Determination of Planck

8/10/2019 Determination of Planck

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De te rmi na ti on of Pl an ck s constant and photoelectriceffectC. PolisseniFebruary 5, 2012AbstractW e r e p o r t t h e d e t e r m i n a t i o n o f t h e P l a n c k s c o n s t a n t t h r o u g h t h e p h o t o e l e c t r i ceffec t by the Millikansmethod [5]. I-V plots have beenbuilt for four different different

wavelengthsof the photons hitting the potassium Leyboldphotocell. By using different methods,the cutoff v o l t a g e h a s b e e n e s t i m a t e d a n d h e n c e t h e Plancksconstant,h= 6.629950.28043210

34m2k g / s . The non-ideal behaviour of thephotocell has been investigated by measuringits quantumefficiency whose value is approx-imately 0.1% for blue light at 4367nm1 I n t r o d u c t i o nI n 1 8 8 7 E . H e r t z o b s e r v e d t h a t e l e c t r o n s a r e e m i t t e d b y a p a r t i c u l a r m a t e r i a l a s acon se - que nce o f the abso rp t ion o f e le c t r oma gne t i c rad ia tion . Thi s phenomenon know aspho to -e l e c t r i c effect p layed a cruc ia l ro le in the de-veloment of the theory of quantum

m ec ha ni cs a n d t h e d e t e r m i n a t i o n o f P l a n c k s c o n s t a n t . I n 1 9 0 2 P . L e n a r d o b s e r v e dt h i s effec t wi th the appa ra t s shown be low [1 ] . Through th i s expe r imen t Lena rd made thefollowing obser-vations, that cannot be explained by classicalphysics. No dalay was observedbet ween the l i g h t c o m i n g t o t h e c a t h o d e a n d t h e g e n e r a - t i o n o f c u r r e n t . I nt h e c l a s s i c a lp i c t u r e h o w- e v e r , t h e c a t h o d e w a r m s u p b e f o r e t h e e m i s - si on ofelectrons and this implies also a mini-mum light intensity, while this is not the casein Lenardsexper iment. Moreo ver, the stop -p i n g p o t e n t i a l h a s b e e n f o u n d t o b e a f u n c - t i o no f t h e p h o to n f r eq u e nc y o n l y a n d n o t a F i g u r e 1 : L a y o u t o f t h ea p p a r a t u s u s e d b y L e n a r d . L i g h t a b s o r p t i o n i n t h e c a t h o d e a l - l ows el ec tr ons totravel to the anode generat-ing a current that it is measured and dependson the applied voltageV.funct ion of the the l ight in tens i ty , whi le in the c lass ica l p ic ture a low intens i ty wouldnot m a k e t h e c a t h o d e h o t e n o u g h t o e m i t e l e c - tr ons. T he sto ppi ng po te ntia l, a lsoknown asc u t o ff v o l t a g e (

V0obeys the fo l lowing re la - t ion :eV

0=h

(1)PlottingeV

0versus

for diffe rent mater ia ls in th e c a th ode l ea ds to a d i ffe r e n t i n t e r c e p t


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b u t t h e s l o p e o f t h e l i n e i s c o n s t a n t a n d i t i s k n o wn a s P l a n c k s c o n s t a n th=6

.631034

Js.2 T h e o r yThe explanat ion of the non-c lass ica l resul tsobta ined by Lenard in h is experiment gaveA.Ens te in the Nobel Pr ize and lead to the dis -covery of the corpuscolar na ture of l ight ,i.e.1

t h e d i s c o v e r y o f p h o t o n s . L i g h t i n f a c t t r a v - e l s i n q u a n t a o f e n e r g yh

a n d i f t h i s e n e r g y i s s ufficient i t can immediately excite an elec-tr on an d pr od uc ecu rr ent . Th e mi ni mu m ex -c i ta t ion energy is the work function, o f t h e o r d e r o f f e w e V , w h i l e t h e m a x i m u m k i n e t i c e n e r g y o f t h e e m i t t e d e l e c t r o n sfo ll ow s fr om energy conservation in Fig. 2:12mev2max=h

