JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Impact of a Wave Packet and an Absorbing Particle

HERBERT E. IVESBell Telephone Laboratories, Inc., New York, New York

(Received January 20, 1944)

INTRODUCTION

THE impact of a wave packet on an absorb-T ing particle presents a more complicated

problem than the previously treated case ofimpact on a reflecting particle.' After impact theradiation no longer exists as such, and it be-comes necessary to study the characteristics ofthe form of energy into which the radiant energyis transformed. This may be done by noting thatthe absorption of radiant energy results in theproduction of heat, which is kinetic energy ofthe atoms or sLbparticles of which the absorbingparticle is composed. Then by considering thechange of mass of these atoms due to theirmotion, the over-all change of mass of the ab-sorbing particle can be obtained, and the equa-tions of motion derived.

(1) STATIONARY ABSORBING PARTICLE

We start by setting up the expression for themomentum of a particle moving as a whole withvelocity v, in which, because of absorbed heat,the constituent atoms are in random motion. Thetotal momentum M will be

- m(viw)M= -

(V-iw)' ,1 __ 1-i

( C2 h

of the particle and those against, and treat theseas all moving with uniform velocities in eachgroup. We designate these two velocities asv+w, and v-w 2. \Ve then rewrite (1) as

1 n(v+wl)M =- [E ___

2 1- (v+wl)"L1-- 2-a

1 M(V-W2)

2 [( V-W2)2]

I

(2)

which we can break down into

1 mvM = - E -

2 (V+W,)2 1-

1

2 I} V

C2

1 mwl+- ___

2 r (V+W,)2 1,1-- C2

(1)

where m is the rest mass of one of the con-stituent atoms and v-4w is its velocity of heatmotion; the denominator on the right is inaccord with the variation of mass with motionderived in the preceding paper.' The symbol w inthis equation is as yet of undefined value, ex-pressing merely the fact that the atoms havevelocities additional to the velocity v of theparticle as a whole.

We now for simplicity consider the atoms intwo groups, those moving, with respect to thecenter of the particle, in the direction of motion

I H. E. Ives, J. Opt. Soc. Am. 33, 163-166 (1943).

1 MW21 mw

2 -- _

(3)

Now if the particle as a whole is to move withthe velocity v, the summation of the momentaof the atoms forward and back, with reference tothe center of motion of the particle, must bezero, that is, the momentum described by thelast two terms of (3) must be zero, or

W1 W2

[ +W1 ,i; f [ (V -W2)2(4)

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VOLUME 34, NUMBER 4 APRIL, 1944

IMPACT OF A WAVE PACKET

We now seek the values of w1 and w2. We notethat if the particle is brought to rest, withoutescape of any of the heat, the heat motions in thestationary particle will be equal in all directions,their velocities assuming the value wo, so thatw 1=w 2 =wO. We can accordingly put

W1 ==wof(wo, v),

W2 = wof 2 (wo, v),

values in (5) we get

V2

Wo 1--C2

WO( - 1w 1=- _vwo

Wo(1 )

2

(5)

Al and f2 being functions which, when v= O, havethe value unity. Rewriting (4)

wof(wo, v)

[1 (+Wof,(Wo, V))2]i

wof 2(wo, v). (6)

(10)

W2 =- -= __vwo VwoQ- 1---

C2 C2

since, when v=O, wl=w 2=wo, the value of Qmust be unity.2

Inserting these values in (3) we have

VWm

M +C2 [/ (V Wof2(W0 v) ) 2

V C ) 1~

Replacing fl(wo, v) by F and .f2(wo, v) bythis equation becomes

F1 F2[1~~~~ v-woF22]~,[ (V+woF 1 (_ £ 11

By squaring and reduction we get from this

1 1 2vwo

F, F2 v2)

C2 C2--

Mov

This is satisfied by

VwoQ+-

1 C2

F, 11 ---

C2

(9)VwO

Q-1 C2

F 2 v21--

C2

where Q is undetermined. Substituting these

1 -- I1--]£2 J 2 J

, (12)

where Mo is the rest mass of the particle.

