M QM (> R) (20)
Z M=R
Fig.(5) plots the llght-flash frequency as a
function of the threshold R.
RESULTS AND DISCUSSION
The curve in Fig.(5) shows that the fre-
quency of light flashes to be expected from the
direct excitation of the rhodopsln molecule by
the exchange of "visible" vlrtual-quanta is much
lower than the frequencies to be expected from
the absorption of Cerenkov quanta that were cal-
culated in Ref.(1). Even if one assumes that the
astronauts were completely dark-adapted and,
therefore, had thresholds for visual sensations
somewhere between 2 and 14 quanta, examination
of Fig.(5) shows the frequency expected to be
much less than the rates observed by astronauts
on Apollo missions 11,12, 13 and 14.
3O
__ 20
=o 10
R=IO
5 6 7 8 I0 12 14 16 18 20 22 24 26 28
ATOmiC NUMBER Z
30
20
C(_>R)
iO-I
I0"t
10-3
fO-4
I0"B
m/cm =
15gm/cm=_
lO-I I I I I
2 5 I0 20
R
Figure 5. Counting rate versus threshold R.
The importance of the MVQ is tha_ given the
cross-section for a photon of frequency
exciting a molecule producing an effect of
interest, one can calculate the cross-section for
a direct excitation of the molecule by a particle
of charge Z and speed Bc in which the incident
particle loses energy _ • For the retina we
know the geometrical and coincidence requirements
for visual sensations as well as the probability
that each photon will excite a rhodopsln molecule;
this knowledge extends, however, only over a very
narrow band of photon energies. _-q_econtribution
from direct excitation calculated in this paper
is thereby limited to those collisions with
rhodopsin molecules in which the energy lost by
the incident particle corresponds to optical fre-
quencies. The potential contribution from more
energetic collisions is, therefore, ignored but
could be calculated if the cross section for the
analogous photo-excitatlons and photo-lonizatlons
leading to visual sensations were known.
ACKNOWLEDGEMENTS
We would like to thank Mr. John B. McLaughlin
for his help in writing the computer prograum.
Figure 4. Relative contributions to the observed
llght-flash frequencies for threshold
R=I0.
771
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772


Direct stimulation of the retina by the method of virtual-quanta for heavy cosmic-ray nuclei

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