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Radyasyonun (Inmn) TabiatIk Teorileri

Tanecik (emisyon)Teorisi

Dalga Teorisi

Newton (1670):

In bir k kaynandan k hzyla (3.108 m/s) herdorultuda yaylan kk madde tanecikleri olduunu ileri

srmtr. Bu teoriye emisyon (salma, yaym) veya tanecikteorisi denir. Bu teoriye gre farkl renklere trl boyuttakitanecikler neden olmaktadr. Ktleleri ok kk hzlar okbyk olduundan bu tanecikler yerekimi etkisi ileyollarndan sapmazlar. Bylece arlkszm gibi hareketettiklerini kabul eden bu teori ile n bir doru zerindeyaylmasnbaarl bir ekilde izah eder.

Ayrca elik bir bilyenin elik bir levha zerinde yansmasgibi nda yansyabileceini aklar. Yine elik bilyelerleyaplan uygun deneylerle n krlmas izahedilebilmektedir.

Huygens (1670):

In dalga tabiatnda olduunu ve dalga yzeyinin btnnoktalarnn elemanter kk dalgacklar meydana getirdiini

dnerek n yansma ve krlma kanunlarn yenidentretmitir.

Newton bu teoriye bir akkan iinde oluandalgalarn nlerinerastlayan engellerin yan kenarlarn dnerek geometrik glgead verilen szmalarn (difraksiyon veya krnm olay)gereke gstererek itiraz etmitir. Bu teori ile n keskinglgeler oluturmasn aklayamamas 1 asr kadar dalgateorisinin gzden dmesine neden oldu.

ngiliz Dr. Young ve Fransz Fresnel (1788-1827) yaptklaralmalarda interferans (giriim) olaylarn ancak dalga teorisiile aklayabildilernk k+nkaranlkolabilecei ancakdalga teorisi ile aklanabilmekteydi.

Ayrca Fresnel k dalgalaryla su dalgalarnn arasndakiglge oluumu(krnm)bakmndan grlen farknkaynandalga boylarnn ok farkl olmasndan kaynaklandn ispatetti. Bylece n dorusal yaylmn baarl bir ekildeaklad. Huygensingrn zor durumda brakan ve gzdendmesine yol aan neden ortadan kaldrlm oldu.


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Krnm Giriim


3/43

Huygens n polarizasyonunu n ses dalgalar gibi boyuna dalgalar olduunudndndenaklayamamt. Ancak Fresnel k titreimlerinin enine olduunu kabul ederekizah etmeyi de baard.

Foucault (Fuko) tarafndan sudaki k hznn deneysel olarak llmesi tanecik teorisinin

btnyle iflasna yol at nk tanecik teorisiyle her ne kadar krlma alarnn oran sabitkalsada youn ortamdaki khznn daha byk olmas gerekmekteydi. Ancak deneyler sudakikhznn havadakinden daha dkolduunugstermitir.

Bylece IIINENNE YAYILAN DALGALAR olduu sonucuna varld.

Maxwell 1861-1864 yllarnda tamamen teorik olarak khzyla hareket eden dalga denklemlerinitrettti. Bu denklemler ayrca n elektrik ve manyetik bileenleri olduunu ve birbirine dikdzlemlerde sins erilerine benzer ekildeyayldklar sonucuna vard.


4/43

1888 ylnda Hertz Maxwellin elektromanyetik dalgalarn bir elektrik titreim devresi kullanaraklaboratuvarnda elde etmi ve bu dalgalarn yansma,krlma, odaklanma, giriim vb. zellikleresahip olduunugstermitir. Ayrca bu dalgalarnhzlarnltnde teorik yoldan bulunan khznaeitolduunu grd.

Artkn dalga tabiatndaolduu 19. yydaphe gtrmez bir gerek olarak kabul edildi. 1900l yllarda bir takm deneylerin klasik fizik kanunlar ve elektromanyetik teoriyleaklanamad grld. O gne kadar elde edilen bilimsel birikim o gn iin yeni olan olaylaraklayamadndan dolayilerkarmayabalad ve o gne kadar kesin doruolduunainanlanfizik kanunlarnn mkemmel olmadnnfarknavarld.