2 Note that we have incidentally derived, from non-relativistic considerations (for this particular problem), theEinstein formula for the composition of velocities, for if

W(1 _H2) wo(1 _ 2)

thenv+wo=v± c Equa-1+ VWO +'W J2 + c2

C C2 C2

tion (14) derived below has in fact been derived elsewhere[Becker, Theorie der Electricitdt (Teubner, Leipzig, 1933),Vol. 2, p. 348] by assuming that the atoms in the absorb-ing body move so that their velocities conform to theEinstein formula.

1

2

F2,

(7)

or

M= (8)

my

V2 lt- W° 2 L1-- I 1--I

223

I

V2 V'

Wo I_ Wo I---C2 C2

V2 i WO 2 1I_ I -C2 C2

M I- VWO)v

1 C2+ 57 +0 (11)

2 V2 - WO 2 1

1 - - I -C2 C2

HERBERT E. IVES

We can rewrite this in the form

Mov MOV£2]+ 1]. (13)

plv2] [lva 2 -WO2

Observing that the quantity in brackets is thekinetic energy of the atoms clue to their heatmotion' divided by jMloc2 , and calling this heatenergy H, we have finally for the momentum ofthe absorbing particle,

H\(l 0o+--)v

M= v . (14)

[1--sc22

From the momentum we can by integrationget the kinetic energy of the absorbing particleafter the impact, which is

(15)

We now consider the relation between theenergy of the wave packet incident on theparticle and the energy imparted to the particle.If Es is the energy density from the (stationary)source at the particle, and At the duration ofemission by the source, we have

Energy content of wave packet=E(cAt).

Now this becomes transformed into the kineticenergy of the particle [Eq. (15)], plus the heatabsorbed by the particle. The heat absorbed bythe particle in the impact is equal to H, theaverage rate of heating during the impact, timest, the duration of the impact, or It. We havethen from the conservation of energy

E,(cAt) = [Io+- 1c2 [11 +t.c2 Llv2 1

(16)

This, by rearrangement, can be written

HtMO+-

(cAt) c2Es- -+ MO = -- ,

c2 v2

c2

(17)

which makes (16) also a statement of the conser-vation of mass upon the acceptance of E,[(cAt)/C2]as the mass equivalent of the packet of radiation.

If in the impact we consider E8 [(cAt)/c 2 ] as amass, and proceed to the next step of assigningto it a momentum, by forming the product ofits mass by its velocity (which is c) we can setup from (14) and (17) the equation for theconservation of momentum in the impact, which is

HtMgO+- V

(cAt) c2- =- _c - v2 1

c2J

We have now (accepting these assumptions) allthe necessary material for finding (1) the velocityattained by the absorbing particle after the im-pact, and (2) the heat absorbed by the particlein the impact.

To obtain (1) we substitute the value of[Mo+ (IIt/c 2 ) /[1-(v 2/ 2)]3 from (18) in (17)which gives

(cAt) (cAt)E, =+Mo=E-

c2 cv

orE,(cAt)

v= - ~c. (19)E,(cAt) + Moc2

To obtain (2) we substitute the value ofi1lo+(I'It/c2 ) from (18) in (16) which gives

(cat)E,(cAt) = E -

cv

3 Strictly speaking it is the kinetic energy when theparticle is brought to rest, but since we have postulatedthat wo is the value of velocity when the particle is broughtto rest without escape of heat, this value of kinetic energyalso applies to the particle in motion.

rv2 1 I Xl 1-- c2 - 1 + t

L 62_ [ i

(20)

224

H , i= MO+- c 2 -1 .

c2 2 -1v 3

1--c -

IMPACT OF A WAVE PACKET

or

Ht=E, _Es(ct)[t- I])

(2) SYSTEM IN UNIFORM MOTION

An important problem now to be investigatedis the result to be expected if the impact experi-ment is done on a platform moving with theuniform velocity V. Specifically, will the endvelocity of the particle appear to be the sameunder this condition, and will the heat absorbedby the particle be the same? The changes intro-duced by this motion are two in number. First,the energy in the wave packet emitted in theapparent time At will be altered. Second, themeasurement of velocities will be altered bythe changes in clock rates and in measuring rods.

