R adiati on S topping P owe r , D am age C ascade s, D isplacem e nt and the D P A

L ear n in g O b ject ives

• Pre di c t st oppi ng pow e r of ra di a t i on a s func t i ons of m a t e ri a l , type , e ne r gy of ra di a t i on

• Co n cep t u al i ze r ad i at i o n d am ag e cas cad es , s t ag es , a nd e vol ut i on i n t i m e

• E s tim a te th e q u a n tita tiv e d isp la c e m e n t r a te s f r o m ra di a t i on, a nd de fi ne t he D P A

• T ra c k t he bui l dup of ra di a t i on poi nt de fe c t s a s func t i ons of t e m pe ra t ure , de fe c t c onc e nt ra t i on

Bu ild in g Up t o Rad iat io n

E ffe c ts

S ho r t & Y i p. C u rre n t O p in io n s in S o lid S t a t e M a t e r ia l S c ie n c e ( 2015)

… w e ’ l l be he r e n e x t w e e k!

Courtesy of Elsevier, Inc., http://www.sciencedirect.com . Used with permission.

Source: Short, M. , and S. Yip. " Materials A ging at the M esoscale: Kinetics of T hermal, S tress, R adiation A ctivations ." Current Opinion in Solid State and Materials Science 19, no. 4 (2015): 245-52.

22. 14 – Nu clear M ater ials Slid e 3

S t o p p in g P o w er

• M o r e en er g et i c p ar t i cl es do m ore da m a ge … to a poi nt

• … but how m uc h?

• C ha r ge vs. no c ha r ge ?

• W ha t a bout da m a ge vs. m ean f r ee p at h ?

h t tp: / / w w w .s r i m .o r g

Courtesy of James F. Ziegler. Used with permission.

St o p p i n g p o w e r o f p r o t o n s i n i r o n

Co u lo m b ic/Nu clear S t o p p in g P o w er

• St oppi ng P ow e r i s de fi ne d a s di f fe re nt i a l e ne r gy l oss a s a func t i on of e ne r gy:

𝑁 ∗ 𝑆

𝜕 𝐸

= −

𝜕𝑥

• Se pa ra bl e c om pone nt s due t o nuc l e a r (s c re e ne d nuc l e us C oul om bi c ), e l e c t roni c , a nd ra di a t i ve t e rm s:

𝑁 ∗ 𝑆

= −

𝑛𝑢𝑐 𝑙 .

−

𝑒 𝑙 𝑒 𝑐 .

−

𝑟 𝑎𝑑 .

R a nge

0

𝑆

𝐸

• Int e gra t e i nve rs e of s t oppi ng

So u r c e: W i k i med i a C o mmo n s

pow e r ove r t he e ne r gy

ra nge of t he pa rt i c l e : 𝑅 𝑎𝑛𝑔 𝑒

• N o t a ll p a r tic le s h a v e i d en t i cal r an g e, str a g g lin g d e sc r ib e s th is v a r ia tio n

= ∫ 𝐸 𝑚𝑎 𝑥 1

𝑑 𝐸

R a n ge

This image is in the public domain.

S t o p p in g P o w er Co m p o n en t s

H . P a u l. A I P C onf . P r oc . 1525 : 309 ( 2013)

• N uc l e a r stoppi ng pow e r: First a ssum e C oul om bi c n u c le u s in te r a c tio n s , d e s c r ib e in te r a to m ic p o te n tia l:

𝑉

= 𝑍𝑍 1 𝑍𝑍 2 𝗌 2

𝑟

(1/ r de pe nde nc e )

• Pos i t i ve nuc l e us sc re e ne d by ne ga t i ve e l e c t ron c l oud:

𝑍 1 𝑍 2 𝜀𝜀 2

𝑉

=

4 𝜋 𝜀𝜀 0 𝑟

𝑒

E ff e c t i ve s c r e e n i n g r ad i u s

S toppi ng P ow e r C ompone nts

P r e t t y gr a phs b y D esm o s Gr a pher ( w w w . desm o s. o r g )

𝑉 𝑟 = 𝑍𝑍 1 𝑍𝑍 2 𝗌 2 ( Cou l om b )

𝑟

𝑉 𝑟 =

𝑍𝑍 1 𝑍𝑍 2 𝗌 2 𝑒

4𝜋 𝗌 0 𝑟

−𝑟𝑟

𝑎 ( S c r eened)

S toppi ng P ow e r C ompone nts

D ec r ea si ng sc r eeni ng s t r eng t h

U nsc r eened po t en t i a l

S t o p p in g P o w er Co m p o n en t s

• A s su m e

≈ 0

𝑟 𝑎𝑑 .