Siyah Cisim Imas

Fotoelektrik OlayCompton Olay aklanamad


5/43

Termal Radyasyon ve Planck Denklemi

Bu blmde klasik fiziinaklayamad baz olaylar ve ilgilideneyler zerinde duracaz.

Bu deneyler sonucu

Inmn kuantl olduu grlmtr!


6/43

_Siyah Cisim Imas_(blackbody radiation veya cavity radiation)

--- Gzlemler ---

Bir metal ubuk stlnca neler olur ?

Scaklk ykseldike renkte nasl bir deiim oluur ?

Bunzen bek alevinin scakl hakknda ne biliyorsunuz ?


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Scak cisimler k yayarlarRenk nce krmz sonra sar sonra daha parlak sarolur


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Cisimler niin ma yapar?

Cisimler atomlardan meydana gelmitir

Mutlak sfrdan yksek scaklktaki btncisimler elektromanyetik ma yaparlars enerjisi

Titreen atomlar ma yaparlar.

Cisimlerin top-yay modeli

Is, molekl hareketlerinin (teleme, dnme, titreim)

ortalama kinetik enerjisinden kaynaklanr


9/43

yaydklar radyasyon daha ksa dalga boyuna kayar(red white blue)

--- Tanmlar---

Siyah cisim:

zerine den btn mkemmel bir ekilde

absorbe eden bir cisimdir. zerine den btn sourduu iin kara grnr veya grnmez (karadelikler gibi..)


10/43

The failures of Classical PhysicsSiyah cisim var mdr?

Nasl yaplabilir ?

T

Ima Gc (R); deal bir siyah cisim olarak dnlen

kk deliin birim alanndan birim zamanda yaylan enerjimiktarnama gc denir. Birimi J/m2.s-1

Enerji Younluu ( ); Siyah cismin boluundaki (cavity)enerji miktarnnboluk hacmine orandr. Birimi J/m3


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Ima nasl gzlemlenir ?

Ima monokromatik mi yoksa polikromatik

midir ?


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Gnein yzey scakl 5800 K ve scak bir odun atei ise yaklak800 oK dir. Cisimler scaklnn 4 kuvveti ile orantl olarak mayapmaktadr.


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Radiated Power from Blackbody When the temperature of a blackbody radiator increases, the overall

radiated energy increases and the peak of the radiation curvemoves to shorter wavelengths. When the maximum is evaluated

from the Planck radiation formula, the product of the peakwavelength and the temperature is found to be a constant.
http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/mod6.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/mod6.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html


14/43

Wien yasas:

Experimental observation

As the temperature raised, the peak in the

energy output shifts to shorter

wavelengths. Wien displacement law

Stefan-Boltzmann law

Wihelm Wien2max 5

1cT Kcm44.12 c

4/ aTVE 4TM

mK10.90,2 3max T


15/43

Zero amplitude

at boundary

L

wavelength

=c/ , = # of

cycles per sec

Standing Wave


16/43

Imann klasik fizik yorumuRayleigh-Jeans kanunu

3

28

c

Birim hacimde birim frekanstakimodlarn says:

standing waves originate from harmonically oscillatingcharges in cavity wall

two degrees of freedom (potential and kinetic energy)Average energy per degree of freedom: kT(equipartition)

average energy per standing wave: kTkT212

dc

kTdu

2

3

8)( total energy per unit volume:

Lord Rayleigh

Enerji dalmn aklamak zere ilk ne srlen teoridir veklasik mekaniin enerjinin eblm prensibini kullanmtr.


