{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Da Silva, Angela Jane","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-09-09T22:31:12Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1988","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Applied Science - MASc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"In recent years, there has been increasing interest in the idea of using a superheated superconducting colloid (SSC) as a detector for neutrinos and dark matter candidates. The primary objective of this work has been to investigate the basic properties of an SSC, consisting of 7 \u03bcm radius tin grains imbedded in epoxy, using a pumped \u2074He cryostat with a low vibrational noise RF-S\uff31UID readout system. The superheating-supercooling hysteresis curves of the colloid have been measured in applied magnetic fields ranging from 3.1 x 10\u207b\u2074T to 1.4 x 10\u207b\u00b2T. The superconducting to normal phase transition in individual grains inside the colloid has been observed and the measured signal size is in reasonable agreement with the calculated values. Finally, it was demonstrated that the colloid could withstand up to 20Mrad of [omitted]-radiation without incurring a significant change in its superconducting-normal phase transition.\r\nA new type of sample, consisting of a planar array of 1 \u03bcm thick metal squares deposited on a mylar substrate, was developed. Both indium and tin were used as a. fabrication material. The characteristics of such samples were investigated, again using the pumped \u2074He cryostat. The full Meissner effect was only observed for applied magnetic fields less than 5 x 10\u207b\u2075T. For higher applied fields, the samples behaved like type-[omitted] superconductors in the mixed state regime, exhibiting flux penetration and trapping.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/28374?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"i I N V E S T I G A T I O N O F A S U P E R H E A T E D S U P E R C O N D U C T I N G C O L L O I D B y Angela Jane D a Silva B . A . Sc., University of Br i t i sh Columbia , 1986 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A October 1988 \u00a9 Angela Jane D a Silva, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P\u00abHSK^> The University of British Columbia Vancouver, Canada Date OCT- f3> \/ 88 DE-6 (2\/88) A b s t r a c t In recent years, there has been increas ing interest in the idea of us ing a superheated s u p e r c o n d u c t i n g c o l l o i d ( S S C ) as a detector for neut r inos and dark mat te r candidates . T h e p r i m a r y ob jec t ive of th i s work has been t o invest igate the basic propert ies of an S S C , cons i s t ing of 7 um r ad ius t i n grains i m b e d d e d in epoxy, us ing a p u m p e d 4 H e cryostat w i t h a low v i b r a t i o n a l noise R F - S O U I D readout sys tem. T h e superheat ing-supercool ing hysteresis curves of the c o l l o i d have been measured in app l ied magnet i c fields ranging f r o m 3.1 x 1 0 \" 4 T to 1.4 x 1 0 ~ 2 T . T h e s u p e r c o n d u c t i n g to n o r m a l phase t r ans i t i on in i n d i v i d u a l grains ins ide the co l lo id has been observed and the measured s ignal size is in reasonable agreement w i t h the ca lcu la ted values. F i n a l l y , i t was demons t ra ted that the c o l l o i d could w i t h s t a n d up to 2 0 M r a d of i - r a d i a t i o n w i t h o u t i n c u r r i n g a significant change in its supeuondu< t i n g - n