For the first of these changes we note thatthe emission time becomes At/[l - (V 2 /c2)]I; theenergy density at the point of impact, takinginto account the Fitzgerald contraction, and thechange of frequency of the source, is

V,

I+-C

E=E,,V

1--C

(21)

where Es is the energy density due to the pointsource in a stationary system of the same meas-ured dimensions ;4 the length of the packet isreduced in the ratio [1- (V/c)]: 1. The energycontent of the packet emitted in apparent timeAt is the product of these, or

(1V+l+c

E,(cAt)

V c2

rate with motion the absolute velocity whichwill, on the moving platform, be measured as v,is (V+v)/[l+( V/c 2)].

Our procedure is now to substitute the changedenergy content in the equations of motion for amoving platform, and find the resultant velocity.We do this in the expression for the conservationof momentum, where the expression correspond-ing to (18) becomes

(cAt)Es,-

C2

c+-_(i + I) M 0 V

[I; [1i- _r

tMo+-) VI

= ,~~ (23)V/2 1

1--_C2

where v' is the velocity of the particle after theimpact, and H' the heat absorbed by theparticle. Now by the conservation of mass

C2 (CAt)(c MO

C2

ClVl 2 Ll-22 1l-2]'

also, from (19)

(cAt) v(

Making these substitutions in (23) we obtain(22)

For the second of these changes we note thatbecause of the Fitzgerald contraction and the(experimentally established) variation of clock

4 Abraham, Theorie der Elekirizitat (B. G. Teubner,Leipzig, 1905), Vol. 2, p. 383, Eq. (245).

+y-) -- V

+-c cv

(1±-) (1--)+

C2 CV

V+vvV (24)V

1+-C2

225

v'=

HERBERT E. IVES

showing that the acquired velocity is that whichwill, on the moving platform, measure the same asthat acquired on the stationary platform.

To investigate the heat absorbed by theparticle when the impact occurs on a movingplatform, which we have designated 11T1', wesubstitute in (23) the value we have just foundfor v', together with the value of lo from (19).From this operation we obtain

H't'=E(cAt)1--1-[1I_ )J (25)Vt c21)

which is identical with (20), showing that theheat absorbed by the particle is the same as forthe case of the stationary platform.

(3) PRESSURE OF RADIATION ON ANABSORBING PARTICLE

for H't' this becomes

E.,(cAt) V cPt = +-c

c2[1 L2]i [4 [ Qj

v2] V~ c- _(t 2 I .

c2

(27)

Proceeding at once to the case of co-movingsource and target we have that v40, so that

E,(cAt)Pt = -

c 1--c2

(28)

andSince the velocity acquired by an absorbing

particle, and the heat absorbed, are the same forthe cases of the stationary and moving plat-forms, it is reasonable to infer that the pressureof radiation is likewise the same in both cases.The matter should, however, be looked intomore closely, since the commonly given valuesfor radiation pressure on a moving absorber leadto a different conclusion.

We arrive at a straightforward solution for thepressure by assuming that the pressure is thequotient of the change of momentum of theincident energy by the time; as expressed in theequation, applying to the moving platform.

V\

(cAt) c)Pt = Es

c2 V2-

c2

iIt( V+v)

V2 -I V2c2 1 -- 1jj~ U4J

Substituting the valueE"(C t){ I- ' -(' - -Y J

(26)

'Att-

V2- i

c2

so thatP=E,. (29)

Remembering that the energy density E in thiscase is E[1+(V/c)]/[1-(V/c)] we have therelation for the pressure on a moving absorbingsurface

I = EcP=EVI+-

(30)

which is to be contrasted with the valueP=E[1-(V/c)] uniformly given by previousinvestigators;5 which, as pointed out in a pre-vious paper, calls for a variation of radiationpressure with motion of the system.

We can arrive at this same value after ananalysis of the ultimate fate of the incidentenergy. This energy does three things:

I This value for the pressure on a moving absorbingtarget was obtained by the writer in a previous paper[H. E. Ives, J. Opt. Soc. Am. 32, 32 (1942), in whichpertinent references are given] but without the detailedanalysis provided by the present paper.

226

IMPACT OF A WAVE PACKET

(1) It puts the constituent atoms of theparticle into motion, showing as a rise in tem-perature of the particle.

(2) It imparts kinetic energy to the particledue to the increased mass of the moving atoms.