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Formln tretilmesi

Rayleigh and Jeans kapal bir kavite iindeki enerji younluunu hesaplamak

istediMetal bir kutu iindeki radyasyon gz nne alndnda radyasyonun duran

dalga (kararl dalga) eklinde olmas gerektii bilinmektedir. Ayn zamanda bu

kararl dalgalar ancak belirli bir dalga boylarna sahip olabilirler (belirlifarkl frakanslar)

Belirli bir scaklkta (T=sbt) ile +dfrekanslar aralnda birim hacimdekienerji miktar

ile verilebilir.

hacim

)erji/dalgasayisi)(en(dalga)( dT

dc

VdN

2

3

8)(:sayisidalga

Dalgalarn farkl sayda

olabilecekleri dikkate alnmtr Enerjinin istenen her deerde olabilecei n grlmtr

enerji/dalga: = kT

(enerjinin e blm kanunu kullanldnda )

d

c

dT 3

28)(


18/43

T() ile verilebilir mi ?

= c/olduu bilindiine gre diferansiyeli d = (c/2)d

e eittir. Bylece frekans ve dalga boyu arasnda geiyaplabilir:

T()d = T()d

buradan T() T()d/d T()c/2

elde edilir.Frekansa bal olarak verilen enerji younluu dalga boyuna

balanacak olursa:

dc

c

cd

c

cd

d

c

d

T

T

223

2

23

2

3

2

88)(

8)(


19/43

Teori uzun dalga boylarnda iyi sonular verirken ksa dalga boylarndadeney sonularndan olduka sapmaktadr.

Hatta bu olaya ultraviyole felaketi (Ultraviolet Catastrophe) adverilmitir. nk eer teori doru olsayd. Oda scaklnda soukcisimler bile grnr ve ultraviyole blgede ma yapacaklard. Yanihepimiz kzartmaolacaktk !!

dkTdu4

8)( 4

1

4)(

8

kT


20/43

PlanckKanunu

Energies are limited to discrete value

Quantization of energy

Plancks distribution

At high frequencies approaches the Rayleigh-Jeanslaw

The Plancks distribution also follows Stefan-Boltzmanns Las

Max Planck

,...2,1,0, nnhE

ddE)1(

8/5

kThc

e

hc

kThc

kThce kThc

1....)1()1( /


21/43

blackbody radiation quantum mechanical

average energy per standing wave:

1

/ kThe

h

M. Planck (1900):1

8)(

/

3

3

kThe

d

c

hdu

(fit to the data)

Js10626.6

34h

Plancks constant

energy of a single oscillator: ,...4,3,2,1 nhn

quantization of energy!

Note: both Stefans Law &


22/43

Planck radiation law

blackbody radiation, photons (bosons) in a cubic cavity:standing wave condition in all three directions x,yand z

d

ec

hfGhdu

dc

dG

d

c

Ldg

Lnnn

Ln

kTh

zyx

zyx

/3

3

2

3

2

3

3

2222

,,

18)()()(

8)(

8)(

2

...4,3,2,1,02

(number of standing waves)

(standing waves in cubic cavity)

(density of standing waves)

Note: both Stefans Law &

Wiens Displacement law

Can Be derived from the

Plank formula


23/43

Finding the Blackbody Peak From the Planck radiation formula

the evaluation of the maximum yields Wien's displacement law To find the peak of theblackbody radiation curve, we take the derivative:

Simplifying gives the maximum condition:
http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/wien.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/wien.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html


24/43

Cavity Modes

A mode for an electromagnetic wave in a cavity must satisfy thecondition of zero electric field at the wall. If the mode is of shorterwavelength, there are more ways you can fit it into the cavity to meet

that condition. Careful analysis by Rayleigh and Jeans showed thatthe number of modes was proportional to the frequency squared.


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Discovery of Cosmic Background Radiation In 1965 Arno A. Penzias and Robert W. Wilson of Bell Laboratories

were testing a sensitive horn antenna which was designed for

detecting low levels of microwave radiation. They discovered a lowlevel of microwave background "noise", like the low level of electricalnoise which might produce "snow" on a television screen. Afterunsuccessful attempts to eliminate it, they pointed their antenna toanother part of the sky to check whether the "noise" was comingfrom space, and got the same kind of signal. Being persuaded thatthe noise was in their instrument, they took other, more

sophisticated steps to eliminate the noise, such as cooling theirdetector to low temperatures. Finding no explanations for the origin of the noise, they finally

concluded that it was indeed coming from space, but that it was thesame from all directions. It was a distribution of microwave radiationwhich matched a blackbody curve for a radiator at about 2.7 Kelvins.