o r m a l phase t r a n s i t i o n . A new type of s ample , cons i s t ing of a p l a n a r a r ray of 1 um th ick meta l squares depos i ted on a m y l a r substrate , was deve loped . B o t h i n d i u m and t i n were used as a. f a b r i c a t i o n m a t e r i a l . T h e character i s t ics of such samples were inves t igated , again using the p u m p e d 4 H e cryos ta t . T h e f u l l Mei s sner effect was o n l y observed for appl ied magnet ic fields less t h a n 5 > ] 0 \" ' T . F o r higher a p p l i e d fields, the samples behaved l ike type-U superconductor s in the m i x e d state regime, e x h i b i t i n g flux p e n e t r a t i o n and t r a p p i n g . n T a b l e of C o n t e n t s A b s t r a c t i i L i s t of Tab le s v L i s t of F i g u r e s v i A c k n o w l e d g e m e n t v i i i 1 I n t r o d u c t i o n 1 1.1 H i s t o r i c a l P e r s p e c t i v e 1 1.2 Superhea ted S u p e r c o n d u c t i n g C o l l o i d 2 1.3 Present W o r k 4 1.4 Thes i s O u t l i n e 4 2 T h e o r y 6 2.1 E n e r g y D e p o s i t i o n 6 2.2 S i g n a l Size 11 3 A p p a r a t u s 3.1 C r y o s t a t IS 3.2 1 K e l v i n Pot 21 3.3 T h e r m o m e t r y 24 3.4 S u p e r c o n d u c t i n g M a g n e t s 25 3.5 S Q U I D R e a d o u t S y s t e m 26 i i i 3.6 T e m p e r a t u r e C o n t r o l 31 3.7 D a t a A q u i s i t i o n 32 4 E x p e r i m e n t s - C o l l o i d S a m p l e 33 4.1 S a m p l e P r e p a r a t i o n 33 4.2 C o o l - D o w n P r o c e d u r e 37 4.3 E x p e r i m e n t a l Resu l t s 38 4.3.1 Hys teres i s Curves 38 4.3.2 R a d i a t i o n Test 43 4.3.3 R a d i a t i o n Hardness 47 5 E x p e r i m e n t s - P l a n a r A r r a y s 49 5.1 F i l m T h i c k n e s s 49 5.2 S a m p l e P r e p a r a t i o n 51 5.3 I n d i u m A r r a y - E x p e r i m e n t a l Resu l t s 53 5.3.1 Hysteres i s Curves 53 5.3.2 T h e r m a l l y A c t i v a t e d F l u x m o t i o n 55 5.3.3 D i s cus s ion 55 5.4 T i n array - E x p e r i m e n t a l Resu l t s 56 6 C o n c l u s i o n s 59 A F L U X P r o g r a m L i s t i n g 61 B H Y S T E R E S I S P r o g r a m L i s t i n g 64 B i b l i o g r a p h y 67 iv i Lis t of Tab les 5.1 T a b l e of some relevant p h y s i c a l propert ies of three m e t a l l i c superconduc-tors 50 v L i s t o f F i g u r e s 1.1 P h a s e d i a g r a m for a type-1 s u p e r c o n d u c t o r . So l id l ine : superheated t ran-s i t i o n ; dashed l ine : t h e r m o d y n a m i c t r a n s i t i o n ; cha in d o t t e d l ine : super-cooled t r a n s i t i o n 2.2 Difference in the free energy between the n o r m a l and s u p e r c o n d u c t i n g s tate as a f u n c t i o n of the s u p e r c o n d u c t i n g order parameter i l l u s t r a t i n g the energy bar r ie r p r o t e c t i n g the superheated s u p e r c o n d u c t i n g state. . . 10 2.3 C o - o r d i n a t e sys tem for s ignal size c a l c u l a t i o n 12 2.4 F l u x t r ans former 15 2.5 E x p e c t e d s ignal size due to a 7 p.m rad ius g r a i n , at a pos i t ion ( R , z ) . f l i p p i n g in an app l i ed field of 0 .01T . P a r a m e t e r R is the rad ia l p o s i t i o n of the gra in w i t h respect to the centre of the coi l and z is the ver t i ca l pos i t ion of the g r a i n w i t h respect to the b o t t o m of the c o i l . So l id l ine : R = 1 .0mm; c h a i n d a s h e d l ine : R = 1 .2mm; c h a i n d o t t e d l ine : R = 1 .4mm 17 3.6 S c h e m a t i c v iew of the cryosta t 19 3.7 L o w e r c ryos ta t