The Sputtering Prblem James A Glackin, James V. Mathesn I prpse t cnsider first the varius elements f the subject, next its varius parts r sectins, and finally the whle in its internal structure. In ther wrds, I shall prceed frm the simple t the cmplex. Carl Vn Clausewitz Intrductin Gun tubes prduced fr Army weapns systems, such as tanks and artillery pieces, are designed t fire a specific number f runds befre they are cnsidered unsafe and must be replaced. The exact rund cunt fr a gun tube is s critical that detailed recrds are kept fr each tube in every Armr and Artillery unit in the Army. Frequent replacement f gun tubes is undesirable nt nly fr bvius fiscal reasns, but als due t the manpwer and firepwer drain it places n a unit in cmbat. One methd f increasing the number f runds a gun tube can safely fire is t place a metal lining inside the tube. One f the techniques fr placing a lining inside a gun tube is called sputtering. Sputtering is a prcess where a cylindrical metal bar is inserted inside a gun tube, centered, anchred in place, and cnnected t a pwer surce. The tube and rd are placed inside a chamber where all air is evacuated and replaced with a nble gas. When electricity is run thrugh the surce rd, a magnetic field develps and atms frm the metal surce rd are brken away and literally fly acrss the gap frm the surce rd t the inside f the gun tube. During flight, the metal atms cllide with atms f the nble gas that separate the surce rd frm the inside wall f the gun tube. Each cllisin with a nble gas atm causes the metal atm t change the directin f its flight and lse energy. In develping the sputtering prcess, the Army used three mathematical mdels t create cmputer simulatins f the sputtering prcess. The cmputer simulatins were used t quickly and inexpensively predict the effects f different materials and experimental cnditins n the sputtering results. One mdel was used t simulate initial directin and energy f a sputtered atm. A secnd mdel was used t analyze the effects f an atm clliding with the inside f the gun tube wall. This chapter deals with the third mathematical mdel, which was develped t analyze the flight f a metal atm frm the surce rd t the inside f the gun tube. In rder t mdel this prtin f the prblem, it is necessary t analyze the flight f an atm frm the surce rd (cathde) t the inside f the gun tube (ande) in 8
small steps. Each step represents an imprtant event in the flight. The first event is when the atm breaks away frm the cathde. The initial velcity vectr f the atm as a result f this event is given as an initial cnditin. The secnd event is determining the lcatin f the sputtered atm s cllisin with a neutral gas atm. Lcating a cllisin site invlves determining a free path length fr the atm and applying the free path length alng the directin f the initial velcity vectr. The third step is examining the effects f the cllisin n the energy and flight path f the sputtered atm. T simplify this step, the cllisin is examined in a center f mass frame f reference. The furth and final step in the prcess invlves determining the lcatin and energy f the metal atm when it impacts with the inside wall f the gun tube (ande). Because a sputtered atm might cllide with multiple neutral gas atms during its flight frm the surce rd t the wall f the gun tube, steps tw and three may be repeated several times fr each atm befre calculating the terminal effects in step fur. Gemetry We begin with an explanatin f the gemetry f ur mdel. The gun tube will be treated as a hllw right circular cylinder with a radius equal t R. The surce rd is als a right circular cylinder with a radius equal t r. Bth cylinders are centered n the z axis s that the straight line distance frm the surce rd t the inside f the gun tube is equal t R-r. (See Figure 1) Figure 1 88
Step 1 - Initial Velcity Vectr We will use vectrs in spherical crdinates t describe the flight f the sputtered metal atm. We use the angle θ t describe the rtatinal angle in the x,y plane and the angle φ t describe the elevatin f the atm abve the x,y plane. Thus a unit directin vectr fr the sputtered atm will be f the frm: Directin = cs( θ) sin( φ ), sin( θ) sin( φ ), cs( φ ) Figure The unit directin vectr is then multiplied by the initial speed t generate an initial velcity vectr fr the atm. Step - Lcating Cllisin Sites The lcatin f an atm s cllisin is determined by mving a specific distance alng the directin f its velcity vectr. The distance mved is called the free path length. The free path length changes fr each atm after each cllisin. The free path length is based n a distributin abut an atm s mean free path length. The mean free path length fr an atm is the average distance that the atm will travel befre it has a cllisin with a neutral gas atm. The mean free path length fr all sputtered atms f the same element remains cnstant as lng as the density f the neutral gas remains unchanged. The mean free path length is calculated by examining the relatin between the density f the neutral gas and the distance a sputtered atm wuld travel in a unit f time if it were allwed t travel unimpeded. The radius f the sputtered atm and the distance it travels in a unit f time are used t create a cylinder thrugh which an atm travels during the unit f time. The vlume f the cylinder is calculated and multiplied by the density f the neutral gas t determine the average number f atms that ccupy a cylinder f that size. The 89