𝛾𝛾 =

• N uc l e a r stoppi ng pow e r form ul a ( W a s p. 47):

𝑛𝑢𝑐 𝑙 .

𝑁 𝜋 𝑍 1 𝑍 2 𝜀𝜀 4

=

𝐸 𝑖

𝑀 1 ln

𝑀 2

𝛾𝛾 𝐸 𝑖

𝜀𝜀 2 𝛾𝛾 𝐸 2

4 𝐸 𝑖

𝑛𝑢𝑐 𝑙 .

𝑁 𝜋 𝑍 1 𝑍 2 𝜀𝜀 4

=

𝐸 𝑖

𝑀 1 ln

𝑀 2

S t o p p in g P o w er Co m p o n en t s

• N ow t urn t o e l e c t roni c st oppi ng. T he B e t he - Bl o ch f o r m u la d e s c r ib e s th is w e ll:

4 𝜋 𝑘 2 𝑍 2 𝜀𝜀 4 𝑛 𝑒

−

= 0

𝑚 𝑒 𝑐 2 𝛽 2

ln

− 𝛽 2

𝛽 = 𝑣 𝑖 𝑜 𝑛

𝑐

; 𝑛 𝑒

= 𝑒 𝑙 𝑒𝑐 𝑡 𝑟 𝑜 𝑛 𝑑 𝑒 𝑛 𝑠 𝑖 𝑡 𝑦

• I i s t he m e a n e xc i t a t i on e ne r gy of t he m e di um

S toppi ng P ow e r C ompone nts

• I i s t he m e a n e xc i t a t i on e ne r gy of t he m e di um

Relat ive S t o p p in g Po w er s

• Pl ot / c om pa re S e /S n

𝑆

2 𝑀

ln 𝛾𝛾 𝑒 𝐸 𝑖

𝑒

𝑆 𝑛

= 2 𝐼

𝑚 𝑒 𝑍 2 ln

• E l e c t roni c st oppi ng pow e r t a ke s ove r by fa c t ors of 10 2 - 10 4 for hi gh e ne r gy i ons…

• … w ha t a bout ne ut rons ?

Relat ive S t o p p in g Po w er s

H . P a u l. A I P C onf . P r oc . 1525 : 309 ( 2013)

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Relat ive S t o p p in g Po w er s

W as, p . 84

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Relat ive S t o p p in g Po w er s

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W as, p . 84

• W h at d o es t h i s say ab o u t :

• E ne r gy de pos ition vs. e ne r gy a t high - E?

• S am e at l o w - E?

• W h en i s t h e m o st d am ag e d o n e t o a m at er i al ?

• Ex p l ai n d am ag e r at es v s. r an g es o f h eav y i o n s & f ast ne ut r ons ?

• …

• W ha t a bout t he r m a l ne ut r ons ?

It All S t ar t s w it h F r en kel P air s

• Fre nke l pa i r – p er f ect v acan cy / i n t e r stiti a l c om bi na t i on

• Produc e d ve ry w e l l by e l e c t ron ra di a t i on

V ac an c y

I n t e r s titia l

T h e Dam ag e Cascad e

• Fr e nke l p air s d o n ’ t s tay th at w ay !

• M a ny i de a s a bout how “d am ag e c ascad e ” ev o lv es

• C al l ed “cascad e” d u e t o s ubs e que nt , c ont i nui ng d am ag e ef f ect s

• W h at ’ s w r o n g w ith t h is

W a s , p . 128

p ictu r e?