assembly 22 3.8 Set-up for flow i m p e d a n c e tests 23 3.9 S Q U I D m o u n t i n g box . (a) top p la te ( ins ide v iew) and m o u n t i n g bracket ; (b) S Q U I D c o m p a r t m e n t 27 3.10 P o w e r spec t r a of S Q U I D s ignal for an app l i ed field of 0.03 T . (a) is w i t h o u t m o u n t i n g box i n s t a l l e d ; (b) is w i t h m o u n t i n g box in s t a l l ed 29 v i 4.11 G r a i n size d i s t r i b u t i o n 34 4.12 Hysteres i s curves for co l lo id sample , (a) is in an app l i ed field of 9.3 x 1 0 - 4 T a n d (b) is in an a p p l i e d field of 9.3 x 1 C T 3 T 39 4.13 B - T phase d i a g r a m of the co l lo id sample . C i r c l e s : superhea t ing t r a n s i t i o n ; squares: supercoo l ing t r a n s i t i o n 41 4.14 W i d t h of the superhea t ing phase t r a n s i t i o n of the co l lo id sample as a f u n c t i o n of app l i ed field 42 4.15 T o t a l s ignal change for the s u p e r c o n d u c t i n g to n o r m a l t r a n s i t i o n in the c o l l o i d sample as a func t ion of app l ied field 44 4.16 (a) S Q U I D s ignal showing single gra in flips f r o m the superheated co l lo id under i r r a d i a t i o n of 140keV 7-ra .ys . T h e a p p l i e d field is 0 .014T and the t e m p e r a t u r e is 3 .09K . (b) P o s i t i o n in the superhea t ing t r a n s i t i o n where the co l lo id was prepared for measurement shown in (a) 46 4.17 S u p e r h e a t i n g transi t Ion curve for the co l lo id sample in an app l i ed field of 9 . 3 x l ( T 3 T . C i r c l e : before 7 - r a d i a t i o n ; square: after 8.3 M r a d 7 - r a d i a t i o n ; t r i a n g l e : after 20 M r a d 7 - r a d i a t i o n 48 5.18 Hysteres i s curves for the i n d i u m array, (a) is i n the earth's magnet i c field a n d (b) is in an a p p l i e d field of 1.6 x 1 0 ~ 3 T . C i rc l e s : t e m p e r a t u r e increas ing ( superhea t ing t r a n s i t i o n ) ; squares: t e m p e r a t u r e decreasing ( supercoo l ing t r a n s i t i o n ) 54 5.19 Hysteres i s curves for t i n array, (a) is i n the earth 's magnet ic field and (b) is in an a p p l i e d field of 3.8 x 1 0 _ J T . C i rc l e s : t e m p e r a t u r e increas ing ( superhea t ing t r a n s i t i o n ) ; squares: t e m p e r a t u r e decreas ing ( supercoo l ing t r a n s i t i o n ) 57 vn A c k n o w l e d g e m e n t I w o u l d l ike to t h a n k m y superv i sor . D r . B r i a n T u r r e l l , for his suppor t a n d encouragement t h r o u g h o u t this w o r k . H e was a lways avai lable to discuss new ideas and results . A l s o . I w o u l d l ike to express m y apprec i a t ion to M a r k Le G r o s , for his invaluable ass i s tance w i t h m a n y aspects of the design and c o n s t r u c t i o n of the cryos ta t , as well as hi s ass istance in p e r f o r m i n g the e x p e r i m e n t a l measurements . In a d d i t i o n , I w o u l d l ike to take this o p p u r t u n i t y to t h a n k D r . A n d r z e j K o t l i c k i for t a k i n g the t i m e to read m y thesis and for offering several suggestions to improve i t , the t e c h n i c a l staff of the depar tment of phys ics who were always w i l l i n g to offer advice and ass is tance. D r . A n d r z e j D r u k i e r , whose un ique p o i n t of view p r o d u c e d m a n y interest ing d i scuss ions , and the Science C o u n c i l of B r i t i s h C o l u m b i a , for the i r financial assistance. F i n a l l y , I w o u l d l ike to t h a n k m y husband for his constant support