length f the cylinder is then divided by the number f atms that ccupy the cylinder and the result is a reasnable apprximatin fr the length f the mean free path (See Figure 3). 1 Figure 3 D This methd yields the fllwing equatin: L =. Where L is the mean π ρ r free path length, D is the distance traveled, ρ is the density f the neutral gas, and r is the radius f the sputtered metal atm. Once the mean free path is determined, the graph in figure 4 is created using x n the equatin N e L = t define the curve. The rati n represents the N percentage f free path lengths that are at least as lng as the crrespnding number f mean free path lengths frm the graph. In the graph, free path length is measured in terms f the number f mean free path lengths a sputtered atm will travel prir t a cllisin (DIST/L). Figure 4 The free path length fr a specific atm at a specific mment in time is then created by applying a unifrm distributin t the n/n axis (randmly chsing a number between 0 and 1) and then finding the crrespnding DIST/L crdinate 90
n the graph in figure 4. The DIST/L term is multiplied by the atm s mean free path length t yield the distance t the atm s next cllisin. Nw that the distance t the cllisin has been determined, the actual lcatin f the cllisin is fund by multiplying distance t cllisin by a unit vectr in the directin f the sputtered atm s velcity vectr and adding the resulting vectr t the sputtered atm s ld psitin vectr. Step 3 - Effects f a cllisin Mving t a Center f Mass Frame f Reference When a sputtered atm cllides with a neutral gas atm, the sputtered atm will change directin and speed. The analysis f a cllisin is best made in a center f mass frame f reference. A center f mass frame f reference means that the rigin f the crdinate system used in making calculatins is fixed n the center f mass f the tw atms invlved in the cllisin. Thus the center f mass des nt mve (as it wuld in the labratry frame f reference). Using the center f mass frame f reference makes the calculatin f the changes in directin and speed f the sputtered atm less cmplicated because the changes are measured against a statinary pint. Change in directin Since we are wrking in spherical crdinates, the directin change frced upn a sputtered atm as the result f a cllisin with a neutral gas atm is best examined in terms f an angle f rtatin and an angle f deflectin. The angle f rtatin is determined by examining the cllisin frm directly behind the sputtered atm. The key t the cllisin effects is the relative psitins f the centers f mass f the sputtered atm and the neutral gas atm at the mment f impact. In figure 5, we see that a circle f radius equal t the sum f the radii f the neutral gas atm and the sputtered metal atm (b max ) can be drawn centered n the neutral gas atm. In rder fr a cllisin t ccur, the center f mass f the sputtered metal atm must be lcated within the circle f radius b max. 91
Figure 5 We will assume that the relative psitins f the centers f mass at the mment f cllisin are cmpletely randm. Therefre, we need nly apply a unifrm distributin ver the circle f radius b max t randmly determine the lcatins f the centers f mass f the tw atms. Measuring the angle f rtatin, θ r, then becmes a simple trignmetry prblem. Calculatin f the angle f deflectin, χ, is made by examining the cllisin in tw dimensins as viewed frm a psitin rthgnal t bth the pre cllisin and pst cllisin velcity vectrs f the sputtered atm (figure ). Figure (pre cllisin) The distance, b (measured in the previus step) is the cntrlling parameter in the determinatin f χ. Using trignmetry, the incming angle fr the cllisin, θ m, is defined by the equatin θ m = sin 1 b b max. 9
The angle f interest, χ, is equal t π minus tw times θ m in the center f mass frame f reference. 3 Figure (pst cllisin) Once the angles θ r and χ are determined, spherical crdinates are used t create a unit vectr fr the directin f the sputtered atm in the revised new system. The unit directin vectr is: sin( χ )cs( θ ),sin( χ)sin( θ ),cs( θ ). The unit directin vectr must then underg a linear transfrmatin (rtatin abut rigin) s that it can be used in the standard crdinate system in the center f mass frame f reference. The revised crdinate system is aligned s that its z axis is n the pre-cllisin path. As a result, the parameters fr the transfrmatin, α and β, cme frm measuring the angles f the pre-cllisin directin f the sputtered atm in the standard system f crdinates (Figure 8). r r r Figure 8 93