=eV

0( 2 ) A p h o t o n o f e n e r g yh

ca n ex ci te an e l ec -Figure 2: E lec tron photoemiss ion in the ca th-ode . Th e hi ghe s t

occ up i ed s t a t e i s th e f e r mi level,EF


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, and the energy required to excite anelectron from there to outside the potential isthe workfunct ion. t ron in one of the occupied s ta tes inc ludingthe Fermi level , wi th the condi t ionthath

isbigger than the work function

. H e r e a f r e e electron gas model has been assumed (see be-low), with the electrons confinedin apotentialwe ll of de pthV

.T he wo rk f un ct io n p la ys a n i mp or ta nt r ol ei n d e t e r m i n i n g t h e p r o p e r t i e s o f am a t e r i a l w i th a d e fi n e d c o n d u c t i o n b a n d a n d t h e c u t off v o l t a g e . I n t h i s a n a l y s i s ,t h e c o n d u c t i o n b a n d o f t h e c a t h o d e w a s a p p r o x i m a t e d t oa f r e e e l e c t r o n g a s . T h e L e y b o l d p h o t o c e l li s i n f a c t c o n s t r u c t e d w i t h a p o t a s s i u m p h o - t o e mi t te r ( c at ho de ) a nd t h e o u te re l e c t r o n s i n t h e c o n d u c t i o n b a n d a r e n o t s i g n i fi c a n t l y p e r t u r b e d b y t h e n u c l e i i n t h i sm a t e r i a l [ 2 ] . T h e d e n s i t y o f s t a t e s o f a 3 d i m e n s i o n a l f r e e e le ct ro n g as i s

D(

E

) =L322h3(2m)32

E

(3)where E denotes the ener gy andL3the volumeof the system, assuming quadratic symmetry.It is po ss ib le th er ef or e to de ri vethe numberof avaiable s tates as

N

=

EF

V

D(

E

)dE

=A(

E

32F

(


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V

U

)32) ( 4 ) w h e r e A i s a c o n s t a n t a n d V i s t h e d e p t h o f the potential and in

the pa r t i cu la r expe r imen the re desc r ibed i s the app l i ed vo l t age to theca thode -anodesystem. Since the current isproportional to the number of avaiable s tates[2], we

obtain

the following expressionI

V

32(5)On the other hand, if we consider more factorsthat can modify the photo current, such asbyN . M o r t o n a n d J . A b r a h a m [ 3 ] , w e fi n d t h e current to be proportional toV

2.The quantum efficiency of the Leybold pho-tocell has also been studied in the experiment.It isdefined a s the ra tio between the p roducedcur re n t and the pho ton s emi t t e d h i t t in g

thecathode, or Q.E.Q.E. =

Ele ct ron s/ Sec ond Ph oton /Se co nd(6)or in mathematical ter ms,Q.E. =IcheP(7)w h e r e I i s t h e c u r r e n t , c t h e s p e e d o f l i g h t i n v a c u u m , h t h e P l a n c k s c o n s t a n t , et h e e l e c - t r o n s c h a r g e , P t h e p o w e r a n dthe wave-lenght of light [4].3 M e t h o d

T h e l a y o u t o f t h e e x p e r i m e n t c a n b e f o u n d i n F i g . 3 . P h o t o n s w e r ee m i t t e d b y a m e r - c u r y l a mp th a t h ad s p e c i fi c e mi s si o n w a ve - l e n g t h s( 1 ) . L i g h t w a s t h e n c o l l e c t e d b y a l e n s w it h me a su r ed f oc al le ng t h o f a pp r ox i -m a t e l y 2 0 c m ( 2 ) a n d t h e n a s p e c i fi c w a v e - l e n g t h w a s s e l e ct e d u si n gi n t e r f e r e n c e fi l t e r s 2


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Figure 3: Setup of the experiment for measur-ing the current produced by the movementof photons due to their excitation.(3). Four filters we re ava iable and the t rans -m i s s i o n p e a ki s g a u s s i a n d i s t r i b u t e d . T h e b a nd wi d th F WH M a n d t he t ra n sm is s i o n c o -effi c i e n t s d e p e n d o n t h e f r e q u a n c y a s i n t h e t ab le b el ow. .( n m ) F W H M ( n m ) T r a n s . ( % ) V i o l e t 40 5 9 . 2 4 0 B l u e 4 36 7 . 4 4 5 G r e e n 5