and wil l ingness to discuss a l l ideas. v m C h a p t e r 1 In t roduct ion 1.1 H i s t o r i c a l Perspec t ive T w e n t y years ago, the idea of u s ing superheated s u p e r c o n d u c t i n g granules for par t ic le de tec t ion was put f o r t h by a g roup f rom O r s a y T h e y were the first to report the o b s e r v a t i o n of a \/3-ray s t imula ted t r ans i t ion between the superheated superconduc t ing state a n d the n o r m a l state. In the i r exper iment , it was demons t ra ted tha t 1 um radius m e r c u r y gra ins , doped w i t h rad ioac t ive g o l d , were sensi t ive to the \/ i-rays e m i t t e d f rom the g o l d . W i t h ev idence for the i rrevers ible s u p e r c o n d u c t i n g to n o r m a l phase t r ans i t i on u n d e r e l ementary pa r t i c l e interact ions , they suggested that a c o l l o i d of such part ic les c o u l d po s s ib ly be used as a nuclear par t ic le detector . S ince that t i m e , m a n y ideas a n d exper iments i n v o l v i n g superheated superconduc t ing granules have been repor ted . In 1972, D r u k i e r and V a l e t t e '\"^ suggested that a co l lo id of superheated s u p e r c o n d u c t i n g grains (SSC') cou ld be used as a detector for charged par t i -cles. M o t i v a t e d by the poss ible app l i ca t ion to charged par t ic le d e t e c t i o n . B l o t et. a l . p e r f o r m e d a series of exper iments us ing e lectron beams to induce the superconduct ing to n o r m a l t r a n s i t i o n in meta l l i c spheres. T h e fo l lowing year , D r u k i e r et. a l . ^ suggested tha t a SSC' cou ld also be used as a t r ans i t i on r a d i a t i o n detector . In 1978, B e h a r et. a l . deve loped a \" g a m m a c a m e r a \" u s ing a co l lec t ion of superheated s u p e r c o n d u c t i n g granules and a square array of U-shaped p i ck-up coi ls . T h a t same year, the SSC' was proposed as a poss ib le n e u t r o n detector M o r e recently , the SSC' has been suggested as a neut ra l 1 Chapter 1. Introduction 2 current n e u t r i n o detector a m a g n e t i c m o n o p o l e detector a n d a co ld dark m a t t e r de tec tor & I 1 0 l . 1.2 S u p e r h e a t e d S u p e r c o n d u c t i n g C o l l o i d T h e superheated s u p e r c o n d u c t i n g co l lo id ( S S C ) consists of s u p e r c o n d u c t i n g grains , a few m i c r o n s i n size, i m b e d d e d in a su i tab le d ie lectr ic m a t e r i a l . If the grains are made of t y p e -1 s u p e r c o n d u c t o r , they can exh ib i t metas tab le states such as superhea t ing a n d s u p e r c o o l i n g . Such states were observed e x p e r i m e n t a l l y over t h i r t y years ago in t h i n s u p e r c o n d u c t i n g rods and la ter , in col lect ions of s u p e r c o n d u c t i n g spheres S u p e r c o o l i n g occurs w h e n a s u p e r c o n d u c t i n g sample remains n o r m a l for a t emp era ture be low T,-. the t h e r m o d y n a m i c c r i t i c a l t e m p e r a t u r e , but above T$c. the supercooled t r a n -s i t ion t e m p e r a t u r e . S i m i l a r l y , superheat ing occurs w h e n the s u p e r c o n d u c t i n g state per-sists for a t e m p e r a t u r e above Tr- but be low T$H\u2022 the superheated t r a n s i t i o n t empera ture . T h e phase d i a g r a m is shown in F igure 1.1. For the SSC' to act as a detector , the co l lo id is p laced in a u n i f o r m magnet ic field and the grains are m a i n t a i n e d just be low the