The linear transfrmatin aligns the pre cllisin path f the sputtered atm with the standard z axis and thus brings the pst cllisin directin vectr int the standard system f crdinates. The matrix fr this transfrmatin is: cs( α)cs( C = sin( β ) sin( α)cs( cs( α )sin( cs( sin( α)sin( sin( α) 0 cs( α ) The pst cllisin directin vectr in the revised system f crdinates is multiplied by C t yield the pst cllisin directin vectr in the standard crdinate system. 4,5 xrevised, yrevised, zrevised C = x, y, z Change in Speed The utging speed f a sputtered atm equals the incming speed f the sputtered atm in the center f mass frame f reference. The laws f cnservatin f mmentum and cnservatin f energy can used t demnstrate why this is true. The prf f this cncept is left as an exercise fr the reader. See exercise. Returning t a Nrmal Frame f Reference Nw that the speed and directin fr the pst cllisin sputtered atm have been determined, the pst cllisin unit directin vectr fr the sputtered atm is multiplied by the pst cllisin speed f the sputtered atm t find a new velcity vectr in the standard crdinate system in the center f mass frame f reference. T find the pst cllisin velcity vectr in the labratry frame f reference, the vectr v cm is added t the pst cllisin velcity vectr in the center f mass frame f reference. Figure 9 94
Figure 9 represents the view f the cllisin in the center f mass frame f reference. The speed f the sputtered metal atm remains unchanged and therefre, the velcity vectrs befre and after cllisin are f the same magnitude. Thus the cllisin in the center f mass frame f reference effects nly the directin f the sputtered metal atm. The lss f velcity is accunted fr when the system is mved back t the labratry frame f reference. Mvement frm the center f mass frame f reference t the labratry frame f reference is accmplished by adding the velcity vectr fr the center f mass (VCM) t the pst cllisin velcity vectr as depicted in Figure. Figure Step 4 - Determining the Impact Lcatin and Energy f a Sputtered Atm Determining the lcatin where an atm impacts the inside wall f the gun tube is dne by keeping track f its psitin vectr thrughut its multiple mvements and cllisins. When the radial mvement f the atm equals the inside radius f the gun tube, its psitin vectr crdinates are used t plt its lcatin n the gun tube wall. The height f the atm as it strikes the gun tube is equal t its z crdinate at the time f impact. The radial lcatin is calculated using the equatin: x crdinate Radial lcatin = cs 1 gun tube radius The kinetic energy f the atm at the mment f impact with the gun tube is calculated using 1 mass ( speed) where the speed is equal t the magnitude f the velcity vectr at the mment f impact. Example: The flight f ne atm In this example we will fllw the flight f ne atm frm the surce rd (r = mm) t the inside f the gun tube wall (R = 50 mm) and cmpute the terminal 95
effects f the atm at the gun tube wall. We will assume several cnditins that wuld actually be cmputed using randm numbers and prbabilistic mdels. Step ne is t find an initial velcity vectr. The initial speed is estimated based n the energy and mass f the metal atm. Let s start with an initial speed f 0,000,000 mm/sec. Assume initial launch angles f θ = 5, φ = 80. This yields an initial velcity vectr f r 0000000 V = cs(5 )sin(80 ), sin(5 )sin(80 1.85, 8.34, 3.43 ), cs(80 mm/sec ) Step tw is t find the lcatin f a cllisin. The cllisin lcatin is based n an initial lcatin f, 0, 000 mm (this lcatin will place the atm n the utside wall f the surce rd). We randmly generate a free path length based n the expnential distributin f free path lengths, figure 4. Assuming a free path length f 30 mm we can cmpute the time it takes until a cllisin. time dist = speed = 30 = 1.5 sec 0000000 Nw we can easily determine the lcatin f the cllisin using vectrs. Cllisin lcatin = initial lcatin + velcity(time) Cllisin lcatin = 3.8, 1.49, 005 Lking at the x and y cmpnents f the cllisin lcatin vectr, we can determine if ur cllisin wuld be utside the gun tube wall radius = 3.8 + 1.49 = 38. 84 mm Our inside gun tube radius was 50 mm, thus this cllisin is befre we reach the gun tube wall. Step three is t determine the velcity f the center f mass and the effects f the cllisin in the center f mass frame f reference. When determining the velcity f the center f mass, ur mdel nly cnsiders the sputtered atm t be mving. Therefre, the velcity f the center f mass is in the same directin as the velcity vectr, but has a different speed. If the neutral gas atm has a mass f 50 Atmic Mass Units (AMU) and the sputtered atm has a mass f 00 AMU 9