4 6 8 . 8 5 0 Y e l l o w 57 8 9 . 8 5 0 A f t e r p a s s i n g t h r o u g h a i n t e r f e r e n c efi l t e r , light was collected by another lens with10cmf o c a l l e n g t h ( 4 ) a n d t h e p h o t o n s w e r e c o l - l i ma t ed i n t o t he L ey bo ldp h o t o c e l l i n o r d e r t o e x c i t e e l e c t r o n s f r o m t h e c a t h o d e ( e m i t -t e r ) ( 5 ) . I n a h i g h v a c u u m , e l e c t r o n s w o u l d tr avel fr om t he p ot assi um e mi tt er to t hepla t-i n u m w i r e l o o p c o l l e c t o r ( a n o d e ) a n d h e n c e p r o d u c e a c u r r e n t t h a t w a sm e a s u r e d b y a Kei th ley P icoammeter Mode l 485 (7) . A var i-ab le vo l t age sou rce (6 ) wasal so ad op te d ac -cording to vary the vol tage across the appara-tu s an d mea sur e th e c ut offv o l t a g eV

0, pointa t w h i c h n o c u r r e n t w o u l d b e a b l e t o p a s s t h r ou gh th e s ys te m . T he

p i c o a m m e t e r p r e - s e n t e d b i g fl u c t u a t i o n s s i n c e t h e l o w c u r r e n t w a s o f t h e o r d e r o f


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n A mp a n d t hi s wa s t ak e n into account when e stimating t he error.To demonstrate that the cut offpotentialV

0is independent on the light intensity and it is afunction only of the photonswavelength, neu-tral densityfilterswere adopted. These filterscorrespondedto a transmission percentage of 30% , 50 %,

70% and 90% and wer e p la ced in f ron t o f the mercuryla mp. M o r e o v e r , i n o r d e r t o m e a s u r e t h e p o w e r o f t h e e m i t t e d l i g h t a n d h e n c ees t ima te the quantum efficiency of the Leybold photocell, apowermeter was adopted. Thismethod how-ev e r d i d n t s e e m t o g i v e a n e s t i m a t e o f t h e c or re c t effi c i e n c y a s i tw i l l b e l a t e r sho wn i n the resul ts .4 R e s u l t s a n d A n a l y s i s4 . 1 P l a n c k C o n s t a n tE s t i m a t i n g P l a n c k s c o n s t a n t t h r o u g h t h e p ho to e l ec tr i c effe c t i m p l i e s t h e c o n st r u c t i o n o f a p l o t o f t h e c u t o ff v o l t a g eV

0vs th e p ho -ton frequency

. T h e c u r v e o b t a i n e d s h o u l d b e a s t r a i g h t l i n e a s i n E q . 1 , w i t h s l o p e p r o - p o r t i o n a lto the Plancksconstanth. I n o r - d er t o m e a s u r e t h e c u t off p o t e n t i a l v a r i o u r p l o t s h a v e b e e n g e n e r a t e dm ea s ur i ng , a s b yL e n a r d [ 1 ] , t h e c u r r e n t a t a f u n c t i o n o f t h e a p pl i ed v ol t ag eb e t w e e n c a t h o d e a n d a n o d e . A s i t i s p o s s i b l e t o n o t e i n F i g . 4 h o w e v e r , thebehavio ur of the cur rent is sl ight ly differ-ent from the prediction. In fact, as thevol ta ge b e c o m e s n e g a t i v e , t h e c u r r e n t d o e s n o t g o t o z e r o , b u t i t r e a c h e sa ne ga t i ve va lu e . Gi ve nthe non-ideal behaviour of the photocell , var- iou s met ho ds hav eb e e n a d o p t e d t o m e a s u r e th e cut offvoltage.Figure 4: Current as a function of t heappliedvoltage for various mercury lines. An anoma-lous behaviour was observed since thecurr en tdoes not drop to zero but reaches nevative val-ues.3


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Determination of Planck