ir superheated s u p e r c o n d u c t i n g to n o r m a l t r a n s i t i o n t empera ture ( p o s i t i o n X in the phase d i a g r a m i l lu s t r a ted in F i g u r e 1.1). S ince the gra ins are s u p e r c o n d u c t i n g , they expel the magnet ic flux v i a the Mei s sner effect. W h e n an i n c o m i n g pa r t i c l e scatters e la s t i ca l ly off of a nucleus in one of the grains , i t i m p a r t s a smal l recoi l energy to the nucleus it scatters f r o m . For low enough recoi l energies 2 0 e V for t i n ) , the nucleus w i l l be b o u n d i n its l a t t i ce site and the energy w i l l be d i s s ipated as heat . For larger recoi l energies, part of the energy m a y go i n t o p r o d u c i n g permanent defects i n the l a t t i ce ra ther t h a n p r o d u c i n g heat . T h e majority- of the energy, however , w i l l cont inue to be d i s s ipated as heat ' '1. If th i s s m a l l a m o u n t of heat is sufficient, it can break C o o p e r pairs , r e su l t ing i n the de s t ruc t ion of the s u p e r c o n d u c t i n g Chapter 1. Introduction 3 F i g u r e 1.1: P h a s e d i a g r a m for a type-1 s u p e r c o n d u c t o r . S o l i d l ine : superheated t rans i -t i o n : dashed l ine : t h e r m o d y n a m i c t r a n s i t i o n ; c h a i n d o t t e d l i n e : supercooled t r a n s i t i o n . Chapter 1. Introduction 4 state at t h e surface of the g ra in . T h i s leads t o the decay of the screening currents at the surface a n d the p r o p o g a t i o n of the n o r m a l zone in to the inter ior of the g ra in , a l l owing the m a g n e t i c flux t o penetra te in to the g r a i n . T h i s change in m a g n e t i c permeab i l i ty can be measured u s i n g a S Q U I D magnetometer . Since the g r a i n was i n a metastable superheated state, the change of state is p e r m a n e n t . In this way, a gra in \" f l i p \" can be used to detect an event . 1.3 Present Work T h e p r i m a r y interest of this thesis work has been to s tudy the basic propert ies of a co l lo id cons i s t ing of t i n grains i m b e d d e d in epoxy. In order to do th i s , it was necessary to cons t ruct a. p u m p e d 4 H e cryosta t w i t h an R F S Q U I D readout sys tem. In cons t ruc t ing the c ryos ta t . care was taken in t r y i n g to m i n i m i z e v i b r a t i o n a l noise. A c o l l o i d sample w7as prepared by m i x i n g 7 \/m? radius t in grains i n epoxy. T h e phase d i a g r a m of the c o l l o i d was then m a p p e d out . T h e sens i t iv i ty of i n d i v i d u a l grams to low energy -.-rays was d e m o n s t r a t e d . In a d d i t i o n , the r a d i a t i o n hardness of the col lo id has been d e m o n s t r a t e d . In an attempt, to improve the character i s t ics achieved w i t h the co l lo id , a new type of sample was deve loped . A pla.nar array of superconduc t ing squares wa.s produced using t h i n fi lms of t i n and i n d i u m depos i ted on to m y l a r and s t a n d a r d photo l i thography technology. T h e character i s t ic s of such samples were then s tud ied . 