mass( sputtered ) V v cm = V v mass( sputtered) + mass( neutral gas) Fr ur example we have v V cm v 00 = V = 50+ 00 1.48,.59,.8 We nw determine the effects f the cllisin. Assume cllisin parameters f θr = and χ = 15. The pre cllisin velcity vectr f the sputtered atm, as v v seen in the center f mass frame f reference, is V V. The pst cllisin velcity vectr has the same magnitude(speed), but it s directin is determined using spherical crdinates and a crdinate system riented with the z axis pinting parallel t ur velcity vectr. cm V ' V ' V ' x y z = speed sin( χ) cs( θ = speed sin( χ) sin( θ = speed cs( χ) = 4 r r ) = 4 ) = 4 sin(15 ) cs( ) = 1.00 5 sin(15 ) sin( ) = 1.98 sin(15 ) = 3.84 We must nw perfrm a rtatin abut the rigin t mve us t the standard x-yz axis. The angles we rtate are determined frm ur velcity vectr, v z cmpnent f V α = arccs v magnitude f Vcm Fr ur example we have: cm v y cmpnent f V β = arctan v x cmpnent f V 5.94 arccs 1.39 radians 4 = 1.5 α = β = arctan 0.433 radians 3.50 = Nw t cmplete the rtatin we multiply the vectr V v ' by the directin csine matrix. The directin csine matrix, C, is determined by taking the csine f the angle between the standard x-y-z axis and the new axis (z-axis pinting in the directin f V v ). cs( α)cs( cs( α )sin( sin( α) C = sin( β ) cs( 0 sin( α)cs( sin( α)sin( cs( α ) Thus we have cm cm 9
1.00,1.98 5, 3.84 cs1.39cs0.433 sin 0.433 sin1.39cs0.433 cs1.39sin 0.433 cs0.433 sin1.39sin 0.433 sin1.39 0 cs1.39 = 3.533,1.84, 3.331 5 This is ur pst cllisin velcity vectr in the center f mass frame f reference in the standard x-y-z crdinate system. Nw t return t the labratry frame f reference we simply add V v cm. 3.533,1.84 5, 3.331 = + 1.48,.59,.8 1.81, 8.505,.445 Repeat step tw. We must nw find the time until ur next cllisin. We again randmly generate a free path length, say 43 mm fr this iteratin. We find the time until ur next cllisin. Our pst cllisin speed is 1.98 mm/sec. time dist = speed 43 = =.1 sec 1.98 We nw determine the lcatin f the nd cllisin using vectrs. nd Cllisin lcatin = 1 st Cllisin lcatin + velcity(time) Cllisin lcatin = 5.8, 30.8, 0 Lking at the x and y cmpnents f this we can determine if ur cllisin wuld be utside the gun tube wall radius = 5.8 + 30.8 = 81. 3 mm Since ur inside gun tube radius was 50 mm, this cllisin cannt ccur because the atm will impact the inside f the gun tube wall prir t reaching the cllisin lcatin. Step fur is t calculate the terminal effects. We have already determined the arrival velcity 1.81, 8.505,.445 and speed 1.98 mm/sec. Nw we must determine the arrival energy. 1 Energy = mv = 4.1 EV 98
Exercises 1. Using dimensinal analysis, find the cnversin factr used t cnvert the terminal energy f the sputtered atm frm mm sec AMU t electrn vlts.. We have stated that in the center f mass frame f reference the pre-cllisin speed and the pst cllisin speed are equal. Prve this statement using cnservatin f energy and cnservatin f mmentum. Hint: Mmentum is cnserved as a vectr quantity. Nte: the mmentum in the y cmpnent was zer prir t the cllisin and therefre must be zer after the cllisin. In this sketch u 1 is the pre cllisin speed f the metal atm in the labratry frame f reference. v 1 and v are the pst cllisin speeds f mass 1 and respectively. cs( α)cs( cs( α )sin( sin( α) 3. Verify the transfrmatin matrix C = sin( β ) cs( 0 by sin( α)cs( sin( α)sin( cs( α ) breaking the transfrmatin int tw steps. A rtatin f angle α and a rtatin f angle β. These tw transfrmatin matrices A and B are then multiplied t yield C. Discuss the difference if any caused by the rder f perfrming the rtatins (What is the relatinship between AB and BA). 99
4. Write an algrithm t find the impact lcatin f the metal atm against the gun tube wall. 5. Cnstruct the flight f an atm frm launch t wall impact. Use the fllwing parameters. r = mm R = 0 mm initial speed = 0,000,000 mm / sec launch angles θ = 0 free path length = 5, 54, and 3 Neutral gas = 50 AMU Metal gas = 0 AMU Cllisin parameters φ = 5 θ = 30 θ = 15 χ = 0 χ = 5 References Clausewitz, Carl Vn. On War. Edited and Translated by Michael Hward and Bernard Brdie, Princetn, New Jersey, Princetn University Press, 19. [1] J. D. Cbine, Gaseus Cnductrs, Thery and Engineering Applicatins. New Yrk: Dver Publicatins, 1958, pp.0-1. [] J. D. Cbine, Gaseus Cnductrs, Thery and Engineering Applicatins. New Yrk: Dver Publicatins, 1958, p5. [3] J. O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Mlecular Thery f Gasses and Liquids. New Yrk: Jhn Wiley and Sns, 1954, p.50. [4] F. H. Miller, Analytic Gemetry and Calculus, A Unified Treatment. New Yrk: Jhn Wiley and Sns, 1949, pp. 49-495. [5] W. H. Beyer, CRC Standard Mathematical Tables, th Editin. Bca Ratn, Flrida: CRC Press, Inc. 1981, p. 1. 0