(3) It does work, the product of pressure timesdistance, on the original mass of the particle.

According to this analysis we can obtain thepressure from the change of momentum of theoriginal mass of the particle, divided by the timeof impact, or

AM 1 Mo(V+v) MOVP= =-1 - [

t tlt_21[_s1 V 1Z21Mov+ V( - -- ) I

-. (31)

being performed on a moving platform, is

(1+) IVIoC2

Es(cAt) + + -[V2i [V 2j

H'rt' - V+v lAfo+-J-C2 --

(AIO+""')C2~C

(32)F ( TJ. 2 13

Grouping terms in Mo and H't'

LV2

2 V2 1

I_-1--C2- C2

Expanding [1 - (v2/c2 )]', substituting the valueof Mo from (19), and proceeding to the caseof the co-moving source and particle, whent = At/[1 - (v2/ 2)], we get again,

P = E8.

We have now all the factors to work outexactly the allocation of the energy incident onan absorbing particle or surface. The equationfor the conservation of energy in the impact ofa wave packet on a particle, the experiment

( 1 +-Y)E 8(cAt) = JVI

I 2 ] -

(33)

Substituting the value of Mo from (19), and thevalue of H't' from (25), expanding [1 - (v2/c2 )]I,and proceeding to the case of v-0 we get

E(c/\tV2- t2

1--c2

11

Energy inwave packet.

E 8(cAt)

c211

Pressure Xtime X veloc-ity or me-chanical workdone byradiation onparticle.

E 8(cAt) v1

-- Ic2

11

Kinetic en-ergy added toparticle byheat motion.

+ E(cAt).

11Heat motionof atoms,causing riseof tempera-ture.

(34)

227

HERBERT E. IVES

If we now divide through by the timeAt[l -( V2/c2)]-j we obtain the relation betweenthe rate of flow of energy into the particle andthe rate of distribution of this energy among itsvarious tasks, as follows:

-(V2 VEs,(c+ V) = PV+Ec 1--J

C2

V2] 11--.-C 2

(35)

The last term indicates that the rate of heating(rise of temperature) is less than for the station-ary platform by the factor [1 -( V2/c2)]i, but thereciprocal of this is exactly the factor by whichthe unit of time on the moving body is increased,so that the rate of heating will be observed thesame. In short, the measurable quantities, pres-sure and rate of heating, appear invariant withthe motion of the platform.

The last term of (35), giving the rate of heat-ing, may be rewritten, in terms of V the velocityof the source, and VT the velocity of the target,as follows:

1+V.'1 VT'1+- 1--£ C FVT 2]

II=ESc c I -I. (36)V., VT LC 2

From this the heating with either source ortarget alone moving can be derived. If the targetis stationary this gives

V'1+-

CI = Ec V

1--Ci

or, if the source is moving away from the target

1+-(37)'

If the target is moving away from the source

H=E~c~- [E CI (38)

so that if the rate of heating is measured on thetarget, it is the same in each case.6

(4) DISCUSSION

In this study, and in the paper immediatelypreceding,' we have applied the assumptions ofconservation of energy and momentum to themass equivalent of radiation. These assumptionshave led to the variation of mass with velocity inthe ratio 1/[1 -(v 2/c2)]I: 1, to values for thevelocity and heat acquired by an absorbingparticle under the impact of radiation, and forthe pressure of radiation on absorbing targets.It is found that these assumptions lead to com-plete invariance of all measurable quantities(absorbed heat, rate of heating, velocity, pres-sure) with velocity of the system.7

It is to be recognized, however, that while theresults obtained are self-consistent, the values ofthe various quantities, in particular of pressureon a moving absorbing target, are at significantvariance with the values obtained by otherapproaches.

6 There is also involved in this analysis (in evaluatingthe energy density from the source at the target) theconvention that the distance between source and targetis always measured at the target, on a scale attached tothe source and experiencing the Fitzgerald contractionproper to the velocity of the source.

I Since, as pointed out by Epstein Am. J. Phys. 10, 1(1942)] the Fitzgerald contraction has been derived froma simple law of attraction, and the variation of clock ratewith velocity follows from the variation of mass withvelocity, there appears to be no need of a principle ofrelativity to derive the general invariance of opticalphenomena with motion.

228