1.4 Thesis Outline In chapter 2. two i m p o r t a n t questions i n v o l v i n g the v i a b i l i t y of the co l lo id as a detector are discussed; how m u c h energy is required to flip a gra in and is a g r a i n flip detectable . In chapter 3, a de ta i l ed descr ip t ion of the a p p a r a t u s used for the exper iments is presented. Chapter 1. Introduction 5 C h a p t e r 4 discusses the p r e p a r t i o n of the co l lo id sample and the subsequent experiments p e r f o r m e d w i t h i t . C h a p t e r 5 discusses the p repara t ion of the p l a n a r a r ray samples a n d the subsequent e x p e r i m e n t s per formed w i t h t h e m . C h a p t e r 6 presents a s u m m a r y of resul t s and conclus ions . C h a p t e r 2 T h e o r y In order to d e t e r m i n e whether the co l lo id w o u l d be useful as a detector , two i m p o r t a n t factors must be cons idered . F i r s t l y , the energy necessary to induce a phase t r a n s i t i o n in a. g ra in must be c a l c u l a t e d . Secondly , it must be demons t ra ted that a gra in t r a n s i t i o n is. in fact , detectable . These two factors w i l l be discussed in de ta i l in this chapter . 2.1 E n e r g y D e p o s i t i o n T o ca l cu la te the energy depos i ted in the detector (g ra in ) , consider an elasitic. co l l i s ion between the i n c o m i n g par t i c l e and a t i n nucleus . T o e s t imate the recoi l energy depos i ted in the t in nucleus , a s imp le m o d e l a s suming tha t the nucleus is o r i g i n a l l y at rest is used. W i t h th i s m o d e l , and conservat ion of energy a n d m o m e n t u m , the recoi l energy is given where ni and v are the mass and i n i t i a l ve loc i ty ol the i n c o m i n g par t i c l e and M is the mass of the t i n nucleus . In order for the gra in to operate as a detector , this recoi l energy must be large enough to cause the s u p e r c o n d u c t i n g g r a i n to f l ip in to the n o r m a l state. In the \" u n i f o r m hea t ing m o d e l \" , th i s means that the energy depos i ted must raise the t e m p e r a t u r e of the gra in by an a m o u n t A T , where A T is the difference between the gra in t e m p e r a t u r e p r io r to the co l l i s ion and the t e m p e r a t u r e of the s u p e r c o n d u c t i n g to n o r m a l t r a n s i t i o n as shown in F i g u r e 1.1. bv (2.1) 6 Chapter 2. Theory 7 T o convert the depos i ted energy i n t o a t e m p e r a t u r e change, one must consider the heat capac i ty of t h e g ra in . N e a r the t r a n s i t i o n t e m p e r a t u r e , the heat capac i ty of the n o r m a l state is less t h a n that i n the s u p e r c o n d u c t i n g state. T h u s , if the n o r m a l state heat c a p a c i t y is u sed , the ac tua l t e m p e r a t u r e increase w i l l be overes t imated . T h e heat c a p a c i t y of a m e t a l is g iven b y Cv = -yT -r AT 3 (2.2) where 7 = U-D{iF)k% D((p) is the dens i ty of states at the F e r m i energy A 7 is the n u m b e r of a toms a n d 9 is the D e b y e t e m p e r a t u r e . T h e first t e r m is the c o n t r i b u t i o n due to the electrons w h i l e the second t e r m is the c o n t r i b u t i o n due to the phonons . For t i n , 7 - 1.78 x l O ^ J m o ] - 1 K ~ - and A - 2.43 x 1 0 ~ 4 J m o l - 1 K - 4 ' 1 5 J . For temperat ures of the order of 1 K . the first, t e r m dominates so ( C V ) s n - 1-8 x 1 0 - 3 T [ J m o l ^ K - 1 ] (2.3) F o r a t i n gra in of v o l u m e V\" [m 3 ] , the heat capac i ty is then CSn-9ra,r, ^ 1.8 X l O \" 3 ? 