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O r i gi na l c o nc ep t i o n o f da m a g e c a sc a de, sho w i ng pa t h o f F r enk el p a ir p r o d u c tio n

Dam ag e Cascad es Revisit ed

W a s , p . 128

• M a ny m ore form s of da m a ge a re possi bl e

• S i ngl e va canc i es & int e rs ti t ia l s n ot a lw a ys e ne r ge t i c a l l y f a vor a bl e

• F r e nke l pa i r s don’ t e xpl a i n obs e r ve d

da m a ge

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R e v i sed da m a g e c a sc a de a c c o u n tin g f o r c r y s t a llin ity

Dam ag e Cascad es Revisit ed

W a s , p . 128

P rim a r y K n o c k - o n A t o m ( P K A)

C o llis i o ns c a n k no c k a t o m s i n c l o se - pa c k ed

d ir e c tio n s

S t a b le , e n e r g e tic a lly fa v o r a b l e , fa s t - mo vi n g de f ec t s

H o w s t a bl e w o ul d t hi s be i n t he l o ng t er m ?

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Cascad e S t ag es – Ballist ics

• P u t s im p l y , a to m s g e t knoc ke d a round

• N o tim e to r e la x !

• ~ 10M e V ne ut rons m ove how fa st ?

• H ow l ong t o m ove one l at t i ce p ar am et er ?

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Cascad e S t ag es – T h er m al S p ike

• T em p er at u r e r i s es ve ry l oc a l l y for a ve ry s hort t i m e

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K . O. T r ach en ko , M . T . D o v e . E . K . H . S a l je . J. P h ys. C onde ns . Ma tte r , 13: 19 47 ( 2001 )

Cascad e S t ag es – Q u en ch

• H eat i s co n d u ct ed aw ay EX TREM E L Y q u i ck l y

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K . O. T r ach en ko , M . T . D o v e . E . K . H . S a l je . J. P h ys. C ond. Ma tte r , 13: 194 7 ( 2001 )

Cascad e S t ag es – An n eal

D . S . Ai d h y e t a l. S c r i p ta Ma te r . , 60( 8) : 69 1 ( 2009 )

• M os t da m a ge “a nne a l s ” out , or re c om bi ne s/ ge t s sunk a w a y

• F or ne ut r ons & i ons , al m os t al l da m age an ne al s!

𝑡 = 0 𝑛𝑠 𝑡 = 2 . 45 𝑛𝑠

Courtesy of Elsevier, Inc., http://www.sciencedirect.com . Used with permission. Source: Aidhy, D. S. " Kinetically Driven Point-Defect Clustering in Irradiated MgO by Molecular-Dynamics Simulation ." Scripta Materialia 60, no. 8 (2009): 691-4.

S i m ul a t ed a nnea l i ng o f F r enk el pa i r s i n Mg O a t 1000K

T yp es o f Rad iat io n

• D i f fe re nt ra di a t i on produc e s di f fe re nt c a sc a de s

M a s s & C ha r g e

S t o ppi n g M ec ha ni sm

I nc r ea si ng mas s , s ame

c h a rg e

A ll e le c tr o n ic Mo s tly e le c tr o n ic

Mo s tly n u c le a r , s o m e c o u lo m b ic

M o der a t e m a ss, no c h a rg e

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E n tir e ly n u c le a r

S i mul a ti on M e thods – B C A

W a s , p . 134

• B in a r y C o llisio n A pproxi m a t i on

• U ses i nter at om i c pot e nt ia l s (lik e M D ) t o a l l ow a t om s t o m ove

• D oe s n ot rest r i ct c rys ta ll ini t y

• C r eat es co l l i s i on cascade s p r et t y w el l !

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S i mul a ti on M e thods – MD

• Sol ve 𝐹 = 𝑚 𝑎 for e ve ry pa i r of a t om s

• Int e rat om i c pot e nt i al s a re t he ke y t o in te r a c tio n s

• R ight : MD si m ulat i on o f 1ke V cascade i n i r on a t 100K

W a s , p . 139

0. 18

ps

9. 5

ps

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S im u lat io n M et h o d s – MD

ht t p: / / w w w . c m bi . r u. nl / r e doc k / i m a ge s / Le nna r dJ one s . png

• I n te r a to m ic p o te n tia ls ar e t he ke y t o i nt e ra c t i ons

• A ttr a c tiv e & r e p u ls iv e t e r m s

• L e nna r d - Jon e s ( L J) p o te n tia l w i de l y us e d

Courtesy of Bo Hanssen & Sander Jans. Used with permission.