After all their efforts to eliminate the "noise" signal, they found that agroup at Princeton had predicted that there would be a residualmicrowave background radiation left over from the Big Bang andwere planning an experiment to try to detect it. Penzias and Wilsonwere awarded the Nobel Prize in 1978 for their discovery.
http://hyperphysics.phy-astr.gsu.edu/hbase/bkg3k.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/bkg3k.html


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27/43

Thermometry

Planck law gives the radiancy as a function ofTand

Resistance is adjusted until filament is invisible against source background

IfT1 is taken as a reference, T2 can be determined

1)/exp(

1)/exp(

2

1

kThc

kThcR

For monochromatic radiation of wavelength , the ratio of spectral intensities

is given by


28/43

Heat Capacities

Dulong Petits Law

The molar heat capacities of monoatomic solids are the same , close

to 25 J/mol. K

Can be justified using classical mechanics

Mean energy of an atom oscillates about its mean position of solid iskT

Unfortunately, at low T the value

approaches to zero

RTkTNU Am 33

KJ/mol9.243

R

T

UC

V

mv


29/43

Einstein and Debye Formula

Einstein used the hypothesis that energy of

oscillation is confined to discrete value

Debye later refined Einstein formula taking into

account that atoms are not oscillating at the same

frequency.

23RfCv

12/

2/

T

T

E

E

E

e

e

Tf

RfCv 3 dxe

exTf

T

x

x

D

D

/

0 2

43

)1(3


30/43

Einstein and Debyes Theory


31/43

overviewstatistical distributions general considerations

Maxwell-BoltzmannBose-EinsteinFermi-Dirac

Maxwell-Boltzmann statisticsMaxwell-Boltzmann distributionenergies in an ideal gas

equipartition of energyquantum statistics

fermions and bosonsBose-Einstein and Fermi-Dirac distribution

comparison of the three statistical distributions

applicationsPlanck radiation lawspecific heats of solidsfree electrons in a metaldying stars


32/43

specific heats of solids

question: internal energy of solids, specific heat

vc : energy needed to raise T of 1 kmol of solid by 1 K at constant V

internal energy resides in vibrations of the solids constituents

vibration of classical 1-D harmonic oscillator:

solid: 3 perpendicular modes of vibration 3 harmonic oscillators

total energy of a solid:

kT

RTkTNE 33 0

RT

Ec 3

V

v

(Dulong Petit)


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problem: deviation from Dulong-Petit!


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solution by Einstein: use average energy of a harmonic oscillator

1/

kTh

e

h

Ee

NhN

kTh

1

33

/

for N oscillators and 3 dimensions:

2/

/2

V

v)1(

3

kTh

kTh

e

e

kT

hR

T

Ec


35/43


refinement by Debye: elastic standing waves of the whole body

quantum of acoustic energy phonons

number of possible standing waves in a body equals 3N

phonons are bosons

BE statistics lead to good agreement with experimentalspecific heats


36/43









37/43

free electrons in a metal

about 1 free electron/atom in a metal3 additional degrees of freedom

specific heat should be

RRRRT

E

c 32

9

2

3

2

6

Vv

why do the electrons not contribute to the specific heat?

dhmdg 3

2/3V28

)(

(number of electron states)


38/43


deh

mdn

kTF

1

1V28)(

/)(3

2/3

(electron energy distribution)

V

V8

32

3/22

NNmhF

: electron density

Fermi energy


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T=0

T>>0

EF


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n

n


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only electrons close close to F contribute to cVelectrons more than kT from F cannot be excitedstates above the low lying electrons are already filled

RkT

c

F

e

2

2

v

low T: cve becomes important, since cDebye scales with T3

high T: cve continues to rise when cDebye already reached 3R


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