1 ) - 1 1 0 T V \" [ J K _ 1 ] (2.4) (2.5) w h e r e p = 7.28 x 1 0 f a g m - 3 is the dens i ty of t i n and MQ = 118.69 g m o l - 1 is i ts a tomic w e i g h t . So the t e m p e r a t u r e rise due to the c o l l i s i o n , a s suming a u n i f o r m h e a t i n g mode l , Chapter 2. Theory 8 is g iven by a rr, ETecoll Erecc,H = r ~~\" = 11 n r v ( ' C-'Sn \u2014 grain 1 lVl \\ Since every low tempera ture apparatus has finite t empera ture s t ab i l i ty , AT must be greater t h a n t h a t t empera tu re s t a b i l i t y in order to insure t h a t any gra in flip is due to p a r t i c l e in te rac t ions r a ther t h a n t e m p e r a t u r e f luctuat ions . If, for th i s reason, a temper-a ture j u m p of A T > 10 m K is r equ i red , then f r o m E q u a t i o n 2.6 one can see that if the reco i l energy is of the order of 1.6 x 1 0 _ l b J (1 k e V ) , a t i n g ra in of rad ius 3 pm cou ld be f l i pped at an o p e r a t i n g t empera ture of 1.5 K (obta inable in a s t andard p u m p e d l i q u i d 4 H e cryos ta t ) . For much lower recoi l energies, e i ther smaller grains a n d \/ o r lower operat-i n g t empera ture s are necessary in order for the deposited energy to be sufficient to cause a change of s tate in the g ra in . It shou ld be noted t h a t the u n i f o r m hea t ing mode l p r o b a b l y gives a conservative e s t imate of the deposi ted energy requi red since it assumes that the entire grain must be heated before the t r a n s i t i o n takes place ' ^ v T h a t is. it assumes that the propagat ion of heat is faster t h a n the propaga t ion of the n o r m a l state and the nue lea t ion of the phase t r a n s i t i o n a lways starts at the equator , where the magnet ic field is the strongest. It m a y be, however , that the n o r m a l zone created by the energy deposit m a y be sufficient to nucleate the t r ans i t i on before the heat spreads through the entire g ra in . T h i s is charac ter i s t i c of the meta s t ab i l i ty of the superheated state, where an energy barr ier protect s the s u p e r c o n d u c t i n g state at the surface of the g ra in . T h e energy barr ier can be seen by cons ider ing F, the difference in the free energy per u n i t surface between the n o r m a l and s u p e r c o n d u c t i n g states. U s i n g the G m z b u r g - L a n d a . u equat ions , F for a type-1 superconduc tor can be a p p r o x i m a t e d by F - = ^ + ^\/Mli{ 2_ _ f{0) + 1 f ( 0 ) 3 } ( 2 7 ) S T T \/ ( 0 ) 8TT ^ 3 J 3 ' M ; ; ' w here Chapter 2. Theory 9 H is the app l i ed magnet ic field HC is the t h e r m o d y n a m i c c r i t i c a l field A is the L o n d o n pene t ra t ion d e p t h \u00a3 is the coherence length a,nd \/ ( 0 ) is the s u p e r c o n d u c t i n g order parameter . C o n s i d e r i n g F as a funct ion of \/ ( 0 ) , F has \u2022 an abso lute m i n i m u m at \/ ( 0 ) = 0, cor re spond ing to the n o r m a l state \u2022 a l o c a l m a x i m u m at f(0) \u2014 fmax - y ^ r * where v = , \/ l \u2014 and HSH is the s u p e r h e a t i n g c r i t i c a l field \u2022 a l o c a l m i n i m u m at \/ ( 0 ) = \/ w \u201e = y^r co r re spond ing to the superheated state. So the superheated state is protected by an energy barr ier per u n i t surface given by ^ A F = F(fmax)- F ( \/ \u201e \u201e \u2022 \u201e ) 4 v ^ \/ : \/ \/ H%H\\ - \u2014 ( i - v 1 -x2)(^ + . \\ - v 1 ( 2 ' 8 ) and i l l u s t r a t e d in F i g u r e 2.2. T h u s , if the energy deposi t is sufficient to overcome this bar r i e r , i t can nucleate the s u p e r c o n d u c t i n g to n o r m a l t r a n s i t i o n before the heat spreads t h r o u g h o u t the ent i re g ra in . E v i d e n c e of such a loca l h e a t i n g m e c h a n i s m has, in fact , been observed by Gonza lez -Mes t re s a n d P e r r e t - C a l l i x ' ^ l in the i r exper iment s w i t h re la t ive ly large (4bum < q< < 63\/7??!) t in grains i r r a d i a t e d w i t h 5 . 