L e n n ar d - Jo n e s P o t e n t i al

S im u lat io n M et h o d s – MD

J . Y u e t a l. J . M at er . C h em . , 19: 39 2 3 - 39 3 0 ( 2009)

• I n te r a to m ic p o te n tia ls ar e t he ke y t o i nt e ra c t i ons

• A ttr a c tiv e & r e p u ls iv e t e r m s

• L e nna r d- Jo n es ( L J) pot e nt i a l w i de l y us e d

© Royal Society of Chemistry. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ .

A t t r a c t i v e t er m s o f sel ec t ed po t en t i a l s ( d o t s ) , an d L J - m o d i f i e d v e r s i o n (l i n e s )

S im u lat io n M et h o d s – MD

A video is played in class to demonstrate the concept.

h ttp : / / w w w - pe r s ona l . um i c h. e du/ ~ gs w / m ov i e s . ht m l

22. 1 4 – Nu clear M ater ials Slid e 30

S im u lat io n M et h o d s – MD

A video is played in class to demonstrate the concept.

h ttp : / / w w w - pe r s ona l . um i c h. e du/ ~ gs w / m ov i e s . ht m l

S i mul a ti on M e thods – MC

• W i t h pre - de t e rm i ne d • Ex am p l e: TRI M di st ri but i o ns for s om e • R a ndom l y c hoos e

f eat u r es scat t er i ng an gles, ne w

• “R oll t he dice” t o sam ple f r om di s tribut i ons

• L e t r a ndom num be r s de t e r m i ne w he r e t hi ngs m ove a nd c ha nge

m ean f r ee p at hs

S i mul a ti on M e thods – MC

h ttp : / / w w w - pe r s ona l . um i c h. e du/ ~ gs w / m ov i e s . ht m l

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S i mul a ti on M e thods – R a te The or y

C . J . Or t iz, M . J . C a tu r l a . J . C om put e r - A i d ed M at er i al s D esi g n 14: 17 1 - 18 1 ( 2007)

• A s su m e r a te - c ont rol l e d e qua t i ons for de fe c t m i gra t i on, c l ust e ri ng

• O ft e n e m pl oys “m e a n fi e l d t he ory”

• G l os ses ove r de t ai l s t o accel er a t e t i m e

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S i mul a ti on M e thods – R a te The or y

C . J . Or t iz, M . J . C a tu r l a . J . C om put e r - A i d ed M at er i al s D esi g n 14: 17 1 - 18 1 ( 2007)

• H ow good i s t he a pproxi m a t i on? Is i t w ort h i t ?

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Dam ag e Cascad es – S umma r y

W a s , p . 140

• S p a n s m u ltip le tim e sc a le s

• S et s t he st age f o r de f ect m i grat i on t o hi ghe r l engt h scal es

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• S im ul a tion a ls o re qui re s m ul tis c a le m e t hods w i t h m uc h c re a tivi t y to g e t rig ht !

Disp lacem en t T h eo r y

• D e fi ne a ra t e of a t om i c di spl a c e m e nt s usi ng fl ux:

M a x i mu m e n e r g y a v ai l ab l e E n e r g y d e p e n d e n t f l u x d i s t r i b u t i o n

𝐸 𝑚𝑎 𝑥

𝑅 = �

0

𝑁 ∗ Φ

∗ 𝜎 𝐷

𝑑 𝐸 𝑖

𝑑 𝑖 𝑠𝑝 𝑙 .

R e a ct i o n r a t e 𝑚 3 − 𝑠𝑒 𝑐 M a t e r i al n u mb e r d e n s i t y

𝑎 𝑡 𝑜 𝑚𝑠

𝑚 3 D i spl a c em en t c r o ss sec t i o n

Disp lacem en t T h eo r y

• D e fi ne a ra t e of a t om i c di spl a c e m e nt s usi ng fl ux:

𝐸 𝑚𝑎 𝑥

𝑅 = �

0

𝑁 ∗ Φ

∗ 𝜎 𝐷

𝑑 𝐸 𝑖

𝑅 𝐷𝑃 𝐴

= = �

𝑁 𝑠 𝑒𝑐

𝐸 𝑚𝑎 𝑥

Φ

∗ 𝜎 𝐷

𝑑 𝐸 𝑖

0

Disp lacem en t T h eo r y

• D e fi ne a ra t e of a t om i c di spl a c e m e nt s usi ng fl ux:

𝐷 𝑃𝐴

= �

𝑠 𝑒𝑐

𝐸 𝑚𝑎 𝑥

Φ

∗ 𝜎 𝐷

𝑑 𝐸 𝑖

0

K now n or pre - de t e rm i ne d O nl y unknow n qua nt i t y

• D e ve l op e xpre s si on for di spl a c e m e nt c ross s e c t i on

Disp lacem en t T h eo r y

• D e fi ne a ra t e of a t om i c di spl a c e m e nt s usi ng fl ux:

𝐷 𝑃𝐴

= �

𝑠𝑒 𝑐

𝐸 𝑚𝑎 𝑥

Φ

∗ 𝜎 𝐷

𝑑 𝐸 𝑖

0

𝑇 𝑚𝑎 𝑥

P r o ba bi l i t y t ha t a n a t o m di spl a c ed b y a p a rtic le with e n e r g y E i le a v e s with r e c o il ener gy T ( di f f er en t i a l ener gy t r a ns f er c r o ss sec t i o n)

𝜎 𝐷

= �

𝑇 𝑚𝑖 𝑛

𝜎 𝐸 𝑖 , 𝑇

𝑑 𝑇

N u m b e r of a t om i c d isp la c e m e n t s

• T i s t he PK A (di spl a c e d a t om ) re c oi l e ne r gy

f r o m a P KA w i th e n e r gy T

Disp lacem en t T h eo r y

𝜎 𝐷

𝑇 𝑚𝑎 𝑥

= �

𝑇 𝑚𝑖 𝑛

𝜎 𝐸 𝑖 , 𝑇

𝑑 𝑇

• A ssum e t he re i s som e t hre s hol d e ne r gy (E d ) be l ow w hi c h a di s pl a c e m e nt doe s not oc c ur:

𝑣 𝑇

= 0;

𝑇 <

𝐸 𝑑

• O th e r w is e a d is p la c e m e n t w ill o c c u r :

𝑣 𝑇 = 1; 𝑇 ≥ 𝐸 𝑑

Disp lacem en t T h eo r y

W a s , p. 74

𝜎 𝐷

𝑇 𝑚𝑎 𝑥

= �

𝑇 𝑚𝑖 𝑛

𝜎

𝑑 𝑇

𝐸 𝑖 , 𝑇

𝑣 𝑇

• Sha rp di spl a c e m e nt t hre shol d m ode l :

• W ha t d oe s th is m o d el n eg l ect ?

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Disp lacem en t T h eo r y

W a s , p. 75

𝜎 𝐷

𝑇 𝑚𝑎 𝑥

= �

𝑇 𝑚𝑖 𝑛

𝜎

𝑑 𝑇

𝐸 𝑖 , 𝑇

𝑣 𝑇

• A dd s om e s m oot hne ss t o t hi s func t i on:

• A c c ount s for a t o mi c v ib r a tio n s, im p u r itie s

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Disp lacem en t T h eo r y

• W ha t i s t hi s t hre shol d e ne r gy?

• L et ’ s est i m at e?

1. E ne r gy t o bre a k m e t a l s urfa c e bonds: ~ 5e V

2. Shi ft re m ove d a t om t o t he i nt e ri or: x2

3. S tu f f a to m in a n in te r stitia l s it e , a ss u m e n o tim e to r e la x th e la ttic e : x 2

4. D isp la c e m e n t isn ’ t in th e e a s ie s t d ir e c tio n : x 2

Disp lacem en t T h eo r y

W a s , p. 83

• W h a t is th is t hre shol d e ne r gy?

• N ot i c e a ny pa t t e rns i n t h e d at a?

• C rys ta l s truc tur e ?

• M e l t i ng poi nt ?