5 M e V o-pa.rtic.les. However , as the par t i c l e size is r e d u c e d , the size of the n u c l e a t i o n centre approaches the size of the gra in a n d the two m o d e l s converge. F i g u r e 2.2: Dif ference in the free energy between the n o r m a l a n d s u p e r c o n d u c t i n g state as a f u n c t i o n of the s u p e r c o n d u c t i n g order pa rameter i l l u s t r a t i n g the energy barr ier p r o t e c t i n g the superheated s u p e r c o n d u c t i n g state . .Chapter 2. Theory 11 2 .2 S i g n a l S i z e It is c e r t a i n l y of interest to ca lcu la te the expected signal size due to a single gra in flip. It is necessary, in fact , to show tha t the s ignal due to a s ingle g ra in flip can indeed be detec ted . It is wel l k n o w n t h a t the d i p o l e field due to a. magnet i c m o m e n t , ??*?., is g iven by B = \u2014 V ( 7 7 l - - ) ( M K S ) (2.9) For j?? d i rec ted a long the k ax i s , th i s can be r e w r i t t e n in spher ica l co-ordinates as -< 77? _ COsd \u00a3 = \u2014 V ( \u2014 ) (2.10) 47T ?\"-- 77? B = r ( 2 c o s 0 r + sin0 0) (2.11) 4?T7^ where ??? = |?n|. For a s u p e r c o n d u c t i n g sphere of rad ius a in an app l i ed m a g n e t i c f ie ld , H,. the mag-netic m o m e n t of the sphere is g iven by ??\"? = -2irfj.oHca 3 (2.12) T h u s , for H, \u2014 H,k\\ as in F i g u r e 2.3, the d ipo le field due to the s u p e r c o n d u c t i n g sphere is B = --n\u201eHt \u2014 ( 2 c o s 0 r + sin9 9) (2.13) For a loop of rad ius R, centred a b o u t the or ig in in the x y - p l a n e , (ie. \u00b1 to He), the abso lute value of the flux go ing t h r o u g h the loop due to the d ipo le at the or ig in is equal to the absolute value of the f lux outs ide the loop i n the x y - p l a n e , w h i c h is g iven by * = \/ \/ B(6=-)-da .1^=0 Jr=R 2 = --u0Hea 3 d4> drr U-l(-k)-(k) (2.14) 2 Jo Jn r 6 Chapter 2. Theory 12 F i g u r e 2.3: C o - o r d i n a t e sys tem for s ignal size c a l c u l a t i o n . Chapter 2. Theory 13 s ince, in the x y - p l a n e , 0 = ^ and 0 = \u2014 k. E v a l u a t i n g gives o 3 ^ = Ku0He- (2.15) So the flux t h r o u g h the loop clue to a s u p e r c o n d u c t i n g sphere at the o r i g i n is given by 3 $(o,o) = -nu0He\u2014 (2.16) For a 5 um r ad ius g r a i n , in a 2 .5cm radius loop and an app l i ed field of 0 .01T, flipping f r o m the s u p e r c o n d u c t i n g state to the norma] state, A $ ( 0 , o ) = 1-57 x 1 0 \" 1 6 W b = 7.6 x 1 0 _ 2 * 0 (2.17) where g = 2.07 x 1 0 ~ 1 5 W b is the flux q u a n t a . S ince the re so lut ion of an R F S Q U I D is < iQ~- J,ty0, such a flux change cou ld be detected . Hence , the detect ion of single grain fl ips is. in p r i n c i p l e , poss ible . T o ca l cu la te the s ignal expected f rom a gra in ins ide the c o l l o i d , one must also consider gra ins not loca ted at the centre of the p i c k - u p loop . To do t h i s , consider V ( ^ f ^ ) in car tes ian co-ordinates . M a k i n g thi s change in co-ordinate sy s tem, V ( - ~ ) = ( x - + y\" + z2)-5\/2{-3xzi - Zyzj -f (x 2 + y 2 - 2z 2)k) (2.18) r-W i t h o u t loss of genera l i ty , one can consider a. g r a i n located at (0, , C] ) . T h e n , reca l l ing that the a p p l i e d field is in the k d i r e c t i o n and z = 0 i n the x y - p l a n e , ^(v..--!) = ~7^oHea 3\/ d<\\> I drr \u2014 (2.19) 2 .'o Jo {r- cos-
sQ, due to the flux,